research article cloud particles evolution...
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Research ArticleCloud Particles Evolution Algorithm
Wei Li Lei Wang Qiaoyong Jiang Xinhong Hei and Bin Wang
School of Computer Science and Engineering Xirsquoan University of Technology Xirsquoan 710048 China
Correspondence should be addressed to Lei Wang wangleeei163com
Received 10 November 2014 Revised 13 January 2015 Accepted 13 January 2015
Academic Editor Jyh-Hong Chou
Copyright copy 2015 Wei Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Many evolutionary algorithms have been paid attention to by the researchers and have been applied to solve optimization problemsThis paper presents a new optimization method called cloud particles evolution algorithm (CPEA) to solve optimization problemsbased on cloud formation process and phase transformation of natural substance The cloud is assumed to have three states inthe proposed algorithm Gaseous state represents the global exploration Liquid state represents the intermediate process fromthe global exploration to the local exploitation Solid state represents the local exploitation The cloud is composed of descript andindependent particles in this algorithmThe cloud particles use phase transformation of three states to realize the global explorationand the local exploitation in the optimization process Moreover the cloud particles not only realize the survival of the fittestthrough competition mechanism but also ensure the diversity of the cloud particles by reciprocity mechanismThe effectiveness ofthe algorithm is validated upon different benchmark problems The proposed algorithm is compared with a number of other well-known optimization algorithms and the experimental results show that cloud particles evolution algorithm has a higher efficiencythan some other algorithms
1 Introduction
Many real-world problems which are classified as global opti-mization problems are very complex and are quite difficult tosolve Various optimization algorithms are developed to solvethese problems Most of these algorithms learn and imitatea variety of intelligent behavior in nature These algorithmsexploit a set of potential solutions and detect the optimalsolution through cooperation and competition among theindividuals of the population [1] For example Genetic algo-rithm [2] proposed by Professor Holland is a computationalmodel of biological evolutionwhich imitates natural selectionand genetic mechanism Ant colony algorithm [3] is a heuris-tic bionic optimization algorithmwhich imitates the foragingbehavior of the ant colony In the ant colony algorithman individual is called an artificial ant The artificial antssearch probabilistically in the solution space to create can-didate solutions The evaluating and the updating of thesecandidate solutions are based on the pheromone associatedwith each one of them The candidate solution with themaximum amount of pheromone is considered to be theoptimal solution of the problem Particle swarm optimization(PSO) [4] is a bionic intelligent computing method proposed
by Kennedy and Eberhart which imitates the flying and theforaging behavior of birds In PSO the optimal solution iscalled a leader The leader guides each particle to searchBesides the particles cooperate with each other to realizeexploration These behaviors are directed by a combinationof swarmrsquos previous best (gBest) and their own previous best(pBest) [5] The artificial bee colony algorithm (ABC) [6 7]is a nonderivative optimization method based on self-organ-izational models of bee and swarm intelligence In ABC beesare divided into three groups which are called ldquoemployedrdquobees ldquoonlookerrdquo bees and ldquoscoutsrdquo bees ldquoOnlookersrdquo beeslook for food sources Food source is the search goal of beesldquoEmployedrdquo bees work on food source In addition ldquoscoutsrdquobees search for the new food source near the hive Artificialimmune system (AIS) [8] is an adaptive system inspired bythe immunologywhich imitates human immunological func-tions principles and models to solve complex problems InAIS the external organisms are called antigen and the inter-nal organisms are called antibodies Immune clonal oper-ation provides multistrategy conditions for recombinationand mutation [9] The clonal selection selects the excellentantibodies from the subpopulation generated by clonal pro-liferation
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 434831 21 pageshttpdxdoiorg1011552015434831
2 Mathematical Problems in Engineering
This paper which is inspired by the natural phenomenaof cloud formation presents a novel optimization algorithmto solve optimization problems called cloud particles evolu-tion algorithm (CPEA) The proposed algorithm uses phasetransformation mechanism to imitate the formation anddiversification of the cloud to solve the optimization prob-lems It is well known that the species evolution is influencedby individual fitness living environment competition andcooperation between the individuals Therefore the algo-rithm introduces competition and reciprocal mechanism inthe evolution process so as to strengthen the viability of thecloud particlesThe performance of the cloud particles evolu-tion algorithm (CPEA) is tested on the classical optimizationproblems The results are compared with other algorithmsin terms of best function value and the number of functionevaluations
The remainder of this paper is organized as follows InSection 2 the proposed CPEA and the concepts are intro-duced in detail In Section 3 the performance of the proposedalgorithm is validated by different optimization problemsThe conclusion and future research are drawn in Section 4
2 Cloud Particles Evolution Algorithm
21 Cloud Formation The idea of the proposed CPEA isinspired from cloud formation The state of the cloud parti-clesrsquo changing process [10ndash12] is shown in Figure 1
After the wet air rises it will go through a set of changesbecause of some certain reasons temperature and pressureThen the water vapor is formedWater vapor plays an impor-tant role in this process When the water vapor is saturatedit will adsorb on the cloud condensation nuclei (CNN) andthe initial cloud particles are formed Cloud particles arecondensed and sublimated continuously by absorbing watervapor around the cloud In the process of the condensationthe cloud particles are getting closer and colliding Then thelarger cloud particles are formed Ice phase is generated whenthe temperature is below 0∘C at this time The emergence ofice crystals damages the stable status of cloud phase structureIce crystals grow up quickly to be the snow crystals becauseof the condensation of the water vapor On the one handwater vapor transfers to ice crystals and ice crystals beginto grow up as supercooled water droplets evaporate On theother hand the supercooledwater dropletswill be frozen afterthe colliding to make the snow particles grow bigger If thesupercooledwater vapor which has been collided is toomanythe snow crystals will be transferred to be solid substancesThen the solid substances such as snow and hail will bedropped If the solid substances descent in the area which hasa higher temperature over 0∘C they will melt into raindropThewater in our planet will be irradiated by the sunshine andit will become the water vapor again in the air The cloud isformed and changed like this
22 Phase and Phase Transformation There are various kindsof substance in nature Substances commonly exist in threestates solid liquid and gaseous Figure 2 is a simple diagramto show howwater transforms among the three states Evapo-ration condensation andmelting are the threeways for water
Raindroplets
Cloudparticles
Snowpellets
cloudparticles
Large
Figure 1 Cloud particles transformation process
Depos
ition
Subli
mati
on
Melting
Freezing
Evap
orati
onCon
dens
ation
Figure 2 Three phases of water
transforming In order to describe the different forms of thesubstance the phase can be used as a symbol for states of sub-stances In thermodynamics phase is defined as some partswhich have the same quality [13] A phase transformation isthe transition which is from one phase to another in thermo-dynamics
Phase transformation is relevant to Gibbs free energyIn thermodynamics the Gibbs free energy [14] is a ther-modynamic potential that measures the ldquousefulrdquo or process-initiating work obtainable from a thermodynamic system ata constant temperature and pressure
23 Population Evolution Mechanism Darwin pointed outthat mutation and natural selection are the two importantfactors for the evolution which explained many natural phe-nomena successfully [15] Nowak from Harvard University
Mathematical Problems in Engineering 3
1 21
3
21
3
2
4
65
78
Cooperators
Kin selection Direct reciprocity Indirect reciprocity Network reciprocity Group selection
Defectors
1 r
Figure 3 Five mechanisms of cooperation evolution [16]
ranked the cooperation as the third important factor of thespecies evolution Evolution is a unity of the cooperationand the opposition [16] Every gene cell and organism ofthe individual should strengthen its own evolution process atthe expense of beating its rivals Therefore individuals oftenconflict with each other because of limited resources [17]However the complex life organizational structure in differ-ent levels is produced in the process of evolution which indi-cates that the cooperation exists in the evolution ThereforeNowak summarizes five mechanisms for the cooperationevolution such as kin selection direct reciprocity indirectreciprocity network reciprocity and group selection (shownin Figure 3)
Haldane believes that the altruism exists in the evolution[18] According to the study of the social insect Hamilton[19] used mathematic model for quantitative description anddefined the Hamilton rules The rule states natural selectionfavors cooperation if the donor and the recipient have kinrelationships Direct reciprocity requires several repeatedencounters between two individuals which can offer helpwith each other The direct reciprocity theory framework isrepeated PrisonerrsquosDilemmaThe classical game strategy is titfor tat TFT [20] The interactions among individuals innature are often asymmetrical and fleeting Therefore indi-rect reciprocity is a more prevalent reciprocity form Indirectreciprocity refers to establishing a good reputation by helpingothers which will be rewarded by others later In Indirectreciprocity one individual acts as donor while the other actsas recipient Indirect reciprocity will favor the cooperationevolution if 119902 the probability of one individualrsquos reputationexceeds the cost-to-benefit ratio of the altruistic act namely119902 gt 119888119887 [21] Network reciprocity is a new form of reciprocityCooperators can form network clusters in order to help eachother The cooperation mechanism of this spatial reciprocityis called network reciprocity Group selection is a minimaliststochastic model [22] The population is divided into severalgroups Cooperators help each other in their own groupswhile defectors do not helpThe pure cooperator groups growfaster than the pure defector groups while the defector groupsreproduce faster than the cooperator groups in any mixedgroup Therefore defectors are often selected in the lowerlevel (within groups) while cooperators are often selected inthe higher (between groups) level
0 2 4 6 8 100
2
4
6
8
10Gaseous phase
Cloud particle
Figure 4 Cloud gaseous phase
24 Description of the Proposed Algorithm The main goal ofthe cloud particles evolution system is to realize the evolutionof the cloud particles through phase transformation In cloudparticles evolution model cloud particles have three phasescloud gaseous phase cloud liquid phase and cloud solidphase Cloud gaseous phase indicates gaseous cloud particleswhich are distributed in the entire search space Cloud liquidphase refers to the rain in nature Cloud solid phase representshailstones in nature
The proposed algorithm which is similar to other meta-heuristic algorithms begins with an initial population calledthe cloud particles At the beginning let the initial state becloud gaseous phase which is composed of many cloud parti-cles (shown in Figure 4) The best individual (best cloud par-ticle) is chosen as a seed or nucleus Then the cloud particlesare going to have a condensational growth or collisions andcoalescence growth (shown in Figure 5) If the populationevolves successfully the cloud particles are going to changefrom cloud gaseous state to liquid state by condensationCloud changes from gaseous phase to cloud liquid phaseThisprocess is called phase transformation When cloud particlesare in liquid state cloud particles transform from globalexploration to local exploitation gradually In this processaccording to the idea of the indirect reciprocity mentionedbefore if 119902 gt 119888119887 the algorithm will favor the cooperationevolution or it will change from the liquid phase to solid phaseby phase transformation When cloud particles are in solid
4 Mathematical Problems in Engineering
0 2 4 6 8 100
2
4
6
8
10Liquid phase
Collision coalescence
Condensation sublimation
Figure 5 Cloud liquid phase
2 3 4 5 6 7 83
4
5
6
7
8Solid phase
Hailstones
Hailstones
Figure 6 Cloud solid phase
state (shown in Figure 6) the cloud particles condense intoice crystals at the beginning and then ice crystals curdle intohailstones The hailstone is the best solution
It can be seen from the biological evolutionary behaviorthat the individuals compete to survive However competi-tive ability is always asymmetry because the different individ-ual has different fitness Usually it is easier for disadvantagedindividuals to compete with the individuals in the same sit-uation while they seldom compete with the advantaged indi-viduals [23] Therefore competition and cooperation whichare two different survival strategies for living organisms areformed in order to adapt to the environment In the CPEAmodel according to the natural phenomena that the pop-ulation develops cooperatively when they evolve these twostrategies can be shown as followsThe populations carry outthe competitive evolution in cloud gaseous phase In cloudliquid phase the populations employ competitivemechanismfor improving local exploitation ability and employ reciprocalmechanism to ensure the diversity of the population Cloudsolid phase indicates the algorithm has found the optimalsolution area therefore the algorithm realizes fast conver-gence by competitive evolution The optimization process ofCPEA model is shown in Figure 7
In CPEA the initialization phase transformation drivingforce condensation operation reciprocity operation andsolidification operation are described as follows
Liquidphase
Solidphase
Optimizationsolution
Globalexploration
Conversionstate Local
exploitation
Gaseousphase
Reciprocalevolution
GlobalLocal
Competitiveevolution
Successfulevolution
Successfulevolution
Successful evolution
Phasetransition
Phase transition
Figure 7 Optimization process of CPEA model
241 Initialization When the metaheuristic methods areused to solve the optimization problem one population isformed at first Then the variables which are involved in theoptimization problem can be represented as an array In thisalgorithm one single solution is called one ldquocloud particlerdquoThe cloud particles have three parameters [24] They areExpectation (119864
119909) Entropy (119864
119899) and Hyperentropy (119867
119890) 119864119909
is the expectation values of the distribution for all cloud par-ticles in the domain 119864
119899is the range of domain which can be
accepted by linguistic values (qualitative concept) In anotherword 119864
119899is ambiguity119867
119890is the dispersion degree of entropy
(119864119899) That is to say119867
119890is the entropy of entropy In addition
119867119890can be also defined as the uncertaintymeasure for entropy
In 119898 dimensional optimization problems a cloud particle isan array of 1 times 119898 This array is defined as follows
Cloud particle = [1199091 1199092 119909
119898] (1)
To start the optimization algorithm amatrix of cloud par-ticles which size is119873times119898 is generated by cloud generator (iepopulation of cloud particles) Cloud particles generation isdescribed as follows Φ(120583 120575) is the normal random variablewhich has an expectation 120583 and a variance 120575119873 is the numberof cloud particle 119883 is the cloud particle generated by cloudgenerator Consider
119878 = 119904119894| Φ (119864
119899 119867119890) 119894 = 1 sdot sdot sdot 119873
119883 = 119909119894| Φ (119864
119909 119904119894) 119904119894isin 119878 119894 = 1 sdot sdot sdot 119873
(2)
The matrix 119883 generated by cloud generator is given as(rows and column are the number of population and thenumber of design variables resp)
Population of cloud particles
=
[[[[[[
[
Cloudparticle1
Cloudparticle2
Cloudparticle119873
]]]]]]
]
=
[[[[[[[
[
1198831
1
1198831
2
1198831
3
sdot sdot sdot 1198831
119898
1198832
1
1198832
2
1198832
3
sdot sdot sdot 1198832
119898
119883119873
1
119883119873
2
119883119873
3
sdot sdot sdot 119883119873
119898
]]]]]]]
]
(3)
Mathematical Problems in Engineering 5
Each of the decision variable values (1199091 1199092 119909
119898) can
represent real values for continuous or discrete problemsThefitness of a cloud particle is calculated by the evaluation ofobjective function (119865) given as
119865119894= 119891 (119909
1
1
1199091
2
1199091
119894
) 119894 = 1 2 3 119898 (4)
In the function above119873 and 119898 are the number of cloudparticles and the number of decision variablesThe cloud par-ticles which number is119873 are generated by the cloud genera-torThe cloud particle which has the optimal value is selectedas the nucleus The total population consists of119873
119901subpopu-
lations Each subpopulation of cloud particles selects a cloudparticle as the nucleus Therefore there are119873
119901nuclei
242 Phase Transformation Driving Force The phase trans-formation driving force (PT) can reflect the evolution extentof the population In the proposed algorithm phase trans-formation driving force is the factor of deciding phase trans-form During the evolution the energy possessed by a cloudparticle is called its evolution potential Phase transformationdriving force is the evolution potential energy differencebetween the cloud particles of new generation and the cloudparticles of old generationThe phase transformation drivingforce is defined as follows
EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897
119894119894 = 1 2 3 119892119890119899 (5)
PT = radic(EPgenindexminus1 minus EP)
2
+ (EPgenindex minus EP)2
2
EP =EPgenindexminus1 + EPgenindex
2
(6)
The cloud evolution system realizes phase transformationif PT le 120575 EP represents the evolution potential of cloudparticles gen represents the maximum generation of thealgorithm bestindividual represents the best solution of thecurrent generation genindex is the number of the currentevolution
243 Condensation Operation The population realizes theglobal exploration in cloud gaseous phase by condensationoperation The population takes the best individual as thenucleus in the process of the condensation operation Inthe cloud gaseous phase the condensation growth space ofcloud particle is calculated by the following equations cd isthe condensation factor generation is the number of currentevolution 119864
119899119894is the entropy of the 119894th subpopulation 119867
119890119894is
hyperentropy of the 119894th subpopulation Consider
119864119899119894=
10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816
radic119892119890119899119890119903119886119905119894119900119899 (7)
119867119890119894=119864119899119894
cd (8)
244 Reciprocity Operation If the evolution potential ofcloud particles between the two generations is less than a
given threshold in cloud liquid phase the algorithm willdetermine whether to employ the reciprocal evolution If thereciprocal evolution condition is met the reciprocal evolu-tion is employed Otherwise the algorithm carries out phasetransformation In reciprocity operation the probability ofone individualrsquos reputation is calculated by 119906 119906 isin [0 1] isa uniformly random number The cost-to-benefit ratio of thealtruistic act is (1minus0618) asymp 038 and 0618 is the golden ratio119873119901represents the number of the subpopulations 119899119906119888119897119890119906119904
119894is
the nucleus of the 119894th subpopulation Consider
119899119906119888119897119890119906119904119873119901=sum119873119901minus1
119894=1
119899119906119888119897119890119906119904119894
119873119901minus 1
(9)
Generally the optima value will act as the nucleus How-ever the nucleus of the last subpopulation is the mean of theoptima values coming from the rest of the subpopulations inreciprocity operation The purpose of the reciprocity mech-anism is to ensure the diversity of the cloud particles Forexample circle represents the location of the global optimumand pentagram represents the location of the local optima inFigure 8 Figure 8(a) displays the position of nucleus beforereciprocity operator Figure 8(b) shows the individuals gen-erated by cloud generator based on Figure 8(a) Figure 8(c)shows the position of nucleus with reciprocity operator basedon Figure 8(a) Figure 8(d) shows the individuals generatedby cloud generator based on Figure 8(c) By comparingFigure 8(b) with Figure 8(d) it can be seen that the distri-bution of the individuals in Figure 8(d) is more widely thanthose in Figure 8(b)
245 Solidification Operation The algorithm realizes con-vergence operation in cloud solid phase by solidificationoperation sf is the solidification factor
119864119899119894119895=
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816
sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895
10038161003816100381610038161003816gt 10minus2
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816 otherwise
(10)
119867119890119894119895=
119864119899119894119895
100 (11)
where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873
119901represents the
number of the subpopulations 119863 represents the number ofdimensions
The pseudocode of CPEA is illustrated in Algorithms 12 3 and 4 119873
119901represents the number of the subpopulation
Childnum represents the number of individuals of everysubpopulation The search space is [minus119903 119903]
3 Experiments and Discussions
In this section the performance of the proposed CPEA istested with a comprehensive set of benchmark functionswhich include 40 different global optimization problemsand CEC2013 benchmark problems The definition of thebenchmark functions and their global optimum(s) are listedin the Appendix The benchmark functions are chosen from
6 Mathematical Problems in Engineering
2 25 3 35 4 45 5044046048
05052054056058
06
NucleusLocal optima valueGlobal optima value
x2
x1
(a)
IndividualsLocal optima valueGlobal optima value
2 25 3 35 4 45 504
045
05
055
06
065
x2
x1
(b)
25 26 27 28 29 3 31044
046
048
05
052
054
056
NucleusLocal optima valueGlobal optima value
x2
x1
(c)
IndividualsLocal optima valueGlobal optima value
22 23 24 25 26 27 28 29 3 31 3204
045
05
055
06
x2
x1
(d)
Figure 8 Reciprocity operator
(1) if State == 0 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) if PT le 120575 then(5) Phase Transformation from gaseous to liquid(6) 119873
119901
= lceil119873119901
2rceil
(7) Childnum = Childnum times 2(8) Select the nucleus of each sub-population(9) 120575 = 12057510
(10) State = 1(11) else(12) Condensation Operation(13) for 119894 = 1119873
119875
(14) Update 119864119899119894for each sub-population according to (7)
(15) Update119867119890119894for each sub-population according to (8)
(16) endfor(17) endif(18) endif
State = 0 Cloud Gaseous Phase State = 1 Cloud Liquid Phase State = 2 Cloud Solid Phase
Algorithm 1 Evolutionary algorithm of the cloud gaseous state
Mathematical Problems in Engineering 7
(1) if State == 1 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) for 119894 = 1119873
119901
(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904
119894119895
| gt 1
(7) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|20
(8) elseif |119899119906119888119897119890119906119904119894119895
| gt 01
(9) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|3
(10) else(11) 119864
119899119894119895= |119899119906119888119897119890119906119904
119894119895
|
(12) endif(13) endfor(14) endfor(15) for 119894 = 1119873
119901
(16) Update119867119890119894for each sub-population according to (8)
(17) endfor(18) if PT le 120575 then(19) if rand gt 119888119887 then(20) Reciprocity Operation according to (9) Reciprocity Operation(21) else(22) Phase Transformation from liquid to solid(23) 119873
119901
= lceil119873119901
2rceil
(24) Childnum = Childnum times 2(25) Select the nucleus of each sub-population(26) 120575 = 120575100
(27) State = 2(28) endif(29) endif(30) endif
Algorithm 2 Evolutionary algorithm of the cloud liquid state
(1) if State == 2 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) Solidification Operation(5) if PT le 120575 then(6) sf = sf times 2(7) 120575 = 12057510
(8) endif(9) for 119894 = 1119873
119875
(10) for 119895 = 1119863(11) Update 119864
119899119894119895according to (10)
(12) Update119867119890119894119895according to (11)
(13) endfor(14) endfor(15) endif
Algorithm 3 Evolutionary algorithm of the cloud solid state
Ali et al [25] and Rahnamayan et al [26]They are unimodalfunctions multimodal functions and rotate functions Allthe algorithms are implemented on the same machine witha T2450 24GHz CPU 15 GB memory and windows XPoperating system with Matlab R2009b
31 Parameter Study In this section the setting basis forinitial values of some parameters involved in the cloud parti-cles evolution algorithm model is described Firstly we hopethat all the cloud particles which are generated by the cloudgenerator can fall into the search space as much as possible
8 Mathematical Problems in Engineering
(1) Initialize119863 (number of dimensions)119873119901
= 5 Childnum = 30 119864119899
=23r119867119890
= 119864119899
1000(2) for 119894 = 1119873
119901
(3) for 119895 = 1119863(4) if 119895 is even(5) 119864
119909119894119895= 1199032
(6) else(7) 119864
119909119894119895= minus1199032
(8) end if(9) end for(10) end for(11) Use cloud generator to produce cloud particles(12) Evaluate Subpopulation(13) Selectnucleus(14) while requirements are not satisfied do(15) Use cloud generator to produce cloud particles(16) Evaluate Subpopulation(17) Selectnucleus(18) Evolutionary algorithm of the cloud gaseous state(19) Evolutionary algorithm of the cloud liquid state(20) Evolutionary algorithm of the cloud solid state(21) end while
Algorithm 4 Cloud particles evolution algorithm CPEA
1 2 3 4 5 6 7 8 9 10965
970
975
980
985
990
995
1000
Number of run times
Valid
clou
d pa
rtic
les
1 2 3 4 5 6 7 8 9 10810
820
830
840
850
860
870
Number of run times
Valid
clou
d pa
rtic
les
En = (13)r
En = (12)r
En = (23)r
En = (34)rEn = (45)r
En = r
Figure 9 Valid cloud particles of different 119864119899
Secondly the diversity of the population can be ensured effec-tively if the cloud particles are distributed in the search spaceuniformly Therefore some experiments are designed toanalyze the feasibility of the selected scheme in order toillustrate the rationality of the selected parameters
311 Definition of 119864119899 Let the search range 119878 be [minus119903 119903] and 119903
is the search radius Set 119903 = 2 1198641199091= minus23 119864
1199092= 23 119867
119890=
0001119864119899 Five hundred cloud particles are generated in every
experiment The experiment is independently simulated 30times The cloud particle is called a valid cloud particle if itfalls into the search space Figure 9 is a graphwhich shows thenumber of the valid cloud particles generated with different119864119899 and Figure 10 is a graph which shows the distribution of
valid cloud particles generated with different 119864119899
It can be seen from Figure 9 that the number of thevalid cloud particles reduces as the 119864
119899increases The average
number of valid cloud particles is basically the same when119864119899is (13)119903 (12)119903 and (23)119903 respectively However the
number of valid cloud particles is relatively fewer when 119864119899is
(34)119903 (45)119903 and 119903 It can be seen from Figure 10 that all thecloud particles fall into the research space when 119864
119899is (13)119903
(12)119903 and (23)119903 respectively However the cloud particlesconcentrate in a certain area of the search space when 119864
119899is
(13)119903 or (12)119903 By contrast the cloud particles are widelydistributed throughout all the search space [minus2 2] when 119864
119899
is (23)119903 Therefore set 119864119899= (23)119903 The dashed rectangle in
Figure 10 represents the decision variables space
312 Definition of 119867119890 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches119891 stopwith different119867
119890 Let the search space
Mathematical Problems in Engineering 9
0 05 1 15 2 25
012
0 05 1 15 2
012
0 05 1 15 2
005
115
2
0 1 2
01234
0 1 2
0123
0 1 2 3
012345
En = (13)r En = (12)r En = (23)r
En = (34)r En = (45)r En = r
x1 x1
x1 x1 x1
x1
x2
x2
x2
x2
x2
x2
minus1
minus2
minus2 minus15 minus1 minus05
minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1
minus2 minus15 minus1 minus05 minus2minus2
minus15
minus15
minus1
minus1
minus05
minus05
minus3
minus1minus2minus3
minus1
minus2
minus1minus2
minus1
minus2
minus3
Figure 10 Distribution of cloud particles in different 119864119899
1 2 3 4 5 6 76000
8000
10000
12000
NFF
Es
Ackley function (D = 100)
He
(a)
1 2 3 4 5 6 73000
4000
5000
6000
7000N
FFEs
Sphere function (D = 100)
He
(b)
Figure 11 NFES of CPEA in different119867119890
be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =
10minus3 1198671198901= 119864119899 1198671198902= 01119864
119899 1198671198903= 001119864
119899 1198671198904= 0001119864
119899
1198671198905= 00001119864
119899 1198671198906= 000001119864
119899 and 119867
1198907= 0000001119864
119899
The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867
119890is equal
to 0001119864119899 For Sphere Function the number of fitness eval-
uations is less when 119867119890is equal to 0001119864
119899 Therefore 119867
119890is
equal to 0001119864119899in the cloud particles evolution algorithm
313 Definition of 119864119909 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864
119909being equal to 119903 (12)119903
(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10
minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864
119909
is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864
119909is equal to (12)119903 or (13)119903There-
fore 119864119909is equal to (12)119903 in the cloud particles evolution
algorithm
314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically
119886 + 119887
119886=119886
119887
def= 120593 (12)
in which 120593 represents the golden ration Its value is
120593 =1 + radic5
2= 16180339887
1
120593= 120593 minus 1 = 0618 sdot sdot sdot
(13)
The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
This paper which is inspired by the natural phenomenaof cloud formation presents a novel optimization algorithmto solve optimization problems called cloud particles evolu-tion algorithm (CPEA) The proposed algorithm uses phasetransformation mechanism to imitate the formation anddiversification of the cloud to solve the optimization prob-lems It is well known that the species evolution is influencedby individual fitness living environment competition andcooperation between the individuals Therefore the algo-rithm introduces competition and reciprocal mechanism inthe evolution process so as to strengthen the viability of thecloud particlesThe performance of the cloud particles evolu-tion algorithm (CPEA) is tested on the classical optimizationproblems The results are compared with other algorithmsin terms of best function value and the number of functionevaluations
The remainder of this paper is organized as follows InSection 2 the proposed CPEA and the concepts are intro-duced in detail In Section 3 the performance of the proposedalgorithm is validated by different optimization problemsThe conclusion and future research are drawn in Section 4
2 Cloud Particles Evolution Algorithm
21 Cloud Formation The idea of the proposed CPEA isinspired from cloud formation The state of the cloud parti-clesrsquo changing process [10ndash12] is shown in Figure 1
After the wet air rises it will go through a set of changesbecause of some certain reasons temperature and pressureThen the water vapor is formedWater vapor plays an impor-tant role in this process When the water vapor is saturatedit will adsorb on the cloud condensation nuclei (CNN) andthe initial cloud particles are formed Cloud particles arecondensed and sublimated continuously by absorbing watervapor around the cloud In the process of the condensationthe cloud particles are getting closer and colliding Then thelarger cloud particles are formed Ice phase is generated whenthe temperature is below 0∘C at this time The emergence ofice crystals damages the stable status of cloud phase structureIce crystals grow up quickly to be the snow crystals becauseof the condensation of the water vapor On the one handwater vapor transfers to ice crystals and ice crystals beginto grow up as supercooled water droplets evaporate On theother hand the supercooledwater dropletswill be frozen afterthe colliding to make the snow particles grow bigger If thesupercooledwater vapor which has been collided is toomanythe snow crystals will be transferred to be solid substancesThen the solid substances such as snow and hail will bedropped If the solid substances descent in the area which hasa higher temperature over 0∘C they will melt into raindropThewater in our planet will be irradiated by the sunshine andit will become the water vapor again in the air The cloud isformed and changed like this
22 Phase and Phase Transformation There are various kindsof substance in nature Substances commonly exist in threestates solid liquid and gaseous Figure 2 is a simple diagramto show howwater transforms among the three states Evapo-ration condensation andmelting are the threeways for water
Raindroplets
Cloudparticles
Snowpellets
cloudparticles
Large
Figure 1 Cloud particles transformation process
Depos
ition
Subli
mati
on
Melting
Freezing
Evap
orati
onCon
dens
ation
Figure 2 Three phases of water
transforming In order to describe the different forms of thesubstance the phase can be used as a symbol for states of sub-stances In thermodynamics phase is defined as some partswhich have the same quality [13] A phase transformation isthe transition which is from one phase to another in thermo-dynamics
Phase transformation is relevant to Gibbs free energyIn thermodynamics the Gibbs free energy [14] is a ther-modynamic potential that measures the ldquousefulrdquo or process-initiating work obtainable from a thermodynamic system ata constant temperature and pressure
23 Population Evolution Mechanism Darwin pointed outthat mutation and natural selection are the two importantfactors for the evolution which explained many natural phe-nomena successfully [15] Nowak from Harvard University
Mathematical Problems in Engineering 3
1 21
3
21
3
2
4
65
78
Cooperators
Kin selection Direct reciprocity Indirect reciprocity Network reciprocity Group selection
Defectors
1 r
Figure 3 Five mechanisms of cooperation evolution [16]
ranked the cooperation as the third important factor of thespecies evolution Evolution is a unity of the cooperationand the opposition [16] Every gene cell and organism ofthe individual should strengthen its own evolution process atthe expense of beating its rivals Therefore individuals oftenconflict with each other because of limited resources [17]However the complex life organizational structure in differ-ent levels is produced in the process of evolution which indi-cates that the cooperation exists in the evolution ThereforeNowak summarizes five mechanisms for the cooperationevolution such as kin selection direct reciprocity indirectreciprocity network reciprocity and group selection (shownin Figure 3)
Haldane believes that the altruism exists in the evolution[18] According to the study of the social insect Hamilton[19] used mathematic model for quantitative description anddefined the Hamilton rules The rule states natural selectionfavors cooperation if the donor and the recipient have kinrelationships Direct reciprocity requires several repeatedencounters between two individuals which can offer helpwith each other The direct reciprocity theory framework isrepeated PrisonerrsquosDilemmaThe classical game strategy is titfor tat TFT [20] The interactions among individuals innature are often asymmetrical and fleeting Therefore indi-rect reciprocity is a more prevalent reciprocity form Indirectreciprocity refers to establishing a good reputation by helpingothers which will be rewarded by others later In Indirectreciprocity one individual acts as donor while the other actsas recipient Indirect reciprocity will favor the cooperationevolution if 119902 the probability of one individualrsquos reputationexceeds the cost-to-benefit ratio of the altruistic act namely119902 gt 119888119887 [21] Network reciprocity is a new form of reciprocityCooperators can form network clusters in order to help eachother The cooperation mechanism of this spatial reciprocityis called network reciprocity Group selection is a minimaliststochastic model [22] The population is divided into severalgroups Cooperators help each other in their own groupswhile defectors do not helpThe pure cooperator groups growfaster than the pure defector groups while the defector groupsreproduce faster than the cooperator groups in any mixedgroup Therefore defectors are often selected in the lowerlevel (within groups) while cooperators are often selected inthe higher (between groups) level
0 2 4 6 8 100
2
4
6
8
10Gaseous phase
Cloud particle
Figure 4 Cloud gaseous phase
24 Description of the Proposed Algorithm The main goal ofthe cloud particles evolution system is to realize the evolutionof the cloud particles through phase transformation In cloudparticles evolution model cloud particles have three phasescloud gaseous phase cloud liquid phase and cloud solidphase Cloud gaseous phase indicates gaseous cloud particleswhich are distributed in the entire search space Cloud liquidphase refers to the rain in nature Cloud solid phase representshailstones in nature
The proposed algorithm which is similar to other meta-heuristic algorithms begins with an initial population calledthe cloud particles At the beginning let the initial state becloud gaseous phase which is composed of many cloud parti-cles (shown in Figure 4) The best individual (best cloud par-ticle) is chosen as a seed or nucleus Then the cloud particlesare going to have a condensational growth or collisions andcoalescence growth (shown in Figure 5) If the populationevolves successfully the cloud particles are going to changefrom cloud gaseous state to liquid state by condensationCloud changes from gaseous phase to cloud liquid phaseThisprocess is called phase transformation When cloud particlesare in liquid state cloud particles transform from globalexploration to local exploitation gradually In this processaccording to the idea of the indirect reciprocity mentionedbefore if 119902 gt 119888119887 the algorithm will favor the cooperationevolution or it will change from the liquid phase to solid phaseby phase transformation When cloud particles are in solid
4 Mathematical Problems in Engineering
0 2 4 6 8 100
2
4
6
8
10Liquid phase
Collision coalescence
Condensation sublimation
Figure 5 Cloud liquid phase
2 3 4 5 6 7 83
4
5
6
7
8Solid phase
Hailstones
Hailstones
Figure 6 Cloud solid phase
state (shown in Figure 6) the cloud particles condense intoice crystals at the beginning and then ice crystals curdle intohailstones The hailstone is the best solution
It can be seen from the biological evolutionary behaviorthat the individuals compete to survive However competi-tive ability is always asymmetry because the different individ-ual has different fitness Usually it is easier for disadvantagedindividuals to compete with the individuals in the same sit-uation while they seldom compete with the advantaged indi-viduals [23] Therefore competition and cooperation whichare two different survival strategies for living organisms areformed in order to adapt to the environment In the CPEAmodel according to the natural phenomena that the pop-ulation develops cooperatively when they evolve these twostrategies can be shown as followsThe populations carry outthe competitive evolution in cloud gaseous phase In cloudliquid phase the populations employ competitivemechanismfor improving local exploitation ability and employ reciprocalmechanism to ensure the diversity of the population Cloudsolid phase indicates the algorithm has found the optimalsolution area therefore the algorithm realizes fast conver-gence by competitive evolution The optimization process ofCPEA model is shown in Figure 7
In CPEA the initialization phase transformation drivingforce condensation operation reciprocity operation andsolidification operation are described as follows
Liquidphase
Solidphase
Optimizationsolution
Globalexploration
Conversionstate Local
exploitation
Gaseousphase
Reciprocalevolution
GlobalLocal
Competitiveevolution
Successfulevolution
Successfulevolution
Successful evolution
Phasetransition
Phase transition
Figure 7 Optimization process of CPEA model
241 Initialization When the metaheuristic methods areused to solve the optimization problem one population isformed at first Then the variables which are involved in theoptimization problem can be represented as an array In thisalgorithm one single solution is called one ldquocloud particlerdquoThe cloud particles have three parameters [24] They areExpectation (119864
119909) Entropy (119864
119899) and Hyperentropy (119867
119890) 119864119909
is the expectation values of the distribution for all cloud par-ticles in the domain 119864
119899is the range of domain which can be
accepted by linguistic values (qualitative concept) In anotherword 119864
119899is ambiguity119867
119890is the dispersion degree of entropy
(119864119899) That is to say119867
119890is the entropy of entropy In addition
119867119890can be also defined as the uncertaintymeasure for entropy
In 119898 dimensional optimization problems a cloud particle isan array of 1 times 119898 This array is defined as follows
Cloud particle = [1199091 1199092 119909
119898] (1)
To start the optimization algorithm amatrix of cloud par-ticles which size is119873times119898 is generated by cloud generator (iepopulation of cloud particles) Cloud particles generation isdescribed as follows Φ(120583 120575) is the normal random variablewhich has an expectation 120583 and a variance 120575119873 is the numberof cloud particle 119883 is the cloud particle generated by cloudgenerator Consider
119878 = 119904119894| Φ (119864
119899 119867119890) 119894 = 1 sdot sdot sdot 119873
119883 = 119909119894| Φ (119864
119909 119904119894) 119904119894isin 119878 119894 = 1 sdot sdot sdot 119873
(2)
The matrix 119883 generated by cloud generator is given as(rows and column are the number of population and thenumber of design variables resp)
Population of cloud particles
=
[[[[[[
[
Cloudparticle1
Cloudparticle2
Cloudparticle119873
]]]]]]
]
=
[[[[[[[
[
1198831
1
1198831
2
1198831
3
sdot sdot sdot 1198831
119898
1198832
1
1198832
2
1198832
3
sdot sdot sdot 1198832
119898
119883119873
1
119883119873
2
119883119873
3
sdot sdot sdot 119883119873
119898
]]]]]]]
]
(3)
Mathematical Problems in Engineering 5
Each of the decision variable values (1199091 1199092 119909
119898) can
represent real values for continuous or discrete problemsThefitness of a cloud particle is calculated by the evaluation ofobjective function (119865) given as
119865119894= 119891 (119909
1
1
1199091
2
1199091
119894
) 119894 = 1 2 3 119898 (4)
In the function above119873 and 119898 are the number of cloudparticles and the number of decision variablesThe cloud par-ticles which number is119873 are generated by the cloud genera-torThe cloud particle which has the optimal value is selectedas the nucleus The total population consists of119873
119901subpopu-
lations Each subpopulation of cloud particles selects a cloudparticle as the nucleus Therefore there are119873
119901nuclei
242 Phase Transformation Driving Force The phase trans-formation driving force (PT) can reflect the evolution extentof the population In the proposed algorithm phase trans-formation driving force is the factor of deciding phase trans-form During the evolution the energy possessed by a cloudparticle is called its evolution potential Phase transformationdriving force is the evolution potential energy differencebetween the cloud particles of new generation and the cloudparticles of old generationThe phase transformation drivingforce is defined as follows
EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897
119894119894 = 1 2 3 119892119890119899 (5)
PT = radic(EPgenindexminus1 minus EP)
2
+ (EPgenindex minus EP)2
2
EP =EPgenindexminus1 + EPgenindex
2
(6)
The cloud evolution system realizes phase transformationif PT le 120575 EP represents the evolution potential of cloudparticles gen represents the maximum generation of thealgorithm bestindividual represents the best solution of thecurrent generation genindex is the number of the currentevolution
243 Condensation Operation The population realizes theglobal exploration in cloud gaseous phase by condensationoperation The population takes the best individual as thenucleus in the process of the condensation operation Inthe cloud gaseous phase the condensation growth space ofcloud particle is calculated by the following equations cd isthe condensation factor generation is the number of currentevolution 119864
119899119894is the entropy of the 119894th subpopulation 119867
119890119894is
hyperentropy of the 119894th subpopulation Consider
119864119899119894=
10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816
radic119892119890119899119890119903119886119905119894119900119899 (7)
119867119890119894=119864119899119894
cd (8)
244 Reciprocity Operation If the evolution potential ofcloud particles between the two generations is less than a
given threshold in cloud liquid phase the algorithm willdetermine whether to employ the reciprocal evolution If thereciprocal evolution condition is met the reciprocal evolu-tion is employed Otherwise the algorithm carries out phasetransformation In reciprocity operation the probability ofone individualrsquos reputation is calculated by 119906 119906 isin [0 1] isa uniformly random number The cost-to-benefit ratio of thealtruistic act is (1minus0618) asymp 038 and 0618 is the golden ratio119873119901represents the number of the subpopulations 119899119906119888119897119890119906119904
119894is
the nucleus of the 119894th subpopulation Consider
119899119906119888119897119890119906119904119873119901=sum119873119901minus1
119894=1
119899119906119888119897119890119906119904119894
119873119901minus 1
(9)
Generally the optima value will act as the nucleus How-ever the nucleus of the last subpopulation is the mean of theoptima values coming from the rest of the subpopulations inreciprocity operation The purpose of the reciprocity mech-anism is to ensure the diversity of the cloud particles Forexample circle represents the location of the global optimumand pentagram represents the location of the local optima inFigure 8 Figure 8(a) displays the position of nucleus beforereciprocity operator Figure 8(b) shows the individuals gen-erated by cloud generator based on Figure 8(a) Figure 8(c)shows the position of nucleus with reciprocity operator basedon Figure 8(a) Figure 8(d) shows the individuals generatedby cloud generator based on Figure 8(c) By comparingFigure 8(b) with Figure 8(d) it can be seen that the distri-bution of the individuals in Figure 8(d) is more widely thanthose in Figure 8(b)
245 Solidification Operation The algorithm realizes con-vergence operation in cloud solid phase by solidificationoperation sf is the solidification factor
119864119899119894119895=
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816
sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895
10038161003816100381610038161003816gt 10minus2
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816 otherwise
(10)
119867119890119894119895=
119864119899119894119895
100 (11)
where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873
119901represents the
number of the subpopulations 119863 represents the number ofdimensions
The pseudocode of CPEA is illustrated in Algorithms 12 3 and 4 119873
119901represents the number of the subpopulation
Childnum represents the number of individuals of everysubpopulation The search space is [minus119903 119903]
3 Experiments and Discussions
In this section the performance of the proposed CPEA istested with a comprehensive set of benchmark functionswhich include 40 different global optimization problemsand CEC2013 benchmark problems The definition of thebenchmark functions and their global optimum(s) are listedin the Appendix The benchmark functions are chosen from
6 Mathematical Problems in Engineering
2 25 3 35 4 45 5044046048
05052054056058
06
NucleusLocal optima valueGlobal optima value
x2
x1
(a)
IndividualsLocal optima valueGlobal optima value
2 25 3 35 4 45 504
045
05
055
06
065
x2
x1
(b)
25 26 27 28 29 3 31044
046
048
05
052
054
056
NucleusLocal optima valueGlobal optima value
x2
x1
(c)
IndividualsLocal optima valueGlobal optima value
22 23 24 25 26 27 28 29 3 31 3204
045
05
055
06
x2
x1
(d)
Figure 8 Reciprocity operator
(1) if State == 0 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) if PT le 120575 then(5) Phase Transformation from gaseous to liquid(6) 119873
119901
= lceil119873119901
2rceil
(7) Childnum = Childnum times 2(8) Select the nucleus of each sub-population(9) 120575 = 12057510
(10) State = 1(11) else(12) Condensation Operation(13) for 119894 = 1119873
119875
(14) Update 119864119899119894for each sub-population according to (7)
(15) Update119867119890119894for each sub-population according to (8)
(16) endfor(17) endif(18) endif
State = 0 Cloud Gaseous Phase State = 1 Cloud Liquid Phase State = 2 Cloud Solid Phase
Algorithm 1 Evolutionary algorithm of the cloud gaseous state
Mathematical Problems in Engineering 7
(1) if State == 1 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) for 119894 = 1119873
119901
(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904
119894119895
| gt 1
(7) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|20
(8) elseif |119899119906119888119897119890119906119904119894119895
| gt 01
(9) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|3
(10) else(11) 119864
119899119894119895= |119899119906119888119897119890119906119904
119894119895
|
(12) endif(13) endfor(14) endfor(15) for 119894 = 1119873
119901
(16) Update119867119890119894for each sub-population according to (8)
(17) endfor(18) if PT le 120575 then(19) if rand gt 119888119887 then(20) Reciprocity Operation according to (9) Reciprocity Operation(21) else(22) Phase Transformation from liquid to solid(23) 119873
119901
= lceil119873119901
2rceil
(24) Childnum = Childnum times 2(25) Select the nucleus of each sub-population(26) 120575 = 120575100
(27) State = 2(28) endif(29) endif(30) endif
Algorithm 2 Evolutionary algorithm of the cloud liquid state
(1) if State == 2 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) Solidification Operation(5) if PT le 120575 then(6) sf = sf times 2(7) 120575 = 12057510
(8) endif(9) for 119894 = 1119873
119875
(10) for 119895 = 1119863(11) Update 119864
119899119894119895according to (10)
(12) Update119867119890119894119895according to (11)
(13) endfor(14) endfor(15) endif
Algorithm 3 Evolutionary algorithm of the cloud solid state
Ali et al [25] and Rahnamayan et al [26]They are unimodalfunctions multimodal functions and rotate functions Allthe algorithms are implemented on the same machine witha T2450 24GHz CPU 15 GB memory and windows XPoperating system with Matlab R2009b
31 Parameter Study In this section the setting basis forinitial values of some parameters involved in the cloud parti-cles evolution algorithm model is described Firstly we hopethat all the cloud particles which are generated by the cloudgenerator can fall into the search space as much as possible
8 Mathematical Problems in Engineering
(1) Initialize119863 (number of dimensions)119873119901
= 5 Childnum = 30 119864119899
=23r119867119890
= 119864119899
1000(2) for 119894 = 1119873
119901
(3) for 119895 = 1119863(4) if 119895 is even(5) 119864
119909119894119895= 1199032
(6) else(7) 119864
119909119894119895= minus1199032
(8) end if(9) end for(10) end for(11) Use cloud generator to produce cloud particles(12) Evaluate Subpopulation(13) Selectnucleus(14) while requirements are not satisfied do(15) Use cloud generator to produce cloud particles(16) Evaluate Subpopulation(17) Selectnucleus(18) Evolutionary algorithm of the cloud gaseous state(19) Evolutionary algorithm of the cloud liquid state(20) Evolutionary algorithm of the cloud solid state(21) end while
Algorithm 4 Cloud particles evolution algorithm CPEA
1 2 3 4 5 6 7 8 9 10965
970
975
980
985
990
995
1000
Number of run times
Valid
clou
d pa
rtic
les
1 2 3 4 5 6 7 8 9 10810
820
830
840
850
860
870
Number of run times
Valid
clou
d pa
rtic
les
En = (13)r
En = (12)r
En = (23)r
En = (34)rEn = (45)r
En = r
Figure 9 Valid cloud particles of different 119864119899
Secondly the diversity of the population can be ensured effec-tively if the cloud particles are distributed in the search spaceuniformly Therefore some experiments are designed toanalyze the feasibility of the selected scheme in order toillustrate the rationality of the selected parameters
311 Definition of 119864119899 Let the search range 119878 be [minus119903 119903] and 119903
is the search radius Set 119903 = 2 1198641199091= minus23 119864
1199092= 23 119867
119890=
0001119864119899 Five hundred cloud particles are generated in every
experiment The experiment is independently simulated 30times The cloud particle is called a valid cloud particle if itfalls into the search space Figure 9 is a graphwhich shows thenumber of the valid cloud particles generated with different119864119899 and Figure 10 is a graph which shows the distribution of
valid cloud particles generated with different 119864119899
It can be seen from Figure 9 that the number of thevalid cloud particles reduces as the 119864
119899increases The average
number of valid cloud particles is basically the same when119864119899is (13)119903 (12)119903 and (23)119903 respectively However the
number of valid cloud particles is relatively fewer when 119864119899is
(34)119903 (45)119903 and 119903 It can be seen from Figure 10 that all thecloud particles fall into the research space when 119864
119899is (13)119903
(12)119903 and (23)119903 respectively However the cloud particlesconcentrate in a certain area of the search space when 119864
119899is
(13)119903 or (12)119903 By contrast the cloud particles are widelydistributed throughout all the search space [minus2 2] when 119864
119899
is (23)119903 Therefore set 119864119899= (23)119903 The dashed rectangle in
Figure 10 represents the decision variables space
312 Definition of 119867119890 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches119891 stopwith different119867
119890 Let the search space
Mathematical Problems in Engineering 9
0 05 1 15 2 25
012
0 05 1 15 2
012
0 05 1 15 2
005
115
2
0 1 2
01234
0 1 2
0123
0 1 2 3
012345
En = (13)r En = (12)r En = (23)r
En = (34)r En = (45)r En = r
x1 x1
x1 x1 x1
x1
x2
x2
x2
x2
x2
x2
minus1
minus2
minus2 minus15 minus1 minus05
minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1
minus2 minus15 minus1 minus05 minus2minus2
minus15
minus15
minus1
minus1
minus05
minus05
minus3
minus1minus2minus3
minus1
minus2
minus1minus2
minus1
minus2
minus3
Figure 10 Distribution of cloud particles in different 119864119899
1 2 3 4 5 6 76000
8000
10000
12000
NFF
Es
Ackley function (D = 100)
He
(a)
1 2 3 4 5 6 73000
4000
5000
6000
7000N
FFEs
Sphere function (D = 100)
He
(b)
Figure 11 NFES of CPEA in different119867119890
be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =
10minus3 1198671198901= 119864119899 1198671198902= 01119864
119899 1198671198903= 001119864
119899 1198671198904= 0001119864
119899
1198671198905= 00001119864
119899 1198671198906= 000001119864
119899 and 119867
1198907= 0000001119864
119899
The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867
119890is equal
to 0001119864119899 For Sphere Function the number of fitness eval-
uations is less when 119867119890is equal to 0001119864
119899 Therefore 119867
119890is
equal to 0001119864119899in the cloud particles evolution algorithm
313 Definition of 119864119909 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864
119909being equal to 119903 (12)119903
(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10
minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864
119909
is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864
119909is equal to (12)119903 or (13)119903There-
fore 119864119909is equal to (12)119903 in the cloud particles evolution
algorithm
314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically
119886 + 119887
119886=119886
119887
def= 120593 (12)
in which 120593 represents the golden ration Its value is
120593 =1 + radic5
2= 16180339887
1
120593= 120593 minus 1 = 0618 sdot sdot sdot
(13)
The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
1 21
3
21
3
2
4
65
78
Cooperators
Kin selection Direct reciprocity Indirect reciprocity Network reciprocity Group selection
Defectors
1 r
Figure 3 Five mechanisms of cooperation evolution [16]
ranked the cooperation as the third important factor of thespecies evolution Evolution is a unity of the cooperationand the opposition [16] Every gene cell and organism ofthe individual should strengthen its own evolution process atthe expense of beating its rivals Therefore individuals oftenconflict with each other because of limited resources [17]However the complex life organizational structure in differ-ent levels is produced in the process of evolution which indi-cates that the cooperation exists in the evolution ThereforeNowak summarizes five mechanisms for the cooperationevolution such as kin selection direct reciprocity indirectreciprocity network reciprocity and group selection (shownin Figure 3)
Haldane believes that the altruism exists in the evolution[18] According to the study of the social insect Hamilton[19] used mathematic model for quantitative description anddefined the Hamilton rules The rule states natural selectionfavors cooperation if the donor and the recipient have kinrelationships Direct reciprocity requires several repeatedencounters between two individuals which can offer helpwith each other The direct reciprocity theory framework isrepeated PrisonerrsquosDilemmaThe classical game strategy is titfor tat TFT [20] The interactions among individuals innature are often asymmetrical and fleeting Therefore indi-rect reciprocity is a more prevalent reciprocity form Indirectreciprocity refers to establishing a good reputation by helpingothers which will be rewarded by others later In Indirectreciprocity one individual acts as donor while the other actsas recipient Indirect reciprocity will favor the cooperationevolution if 119902 the probability of one individualrsquos reputationexceeds the cost-to-benefit ratio of the altruistic act namely119902 gt 119888119887 [21] Network reciprocity is a new form of reciprocityCooperators can form network clusters in order to help eachother The cooperation mechanism of this spatial reciprocityis called network reciprocity Group selection is a minimaliststochastic model [22] The population is divided into severalgroups Cooperators help each other in their own groupswhile defectors do not helpThe pure cooperator groups growfaster than the pure defector groups while the defector groupsreproduce faster than the cooperator groups in any mixedgroup Therefore defectors are often selected in the lowerlevel (within groups) while cooperators are often selected inthe higher (between groups) level
0 2 4 6 8 100
2
4
6
8
10Gaseous phase
Cloud particle
Figure 4 Cloud gaseous phase
24 Description of the Proposed Algorithm The main goal ofthe cloud particles evolution system is to realize the evolutionof the cloud particles through phase transformation In cloudparticles evolution model cloud particles have three phasescloud gaseous phase cloud liquid phase and cloud solidphase Cloud gaseous phase indicates gaseous cloud particleswhich are distributed in the entire search space Cloud liquidphase refers to the rain in nature Cloud solid phase representshailstones in nature
The proposed algorithm which is similar to other meta-heuristic algorithms begins with an initial population calledthe cloud particles At the beginning let the initial state becloud gaseous phase which is composed of many cloud parti-cles (shown in Figure 4) The best individual (best cloud par-ticle) is chosen as a seed or nucleus Then the cloud particlesare going to have a condensational growth or collisions andcoalescence growth (shown in Figure 5) If the populationevolves successfully the cloud particles are going to changefrom cloud gaseous state to liquid state by condensationCloud changes from gaseous phase to cloud liquid phaseThisprocess is called phase transformation When cloud particlesare in liquid state cloud particles transform from globalexploration to local exploitation gradually In this processaccording to the idea of the indirect reciprocity mentionedbefore if 119902 gt 119888119887 the algorithm will favor the cooperationevolution or it will change from the liquid phase to solid phaseby phase transformation When cloud particles are in solid
4 Mathematical Problems in Engineering
0 2 4 6 8 100
2
4
6
8
10Liquid phase
Collision coalescence
Condensation sublimation
Figure 5 Cloud liquid phase
2 3 4 5 6 7 83
4
5
6
7
8Solid phase
Hailstones
Hailstones
Figure 6 Cloud solid phase
state (shown in Figure 6) the cloud particles condense intoice crystals at the beginning and then ice crystals curdle intohailstones The hailstone is the best solution
It can be seen from the biological evolutionary behaviorthat the individuals compete to survive However competi-tive ability is always asymmetry because the different individ-ual has different fitness Usually it is easier for disadvantagedindividuals to compete with the individuals in the same sit-uation while they seldom compete with the advantaged indi-viduals [23] Therefore competition and cooperation whichare two different survival strategies for living organisms areformed in order to adapt to the environment In the CPEAmodel according to the natural phenomena that the pop-ulation develops cooperatively when they evolve these twostrategies can be shown as followsThe populations carry outthe competitive evolution in cloud gaseous phase In cloudliquid phase the populations employ competitivemechanismfor improving local exploitation ability and employ reciprocalmechanism to ensure the diversity of the population Cloudsolid phase indicates the algorithm has found the optimalsolution area therefore the algorithm realizes fast conver-gence by competitive evolution The optimization process ofCPEA model is shown in Figure 7
In CPEA the initialization phase transformation drivingforce condensation operation reciprocity operation andsolidification operation are described as follows
Liquidphase
Solidphase
Optimizationsolution
Globalexploration
Conversionstate Local
exploitation
Gaseousphase
Reciprocalevolution
GlobalLocal
Competitiveevolution
Successfulevolution
Successfulevolution
Successful evolution
Phasetransition
Phase transition
Figure 7 Optimization process of CPEA model
241 Initialization When the metaheuristic methods areused to solve the optimization problem one population isformed at first Then the variables which are involved in theoptimization problem can be represented as an array In thisalgorithm one single solution is called one ldquocloud particlerdquoThe cloud particles have three parameters [24] They areExpectation (119864
119909) Entropy (119864
119899) and Hyperentropy (119867
119890) 119864119909
is the expectation values of the distribution for all cloud par-ticles in the domain 119864
119899is the range of domain which can be
accepted by linguistic values (qualitative concept) In anotherword 119864
119899is ambiguity119867
119890is the dispersion degree of entropy
(119864119899) That is to say119867
119890is the entropy of entropy In addition
119867119890can be also defined as the uncertaintymeasure for entropy
In 119898 dimensional optimization problems a cloud particle isan array of 1 times 119898 This array is defined as follows
Cloud particle = [1199091 1199092 119909
119898] (1)
To start the optimization algorithm amatrix of cloud par-ticles which size is119873times119898 is generated by cloud generator (iepopulation of cloud particles) Cloud particles generation isdescribed as follows Φ(120583 120575) is the normal random variablewhich has an expectation 120583 and a variance 120575119873 is the numberof cloud particle 119883 is the cloud particle generated by cloudgenerator Consider
119878 = 119904119894| Φ (119864
119899 119867119890) 119894 = 1 sdot sdot sdot 119873
119883 = 119909119894| Φ (119864
119909 119904119894) 119904119894isin 119878 119894 = 1 sdot sdot sdot 119873
(2)
The matrix 119883 generated by cloud generator is given as(rows and column are the number of population and thenumber of design variables resp)
Population of cloud particles
=
[[[[[[
[
Cloudparticle1
Cloudparticle2
Cloudparticle119873
]]]]]]
]
=
[[[[[[[
[
1198831
1
1198831
2
1198831
3
sdot sdot sdot 1198831
119898
1198832
1
1198832
2
1198832
3
sdot sdot sdot 1198832
119898
119883119873
1
119883119873
2
119883119873
3
sdot sdot sdot 119883119873
119898
]]]]]]]
]
(3)
Mathematical Problems in Engineering 5
Each of the decision variable values (1199091 1199092 119909
119898) can
represent real values for continuous or discrete problemsThefitness of a cloud particle is calculated by the evaluation ofobjective function (119865) given as
119865119894= 119891 (119909
1
1
1199091
2
1199091
119894
) 119894 = 1 2 3 119898 (4)
In the function above119873 and 119898 are the number of cloudparticles and the number of decision variablesThe cloud par-ticles which number is119873 are generated by the cloud genera-torThe cloud particle which has the optimal value is selectedas the nucleus The total population consists of119873
119901subpopu-
lations Each subpopulation of cloud particles selects a cloudparticle as the nucleus Therefore there are119873
119901nuclei
242 Phase Transformation Driving Force The phase trans-formation driving force (PT) can reflect the evolution extentof the population In the proposed algorithm phase trans-formation driving force is the factor of deciding phase trans-form During the evolution the energy possessed by a cloudparticle is called its evolution potential Phase transformationdriving force is the evolution potential energy differencebetween the cloud particles of new generation and the cloudparticles of old generationThe phase transformation drivingforce is defined as follows
EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897
119894119894 = 1 2 3 119892119890119899 (5)
PT = radic(EPgenindexminus1 minus EP)
2
+ (EPgenindex minus EP)2
2
EP =EPgenindexminus1 + EPgenindex
2
(6)
The cloud evolution system realizes phase transformationif PT le 120575 EP represents the evolution potential of cloudparticles gen represents the maximum generation of thealgorithm bestindividual represents the best solution of thecurrent generation genindex is the number of the currentevolution
243 Condensation Operation The population realizes theglobal exploration in cloud gaseous phase by condensationoperation The population takes the best individual as thenucleus in the process of the condensation operation Inthe cloud gaseous phase the condensation growth space ofcloud particle is calculated by the following equations cd isthe condensation factor generation is the number of currentevolution 119864
119899119894is the entropy of the 119894th subpopulation 119867
119890119894is
hyperentropy of the 119894th subpopulation Consider
119864119899119894=
10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816
radic119892119890119899119890119903119886119905119894119900119899 (7)
119867119890119894=119864119899119894
cd (8)
244 Reciprocity Operation If the evolution potential ofcloud particles between the two generations is less than a
given threshold in cloud liquid phase the algorithm willdetermine whether to employ the reciprocal evolution If thereciprocal evolution condition is met the reciprocal evolu-tion is employed Otherwise the algorithm carries out phasetransformation In reciprocity operation the probability ofone individualrsquos reputation is calculated by 119906 119906 isin [0 1] isa uniformly random number The cost-to-benefit ratio of thealtruistic act is (1minus0618) asymp 038 and 0618 is the golden ratio119873119901represents the number of the subpopulations 119899119906119888119897119890119906119904
119894is
the nucleus of the 119894th subpopulation Consider
119899119906119888119897119890119906119904119873119901=sum119873119901minus1
119894=1
119899119906119888119897119890119906119904119894
119873119901minus 1
(9)
Generally the optima value will act as the nucleus How-ever the nucleus of the last subpopulation is the mean of theoptima values coming from the rest of the subpopulations inreciprocity operation The purpose of the reciprocity mech-anism is to ensure the diversity of the cloud particles Forexample circle represents the location of the global optimumand pentagram represents the location of the local optima inFigure 8 Figure 8(a) displays the position of nucleus beforereciprocity operator Figure 8(b) shows the individuals gen-erated by cloud generator based on Figure 8(a) Figure 8(c)shows the position of nucleus with reciprocity operator basedon Figure 8(a) Figure 8(d) shows the individuals generatedby cloud generator based on Figure 8(c) By comparingFigure 8(b) with Figure 8(d) it can be seen that the distri-bution of the individuals in Figure 8(d) is more widely thanthose in Figure 8(b)
245 Solidification Operation The algorithm realizes con-vergence operation in cloud solid phase by solidificationoperation sf is the solidification factor
119864119899119894119895=
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816
sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895
10038161003816100381610038161003816gt 10minus2
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816 otherwise
(10)
119867119890119894119895=
119864119899119894119895
100 (11)
where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873
119901represents the
number of the subpopulations 119863 represents the number ofdimensions
The pseudocode of CPEA is illustrated in Algorithms 12 3 and 4 119873
119901represents the number of the subpopulation
Childnum represents the number of individuals of everysubpopulation The search space is [minus119903 119903]
3 Experiments and Discussions
In this section the performance of the proposed CPEA istested with a comprehensive set of benchmark functionswhich include 40 different global optimization problemsand CEC2013 benchmark problems The definition of thebenchmark functions and their global optimum(s) are listedin the Appendix The benchmark functions are chosen from
6 Mathematical Problems in Engineering
2 25 3 35 4 45 5044046048
05052054056058
06
NucleusLocal optima valueGlobal optima value
x2
x1
(a)
IndividualsLocal optima valueGlobal optima value
2 25 3 35 4 45 504
045
05
055
06
065
x2
x1
(b)
25 26 27 28 29 3 31044
046
048
05
052
054
056
NucleusLocal optima valueGlobal optima value
x2
x1
(c)
IndividualsLocal optima valueGlobal optima value
22 23 24 25 26 27 28 29 3 31 3204
045
05
055
06
x2
x1
(d)
Figure 8 Reciprocity operator
(1) if State == 0 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) if PT le 120575 then(5) Phase Transformation from gaseous to liquid(6) 119873
119901
= lceil119873119901
2rceil
(7) Childnum = Childnum times 2(8) Select the nucleus of each sub-population(9) 120575 = 12057510
(10) State = 1(11) else(12) Condensation Operation(13) for 119894 = 1119873
119875
(14) Update 119864119899119894for each sub-population according to (7)
(15) Update119867119890119894for each sub-population according to (8)
(16) endfor(17) endif(18) endif
State = 0 Cloud Gaseous Phase State = 1 Cloud Liquid Phase State = 2 Cloud Solid Phase
Algorithm 1 Evolutionary algorithm of the cloud gaseous state
Mathematical Problems in Engineering 7
(1) if State == 1 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) for 119894 = 1119873
119901
(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904
119894119895
| gt 1
(7) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|20
(8) elseif |119899119906119888119897119890119906119904119894119895
| gt 01
(9) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|3
(10) else(11) 119864
119899119894119895= |119899119906119888119897119890119906119904
119894119895
|
(12) endif(13) endfor(14) endfor(15) for 119894 = 1119873
119901
(16) Update119867119890119894for each sub-population according to (8)
(17) endfor(18) if PT le 120575 then(19) if rand gt 119888119887 then(20) Reciprocity Operation according to (9) Reciprocity Operation(21) else(22) Phase Transformation from liquid to solid(23) 119873
119901
= lceil119873119901
2rceil
(24) Childnum = Childnum times 2(25) Select the nucleus of each sub-population(26) 120575 = 120575100
(27) State = 2(28) endif(29) endif(30) endif
Algorithm 2 Evolutionary algorithm of the cloud liquid state
(1) if State == 2 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) Solidification Operation(5) if PT le 120575 then(6) sf = sf times 2(7) 120575 = 12057510
(8) endif(9) for 119894 = 1119873
119875
(10) for 119895 = 1119863(11) Update 119864
119899119894119895according to (10)
(12) Update119867119890119894119895according to (11)
(13) endfor(14) endfor(15) endif
Algorithm 3 Evolutionary algorithm of the cloud solid state
Ali et al [25] and Rahnamayan et al [26]They are unimodalfunctions multimodal functions and rotate functions Allthe algorithms are implemented on the same machine witha T2450 24GHz CPU 15 GB memory and windows XPoperating system with Matlab R2009b
31 Parameter Study In this section the setting basis forinitial values of some parameters involved in the cloud parti-cles evolution algorithm model is described Firstly we hopethat all the cloud particles which are generated by the cloudgenerator can fall into the search space as much as possible
8 Mathematical Problems in Engineering
(1) Initialize119863 (number of dimensions)119873119901
= 5 Childnum = 30 119864119899
=23r119867119890
= 119864119899
1000(2) for 119894 = 1119873
119901
(3) for 119895 = 1119863(4) if 119895 is even(5) 119864
119909119894119895= 1199032
(6) else(7) 119864
119909119894119895= minus1199032
(8) end if(9) end for(10) end for(11) Use cloud generator to produce cloud particles(12) Evaluate Subpopulation(13) Selectnucleus(14) while requirements are not satisfied do(15) Use cloud generator to produce cloud particles(16) Evaluate Subpopulation(17) Selectnucleus(18) Evolutionary algorithm of the cloud gaseous state(19) Evolutionary algorithm of the cloud liquid state(20) Evolutionary algorithm of the cloud solid state(21) end while
Algorithm 4 Cloud particles evolution algorithm CPEA
1 2 3 4 5 6 7 8 9 10965
970
975
980
985
990
995
1000
Number of run times
Valid
clou
d pa
rtic
les
1 2 3 4 5 6 7 8 9 10810
820
830
840
850
860
870
Number of run times
Valid
clou
d pa
rtic
les
En = (13)r
En = (12)r
En = (23)r
En = (34)rEn = (45)r
En = r
Figure 9 Valid cloud particles of different 119864119899
Secondly the diversity of the population can be ensured effec-tively if the cloud particles are distributed in the search spaceuniformly Therefore some experiments are designed toanalyze the feasibility of the selected scheme in order toillustrate the rationality of the selected parameters
311 Definition of 119864119899 Let the search range 119878 be [minus119903 119903] and 119903
is the search radius Set 119903 = 2 1198641199091= minus23 119864
1199092= 23 119867
119890=
0001119864119899 Five hundred cloud particles are generated in every
experiment The experiment is independently simulated 30times The cloud particle is called a valid cloud particle if itfalls into the search space Figure 9 is a graphwhich shows thenumber of the valid cloud particles generated with different119864119899 and Figure 10 is a graph which shows the distribution of
valid cloud particles generated with different 119864119899
It can be seen from Figure 9 that the number of thevalid cloud particles reduces as the 119864
119899increases The average
number of valid cloud particles is basically the same when119864119899is (13)119903 (12)119903 and (23)119903 respectively However the
number of valid cloud particles is relatively fewer when 119864119899is
(34)119903 (45)119903 and 119903 It can be seen from Figure 10 that all thecloud particles fall into the research space when 119864
119899is (13)119903
(12)119903 and (23)119903 respectively However the cloud particlesconcentrate in a certain area of the search space when 119864
119899is
(13)119903 or (12)119903 By contrast the cloud particles are widelydistributed throughout all the search space [minus2 2] when 119864
119899
is (23)119903 Therefore set 119864119899= (23)119903 The dashed rectangle in
Figure 10 represents the decision variables space
312 Definition of 119867119890 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches119891 stopwith different119867
119890 Let the search space
Mathematical Problems in Engineering 9
0 05 1 15 2 25
012
0 05 1 15 2
012
0 05 1 15 2
005
115
2
0 1 2
01234
0 1 2
0123
0 1 2 3
012345
En = (13)r En = (12)r En = (23)r
En = (34)r En = (45)r En = r
x1 x1
x1 x1 x1
x1
x2
x2
x2
x2
x2
x2
minus1
minus2
minus2 minus15 minus1 minus05
minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1
minus2 minus15 minus1 minus05 minus2minus2
minus15
minus15
minus1
minus1
minus05
minus05
minus3
minus1minus2minus3
minus1
minus2
minus1minus2
minus1
minus2
minus3
Figure 10 Distribution of cloud particles in different 119864119899
1 2 3 4 5 6 76000
8000
10000
12000
NFF
Es
Ackley function (D = 100)
He
(a)
1 2 3 4 5 6 73000
4000
5000
6000
7000N
FFEs
Sphere function (D = 100)
He
(b)
Figure 11 NFES of CPEA in different119867119890
be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =
10minus3 1198671198901= 119864119899 1198671198902= 01119864
119899 1198671198903= 001119864
119899 1198671198904= 0001119864
119899
1198671198905= 00001119864
119899 1198671198906= 000001119864
119899 and 119867
1198907= 0000001119864
119899
The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867
119890is equal
to 0001119864119899 For Sphere Function the number of fitness eval-
uations is less when 119867119890is equal to 0001119864
119899 Therefore 119867
119890is
equal to 0001119864119899in the cloud particles evolution algorithm
313 Definition of 119864119909 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864
119909being equal to 119903 (12)119903
(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10
minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864
119909
is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864
119909is equal to (12)119903 or (13)119903There-
fore 119864119909is equal to (12)119903 in the cloud particles evolution
algorithm
314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically
119886 + 119887
119886=119886
119887
def= 120593 (12)
in which 120593 represents the golden ration Its value is
120593 =1 + radic5
2= 16180339887
1
120593= 120593 minus 1 = 0618 sdot sdot sdot
(13)
The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
0 2 4 6 8 100
2
4
6
8
10Liquid phase
Collision coalescence
Condensation sublimation
Figure 5 Cloud liquid phase
2 3 4 5 6 7 83
4
5
6
7
8Solid phase
Hailstones
Hailstones
Figure 6 Cloud solid phase
state (shown in Figure 6) the cloud particles condense intoice crystals at the beginning and then ice crystals curdle intohailstones The hailstone is the best solution
It can be seen from the biological evolutionary behaviorthat the individuals compete to survive However competi-tive ability is always asymmetry because the different individ-ual has different fitness Usually it is easier for disadvantagedindividuals to compete with the individuals in the same sit-uation while they seldom compete with the advantaged indi-viduals [23] Therefore competition and cooperation whichare two different survival strategies for living organisms areformed in order to adapt to the environment In the CPEAmodel according to the natural phenomena that the pop-ulation develops cooperatively when they evolve these twostrategies can be shown as followsThe populations carry outthe competitive evolution in cloud gaseous phase In cloudliquid phase the populations employ competitivemechanismfor improving local exploitation ability and employ reciprocalmechanism to ensure the diversity of the population Cloudsolid phase indicates the algorithm has found the optimalsolution area therefore the algorithm realizes fast conver-gence by competitive evolution The optimization process ofCPEA model is shown in Figure 7
In CPEA the initialization phase transformation drivingforce condensation operation reciprocity operation andsolidification operation are described as follows
Liquidphase
Solidphase
Optimizationsolution
Globalexploration
Conversionstate Local
exploitation
Gaseousphase
Reciprocalevolution
GlobalLocal
Competitiveevolution
Successfulevolution
Successfulevolution
Successful evolution
Phasetransition
Phase transition
Figure 7 Optimization process of CPEA model
241 Initialization When the metaheuristic methods areused to solve the optimization problem one population isformed at first Then the variables which are involved in theoptimization problem can be represented as an array In thisalgorithm one single solution is called one ldquocloud particlerdquoThe cloud particles have three parameters [24] They areExpectation (119864
119909) Entropy (119864
119899) and Hyperentropy (119867
119890) 119864119909
is the expectation values of the distribution for all cloud par-ticles in the domain 119864
119899is the range of domain which can be
accepted by linguistic values (qualitative concept) In anotherword 119864
119899is ambiguity119867
119890is the dispersion degree of entropy
(119864119899) That is to say119867
119890is the entropy of entropy In addition
119867119890can be also defined as the uncertaintymeasure for entropy
In 119898 dimensional optimization problems a cloud particle isan array of 1 times 119898 This array is defined as follows
Cloud particle = [1199091 1199092 119909
119898] (1)
To start the optimization algorithm amatrix of cloud par-ticles which size is119873times119898 is generated by cloud generator (iepopulation of cloud particles) Cloud particles generation isdescribed as follows Φ(120583 120575) is the normal random variablewhich has an expectation 120583 and a variance 120575119873 is the numberof cloud particle 119883 is the cloud particle generated by cloudgenerator Consider
119878 = 119904119894| Φ (119864
119899 119867119890) 119894 = 1 sdot sdot sdot 119873
119883 = 119909119894| Φ (119864
119909 119904119894) 119904119894isin 119878 119894 = 1 sdot sdot sdot 119873
(2)
The matrix 119883 generated by cloud generator is given as(rows and column are the number of population and thenumber of design variables resp)
Population of cloud particles
=
[[[[[[
[
Cloudparticle1
Cloudparticle2
Cloudparticle119873
]]]]]]
]
=
[[[[[[[
[
1198831
1
1198831
2
1198831
3
sdot sdot sdot 1198831
119898
1198832
1
1198832
2
1198832
3
sdot sdot sdot 1198832
119898
119883119873
1
119883119873
2
119883119873
3
sdot sdot sdot 119883119873
119898
]]]]]]]
]
(3)
Mathematical Problems in Engineering 5
Each of the decision variable values (1199091 1199092 119909
119898) can
represent real values for continuous or discrete problemsThefitness of a cloud particle is calculated by the evaluation ofobjective function (119865) given as
119865119894= 119891 (119909
1
1
1199091
2
1199091
119894
) 119894 = 1 2 3 119898 (4)
In the function above119873 and 119898 are the number of cloudparticles and the number of decision variablesThe cloud par-ticles which number is119873 are generated by the cloud genera-torThe cloud particle which has the optimal value is selectedas the nucleus The total population consists of119873
119901subpopu-
lations Each subpopulation of cloud particles selects a cloudparticle as the nucleus Therefore there are119873
119901nuclei
242 Phase Transformation Driving Force The phase trans-formation driving force (PT) can reflect the evolution extentof the population In the proposed algorithm phase trans-formation driving force is the factor of deciding phase trans-form During the evolution the energy possessed by a cloudparticle is called its evolution potential Phase transformationdriving force is the evolution potential energy differencebetween the cloud particles of new generation and the cloudparticles of old generationThe phase transformation drivingforce is defined as follows
EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897
119894119894 = 1 2 3 119892119890119899 (5)
PT = radic(EPgenindexminus1 minus EP)
2
+ (EPgenindex minus EP)2
2
EP =EPgenindexminus1 + EPgenindex
2
(6)
The cloud evolution system realizes phase transformationif PT le 120575 EP represents the evolution potential of cloudparticles gen represents the maximum generation of thealgorithm bestindividual represents the best solution of thecurrent generation genindex is the number of the currentevolution
243 Condensation Operation The population realizes theglobal exploration in cloud gaseous phase by condensationoperation The population takes the best individual as thenucleus in the process of the condensation operation Inthe cloud gaseous phase the condensation growth space ofcloud particle is calculated by the following equations cd isthe condensation factor generation is the number of currentevolution 119864
119899119894is the entropy of the 119894th subpopulation 119867
119890119894is
hyperentropy of the 119894th subpopulation Consider
119864119899119894=
10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816
radic119892119890119899119890119903119886119905119894119900119899 (7)
119867119890119894=119864119899119894
cd (8)
244 Reciprocity Operation If the evolution potential ofcloud particles between the two generations is less than a
given threshold in cloud liquid phase the algorithm willdetermine whether to employ the reciprocal evolution If thereciprocal evolution condition is met the reciprocal evolu-tion is employed Otherwise the algorithm carries out phasetransformation In reciprocity operation the probability ofone individualrsquos reputation is calculated by 119906 119906 isin [0 1] isa uniformly random number The cost-to-benefit ratio of thealtruistic act is (1minus0618) asymp 038 and 0618 is the golden ratio119873119901represents the number of the subpopulations 119899119906119888119897119890119906119904
119894is
the nucleus of the 119894th subpopulation Consider
119899119906119888119897119890119906119904119873119901=sum119873119901minus1
119894=1
119899119906119888119897119890119906119904119894
119873119901minus 1
(9)
Generally the optima value will act as the nucleus How-ever the nucleus of the last subpopulation is the mean of theoptima values coming from the rest of the subpopulations inreciprocity operation The purpose of the reciprocity mech-anism is to ensure the diversity of the cloud particles Forexample circle represents the location of the global optimumand pentagram represents the location of the local optima inFigure 8 Figure 8(a) displays the position of nucleus beforereciprocity operator Figure 8(b) shows the individuals gen-erated by cloud generator based on Figure 8(a) Figure 8(c)shows the position of nucleus with reciprocity operator basedon Figure 8(a) Figure 8(d) shows the individuals generatedby cloud generator based on Figure 8(c) By comparingFigure 8(b) with Figure 8(d) it can be seen that the distri-bution of the individuals in Figure 8(d) is more widely thanthose in Figure 8(b)
245 Solidification Operation The algorithm realizes con-vergence operation in cloud solid phase by solidificationoperation sf is the solidification factor
119864119899119894119895=
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816
sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895
10038161003816100381610038161003816gt 10minus2
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816 otherwise
(10)
119867119890119894119895=
119864119899119894119895
100 (11)
where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873
119901represents the
number of the subpopulations 119863 represents the number ofdimensions
The pseudocode of CPEA is illustrated in Algorithms 12 3 and 4 119873
119901represents the number of the subpopulation
Childnum represents the number of individuals of everysubpopulation The search space is [minus119903 119903]
3 Experiments and Discussions
In this section the performance of the proposed CPEA istested with a comprehensive set of benchmark functionswhich include 40 different global optimization problemsand CEC2013 benchmark problems The definition of thebenchmark functions and their global optimum(s) are listedin the Appendix The benchmark functions are chosen from
6 Mathematical Problems in Engineering
2 25 3 35 4 45 5044046048
05052054056058
06
NucleusLocal optima valueGlobal optima value
x2
x1
(a)
IndividualsLocal optima valueGlobal optima value
2 25 3 35 4 45 504
045
05
055
06
065
x2
x1
(b)
25 26 27 28 29 3 31044
046
048
05
052
054
056
NucleusLocal optima valueGlobal optima value
x2
x1
(c)
IndividualsLocal optima valueGlobal optima value
22 23 24 25 26 27 28 29 3 31 3204
045
05
055
06
x2
x1
(d)
Figure 8 Reciprocity operator
(1) if State == 0 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) if PT le 120575 then(5) Phase Transformation from gaseous to liquid(6) 119873
119901
= lceil119873119901
2rceil
(7) Childnum = Childnum times 2(8) Select the nucleus of each sub-population(9) 120575 = 12057510
(10) State = 1(11) else(12) Condensation Operation(13) for 119894 = 1119873
119875
(14) Update 119864119899119894for each sub-population according to (7)
(15) Update119867119890119894for each sub-population according to (8)
(16) endfor(17) endif(18) endif
State = 0 Cloud Gaseous Phase State = 1 Cloud Liquid Phase State = 2 Cloud Solid Phase
Algorithm 1 Evolutionary algorithm of the cloud gaseous state
Mathematical Problems in Engineering 7
(1) if State == 1 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) for 119894 = 1119873
119901
(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904
119894119895
| gt 1
(7) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|20
(8) elseif |119899119906119888119897119890119906119904119894119895
| gt 01
(9) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|3
(10) else(11) 119864
119899119894119895= |119899119906119888119897119890119906119904
119894119895
|
(12) endif(13) endfor(14) endfor(15) for 119894 = 1119873
119901
(16) Update119867119890119894for each sub-population according to (8)
(17) endfor(18) if PT le 120575 then(19) if rand gt 119888119887 then(20) Reciprocity Operation according to (9) Reciprocity Operation(21) else(22) Phase Transformation from liquid to solid(23) 119873
119901
= lceil119873119901
2rceil
(24) Childnum = Childnum times 2(25) Select the nucleus of each sub-population(26) 120575 = 120575100
(27) State = 2(28) endif(29) endif(30) endif
Algorithm 2 Evolutionary algorithm of the cloud liquid state
(1) if State == 2 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) Solidification Operation(5) if PT le 120575 then(6) sf = sf times 2(7) 120575 = 12057510
(8) endif(9) for 119894 = 1119873
119875
(10) for 119895 = 1119863(11) Update 119864
119899119894119895according to (10)
(12) Update119867119890119894119895according to (11)
(13) endfor(14) endfor(15) endif
Algorithm 3 Evolutionary algorithm of the cloud solid state
Ali et al [25] and Rahnamayan et al [26]They are unimodalfunctions multimodal functions and rotate functions Allthe algorithms are implemented on the same machine witha T2450 24GHz CPU 15 GB memory and windows XPoperating system with Matlab R2009b
31 Parameter Study In this section the setting basis forinitial values of some parameters involved in the cloud parti-cles evolution algorithm model is described Firstly we hopethat all the cloud particles which are generated by the cloudgenerator can fall into the search space as much as possible
8 Mathematical Problems in Engineering
(1) Initialize119863 (number of dimensions)119873119901
= 5 Childnum = 30 119864119899
=23r119867119890
= 119864119899
1000(2) for 119894 = 1119873
119901
(3) for 119895 = 1119863(4) if 119895 is even(5) 119864
119909119894119895= 1199032
(6) else(7) 119864
119909119894119895= minus1199032
(8) end if(9) end for(10) end for(11) Use cloud generator to produce cloud particles(12) Evaluate Subpopulation(13) Selectnucleus(14) while requirements are not satisfied do(15) Use cloud generator to produce cloud particles(16) Evaluate Subpopulation(17) Selectnucleus(18) Evolutionary algorithm of the cloud gaseous state(19) Evolutionary algorithm of the cloud liquid state(20) Evolutionary algorithm of the cloud solid state(21) end while
Algorithm 4 Cloud particles evolution algorithm CPEA
1 2 3 4 5 6 7 8 9 10965
970
975
980
985
990
995
1000
Number of run times
Valid
clou
d pa
rtic
les
1 2 3 4 5 6 7 8 9 10810
820
830
840
850
860
870
Number of run times
Valid
clou
d pa
rtic
les
En = (13)r
En = (12)r
En = (23)r
En = (34)rEn = (45)r
En = r
Figure 9 Valid cloud particles of different 119864119899
Secondly the diversity of the population can be ensured effec-tively if the cloud particles are distributed in the search spaceuniformly Therefore some experiments are designed toanalyze the feasibility of the selected scheme in order toillustrate the rationality of the selected parameters
311 Definition of 119864119899 Let the search range 119878 be [minus119903 119903] and 119903
is the search radius Set 119903 = 2 1198641199091= minus23 119864
1199092= 23 119867
119890=
0001119864119899 Five hundred cloud particles are generated in every
experiment The experiment is independently simulated 30times The cloud particle is called a valid cloud particle if itfalls into the search space Figure 9 is a graphwhich shows thenumber of the valid cloud particles generated with different119864119899 and Figure 10 is a graph which shows the distribution of
valid cloud particles generated with different 119864119899
It can be seen from Figure 9 that the number of thevalid cloud particles reduces as the 119864
119899increases The average
number of valid cloud particles is basically the same when119864119899is (13)119903 (12)119903 and (23)119903 respectively However the
number of valid cloud particles is relatively fewer when 119864119899is
(34)119903 (45)119903 and 119903 It can be seen from Figure 10 that all thecloud particles fall into the research space when 119864
119899is (13)119903
(12)119903 and (23)119903 respectively However the cloud particlesconcentrate in a certain area of the search space when 119864
119899is
(13)119903 or (12)119903 By contrast the cloud particles are widelydistributed throughout all the search space [minus2 2] when 119864
119899
is (23)119903 Therefore set 119864119899= (23)119903 The dashed rectangle in
Figure 10 represents the decision variables space
312 Definition of 119867119890 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches119891 stopwith different119867
119890 Let the search space
Mathematical Problems in Engineering 9
0 05 1 15 2 25
012
0 05 1 15 2
012
0 05 1 15 2
005
115
2
0 1 2
01234
0 1 2
0123
0 1 2 3
012345
En = (13)r En = (12)r En = (23)r
En = (34)r En = (45)r En = r
x1 x1
x1 x1 x1
x1
x2
x2
x2
x2
x2
x2
minus1
minus2
minus2 minus15 minus1 minus05
minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1
minus2 minus15 minus1 minus05 minus2minus2
minus15
minus15
minus1
minus1
minus05
minus05
minus3
minus1minus2minus3
minus1
minus2
minus1minus2
minus1
minus2
minus3
Figure 10 Distribution of cloud particles in different 119864119899
1 2 3 4 5 6 76000
8000
10000
12000
NFF
Es
Ackley function (D = 100)
He
(a)
1 2 3 4 5 6 73000
4000
5000
6000
7000N
FFEs
Sphere function (D = 100)
He
(b)
Figure 11 NFES of CPEA in different119867119890
be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =
10minus3 1198671198901= 119864119899 1198671198902= 01119864
119899 1198671198903= 001119864
119899 1198671198904= 0001119864
119899
1198671198905= 00001119864
119899 1198671198906= 000001119864
119899 and 119867
1198907= 0000001119864
119899
The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867
119890is equal
to 0001119864119899 For Sphere Function the number of fitness eval-
uations is less when 119867119890is equal to 0001119864
119899 Therefore 119867
119890is
equal to 0001119864119899in the cloud particles evolution algorithm
313 Definition of 119864119909 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864
119909being equal to 119903 (12)119903
(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10
minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864
119909
is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864
119909is equal to (12)119903 or (13)119903There-
fore 119864119909is equal to (12)119903 in the cloud particles evolution
algorithm
314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically
119886 + 119887
119886=119886
119887
def= 120593 (12)
in which 120593 represents the golden ration Its value is
120593 =1 + radic5
2= 16180339887
1
120593= 120593 minus 1 = 0618 sdot sdot sdot
(13)
The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Each of the decision variable values (1199091 1199092 119909
119898) can
represent real values for continuous or discrete problemsThefitness of a cloud particle is calculated by the evaluation ofobjective function (119865) given as
119865119894= 119891 (119909
1
1
1199091
2
1199091
119894
) 119894 = 1 2 3 119898 (4)
In the function above119873 and 119898 are the number of cloudparticles and the number of decision variablesThe cloud par-ticles which number is119873 are generated by the cloud genera-torThe cloud particle which has the optimal value is selectedas the nucleus The total population consists of119873
119901subpopu-
lations Each subpopulation of cloud particles selects a cloudparticle as the nucleus Therefore there are119873
119901nuclei
242 Phase Transformation Driving Force The phase trans-formation driving force (PT) can reflect the evolution extentof the population In the proposed algorithm phase trans-formation driving force is the factor of deciding phase trans-form During the evolution the energy possessed by a cloudparticle is called its evolution potential Phase transformationdriving force is the evolution potential energy differencebetween the cloud particles of new generation and the cloudparticles of old generationThe phase transformation drivingforce is defined as follows
EP119894= 119887119890119904119905119894119899119889119894V119894119889119906119886119897
119894119894 = 1 2 3 119892119890119899 (5)
PT = radic(EPgenindexminus1 minus EP)
2
+ (EPgenindex minus EP)2
2
EP =EPgenindexminus1 + EPgenindex
2
(6)
The cloud evolution system realizes phase transformationif PT le 120575 EP represents the evolution potential of cloudparticles gen represents the maximum generation of thealgorithm bestindividual represents the best solution of thecurrent generation genindex is the number of the currentevolution
243 Condensation Operation The population realizes theglobal exploration in cloud gaseous phase by condensationoperation The population takes the best individual as thenucleus in the process of the condensation operation Inthe cloud gaseous phase the condensation growth space ofcloud particle is calculated by the following equations cd isthe condensation factor generation is the number of currentevolution 119864
119899119894is the entropy of the 119894th subpopulation 119867
119890119894is
hyperentropy of the 119894th subpopulation Consider
119864119899119894=
10038161003816100381610038161198991199061198881198971198901199061199041198941003816100381610038161003816
radic119892119890119899119890119903119886119905119894119900119899 (7)
119867119890119894=119864119899119894
cd (8)
244 Reciprocity Operation If the evolution potential ofcloud particles between the two generations is less than a
given threshold in cloud liquid phase the algorithm willdetermine whether to employ the reciprocal evolution If thereciprocal evolution condition is met the reciprocal evolu-tion is employed Otherwise the algorithm carries out phasetransformation In reciprocity operation the probability ofone individualrsquos reputation is calculated by 119906 119906 isin [0 1] isa uniformly random number The cost-to-benefit ratio of thealtruistic act is (1minus0618) asymp 038 and 0618 is the golden ratio119873119901represents the number of the subpopulations 119899119906119888119897119890119906119904
119894is
the nucleus of the 119894th subpopulation Consider
119899119906119888119897119890119906119904119873119901=sum119873119901minus1
119894=1
119899119906119888119897119890119906119904119894
119873119901minus 1
(9)
Generally the optima value will act as the nucleus How-ever the nucleus of the last subpopulation is the mean of theoptima values coming from the rest of the subpopulations inreciprocity operation The purpose of the reciprocity mech-anism is to ensure the diversity of the cloud particles Forexample circle represents the location of the global optimumand pentagram represents the location of the local optima inFigure 8 Figure 8(a) displays the position of nucleus beforereciprocity operator Figure 8(b) shows the individuals gen-erated by cloud generator based on Figure 8(a) Figure 8(c)shows the position of nucleus with reciprocity operator basedon Figure 8(a) Figure 8(d) shows the individuals generatedby cloud generator based on Figure 8(c) By comparingFigure 8(b) with Figure 8(d) it can be seen that the distri-bution of the individuals in Figure 8(d) is more widely thanthose in Figure 8(b)
245 Solidification Operation The algorithm realizes con-vergence operation in cloud solid phase by solidificationoperation sf is the solidification factor
119864119899119894119895=
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816
sf if 10038161003816100381610038161003816119899119906119888119897119890119906119904119894119895
10038161003816100381610038161003816gt 10minus2
10038161003816100381610038161003816119899119906119888119897119890119906119904
119894119895
10038161003816100381610038161003816 otherwise
(10)
119867119890119894119895=
119864119899119894119895
100 (11)
where 119894 = 1 2 119873119901and 119895 = 1 2 119863119873
119901represents the
number of the subpopulations 119863 represents the number ofdimensions
The pseudocode of CPEA is illustrated in Algorithms 12 3 and 4 119873
119901represents the number of the subpopulation
Childnum represents the number of individuals of everysubpopulation The search space is [minus119903 119903]
3 Experiments and Discussions
In this section the performance of the proposed CPEA istested with a comprehensive set of benchmark functionswhich include 40 different global optimization problemsand CEC2013 benchmark problems The definition of thebenchmark functions and their global optimum(s) are listedin the Appendix The benchmark functions are chosen from
6 Mathematical Problems in Engineering
2 25 3 35 4 45 5044046048
05052054056058
06
NucleusLocal optima valueGlobal optima value
x2
x1
(a)
IndividualsLocal optima valueGlobal optima value
2 25 3 35 4 45 504
045
05
055
06
065
x2
x1
(b)
25 26 27 28 29 3 31044
046
048
05
052
054
056
NucleusLocal optima valueGlobal optima value
x2
x1
(c)
IndividualsLocal optima valueGlobal optima value
22 23 24 25 26 27 28 29 3 31 3204
045
05
055
06
x2
x1
(d)
Figure 8 Reciprocity operator
(1) if State == 0 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) if PT le 120575 then(5) Phase Transformation from gaseous to liquid(6) 119873
119901
= lceil119873119901
2rceil
(7) Childnum = Childnum times 2(8) Select the nucleus of each sub-population(9) 120575 = 12057510
(10) State = 1(11) else(12) Condensation Operation(13) for 119894 = 1119873
119875
(14) Update 119864119899119894for each sub-population according to (7)
(15) Update119867119890119894for each sub-population according to (8)
(16) endfor(17) endif(18) endif
State = 0 Cloud Gaseous Phase State = 1 Cloud Liquid Phase State = 2 Cloud Solid Phase
Algorithm 1 Evolutionary algorithm of the cloud gaseous state
Mathematical Problems in Engineering 7
(1) if State == 1 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) for 119894 = 1119873
119901
(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904
119894119895
| gt 1
(7) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|20
(8) elseif |119899119906119888119897119890119906119904119894119895
| gt 01
(9) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|3
(10) else(11) 119864
119899119894119895= |119899119906119888119897119890119906119904
119894119895
|
(12) endif(13) endfor(14) endfor(15) for 119894 = 1119873
119901
(16) Update119867119890119894for each sub-population according to (8)
(17) endfor(18) if PT le 120575 then(19) if rand gt 119888119887 then(20) Reciprocity Operation according to (9) Reciprocity Operation(21) else(22) Phase Transformation from liquid to solid(23) 119873
119901
= lceil119873119901
2rceil
(24) Childnum = Childnum times 2(25) Select the nucleus of each sub-population(26) 120575 = 120575100
(27) State = 2(28) endif(29) endif(30) endif
Algorithm 2 Evolutionary algorithm of the cloud liquid state
(1) if State == 2 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) Solidification Operation(5) if PT le 120575 then(6) sf = sf times 2(7) 120575 = 12057510
(8) endif(9) for 119894 = 1119873
119875
(10) for 119895 = 1119863(11) Update 119864
119899119894119895according to (10)
(12) Update119867119890119894119895according to (11)
(13) endfor(14) endfor(15) endif
Algorithm 3 Evolutionary algorithm of the cloud solid state
Ali et al [25] and Rahnamayan et al [26]They are unimodalfunctions multimodal functions and rotate functions Allthe algorithms are implemented on the same machine witha T2450 24GHz CPU 15 GB memory and windows XPoperating system with Matlab R2009b
31 Parameter Study In this section the setting basis forinitial values of some parameters involved in the cloud parti-cles evolution algorithm model is described Firstly we hopethat all the cloud particles which are generated by the cloudgenerator can fall into the search space as much as possible
8 Mathematical Problems in Engineering
(1) Initialize119863 (number of dimensions)119873119901
= 5 Childnum = 30 119864119899
=23r119867119890
= 119864119899
1000(2) for 119894 = 1119873
119901
(3) for 119895 = 1119863(4) if 119895 is even(5) 119864
119909119894119895= 1199032
(6) else(7) 119864
119909119894119895= minus1199032
(8) end if(9) end for(10) end for(11) Use cloud generator to produce cloud particles(12) Evaluate Subpopulation(13) Selectnucleus(14) while requirements are not satisfied do(15) Use cloud generator to produce cloud particles(16) Evaluate Subpopulation(17) Selectnucleus(18) Evolutionary algorithm of the cloud gaseous state(19) Evolutionary algorithm of the cloud liquid state(20) Evolutionary algorithm of the cloud solid state(21) end while
Algorithm 4 Cloud particles evolution algorithm CPEA
1 2 3 4 5 6 7 8 9 10965
970
975
980
985
990
995
1000
Number of run times
Valid
clou
d pa
rtic
les
1 2 3 4 5 6 7 8 9 10810
820
830
840
850
860
870
Number of run times
Valid
clou
d pa
rtic
les
En = (13)r
En = (12)r
En = (23)r
En = (34)rEn = (45)r
En = r
Figure 9 Valid cloud particles of different 119864119899
Secondly the diversity of the population can be ensured effec-tively if the cloud particles are distributed in the search spaceuniformly Therefore some experiments are designed toanalyze the feasibility of the selected scheme in order toillustrate the rationality of the selected parameters
311 Definition of 119864119899 Let the search range 119878 be [minus119903 119903] and 119903
is the search radius Set 119903 = 2 1198641199091= minus23 119864
1199092= 23 119867
119890=
0001119864119899 Five hundred cloud particles are generated in every
experiment The experiment is independently simulated 30times The cloud particle is called a valid cloud particle if itfalls into the search space Figure 9 is a graphwhich shows thenumber of the valid cloud particles generated with different119864119899 and Figure 10 is a graph which shows the distribution of
valid cloud particles generated with different 119864119899
It can be seen from Figure 9 that the number of thevalid cloud particles reduces as the 119864
119899increases The average
number of valid cloud particles is basically the same when119864119899is (13)119903 (12)119903 and (23)119903 respectively However the
number of valid cloud particles is relatively fewer when 119864119899is
(34)119903 (45)119903 and 119903 It can be seen from Figure 10 that all thecloud particles fall into the research space when 119864
119899is (13)119903
(12)119903 and (23)119903 respectively However the cloud particlesconcentrate in a certain area of the search space when 119864
119899is
(13)119903 or (12)119903 By contrast the cloud particles are widelydistributed throughout all the search space [minus2 2] when 119864
119899
is (23)119903 Therefore set 119864119899= (23)119903 The dashed rectangle in
Figure 10 represents the decision variables space
312 Definition of 119867119890 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches119891 stopwith different119867
119890 Let the search space
Mathematical Problems in Engineering 9
0 05 1 15 2 25
012
0 05 1 15 2
012
0 05 1 15 2
005
115
2
0 1 2
01234
0 1 2
0123
0 1 2 3
012345
En = (13)r En = (12)r En = (23)r
En = (34)r En = (45)r En = r
x1 x1
x1 x1 x1
x1
x2
x2
x2
x2
x2
x2
minus1
minus2
minus2 minus15 minus1 minus05
minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1
minus2 minus15 minus1 minus05 minus2minus2
minus15
minus15
minus1
minus1
minus05
minus05
minus3
minus1minus2minus3
minus1
minus2
minus1minus2
minus1
minus2
minus3
Figure 10 Distribution of cloud particles in different 119864119899
1 2 3 4 5 6 76000
8000
10000
12000
NFF
Es
Ackley function (D = 100)
He
(a)
1 2 3 4 5 6 73000
4000
5000
6000
7000N
FFEs
Sphere function (D = 100)
He
(b)
Figure 11 NFES of CPEA in different119867119890
be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =
10minus3 1198671198901= 119864119899 1198671198902= 01119864
119899 1198671198903= 001119864
119899 1198671198904= 0001119864
119899
1198671198905= 00001119864
119899 1198671198906= 000001119864
119899 and 119867
1198907= 0000001119864
119899
The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867
119890is equal
to 0001119864119899 For Sphere Function the number of fitness eval-
uations is less when 119867119890is equal to 0001119864
119899 Therefore 119867
119890is
equal to 0001119864119899in the cloud particles evolution algorithm
313 Definition of 119864119909 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864
119909being equal to 119903 (12)119903
(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10
minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864
119909
is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864
119909is equal to (12)119903 or (13)119903There-
fore 119864119909is equal to (12)119903 in the cloud particles evolution
algorithm
314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically
119886 + 119887
119886=119886
119887
def= 120593 (12)
in which 120593 represents the golden ration Its value is
120593 =1 + radic5
2= 16180339887
1
120593= 120593 minus 1 = 0618 sdot sdot sdot
(13)
The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
2 25 3 35 4 45 5044046048
05052054056058
06
NucleusLocal optima valueGlobal optima value
x2
x1
(a)
IndividualsLocal optima valueGlobal optima value
2 25 3 35 4 45 504
045
05
055
06
065
x2
x1
(b)
25 26 27 28 29 3 31044
046
048
05
052
054
056
NucleusLocal optima valueGlobal optima value
x2
x1
(c)
IndividualsLocal optima valueGlobal optima value
22 23 24 25 26 27 28 29 3 31 3204
045
05
055
06
x2
x1
(d)
Figure 8 Reciprocity operator
(1) if State == 0 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) if PT le 120575 then(5) Phase Transformation from gaseous to liquid(6) 119873
119901
= lceil119873119901
2rceil
(7) Childnum = Childnum times 2(8) Select the nucleus of each sub-population(9) 120575 = 12057510
(10) State = 1(11) else(12) Condensation Operation(13) for 119894 = 1119873
119875
(14) Update 119864119899119894for each sub-population according to (7)
(15) Update119867119890119894for each sub-population according to (8)
(16) endfor(17) endif(18) endif
State = 0 Cloud Gaseous Phase State = 1 Cloud Liquid Phase State = 2 Cloud Solid Phase
Algorithm 1 Evolutionary algorithm of the cloud gaseous state
Mathematical Problems in Engineering 7
(1) if State == 1 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) for 119894 = 1119873
119901
(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904
119894119895
| gt 1
(7) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|20
(8) elseif |119899119906119888119897119890119906119904119894119895
| gt 01
(9) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|3
(10) else(11) 119864
119899119894119895= |119899119906119888119897119890119906119904
119894119895
|
(12) endif(13) endfor(14) endfor(15) for 119894 = 1119873
119901
(16) Update119867119890119894for each sub-population according to (8)
(17) endfor(18) if PT le 120575 then(19) if rand gt 119888119887 then(20) Reciprocity Operation according to (9) Reciprocity Operation(21) else(22) Phase Transformation from liquid to solid(23) 119873
119901
= lceil119873119901
2rceil
(24) Childnum = Childnum times 2(25) Select the nucleus of each sub-population(26) 120575 = 120575100
(27) State = 2(28) endif(29) endif(30) endif
Algorithm 2 Evolutionary algorithm of the cloud liquid state
(1) if State == 2 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) Solidification Operation(5) if PT le 120575 then(6) sf = sf times 2(7) 120575 = 12057510
(8) endif(9) for 119894 = 1119873
119875
(10) for 119895 = 1119863(11) Update 119864
119899119894119895according to (10)
(12) Update119867119890119894119895according to (11)
(13) endfor(14) endfor(15) endif
Algorithm 3 Evolutionary algorithm of the cloud solid state
Ali et al [25] and Rahnamayan et al [26]They are unimodalfunctions multimodal functions and rotate functions Allthe algorithms are implemented on the same machine witha T2450 24GHz CPU 15 GB memory and windows XPoperating system with Matlab R2009b
31 Parameter Study In this section the setting basis forinitial values of some parameters involved in the cloud parti-cles evolution algorithm model is described Firstly we hopethat all the cloud particles which are generated by the cloudgenerator can fall into the search space as much as possible
8 Mathematical Problems in Engineering
(1) Initialize119863 (number of dimensions)119873119901
= 5 Childnum = 30 119864119899
=23r119867119890
= 119864119899
1000(2) for 119894 = 1119873
119901
(3) for 119895 = 1119863(4) if 119895 is even(5) 119864
119909119894119895= 1199032
(6) else(7) 119864
119909119894119895= minus1199032
(8) end if(9) end for(10) end for(11) Use cloud generator to produce cloud particles(12) Evaluate Subpopulation(13) Selectnucleus(14) while requirements are not satisfied do(15) Use cloud generator to produce cloud particles(16) Evaluate Subpopulation(17) Selectnucleus(18) Evolutionary algorithm of the cloud gaseous state(19) Evolutionary algorithm of the cloud liquid state(20) Evolutionary algorithm of the cloud solid state(21) end while
Algorithm 4 Cloud particles evolution algorithm CPEA
1 2 3 4 5 6 7 8 9 10965
970
975
980
985
990
995
1000
Number of run times
Valid
clou
d pa
rtic
les
1 2 3 4 5 6 7 8 9 10810
820
830
840
850
860
870
Number of run times
Valid
clou
d pa
rtic
les
En = (13)r
En = (12)r
En = (23)r
En = (34)rEn = (45)r
En = r
Figure 9 Valid cloud particles of different 119864119899
Secondly the diversity of the population can be ensured effec-tively if the cloud particles are distributed in the search spaceuniformly Therefore some experiments are designed toanalyze the feasibility of the selected scheme in order toillustrate the rationality of the selected parameters
311 Definition of 119864119899 Let the search range 119878 be [minus119903 119903] and 119903
is the search radius Set 119903 = 2 1198641199091= minus23 119864
1199092= 23 119867
119890=
0001119864119899 Five hundred cloud particles are generated in every
experiment The experiment is independently simulated 30times The cloud particle is called a valid cloud particle if itfalls into the search space Figure 9 is a graphwhich shows thenumber of the valid cloud particles generated with different119864119899 and Figure 10 is a graph which shows the distribution of
valid cloud particles generated with different 119864119899
It can be seen from Figure 9 that the number of thevalid cloud particles reduces as the 119864
119899increases The average
number of valid cloud particles is basically the same when119864119899is (13)119903 (12)119903 and (23)119903 respectively However the
number of valid cloud particles is relatively fewer when 119864119899is
(34)119903 (45)119903 and 119903 It can be seen from Figure 10 that all thecloud particles fall into the research space when 119864
119899is (13)119903
(12)119903 and (23)119903 respectively However the cloud particlesconcentrate in a certain area of the search space when 119864
119899is
(13)119903 or (12)119903 By contrast the cloud particles are widelydistributed throughout all the search space [minus2 2] when 119864
119899
is (23)119903 Therefore set 119864119899= (23)119903 The dashed rectangle in
Figure 10 represents the decision variables space
312 Definition of 119867119890 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches119891 stopwith different119867
119890 Let the search space
Mathematical Problems in Engineering 9
0 05 1 15 2 25
012
0 05 1 15 2
012
0 05 1 15 2
005
115
2
0 1 2
01234
0 1 2
0123
0 1 2 3
012345
En = (13)r En = (12)r En = (23)r
En = (34)r En = (45)r En = r
x1 x1
x1 x1 x1
x1
x2
x2
x2
x2
x2
x2
minus1
minus2
minus2 minus15 minus1 minus05
minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1
minus2 minus15 minus1 minus05 minus2minus2
minus15
minus15
minus1
minus1
minus05
minus05
minus3
minus1minus2minus3
minus1
minus2
minus1minus2
minus1
minus2
minus3
Figure 10 Distribution of cloud particles in different 119864119899
1 2 3 4 5 6 76000
8000
10000
12000
NFF
Es
Ackley function (D = 100)
He
(a)
1 2 3 4 5 6 73000
4000
5000
6000
7000N
FFEs
Sphere function (D = 100)
He
(b)
Figure 11 NFES of CPEA in different119867119890
be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =
10minus3 1198671198901= 119864119899 1198671198902= 01119864
119899 1198671198903= 001119864
119899 1198671198904= 0001119864
119899
1198671198905= 00001119864
119899 1198671198906= 000001119864
119899 and 119867
1198907= 0000001119864
119899
The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867
119890is equal
to 0001119864119899 For Sphere Function the number of fitness eval-
uations is less when 119867119890is equal to 0001119864
119899 Therefore 119867
119890is
equal to 0001119864119899in the cloud particles evolution algorithm
313 Definition of 119864119909 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864
119909being equal to 119903 (12)119903
(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10
minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864
119909
is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864
119909is equal to (12)119903 or (13)119903There-
fore 119864119909is equal to (12)119903 in the cloud particles evolution
algorithm
314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically
119886 + 119887
119886=119886
119887
def= 120593 (12)
in which 120593 represents the golden ration Its value is
120593 =1 + radic5
2= 16180339887
1
120593= 120593 minus 1 = 0618 sdot sdot sdot
(13)
The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
(1) if State == 1 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) for 119894 = 1119873
119901
(5) for 119895 = 1119863(6) if |119899119906119888119897119890119906119904
119894119895
| gt 1
(7) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|20
(8) elseif |119899119906119888119897119890119906119904119894119895
| gt 01
(9) 119864119899119894119895= |119899119906119888119897119890119906119904
119894119895
|3
(10) else(11) 119864
119899119894119895= |119899119906119888119897119890119906119904
119894119895
|
(12) endif(13) endfor(14) endfor(15) for 119894 = 1119873
119901
(16) Update119867119890119894for each sub-population according to (8)
(17) endfor(18) if PT le 120575 then(19) if rand gt 119888119887 then(20) Reciprocity Operation according to (9) Reciprocity Operation(21) else(22) Phase Transformation from liquid to solid(23) 119873
119901
= lceil119873119901
2rceil
(24) Childnum = Childnum times 2(25) Select the nucleus of each sub-population(26) 120575 = 120575100
(27) State = 2(28) endif(29) endif(30) endif
Algorithm 2 Evolutionary algorithm of the cloud liquid state
(1) if State == 2 then(2) Calculate the evolution potential of cloud particles according to (5)(3) Calculate the phase transformation driving force according to (6)(4) Solidification Operation(5) if PT le 120575 then(6) sf = sf times 2(7) 120575 = 12057510
(8) endif(9) for 119894 = 1119873
119875
(10) for 119895 = 1119863(11) Update 119864
119899119894119895according to (10)
(12) Update119867119890119894119895according to (11)
(13) endfor(14) endfor(15) endif
Algorithm 3 Evolutionary algorithm of the cloud solid state
Ali et al [25] and Rahnamayan et al [26]They are unimodalfunctions multimodal functions and rotate functions Allthe algorithms are implemented on the same machine witha T2450 24GHz CPU 15 GB memory and windows XPoperating system with Matlab R2009b
31 Parameter Study In this section the setting basis forinitial values of some parameters involved in the cloud parti-cles evolution algorithm model is described Firstly we hopethat all the cloud particles which are generated by the cloudgenerator can fall into the search space as much as possible
8 Mathematical Problems in Engineering
(1) Initialize119863 (number of dimensions)119873119901
= 5 Childnum = 30 119864119899
=23r119867119890
= 119864119899
1000(2) for 119894 = 1119873
119901
(3) for 119895 = 1119863(4) if 119895 is even(5) 119864
119909119894119895= 1199032
(6) else(7) 119864
119909119894119895= minus1199032
(8) end if(9) end for(10) end for(11) Use cloud generator to produce cloud particles(12) Evaluate Subpopulation(13) Selectnucleus(14) while requirements are not satisfied do(15) Use cloud generator to produce cloud particles(16) Evaluate Subpopulation(17) Selectnucleus(18) Evolutionary algorithm of the cloud gaseous state(19) Evolutionary algorithm of the cloud liquid state(20) Evolutionary algorithm of the cloud solid state(21) end while
Algorithm 4 Cloud particles evolution algorithm CPEA
1 2 3 4 5 6 7 8 9 10965
970
975
980
985
990
995
1000
Number of run times
Valid
clou
d pa
rtic
les
1 2 3 4 5 6 7 8 9 10810
820
830
840
850
860
870
Number of run times
Valid
clou
d pa
rtic
les
En = (13)r
En = (12)r
En = (23)r
En = (34)rEn = (45)r
En = r
Figure 9 Valid cloud particles of different 119864119899
Secondly the diversity of the population can be ensured effec-tively if the cloud particles are distributed in the search spaceuniformly Therefore some experiments are designed toanalyze the feasibility of the selected scheme in order toillustrate the rationality of the selected parameters
311 Definition of 119864119899 Let the search range 119878 be [minus119903 119903] and 119903
is the search radius Set 119903 = 2 1198641199091= minus23 119864
1199092= 23 119867
119890=
0001119864119899 Five hundred cloud particles are generated in every
experiment The experiment is independently simulated 30times The cloud particle is called a valid cloud particle if itfalls into the search space Figure 9 is a graphwhich shows thenumber of the valid cloud particles generated with different119864119899 and Figure 10 is a graph which shows the distribution of
valid cloud particles generated with different 119864119899
It can be seen from Figure 9 that the number of thevalid cloud particles reduces as the 119864
119899increases The average
number of valid cloud particles is basically the same when119864119899is (13)119903 (12)119903 and (23)119903 respectively However the
number of valid cloud particles is relatively fewer when 119864119899is
(34)119903 (45)119903 and 119903 It can be seen from Figure 10 that all thecloud particles fall into the research space when 119864
119899is (13)119903
(12)119903 and (23)119903 respectively However the cloud particlesconcentrate in a certain area of the search space when 119864
119899is
(13)119903 or (12)119903 By contrast the cloud particles are widelydistributed throughout all the search space [minus2 2] when 119864
119899
is (23)119903 Therefore set 119864119899= (23)119903 The dashed rectangle in
Figure 10 represents the decision variables space
312 Definition of 119867119890 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches119891 stopwith different119867
119890 Let the search space
Mathematical Problems in Engineering 9
0 05 1 15 2 25
012
0 05 1 15 2
012
0 05 1 15 2
005
115
2
0 1 2
01234
0 1 2
0123
0 1 2 3
012345
En = (13)r En = (12)r En = (23)r
En = (34)r En = (45)r En = r
x1 x1
x1 x1 x1
x1
x2
x2
x2
x2
x2
x2
minus1
minus2
minus2 minus15 minus1 minus05
minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1
minus2 minus15 minus1 minus05 minus2minus2
minus15
minus15
minus1
minus1
minus05
minus05
minus3
minus1minus2minus3
minus1
minus2
minus1minus2
minus1
minus2
minus3
Figure 10 Distribution of cloud particles in different 119864119899
1 2 3 4 5 6 76000
8000
10000
12000
NFF
Es
Ackley function (D = 100)
He
(a)
1 2 3 4 5 6 73000
4000
5000
6000
7000N
FFEs
Sphere function (D = 100)
He
(b)
Figure 11 NFES of CPEA in different119867119890
be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =
10minus3 1198671198901= 119864119899 1198671198902= 01119864
119899 1198671198903= 001119864
119899 1198671198904= 0001119864
119899
1198671198905= 00001119864
119899 1198671198906= 000001119864
119899 and 119867
1198907= 0000001119864
119899
The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867
119890is equal
to 0001119864119899 For Sphere Function the number of fitness eval-
uations is less when 119867119890is equal to 0001119864
119899 Therefore 119867
119890is
equal to 0001119864119899in the cloud particles evolution algorithm
313 Definition of 119864119909 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864
119909being equal to 119903 (12)119903
(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10
minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864
119909
is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864
119909is equal to (12)119903 or (13)119903There-
fore 119864119909is equal to (12)119903 in the cloud particles evolution
algorithm
314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically
119886 + 119887
119886=119886
119887
def= 120593 (12)
in which 120593 represents the golden ration Its value is
120593 =1 + radic5
2= 16180339887
1
120593= 120593 minus 1 = 0618 sdot sdot sdot
(13)
The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
(1) Initialize119863 (number of dimensions)119873119901
= 5 Childnum = 30 119864119899
=23r119867119890
= 119864119899
1000(2) for 119894 = 1119873
119901
(3) for 119895 = 1119863(4) if 119895 is even(5) 119864
119909119894119895= 1199032
(6) else(7) 119864
119909119894119895= minus1199032
(8) end if(9) end for(10) end for(11) Use cloud generator to produce cloud particles(12) Evaluate Subpopulation(13) Selectnucleus(14) while requirements are not satisfied do(15) Use cloud generator to produce cloud particles(16) Evaluate Subpopulation(17) Selectnucleus(18) Evolutionary algorithm of the cloud gaseous state(19) Evolutionary algorithm of the cloud liquid state(20) Evolutionary algorithm of the cloud solid state(21) end while
Algorithm 4 Cloud particles evolution algorithm CPEA
1 2 3 4 5 6 7 8 9 10965
970
975
980
985
990
995
1000
Number of run times
Valid
clou
d pa
rtic
les
1 2 3 4 5 6 7 8 9 10810
820
830
840
850
860
870
Number of run times
Valid
clou
d pa
rtic
les
En = (13)r
En = (12)r
En = (23)r
En = (34)rEn = (45)r
En = r
Figure 9 Valid cloud particles of different 119864119899
Secondly the diversity of the population can be ensured effec-tively if the cloud particles are distributed in the search spaceuniformly Therefore some experiments are designed toanalyze the feasibility of the selected scheme in order toillustrate the rationality of the selected parameters
311 Definition of 119864119899 Let the search range 119878 be [minus119903 119903] and 119903
is the search radius Set 119903 = 2 1198641199091= minus23 119864
1199092= 23 119867
119890=
0001119864119899 Five hundred cloud particles are generated in every
experiment The experiment is independently simulated 30times The cloud particle is called a valid cloud particle if itfalls into the search space Figure 9 is a graphwhich shows thenumber of the valid cloud particles generated with different119864119899 and Figure 10 is a graph which shows the distribution of
valid cloud particles generated with different 119864119899
It can be seen from Figure 9 that the number of thevalid cloud particles reduces as the 119864
119899increases The average
number of valid cloud particles is basically the same when119864119899is (13)119903 (12)119903 and (23)119903 respectively However the
number of valid cloud particles is relatively fewer when 119864119899is
(34)119903 (45)119903 and 119903 It can be seen from Figure 10 that all thecloud particles fall into the research space when 119864
119899is (13)119903
(12)119903 and (23)119903 respectively However the cloud particlesconcentrate in a certain area of the search space when 119864
119899is
(13)119903 or (12)119903 By contrast the cloud particles are widelydistributed throughout all the search space [minus2 2] when 119864
119899
is (23)119903 Therefore set 119864119899= (23)119903 The dashed rectangle in
Figure 10 represents the decision variables space
312 Definition of 119867119890 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches119891 stopwith different119867
119890 Let the search space
Mathematical Problems in Engineering 9
0 05 1 15 2 25
012
0 05 1 15 2
012
0 05 1 15 2
005
115
2
0 1 2
01234
0 1 2
0123
0 1 2 3
012345
En = (13)r En = (12)r En = (23)r
En = (34)r En = (45)r En = r
x1 x1
x1 x1 x1
x1
x2
x2
x2
x2
x2
x2
minus1
minus2
minus2 minus15 minus1 minus05
minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1
minus2 minus15 minus1 minus05 minus2minus2
minus15
minus15
minus1
minus1
minus05
minus05
minus3
minus1minus2minus3
minus1
minus2
minus1minus2
minus1
minus2
minus3
Figure 10 Distribution of cloud particles in different 119864119899
1 2 3 4 5 6 76000
8000
10000
12000
NFF
Es
Ackley function (D = 100)
He
(a)
1 2 3 4 5 6 73000
4000
5000
6000
7000N
FFEs
Sphere function (D = 100)
He
(b)
Figure 11 NFES of CPEA in different119867119890
be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =
10minus3 1198671198901= 119864119899 1198671198902= 01119864
119899 1198671198903= 001119864
119899 1198671198904= 0001119864
119899
1198671198905= 00001119864
119899 1198671198906= 000001119864
119899 and 119867
1198907= 0000001119864
119899
The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867
119890is equal
to 0001119864119899 For Sphere Function the number of fitness eval-
uations is less when 119867119890is equal to 0001119864
119899 Therefore 119867
119890is
equal to 0001119864119899in the cloud particles evolution algorithm
313 Definition of 119864119909 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864
119909being equal to 119903 (12)119903
(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10
minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864
119909
is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864
119909is equal to (12)119903 or (13)119903There-
fore 119864119909is equal to (12)119903 in the cloud particles evolution
algorithm
314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically
119886 + 119887
119886=119886
119887
def= 120593 (12)
in which 120593 represents the golden ration Its value is
120593 =1 + radic5
2= 16180339887
1
120593= 120593 minus 1 = 0618 sdot sdot sdot
(13)
The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
0 05 1 15 2 25
012
0 05 1 15 2
012
0 05 1 15 2
005
115
2
0 1 2
01234
0 1 2
0123
0 1 2 3
012345
En = (13)r En = (12)r En = (23)r
En = (34)r En = (45)r En = r
x1 x1
x1 x1 x1
x1
x2
x2
x2
x2
x2
x2
minus1
minus2
minus2 minus15 minus1 minus05
minus4 minus3 minus2 minus1 minus4 minus3 minus2 minus1 minus4minus5 minus3 minus2 minus1
minus2 minus15 minus1 minus05 minus2minus2
minus15
minus15
minus1
minus1
minus05
minus05
minus3
minus1minus2minus3
minus1
minus2
minus1minus2
minus1
minus2
minus3
Figure 10 Distribution of cloud particles in different 119864119899
1 2 3 4 5 6 76000
8000
10000
12000
NFF
Es
Ackley function (D = 100)
He
(a)
1 2 3 4 5 6 73000
4000
5000
6000
7000N
FFEs
Sphere function (D = 100)
He
(b)
Figure 11 NFES of CPEA in different119867119890
be [minus32 32] (Ackley) and [minus100 100] (Sphere) 119891 stop =
10minus3 1198671198901= 119864119899 1198671198902= 01119864
119899 1198671198903= 001119864
119899 1198671198904= 0001119864
119899
1198671198905= 00001119864
119899 1198671198906= 000001119864
119899 and 119867
1198907= 0000001119864
119899
The reported values are the average of the results for 50 inde-pendent runs As shown in Figure 11 for Ackley Functionthe number of fitness evaluations is least when 119867
119890is equal
to 0001119864119899 For Sphere Function the number of fitness eval-
uations is less when 119867119890is equal to 0001119864
119899 Therefore 119867
119890is
equal to 0001119864119899in the cloud particles evolution algorithm
313 Definition of 119864119909 Ackley Function (multimode) and
Sphere Function (unimode) are selected to calculate the num-ber of fitness evaluations (NFES) of the algorithm when thesolution reaches 119891 stop with 119864
119909being equal to 119903 (12)119903
(13)119903 (14)119903 and (15)119903 respectively Let the search spacebe [minus119903 119903] 119891 stop = 10
minus3 As shown in Figure 12 for AckleyFunction the number of fitness evaluations is least when 119864
119909
is equal to (12)119903 For Sphere Function the number of fitnessevaluations is less when119864
119909is equal to (12)119903 or (13)119903There-
fore 119864119909is equal to (12)119903 in the cloud particles evolution
algorithm
314 Definition of 119888119887 Many things in nature have themathematical proportion relationship between their parts Ifthe substance is divided into two parts the ratio of the wholeand the larger part is equal to the ratio of the larger part to thesmaller part this ratio is defined as golden ratio expressedalgebraically
119886 + 119887
119886=119886
119887
def= 120593 (12)
in which 120593 represents the golden ration Its value is
120593 =1 + radic5
2= 16180339887
1
120593= 120593 minus 1 = 0618 sdot sdot sdot
(13)
The golden section has a wide application not only inthe fields of painting sculpture music architecture andmanagement but also in nature Studies have shown that plantleaves branches or petals are distributed in accordance withthe golden ratio in order to survive in the wind frost rainand snow It is the evolutionary result or the best solution
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
0
1
2
3
4
5
6
NFF
Es
Ackley function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(a)
0
1
2
3
NFF
Es
Sphere function (D = 100)
(13)r(12)r (15)r(14)r
Ex
r
times104
(b)
Figure 12 NFES of CPEA in different 119864119909
which adapts its own growth after hundreds of millions ofyears of long-term evolutionary process
In this model competition and reciprocity of the pop-ulation depend on the fitness and are a dynamic relation-ship which constantly adjust and change With competitionand reciprocity mechanism the convergence speed and thepopulation diversity are improved Premature convergence isavoided and large search space and complex computationalproblems are solved Indirect reciprocity is selected in thispaper Here set 119888119887 = (1 minus 0618) asymp 038
32 Experiments Settings For fair comparison we set theparameters to be fixed For CPEA population size = 5119888ℎ119894119897119889119899119906119898 = 30 119864
119899= (23)119903 (the search space is [minus119903 119903])
119867119890= 1198641198991000 119864
119909= (12)119903 cd = 1000 120575 = 10
minus2 and sf =1000 For CMA-ES the parameter values are chosen from[27] The parameters of JADE come from [28] For PSO theparameter values are chosen from [29] The parameters ofDE are set to be 119865 = 03 and CR = 09 used in [30] ForABC the number of colony size NP = 20 and the number offood sources FoodNumber = NP2 and limit = 100 A foodsource will not be exploited anymore and is assumed to beabandoned when limit is exceeded for the source For CEBApopulation size = 10 119888ℎ119894119897119889119899119906119898 = 50 119864
119899= 1 and 119867
119890=
1198641198996 The parameters of PSO-cf-Local and FDR-PSO come
from [31 32]
33 Comparison Strategies Five comparison strategies comefrom the literature [33] and literature [26] to evaluate thealgorithms The comparison strategies are described as fol-lows
Successful Run A run during which the algorithm achievesthe fixed accuracy level within the max number of functionevaluations for the particular dimension is shown as follows
SR = successful runstotal runs
(14)
Error The error of a solution 119909 is defined as 119891(119909) minus 119891(119909lowast)where 119909
lowast is the global minimum of the function The
minimum error is written when the max number of functionevaluations is reached in 30 runs Both the average andstandard deviation of the error values are calculated
Convergence Graphs The convergence graphs show themedian performance of the total runs with termination byeither the max number of function evaluations or the termi-nation error value
Number of Function Evaluations (NFES) The number offunction evaluations is recorded when the value-to-reach isreached The average and standard deviation of the numberof function evaluations are calculated
Acceleration Rate (AR) The acceleration rate is defined asfollows based on the NFES for the two algorithms
AR =NFES
1198861
NFES1198862
(15)
in which AR gt 1 means Algorithm 2 (1198862) is faster thanAlgorithm 1 (1198861)
The average acceleration rate (ARave) and the averagesuccess rate (SRave) over 119899 test functions are calculated asfollows
ARave =1
119899
119899
sum
119894=1
AR119894
SRave =1
119899
119899
sum
119894=1
SR119894
(16)
34 Comparison of CMA-ES DE RES LOS ABC CEBA andCPEA This section compares CMA-ES DE RES LOS ABCCEBA [34] and CPEA in terms of mean number of functionevaluations for successful runs Results of CMA-ES DE RESand LOS are taken from literature [35] Each run is stoppedand regarded as successful if the function value is smaller than119891 stop
As shown in Table 1 CPEA can reach 119891 stop faster thanother algorithms with respect to119891
07and 11989117 However CPEA
converges slower thanCMA-ESwith respect to11989118When the
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table 1 Average number of function evaluations to reach 119891 stop of CPEA versus CMA-ES DE RES LOS ABC and CEBA on 11989107
11989117
11989118
and 119891
33
function
119865 119891 stop init 119863 CMA-ES DE RES LOS ABC CEBA CPEA
11989107
1119890 minus 3 [minus600 600]11986320 3111 8691 mdash mdash 12463 94412 295030 4455 11410 21198905 mdash 29357 163545 4270100 12796 31796 mdash mdash 34777 458102 7260
11989117
09 [minus512 512]119863 20 68586 12971 mdash 921198904 11183 30436 2850
DE [minus600 600]119863 30 147416 20150 101198905 231198905 26990 44138 6187100 1010989 73620 mdash mdash 449404 60854 10360
11989118
1119890 minus 3 [minus30 30]11986320 2667 mdash mdash 601198904 10183 64400 333030 3701 12481 111198905 931198904 17810 91749 4100100 11900 36801 mdash mdash 59423 205470 12248
11989133
09 [minus512 512]119863 30 152000 1251198906 mdash mdash mdash 50578 13100100 240899 5522 mdash mdash mdash 167324 14300
The bold entities indicate the best results obtained by the algorithm119863 represents dimension
dimension is 30 CPEA can reach 119891 stop relatively fast withrespect to119891
33 Nonetheless CPEA converges slower while the
dimension increases
35 Comparison of CMA-ES DE PSO ABC CEBA andCPEA This section compares the error of a solution of CMA-ES DE PSO ABC [36] CEBA and CPEA under a givenNFES The best solution of every single function in differentalgorithms is highlighted with boldface
As shown in Table 2 CPEA obtains better results in func-tion119891
0111989102119891041198910511989107119891091198911011989111 and119891
12while it obtains
relatively bad results in other functions The reason for thisphenomenon is that the changing rule of119864
119899is set according to
the priori information and background knowledge instead ofadaptive mechanism in CPEATherefore the results of somefunctions are relatively worse Besides 119864
119899which can be opti-
mized in order to get better results is an important controllingparameter in the algorithm
36 Comparison of JADE DE PSO ABC CEBA and CPEAThis section compares JADE DE PSO ABC CEBA andCPEA in terms of SR and NFES for most functions at119863 = 30
except11989103(119863 = 2)The termination criterion is to find a value
smaller than 10minus8 The best result of different algorithms foreach function is highlighted in boldface
As shown in Table 3 CPEA can reach the specified pre-cision successfully except for function 119891
14which successful
run is 96 In addition CPEA can reach the specified precisionfaster than other algorithms in function 119891
01 11989102 11989104 11989111 11989114
and 11989117 For other functions CPEA spends relatively long
time to reach the precision The reason for this phenomenonis that for some functions the changing rule of 119864
119899given in
the algorithm makes CPEA to spend more time to get rid ofthe local optima and converge to the specified precision
As shown in Table 4 the solution obtained by CPEA iscloser to the best solution compared with other algorithmsexcept for functions 119891
13and 119891
20 The reason is that the phase
transformation mechanism introduced in the algorithm canimprove the convergence of the population greatly Besides
the reciprocity mechanism of the algorithm can improve theadaptation and the diversity of the population in case thealgorithm falls into the local optimum Therefore the algo-rithm can obtain better results in these functions
As shown in Table 5 CPEA can find the optimal solutionfor the test functions However the stability of CPEA is worsethan JADE and DE (119891
25 11989126 11989127 11989128
and 11989131) This result
shows that the setting of119867119890needs to be further improved
37 Comparison of CMAES JADE DE PSO ABC CEBA andCPEA The convergence curves of different algorithms onsome selected functions are plotted in Figures 13 and 14 Thesolutions obtained by CPEA are better than other algorithmswith respect to functions 119891
01 11989111 11989116 11989132 and 119891
33 However
the solution obtained by CPEA is worse than CMA-ES withrespect to function 119891
04 The solution obtained by CPEA is
worse than JADE for high dimensional problems such asfunction 119891
05and function 119891
07 The reason is that the setting
of initial population of CPEA is insufficient for high dimen-sional problems
38 Comparison of DE ODE RDE PSO ABC CEBA andCPEA The algorithms DE ODE RDE PSO ABC CBEAand CPEA are compared in terms of convergence speed Thetermination criterion is to find error values smaller than 10minus8The results of solving the benchmark functions are given inTables 6 and 7 Data of all functions in DE ODE and RDEare from literature [26]The best result ofNEFS and the SR foreach function are highlighted in boldfaceThe average successrates and the average acceleration rate on test functions areshown in the last row of the tables
It can be seen fromTables 6 and 7 that CPEA outperformsDE ODE RDE PSO ABC andCBEA on 38 of the functionsCPEAperforms better than other algorithms on 23 functionsIn Table 6 the average AR is 098 086 401 636 131 and1858 for ODE RDE PSO ABC CBEA and CPEA respec-tively In Table 7 the average AR is 127 088 112 231 046and 456 for ODE RDE PSO ABC CBEA and CPEArespectively In Table 6 the average SR is 096 098 096 067
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
Table 2 Comparison on the ERROR values of CPEA versus CMA-ES DE PSO ABC and CEBA on function 11989101
sim11989113
119865 NFES 119863CMA-ESmeanSD
DEmeanSD
PSOmeanSD
ABCmeanSD
CEBAmeanSD
CPEAmeanSD
11989101
30 k 30 587e minus 29156e minus 29
320e minus 07251e minus 07
544e minus 12156e minus 11
971e minus 16226e minus 16
174e minus 02155e minus 02
319e minus 33621e minus 33
30 k 50 102e minus 28187e minus 29
295e minus 03288e minus 03
393e minus 03662e minus 02
235e minus 08247e minus 08
407e minus 02211e minus 02
109e minus 48141e minus 48
11989102
10 k 30 150e minus 05366e minus 05
230e + 00178e + 00
624e minus 06874e minus 05
000e + 00000e + 00
616e + 00322e + 00
000e + 00000e + 00
50 k 50 000e + 00000e + 00
165e + 00205e + 00
387e + 00265e + 00
000e + 00000e + 00
608e + 00534e + 00
000e + 00000e + 00
11989103
10 k 2 265e minus 30808e minus 31
000e + 00000e + 00
501e minus 14852e minus 14
483e minus 04140e minus 03
583e minus 04832e minus 04
152e minus 13244e minus 13
11989104
100 k 30 828e minus 27169e minus 27
247e minus 03250e minus 03
169e minus 02257e minus 02
136e + 00193e + 00
159e + 00793e + 00
331e minus 14142e minus 13
100 k 50 229e minus 06222e minus 06
737e minus 01193e minus 01
501e + 00388e + 00
119e + 02454e + 01
313e + 02394e + 02
398e minus 07879e minus 07
11989105
30 k 30 819e minus 09508e minus 09
127e minus 12394e minus 12
866e minus 34288e minus 33
800e minus 13224e minus 12
254e minus 09413e minus 09
211e minus 37420e minus 36
50 k 50 268e minus 08167e minus 09
193e minus 13201e minus 13
437e minus 54758e minus 54
954e minus 12155e minus 11
134e minus 15201e minus 15
107e minus 23185e minus 23
11989106
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
360e minus 09105e minus 08
11e minus 0382e minus 04
771e minus 09452e minus 09
11989107
30 k 30 188e minus 12140e minus 12
308e minus 06541e minus 06
164e minus 09285e minus 08
154e minus 09278e minus 08
207e minus 06557e minus 05
532e minus 16759e minus 16
50 k 50 000e + 00000e + 00
328e minus 05569e minus 05
179e minus 02249e minus 02
127e minus 09221e minus 09
949e minus 03423e minus 03
000e + 00000e + 00
11989108
30 k 3 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
280e minus 03213e minus 03
385e minus 06621e minus 06
11989109
30 k 2 000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
000e + 00000e + 00
11989110
5 k 2 699e minus 47221e minus 46
128e minus 2639e minus 26
659e minus 07771e minus 07
232e minus 04260e minus 04
824e minus 08137e minus 07
829e minus 59342e minus 58
11989111
30 k 30 123e minus 04182e minus 04
374e minus 01291e minus 01
478e minus 01194e minus 01
239e + 00327e + 00
142e + 00813e + 00
570e minus 40661e minus 39
50 k 50 558e minus 01155e minus 01
336e + 00155e + 00
315e + 01111e + 01
571e + 01698e + 00
205e + 01940e + 00
233e minus 05358e minus 05
11989112
30 k 30 319e minus 14145e minus 14
120e minus 08224e minus 08
215e minus 05518e minus 04
356e minus 10122e minus 10
193e minus 07645e minus 08
233e minus 23250e minus 22
50 k 50 141e minus 14243e minus 14
677e minus 05111e minus 05
608e minus 03335e minus 03
637e minus 10683e minus 10
976e minus 03308e minus 03
190e minus 72329e minus 72
11989113
30 k 30 394e minus 11115e minus 10
134e minus 31321e minus 31
134e minus 31000e + 00
657e minus 04377e minus 03
209e minus 02734e minus 02
120e minus 12282e minus 12
50 k 50 308e minus 03380e minus 03
134e minus 31000e + 00
134e minus 31000e + 00
141e minus 02929e minus 03
422e minus 02253e minus 02
236e minus 12104e minus 12
ldquoSDrdquo stands for standard deviation
079 089 and 099 forDEODERDE PSOABCCBEA andCPEA respectively In Table 7 the average SR is 084 084084 073 089 077 and 098 for DE ODE RDE PSO ABCCBEA and CPEA respectively These results clearly demon-strate that CPEA has better performance
39 Test on the CEC2013 Benchmark Problems To furtherverify the performance of CPEA a set of recently proposedCEC2013 rotated benchmark problems are used In the exper-iment 119905-test [37] has been carried out to show the differences
between CPEA and the other algorithm ldquoGeneral merit overcontenderrdquo displays the difference between the number ofbetter results and the number of worse results which is usedto give an overall comparison between the two algorithmsTable 8 gives the information about the average error stan-dard deviation and 119905-value of 30 runs of 7 algorithms over300000 FEs on 20 test functionswith119863 = 10The best resultsamong the algorithms are shown in bold
Table 8 consists of unimodal problems (11986501ndash11986505) and
basic multimodal problems (11986506ndash11986520) From the results of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
Table 3 Comparison on the SR and the NFES of CPEA versus JADE DE PSO ABC and CEBA for each function
119865
11989101
meanSD
11989102
meanSD
11989103
meanSD
11989104
meanSD
11989107
meanSD
11989111
meanSD
11989114
meanSD
11989116
meanSD
11989117
meanSD
JADE SR 1 1 1 1 1 098 094 1 1
NFES 28e + 460e + 2
10e + 435e + 2
57e + 369e + 2
69e + 430e + 3
33e + 326e + 2
66e + 442e + 3
40e + 421e + 3
13e + 338e + 2
13e + 518e + 3
DE SR 1 1 1 1 096 1 094 1 0
NFES 33e + 466e + 2
11e + 423e + 3
16e + 311e + 2
19e + 512e + 4
34e + 416e + 3
14e + 510e + 4
26e + 416e + 4
21e + 481e + 3 mdash
PSO SR 1 1 1 0 086 0 0 1 0
NFES 29e + 475e + 2
16e + 414e + 3
77e + 396e + 2 mdash 28e + 4
98e + 2 mdash mdash 18e + 472e + 2 mdash
ABC SR 1 1 1 0 096 0 0 1 1
NFES 17e + 410e + 3
63e + 315e + 3
16e + 426e + 3 mdash 19e + 4
23e + 3 mdash mdash 65e + 339e + 2
39e + 415e + 4
CEBA SR 1 1 1 0 1 1 096 1 1
NFES 13e + 558e + 3
59e + 446e + 3
42e + 461e + 3 mdash 22e + 5
12e + 413e + 558e + 3
29e + 514e + 4
54e + 438e + 3
11e + 573e + 3
CPEA SR 1 1 1 1 1 1 096 1 1
NFES 43e + 351e + 2
45e + 327e + 2
57e + 330e + 3
61e + 426e + 3
77e + 334e + 2
23e + 434e + 3
34e + 416e + 3
24e + 343e + 2
10e + 413e + 3
Table 4 Comparison on the ERROR values of CPEA versus JADE DE PSO and ABC at119863 = 50 except 11989121
(119863 = 2) and 11989122
(119863 = 2) for eachfunction
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989113
40 k 134e minus 31000e + 00 134e minus 31000e + 00 135e minus 31000e + 00 352e minus 03633e minus 02 364e minus 12468e minus 1211989115
40 k 918e minus 11704e minus 11 294e minus 05366e minus 05 181e minus 09418e minus 08 411e minus 14262e minus 14 459e minus 53136e minus 5211989116
40 k 115e minus 20142e minus 20 662e minus 08124e minus 07 691e minus 19755e minus 19 341e minus 15523e minus 15 177e minus 76000e + 0011989117
200 k 671e minus 07132e minus 07 258e minus 01615e minus 01 673e + 00206e + 00 607e minus 09185e minus 08 000e + 00000e + 0011989118
50 k 116e minus 07427e minus 08 389e minus 04210e minus 04 170e minus 09313e minus 08 293e minus 09234e minus 08 888e minus 16000e + 0011989119
40 k 114e minus 02168e minus 02 990e minus 03666e minus 03 900e minus 06162e minus 05 222e minus 05195e minus 04 164e minus 64513e minus 6411989120
40 k 000e + 00000e + 00 000e + 00000e + 00 140e minus 10281e minus 10 438e minus 03333e minus 03 121e minus 10311e minus 0911989121
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 106e minus 17334e minus 17 000e + 00000e + 0011989122
30 k 128e minus 45186e minus 45 499e minus 164000e + 00 263e minus 51682e minus 51 662e minus 18584e minus 18 000e + 00000e + 0011989123
30 k 000e + 00000e + 00 000e + 00000e + 00 000e + 00000e + 00 000e + 00424e minus 04 000e + 00000e + 00
Table 5 Comparison on the optimum solution of CPEA versus JADE DE PSO and ABC for each function
119865 NFES JADE meanSD DE meanSD PSO meanSD ABC meanSD CPEA meanSD11989108
40 k minus386278000e + 00 minus386278000e + 00 minus386278620119890 minus 17 minus386278000e + 00 minus386278000e + 0011989124
40 k minus331044360119890 minus 02 minus324608570119890 minus 02 minus325800590119890 minus 02 minus333539000119890 + 00 minus333539303e minus 0611989125
40 k minus101532400119890 minus 14 minus101532420e minus 16 minus579196330119890 + 00 minus101532125119890 minus 13 minus101532158119890 minus 0911989126
40 k minus104029940119890 minus 16 minus104029250e minus 16 minus714128350119890 + 00 minus104029177119890 minus 15 minus104029194119890 minus 1011989127
40 k minus105364810119890 minus 12 minus105364710e minus 16 minus702213360119890 + 00 minus105364403119890 minus 05 minus105364202119890 minus 0911989128
40 k 300000110119890 minus 15 30000064e minus 16 300000130119890 minus 15 300000617119890 minus 05 300000201119890 minus 1511989129
40 k 255119890 minus 20479119890 minus 20 000e + 00000e + 00 210119890 minus 03428119890 minus 03 114119890 minus 18162119890 minus 18 000e + 00000e + 0011989130
40 k minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 00 minus23458000e + 0011989131
40 k 478119890 minus 31107119890 minus 30 359e minus 55113e minus 54 122119890 minus 36385119890 minus 36 140119890 minus 11138119890 minus 11 869119890 minus 25114119890 minus 24
11989132
40 k 149119890 minus 02423119890 minus 03 346119890 minus 02514119890 minus 03 658119890 minus 02122119890 minus 02 295119890 minus 01113119890 minus 01 570e minus 03358e minus 03Data of functions 119891
08 11989124 11989125 11989126 11989127 and 119891
28in JADE DE and PSO are from literature [28]
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es
CPEACMAESDE
PSOABCCEBA
1010
100
10minus10
10minus20
10minus30
f01
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es
1010
100
10minus10
10minus20
f04
CPEACMAESDE
PSOABCCEBA
0 2000 4000 6000 8000 10000NFES
Mea
n of
func
tion
valu
es 1010
10minus10
10minus30
10minus50
f16
CPEACMAESDE
PSOABCCEBA
0 6000 12000 18000 24000 30000NFES
Mea
n of
func
tion
valu
es 103
101
10minus1
10minus3
f32
Figure 13 Convergence graph for test functions 11989101
11989104
11989116
and 11989132
(119863 = 30)
Table 8 we can see that PSO ABC PSO-FDR PSO-cf-Localand CPEAwork well for 119865
01 where PSO-FDR PSO-cf-Local
and CPEA can always achieve the optimal solution in eachrun CPEA outperforms other algorithms on119865
01119865021198650311986507
11986508 11986510 11986512 11986515 11986516 and 119865
18 the cause of the outstanding
performance may be because of the phase transformationmechanism which leads to faster convergence in the latestage of evolution
4 Conclusions
This paper presents a new optimization method called cloudparticles evolution algorithm (CPEA)The fundamental con-cepts and ideas come from the formation of the cloud andthe phase transformation of the substances in nature In thispaper CPEA solves 40 benchmark optimization problemswhich have difficulties in discontinuity nonconvexity mul-timode and high-dimensionality The evolutionary processof the cloud particles is divided into three states the cloudgaseous state the cloud liquid state and the cloud solid stateThe algorithm realizes the change of the state of the cloudwith the phase transformation driving force according tothe evolutionary process The algorithm reduces the popu-lation number appropriately and increases the individualsof subpopulations in each phase transformation ThereforeCPEA can improve the whole exploitation ability of thepopulation greatly and enhance the convergence There is a
larger adjustment for 119864119899to improve the exploration ability
of the population in melting or gasification operation Whenthe algorithm falls into the local optimum it improves thediversity of the population through the indirect reciprocitymechanism to solve this problem
As we can see from experiments it is helpful to improvethe performance of the algorithmwith appropriate parametersettings and change strategy in the evolutionary process Aswe all know the benefit is proportional to the risk for solvingthe practical engineering application problemsGenerally thebetter the benefit is the higher the risk isWe always hope thatthe benefit is increased and the risk is reduced or controlledat the same time In general the adaptive system adapts tothe dynamic characteristics change of the object and distur-bance by correcting its own characteristics Therefore thebenefit is increased However the adaptive mechanism isachieved at the expense of computational efficiency Conse-quently the settings of parameter should be based on theactual situation The settings of parameters should be basedon the prior information and background knowledge insome certain situations or based on adaptive mechanism forsolving different problems in other situations The adaptivemechanismwill not be selected if the static settings can ensureto generate more benefit and control the risk Inversely if theadaptivemechanism can only ensure to generatemore benefitwhile it cannot control the risk we will not choose it Theadaptive mechanism should be selected only if it can not only
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
0 5000 10000 15000 20000 25000 30000NFES
Mea
n of
func
tion
valu
es
CPEAJADEDE
PSOABCCEBA
100
10minus5
10minus10
10minus15
10minus20
10minus25
f05
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
f11
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es
104
102
100
10minus2
10minus4
10minus6
f07
CPEAJADEDE
PSOABCCEBA
0 10000 20000 30000 40000 50000NFES
Mea
n of
func
tion
valu
es 102
10minus2
10minus10
10minus14
10minus6f33
Figure 14 Convergence graph for test functions 11989105
11989111
11989133
(119863 = 50) and 11989107
(119863 = 100)
increase the benefit but also reduce the risk In CPEA 119864119899and
119867119890are statically set according to the priori information and
background knowledge As we can see from experiments theresults from some functions are better while the results fromthe others are relatively worse among the 40 tested functions
As previously mentioned parameter setting is an impor-tant focus Then the next step is the adaptive settings of 119864
119899
and 119867119890 Through adaptive behavior these parameters can
automatically modify the calculation direction according tothe current situation while the calculation is running More-over possible directions for future work include improvingthe computational efficiency and the quality of solutions ofCPEA Finally CPEA will be applied to solve large scaleoptimization problems and engineering design problems
Appendix
Test Functions
11989101
Sphere model 11989101(119909) = sum
119899
119894=1
1199092
119894
[minus100 100]min(119891
01) = 0
11989102
Step function 11989102(119909) = sum
119899
119894=1
(lfloor119909119894+ 05rfloor)
2 [minus100100] min(119891
02) = 0
11989103
Beale function 11989103(119909) = [15 minus 119909
1(1 minus 119909
2)]2
+
[225minus1199091(1minus119909
2
2
)]2
+[2625minus1199091(1minus119909
3
2
)]2 [minus45 45]
min(11989103) = 0
11989104
Schwefelrsquos problem 12 11989104(119909) = sum
119899
119894=1
(sum119894
119895=1
119909119895)2
[minus65 65] min(119891
04) = 0
11989105Sum of different power 119891
05(119909) = sum
119899
119894=1
|119909119894|119894+1 [minus1
1] min(11989105) = 119891(0 0) = 0
11989106
Easom function 11989106(119909) =
minus cos(1199091) cos(119909
2) exp(minus(119909
1minus 120587)2
minus (1199092minus 120587)2
) [minus100100] min(119891
06) = minus1
11989107
Griewangkrsquos function 11989107(119909) = sum
119899
119894=1
(1199092
119894
4000) minus
prod119899
119894=1
cos(119909119894radic119894) + 1 [minus600 600] min(119891
07) = 0
11989108
Hartmann function 11989108(119909) =
minussum4
119894=1
119886119894exp(minussum3
119895=1
119860119894119895(119909119895minus 119875119894119895)2
) [0 1] min(11989108) =
minus386278214782076 Consider119886 = [1 12 3 32]
119860 =[[[
[
3 10 30
01 10 35
3 10 30
01 10 35
]]]
]
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
Table6Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989101
3087748
147716
118
3115096
1076
20290
1432
18370
1477
133381
1066
5266
11666
11989102
3041588
123124
118
054316
1077
11541
1360
6247
1666
58416
1071
2966
11402
11989103
24324
14776
1090
4904
1088
6110
1071
18050
1024
38278
1011
4350
1099
11989104
20177880
1168680
110
5231152
1077
mdash0
mdashmdash
0mdash
mdash0
mdash1106
81
1607
11989105
3025140
18328
1301
29096
1086
5910
1425
6134
1410
18037
113
91700
11479
11989106
28016
166
081
121
8248
1097
7185
1112
2540
71
032
66319
1012
5033
115
911989107
30113428
169342
096
164
149744
1076
20700
084
548
20930
096
542
209278
1054
9941
11141
11989108
33376
13580
1094
3652
1092
4090
1083
1492
1226
206859
1002
31874
1011
11989109
25352
144
681
120
6292
1085
2830
118
91077
1497
9200
1058
3856
113
911989110
23608
13288
110
93924
1092
5485
1066
10891
1033
6633
1054
1920
118
811989111
30385192
1369104
110
4498200
1077
mdash0
mdashmdash
0mdash
142564
1270
1266
21
3042
11989112
30187300
1155636
112
0244396
1076
mdash0
mdash25787
172
3157139
1119
10183
11839
11989113
30101460
17040
81
144
138308
1073
6512
11558
mdash0
mdashmdash
0mdash
4300
12360
11989114
30570290
028
72250
088
789
285108
1200
mdash0
mdashmdash
0mdash
292946
119
531112
098
1833
11989115
3096
488
153304
118
1126780
1076
18757
1514
15150
1637
103475
1093
6466
11492
11989116
21016
1996
110
21084
1094
428
1237
301
1338
3292
1031
610
116
711989117
10328844
170389
076
467
501875
096
066
mdash0
mdash7935
14144
48221
1682
3725
188
28
11989118
30169152
198296
117
2222784
1076
27250
092
621
32398
1522
1840
411
092
1340
01
1262
11989119
3041116
41
337532
112
2927230
024
044
mdash0
mdash156871
1262
148014
1278
6233
165
97
Ave
096
098
098
096
086
067
401
079
636
089
131
099
1858
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 17
Table7Com
paris
onof
DE
ODE
RDE
PSOA
BCC
EBAand
CPEA
Theb
estresultfor
each
case
ishigh
lighted
inbo
ldface
119865119863
DE
ODE
AR
ODE
RDE
AR
RDE
PSO
AR
PSO
ABC
AR
ABC
CEBA
AR
CEBA
CPEA
AR
CPEA
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
NFE
SSR
11989120
52163
088
2024
110
72904
096
074
11960
1018
mdash0
mdash58252
1004
4216
092
051
11989121
27976
15092
115
68332
1096
8372
098
095
15944
1050
11129
1072
1025
177
811989122
23780
13396
1111
3820
1098
5755
1066
2050
118
46635
1057
855
1442
11989123
1019528
115704
112
423156
1084
9290
1210
5845
1334
17907
110
91130
11728
11989124
6mdash
0mdash
0mdash
mdash0
mdash7966
094
mdash30
181
mdashmdash
0mdash
42716
098
mdash11989125
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
4280
1mdash
mdash0
mdash5471
094
mdash11989126
4mdash
0mdash
0mdash
mdash0
mdashmdash
0mdash
8383
1mdash
mdash0
mdash5391
096
mdash11989127
44576
145
001
102
5948
1077
mdash0
mdash4850
1094
mdash0
mdash5512
092
083
11989128
24780
146
841
102
4924
1097
6770
1071
6228
1077
14248
1034
4495
110
611989129
28412
15772
114
610860
1077
mdash0
mdash1914
1440
37765
1022
4950
117
011989130
23256
13120
110
43308
1098
3275
1099
1428
1228
114261
1003
3025
110
811989131
4110
81
1232
1090
1388
1080
990
111
29944
1011
131217
1001
2662
1042
11989134
25232
15956
092
088
5768
1091
3495
115
01852
1283
9926
1053
4370
112
011989135
538532
116340
1236
46080
1084
11895
1324
25303
115
235292
110
920
951
1839
11989136
27888
14272
118
59148
1086
10910
1072
8359
1094
13529
1058
1340
1589
11989137
25284
14728
1112
5488
1097
7685
1069
1038
1509
9056
1058
1022
1517
11989138
25280
14804
1110
5540
1095
7680
1069
1598
1330
8905
1059
990
1533
11989139
1037824
124206
115
646
800
1080
mdash0
mdashmdash
0mdash
153161
1025
38420
094
098
11989140
22976
12872
110
33200
1093
2790
110
766
61
447
9032
1033
3085
1096
Ave
084
084
127
084
088
073
112
089
231
077
046
098
456
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Mathematical Problems in Engineering
Table 8 Results for CEC2013 benchmark functions over 30 independent runs
Function Result DE PSO ABC CEBA PSO-FDR PSO-cf-Local CPEA
11986501
119865meanSD
164e minus 02(+)398e minus 02
151e minus 14(asymp)576e minus 14
151e minus 14(asymp)576e minus 14
499e + 02(+)146e + 02
000e + 00(asymp)000e + 00
000e + 00(asymp)000e + 00
000e+00000e + 00
11986502
119865meanSD
120e + 05(+)150e + 05
173e + 05(+)202e + 05
331e + 06(+)113e + 06
395e + 06(+)160e + 06
437e + 04(+)376e + 04
211e + 04(asymp)161e + 04
182e + 04112e + 04
11986503
119865meanSD
175e + 06(+)609e + 05
463e + 05(+)877e + 05
135e + 07(+)130e + 07
679e + 08(+)327e + 08
138e + 06(+)189e + 06
292e + 04(+)710e + 04
491e + 00252e + 01
11986504
119865meanSD
130e + 03(+)143e + 03
197e + 02(minus)111e + 02
116e + 04(+)319e + 03
625e + 03(+)225e + 03
154e + 02(minus)160e + 02
818e + 02(asymp)526e + 02
726e + 02326e + 02
11986505
119865meanSD
138e minus 01(+)362e minus 01
530e minus 14(minus)576e minus 14
162e minus 13(minus)572e minus 14
115e + 02(+)945e + 01
757e minus 15(minus)288e minus 14
000e + 00(minus)000e + 00
772e minus 05223e minus 05
11986506
119865meanSD
630e + 00(asymp)447e + 00
621e + 00(asymp)436e + 00
151e + 00(minus)290e + 00
572e + 01(+)194e + 01
528e + 00(asymp)492e + 00
926e + 00(+)251e + 00
621e + 00480e + 00
11986507
119865meanSD
725e minus 01(+)193e + 00
163e + 00(+)189e + 00
408e + 01(+)118e + 01
401e + 01(+)531e + 00
189e + 00(+)194e + 00
687e minus 01(+)123e + 00
169e minus 03305e minus 03
11986508
119865meanSD
202e + 01(asymp)633e minus 02
202e + 01(asymp)688e minus 02
204e + 01(+)813e minus 02
202e + 01(asymp)586e minus 02
202e + 01(asymp)646e minus 02
202e + 01(asymp)773e minus 02
202e + 01462e minus 02
11986509
119865meanSD
103e + 00(minus)841e minus 01
297e + 00(asymp)126e + 00
530e + 00(+)855e minus 01
683e + 00(+)571e minus 01
244e + 00(asymp)137e + 00
226e + 00(asymp)862e minus 01
232e + 00129e + 00
11986510
119865meanSD
564e minus 02(asymp)411e minus 02
388e minus 01(+)290e minus 01
142e + 00(+)327e minus 01
624e + 01(+)140e + 01
339e minus 01(+)225e minus 01
112e minus 01(+)553e minus 02
535e minus 02281e minus 02
11986511
119865meanSD
729e minus 01(minus)863e minus 01
563e minus 01(minus)724e minus 01
000e + 00(minus)000e + 00
456e + 01(+)589e + 00
663e minus 01(minus)657e minus 01
122e + 00(minus)129e + 00
795e + 00330e + 00
11986512
119865meanSD
531e + 00(asymp)227e + 00
138e + 01(+)658e + 00
280e + 01(+)822e + 00
459e + 01(+)513e + 00
138e + 01(+)480e + 00
636e + 00(+)219e + 00
500e + 00121e + 00
11986513
119865meanSD
774e + 00(minus)459e + 00
189e + 01(asymp)614e + 00
339e + 01(+)986e + 00
468e + 01(+)805e + 00
153e + 01(asymp)607e + 00
904e + 00(minus)400e + 00
170e + 01872e + 00
11986514
119865meanSD
283e + 00(minus)336e + 00
482e + 01(minus)511e + 01
146e minus 01(minus)609e minus 02
974e + 02(+)112e + 02
565e + 01(minus)834e + 01
200e + 02(minus)100e + 02
317e + 02153e + 02
11986515
119865meanSD
304e + 02(asymp)203e + 02
688e + 02(+)291e + 02
735e + 02(+)194e + 02
721e + 02(+)165e + 02
638e + 02(+)192e + 02
390e + 02(+)124e + 02
302e + 02115e + 02
11986516
119865meanSD
956e minus 01(+)142e minus 01
713e minus 01(+)186e minus 01
893e minus 01(+)191e minus 01
102e + 00(+)114e minus 01
760e minus 01(+)238e minus 01
588e minus 01(+)259e minus 01
447e minus 02410e minus 02
11986517
119865meanSD
145e + 01(asymp)411e + 00
874e + 00(minus)471e + 00
855e + 00(minus)261e + 00
464e + 01(+)721e + 00
132e + 01(asymp)544e + 00
137e + 01(asymp)115e + 00
143e + 01162e + 00
11986518
119865meanSD
229e + 01(+)549e + 00
231e + 01(+)104e + 01
401e + 01(+)584e + 00
548e + 01(+)662e + 00
210e + 01(+)505e + 00
185e + 01(asymp)564e + 00
174e + 01465e + 00
11986519
119865meanSD
481e minus 01(asymp)188e minus 01
544e minus 01(asymp)190e minus 01
622e minus 02(minus)285e minus 02
717e + 00(+)121e + 00
533e minus 01(asymp)158e minus 01
461e minus 01(asymp)128e minus 01
468e minus 01169e minus 01
11986520
119865meanSD
181e + 00(minus)777e minus 01
273e + 00(asymp)718e minus 01
340e + 00(+)240e minus 01
308e + 00(+)219e minus 01
210e + 00(minus)525e minus 01
254e + 00(asymp)342e minus 01
272e + 00583e minus 01
Generalmerit overcontender
3 3 7 19 3 3
ldquo+rdquo ldquominusrdquo and ldquoasymprdquo denote that the performance of CPEA (119863 = 10) is better than worse than and similar to that of other algorithms
119875 =
[[[[[
[
036890 011700 026730
046990 043870 074700
01091 08732 055470
003815 057430 088280
]]]]]
]
(A1)
11989109
Six Hump Camel back function 11989109(119909) = 4119909
2
1
minus
211199094
1
+(13)1199096
1
+11990911199092minus41199094
2
+41199094
2
[minus5 5] min(11989109) =
minus10316
11989110
Matyas function (dimension = 2) 11989110(119909) =
026(1199092
1
+ 1199092
2
) minus 04811990911199092 [minus10 10] min(119891
10) = 0
11989111
Zakharov function 11989111(119909) = sum
119899
119894=1
1199092
119894
+
(sum119899
119894=1
05119894119909119894)2
+ (sum119899
119894=1
05119894119909119894)4 [minus5 10] min(119891
11) = 0
11989112
Schwefelrsquos problem 222 11989112(119909) = sum
119899
119894=1
|119909119894| +
prod119899
119894=1
|119909119894| [minus10 10] min(119891
12) = 0
11989113
Levy function 11989113(119909) = sin2(3120587119909
1) + sum
119899minus1
119894=1
(119909119894minus
1)2
times (1 + sin2(3120587119909119894+1)) + (119909
119899minus 1)2
(1 + sin2(2120587119909119899))
[minus10 10] min(11989113) = 0
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 19
11989114Schwefelrsquos problem 221 119891
14(119909) = max|119909
119894| 1 le 119894 le
119899 [minus100 100] min(11989114) = 0
11989115
Axis parallel hyperellipsoid 11989115(119909) = sum
119899
119894=1
1198941199092
119894
[minus512 512] min(119891
15) = 0
11989116De Jongrsquos function 4 (no noise) 119891
16(119909) = sum
119899
119894=1
1198941199094
119894
[minus128 128] min(119891
16) = 0
11989117
Rastriginrsquos function 11989117(119909) = sum
119899
119894=1
(1199092
119894
minus
10 cos(2120587119909119894) + 10) [minus512 512] min(119891
17) = 0
11989118
Ackleyrsquos path function 11989118(119909) =
minus20 exp(minus02radicsum119899119894=1
1199092119894
119899) minus exp(sum119899119894=1
cos(2120587119909119894)119899) +
20 + 119890 [minus32 32] min(11989118) = 0
11989119Alpine function 119891
19(119909) = sum
119899
119894=1
|119909119894sin(119909119894) + 01119909
119894|
[minus10 10] min(11989119) = 0
11989120
Pathological function 11989120(119909) = sum
119899minus1
119894
(05 +
(sin2radic(1001199092119894
+ 1199092119894+1
) minus 05)(1 + 0001(1199092
119894
minus 2119909119894119909119894+1+
1199092
119894+1
)2
)) [minus100 100] min(11989120) = 0
11989121
Schafferrsquos function 6 (Dimension = 2) 11989121(119909) =
05+(sin2radic(11990921
+ 11990922
)minus05)(1+001(1199092
1
+1199092
2
)2
) [minus1010] min(119891
21) = 0
11989122
Camel Back-3 Three Hump Problem 11989122(119909) =
21199092
1
minus1051199094
1
+(16)1199096
1
+11990911199092+1199092
2
[minus5 5]min(11989122) = 0
11989123
Exponential Problem 11989123(119909) =
minus exp(minus05sum119899119894=1
1199092
119894
) [minus1 1] min(11989123) = minus1
11989124
Hartmann function 2 (Dimension = 6) 11989124(119909) =
minussum4
119894=1
119886119894exp(minussum6
119895=1
119861119894119895(119909119895minus 119876119894119895)2
) [0 1]
119886 = [1 12 3 32]
119861 =
[[[[[
[
10 3 17 305 17 8
005 17 10 01 8 14
3 35 17 10 17 8
17 8 005 10 01 14
]]]]]
]
119876 =
[[[[[
[
01312 01696 05569 00124 08283 05886
02329 04135 08307 03736 01004 09991
02348 01451 03522 02883 03047 06650
04047 08828 08732 05743 01091 00381
]]]]]
]
(A2)
min(11989124) = minus333539215295525
11989125
Shekelrsquos Family 5 119898 = 5 min(11989125) = minus101532
11989125(119909) = minussum
119898
119894=1
[(119909119894minus 119886119894)(119909119894minus 119886119894)119879
+ 119888119894]minus1
119886 =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
4 1 8 6 3 2 5 8 6 7
4 1 8 6 7 9 5 1 2 36
4 1 8 6 3 2 3 8 6 7
4 1 8 6 7 9 3 1 2 36
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119879
(A3)
119888 =100381610038161003816100381601 02 02 04 04 06 03 07 05 05
1003816100381610038161003816
119879
Table 9
119894 119886119894
119887119894
119888119894
119889119894
1 05 00 00 012 12 10 00 053 10 00 minus05 054 10 minus05 00 055 12 00 10 05
11989126Shekelrsquos Family 7119898 = 7 min(119891
26) = minus104029
11989127
Shekelrsquos Family 10 119898 = 10 min(11989127) = 119891(4 4 4
4) = minus10536411989128Goldstein and Price [minus2 2] min(119891
28) = 3
11989128(119909) = [1 + (119909
1+ 1199092+ 1)2
sdot (19 minus 141199091+ 31199092
1
minus 141199092+ 611990911199092+ 31199092
2
)]
times [30 + (21199091minus 31199092)2
sdot (18 minus 321199091+ 12119909
2
1
+ 481199092minus 36119909
11199092+ 27119909
2
2
)]
(A4)
11989129
Becker and Lago Problem 11989129(119909) = (|119909
1| minus 5)2
minus
(|1199092| minus 5)2 [minus10 le 10] min(119891
29) = 0
11989130Hosaki Problem119891
30(119909) = (1minus8119909
1+71199092
1
minus(73)1199093
1
+
(14)1199094
1
)1199092
2
exp(minus1199092) 0 le 119909
1le 5 0 le 119909
2le 6
min(11989130) = minus23458
11989131
Miele and Cantrell Problem 11989131(119909) =
(exp(1199091) minus 1199092)4
+100(1199092minus 1199093)6
+(tan(1199093minus 1199094))4
+1199098
1
[minus1 1] min(119891
31) = 0
11989132Quartic function that is noise119891
32(119909) = sum
119899
119894=1
1198941199094
119894
+
random[0 1) [minus128 128] min(11989132) = 0
11989133
Rastriginrsquos function A 11989133(119909) = 119865Rastrigin(119911)
[minus512 512] min(11989133) = 0
119911 = 119909lowast119872 119863 is the dimension119872 is a119863times119863orthogonalmatrix11989134
Multi-Gaussian Problem 11989134(119909) =
minussum5
119894=1
119886119894exp(minus(((119909
1minus 119887119894)2
+ (1199092minus 119888119894)2
)1198892
119894
)) [minus2 2]min(119891
34) = minus129695 (see Table 9)
11989135
Inverted cosine wave function (Masters)11989135(119909) = minussum
119899minus1
119894=1
(exp((minus(1199092119894
+ 1199092
119894+1
+ 05119909119894119909119894+1))8) times
cos(4radic1199092119894
+ 1199092119894+1
+ 05119909119894119909119894+1)) [minus5 le 5] min(119891
35) =
minus119899 + 111989136Periodic Problem 119891
36(119909) = 1 + sin2119909
1+ sin2119909
2minus
01 exp(minus11990921
minus 1199092
2
) [minus10 10] min(11989136) = 09
11989137
Bohachevsky 1 Problem 11989137(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) minus 04 cos(4120587119909
2) + 07 [minus50 50]
min(11989137) = 0
11989138
Bohachevsky 2 Problem 11989138(119909) = 119909
2
1
+ 21199092
2
minus
03 cos(31205871199091) cos(4120587119909
2)+03 [minus50 50]min(119891
38) = 0
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
20 Mathematical Problems in Engineering
11989139
Salomon Problem 11989139(119909) = 1 minus cos(2120587119909) +
01119909 [minus100 100] 119909 = radicsum119899119894=1
1199092119894
min(11989139) = 0
11989140McCormick Problem119891
40(119909) = sin(119909
1+1199092)+(1199091minus
1199092)2
minus(32)1199091+(52)119909
2+1 minus15 le 119909
1le 4 minus3 le 119909
2le
3 min(11989140) = minus19133
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors express their sincere thanks to the reviewers fortheir significant contributions to the improvement of the finalpaper This research is partly supported by the support of theNationalNatural Science Foundation of China (nos 61272283andU1334211) Shaanxi Science and Technology Project (nos2014KW02-01 and 2014XT-11) and Shaanxi Province NaturalScience Foundation (no 2012JQ8052)
References
[1] H-G Beyer and H-P Schwefel ldquoEvolution strategiesmdasha com-prehensive introductionrdquo Natural Computing vol 1 no 1 pp3ndash52 2002
[2] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence A Bradford Book 1992
[3] E Bonabeau M Dorigo and G Theraulaz ldquoInspiration foroptimization from social insect behaviourrdquoNature vol 406 no6791 pp 39ndash42 2000
[4] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[5] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization I Background and developmentrdquo NaturalComputing vol 6 no 4 pp 467ndash484 2007
[6] B Basturk and D Karaboga ldquoAn artifical bee colony(ABC)algorithm for numeric function optimizationrdquo in Proceedings ofthe IEEE Swarm Intelligence Symposium pp 12ndash14 IndianapolisIndiana USA 2006
[7] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007
[8] L N De Castro and J Timmis Artifical Immune Systems ANew Computational Intelligence Approach Springer BerlinGermany 2002
[9] MG GongH F Du and L C Jiao ldquoOptimal approximation oflinear systems by artificial immune responserdquo Science in ChinaSeries F Information Sciences vol 49 no 1 pp 63ndash79 2006
[10] S Twomey ldquoThe nuclei of natural cloud formation part II thesupersaturation in natural clouds and the variation of clouddroplet concentrationrdquo Geofisica Pura e Applicata vol 43 no1 pp 243ndash249 1959
[11] F A Wang Cloud Meteorological Press Beijing China 2002(Chinese)
[12] Z C Gu The Foundation of Cloud Mist and PrecipitationScience Press Beijing China 1980 (Chinese)
[13] R C Reid andMModellThermodynamics and Its ApplicationsPrentice Hall New York NY USA 2008
[14] W Greiner L Neise and H Stocker Thermodynamics andStatistical Mechanics Classical Theoretical Physics SpringerNew York NY USA 1995
[15] P Elizabeth ldquoOn the origin of cooperationrdquo Science vol 325no 5945 pp 1196ndash1199 2009
[16] M A Nowak ldquoFive rules for the evolution of cooperationrdquoScience vol 314 no 5805 pp 1560ndash1563 2006
[17] J NThompson and B M Cunningham ldquoGeographic structureand dynamics of coevolutionary selectionrdquo Nature vol 417 no13 pp 735ndash738 2002
[18] J B S Haldane The Causes of Evolution Longmans Green ampCo London UK 1932
[19] W D Hamilton ldquoThe genetical evolution of social behaviourrdquoJournal of Theoretical Biology vol 7 no 1 pp 17ndash52 1964
[20] R Axelrod andWDHamilton ldquoThe evolution of cooperationrdquoAmerican Association for the Advancement of Science Sciencevol 211 no 4489 pp 1390ndash1396 1981
[21] MANowak andK Sigmund ldquoEvolution of indirect reciprocityby image scoringrdquo Nature vol 393 no 6685 pp 573ndash577 1998
[22] A Traulsen and M A Nowak ldquoEvolution of cooperation bymultilevel selectionrdquo Proceedings of the National Academy ofSciences of the United States of America vol 103 no 29 pp10952ndash10955 2006
[23] L Shi and RWang ldquoThe evolution of cooperation in asymmet-ric systemsrdquo Science China Life Sciences vol 53 no 1 pp 1ndash112010 (Chinese)
[24] D Y Li ldquoUncertainty in knowledge representationrdquoEngineeringScience vol 2 no 10 pp 73ndash79 2000 (Chinese)
[25] MM Ali C Khompatraporn and Z B Zabinsky ldquoA numericalevaluation of several stochastic algorithms on selected con-tinuous global optimization test problemsrdquo Journal of GlobalOptimization vol 31 no 4 pp 635ndash672 2005
[26] R S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolutionrdquo IEEETransactions onEvolutionary Computation vol 12 no 1 pp 64ndash79 2008
[27] N Hansen S D Muller and P Koumoutsakos ldquoReducing thetime complexity of the derandomized evolution strategy withcovariancematrix adaptation (CMA-ES)rdquoEvolutionaryCompu-tation vol 11 no 1 pp 1ndash18 2003
[28] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009
[29] I C Trelea ldquoThe particle swarm optimization algorithm con-vergence analysis and parameter selectionrdquo Information Pro-cessing Letters vol 85 no 6 pp 317ndash325 2003
[30] J Brest S Greiner B Boskovic M Mernik and V ZumerldquoSelf-adapting control parameters in differential evolution acomparative study on numerical benchmark problemsrdquo IEEETransactions on Evolutionary Computation vol 10 no 6 pp646ndash657 2006
[31] J Kennedy and R Mendes ldquoPopulation structure and particleswarm performancerdquo in Proceedings of the Congress on Evolu-tionary Computation pp 1671ndash1676 Honolulu Hawaii USAMay 2002
[32] T Peram K Veeramachaneni and C K Mohan ldquoFitness-distance-ratio based particle swarm optimizationrdquo in Proceed-ings of the IEEE Swarm Intelligence Symposium pp 174ndash181Indianapolis Ind USA April 2003
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 21
[33] P N Suganthan N Hansen J J Liang et al ldquoProblem defini-tions and evaluation criteria for the CEC2005 special sessiononreal-parameter optimizationrdquo 2005 httpwwwntuedusghomeEPNSugan
[34] G W Zhang R He Y Liu D Y Li and G S Chen ldquoAnevolutionary algorithm based on cloud modelrdquo Chinese Journalof Computers vol 31 no 7 pp 1082ndash1091 2008 (Chinese)
[35] NHansen and S Kern ldquoEvaluating the CMAevolution strategyon multimodal test functionsrdquo in Parallel Problem Solving fromNaturemdashPPSN VIII vol 3242 of Lecture Notes in ComputerScience pp 282ndash291 Springer Berlin Germany 2004
[36] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009
[37] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquo CAD Computer Aided Designvol 43 no 3 pp 303ndash315 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of