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Research ArticleWolf Pack Algorithm for Unconstrained Global Optimization
Hu-Sheng Wu12 and Feng-Ming Zhang1
1 Materiel Management and Safety Engineering Institute Air Force Engineering University Xirsquoan 710051 China2Materiel Engineering Institute Armed Police Force Engineering University Xirsquoan 710086 China
Correspondence should be addressed to Hu-Sheng Wu wuhusheng0421gmailcom
Received 28 June 2013 Revised 13 January 2014 Accepted 27 January 2014 Published 9 March 2014
Academic Editor Orwa Jaber Housheya
Copyright copy 2014 H-S Wu and F-M Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The wolf pack unites and cooperates closely to hunt for the prey in the Tibetan Plateau which shows wonderful skills and amazingstrategies Inspired by their prey hunting behaviors and distribution mode we abstracted three intelligent behaviors scoutingcalling and besieging and two intelligent rules winner-take-all generation rule of lead wolf and stronger-survive renewing ruleof wolf pack Then we proposed a new heuristic swarm intelligent method named wolf pack algorithm (WPA) Experimentsare conducted on a suit of benchmark functions with different characteristics unimodalmultimodal separablenonseparableand the impact of several distance measurements and parameters on WPA is discussed What is more the compared simulationexperiments with other five typical intelligent algorithms genetic algorithm particle swarm optimization algorithm artificial fishswarm algorithm artificial bee colony algorithm and firefly algorithm show that WPA has better convergence and robustnessespecially for high-dimensional functions
1 Introduction
Global optimization is a hot topic with applications in manyareas such as science economy and engineering Generallyunconstrained global optimization problems can be formu-lated as follows
min or max119891 (119883) 119883 = (1199091 1199092 119909
119899) (1)
where 119891 119877119899 rarr 119877 is a real-valued objective function 119883 isin119877119899 and 119899 is the number of parameters to be optimized
As many real-world problems are becoming increasinglycomplex global optimization especially using traditionalmethods is becoming a challenging task [1] Because ofits great search space high-dimensional global optimizationproblems aremore difficult [2] Fortunately many algorithmsinspired by nature have become powerful tools for theseproblems [3ndash5] Since with long time of biological evolutionand natural selection there are many marvelous swarmintelligence phenomenons in nature which are wonderfuland can give us endless inspiration The remarkable swarmbehavior of animals such as swarming ants schooling fishand flocking birds has for long captivated the attention ofnaturalists and scientists [6] People have developed many
intelligent optimization methods to solve complex globalproblems in recent decades In 1995 inspired by social behav-ior and movement dynamics of birds Kennedy proposedthe particle swarm optimization algorithm (PSO) [7] In1996 inspired by social division and foraging behavior ofant colonies Dorigo proposed the ant colony optimizationalgorithm (ACO) [8] In 2002 inspired by foraging behaviorof fish schools Li proposed the artificial fish swarm algorithm(AFSA) [9] In 2005 motivated by the intelligent foragingbehavior of honeybee swarms Karaboga proposed the arti-ficial bee colony (ABC) algorithm [10] In 2008 based onthe flashing behavior of fireflies Doctor Yang proposed fireflyalgorithm (FA) [11] Researchers even give some conceptionsof swarm intelligent algorithms such as rats herds algorithmmosquito swarms algorithm and dolphins herds algorithm[12] Birds fishes ants and bees do not have any humancomplex intelligence such as logical reasoning and syntheticjudgment but under the same aim food they stand outpowerful swarm intelligence through constantly adaptingenvironment and mutual cooperation which give us manynew ideas for solving complex problems
The wolf pack is marvelous Harsh living environmentand constant evolution for centuries have created their
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 465082 17 pageshttpdxdoiorg1011552014465082
2 Mathematical Problems in Engineering
rigorous organization system and subtle hunting behaviorWolves tactics of Mongolia cavalry in Genghis Khan periodsubmarine tactics of Nazi Admiral Doenitz in World WarII and US military wolves attack system for electroniccountermeasures all highlight great charm of their swarmintelligence [13] proposes a wolf colony algorithm (WCA)to solve the optimization problem But the accuracy andefficiency of WCA are not good enough and easily fall intolocal optima especially for high-dimensional functions Soin this paper we reanalyzed collaborative predation behaviorand prey distribution mode of wolves and proposed a newswarm intelligence algorithm called wolf pack algorithm(WPA) Moreover the efficiency and robustness of the newalgorithm were tested by compared experiments
The remainder of this paper is structured as followsIn Section 2 the predation behaviors and prey distributionof wolves are analyzed In Section 3 WPA is describedSection 4 describes the experimental setup followed byexperimental results and analysis Finally conclusion andfuture work are presented in Section 5
2 System Analyzing of Wolf Pack
Wolves are gregarious animals and have clearly social workdivision There is a lead wolf some elite wolves act as scoutsand some ferociouswolves in awolf packThey cooperatewellwith each other and take their respective responsibility for thesurvival and thriving of wolf pack
Firstly the lead wolf as a leader under the law of thejungle is always the smartest and most ferocious one Itis responsible for commanding the wolves and constantlymaking decision by evaluating surrounding situation andperceiving information from other wolves These can avoidthe wolves in danger and command the wolves to smoothlycapture prey as soon as possible
Secondly the lead wolf sends some elite wolves to huntaround and look for prey in the probable scope Those elitewolves are scoutsTheywalk around and independentlymakedecision according to the concentration of smell left by preyand higher concentration means the prey is closer to thewolves So they always move towards the direction of gettingstronger smell
Thirdly once a scout wolf finds the trace of prey itwill howl and report that to lead wolf Then the lead wolfwill evaluate this situation and make a decision whether tosummon the ferocious wolves to round up the prey or notIf they are summoned the ferocious wolves will move fasttowards the direction of the scout wolf
Fourthly after capturing the prey the prey is not dis-tributed equitably but in an order from the strong to theweakThat is to say that the stronger the wolf is the more thefood it will get is Although this distribution rule will makesome weak wolf dead for lack of food it makes sure that thewolves that have the ability to capture prey getmore food so asto keep being strong and can capture more prey successfullyin the next timeThe rule avoids that thewhole pack starves todeath and ensures its continuance and proliferating In whatfollows the author made detailed description and realizationfor the above intelligent behaviors and rules
3 Wolf Pack Algorithm
31 Some Definitions If the predatory space of the artificialwolves is a 119873times119863 Euclidean space 119873 is the number of wolves119863 is the number of variables The position of one wolf 119894 isa vector X
119894= (1199091198941
1199091198942
119909119894119863
) and 119909119894119889is the 119889th variable
value of the 119894th artificial wolf 119884 = 119891(X) represents theconcentration of preyrsquos smell perceived by artificial wolveswhich is also the objective function value
The distance between two wolves 119901 and 119902 is describedas 119871(119901 119902) Several distance measurements can be selectedaccording to specific problems For example hamming dis-tance can be used in WPA for 0-1 discrete optimizationwhileManhattan distance (MD) and Euclidean distance (ED)can be used in WPA for continuous numerical functionoptimization In this paper we mainly discuss the latterproblem and the selection of distance measurements will bediscussed in Section 421 Moreover because the problemsof maximum value and minimal value can convert to eachother only the maximum value problem is discussed in whatfollows
32 The Description of Intelligent Behaviors and Rules Thecooperation between lead wolf scout wolves and ferociouswolves makes nearly perfect predation while prey distribu-tion from the strong to the weak makes the wolf pack thrivestowards the direction of the prey that it most probably can beable to capture The whole predation behavior of wolf pack isabstracted three intelligent behaviors scouting calling andbesieging behavior and two intelligent rules winner-take-all generating rule for the lead wolf and the stronger-surviverenewing rule for the wolf pack
(1) The winner-take-all generating rule for the lead wolfthe artificial wolf with the best objective function value is leadwolf During each iteration compare the function value of thelead wolf with the best one of other wolves if the value oflead wolf is not better it will be replaced Then the best wolfbecomes lead wolf Rather than acting the three intelligentbehaviors the lead wolf directly goes into the next iterationuntil it is replaced by other better wolf
(2) Scouting behavior S num elite wolves except the leadwolf are considered as the scout wolves they search thesolution in predatory space 119884
119894is the concentration of prey
smell perceived by the scout wolf 119894 119884lead is the concentrationof prey smell perceived by the lead wolf
If 119884119894
gt 119884lead that means the scout wolf is nearer to theprey and probably captures prey so the scout wolf 119894 becomeslead wolf and 119884lead = 119884
119894
If 119884119894
lt 119884lead the scout wolf 119894 respectively takes asteptowards ℎ different directions the step length is 119904119905119890119901
119886 After
taking a step towards the 119901th direction the state of the scoutwolf 119894 is formulated below
119909119901
119894119889= 119909119894119889
+ sin(2120587 times119901
ℎ) times step119889
119886 119901 = 1 2 ℎ (2)
It should be noted that ℎ is different for each wolf becauseof their different seeking ways So ℎ is randomly selected in[ℎmin ℎmax] and it must be an integer 119884
1198940is the concentration
of prey smell perceived by the scout wolf 119894 and 119884119894119901represents
Mathematical Problems in Engineering 3
the one after it took a step towards the 119901th direction Ifmax119884
1198941 1198841198942
119884119894ℎ
gt 1198841198940 the wolf 119894 steps forward and its
position 119883119894is updated Then repeat the above until 119884
119894gt 119884lead
or the maximum number of repetitions 119879max is reached(3)Calling behavior the lead wolf will howl and summon
119872 119899119906119898 ferocious wolves to gather around the prey Here theposition of the lead wolf is considered as the one of the preyso that the ferocious wolves aggregate towards the position ofleadwolf 119904119905119890119901
119887is the step length119892119896
119889is the position of artificial
lead wolf in the 119889th variable space at the 119896th iteration Theposition of the ferocious wolf 119894 in the 119896th iterative calculationis updated according to the following equation
119909119896+1119894119889
= 119909119896119894119889
+ step119889119887
sdot(119892119896119889
minus 119909119896119894119889
)1003816100381610038161003816119892119896
119889minus 119909119896119894119889
1003816100381610038161003816 (3)
This formula consists of two parts the former is thecurrent position of wolf 119894 which represents the foundationfor prey hunting the latter represents the aggregate tendencyof other wolves towards the lead wolf which shows the leadwolf rsquos leadership to the wolf pack
If 119884119894
gt 119884lead the ferocious wolf 119894 becomes lead wolfand 119884lead = 119884
119894 then the wolf 119894 takes the calling behavior If
119884119894
lt 119884lead the ferocious wolf 119894 keeps on aggregating towardsthe lead wolf with a fast speed until 119871(119894 119897) lt 119871near the wolftakes besieging behavior 119871(119894 119897) is the distance between thewolf 119894 and the lead wolf 119897 119871near is the distance determinantcoefficient as a judging condition which determine whetherwolf 119894 changes state from aggregating towards the lead wolfto besieging behavior The different value of 119871near will affectalgorithmic convergence rate There will be a discussion inSection 422
Calling behavior shows information transferring andsharing mechanism in wolf pack and blends the idea of socialcognition
(4) Besieging behavior after large-steps running towardsthe lead wolf the wolves are close to the prey then all wolvesexcept the leadwolf will take besieging behavior for capturingprey Now the position of lead wolf is considered as theposition of prey In particular 119866119896
119889reprensents the position of
prey in the119889th variable space at the 119896th iterationThepositionof wolf 119894 is updated according to the following equation
119909119896+1119894119889
= 119909119896119894119889
+ 120582 sdot step119889119888
sdot10038161003816100381610038161003816119866119896
119889minus 119909119896119894119889
10038161003816100381610038161003816 (4)
120582 is a random number uniformly distributed at theinterval [minus1 1] 119904119905119890119901
119888is the step length of wolf 119894 when it
takes besieging behavior1198841198940is the concentration of prey smell
perceived by the wolf 119894 and 119884119894119896represents the one after it
took this behavior If 1198841198940
lt 119884119894119896 the position X
119894is updated
otherwise it not changedThere are 119904119905119890119901
119886 119904119905119890119901119887 and 119904119905119890119901
119888in the three intelligent
behaviors and the three-step length in 119889th variable spaceshould have the following relationship
step119889119886
=step119889119887
2= 2 sdot step119889
119888= 119878 (5)
119878 is step coefficient and represents the fineness degree ofartificial wolf searching for prey in resolution space
(5) The stronger-survive renewing rule for the wolf packthe prey is distributed from the strong to the weak which willresult in some weak wolves deadThe algorithm will generate119877 wolves while deleting 119877 wolves with bad objective functionvalues Specifically with the help of the lead wolf rsquos huntingexperience in the 119889th variable space position of the 119894th oneof 119877 wolves is defined as follows
119909119894119889
= 119892119889
sdot rand 119894 = 1 2 119877 (6)
119892119889is the position of artificial lead wolf in the 119889th variable
space rand is a random number uniformly distributed at theinterval [minus01 01]
When the value of 119877 is larger it is better for sustainingwolf rsquos diversity and making the algorithm have the abilityto open up new resolution space But if 119877 is too large thealgorithm will nearly be a random search approach Becausethe number and scale of prey captured by wolves are differentin natural word which will lead to different number ofweak wolf dead 119877 is an integer and randomly selected atthe interval [119899(2 lowast 120573) 119899120573] 120573 is the population renewingproportional coefficient
33 Algorithm Description As described in the previoussection WPA has three artificial intelligent behaviors andtwo intelligent rules There are scouting behavior callingbehavior and besieging behavior and winner-take-all rule forgenerating lead wolf and the stronger-survive renewing rulefor wolf pack
Firstly the scouting behavior accelerates the possibilitythat WPA can fully traverse the solution space Secondly thewinner-take-all rule for generating lead wolf and the callingbehavior make the wolves move towards the lead wolf whoseposition is the nearest to the prey and most likely capturingpreyThe winner-take-all rule and calling behavior also makewolves arrive at the neighborhood of the global optimumonly after a few iterations elapsed since the step of wolvesin calling behavior is the largest one Thirdly with a smallstep step
119888 besieging behavior makes WPA algorithm have
the ability to open up new solution space and carefully searchthe global optima in good solution area Fourthly with thehelp of stronger-survive renewing rule for the wolf pack thealgorithm can get several new wolves whose positions arenear the best wolf lead wolf which allows for more latitudeof search space to anchor the global optimum while keepingpopulation diversity in each iteration
All the abovemakeWPA possesses superior performancein accuracy and robustness which will be seen in Section 4
Having discussed all the components ofWPA the impor-tant computation steps are detailed below
Step 1 (initialization) Initialize the following parametersthe initial position of artificial wolf 119894 (X
119894) the number of
the wolves (119873) the maximum number of iterations (119896max)the step coefficient (119878) the distance determinant coefficient(119871near) the maximum number of repetitions in scoutingbehavior (119879max) and the population renewing proportionalcoefficient (120573)
4 Mathematical Problems in Engineering
Table 1 Benchmark functions in experiments
No Functions Formulation Global extremum 119863 C Range1 Rosenbrock 119891() = 100(119909
2minus 11990921)2 + (1 minus 119909
1)2 119891min() = 0 2 UN (minus2048 2048)
2 Colville119891() = 100(1199092
1minus 1199092)2
+ (1199091
minus 1)2
+ (1199093
minus 1)2
+ 90(11990923
minus 1199094)2
+ 101(1199092
minus 1)2
+ (1199094
minus 1)2
+ 198(1199092
minus 1)(1199094
minus 1)119891min() = 0 4 UN (minus10 10)
3 Sphere 119891 () =119863
sum119894=1
1199092119894
119891min() = 0 200 US (minus100 100)
4 Sumsquares 119891 () =119863
sum119894=1
1198941199092119894
119891min() = 0 150 US (minus10 10)
5 Booth 119891() = (1199091
+ 21199092
minus 7)2 + (21199091
+ 1199092
minus 5)2 119891min() = 0 2 MS (minus10 10)
6 Bridge 119891 () =sinradic1199092
1+ 11990922
radic11990921
+ 11990922
+ exp(cos 2120587119909
1+ cos 2120587119909
2
2) minus 07129 119891max() = 30054 2 MN (minus15 15)
7 Ackley 119891() = minus20 exp(minus02radic1
119863
119863
sum119894=1
1199092119894) minus exp(
1
119863
119863
sum119894=1
cos 2120587119909119894) + 20 + 119890 119891min() = 0 50 MN (minus32 32)
8 Griewank 119891() =1
4000
119863
sum119894=1
1199092119894
minus119863
prod119894=1
cos(119909119894
radic119894) + 1 119891min() = 0 100 MN (minus600 600)
119863 dimension C characteristic U unimodal M multimodal S separable N nonseparable
Step 2 The wolf with best function value is considered aslead wolf In practical computation 119878 num = 119872 num =119899 minus 1 which means that wolves except for lead wolf actwith different behavior as different status So here exceptfor lead wolf according to formula (2) the rest of the 119899 minus 1wolves firstly act as the artificial scout wolves to take scoutingbehavior until 119884
119894gt 119884lead or the maximum number of
repetition 119879max is reached and then go to Step 3
Step 3 Except for the lead wolf the rest of the 119899 minus 1 wolvessecondly act as the artificial ferocious wolves and gathertowards the lead wolf according to (3) 119884
119894is the smell
concentration of prey perceived by wolf 119894 if 119884119894
ge 119884lead go toStep 2 otherwise the wolf 119894 continues running until 119871(119894 119897) le119871near then go to Step 4
Step 4 The position of artificial wolves who take besiegingbehavior is updated according to (4)
Step 5 Update the position of lead wolf under the winner-take-all generating rule and update the wolf pack under thepopulation renewing rule according to (6)
Step 6 If the program reaches the precision requirement orthemaximumnumber of iterations the position and functionvalue of lead wolf the problem optimal solution will beoutputted otherwise go to Step 2
So the flow chart of WPA can be shown as Figure 1
4 Experimental Results
The ingredients of the WPA method have been describedin Section 3 In this section the design of experimentsis explained sensitivity analysis of parameters on WPAis explored and the empirical results are reported which
Initialization
Scouting behavior
Yi gt Ylead
Yi gt Ylead
orT gt Tmax
Calling behavior
L(i l) gt Lnear
Besieging behavior
Renew the position of lead wolf
Renew wolf pack
Terminate
Output resultsYes
Yes
Yes
Yes
No
No
No
No
Figure 1 The flow chart of WPA
compare the WPA approach with those of GA PSO ASFAABC and FA
41 Design of the Experiments
411 Benchmark Functions In order to evaluate the perfor-mance of these algorithms eight classical benchmark func-tions are presented inTable 1Though only eight functions areused in this test they are enough to include some differentkinds of problems such as unimodal multimodal regularirregular separable nonseparable and multidimensional
If a function has more than one local optimum thisfunction is calledmultimodalMultimodal functions are usedto test the ability of algorithms to get rid of local minima
Mathematical Problems in Engineering 5
Table 2 The list of various methods used in the paper
Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]
Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]
Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions
In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable
The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1
412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2
For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3
Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms
Table 3 Six criteria and their abbreviations
Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art
Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816
119865lowastlt 120576 119865lowast = 0
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0
(7)
The SR is a percentage value that is calculated as
SR =successful runs
runs (8)
Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance
All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory
42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9
421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution
6 Mathematical Problems in Engineering
Table 4 Sensitivity analysis of distance measurements
Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts
Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053
Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220
Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825
Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109
Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546
Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541
Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884
Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903
ED 0 08350 00167 01181 92 gt135119890 + 4
space Therefore a discussion about their impacts on theperformance of WPA is needed
There are two wolves X119901
= (1199091199011
1199091199012
119909119901119863
) is theposition of wolf 119901X
119902= (1199091199021
1199091199022
119909119902119863
) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space
119871 ED (119901 119902) =119863
sum119889=1
(119909119901119889
minus 119909119902119889
)2
119871MD (119901 119902) =119863
sum119889=1
10038161003816100381610038161003816119909119901119889 minus 119909119902119889
10038161003816100381610038161003816
(9)
The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima
As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness
Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD
422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant
Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case
Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage
Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions
Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only
Mathematical Problems in Engineering 7
Table5Sensitivityanalysisof
stepcoeffi
cient(
119878)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
69119890
minus8
plusmn4
3119890minus
82
7119890minus
8plusmn
35119890
minus8
11119890
minus8
plusmn9
1119890minus
93
2119890minus
9plusmn
27119890
minus9
50119890
minus9
plusmn5
7119890minus
93
2119890minus
9plusmn
37119890
minus9
12119890
minus9
plusmn1
6119890minus
9Colville
13119890
minus7
plusmn7
1119890minus
83
3119890minus
7plusmn
28119890
minus7(90)
26119890
minus7
plusmn1
9119890minus
72
3119890minus
7plusmn
14119890
minus7
35119890
minus7
plusmn2
5119890minus
79
5119890minus
7plusmn
10119890
minus6(80)
14119890
minus6
plusmn1
5119890minus
6(50)
Sphere
23119890
minus14
5plusmn
71119890
minus14
56
6119890minus
152
plusmn2
1119890minus
151
21119890
minus14
6plusmn
45119890
minus14
63
9119890minus
146
plusmn1
2119890minus
145
12119890
minus14
5plusmn
34119890
minus14
51
7119890minus
146
plusmn5
3119890minus
146
22119890
minus14
9plusmn
68119890
minus14
9Sumsquares
98119890
minus14
5plusmn
31119890
minus14
43
1119890minus
146
plusmn8
4119890minus
146
81119890
minus14
7plusmn
26119890
minus14
64
8119890minus
146
plusmn1
0119890minus
145
38119890
minus15
2plusmn
79119890
minus15
23
4119890minus
147
plusmn1
1119890minus
146
12119890
minus14
7plusmn
39119890
minus14
7Bo
oth
54119890
minus7
plusmn3
3119890minus
71
6119890minus
9plusmn
11119890
minus9
32119890
minus11
plusmn1
6119890minus
111
3119890minus
12plusmn
91119890
minus13
13119890
minus13
plusmn1
2119890minus
133
9119890minus
15plusmn
18119890
minus15
12119890
minus16
plusmn5
8119890minus
17Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
025
plusmn0
53(80)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
0plusmn
00
06plusmn
019
(92)
020
plusmn0
42(86)
8 Mathematical Problems in Engineering
Table6Sensitivityanalysisof
distance
determ
inantcoefficient(
119871near)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
44119890
minus8
plusmn6
5119890minus
82
3119890minus
8plusmn
37119890
minus8
34119890
minus9
plusmn4
8119890minus
93
0119890minus
8plusmn
29119890
minus8
19119890
minus8
plusmn2
4119890minus
82
4119890minus
8plusmn
47119890
minus8
29119890
minus8
plusmn5
3119890minus
8Colville
20119890
minus7
plusmn9
9119890minus
82
6119890minus
7plusmn
16119890
minus7
35119890
minus7
plusmn2
6119890minus
72
3119890minus
7plusmn
15119890
minus7
12119890
minus7
plusmn3
4119890minus
82
8119890minus
7plusmn
19119890
minus7
14119890
minus7
plusmn6
9119890minus
8Sphere
68119890
minus14
6plusmn
20119890
minus14
51
9119890minus
146
plusmn6
2119890minus
146
17119890
minus14
5plusmn
43119890
minus14
52
6119890minus
148
plusmn8
3119890minus
148
36119890
minus14
6plusmn
11119890
minus14
53
7119890minus
151
plusmn1
1119890minus
150
53119890
minus14
9plusmn
17119890
minus14
8Sumsquares1
1119890
minus14
7plusmn
34119890
minus14
71
0119890minus
146
plusmn3
3119890minus
146
37119890
minus15
1plusmn
89119890
minus15
16
2119890minus
146
plusmn1
9119890minus
145
62119890
minus15
2plusmn
19119890
minus15
11
22119890
minus14
5plusmn
29119890
minus14
51
3119890minus
148
plusmn4
0119890minus
148
Booth
26119890
minus11
plusmn1
3119890minus
112
9119890minus
11plusmn
19119890
minus11
24119890
minus11
plusmn1
6119890minus
113
1119890minus
11plusmn
18119890
minus01
12
4119890minus
11plusmn
13119890
minus11
31119890
minus11
plusmn2
1119890minus
111
0119890minus
10plusmn
13119890
minus10
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
014
plusmn0
43(90)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
159
119890minus
15plusmn
149
119890minus
158
9119890minus
16plusmn
0Grie
wank
008
plusmn0
26(90)
10119890
minus3
plusmn0
02(96)
0plusmn
00
plusmn0
0plusmn
00
10plusmn
033
(92)
0plusmn
0
Mathematical Problems in Engineering 9
Table7Sensitivityanalysisof
them
axim
umnu
mbero
frepetition
sinscou
tingbehavior
(119879max)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)6
810
1214
1618
Rosenb
rock
24119890
minus8
plusmn2
6119890minus
88
4119890minus
9plusmn
80119890
minus9
13119890
minus8
plusmn1
3119890minus
81
4119890minus
8plusmn
10119890
minus8
20119890
minus8
plusmn1
9119890minus
82
1119890minus
8plusmn
25119890
minus8
12119890
minus8
plusmn8
9119890minus
9Colville
48119890
minus7
plusmn2
2119890minus
73
4119890minus
7plusmn
18119890
minus7
15119890
minus7
plusmn1
2119890minus
73
8119890minus
7plusmn
20119890
minus7
36119890
minus7
plusmn3
7119890minus
7(96)
34119890
minus7
plusmn2
5119890minus
72
6119890minus
7plusmn
15119890
minus7
Sphere
71119890
minus14
7plusmn
22119890
minus14
64
5119890minus
146
plusmn9
0119890minus
146
78119890
minus14
6plusmn
23119890
minus14
51
9119890minus
148
plusmn5
3119890minus
148
57119890
minus14
8plusmn
13119890
minus14
76
9119890minus
145
plusmn2
2119890minus
144
36119890
minus14
7plusmn
11119890
minus14
6Sumsquares
41119890
minus14
6plusmn
13119890
minus14
52
4119890minus
149
plusmn4
8119890minus
149
42119890
minus14
9plusmn
13119890
minus14
88
3119890minus
150
plusmn2
6119890minus
149
85119890
minus14
7plusmn
27119890
minus14
65
4119890minus
146
plusmn9
0119890minus
146
14119890
minus15
1plusmn
44119890
minus15
1Bo
oth
32119890
minus11
plusmn2
9119890minus
114
2119890minus
11plusmn
27119890
minus11
25119890
minus11
plusmn1
5119890minus
112
1119890minus
11plusmn
15119890
minus11
32119890
minus11
plusmn2
5119890minus
112
6119890minus
11plusmn
18119890
minus11
26119890
minus11
plusmn2
7119890minus
11Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
08
9119890minus
16plusmn
0Grie
wank
010
plusmn0
33(92)
0plusmn
01
0119890minus
3plusmn
002
(98)
009
plusmn0
31(88)
0plusmn
00
09plusmn
029
(94)
83119890
minus4
plusmn0
02(98)
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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2 Mathematical Problems in Engineering
rigorous organization system and subtle hunting behaviorWolves tactics of Mongolia cavalry in Genghis Khan periodsubmarine tactics of Nazi Admiral Doenitz in World WarII and US military wolves attack system for electroniccountermeasures all highlight great charm of their swarmintelligence [13] proposes a wolf colony algorithm (WCA)to solve the optimization problem But the accuracy andefficiency of WCA are not good enough and easily fall intolocal optima especially for high-dimensional functions Soin this paper we reanalyzed collaborative predation behaviorand prey distribution mode of wolves and proposed a newswarm intelligence algorithm called wolf pack algorithm(WPA) Moreover the efficiency and robustness of the newalgorithm were tested by compared experiments
The remainder of this paper is structured as followsIn Section 2 the predation behaviors and prey distributionof wolves are analyzed In Section 3 WPA is describedSection 4 describes the experimental setup followed byexperimental results and analysis Finally conclusion andfuture work are presented in Section 5
2 System Analyzing of Wolf Pack
Wolves are gregarious animals and have clearly social workdivision There is a lead wolf some elite wolves act as scoutsand some ferociouswolves in awolf packThey cooperatewellwith each other and take their respective responsibility for thesurvival and thriving of wolf pack
Firstly the lead wolf as a leader under the law of thejungle is always the smartest and most ferocious one Itis responsible for commanding the wolves and constantlymaking decision by evaluating surrounding situation andperceiving information from other wolves These can avoidthe wolves in danger and command the wolves to smoothlycapture prey as soon as possible
Secondly the lead wolf sends some elite wolves to huntaround and look for prey in the probable scope Those elitewolves are scoutsTheywalk around and independentlymakedecision according to the concentration of smell left by preyand higher concentration means the prey is closer to thewolves So they always move towards the direction of gettingstronger smell
Thirdly once a scout wolf finds the trace of prey itwill howl and report that to lead wolf Then the lead wolfwill evaluate this situation and make a decision whether tosummon the ferocious wolves to round up the prey or notIf they are summoned the ferocious wolves will move fasttowards the direction of the scout wolf
Fourthly after capturing the prey the prey is not dis-tributed equitably but in an order from the strong to theweakThat is to say that the stronger the wolf is the more thefood it will get is Although this distribution rule will makesome weak wolf dead for lack of food it makes sure that thewolves that have the ability to capture prey getmore food so asto keep being strong and can capture more prey successfullyin the next timeThe rule avoids that thewhole pack starves todeath and ensures its continuance and proliferating In whatfollows the author made detailed description and realizationfor the above intelligent behaviors and rules
3 Wolf Pack Algorithm
31 Some Definitions If the predatory space of the artificialwolves is a 119873times119863 Euclidean space 119873 is the number of wolves119863 is the number of variables The position of one wolf 119894 isa vector X
119894= (1199091198941
1199091198942
119909119894119863
) and 119909119894119889is the 119889th variable
value of the 119894th artificial wolf 119884 = 119891(X) represents theconcentration of preyrsquos smell perceived by artificial wolveswhich is also the objective function value
The distance between two wolves 119901 and 119902 is describedas 119871(119901 119902) Several distance measurements can be selectedaccording to specific problems For example hamming dis-tance can be used in WPA for 0-1 discrete optimizationwhileManhattan distance (MD) and Euclidean distance (ED)can be used in WPA for continuous numerical functionoptimization In this paper we mainly discuss the latterproblem and the selection of distance measurements will bediscussed in Section 421 Moreover because the problemsof maximum value and minimal value can convert to eachother only the maximum value problem is discussed in whatfollows
32 The Description of Intelligent Behaviors and Rules Thecooperation between lead wolf scout wolves and ferociouswolves makes nearly perfect predation while prey distribu-tion from the strong to the weak makes the wolf pack thrivestowards the direction of the prey that it most probably can beable to capture The whole predation behavior of wolf pack isabstracted three intelligent behaviors scouting calling andbesieging behavior and two intelligent rules winner-take-all generating rule for the lead wolf and the stronger-surviverenewing rule for the wolf pack
(1) The winner-take-all generating rule for the lead wolfthe artificial wolf with the best objective function value is leadwolf During each iteration compare the function value of thelead wolf with the best one of other wolves if the value oflead wolf is not better it will be replaced Then the best wolfbecomes lead wolf Rather than acting the three intelligentbehaviors the lead wolf directly goes into the next iterationuntil it is replaced by other better wolf
(2) Scouting behavior S num elite wolves except the leadwolf are considered as the scout wolves they search thesolution in predatory space 119884
119894is the concentration of prey
smell perceived by the scout wolf 119894 119884lead is the concentrationof prey smell perceived by the lead wolf
If 119884119894
gt 119884lead that means the scout wolf is nearer to theprey and probably captures prey so the scout wolf 119894 becomeslead wolf and 119884lead = 119884
119894
If 119884119894
lt 119884lead the scout wolf 119894 respectively takes asteptowards ℎ different directions the step length is 119904119905119890119901
119886 After
taking a step towards the 119901th direction the state of the scoutwolf 119894 is formulated below
119909119901
119894119889= 119909119894119889
+ sin(2120587 times119901
ℎ) times step119889
119886 119901 = 1 2 ℎ (2)
It should be noted that ℎ is different for each wolf becauseof their different seeking ways So ℎ is randomly selected in[ℎmin ℎmax] and it must be an integer 119884
1198940is the concentration
of prey smell perceived by the scout wolf 119894 and 119884119894119901represents
Mathematical Problems in Engineering 3
the one after it took a step towards the 119901th direction Ifmax119884
1198941 1198841198942
119884119894ℎ
gt 1198841198940 the wolf 119894 steps forward and its
position 119883119894is updated Then repeat the above until 119884
119894gt 119884lead
or the maximum number of repetitions 119879max is reached(3)Calling behavior the lead wolf will howl and summon
119872 119899119906119898 ferocious wolves to gather around the prey Here theposition of the lead wolf is considered as the one of the preyso that the ferocious wolves aggregate towards the position ofleadwolf 119904119905119890119901
119887is the step length119892119896
119889is the position of artificial
lead wolf in the 119889th variable space at the 119896th iteration Theposition of the ferocious wolf 119894 in the 119896th iterative calculationis updated according to the following equation
119909119896+1119894119889
= 119909119896119894119889
+ step119889119887
sdot(119892119896119889
minus 119909119896119894119889
)1003816100381610038161003816119892119896
119889minus 119909119896119894119889
1003816100381610038161003816 (3)
This formula consists of two parts the former is thecurrent position of wolf 119894 which represents the foundationfor prey hunting the latter represents the aggregate tendencyof other wolves towards the lead wolf which shows the leadwolf rsquos leadership to the wolf pack
If 119884119894
gt 119884lead the ferocious wolf 119894 becomes lead wolfand 119884lead = 119884
119894 then the wolf 119894 takes the calling behavior If
119884119894
lt 119884lead the ferocious wolf 119894 keeps on aggregating towardsthe lead wolf with a fast speed until 119871(119894 119897) lt 119871near the wolftakes besieging behavior 119871(119894 119897) is the distance between thewolf 119894 and the lead wolf 119897 119871near is the distance determinantcoefficient as a judging condition which determine whetherwolf 119894 changes state from aggregating towards the lead wolfto besieging behavior The different value of 119871near will affectalgorithmic convergence rate There will be a discussion inSection 422
Calling behavior shows information transferring andsharing mechanism in wolf pack and blends the idea of socialcognition
(4) Besieging behavior after large-steps running towardsthe lead wolf the wolves are close to the prey then all wolvesexcept the leadwolf will take besieging behavior for capturingprey Now the position of lead wolf is considered as theposition of prey In particular 119866119896
119889reprensents the position of
prey in the119889th variable space at the 119896th iterationThepositionof wolf 119894 is updated according to the following equation
119909119896+1119894119889
= 119909119896119894119889
+ 120582 sdot step119889119888
sdot10038161003816100381610038161003816119866119896
119889minus 119909119896119894119889
10038161003816100381610038161003816 (4)
120582 is a random number uniformly distributed at theinterval [minus1 1] 119904119905119890119901
119888is the step length of wolf 119894 when it
takes besieging behavior1198841198940is the concentration of prey smell
perceived by the wolf 119894 and 119884119894119896represents the one after it
took this behavior If 1198841198940
lt 119884119894119896 the position X
119894is updated
otherwise it not changedThere are 119904119905119890119901
119886 119904119905119890119901119887 and 119904119905119890119901
119888in the three intelligent
behaviors and the three-step length in 119889th variable spaceshould have the following relationship
step119889119886
=step119889119887
2= 2 sdot step119889
119888= 119878 (5)
119878 is step coefficient and represents the fineness degree ofartificial wolf searching for prey in resolution space
(5) The stronger-survive renewing rule for the wolf packthe prey is distributed from the strong to the weak which willresult in some weak wolves deadThe algorithm will generate119877 wolves while deleting 119877 wolves with bad objective functionvalues Specifically with the help of the lead wolf rsquos huntingexperience in the 119889th variable space position of the 119894th oneof 119877 wolves is defined as follows
119909119894119889
= 119892119889
sdot rand 119894 = 1 2 119877 (6)
119892119889is the position of artificial lead wolf in the 119889th variable
space rand is a random number uniformly distributed at theinterval [minus01 01]
When the value of 119877 is larger it is better for sustainingwolf rsquos diversity and making the algorithm have the abilityto open up new resolution space But if 119877 is too large thealgorithm will nearly be a random search approach Becausethe number and scale of prey captured by wolves are differentin natural word which will lead to different number ofweak wolf dead 119877 is an integer and randomly selected atthe interval [119899(2 lowast 120573) 119899120573] 120573 is the population renewingproportional coefficient
33 Algorithm Description As described in the previoussection WPA has three artificial intelligent behaviors andtwo intelligent rules There are scouting behavior callingbehavior and besieging behavior and winner-take-all rule forgenerating lead wolf and the stronger-survive renewing rulefor wolf pack
Firstly the scouting behavior accelerates the possibilitythat WPA can fully traverse the solution space Secondly thewinner-take-all rule for generating lead wolf and the callingbehavior make the wolves move towards the lead wolf whoseposition is the nearest to the prey and most likely capturingpreyThe winner-take-all rule and calling behavior also makewolves arrive at the neighborhood of the global optimumonly after a few iterations elapsed since the step of wolvesin calling behavior is the largest one Thirdly with a smallstep step
119888 besieging behavior makes WPA algorithm have
the ability to open up new solution space and carefully searchthe global optima in good solution area Fourthly with thehelp of stronger-survive renewing rule for the wolf pack thealgorithm can get several new wolves whose positions arenear the best wolf lead wolf which allows for more latitudeof search space to anchor the global optimum while keepingpopulation diversity in each iteration
All the abovemakeWPA possesses superior performancein accuracy and robustness which will be seen in Section 4
Having discussed all the components ofWPA the impor-tant computation steps are detailed below
Step 1 (initialization) Initialize the following parametersthe initial position of artificial wolf 119894 (X
119894) the number of
the wolves (119873) the maximum number of iterations (119896max)the step coefficient (119878) the distance determinant coefficient(119871near) the maximum number of repetitions in scoutingbehavior (119879max) and the population renewing proportionalcoefficient (120573)
4 Mathematical Problems in Engineering
Table 1 Benchmark functions in experiments
No Functions Formulation Global extremum 119863 C Range1 Rosenbrock 119891() = 100(119909
2minus 11990921)2 + (1 minus 119909
1)2 119891min() = 0 2 UN (minus2048 2048)
2 Colville119891() = 100(1199092
1minus 1199092)2
+ (1199091
minus 1)2
+ (1199093
minus 1)2
+ 90(11990923
minus 1199094)2
+ 101(1199092
minus 1)2
+ (1199094
minus 1)2
+ 198(1199092
minus 1)(1199094
minus 1)119891min() = 0 4 UN (minus10 10)
3 Sphere 119891 () =119863
sum119894=1
1199092119894
119891min() = 0 200 US (minus100 100)
4 Sumsquares 119891 () =119863
sum119894=1
1198941199092119894
119891min() = 0 150 US (minus10 10)
5 Booth 119891() = (1199091
+ 21199092
minus 7)2 + (21199091
+ 1199092
minus 5)2 119891min() = 0 2 MS (minus10 10)
6 Bridge 119891 () =sinradic1199092
1+ 11990922
radic11990921
+ 11990922
+ exp(cos 2120587119909
1+ cos 2120587119909
2
2) minus 07129 119891max() = 30054 2 MN (minus15 15)
7 Ackley 119891() = minus20 exp(minus02radic1
119863
119863
sum119894=1
1199092119894) minus exp(
1
119863
119863
sum119894=1
cos 2120587119909119894) + 20 + 119890 119891min() = 0 50 MN (minus32 32)
8 Griewank 119891() =1
4000
119863
sum119894=1
1199092119894
minus119863
prod119894=1
cos(119909119894
radic119894) + 1 119891min() = 0 100 MN (minus600 600)
119863 dimension C characteristic U unimodal M multimodal S separable N nonseparable
Step 2 The wolf with best function value is considered aslead wolf In practical computation 119878 num = 119872 num =119899 minus 1 which means that wolves except for lead wolf actwith different behavior as different status So here exceptfor lead wolf according to formula (2) the rest of the 119899 minus 1wolves firstly act as the artificial scout wolves to take scoutingbehavior until 119884
119894gt 119884lead or the maximum number of
repetition 119879max is reached and then go to Step 3
Step 3 Except for the lead wolf the rest of the 119899 minus 1 wolvessecondly act as the artificial ferocious wolves and gathertowards the lead wolf according to (3) 119884
119894is the smell
concentration of prey perceived by wolf 119894 if 119884119894
ge 119884lead go toStep 2 otherwise the wolf 119894 continues running until 119871(119894 119897) le119871near then go to Step 4
Step 4 The position of artificial wolves who take besiegingbehavior is updated according to (4)
Step 5 Update the position of lead wolf under the winner-take-all generating rule and update the wolf pack under thepopulation renewing rule according to (6)
Step 6 If the program reaches the precision requirement orthemaximumnumber of iterations the position and functionvalue of lead wolf the problem optimal solution will beoutputted otherwise go to Step 2
So the flow chart of WPA can be shown as Figure 1
4 Experimental Results
The ingredients of the WPA method have been describedin Section 3 In this section the design of experimentsis explained sensitivity analysis of parameters on WPAis explored and the empirical results are reported which
Initialization
Scouting behavior
Yi gt Ylead
Yi gt Ylead
orT gt Tmax
Calling behavior
L(i l) gt Lnear
Besieging behavior
Renew the position of lead wolf
Renew wolf pack
Terminate
Output resultsYes
Yes
Yes
Yes
No
No
No
No
Figure 1 The flow chart of WPA
compare the WPA approach with those of GA PSO ASFAABC and FA
41 Design of the Experiments
411 Benchmark Functions In order to evaluate the perfor-mance of these algorithms eight classical benchmark func-tions are presented inTable 1Though only eight functions areused in this test they are enough to include some differentkinds of problems such as unimodal multimodal regularirregular separable nonseparable and multidimensional
If a function has more than one local optimum thisfunction is calledmultimodalMultimodal functions are usedto test the ability of algorithms to get rid of local minima
Mathematical Problems in Engineering 5
Table 2 The list of various methods used in the paper
Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]
Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]
Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions
In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable
The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1
412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2
For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3
Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms
Table 3 Six criteria and their abbreviations
Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art
Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816
119865lowastlt 120576 119865lowast = 0
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0
(7)
The SR is a percentage value that is calculated as
SR =successful runs
runs (8)
Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance
All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory
42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9
421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution
6 Mathematical Problems in Engineering
Table 4 Sensitivity analysis of distance measurements
Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts
Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053
Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220
Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825
Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109
Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546
Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541
Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884
Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903
ED 0 08350 00167 01181 92 gt135119890 + 4
space Therefore a discussion about their impacts on theperformance of WPA is needed
There are two wolves X119901
= (1199091199011
1199091199012
119909119901119863
) is theposition of wolf 119901X
119902= (1199091199021
1199091199022
119909119902119863
) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space
119871 ED (119901 119902) =119863
sum119889=1
(119909119901119889
minus 119909119902119889
)2
119871MD (119901 119902) =119863
sum119889=1
10038161003816100381610038161003816119909119901119889 minus 119909119902119889
10038161003816100381610038161003816
(9)
The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima
As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness
Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD
422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant
Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case
Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage
Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions
Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only
Mathematical Problems in Engineering 7
Table5Sensitivityanalysisof
stepcoeffi
cient(
119878)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
69119890
minus8
plusmn4
3119890minus
82
7119890minus
8plusmn
35119890
minus8
11119890
minus8
plusmn9
1119890minus
93
2119890minus
9plusmn
27119890
minus9
50119890
minus9
plusmn5
7119890minus
93
2119890minus
9plusmn
37119890
minus9
12119890
minus9
plusmn1
6119890minus
9Colville
13119890
minus7
plusmn7
1119890minus
83
3119890minus
7plusmn
28119890
minus7(90)
26119890
minus7
plusmn1
9119890minus
72
3119890minus
7plusmn
14119890
minus7
35119890
minus7
plusmn2
5119890minus
79
5119890minus
7plusmn
10119890
minus6(80)
14119890
minus6
plusmn1
5119890minus
6(50)
Sphere
23119890
minus14
5plusmn
71119890
minus14
56
6119890minus
152
plusmn2
1119890minus
151
21119890
minus14
6plusmn
45119890
minus14
63
9119890minus
146
plusmn1
2119890minus
145
12119890
minus14
5plusmn
34119890
minus14
51
7119890minus
146
plusmn5
3119890minus
146
22119890
minus14
9plusmn
68119890
minus14
9Sumsquares
98119890
minus14
5plusmn
31119890
minus14
43
1119890minus
146
plusmn8
4119890minus
146
81119890
minus14
7plusmn
26119890
minus14
64
8119890minus
146
plusmn1
0119890minus
145
38119890
minus15
2plusmn
79119890
minus15
23
4119890minus
147
plusmn1
1119890minus
146
12119890
minus14
7plusmn
39119890
minus14
7Bo
oth
54119890
minus7
plusmn3
3119890minus
71
6119890minus
9plusmn
11119890
minus9
32119890
minus11
plusmn1
6119890minus
111
3119890minus
12plusmn
91119890
minus13
13119890
minus13
plusmn1
2119890minus
133
9119890minus
15plusmn
18119890
minus15
12119890
minus16
plusmn5
8119890minus
17Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
025
plusmn0
53(80)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
0plusmn
00
06plusmn
019
(92)
020
plusmn0
42(86)
8 Mathematical Problems in Engineering
Table6Sensitivityanalysisof
distance
determ
inantcoefficient(
119871near)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
44119890
minus8
plusmn6
5119890minus
82
3119890minus
8plusmn
37119890
minus8
34119890
minus9
plusmn4
8119890minus
93
0119890minus
8plusmn
29119890
minus8
19119890
minus8
plusmn2
4119890minus
82
4119890minus
8plusmn
47119890
minus8
29119890
minus8
plusmn5
3119890minus
8Colville
20119890
minus7
plusmn9
9119890minus
82
6119890minus
7plusmn
16119890
minus7
35119890
minus7
plusmn2
6119890minus
72
3119890minus
7plusmn
15119890
minus7
12119890
minus7
plusmn3
4119890minus
82
8119890minus
7plusmn
19119890
minus7
14119890
minus7
plusmn6
9119890minus
8Sphere
68119890
minus14
6plusmn
20119890
minus14
51
9119890minus
146
plusmn6
2119890minus
146
17119890
minus14
5plusmn
43119890
minus14
52
6119890minus
148
plusmn8
3119890minus
148
36119890
minus14
6plusmn
11119890
minus14
53
7119890minus
151
plusmn1
1119890minus
150
53119890
minus14
9plusmn
17119890
minus14
8Sumsquares1
1119890
minus14
7plusmn
34119890
minus14
71
0119890minus
146
plusmn3
3119890minus
146
37119890
minus15
1plusmn
89119890
minus15
16
2119890minus
146
plusmn1
9119890minus
145
62119890
minus15
2plusmn
19119890
minus15
11
22119890
minus14
5plusmn
29119890
minus14
51
3119890minus
148
plusmn4
0119890minus
148
Booth
26119890
minus11
plusmn1
3119890minus
112
9119890minus
11plusmn
19119890
minus11
24119890
minus11
plusmn1
6119890minus
113
1119890minus
11plusmn
18119890
minus01
12
4119890minus
11plusmn
13119890
minus11
31119890
minus11
plusmn2
1119890minus
111
0119890minus
10plusmn
13119890
minus10
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
014
plusmn0
43(90)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
159
119890minus
15plusmn
149
119890minus
158
9119890minus
16plusmn
0Grie
wank
008
plusmn0
26(90)
10119890
minus3
plusmn0
02(96)
0plusmn
00
plusmn0
0plusmn
00
10plusmn
033
(92)
0plusmn
0
Mathematical Problems in Engineering 9
Table7Sensitivityanalysisof
them
axim
umnu
mbero
frepetition
sinscou
tingbehavior
(119879max)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)6
810
1214
1618
Rosenb
rock
24119890
minus8
plusmn2
6119890minus
88
4119890minus
9plusmn
80119890
minus9
13119890
minus8
plusmn1
3119890minus
81
4119890minus
8plusmn
10119890
minus8
20119890
minus8
plusmn1
9119890minus
82
1119890minus
8plusmn
25119890
minus8
12119890
minus8
plusmn8
9119890minus
9Colville
48119890
minus7
plusmn2
2119890minus
73
4119890minus
7plusmn
18119890
minus7
15119890
minus7
plusmn1
2119890minus
73
8119890minus
7plusmn
20119890
minus7
36119890
minus7
plusmn3
7119890minus
7(96)
34119890
minus7
plusmn2
5119890minus
72
6119890minus
7plusmn
15119890
minus7
Sphere
71119890
minus14
7plusmn
22119890
minus14
64
5119890minus
146
plusmn9
0119890minus
146
78119890
minus14
6plusmn
23119890
minus14
51
9119890minus
148
plusmn5
3119890minus
148
57119890
minus14
8plusmn
13119890
minus14
76
9119890minus
145
plusmn2
2119890minus
144
36119890
minus14
7plusmn
11119890
minus14
6Sumsquares
41119890
minus14
6plusmn
13119890
minus14
52
4119890minus
149
plusmn4
8119890minus
149
42119890
minus14
9plusmn
13119890
minus14
88
3119890minus
150
plusmn2
6119890minus
149
85119890
minus14
7plusmn
27119890
minus14
65
4119890minus
146
plusmn9
0119890minus
146
14119890
minus15
1plusmn
44119890
minus15
1Bo
oth
32119890
minus11
plusmn2
9119890minus
114
2119890minus
11plusmn
27119890
minus11
25119890
minus11
plusmn1
5119890minus
112
1119890minus
11plusmn
15119890
minus11
32119890
minus11
plusmn2
5119890minus
112
6119890minus
11plusmn
18119890
minus11
26119890
minus11
plusmn2
7119890minus
11Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
08
9119890minus
16plusmn
0Grie
wank
010
plusmn0
33(92)
0plusmn
01
0119890minus
3plusmn
002
(98)
009
plusmn0
31(88)
0plusmn
00
09plusmn
029
(94)
83119890
minus4
plusmn0
02(98)
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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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
Journal of
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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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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
the one after it took a step towards the 119901th direction Ifmax119884
1198941 1198841198942
119884119894ℎ
gt 1198841198940 the wolf 119894 steps forward and its
position 119883119894is updated Then repeat the above until 119884
119894gt 119884lead
or the maximum number of repetitions 119879max is reached(3)Calling behavior the lead wolf will howl and summon
119872 119899119906119898 ferocious wolves to gather around the prey Here theposition of the lead wolf is considered as the one of the preyso that the ferocious wolves aggregate towards the position ofleadwolf 119904119905119890119901
119887is the step length119892119896
119889is the position of artificial
lead wolf in the 119889th variable space at the 119896th iteration Theposition of the ferocious wolf 119894 in the 119896th iterative calculationis updated according to the following equation
119909119896+1119894119889
= 119909119896119894119889
+ step119889119887
sdot(119892119896119889
minus 119909119896119894119889
)1003816100381610038161003816119892119896
119889minus 119909119896119894119889
1003816100381610038161003816 (3)
This formula consists of two parts the former is thecurrent position of wolf 119894 which represents the foundationfor prey hunting the latter represents the aggregate tendencyof other wolves towards the lead wolf which shows the leadwolf rsquos leadership to the wolf pack
If 119884119894
gt 119884lead the ferocious wolf 119894 becomes lead wolfand 119884lead = 119884
119894 then the wolf 119894 takes the calling behavior If
119884119894
lt 119884lead the ferocious wolf 119894 keeps on aggregating towardsthe lead wolf with a fast speed until 119871(119894 119897) lt 119871near the wolftakes besieging behavior 119871(119894 119897) is the distance between thewolf 119894 and the lead wolf 119897 119871near is the distance determinantcoefficient as a judging condition which determine whetherwolf 119894 changes state from aggregating towards the lead wolfto besieging behavior The different value of 119871near will affectalgorithmic convergence rate There will be a discussion inSection 422
Calling behavior shows information transferring andsharing mechanism in wolf pack and blends the idea of socialcognition
(4) Besieging behavior after large-steps running towardsthe lead wolf the wolves are close to the prey then all wolvesexcept the leadwolf will take besieging behavior for capturingprey Now the position of lead wolf is considered as theposition of prey In particular 119866119896
119889reprensents the position of
prey in the119889th variable space at the 119896th iterationThepositionof wolf 119894 is updated according to the following equation
119909119896+1119894119889
= 119909119896119894119889
+ 120582 sdot step119889119888
sdot10038161003816100381610038161003816119866119896
119889minus 119909119896119894119889
10038161003816100381610038161003816 (4)
120582 is a random number uniformly distributed at theinterval [minus1 1] 119904119905119890119901
119888is the step length of wolf 119894 when it
takes besieging behavior1198841198940is the concentration of prey smell
perceived by the wolf 119894 and 119884119894119896represents the one after it
took this behavior If 1198841198940
lt 119884119894119896 the position X
119894is updated
otherwise it not changedThere are 119904119905119890119901
119886 119904119905119890119901119887 and 119904119905119890119901
119888in the three intelligent
behaviors and the three-step length in 119889th variable spaceshould have the following relationship
step119889119886
=step119889119887
2= 2 sdot step119889
119888= 119878 (5)
119878 is step coefficient and represents the fineness degree ofartificial wolf searching for prey in resolution space
(5) The stronger-survive renewing rule for the wolf packthe prey is distributed from the strong to the weak which willresult in some weak wolves deadThe algorithm will generate119877 wolves while deleting 119877 wolves with bad objective functionvalues Specifically with the help of the lead wolf rsquos huntingexperience in the 119889th variable space position of the 119894th oneof 119877 wolves is defined as follows
119909119894119889
= 119892119889
sdot rand 119894 = 1 2 119877 (6)
119892119889is the position of artificial lead wolf in the 119889th variable
space rand is a random number uniformly distributed at theinterval [minus01 01]
When the value of 119877 is larger it is better for sustainingwolf rsquos diversity and making the algorithm have the abilityto open up new resolution space But if 119877 is too large thealgorithm will nearly be a random search approach Becausethe number and scale of prey captured by wolves are differentin natural word which will lead to different number ofweak wolf dead 119877 is an integer and randomly selected atthe interval [119899(2 lowast 120573) 119899120573] 120573 is the population renewingproportional coefficient
33 Algorithm Description As described in the previoussection WPA has three artificial intelligent behaviors andtwo intelligent rules There are scouting behavior callingbehavior and besieging behavior and winner-take-all rule forgenerating lead wolf and the stronger-survive renewing rulefor wolf pack
Firstly the scouting behavior accelerates the possibilitythat WPA can fully traverse the solution space Secondly thewinner-take-all rule for generating lead wolf and the callingbehavior make the wolves move towards the lead wolf whoseposition is the nearest to the prey and most likely capturingpreyThe winner-take-all rule and calling behavior also makewolves arrive at the neighborhood of the global optimumonly after a few iterations elapsed since the step of wolvesin calling behavior is the largest one Thirdly with a smallstep step
119888 besieging behavior makes WPA algorithm have
the ability to open up new solution space and carefully searchthe global optima in good solution area Fourthly with thehelp of stronger-survive renewing rule for the wolf pack thealgorithm can get several new wolves whose positions arenear the best wolf lead wolf which allows for more latitudeof search space to anchor the global optimum while keepingpopulation diversity in each iteration
All the abovemakeWPA possesses superior performancein accuracy and robustness which will be seen in Section 4
Having discussed all the components ofWPA the impor-tant computation steps are detailed below
Step 1 (initialization) Initialize the following parametersthe initial position of artificial wolf 119894 (X
119894) the number of
the wolves (119873) the maximum number of iterations (119896max)the step coefficient (119878) the distance determinant coefficient(119871near) the maximum number of repetitions in scoutingbehavior (119879max) and the population renewing proportionalcoefficient (120573)
4 Mathematical Problems in Engineering
Table 1 Benchmark functions in experiments
No Functions Formulation Global extremum 119863 C Range1 Rosenbrock 119891() = 100(119909
2minus 11990921)2 + (1 minus 119909
1)2 119891min() = 0 2 UN (minus2048 2048)
2 Colville119891() = 100(1199092
1minus 1199092)2
+ (1199091
minus 1)2
+ (1199093
minus 1)2
+ 90(11990923
minus 1199094)2
+ 101(1199092
minus 1)2
+ (1199094
minus 1)2
+ 198(1199092
minus 1)(1199094
minus 1)119891min() = 0 4 UN (minus10 10)
3 Sphere 119891 () =119863
sum119894=1
1199092119894
119891min() = 0 200 US (minus100 100)
4 Sumsquares 119891 () =119863
sum119894=1
1198941199092119894
119891min() = 0 150 US (minus10 10)
5 Booth 119891() = (1199091
+ 21199092
minus 7)2 + (21199091
+ 1199092
minus 5)2 119891min() = 0 2 MS (minus10 10)
6 Bridge 119891 () =sinradic1199092
1+ 11990922
radic11990921
+ 11990922
+ exp(cos 2120587119909
1+ cos 2120587119909
2
2) minus 07129 119891max() = 30054 2 MN (minus15 15)
7 Ackley 119891() = minus20 exp(minus02radic1
119863
119863
sum119894=1
1199092119894) minus exp(
1
119863
119863
sum119894=1
cos 2120587119909119894) + 20 + 119890 119891min() = 0 50 MN (minus32 32)
8 Griewank 119891() =1
4000
119863
sum119894=1
1199092119894
minus119863
prod119894=1
cos(119909119894
radic119894) + 1 119891min() = 0 100 MN (minus600 600)
119863 dimension C characteristic U unimodal M multimodal S separable N nonseparable
Step 2 The wolf with best function value is considered aslead wolf In practical computation 119878 num = 119872 num =119899 minus 1 which means that wolves except for lead wolf actwith different behavior as different status So here exceptfor lead wolf according to formula (2) the rest of the 119899 minus 1wolves firstly act as the artificial scout wolves to take scoutingbehavior until 119884
119894gt 119884lead or the maximum number of
repetition 119879max is reached and then go to Step 3
Step 3 Except for the lead wolf the rest of the 119899 minus 1 wolvessecondly act as the artificial ferocious wolves and gathertowards the lead wolf according to (3) 119884
119894is the smell
concentration of prey perceived by wolf 119894 if 119884119894
ge 119884lead go toStep 2 otherwise the wolf 119894 continues running until 119871(119894 119897) le119871near then go to Step 4
Step 4 The position of artificial wolves who take besiegingbehavior is updated according to (4)
Step 5 Update the position of lead wolf under the winner-take-all generating rule and update the wolf pack under thepopulation renewing rule according to (6)
Step 6 If the program reaches the precision requirement orthemaximumnumber of iterations the position and functionvalue of lead wolf the problem optimal solution will beoutputted otherwise go to Step 2
So the flow chart of WPA can be shown as Figure 1
4 Experimental Results
The ingredients of the WPA method have been describedin Section 3 In this section the design of experimentsis explained sensitivity analysis of parameters on WPAis explored and the empirical results are reported which
Initialization
Scouting behavior
Yi gt Ylead
Yi gt Ylead
orT gt Tmax
Calling behavior
L(i l) gt Lnear
Besieging behavior
Renew the position of lead wolf
Renew wolf pack
Terminate
Output resultsYes
Yes
Yes
Yes
No
No
No
No
Figure 1 The flow chart of WPA
compare the WPA approach with those of GA PSO ASFAABC and FA
41 Design of the Experiments
411 Benchmark Functions In order to evaluate the perfor-mance of these algorithms eight classical benchmark func-tions are presented inTable 1Though only eight functions areused in this test they are enough to include some differentkinds of problems such as unimodal multimodal regularirregular separable nonseparable and multidimensional
If a function has more than one local optimum thisfunction is calledmultimodalMultimodal functions are usedto test the ability of algorithms to get rid of local minima
Mathematical Problems in Engineering 5
Table 2 The list of various methods used in the paper
Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]
Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]
Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions
In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable
The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1
412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2
For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3
Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms
Table 3 Six criteria and their abbreviations
Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art
Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816
119865lowastlt 120576 119865lowast = 0
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0
(7)
The SR is a percentage value that is calculated as
SR =successful runs
runs (8)
Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance
All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory
42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9
421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution
6 Mathematical Problems in Engineering
Table 4 Sensitivity analysis of distance measurements
Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts
Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053
Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220
Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825
Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109
Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546
Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541
Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884
Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903
ED 0 08350 00167 01181 92 gt135119890 + 4
space Therefore a discussion about their impacts on theperformance of WPA is needed
There are two wolves X119901
= (1199091199011
1199091199012
119909119901119863
) is theposition of wolf 119901X
119902= (1199091199021
1199091199022
119909119902119863
) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space
119871 ED (119901 119902) =119863
sum119889=1
(119909119901119889
minus 119909119902119889
)2
119871MD (119901 119902) =119863
sum119889=1
10038161003816100381610038161003816119909119901119889 minus 119909119902119889
10038161003816100381610038161003816
(9)
The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima
As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness
Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD
422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant
Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case
Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage
Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions
Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only
Mathematical Problems in Engineering 7
Table5Sensitivityanalysisof
stepcoeffi
cient(
119878)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
69119890
minus8
plusmn4
3119890minus
82
7119890minus
8plusmn
35119890
minus8
11119890
minus8
plusmn9
1119890minus
93
2119890minus
9plusmn
27119890
minus9
50119890
minus9
plusmn5
7119890minus
93
2119890minus
9plusmn
37119890
minus9
12119890
minus9
plusmn1
6119890minus
9Colville
13119890
minus7
plusmn7
1119890minus
83
3119890minus
7plusmn
28119890
minus7(90)
26119890
minus7
plusmn1
9119890minus
72
3119890minus
7plusmn
14119890
minus7
35119890
minus7
plusmn2
5119890minus
79
5119890minus
7plusmn
10119890
minus6(80)
14119890
minus6
plusmn1
5119890minus
6(50)
Sphere
23119890
minus14
5plusmn
71119890
minus14
56
6119890minus
152
plusmn2
1119890minus
151
21119890
minus14
6plusmn
45119890
minus14
63
9119890minus
146
plusmn1
2119890minus
145
12119890
minus14
5plusmn
34119890
minus14
51
7119890minus
146
plusmn5
3119890minus
146
22119890
minus14
9plusmn
68119890
minus14
9Sumsquares
98119890
minus14
5plusmn
31119890
minus14
43
1119890minus
146
plusmn8
4119890minus
146
81119890
minus14
7plusmn
26119890
minus14
64
8119890minus
146
plusmn1
0119890minus
145
38119890
minus15
2plusmn
79119890
minus15
23
4119890minus
147
plusmn1
1119890minus
146
12119890
minus14
7plusmn
39119890
minus14
7Bo
oth
54119890
minus7
plusmn3
3119890minus
71
6119890minus
9plusmn
11119890
minus9
32119890
minus11
plusmn1
6119890minus
111
3119890minus
12plusmn
91119890
minus13
13119890
minus13
plusmn1
2119890minus
133
9119890minus
15plusmn
18119890
minus15
12119890
minus16
plusmn5
8119890minus
17Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
025
plusmn0
53(80)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
0plusmn
00
06plusmn
019
(92)
020
plusmn0
42(86)
8 Mathematical Problems in Engineering
Table6Sensitivityanalysisof
distance
determ
inantcoefficient(
119871near)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
44119890
minus8
plusmn6
5119890minus
82
3119890minus
8plusmn
37119890
minus8
34119890
minus9
plusmn4
8119890minus
93
0119890minus
8plusmn
29119890
minus8
19119890
minus8
plusmn2
4119890minus
82
4119890minus
8plusmn
47119890
minus8
29119890
minus8
plusmn5
3119890minus
8Colville
20119890
minus7
plusmn9
9119890minus
82
6119890minus
7plusmn
16119890
minus7
35119890
minus7
plusmn2
6119890minus
72
3119890minus
7plusmn
15119890
minus7
12119890
minus7
plusmn3
4119890minus
82
8119890minus
7plusmn
19119890
minus7
14119890
minus7
plusmn6
9119890minus
8Sphere
68119890
minus14
6plusmn
20119890
minus14
51
9119890minus
146
plusmn6
2119890minus
146
17119890
minus14
5plusmn
43119890
minus14
52
6119890minus
148
plusmn8
3119890minus
148
36119890
minus14
6plusmn
11119890
minus14
53
7119890minus
151
plusmn1
1119890minus
150
53119890
minus14
9plusmn
17119890
minus14
8Sumsquares1
1119890
minus14
7plusmn
34119890
minus14
71
0119890minus
146
plusmn3
3119890minus
146
37119890
minus15
1plusmn
89119890
minus15
16
2119890minus
146
plusmn1
9119890minus
145
62119890
minus15
2plusmn
19119890
minus15
11
22119890
minus14
5plusmn
29119890
minus14
51
3119890minus
148
plusmn4
0119890minus
148
Booth
26119890
minus11
plusmn1
3119890minus
112
9119890minus
11plusmn
19119890
minus11
24119890
minus11
plusmn1
6119890minus
113
1119890minus
11plusmn
18119890
minus01
12
4119890minus
11plusmn
13119890
minus11
31119890
minus11
plusmn2
1119890minus
111
0119890minus
10plusmn
13119890
minus10
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
014
plusmn0
43(90)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
159
119890minus
15plusmn
149
119890minus
158
9119890minus
16plusmn
0Grie
wank
008
plusmn0
26(90)
10119890
minus3
plusmn0
02(96)
0plusmn
00
plusmn0
0plusmn
00
10plusmn
033
(92)
0plusmn
0
Mathematical Problems in Engineering 9
Table7Sensitivityanalysisof
them
axim
umnu
mbero
frepetition
sinscou
tingbehavior
(119879max)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)6
810
1214
1618
Rosenb
rock
24119890
minus8
plusmn2
6119890minus
88
4119890minus
9plusmn
80119890
minus9
13119890
minus8
plusmn1
3119890minus
81
4119890minus
8plusmn
10119890
minus8
20119890
minus8
plusmn1
9119890minus
82
1119890minus
8plusmn
25119890
minus8
12119890
minus8
plusmn8
9119890minus
9Colville
48119890
minus7
plusmn2
2119890minus
73
4119890minus
7plusmn
18119890
minus7
15119890
minus7
plusmn1
2119890minus
73
8119890minus
7plusmn
20119890
minus7
36119890
minus7
plusmn3
7119890minus
7(96)
34119890
minus7
plusmn2
5119890minus
72
6119890minus
7plusmn
15119890
minus7
Sphere
71119890
minus14
7plusmn
22119890
minus14
64
5119890minus
146
plusmn9
0119890minus
146
78119890
minus14
6plusmn
23119890
minus14
51
9119890minus
148
plusmn5
3119890minus
148
57119890
minus14
8plusmn
13119890
minus14
76
9119890minus
145
plusmn2
2119890minus
144
36119890
minus14
7plusmn
11119890
minus14
6Sumsquares
41119890
minus14
6plusmn
13119890
minus14
52
4119890minus
149
plusmn4
8119890minus
149
42119890
minus14
9plusmn
13119890
minus14
88
3119890minus
150
plusmn2
6119890minus
149
85119890
minus14
7plusmn
27119890
minus14
65
4119890minus
146
plusmn9
0119890minus
146
14119890
minus15
1plusmn
44119890
minus15
1Bo
oth
32119890
minus11
plusmn2
9119890minus
114
2119890minus
11plusmn
27119890
minus11
25119890
minus11
plusmn1
5119890minus
112
1119890minus
11plusmn
15119890
minus11
32119890
minus11
plusmn2
5119890minus
112
6119890minus
11plusmn
18119890
minus11
26119890
minus11
plusmn2
7119890minus
11Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
08
9119890minus
16plusmn
0Grie
wank
010
plusmn0
33(92)
0plusmn
01
0119890minus
3plusmn
002
(98)
009
plusmn0
31(88)
0plusmn
00
09plusmn
029
(94)
83119890
minus4
plusmn0
02(98)
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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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Journal of
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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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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
4 Mathematical Problems in Engineering
Table 1 Benchmark functions in experiments
No Functions Formulation Global extremum 119863 C Range1 Rosenbrock 119891() = 100(119909
2minus 11990921)2 + (1 minus 119909
1)2 119891min() = 0 2 UN (minus2048 2048)
2 Colville119891() = 100(1199092
1minus 1199092)2
+ (1199091
minus 1)2
+ (1199093
minus 1)2
+ 90(11990923
minus 1199094)2
+ 101(1199092
minus 1)2
+ (1199094
minus 1)2
+ 198(1199092
minus 1)(1199094
minus 1)119891min() = 0 4 UN (minus10 10)
3 Sphere 119891 () =119863
sum119894=1
1199092119894
119891min() = 0 200 US (minus100 100)
4 Sumsquares 119891 () =119863
sum119894=1
1198941199092119894
119891min() = 0 150 US (minus10 10)
5 Booth 119891() = (1199091
+ 21199092
minus 7)2 + (21199091
+ 1199092
minus 5)2 119891min() = 0 2 MS (minus10 10)
6 Bridge 119891 () =sinradic1199092
1+ 11990922
radic11990921
+ 11990922
+ exp(cos 2120587119909
1+ cos 2120587119909
2
2) minus 07129 119891max() = 30054 2 MN (minus15 15)
7 Ackley 119891() = minus20 exp(minus02radic1
119863
119863
sum119894=1
1199092119894) minus exp(
1
119863
119863
sum119894=1
cos 2120587119909119894) + 20 + 119890 119891min() = 0 50 MN (minus32 32)
8 Griewank 119891() =1
4000
119863
sum119894=1
1199092119894
minus119863
prod119894=1
cos(119909119894
radic119894) + 1 119891min() = 0 100 MN (minus600 600)
119863 dimension C characteristic U unimodal M multimodal S separable N nonseparable
Step 2 The wolf with best function value is considered aslead wolf In practical computation 119878 num = 119872 num =119899 minus 1 which means that wolves except for lead wolf actwith different behavior as different status So here exceptfor lead wolf according to formula (2) the rest of the 119899 minus 1wolves firstly act as the artificial scout wolves to take scoutingbehavior until 119884
119894gt 119884lead or the maximum number of
repetition 119879max is reached and then go to Step 3
Step 3 Except for the lead wolf the rest of the 119899 minus 1 wolvessecondly act as the artificial ferocious wolves and gathertowards the lead wolf according to (3) 119884
119894is the smell
concentration of prey perceived by wolf 119894 if 119884119894
ge 119884lead go toStep 2 otherwise the wolf 119894 continues running until 119871(119894 119897) le119871near then go to Step 4
Step 4 The position of artificial wolves who take besiegingbehavior is updated according to (4)
Step 5 Update the position of lead wolf under the winner-take-all generating rule and update the wolf pack under thepopulation renewing rule according to (6)
Step 6 If the program reaches the precision requirement orthemaximumnumber of iterations the position and functionvalue of lead wolf the problem optimal solution will beoutputted otherwise go to Step 2
So the flow chart of WPA can be shown as Figure 1
4 Experimental Results
The ingredients of the WPA method have been describedin Section 3 In this section the design of experimentsis explained sensitivity analysis of parameters on WPAis explored and the empirical results are reported which
Initialization
Scouting behavior
Yi gt Ylead
Yi gt Ylead
orT gt Tmax
Calling behavior
L(i l) gt Lnear
Besieging behavior
Renew the position of lead wolf
Renew wolf pack
Terminate
Output resultsYes
Yes
Yes
Yes
No
No
No
No
Figure 1 The flow chart of WPA
compare the WPA approach with those of GA PSO ASFAABC and FA
41 Design of the Experiments
411 Benchmark Functions In order to evaluate the perfor-mance of these algorithms eight classical benchmark func-tions are presented inTable 1Though only eight functions areused in this test they are enough to include some differentkinds of problems such as unimodal multimodal regularirregular separable nonseparable and multidimensional
If a function has more than one local optimum thisfunction is calledmultimodalMultimodal functions are usedto test the ability of algorithms to get rid of local minima
Mathematical Problems in Engineering 5
Table 2 The list of various methods used in the paper
Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]
Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]
Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions
In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable
The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1
412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2
For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3
Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms
Table 3 Six criteria and their abbreviations
Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art
Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816
119865lowastlt 120576 119865lowast = 0
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0
(7)
The SR is a percentage value that is calculated as
SR =successful runs
runs (8)
Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance
All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory
42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9
421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution
6 Mathematical Problems in Engineering
Table 4 Sensitivity analysis of distance measurements
Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts
Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053
Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220
Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825
Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109
Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546
Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541
Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884
Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903
ED 0 08350 00167 01181 92 gt135119890 + 4
space Therefore a discussion about their impacts on theperformance of WPA is needed
There are two wolves X119901
= (1199091199011
1199091199012
119909119901119863
) is theposition of wolf 119901X
119902= (1199091199021
1199091199022
119909119902119863
) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space
119871 ED (119901 119902) =119863
sum119889=1
(119909119901119889
minus 119909119902119889
)2
119871MD (119901 119902) =119863
sum119889=1
10038161003816100381610038161003816119909119901119889 minus 119909119902119889
10038161003816100381610038161003816
(9)
The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima
As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness
Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD
422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant
Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case
Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage
Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions
Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only
Mathematical Problems in Engineering 7
Table5Sensitivityanalysisof
stepcoeffi
cient(
119878)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
69119890
minus8
plusmn4
3119890minus
82
7119890minus
8plusmn
35119890
minus8
11119890
minus8
plusmn9
1119890minus
93
2119890minus
9plusmn
27119890
minus9
50119890
minus9
plusmn5
7119890minus
93
2119890minus
9plusmn
37119890
minus9
12119890
minus9
plusmn1
6119890minus
9Colville
13119890
minus7
plusmn7
1119890minus
83
3119890minus
7plusmn
28119890
minus7(90)
26119890
minus7
plusmn1
9119890minus
72
3119890minus
7plusmn
14119890
minus7
35119890
minus7
plusmn2
5119890minus
79
5119890minus
7plusmn
10119890
minus6(80)
14119890
minus6
plusmn1
5119890minus
6(50)
Sphere
23119890
minus14
5plusmn
71119890
minus14
56
6119890minus
152
plusmn2
1119890minus
151
21119890
minus14
6plusmn
45119890
minus14
63
9119890minus
146
plusmn1
2119890minus
145
12119890
minus14
5plusmn
34119890
minus14
51
7119890minus
146
plusmn5
3119890minus
146
22119890
minus14
9plusmn
68119890
minus14
9Sumsquares
98119890
minus14
5plusmn
31119890
minus14
43
1119890minus
146
plusmn8
4119890minus
146
81119890
minus14
7plusmn
26119890
minus14
64
8119890minus
146
plusmn1
0119890minus
145
38119890
minus15
2plusmn
79119890
minus15
23
4119890minus
147
plusmn1
1119890minus
146
12119890
minus14
7plusmn
39119890
minus14
7Bo
oth
54119890
minus7
plusmn3
3119890minus
71
6119890minus
9plusmn
11119890
minus9
32119890
minus11
plusmn1
6119890minus
111
3119890minus
12plusmn
91119890
minus13
13119890
minus13
plusmn1
2119890minus
133
9119890minus
15plusmn
18119890
minus15
12119890
minus16
plusmn5
8119890minus
17Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
025
plusmn0
53(80)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
0plusmn
00
06plusmn
019
(92)
020
plusmn0
42(86)
8 Mathematical Problems in Engineering
Table6Sensitivityanalysisof
distance
determ
inantcoefficient(
119871near)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
44119890
minus8
plusmn6
5119890minus
82
3119890minus
8plusmn
37119890
minus8
34119890
minus9
plusmn4
8119890minus
93
0119890minus
8plusmn
29119890
minus8
19119890
minus8
plusmn2
4119890minus
82
4119890minus
8plusmn
47119890
minus8
29119890
minus8
plusmn5
3119890minus
8Colville
20119890
minus7
plusmn9
9119890minus
82
6119890minus
7plusmn
16119890
minus7
35119890
minus7
plusmn2
6119890minus
72
3119890minus
7plusmn
15119890
minus7
12119890
minus7
plusmn3
4119890minus
82
8119890minus
7plusmn
19119890
minus7
14119890
minus7
plusmn6
9119890minus
8Sphere
68119890
minus14
6plusmn
20119890
minus14
51
9119890minus
146
plusmn6
2119890minus
146
17119890
minus14
5plusmn
43119890
minus14
52
6119890minus
148
plusmn8
3119890minus
148
36119890
minus14
6plusmn
11119890
minus14
53
7119890minus
151
plusmn1
1119890minus
150
53119890
minus14
9plusmn
17119890
minus14
8Sumsquares1
1119890
minus14
7plusmn
34119890
minus14
71
0119890minus
146
plusmn3
3119890minus
146
37119890
minus15
1plusmn
89119890
minus15
16
2119890minus
146
plusmn1
9119890minus
145
62119890
minus15
2plusmn
19119890
minus15
11
22119890
minus14
5plusmn
29119890
minus14
51
3119890minus
148
plusmn4
0119890minus
148
Booth
26119890
minus11
plusmn1
3119890minus
112
9119890minus
11plusmn
19119890
minus11
24119890
minus11
plusmn1
6119890minus
113
1119890minus
11plusmn
18119890
minus01
12
4119890minus
11plusmn
13119890
minus11
31119890
minus11
plusmn2
1119890minus
111
0119890minus
10plusmn
13119890
minus10
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
014
plusmn0
43(90)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
159
119890minus
15plusmn
149
119890minus
158
9119890minus
16plusmn
0Grie
wank
008
plusmn0
26(90)
10119890
minus3
plusmn0
02(96)
0plusmn
00
plusmn0
0plusmn
00
10plusmn
033
(92)
0plusmn
0
Mathematical Problems in Engineering 9
Table7Sensitivityanalysisof
them
axim
umnu
mbero
frepetition
sinscou
tingbehavior
(119879max)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)6
810
1214
1618
Rosenb
rock
24119890
minus8
plusmn2
6119890minus
88
4119890minus
9plusmn
80119890
minus9
13119890
minus8
plusmn1
3119890minus
81
4119890minus
8plusmn
10119890
minus8
20119890
minus8
plusmn1
9119890minus
82
1119890minus
8plusmn
25119890
minus8
12119890
minus8
plusmn8
9119890minus
9Colville
48119890
minus7
plusmn2
2119890minus
73
4119890minus
7plusmn
18119890
minus7
15119890
minus7
plusmn1
2119890minus
73
8119890minus
7plusmn
20119890
minus7
36119890
minus7
plusmn3
7119890minus
7(96)
34119890
minus7
plusmn2
5119890minus
72
6119890minus
7plusmn
15119890
minus7
Sphere
71119890
minus14
7plusmn
22119890
minus14
64
5119890minus
146
plusmn9
0119890minus
146
78119890
minus14
6plusmn
23119890
minus14
51
9119890minus
148
plusmn5
3119890minus
148
57119890
minus14
8plusmn
13119890
minus14
76
9119890minus
145
plusmn2
2119890minus
144
36119890
minus14
7plusmn
11119890
minus14
6Sumsquares
41119890
minus14
6plusmn
13119890
minus14
52
4119890minus
149
plusmn4
8119890minus
149
42119890
minus14
9plusmn
13119890
minus14
88
3119890minus
150
plusmn2
6119890minus
149
85119890
minus14
7plusmn
27119890
minus14
65
4119890minus
146
plusmn9
0119890minus
146
14119890
minus15
1plusmn
44119890
minus15
1Bo
oth
32119890
minus11
plusmn2
9119890minus
114
2119890minus
11plusmn
27119890
minus11
25119890
minus11
plusmn1
5119890minus
112
1119890minus
11plusmn
15119890
minus11
32119890
minus11
plusmn2
5119890minus
112
6119890minus
11plusmn
18119890
minus11
26119890
minus11
plusmn2
7119890minus
11Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
08
9119890minus
16plusmn
0Grie
wank
010
plusmn0
33(92)
0plusmn
01
0119890minus
3plusmn
002
(98)
009
plusmn0
31(88)
0plusmn
00
09plusmn
029
(94)
83119890
minus4
plusmn0
02(98)
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal 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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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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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
Mathematical Problems in Engineering 5
Table 2 The list of various methods used in the paper
Method Authors and referencesGenetic algorithm (GA) Goldberg [14]Particle swarm optimization algorithm(PSO) Kennedy and Eberhart [7]
Artificial fish school algorithm (ASFA) Li et al [9]Artificial bee colony algorithm (ABC) Karaboga [10]Firefly algorithm (FA) Yang [11]
Another group of test problems is separable or nonseparablefunctions A 119901-variable separable function can be expressedas the sum of 119901 functions of one variable such as Sumsquaresand Rastrigin Nonseparable functions cannot be written inthis form such as Bridge Rosenbrock Ackley andGriewankBecause nonseparable functions have interrelation amongtheir variable these functions are more difficult than theseparable functions
In Table 1 characteristics of each function are given underthe column titled 119862 In this column 119872 means that thefunction is multimodal while 119880 means that the functionis unimodal If the function is separable abbreviation 119878 isused to indicate this specification Letter 119873 refers to that thefunction is nonseparable As seen from Table 1 4 functionsare multimodal 4 functions are unimodal 3 functions areseparable and 5 functions are nonseparable
The variety of functions forms and dimensions makeit possible to fairly assess the robustness of the proposedalgorithms within limit iteration Many of these functionsallow a choice of dimension and an input dimension rangingfrom 2 to 200 for test functions is given Dimensions of theproblems that we used can be found under the column titled119863 Besides initial ranges formulas and global optimumvalues of these functions are also given in Table 1
412 Experimental Settings In this subsection experimentalsettings are given Firstly in order to fully compare the perfor-mance of different algorithms we take the simulation underthe same situation So the values of the common parametersused in each algorithm such as population size and evaluationnumber were chosen to be the same Population size was100 and the maximum evaluation number was 2000 forall algorithms on all functions Additionally we follow theparameter settings in the original paper of GA PSO AFSAABC and FA see Table 2
For each experiment 50 independent runs were con-ducted with different initial random seeds To evaluate theperformance of these algorithms six criteria are given inTable 3
Accelerating convergence speed and avoiding the localoptima have become two important and appealing goals inswarm intelligent search algorithms So as seen in Table 3we adopted criteria best mean and standard deviation toevaluate efficiency and accuracy of algorithms and adoptedcriteria Art Worst and SR to evaluate convergence speedeffectiveness and robustness of six algorithms
Table 3 Six criteria and their abbreviations
Criteria AbbreviationThe best value of optima found in 50 runs BestThe worst value of optima found in 50 runs WorstThe average value of optima found in 50 runs MeanThe standard deviations StdDevThe success rate of the results SRThe average reaching time Art
Specifically speaking SR provides very useful informa-tion about how stable an algorithm is Success is claimed ifan algorithm successfully gets a solution below a prespecifiedthreshold value with the maximum number of functionevaluations [15] So to calculate the success rate an erroraccuracy level 120576 = 10minus6 must be set (120576 = 10minus6 also usedin [16]) Thus we compared the result 119865 with the knownanalytical optima 119865lowast and consider 119865 to be ldquosuccessfulrdquo if thefollowing inequality holds
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816
119865lowastlt 120576 119865lowast = 0
1003816100381610038161003816119865 minus 119865lowast1003816100381610038161003816 lt 120576 119865lowast = 0
(7)
The SR is a percentage value that is calculated as
SR =successful runs
runs (8)
Art is the average value of time once an algorithm gets asolution satisfying the formula (7) in 50-run computationsArt also provides very useful information about how fastan algorithm converges to certain accuracy or under thesame termination criterion which has important practicalsignificance
All algorithms have been tested in Matlab 2008a over thesame Lenovo A4600R computer with a Dual-Core 260GHzprocessor running Windows XP operating system over199Gb of memory
42 Experiments 1 Effect of Distance Measurements and FourParameters on WPA In order to study the effect of twodistance measures and four parameters on WPA differentmeasures and values of parameters were tested on typicalfunctions listed in Table 1 Each experiment WPA algorithmthat runs 50 times on each function and several criteriadescribed in Section 412 are used The experiment is con-ducted with the original coefficients shown in Table 9
421 Effect of Distance Measurements on the Performance ofWPA This subsection will investigate the performance ofdifferent distance measurements using functions with dif-ferent characteristics As is known to all Euclidean distance(ED) and Manhattan distance (MD) are the two most com-mon distance metrics in practical continuous optimizationIn the proposedWPA MD or ED can be adopted to measurethe distance between two wolves in the candidate solution
6 Mathematical Problems in Engineering
Table 4 Sensitivity analysis of distance measurements
Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts
Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053
Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220
Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825
Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109
Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546
Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541
Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884
Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903
ED 0 08350 00167 01181 92 gt135119890 + 4
space Therefore a discussion about their impacts on theperformance of WPA is needed
There are two wolves X119901
= (1199091199011
1199091199012
119909119901119863
) is theposition of wolf 119901X
119902= (1199091199021
1199091199022
119909119902119863
) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space
119871 ED (119901 119902) =119863
sum119889=1
(119909119901119889
minus 119909119902119889
)2
119871MD (119901 119902) =119863
sum119889=1
10038161003816100381610038161003816119909119901119889 minus 119909119902119889
10038161003816100381610038161003816
(9)
The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima
As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness
Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD
422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant
Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case
Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage
Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions
Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only
Mathematical Problems in Engineering 7
Table5Sensitivityanalysisof
stepcoeffi
cient(
119878)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
69119890
minus8
plusmn4
3119890minus
82
7119890minus
8plusmn
35119890
minus8
11119890
minus8
plusmn9
1119890minus
93
2119890minus
9plusmn
27119890
minus9
50119890
minus9
plusmn5
7119890minus
93
2119890minus
9plusmn
37119890
minus9
12119890
minus9
plusmn1
6119890minus
9Colville
13119890
minus7
plusmn7
1119890minus
83
3119890minus
7plusmn
28119890
minus7(90)
26119890
minus7
plusmn1
9119890minus
72
3119890minus
7plusmn
14119890
minus7
35119890
minus7
plusmn2
5119890minus
79
5119890minus
7plusmn
10119890
minus6(80)
14119890
minus6
plusmn1
5119890minus
6(50)
Sphere
23119890
minus14
5plusmn
71119890
minus14
56
6119890minus
152
plusmn2
1119890minus
151
21119890
minus14
6plusmn
45119890
minus14
63
9119890minus
146
plusmn1
2119890minus
145
12119890
minus14
5plusmn
34119890
minus14
51
7119890minus
146
plusmn5
3119890minus
146
22119890
minus14
9plusmn
68119890
minus14
9Sumsquares
98119890
minus14
5plusmn
31119890
minus14
43
1119890minus
146
plusmn8
4119890minus
146
81119890
minus14
7plusmn
26119890
minus14
64
8119890minus
146
plusmn1
0119890minus
145
38119890
minus15
2plusmn
79119890
minus15
23
4119890minus
147
plusmn1
1119890minus
146
12119890
minus14
7plusmn
39119890
minus14
7Bo
oth
54119890
minus7
plusmn3
3119890minus
71
6119890minus
9plusmn
11119890
minus9
32119890
minus11
plusmn1
6119890minus
111
3119890minus
12plusmn
91119890
minus13
13119890
minus13
plusmn1
2119890minus
133
9119890minus
15plusmn
18119890
minus15
12119890
minus16
plusmn5
8119890minus
17Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
025
plusmn0
53(80)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
0plusmn
00
06plusmn
019
(92)
020
plusmn0
42(86)
8 Mathematical Problems in Engineering
Table6Sensitivityanalysisof
distance
determ
inantcoefficient(
119871near)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
44119890
minus8
plusmn6
5119890minus
82
3119890minus
8plusmn
37119890
minus8
34119890
minus9
plusmn4
8119890minus
93
0119890minus
8plusmn
29119890
minus8
19119890
minus8
plusmn2
4119890minus
82
4119890minus
8plusmn
47119890
minus8
29119890
minus8
plusmn5
3119890minus
8Colville
20119890
minus7
plusmn9
9119890minus
82
6119890minus
7plusmn
16119890
minus7
35119890
minus7
plusmn2
6119890minus
72
3119890minus
7plusmn
15119890
minus7
12119890
minus7
plusmn3
4119890minus
82
8119890minus
7plusmn
19119890
minus7
14119890
minus7
plusmn6
9119890minus
8Sphere
68119890
minus14
6plusmn
20119890
minus14
51
9119890minus
146
plusmn6
2119890minus
146
17119890
minus14
5plusmn
43119890
minus14
52
6119890minus
148
plusmn8
3119890minus
148
36119890
minus14
6plusmn
11119890
minus14
53
7119890minus
151
plusmn1
1119890minus
150
53119890
minus14
9plusmn
17119890
minus14
8Sumsquares1
1119890
minus14
7plusmn
34119890
minus14
71
0119890minus
146
plusmn3
3119890minus
146
37119890
minus15
1plusmn
89119890
minus15
16
2119890minus
146
plusmn1
9119890minus
145
62119890
minus15
2plusmn
19119890
minus15
11
22119890
minus14
5plusmn
29119890
minus14
51
3119890minus
148
plusmn4
0119890minus
148
Booth
26119890
minus11
plusmn1
3119890minus
112
9119890minus
11plusmn
19119890
minus11
24119890
minus11
plusmn1
6119890minus
113
1119890minus
11plusmn
18119890
minus01
12
4119890minus
11plusmn
13119890
minus11
31119890
minus11
plusmn2
1119890minus
111
0119890minus
10plusmn
13119890
minus10
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
014
plusmn0
43(90)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
159
119890minus
15plusmn
149
119890minus
158
9119890minus
16plusmn
0Grie
wank
008
plusmn0
26(90)
10119890
minus3
plusmn0
02(96)
0plusmn
00
plusmn0
0plusmn
00
10plusmn
033
(92)
0plusmn
0
Mathematical Problems in Engineering 9
Table7Sensitivityanalysisof
them
axim
umnu
mbero
frepetition
sinscou
tingbehavior
(119879max)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)6
810
1214
1618
Rosenb
rock
24119890
minus8
plusmn2
6119890minus
88
4119890minus
9plusmn
80119890
minus9
13119890
minus8
plusmn1
3119890minus
81
4119890minus
8plusmn
10119890
minus8
20119890
minus8
plusmn1
9119890minus
82
1119890minus
8plusmn
25119890
minus8
12119890
minus8
plusmn8
9119890minus
9Colville
48119890
minus7
plusmn2
2119890minus
73
4119890minus
7plusmn
18119890
minus7
15119890
minus7
plusmn1
2119890minus
73
8119890minus
7plusmn
20119890
minus7
36119890
minus7
plusmn3
7119890minus
7(96)
34119890
minus7
plusmn2
5119890minus
72
6119890minus
7plusmn
15119890
minus7
Sphere
71119890
minus14
7plusmn
22119890
minus14
64
5119890minus
146
plusmn9
0119890minus
146
78119890
minus14
6plusmn
23119890
minus14
51
9119890minus
148
plusmn5
3119890minus
148
57119890
minus14
8plusmn
13119890
minus14
76
9119890minus
145
plusmn2
2119890minus
144
36119890
minus14
7plusmn
11119890
minus14
6Sumsquares
41119890
minus14
6plusmn
13119890
minus14
52
4119890minus
149
plusmn4
8119890minus
149
42119890
minus14
9plusmn
13119890
minus14
88
3119890minus
150
plusmn2
6119890minus
149
85119890
minus14
7plusmn
27119890
minus14
65
4119890minus
146
plusmn9
0119890minus
146
14119890
minus15
1plusmn
44119890
minus15
1Bo
oth
32119890
minus11
plusmn2
9119890minus
114
2119890minus
11plusmn
27119890
minus11
25119890
minus11
plusmn1
5119890minus
112
1119890minus
11plusmn
15119890
minus11
32119890
minus11
plusmn2
5119890minus
112
6119890minus
11plusmn
18119890
minus11
26119890
minus11
plusmn2
7119890minus
11Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
08
9119890minus
16plusmn
0Grie
wank
010
plusmn0
33(92)
0plusmn
01
0119890minus
3plusmn
002
(98)
009
plusmn0
31(88)
0plusmn
00
09plusmn
029
(94)
83119890
minus4
plusmn0
02(98)
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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International Journal of
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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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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Table 4 Sensitivity analysis of distance measurements
Function Global extremum 119863 Distance Best Worst Mean StdDev SR Arts
Rosenbrock 119891min() = 0 2 MD 921119890 minus 11 324119890 minus 8 112119890 minus 8 118119890 minus 8 100 105165ED 426119890 minus 9 271119890 minus 7 127119890 minus 7 681119890 minus 8 100 371053
Colville 119891min() = 0 4 MD 562119890 minus 8 528119890 minus 7 249119890 minus 7 223119890 minus 7 100 468619ED 174119890 minus 7 170119890 minus 6 574119890 minus 7 370119890 minus 7 90 683220
Sphere 119891min() = 0 200 MD 320119890 minus 161 329119890 minus 144 207119890 minus 145 749119890 minus 145 100 115494ED 176119890 minus 160 336119890 minus 143 168119890 minus 144 751119890 minus 144 100 116825
Sumsquares 119891min() = 0 150 MD 156119890 minus 161 309119890 minus 144 179119890 minus 145 695119890 minus 145 100 85565ED 397119890 minus 160 224119890 minus 144 113119890 minus 145 500119890 minus 145 100 87109
Booth 119891min() = 0 2 MD 563119890 minus 12 115119890 minus 10 419119890 minus 11 332119890 minus 11 100 111074ED 108119890 minus 9 264119890 minus 8 116119890 minus 8 693119890 minus 9 100 405546
Bridge 119891max() = 30054 2 MD 30054 30054 30054 456119890 minus 16 100 11093ED 30054 30054 30054 456119890 minus 16 100 19541
Ackley 119891min() = 0 50 MD 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 193648ED 888119890 minus 16 888119890 minus 16 888119890 minus 16 0 100 436884
Griewank 119891min() = 0 100 MD 0 01507 301119890 minus 3 00213 98 gt8771198903
ED 0 08350 00167 01181 92 gt135119890 + 4
space Therefore a discussion about their impacts on theperformance of WPA is needed
There are two wolves X119901
= (1199091199011
1199091199012
119909119901119863
) is theposition of wolf 119901X
119902= (1199091199021
1199091199022
119909119902119863
) is the positionof wolf 119902 and the ED and MD between them can berespectively calculated as formula (9) 119863 is the dimensionnumber of solution space
119871 ED (119901 119902) =119863
sum119889=1
(119909119901119889
minus 119909119902119889
)2
119871MD (119901 119902) =119863
sum119889=1
10038161003816100381610038161003816119909119901119889 minus 119909119902119889
10038161003816100381610038161003816
(9)
The statistical results obtained by WPA after 50-runcomputation are shown in Table 4 Firstly we note that WPAwithEuclidean distance (WPA ED)does not get 100 successrate on Colville (119863 = 4) and Griewank functions (119863 = 100)while WPA with Manhattan distance (WPA MD) does notget 100 success rate on Griewank functions (119863 = 100)which means that WPA ED and WPA MD with originalcoefficients still have the risk of premature convergence tolocal optima
As seen from Table 4 WPA is not very sensitive to twodistance measurements on most functions (RosenbrockSphere Sumsquares Booth and Ackley) and no matterwhich metric is used WPA can always get a good resultwith SR = 100 But for these functions comparing theresults between WPA MD and WPA ED in detail we canfind that WPA MD has shorter average reaching time (ARt)which means faster convergence speed to a certain accuracyThe reason may be that ED has the higher computationalcomplexityMeanwhileWPA MDhas better performance onother four criteria (best worst mean and StdDev) whichmeans better solution accuracy and robustness
Naturally because of its better efficiency precision androbustness WD is more suitable for WPA So the WPAalgorithm used in what follows is WPA MD
422 Effect of Four Parameters on the Performance of WPAIn this subsectionwe investigate the impact of the parameters119878 119871near 119879max and 120573 on the new algorithm 119878 is the stepcoefficient 119871near is the distance determinant coefficient 119879maxis the maximum number of repetitions in scouting behaviorand 120573 is the population renewing proportional coefficientThe parameters selection procedure is performed in a one-factor-at-a-time manner For each sensitivity analysis in thissection only one parameter is varied each time and theremaining parameters are kept at the values suggested by theoriginal estimate listed in Table 9 The interaction relationbetween parameters is assumed unimportant
Each time one of the WPA parameters is varied in a cer-tain interval to see which value within this internal will resultin the best performance Specifically theWPA algorithm alsoruns 50 times on each case
Table 5 shows the sensitivity analysis of the step coef-ficient 119878 All results are shown in the form of Mean plusmnStd (SR) The choice of interval [004 016] used in thisanalysis was motivated by the original Nelder-Mead simplexsearch procedure where a step coefficient greater than 004was suggested for general usage
Meanwhile based on detailed comparison of the resultson Rosenbrock Sphere and Bridge functions step coefficientis not sensitive to WPA and for Booth function there is atendency of better results with larger 119878 From Table 5 it isfound that a step coefficient setting at 012 returns the bestresult which has better Mean small Std and SR = 100 forall functions
Tables 6ndash8 analyze sensitivity of 119871near 119879max and 120573 Gen-erally speaking 119871near 119879max and 120573 are not sensitive to mostfunctions exceptGriewank function sinceGriewanknot only
Mathematical Problems in Engineering 7
Table5Sensitivityanalysisof
stepcoeffi
cient(
119878)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
69119890
minus8
plusmn4
3119890minus
82
7119890minus
8plusmn
35119890
minus8
11119890
minus8
plusmn9
1119890minus
93
2119890minus
9plusmn
27119890
minus9
50119890
minus9
plusmn5
7119890minus
93
2119890minus
9plusmn
37119890
minus9
12119890
minus9
plusmn1
6119890minus
9Colville
13119890
minus7
plusmn7
1119890minus
83
3119890minus
7plusmn
28119890
minus7(90)
26119890
minus7
plusmn1
9119890minus
72
3119890minus
7plusmn
14119890
minus7
35119890
minus7
plusmn2
5119890minus
79
5119890minus
7plusmn
10119890
minus6(80)
14119890
minus6
plusmn1
5119890minus
6(50)
Sphere
23119890
minus14
5plusmn
71119890
minus14
56
6119890minus
152
plusmn2
1119890minus
151
21119890
minus14
6plusmn
45119890
minus14
63
9119890minus
146
plusmn1
2119890minus
145
12119890
minus14
5plusmn
34119890
minus14
51
7119890minus
146
plusmn5
3119890minus
146
22119890
minus14
9plusmn
68119890
minus14
9Sumsquares
98119890
minus14
5plusmn
31119890
minus14
43
1119890minus
146
plusmn8
4119890minus
146
81119890
minus14
7plusmn
26119890
minus14
64
8119890minus
146
plusmn1
0119890minus
145
38119890
minus15
2plusmn
79119890
minus15
23
4119890minus
147
plusmn1
1119890minus
146
12119890
minus14
7plusmn
39119890
minus14
7Bo
oth
54119890
minus7
plusmn3
3119890minus
71
6119890minus
9plusmn
11119890
minus9
32119890
minus11
plusmn1
6119890minus
111
3119890minus
12plusmn
91119890
minus13
13119890
minus13
plusmn1
2119890minus
133
9119890minus
15plusmn
18119890
minus15
12119890
minus16
plusmn5
8119890minus
17Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
025
plusmn0
53(80)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
0plusmn
00
06plusmn
019
(92)
020
plusmn0
42(86)
8 Mathematical Problems in Engineering
Table6Sensitivityanalysisof
distance
determ
inantcoefficient(
119871near)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
44119890
minus8
plusmn6
5119890minus
82
3119890minus
8plusmn
37119890
minus8
34119890
minus9
plusmn4
8119890minus
93
0119890minus
8plusmn
29119890
minus8
19119890
minus8
plusmn2
4119890minus
82
4119890minus
8plusmn
47119890
minus8
29119890
minus8
plusmn5
3119890minus
8Colville
20119890
minus7
plusmn9
9119890minus
82
6119890minus
7plusmn
16119890
minus7
35119890
minus7
plusmn2
6119890minus
72
3119890minus
7plusmn
15119890
minus7
12119890
minus7
plusmn3
4119890minus
82
8119890minus
7plusmn
19119890
minus7
14119890
minus7
plusmn6
9119890minus
8Sphere
68119890
minus14
6plusmn
20119890
minus14
51
9119890minus
146
plusmn6
2119890minus
146
17119890
minus14
5plusmn
43119890
minus14
52
6119890minus
148
plusmn8
3119890minus
148
36119890
minus14
6plusmn
11119890
minus14
53
7119890minus
151
plusmn1
1119890minus
150
53119890
minus14
9plusmn
17119890
minus14
8Sumsquares1
1119890
minus14
7plusmn
34119890
minus14
71
0119890minus
146
plusmn3
3119890minus
146
37119890
minus15
1plusmn
89119890
minus15
16
2119890minus
146
plusmn1
9119890minus
145
62119890
minus15
2plusmn
19119890
minus15
11
22119890
minus14
5plusmn
29119890
minus14
51
3119890minus
148
plusmn4
0119890minus
148
Booth
26119890
minus11
plusmn1
3119890minus
112
9119890minus
11plusmn
19119890
minus11
24119890
minus11
plusmn1
6119890minus
113
1119890minus
11plusmn
18119890
minus01
12
4119890minus
11plusmn
13119890
minus11
31119890
minus11
plusmn2
1119890minus
111
0119890minus
10plusmn
13119890
minus10
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
014
plusmn0
43(90)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
159
119890minus
15plusmn
149
119890minus
158
9119890minus
16plusmn
0Grie
wank
008
plusmn0
26(90)
10119890
minus3
plusmn0
02(96)
0plusmn
00
plusmn0
0plusmn
00
10plusmn
033
(92)
0plusmn
0
Mathematical Problems in Engineering 9
Table7Sensitivityanalysisof
them
axim
umnu
mbero
frepetition
sinscou
tingbehavior
(119879max)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)6
810
1214
1618
Rosenb
rock
24119890
minus8
plusmn2
6119890minus
88
4119890minus
9plusmn
80119890
minus9
13119890
minus8
plusmn1
3119890minus
81
4119890minus
8plusmn
10119890
minus8
20119890
minus8
plusmn1
9119890minus
82
1119890minus
8plusmn
25119890
minus8
12119890
minus8
plusmn8
9119890minus
9Colville
48119890
minus7
plusmn2
2119890minus
73
4119890minus
7plusmn
18119890
minus7
15119890
minus7
plusmn1
2119890minus
73
8119890minus
7plusmn
20119890
minus7
36119890
minus7
plusmn3
7119890minus
7(96)
34119890
minus7
plusmn2
5119890minus
72
6119890minus
7plusmn
15119890
minus7
Sphere
71119890
minus14
7plusmn
22119890
minus14
64
5119890minus
146
plusmn9
0119890minus
146
78119890
minus14
6plusmn
23119890
minus14
51
9119890minus
148
plusmn5
3119890minus
148
57119890
minus14
8plusmn
13119890
minus14
76
9119890minus
145
plusmn2
2119890minus
144
36119890
minus14
7plusmn
11119890
minus14
6Sumsquares
41119890
minus14
6plusmn
13119890
minus14
52
4119890minus
149
plusmn4
8119890minus
149
42119890
minus14
9plusmn
13119890
minus14
88
3119890minus
150
plusmn2
6119890minus
149
85119890
minus14
7plusmn
27119890
minus14
65
4119890minus
146
plusmn9
0119890minus
146
14119890
minus15
1plusmn
44119890
minus15
1Bo
oth
32119890
minus11
plusmn2
9119890minus
114
2119890minus
11plusmn
27119890
minus11
25119890
minus11
plusmn1
5119890minus
112
1119890minus
11plusmn
15119890
minus11
32119890
minus11
plusmn2
5119890minus
112
6119890minus
11plusmn
18119890
minus11
26119890
minus11
plusmn2
7119890minus
11Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
08
9119890minus
16plusmn
0Grie
wank
010
plusmn0
33(92)
0plusmn
01
0119890minus
3plusmn
002
(98)
009
plusmn0
31(88)
0plusmn
00
09plusmn
029
(94)
83119890
minus4
plusmn0
02(98)
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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
Table5Sensitivityanalysisof
stepcoeffi
cient(
119878)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
69119890
minus8
plusmn4
3119890minus
82
7119890minus
8plusmn
35119890
minus8
11119890
minus8
plusmn9
1119890minus
93
2119890minus
9plusmn
27119890
minus9
50119890
minus9
plusmn5
7119890minus
93
2119890minus
9plusmn
37119890
minus9
12119890
minus9
plusmn1
6119890minus
9Colville
13119890
minus7
plusmn7
1119890minus
83
3119890minus
7plusmn
28119890
minus7(90)
26119890
minus7
plusmn1
9119890minus
72
3119890minus
7plusmn
14119890
minus7
35119890
minus7
plusmn2
5119890minus
79
5119890minus
7plusmn
10119890
minus6(80)
14119890
minus6
plusmn1
5119890minus
6(50)
Sphere
23119890
minus14
5plusmn
71119890
minus14
56
6119890minus
152
plusmn2
1119890minus
151
21119890
minus14
6plusmn
45119890
minus14
63
9119890minus
146
plusmn1
2119890minus
145
12119890
minus14
5plusmn
34119890
minus14
51
7119890minus
146
plusmn5
3119890minus
146
22119890
minus14
9plusmn
68119890
minus14
9Sumsquares
98119890
minus14
5plusmn
31119890
minus14
43
1119890minus
146
plusmn8
4119890minus
146
81119890
minus14
7plusmn
26119890
minus14
64
8119890minus
146
plusmn1
0119890minus
145
38119890
minus15
2plusmn
79119890
minus15
23
4119890minus
147
plusmn1
1119890minus
146
12119890
minus14
7plusmn
39119890
minus14
7Bo
oth
54119890
minus7
plusmn3
3119890minus
71
6119890minus
9plusmn
11119890
minus9
32119890
minus11
plusmn1
6119890minus
111
3119890minus
12plusmn
91119890
minus13
13119890
minus13
plusmn1
2119890minus
133
9119890minus
15plusmn
18119890
minus15
12119890
minus16
plusmn5
8119890minus
17Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
025
plusmn0
53(80)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
0plusmn
00
06plusmn
019
(92)
020
plusmn0
42(86)
8 Mathematical Problems in Engineering
Table6Sensitivityanalysisof
distance
determ
inantcoefficient(
119871near)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
44119890
minus8
plusmn6
5119890minus
82
3119890minus
8plusmn
37119890
minus8
34119890
minus9
plusmn4
8119890minus
93
0119890minus
8plusmn
29119890
minus8
19119890
minus8
plusmn2
4119890minus
82
4119890minus
8plusmn
47119890
minus8
29119890
minus8
plusmn5
3119890minus
8Colville
20119890
minus7
plusmn9
9119890minus
82
6119890minus
7plusmn
16119890
minus7
35119890
minus7
plusmn2
6119890minus
72
3119890minus
7plusmn
15119890
minus7
12119890
minus7
plusmn3
4119890minus
82
8119890minus
7plusmn
19119890
minus7
14119890
minus7
plusmn6
9119890minus
8Sphere
68119890
minus14
6plusmn
20119890
minus14
51
9119890minus
146
plusmn6
2119890minus
146
17119890
minus14
5plusmn
43119890
minus14
52
6119890minus
148
plusmn8
3119890minus
148
36119890
minus14
6plusmn
11119890
minus14
53
7119890minus
151
plusmn1
1119890minus
150
53119890
minus14
9plusmn
17119890
minus14
8Sumsquares1
1119890
minus14
7plusmn
34119890
minus14
71
0119890minus
146
plusmn3
3119890minus
146
37119890
minus15
1plusmn
89119890
minus15
16
2119890minus
146
plusmn1
9119890minus
145
62119890
minus15
2plusmn
19119890
minus15
11
22119890
minus14
5plusmn
29119890
minus14
51
3119890minus
148
plusmn4
0119890minus
148
Booth
26119890
minus11
plusmn1
3119890minus
112
9119890minus
11plusmn
19119890
minus11
24119890
minus11
plusmn1
6119890minus
113
1119890minus
11plusmn
18119890
minus01
12
4119890minus
11plusmn
13119890
minus11
31119890
minus11
plusmn2
1119890minus
111
0119890minus
10plusmn
13119890
minus10
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
014
plusmn0
43(90)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
159
119890minus
15plusmn
149
119890minus
158
9119890minus
16plusmn
0Grie
wank
008
plusmn0
26(90)
10119890
minus3
plusmn0
02(96)
0plusmn
00
plusmn0
0plusmn
00
10plusmn
033
(92)
0plusmn
0
Mathematical Problems in Engineering 9
Table7Sensitivityanalysisof
them
axim
umnu
mbero
frepetition
sinscou
tingbehavior
(119879max)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)6
810
1214
1618
Rosenb
rock
24119890
minus8
plusmn2
6119890minus
88
4119890minus
9plusmn
80119890
minus9
13119890
minus8
plusmn1
3119890minus
81
4119890minus
8plusmn
10119890
minus8
20119890
minus8
plusmn1
9119890minus
82
1119890minus
8plusmn
25119890
minus8
12119890
minus8
plusmn8
9119890minus
9Colville
48119890
minus7
plusmn2
2119890minus
73
4119890minus
7plusmn
18119890
minus7
15119890
minus7
plusmn1
2119890minus
73
8119890minus
7plusmn
20119890
minus7
36119890
minus7
plusmn3
7119890minus
7(96)
34119890
minus7
plusmn2
5119890minus
72
6119890minus
7plusmn
15119890
minus7
Sphere
71119890
minus14
7plusmn
22119890
minus14
64
5119890minus
146
plusmn9
0119890minus
146
78119890
minus14
6plusmn
23119890
minus14
51
9119890minus
148
plusmn5
3119890minus
148
57119890
minus14
8plusmn
13119890
minus14
76
9119890minus
145
plusmn2
2119890minus
144
36119890
minus14
7plusmn
11119890
minus14
6Sumsquares
41119890
minus14
6plusmn
13119890
minus14
52
4119890minus
149
plusmn4
8119890minus
149
42119890
minus14
9plusmn
13119890
minus14
88
3119890minus
150
plusmn2
6119890minus
149
85119890
minus14
7plusmn
27119890
minus14
65
4119890minus
146
plusmn9
0119890minus
146
14119890
minus15
1plusmn
44119890
minus15
1Bo
oth
32119890
minus11
plusmn2
9119890minus
114
2119890minus
11plusmn
27119890
minus11
25119890
minus11
plusmn1
5119890minus
112
1119890minus
11plusmn
15119890
minus11
32119890
minus11
plusmn2
5119890minus
112
6119890minus
11plusmn
18119890
minus11
26119890
minus11
plusmn2
7119890minus
11Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
08
9119890minus
16plusmn
0Grie
wank
010
plusmn0
33(92)
0plusmn
01
0119890minus
3plusmn
002
(98)
009
plusmn0
31(88)
0plusmn
00
09plusmn
029
(94)
83119890
minus4
plusmn0
02(98)
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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
8 Mathematical Problems in Engineering
Table6Sensitivityanalysisof
distance
determ
inantcoefficient(
119871near)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)004
006
008
010
012
014
016
Rosenb
rock
44119890
minus8
plusmn6
5119890minus
82
3119890minus
8plusmn
37119890
minus8
34119890
minus9
plusmn4
8119890minus
93
0119890minus
8plusmn
29119890
minus8
19119890
minus8
plusmn2
4119890minus
82
4119890minus
8plusmn
47119890
minus8
29119890
minus8
plusmn5
3119890minus
8Colville
20119890
minus7
plusmn9
9119890minus
82
6119890minus
7plusmn
16119890
minus7
35119890
minus7
plusmn2
6119890minus
72
3119890minus
7plusmn
15119890
minus7
12119890
minus7
plusmn3
4119890minus
82
8119890minus
7plusmn
19119890
minus7
14119890
minus7
plusmn6
9119890minus
8Sphere
68119890
minus14
6plusmn
20119890
minus14
51
9119890minus
146
plusmn6
2119890minus
146
17119890
minus14
5plusmn
43119890
minus14
52
6119890minus
148
plusmn8
3119890minus
148
36119890
minus14
6plusmn
11119890
minus14
53
7119890minus
151
plusmn1
1119890minus
150
53119890
minus14
9plusmn
17119890
minus14
8Sumsquares1
1119890
minus14
7plusmn
34119890
minus14
71
0119890minus
146
plusmn3
3119890minus
146
37119890
minus15
1plusmn
89119890
minus15
16
2119890minus
146
plusmn1
9119890minus
145
62119890
minus15
2plusmn
19119890
minus15
11
22119890
minus14
5plusmn
29119890
minus14
51
3119890minus
148
plusmn4
0119890minus
148
Booth
26119890
minus11
plusmn1
3119890minus
112
9119890minus
11plusmn
19119890
minus11
24119890
minus11
plusmn1
6119890minus
113
1119890minus
11plusmn
18119890
minus01
12
4119890minus
11plusmn
13119890
minus11
31119890
minus11
plusmn2
1119890minus
111
0119890minus
10plusmn
13119890
minus10
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
014
plusmn0
43(90)
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
01
2119890minus
15plusmn
11119890
minus15
89119890
minus16
plusmn0
159
119890minus
15plusmn
149
119890minus
158
9119890minus
16plusmn
0Grie
wank
008
plusmn0
26(90)
10119890
minus3
plusmn0
02(96)
0plusmn
00
plusmn0
0plusmn
00
10plusmn
033
(92)
0plusmn
0
Mathematical Problems in Engineering 9
Table7Sensitivityanalysisof
them
axim
umnu
mbero
frepetition
sinscou
tingbehavior
(119879max)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)6
810
1214
1618
Rosenb
rock
24119890
minus8
plusmn2
6119890minus
88
4119890minus
9plusmn
80119890
minus9
13119890
minus8
plusmn1
3119890minus
81
4119890minus
8plusmn
10119890
minus8
20119890
minus8
plusmn1
9119890minus
82
1119890minus
8plusmn
25119890
minus8
12119890
minus8
plusmn8
9119890minus
9Colville
48119890
minus7
plusmn2
2119890minus
73
4119890minus
7plusmn
18119890
minus7
15119890
minus7
plusmn1
2119890minus
73
8119890minus
7plusmn
20119890
minus7
36119890
minus7
plusmn3
7119890minus
7(96)
34119890
minus7
plusmn2
5119890minus
72
6119890minus
7plusmn
15119890
minus7
Sphere
71119890
minus14
7plusmn
22119890
minus14
64
5119890minus
146
plusmn9
0119890minus
146
78119890
minus14
6plusmn
23119890
minus14
51
9119890minus
148
plusmn5
3119890minus
148
57119890
minus14
8plusmn
13119890
minus14
76
9119890minus
145
plusmn2
2119890minus
144
36119890
minus14
7plusmn
11119890
minus14
6Sumsquares
41119890
minus14
6plusmn
13119890
minus14
52
4119890minus
149
plusmn4
8119890minus
149
42119890
minus14
9plusmn
13119890
minus14
88
3119890minus
150
plusmn2
6119890minus
149
85119890
minus14
7plusmn
27119890
minus14
65
4119890minus
146
plusmn9
0119890minus
146
14119890
minus15
1plusmn
44119890
minus15
1Bo
oth
32119890
minus11
plusmn2
9119890minus
114
2119890minus
11plusmn
27119890
minus11
25119890
minus11
plusmn1
5119890minus
112
1119890minus
11plusmn
15119890
minus11
32119890
minus11
plusmn2
5119890minus
112
6119890minus
11plusmn
18119890
minus11
26119890
minus11
plusmn2
7119890minus
11Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
08
9119890minus
16plusmn
0Grie
wank
010
plusmn0
33(92)
0plusmn
01
0119890minus
3plusmn
002
(98)
009
plusmn0
31(88)
0plusmn
00
09plusmn
029
(94)
83119890
minus4
plusmn0
02(98)
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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
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Journal of
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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
Journal of
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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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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table7Sensitivityanalysisof
them
axim
umnu
mbero
frepetition
sinscou
tingbehavior
(119879max)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)6
810
1214
1618
Rosenb
rock
24119890
minus8
plusmn2
6119890minus
88
4119890minus
9plusmn
80119890
minus9
13119890
minus8
plusmn1
3119890minus
81
4119890minus
8plusmn
10119890
minus8
20119890
minus8
plusmn1
9119890minus
82
1119890minus
8plusmn
25119890
minus8
12119890
minus8
plusmn8
9119890minus
9Colville
48119890
minus7
plusmn2
2119890minus
73
4119890minus
7plusmn
18119890
minus7
15119890
minus7
plusmn1
2119890minus
73
8119890minus
7plusmn
20119890
minus7
36119890
minus7
plusmn3
7119890minus
7(96)
34119890
minus7
plusmn2
5119890minus
72
6119890minus
7plusmn
15119890
minus7
Sphere
71119890
minus14
7plusmn
22119890
minus14
64
5119890minus
146
plusmn9
0119890minus
146
78119890
minus14
6plusmn
23119890
minus14
51
9119890minus
148
plusmn5
3119890minus
148
57119890
minus14
8plusmn
13119890
minus14
76
9119890minus
145
plusmn2
2119890minus
144
36119890
minus14
7plusmn
11119890
minus14
6Sumsquares
41119890
minus14
6plusmn
13119890
minus14
52
4119890minus
149
plusmn4
8119890minus
149
42119890
minus14
9plusmn
13119890
minus14
88
3119890minus
150
plusmn2
6119890minus
149
85119890
minus14
7plusmn
27119890
minus14
65
4119890minus
146
plusmn9
0119890minus
146
14119890
minus15
1plusmn
44119890
minus15
1Bo
oth
32119890
minus11
plusmn2
9119890minus
114
2119890minus
11plusmn
27119890
minus11
25119890
minus11
plusmn1
5119890minus
112
1119890minus
11plusmn
15119890
minus11
32119890
minus11
plusmn2
5119890minus
112
6119890minus
11plusmn
18119890
minus11
26119890
minus11
plusmn2
7119890minus
11Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
12119890
minus15
plusmn1
1119890minus
158
9119890minus
16plusmn
08
9119890minus
16plusmn
0Grie
wank
010
plusmn0
33(92)
0plusmn
01
0119890minus
3plusmn
002
(98)
009
plusmn0
31(88)
0plusmn
00
09plusmn
029
(94)
83119890
minus4
plusmn0
02(98)
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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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Mathematical PhysicsAdvances in
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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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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Table8Sensitivityanalysisof
popu
latio
nrenewingprop
ortio
nalcoefficient(
120573)
Functio
nsMean
plusmnStd(SR)(thed
efaultof
SRis100
)2
34
56
78
Rosenb
rock
10119890
minus8
plusmn9
2119890minus
98
7119890minus
9plusmn
76119890
minus9
12119890
minus8
plusmn1
0119890minus
88
6119890minus
9plusmn
83119890
minus9
14119890
minus8
plusmn1
3119890minus
89
9119890minus
9plusmn
98119890
minus9
11119890
minus8
plusmn1
2119890minus
9Colville
32119890
minus8
plusmn1
8119890
minus8
14119890
minus7
plusmn1
3119890minus
71
2119890minus
7plusmn
59119890
minus8
14119890
minus7
plusmn9
4119890minus
83
0119890minus
7plusmn
69119890
minus8
39119890
minus7
plusmn1
7119890minus
78
6119890minus
7plusmn
40119890
minus7(80)
Sphere
19119890
minus16
6plusmn
05
2119890minus
158
plusmn1
6119890minus
157
29119890
minus15
3plusmn
92119890
minus15
34
3119890minus
149
plusmn1
3119890minus
148
79119890
minus13
9plusmn
25119890
minus13
88
3119890minus
134
plusmn1
8119890minus
133
34119890
minus12
6plusmn
80119890
minus12
6Sumsquares
28119890
minus16
7plusmn
01
4119890minus
157
plusmn4
3119890minus
157
28119890
minus15
5plusmn
45119890
minus15
58
3119890minus
146
plusmn1
8119890minus
145
69119890
minus14
3plusmn
17119890
minus14
25
3119890minus
143
plusmn1
3119890minus
142
33119890
minus12
7plusmn
10119890
minus12
6Bo
oth
81119890
minus11
plusmn1
3119890minus
102
5119890minus
11plusmn
17119890
minus11
19119890
minus11
plusmn1
2119890minus
112
5119890minus
11plusmn
17119890
minus01
12
5119890minus
11plusmn
15119890
minus11
23119890
minus11
plusmn1
5119890minus
112
3119890minus
11plusmn
14119890
minus11
Bridge
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
300
54plusmn
47119890
minus16
Ackley
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
89119890
minus16
plusmn0
Grie
wank
0plusmn
00
plusmn0
0plusmn
00
plusmn0
019
plusmn0
41(86)
0plusmn
01
2119890minus
3plusmn
031
(96)
Mathematical Problems in Engineering 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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 11
Table 9 Best suggestions for WPA parameters
No WPA parameters name Original Best-suggested1 Step coefficient (119878) 008 0122 Distance determinant coefficient (119871near) 012 0083 The maximum number of repetitions in scouting (119879max) 10 84 Population renewal coefficient (120573) 5 2
020
2
0
xy
minusf(xy) minus1000
minus2000
minus3000
minus4000
minus2minus2
minus4minus4
(a)
x
y
0 1 2
0
05
1
15
2
minus05
minus15
minus2minus2
minus1
minus1
(b)
Figure 2 Rosenbrock function (119863 = 2) (a) surface plot and (b) contour lines
is a high-dimensional function for its 100 parameters butalso has very large search space for its interval of [minus600 600]which is hard to optimized
Table 6 illustrates the sensitivity analysis of 119871near andfrom this table it is found that setting 119871near at 008 returns thebest results with the best mean smaller standard deviationsand 100 success rate for all functions
Tables 7-8 indicate that119879max and 120573 respectively setting at8 and 2 return the best results on eight functions
So we summarize the above findings in Table 9 andapply these parameter values in our approach for conductingexperimental comparisons with other algorithms listed inTable 2
43 Experiments 2 WPA versus GA PSO AFSA ABC andFA In this section we compared GA PSO AFSA ABCFA and WPA algorithms on eight functions described inTable 1 Each of the experimentswas repeated for 50 runswithdifferent random seeds and the best worst and mean valuesstandard deviations success rates and average reaching timeare given in Table 10 The best results for each case arehighlighted in boldface
As can clearly be seen from Table 10 when solving theunimodal nonseparable problems (Rosenbrock Colville)although the results of WPA are not good enough as FAor ASFA algorithm WPA also achieves 100 success rateFirstly with respect to Rosenbrock function its surface plotand contour lines are shown in Figure 2
As seen in Figure 2 Rosenbrock function is well knownfor its Rosenbrock valley Global minimum value for thisfunction is 0 and optimum solution is (119909
1 1199092) = (1 1)
But the global optimum is inside a long narrow parabolic-shaped flat valley Since it is difficult to converge to theglobal optimum of this function the variables are stronglydependent and the gradients generally do not point towardsthe optimum this problem is repeatedly used to test theperformance of the algorithms [17] As shown in Table 10PSO AFSA FA and WPA achieve 100 success rate andPSO shows the fastest convergence speed AFSA gets thevalue 110119890 minus 13 with the best accuracy FA also showsgood performance because of its robustness on Rosenbrockfunction
On theColville function its surface plot and contour linesare shown in Figure 3 Colville function also has a narrowcurving valley and it is hard to be optimized if the searchspace cannot be explored properly and the direction changescannot be kept up with Its global minimum value is 0 andoptimum solution is (119909
1 1199092 1199093 1199094) = (1 1 1 1)
Although the best accurate solution is obtained by AFSAWPA outperforms the other algorithms in terms of the worstmean std SR and Art on Colville function
Sphere and Sumsquares are convex unimodal and sepa-rable functions They are all high-dimensional functions fortheir 200 and 150 parameters respectively and the globalminima are all 0 and optimum solution is (119909
1 1199092 119909
119898) =
(0 0 0) Surface plot and contour lines of them arerespectively shown in Figures 4 and 5
As seen from Table 10 when solving the unimodal sep-arable problems we note that WPA outperforms other fivealgorithms both on convergence speed and solution accuracyIn particular WPA offers the highest accuracy and improvesthe precision by about 170 orders ofmagnitude on Sphere and
12 Mathematical Problems in Engineering
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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
Table10Statistic
alresults
of50
runs
obtained
byGAP
SOA
FSAA
BCFAand
WPA
algorithm
s
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Rosenb
rock
119891min
(119909)
=0
2UN
GA
178
119890minus
1000373
000
91000
9210
gt7598
323
PSO
226
119890minus
115
89119890
minus7
107
119890minus
71
30119890
minus7
100
07444
AFS
A110
eminus13
111
119890minus
92
34119890
minus10
262
119890minus
10100
20578
ABC
599
119890minus
6000
998
61119890
minus4
00015
0gt3910
297
FA6
28119890
minus13
629
eminus10
186eminus
10162eminus
10100
3312
56WPA
349
119890minus
112
34119890
minus8
509
119890minus
94
34119890
minus9
100
66333
Colville
119891min
(119909)
=0
4UN
GA
00022
03343
01272
01062
0gt12
2119890+
3PS
O1
29119890
minus6
346
119890minus
45
06119890
minus5
671
119890minus
50
gt114
0869
AFS
A366
eminus8
891
119890minus
73
16119890
minus7
232
119890minus
7100
4018
07ABC
00103
05337
01871
01232
0gt3844193
FA2
41119890
minus7
369
119890minus
56
62119890
minus6
807
119890minus
68
gt3
14119890
+3
WPA
471
119890minus
8372
eminus7
125eminus
7697eminus
8100
27405
4
Sphere
119891min
(119909)
=0
200
US
GA
156
119890+
51
81119890
+5
171
119890+
55
78119890
+3
0gt4
44119890
+4
PSO
10361
15520
12883
01206
0gt2719
201
AFS
A5
12119890
+5
579
119890+
55
51119890
+5
163
119890+
40
gt7
41119890
+3
ABC
000
4112
521
004
4401773
0gt44
29045
FA01432
02327
01865
00199
0gt8
34119890
+3
WPA
149eminus
172
241
eminus165
156eminus
166
0100
61729
Sumsquares
119891min
(119909)
=0
150
US
GA
593
119890+
47
15+
46
63119890
+4
288
119890+
30
gt3
16119890
+4
PSO
397098
911145
559050
104165
0gt2325464
AFS
A1
43119890
+5
179
119890+
51
64119890
+5
958
119890+
30
gt7
36119890
+3
ABC
171
119890minus
500017
199
119890minus
43
36119890
minus4
0gt4351848
FA89920
998861
405721
192743
0gt6
88119890
+3
WPA
268
eminus172
547
eminus166
262eminus
167
0100
65954
Booth
119891min
(119909)
=0
2MS
GA
455
119890minus
114
55119890
minus11
455
119890minus
110
100
12621
PSO
122
119890minus
122
41119890
minus8
280
119890minus
94
52119890
minus9
100
020
79AFS
A3
02119890
minus12
145
119890minus
94
61119890
minus10
408
119890minus
10100
44329
ABC
605
eminus20
141
eminus17
463eminus
18414eminus
18100
04175
FA1
80119890
minus12
439
119890minus
91
18119890
minus9
111
119890minus
9100
379191
WPA
822
119890minus
157
05119890
minus13
121
119890minus
131
19119890
minus13
100
69339
Bridge
119891max
(119909)
=3
0054
2MN
GA
300
54300
54300
541
35119890
minus15
100
01927
PSO
300
54300
54300
544
84119890
minus8
100
009
29AFS
A300
54300
4730052
169
119890minus
412
gt8
01119890
+3
ABC
300
54300
54300
543
59119890
minus15
100
00932
FA300
54300
54300
543
11119890
minus10
100
227230
WPA
300
54300
54300
54358eminus
15100
01742
Ackley
119891min
(119909)
=0
50MN
GA
114570
126095
1216
1202719
0gt10
4119890+
4PS
O004
6917401
06846
06344
0gt1925522
AFS
A2016
0020600
9204229
01009
0gt9
80119890
+3
ABC
200085
200025
200061
00014
0gt5963841
FA00101
00209
00160
00021
0gt4
28119890
+3
WPA
888
eminus16
444
eminus15
110eminus
15852eminus
16100
79476
Mathematical Problems in Engineering 13
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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
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
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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
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
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
Table10C
ontin
ued
Functio
nGlobalextremum
119863C
Algorith
ms
Best
Worst
Mean
StdD
evSR
Art119904
Grie
wank
119891min
(119909)
=0
100
MN
GA
3174
525
3996
376
3634174
172922
0gt2
07119890
+4
PSO
00029
00082
00052
00011
0gt3670
080
AFS
A2
05119890
+3
255
119890+
32
33119890
+3
1096
821
0gt6
51119890
+3
ABC
895
119890minus
7000
432
26119890
minus4
781
119890minus
42
gt6209561
FA000
6800118
000
9100011
0gt5
72119890
+3
WPA
00
00
100
145338
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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
14 Mathematical Problems in Engineering
0
100
10
0
1
xy
minusf(xy)
minus1
minus2
minus3
minus10minus10
times106
(a)
minus5
minus5
minus10minus10
x
y
0 5 10
0
5
10
(b)
Figure 3 Colville function (1199091
= 1199093 1199092
= 1199094) (a) surface plot and (b) contour lines
0100
0
1000
05
1
15
2
xy
minus100minus100
f(xy)
times104
(a)
x
y
0 50 100
0
50
100
minus100minus100
minus50
minus50
(b)
Figure 4 Sphere function (119863 = 2) (a) surface plot and (b) contour lines
minus100
minus200
minus300
minus10minus10
0
100
10
0
xy
minusf(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 5 Sumsquares function (119863 = 2) (a) surface plot and (b) contour lines
Mathematical Problems in Engineering 15
0
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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
100
100
1000
2000
3000
xy
minus10minus10
f(xy)
(a)
x
y
0 5 10
0
5
10
minus10minus10
minus5
minus5
(b)
Figure 6 Booth function (119863 = 2) (a) surface plot and (b) contour lines
0
20
20
1
2
3
xy
minus2minus2
f(xy)
(a)
x
y
minus1 0 1
0
1
05
05
15
15
minus05
minus05
minus1
minus15minus15
(b)
Figure 7 Bridge function (119863 = 2) (a) surface plot and (b) contour lines
Sumsquares functions when compared with the best resultsof the other algorithms
Booth is a multimodal and separable function Its globalminimum value is 0 and optimum solution is (119909
1 1199092) =
(1 3)WhenhandingBooth function ABC can get the closer-to-optimal solution within shorter time Surface plot andcontour lines of Booth are shown in Figure 6
As shown in Figure 6 Booth function has flat surfaces andis difficult for algorithms since the flatness of the functiondoes not give the algorithm any information to direct thesearch process towards the minima SoWPA does not get thebest value as good as ABC but it can also find good solutionand achieve 100 success rate
Bridge and Ackley are multimodal and nonseparablefunctions The global maximum value of Bridge function is30054 and optimum solution is (119909
1 1199092) rarr (0 0)The global
minimumvalue ofAckley function is 0 andoptimumsolutionis (1199091 1199092 119909
119898) = (0 0 0) Surface plot and contour
lines of them are separately shown in Figures 7 and 8
As seen in Figures 7 and 8 the locations of the extremumare regularly distributed and there aremany local extremumsnear the global extremumThedifficult part of finding optimais that algorithms may easily be trapped in local optima ontheir way towards the global optimum or oscillate betweenthese local extremums From Table 10 all algorithms exceptASFA show equal performance and achieve 100 successrate on Bridge function While with respect to Ackley (119863 =50) only WPA achieves 100 success rate and improves theprecision by 13 or 15 orders of magnitude when comparedwith the best results of other algorithms
Otherwise the dimensionality and size of the searchspace are important issues in the problem [18] Griewankfunction an multimodal and nonseparable function has theglobalminimum value of 0 and its corresponding global opti-mum solution is (119909
1 1199092 119909
119898) = (0 0 0) Moreover
the increment in the dimension of function increases thedifficulty Since the number of local optima increases with thedimensionality the function is strongly multimodal Surface
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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
16 Mathematical Problems in Engineering
020
400
50
0
xy
minus10
minus20
minus20
minus30
minus40minus50
minusf(xy)
(a)
minus10
minus10
minus20
minus20
minus30
minus30
x
y
0 10 20 30
0
10
20
30
(b)
Figure 8 Ackley function (119863 = 2) (a) surface plot and (b) contour lines
01000
0
1000
0
xy
minusf(xy)
minus50
minus100
minus150
minus200
minus1000 minus1000
(a)
x
y
0 200 400 600
0
200
400
600
minus200
minus200
minus400
minus400minus600
minus600
(b)
Figure 9 Griewank function (119863 = 2) (a) surface plot and (b) contour lines
plot and contour lines of Griewank function are shown inFigure 9
WPA with optimized coefficients has good performancein high-dimensional functions Griewank function (119863 =100) is a good example In such a great search space as shownin Table 10 other algorithms present serious flaws suchas premature convergence and difficulty to overcome localminima while WPA successfully gets the global optimum 0in 50 runs computation
As is shown in Table 10 SR shows the robustness ofevery algorithm and it means how consistently the algorithmachieves the threshold during all runs performed in theexperiments WPA achieves 100 success rate for functionswith different characteristics which shows its good robust-ness
In the experiments there are 8 functions with variablesranging from 2 to 200 WPA statistically outperforms GA on6 PSO on 5 ASFA on 6 ABC on 6 and FA on 7 of these8 functions Six of the functions on which GA and ABCare unsuccessful are two unimodal nonseparable functions
(Rosenbrock and Colville) and four high-dimensional func-tions (Sphere Sumsquares Ackley and Griewank) PSO andFA are unsuccessful on 1 unimodal nonseparable functionand four high-dimensional functions But WPA is also notperfect enough for all functions there are many problemsthat need to be solved for this new algorithm From Table 10on the Rosenbrock function the accuracy and convergencespeed obtained byWPA are not the best ones So amelioratingWPA inspired by intelligent behaviors of wolves for thesespecial problems is one of our future works However sofar it seems to be difficult to simultaneously achieve bothfast convergence speed and avoiding local optima for everycomplex function [19]
It can be drawn that the efficiency of WPA becomesmuch clearer as the number of variables increases WPAperforms statistically better than the five other state-of-the-art algorithms on high-dimensional functions Nowadayshigh-dimensional problems have been a focus in evolu-tionary computing domain since many recent real-worldproblems (biocomputing data mining design etc) involve
Mathematical Problems in Engineering 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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 17
optimization of a large number of variables [20] It isconvincing that WPA has extensive application in scienceresearch and engineering practices
5 Conclusions
Inspired by the intelligent behaviors of wolves a new swarmintelligent optimizationmethod wolf pack algorithm (WPA)is presented for locating the global optima of continuousunconstrained optimization problems We testify the per-formance of WPA on a suite of benchmark functions withdifferent characteristics and analyze the effect of distancemeasurements and parameters on WPA Compared withPSO ASFA GA ABC and FA WPA is observed to performequally or potentially more powerful Especially for high-dimensional functions such as Sphere (119863 = 200) Sumsquares(119863 = 150) Ackley (119863 = 50) and Griewank (119863 = 100) WPAmay be a better choice sinceWPA possesses superior perfor-mance in terms of accuracy convergence speed stability androbustness
After all WPA is a new attempt and achieves somesuccess for global optimization which can provide new ideasfor solving engineering and science optimization problemsIn future different improvements can be made on theWPA algorithm and tests can be made on more differenttest functions Meanwhile practical applications in areas ofclassification parameters optimization engineering processcontrol and design and optimization of controller would alsobe worth further studying
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011
[2] C Grosan and A Abraham ldquoA novel global optimization tech-nique for high dimensional functionsrdquo International Journal ofIntelligent Systems vol 24 no 4 pp 421ndash440 2009
[3] Y Yang Y Wang X Yuan and F Yin ldquoHybrid chaos optimiza-tion algorithm with artificial emotionrdquo Applied Mathematicsand Computation vol 218 no 11 pp 6585ndash6611 2012
[4] W SGao andC Shao ldquoPseudo-collision in swarmoptimizationalgorithm and solution rain forest algorithmrdquo Acta PhysicaSinica vol 62 no 19 Article ID 190202 pp 1ndash15 2013
[5] Y Celik and E Ulker ldquoAn improved marriage in honeybees optimization algorithm for single objective unconstrainedoptimizationrdquoThe Scientific World Journal vol 2013 Article ID370172 11 pages 2013
[6] E Cuevas D Zaldıvar and M Perez-Cisneros ldquoA swarmoptimization algorithm for multimodal functions and its appli-cation in multicircle detectionrdquo Mathematical Problems inEngineering vol 2013 Article ID 948303 22 pages 2013
[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 Perth Australia December 1995
[8] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992
[9] X-L Li Z-J Shao and J-X Qian ldquoOptimizing methodbased on autonomous animats Fish-swarm Algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002
[10] D Karaboga ldquoAn idea based on honeybee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University KayseriTurkey 2005
[11] X-S Yang ldquoFirefly algorithms formultimodal optimizationrdquo inStochastic Algorithms Foundations andApplications vol 5792 ofLecture Notes in Computer Science pp 169ndash178 Springer BerlinGermany 2009
[12] J A Ruiz-Vanoye O Dıaz-Parra F Cocon et al ldquoMeta-Heuristics algorithms based on the grouping of animals bysocial behavior for the travelling sales problemsrdquo InternationalJournal of Combinatorial Optimization Problems and Informat-ics vol 3 no 3 pp 104ndash123 2012
[13] C-G Liu X-H Yan and C-Y Liu ldquoThe wolf colony algorithmand its applicationrdquo Chinese Journal of Electronics vol 20 no 2pp 212ndash216 2011
[14] D E Goldberg Genetic Algorithms in Search Optimisation andMachine Learning Addison-Wesley Reading Mass USA 1989
[15] S-K S Fan andE Zahara ldquoAhybrid simplex search and particleswarm optimization for unconstrained optimizationrdquo EuropeanJournal ofOperational Research vol 181 no 2 pp 527ndash548 2007
[16] P Caamano F Bellas J A Becerra and R J Duro ldquoEvolution-ary algorithm characterization in real parameter optimizationproblemsrdquo Applied Soft Computing vol 13 no 4 pp 1902ndash19212013
[17] D Ortiz-Boyer C Hervas-Martınez and N Garcıa-PedrajasldquoCIXL2 a crossover operator for evolutionary algorithmsbased on population featuresrdquo Journal of Artificial IntelligenceResearch vol 24 pp 1ndash48 2005
[18] M S Kıran and M Gunduz ldquoA recombination-based hybridi-zation of particle swarm optimization and artificial bee colonyalgorithm for continuous optimization problemsrdquo Applied SoftComputing vol 13 no 4 pp 2188ndash2203 2013
[19] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011
[20] Y F Ren and Y Wu ldquoAn efficient algorithm for high-dime-nsional function optimizationrdquo Soft Computing vol 17 no 6pp 995ndash1004 2013
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
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