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Research ArticleModified Bat Algorithm Based on Leacutevy Flight andOpposition Based Learning
Xian Shan1 Kang Liu2 and Pei-Liang Sun3
1School of Science China University of Petroleum Qingdao 266580 China2College of Mechanical and Electronic Engineering China University of Petroleum Qingdao 266580 China3School of Economics and Management China University of Petroleum Qingdao 266580 China
Correspondence should be addressed to Kang Liu lkzsww163com
Received 14 July 2016 Accepted 25 October 2016
Academic Editor Xiang Li
Copyright copy 2016 Xian Shan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Bat Algorithm (BA) is a swarm intelligence algorithmwhich has been intensively applied to solve academic and real life optimizationproblems However due to the lack of good balance between exploration and exploitation BA sometimes fails at finding globaloptimum and is easily trapped into local optima In order to overcome the premature problem and improve the local searchingability of Bat Algorithm for optimization problems we propose an improved BA called OBMLBA In the proposed algorithm amodified search equation with more useful information from the search experiences is introduced to generate a candidate solutionand Levy Flight random walk is incorporatedwith BA in order to avoid being trapped into local optima Furthermore the conceptof opposition based learning (OBL) is embedded to BA to enhance the diversity and convergence capability To evaluate theperformance of the proposed approach 16 benchmark functions have been employed The results obtained by the experimentsdemonstrate the effectiveness and efficiency of OBMLBA for global optimization problems Comparisons with some other BAvariants and other state-of-the-art algorithms have shown the proposed approach significantly improves the performance of BAPerformances of the proposed algorithm on large scale optimization problems and real world optimization problems are notdiscussed in the paper and it will be studied in the future work
1 Introduction
With the development of natural science and technologya large number of complicated optimization problems arisein a variety of fields such as engineering manufacturinginformation technology and economic management Thesereal world problems usually have an objective functionwith lots of decision variables and constrains appearingwith discontinuous nonconvex features [1] Optimizationmethods are always needed to obtain the optimal solution ofthese problems
Optimization algorithms proposed by researchers can becategorized into two groups deterministic algorithms andstochastic algorithms Deterministic algorithms usually needthe information of gradient They are effective for problemswith one global optimum while they might be invalid forproblemswith several local optima or problemswith gradient
information unavailable Compared with deterministic algo-rithms stochastic algorithms require only the information ofthe objective function [2] They have been successfully usedinmany optimization problems characterized as nondifferen-tiable and nonconvex
Swarm intelligence algorithms regarded as a subset ofstochastic algorithms have been widely used to solve opti-mization problems during the past decadesThey are inspiredby the collective behavior of swarms in nature Researchershave observed such behaviors of animals plants or humansanalyzed the driving force behind the phenomena and thenproposed various types of algorithms For example GeneticAlgorithm (GA) [3] and Differential Evolution (DE) [4] weregeneralized by inspiration of the biological evolution andnatural selection Particle SwarmOptimization (PSO) [5] wasproposed by simulating the social and cognitive behaviorof fish or bird swarms Ant Colony Algorithm (ACO) [6]
Hindawi Publishing CorporationScientific ProgrammingVolume 2016 Article ID 8031560 13 pageshttpdxdoiorg10115520168031560
2 Scientific Programming
and Artificial Bee Colony (ABC) [7] were introduced by theforaging behavior of ants and bees Cat Swarm Algorithm(CSO) [8] Cuckoo Search Algorithm (CS) [9] and BatAlgorithm (BA) [10] were also proposed by inspiration fromthe intelligence features in behaviors or organisms of swarms
Bat Algorithm proposed by Yang in 2010 is a swarmintelligence algorithm inspired by the echolocation behaviorof bats Echolocation is a type of sonar which is used bybats to detect prey hunt and avoid obstacles With thehelp of echolocation not only can the bats be able todetect the distance of the prey but also they can identifythe shape position and angle of the prey [10] Comparedwith other existing swarm intelligence algorithms BA hasmany advantages such as less control parameters excellentglobal optimization ability and implementation simplicitywhich have shown excellent performances in some classesof optimization problems Due to its simplicity convergencespeed and population feature it has been intensively usedto solve academic and real life problems like multiobjectiveoptimization [11] engineering optimization [12] cluster anal-ysis [13] scheduling problems [14] structure optimization[15] image processing [16] manufacturing designing [17]and various other problems
It has been shown that BA is an excellent and powerfulalgorithm for global optimization problems discrete opti-mization problems and constrained optimization problemsHowever due to the lack of good balance between explorationand exploitation in basic BA the algorithm sometimesfails at finding global optimum and is easily trapped intolocal optima Much efforts have been made to improve theperformance of BA These improvements or modificationscan be divided into two categories
Introduction of New Search Mechanisms for Generating Can-didate Solutions Inspired by DE Xie et al [18] presented amodified approachDLBAwith a new solution updatemecha-nism combining the concept of differential operator and LevyFlight trajectory Li and Zhou [19] embedded the complexvalue encoding mechanism into Bat Algorithm to improvethe diversity and convergence performance Gandomi andYang [20] introduced chaotic maps into the populationinitialization and position update equation and proposeda modified algorithm Bahmani-Firouzi and Azizipanah-Abarghooee [21] proposed four improved velocity updateequations in order to achieve a better balance between theexploration and exploitation Jaddi et al [22] divided theswarm population into two subpopulation groups with eachgroup designed to generate solutions according to differentequations respectively Yilmaz and Kucuksille [23] definedtwo modification structures of the velocity equation inspiredby PSO and then hybridized the modified BA with InvasiveWeed Optimization algorithm The velocity equations usedinformation of the best-so-far solution and the neighborhoodsolutions which can effectively enhance the search ability ofBA
Hybridization of BAwith Other Operators For example Khanand Sahai [24] proposed a hybrid approach involving PSOHS and SA for generating new solutions Wang and Guo [25]
introduced a mutation operator into BA using SA He et al[26] developed a hybrid BA which combines SA and Gaussdistribution Sadeghi et al [27] developed a hybrid BA withlocal search based on PSO Lin et al [28] developed a chaoticBA with Levy Flights and parameters estimated by chaoticmapsMeng et al [29] incorporated the batsrsquo habitat selectionand their self-adaptive compensation for Doppler effect inechoes into the basic BA and proposed a new self-adaptivemodified algorithm NBA
The study is not limited to the above two aspects andmorework can be seen in [30ndash34] and so on
As technology and science develop dramatically fastmore and more complicated optimization problems need tobe solved by advanced numerical methods As an impor-tant heuristic optimization algorithm researches on how toimprove the convergence accuracy and efficiency of BA areof great significance to improve the heuristic optimizationtheory and solve the real world optimization problems
Exploration and exploitation are two critical characteris-tics in the updating process of an algorithm Exploration is theability of the algorithm to find the global optimum solutionsin different areas of the search space while exploitation refersto the capability of the algorithm to find better solutions withprevious experience Researches in the literature show thatfor a swarm intelligence algorithm the exploration capabilityshould be employed first so as to search in the whole spacewhile the exploitation capability should be considered laterby improving the quantity of the solution in the local searchprocess [23]
To achieve a better balance between exploration andexploitation behavior of BA it is highly required to modifythe global search approach to explore the search region anddevelop a local search method to exploit in the nonvisitedregions Accordingly modifications of the search equationand local search strategies for BA are studied in the researchA modified algorithm called OBMLBA is proposed in thisstudy The main difference between OBMLBA and other BAsis the interaction behavior between bats In the proposedapproach the bat explores the search space with the help ofthe best-so-far solution and the neighborhood solutions byusing echolocation which can well balance the explorationand exploitation The new population 119875 is generated by usinga new modified position update equation with frequency119891 defined by sinusoidal function Levy Flight random walkis used to exploit the local search space whereas opposi-tion based learning is used to introduce opposition basedsolutions in population OP Individuals of 119875 and OP aremerged together and the best119873 individuals are used as a newpopulation for the next generation The proposed methodhas been tested on 16 benchmark functions Comparisonwith variants of BA and other state-of-the-art algorithmsdemonstrates the effectiveness and superiority of OBMLBAin solving optimization problems
The paper is organized as follows After the introductionthe basic concept of BA is introduced in Section 2 Theproposed OBMLBA is presented in Section 3 Section 4evaluates the performance of the proposed algorithm bycomparing results with other algorithms Finally conclusionsand future work are discussed in Section 5
Scientific Programming 3
(1) Input the parameters(2) Set iter = 1 Initialize the position and velocity of each individual in the population(3) Evaluate the objective function and related parameter values of each individual(4)While iter le iter max do(5) Generate new solutions using Eq (2)simEq (4)(6) if rand(0 1) gt 119903119894(7) Generate a new local solution around the selected solution 119909lowast using Eq (5)(8) end if(9) if rand(0 1) lt 119860 119894 and 119891(119909119894new) lt 119891(119909119894)(10) update the solution loudness and pulse emission rate by Eq (6)(11) end if(12) end while(13) Output the individual with the smallest objective function value in the population
Algorithm 1 Basic Bat Algorithm
2 Basic Bat Algorithm
The optimization problems considered in this paper aresingle-objective unconstrained continuous problems to beminimized
Bat Algorithm is a heuristic algorithm proposed by Yangin 2010 It is based on the echolocation capability of microbats guiding them on their foraging behavior In BA theposition of a bat represents a possible solution of the givenoptimization problemThe position of the food source foundby the 119894th bat can be expressed as 119909119894 = (1199091198941 1199091198942 119909119894119863)Thefitness of 119909119894 corresponds to the quality of the position the batlocates in
21 Initiation At the beginning bats do not know the loca-tion of food sources They generate a randomly distributedpopulation 119875 of119873 solutions where119873 denotes the number ofbats in the population Each solution can be produced withinthe search space as follows
119909119894119895 = 119909min + rand (0 1) (119909max minus 119909min) (1)
where 119894 = 1 2 119873 119895 = 1 2 119863 119909max and 119909min denotethe upper and lower bounds of the solution in dimension119895 respectively rand(0 1) is a uniformly distributed valuegenerated in the range of [0 1]22 Generation of New Solutions Bats modify the solutionbased on the information of the current location and the bestsource food location Bats navigate by adjusting their flyingdirections using their own and other swarm membersrsquo bestexperience to find the optimum of the problem
At this stage for each food source position 119909119894 a newposition was formulated as follows
119891119894 = 119891min + 120573 (119891max minus 119891min) (2)
V119905119894 = V119905minus1119894 + (119909119905119894 minus 119909lowast)119891119894 (3)
119909119905119894 = 119909119905minus1119894 + V119905119894 (4)
where 119894 represents each bat in the population 119894 = 1 2 119873and 119905 represents the 119905th iteration 119909119905119894 and V119905119894 are the position
and velocity components of the 119894th bat in the population at the119905th iteration 119891119894 denotes the pulse frequency that affects thevelocity of the 119894th bat 119891max and 119891min represent the maximumandminimum of119891119894 120573 is a random number between [0 1] 119909lowastis the best position found by the whole population
23 Local Search Once the new solutions are generated thelocal search is invoked by batsrsquo random walk If the pulseemission rate 119903 of the 119894th bat is smaller than a randomnumberselect 119909119894old from the population and generate a new position119909119894new as follows
119909119894new = 119909119894old + 120576119860119905 (5)
where119909119894old represents a solution chosen in current populationby some mechanism and 120576 is a random vector drawn from auniform distribution 119860119905 is the average loudness of all bats atthe time step 11990524 Solutions Loudness and Pulse Emission Rate UpdatingIf a random number is bigger than the loudness 119860119905 and119891(119909119894new) lt 119891(119909119894) accept the new solution 119909119894new At the sametime the loudness 119860119905 is decreased while its pulse emission 119903119894is increased as follows
119860119905+1119894 = 120572119860119905119894119903119905+1119894 = 1199030119894 (1 minus 119890120574119905)
(6)
where 120572 and 120574 are constants The initial loudness 1198600 andinitial pulse emission rate 1199030 are randomly generated numbersin the range of [1 2] and [0 1] respectively
The pseudocode of Bat Algorithm is presented in Algo-rithm 1
3 Enhanced Bat Algorithm
31 Location Update Equation Based on DE Yilmaz andKucuksille [23] pointed that the update equations of velocityin basic BA provide local search only with the guidance ofthe best-so-far solution whichmakes the BAeasily entrapped
4 Scientific Programming
into a local minimum In order to improve the search abilitythey proposed a modified equation
V119905119894 = 120596V119905minus1119894 + (119909119905119894 minus 119909lowast) 1198911198941205771 + (119909119905119894 minus 119909119905119896) 1198911198941205772 (7)
1205771 + 1205772 = 1 (8)
1205771 = 1 + (120577init minus 1) (itermax minus iter)119899(itermax)119899 (9)
where 119909119905119896 is a solution randomly chosen from the populationand 1205771 and 1205772 are learning factors ranging from 0 to 1 120577initis the initial value of 1205771 itermax is the maximal number ofiterations iter is the number of iterations and 119899 is a nonlinearindex
The second term in the modified search equation is afactor affecting the velocity of the solution with guidanceof the best solution 119909lowast which emphasizes exploitation Thethird term is another factor that affects the velocity withhelp of randomly selected solution 119909119905119896 which emphasizesexploration Effects of the best solution and the kth solutionare adjusted by changing the value of 1205771 Experiments resultsindicate that the modified algorithm can outperform BA inmost test functions
Xie et al [18] proposed an improved BA based on Dif-ferential Evolution operatorThe position updating equationswere defined as follows
119909119905+1119894 = 119909119905best + 1198911119894 (1199091199051199031 minus 1199091199051199032) + 1198912119894 (1199091199051199033 minus 1199091199051199034) (10)
where 119909119905best is the global best location in current population119909119905119903119894 (119894 = 1 2 3 4) is a randomly chosen solution in thepopulation 1198911119894 and 1198912119894 are defined as follows
1198911199051119894 = ((1198911min minus 1198911max) 119905119899119905 + 1198911max)1205731 (11)
1198911199052119894 = ((1198912min minus 1198912max) 119905119899119905 + 1198912max)1205732 (12)
where 1198911max and 1198911min are minimum and maximum fre-quency values of 1198911119894 while 1198912max and 1198912min are minimumandmaximum frequency values of1198912119894 119899119905 is a fixed parameter119905 is the number of iterations 120573 isin [0 1] is a random vectordrawing a uniform distribution The modified algorithm hasshown a better performance than classical BA
Inspired by Yilmaz and Zhoursquos methods we propose amodified location updating equation
119909119905+1119894 = 119909119905119894 + 1198911119894 (119909119905best minus 1199091199051199031) + 1198912119894 (1199091199051199032 minus 1199091199051199033) (13)
where 119909119905119903119894 (119894 = 1 2 3) are individuals randomly selected fromthe population 1198911119894 and 1198912119894 are frequencies
As in mutation operation of DE the frequency 119891 is animportant parameter that affects the convergence perfor-mance of the algorithm The value of 119891 changes according tothe predefined formula In basic BA and DLBA 119891 is definedin (2) (11) and (12) respectively Here we use a sinusoidalformulation [35] permitting certain flexibility in changing
the direction of 119891 The frequencies 1198911119894 and 1198912119894 are defined asfollows
1198911119894 = 12 (sin (2120587 lowast 120596 lowast 119905) lowast iter
itermax+ 1) (14)
1198912119894 = 12 (sin (2120587 lowast 120596 lowast 119905) lowast iter
itermax+ 1) (15)
where 120596 represents frequency of the sinusoidal function
32 Local Search Based on Levy Flight Levy Flight which isbased on the flight behavior of animals and insects has beenvariously studied in literatures [36] It is defined as a kind ofnon-Gauss stochastic process with a randomwalk whose stepsizes are based on Levy stable distribution
There are lots of versions of Levy distribution MantegnaAlgorithm [37] has been tested to be the most efficientmethod to generate Levy distribution values It has beenadopted in many evolution algorithms in order to escapefrom local minimum
When generating a new solution 1199091015840119894 for the 119894th solution 119909119894by performing Levy Flight the new candidate 1199091015840119894 is defined asfollows
1199091015840119894 = 119909119894 + 120572 oplus Levy (120582) (16)
where 120572 is a random step size parameter 120582 is a LevyFlight distribution parameter and oplus denotes entry wisemultiplication
In Mantegna Algorithm the step size 119904 can be defined asfollows
119904 = 120572 oplus Levy (120582) sim 001 119906|V|1120573 (119909119895 minus 119909best) (17)
where 119906 and V are obtained from normal distribution that is
119906 sim 119873 (0 1205902119906) V sim 119873(0 1205902V)
(18)
with
120590119906 = Γ (1 + 120573) sin (1205871205732)120573Γ [(1 + 120573) 2] 2(120573minus1)2
1120573
120590V = 1
(19)
Γ (119911) = int 119905119911minus1119890minus119911119889119911 (20)
The pseudocode of Levy Flight Search is shown inAlgorithm 2
33 Opposition Based Learning Opposition based learningwas proposed by Rahnamayan et al [38] in 2005 It has beensuccessfully used in machine learning and evolutionary com-puting [39] For an optimization problem if the candidatesolutions are near the global optima it is easy to find the idealsolutions with fast convergence speed But if the candidate
Scientific Programming 5
(1) Input the optimization function 119891(119909) and 120573(2) Select the individual 119909119894 which is chosen to be modified by Levy Flight(3) Compute 120590119906 using Eq (19)(4) Compute step size s using Eq (17) and Eq (18)(5) Generate a new candidate solution 1199091015840119894 using Eq (16)(6) Calculate119891(1199091015840)(7) if 119891(1199091015840119894 ) lt 119891(119909119894)(8) 119909119894 = 1199091015840119894 (9) end if
Algorithm 2 Pseudocode of Levy Flight Search
(1) Input original population 119875 population size 119878119873 the dimension of solution 119863 optimization function 119891(119909)(2) Generate opposite population OP as follows(3) for 119894 = 1 119878119873(4) for 119895 = 1 119863(5) 119909119894119895 = 119886119894 + 119887119894 minus 119909119894119895(6) end for(7) end for(8) Merge the original population 119875 and opposite population OP together(9) Calculate the fitness of population 119875OP(10) Choose the top half of population 119875OP as the current population
Algorithm 3 Opposition based optimization
solutions are far away from the global optima it will takemore computational efforts to get the required solutions Theconcept of OBL is to consider the candidate solutions andtheir opposite counterpart solutions simultaneously in anoptimization algorithm Then the greedy selection is appliedbetween them according to their objective function valuesWith the help ofOBL the probability of finding global optimais increased
In basic OBL let 119909 = (1199091 1199092 119909119863) be a119863-dimensionalsolution in the search space with component variables 119909119894 isin[119886119894 119887119894]Then the opposite point = (1 2 119863) is definedby its coordinates 1 2 119863 where
119894 = 119886119894 + 119887119894 minus 119909119894 119894 = 1 2 119863 (21)
OBL can be used not only to initialize the populationinitialization but also to improve the population diversityduring the evolutionary process Once the opposite popula-tion generated it is combinedwith the original population forselection Individuals with fitness ranking in the top half of allcandidate solutions will be selected as the current populationThe pseudocode of OBL is given in Algorithm 3
34 Proposed Modified Bat Algorithm Based on the basic BAand the above considerations we propose amodifiedmethodcalled OBMLBA It is clear that local search with LevyFlight can improve the ability of exploitation Incorporatingwith OBL strategy can enhance the population diversityInspired by DLBA the new location update equation takesadvantage of the information of both the global best solution
and randomly chosen solution near the candidate solutionSo the modified BA can get a good balance between theexploration and exploitation Themain steps of the algorithmare summarized in Algorithm 4
4 Experimental Results and Discussion
In order to evaluate the performance of the proposedalgorithm we select 16 standard benchmark functions toinvestigate the efficiency of the proposed approach Thesefunctions are presented in Table 1The function set comprisesunimodal functions and multimodal functions which canbe used to test the convergence speed and local prematureconvergence problem respectively 1198651sim1198655 are continuousunimodal functions 1198657 is a discontinuous step function1198659 is a noisy quartic function 11986510sim11986516 are multimodalfunctions with high dimension These functions are cate-gorized into low-dimension middle-dimension and high-dimension functions according to 119863 = 15 30 and 60respectively
Related parameters of the basic BADLBA andOBMLBAare presented in Table 2 The maximum number of iterationsis set to be 500 1000 and 2000 in case of119863 = 15 30 and 60
Experiments have been carried out on the test functionswith different dimensions The computational results aresummarized in Tables 3ndash5 in terms of the mean error andstandard deviation of the function error values In additionsome convergence characteristics graphs are shown in Fig-ure 1
6 Scientific Programming
(1) Input the parameters(2) Set iter = 1 Initialize the position and velocity of each individual in the population(3) Evaluate the objective function and related parameter values of each individual(4) While iter le iter max do(5) Apply Levy Flight Local Search strategy using Algorithm 2(6) Generate new solutions using Eq (13)(7) if rand(0 1) gt 119903119894(8) Apply the Opposition based learning using Algorithm 3(9) Generate a new local solution around the selected solution 119909lowast using Eq (5)(10) end if(11) if rand(0 1) lt 119860 119894 and 119891(119909119894new) lt 119891(119909119894)(12) update the solution loudness and pulse emission rate(13) end if(14) end while(15) Output the individual with the smallest objective function value in the population
Algorithm 4 Pseudocode of OBMLBA
Table 1 Benchmark problems
Fun Name Scope 119891min
F1 Sphere [minus100 100] 0F2 Elliptic [minus100 100] 0F3 SumSquare [minus10 10] 0F4 SumPower [minus1 1] 0F5 Schwefel222 [minus10 10] 0F6 Schwefel221 [minus100 100] 0F7 Step [minus100 100] 0F8 Exponential [minus128 128] 0F9 Quartic [minus128 128] 0F10 Rosenbrock [minus5 10] 0F11 Rastrigin [minus512 512] 0F12 Griewank [minus600 600] 0F13 Ackley [minus32 32] 0F14 Alpine [minus10 10] 0F15 Levy [minus10 10] 0F16 Weierstrass [minus05 05] 0
As seen from Tables 3ndash5 OBMLBA can get the globaloptimal on 1198911sim1198914 1198917 1198918 and 11989111sim11989113 with 119863 = 15 30and 60 On 1198916 and 11989114 the precision of solutions obtained byOBMLBA is smaller than the order of 1119864 minus 90 and the orderfor 1119864minus 100 respectively Results indicate that OBMLBA canget solutions extremely close to the global optimal values onmost of the functions
Solutions obtained by BAs are compared with resultsobtained by MLBA and OBMLBA As shown in Tables 3ndash5OBMLBA performs significantly better than basic BA LBAand DLBA on 1198911sim1198916 11989114 and 11989115 All algorithms except forbasic BA and LBA perform well on 1198917 1198918 11989111sim11989113 and 11989116OBMLBA is comparable to DLBA and MLBA on functions1198919 and 11989110
Compared with MLBA OBMLBA provides better per-formance on functions 1198911sim1198916 11989114 and 11989115 For 1198919 11989110 and
Table 2 Parameters setting
Parameters BA DLBA OBMLBAPopulation size119873 40Run time 20Loudness 095Pulse emission 085Factors updating 119860 and 120574 09Minimum of frequency 119891119898119894119899 0Maximum of frequency 119891max 1Factors updating frequency mdash mdash 03
11989115 results obtained by MLBA and OBMLBA are quite closeto each other Both MLBA and OBMLBA get the globaloptimum of 1198917 1198918 11989111sim11989113 and 11989116
From the convergence characteristic graphs it can beobviously observed that OBMLBA is the fastest algorithm inall the considered BAs MLBA is slower than OBMLBA butfaster than DLBA LBA and basic BA OBMLBA exhibits thebest accuracy and achieves the fastest convergence speedTheresults indeed indicate the advantage of themodified positionupdate equation and OBL strategy
On the whole algorithms proposed in this study such asMLBA andOBMLBAwork better than other BAs consideredAnd OBMLBA is more effective than MLBA
In Table 6 OBMLBA is compared with some state-of-the-art algorithms such as ABC PSO OCABC CLPSOand OLPSO-G Results of these algorithms are all deriveddirectly from their corresponding literatures [40] Resultsshow that OBMLBA has outperformed other methods on6 of 7 functions OBMLBA performs worse than ABC andOCABC only on Rosenbrock function
According to the analyses above it can be clearly observedthat OBMLBA provides outstanding performance with fastconvergence speed and high convergence accuracy on mostof the test functions
Scientific Programming 7
Table3Re
sults
ofcomparis
onso
fBAso
n15-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
119119864+
04184
119864+03
260119864+
03253
119864+03
445119864minus
54139
119864minus53
230119864minus
69707
119864minus69
36601
Eminus2
040
119865 2705
119864+07
304119864+
07189
119864+07
193119864+
07102
119864minus42
454119864minus
42396
119864minus67
125119864minus
66347
13Eminus2
020
119865 3810
119864+02
139119864+
02132
119864+02
225119864+
02377
119864minus52
169119864minus
51810
119864minus66
256119864minus
65621
79Eminus2
020
119865 4259
119864minus02
150119864minus
02164
119864minus03
235119864minus
03522
119864minus90
227119864minus
89457
119864minus126
12069119864
minus125
112Eminus3
050
119865 5408
119864+01
724119864+
00108
119864+01
143119864+
01373
119864minus28
145119864minus
27175
119864minus33
552119864minus
33117
Eminus1
04524
41Eminus1
04119865 6
501119864+
01316
119864+00
147119864+
01146
119864+01
304119864minus
27100
119864minus26
250119864minus
35785
119864minus35
668Eminus9
5298
692E
minus94
119865 7111
119864+04
213119864+
03136
119864+03
216119864+
030
00
00
0119865 8
152119864+
00443
119864minus01
126119864minus
01262
119864minus01
00
00
00
119865 9265
119864+00
113119864+
00134
119864minus01
270119864minus
01479
119864minus04
431119864minus
04399
119864minus04
282119864minus
04119
Eminus0
4100
Eminus0
4119865 10
589119864+
04202
119864+04
603119864+
02134
119864+03
135119864+
01210
119864minus01
129E+0
1371
Eminus0
1130
119864+01
340119864minus
01119865 11
131119864+
02816
119864+00
781119864+
01448
119864+01
00
00
00
119865 12106
119864+02
224119864+
01166
119864+01
227119864+
010
00
00
0119865 13
183119864+
01488
119864minus01
107119864+
01565
119864+00
00
00
00
119865 14142
119864+01
104119864+
00529
119864+00
526119864+
00182
119864minus27
792119864minus
27567
119864minus35
179119864minus
34891
Eminus1
06368
23Eminus1
05119865 15
152119864+
02113
119864+01
273119864+
01176
119864+01
747119864minus
01300
119864minus01
278Eminus0
1274
Eminus0
1483
119864minus01
526119864minus
01119865 16
168119864+
01160
119864+00
987119864+
00523
119864+00
00
00
00
8 Scientific Programming
Table4Re
sults
ofcomparis
onso
fBAso
n30-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
405119864+
04280
119864+03
380119864+
03680
119864+03
373119864minus
49167
119864minus48
180119864minus
117568
119864minus117
00
119865 2530
119864+08
115119864+
08954
119864+07
835119864+
07296
119864minus50
125119864minus
49320
119864minus117
101119864minus
1160
0119865 3
515119864+
03797
119864+02
729119864+
02121
119864+03
980119864minus
48438
119864minus47
272119864minus
123651
119864minus123
00
119865 4753
119864minus02
297119864minus
02580
119864minus03
139119864minus
02174
119864minus85
778119864minus
85176
119864minus198
00
0119865 5
233119864+
05428
119864+05
758119864+
01178
119864+02
972119864minus
26433
119864minus25
512119864minus
60162
119864minus59
680Eminus2
100
119865 6683
119864+01
350119864+
00176
119864+01
196119864+
01542
119864minus30
111119864minus
29263
119864minus55
833119864minus
55140
Eminus2
050
119865 7371
119864+04
426119864+
03569
119864+03
814119864+
030
00
00
0119865 8
239119864+
01619
119864+00
123119864+
00262
119864+00
00
00
00
119865 9402
119864+01
691119864+
00225
119864+00
519119864+
00285
119864minus04
283119864minus
04230
119864minus04
120119864minus
04625
Eminus0
5421
Eminus0
5119865 10
385119864+
05651
119864+04
259119864+
04615
119864+04
287119864+
01622
119864minus02
283119864+
01216
119864minus01
282E+0
1248
Eminus0
1119865 11
337119864+
02157
119864+01
213119864+
02734
119864+01
00
00
00
119865 12361
119864+02
339119864+
01374
119864+01
597119864+
010
00
00
0119865 13
196119864+
01271
119864minus01
656119864+
00525
119864+00
00
00
00
119865 14426
119864+01
274119864+
00155
119864+01
142119864+
01126
119864minus27
556119864minus
27291
119864minus60
910119864minus
60330
Eminus2
150
119865 15500
119864+02
897119864+
01696
119864+01
604119864+
01901
119864+00
307119864+
00578
119864+00
282119864+
00517
E+0
0240
E+0
0119865 16
413119864+
01117
119864+00
161119864+
01138
119864+01
00
00
00
Scientific Programming 9
Table5Re
sults
ofcomparis
onso
fBAso
n60
-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
103119864+
05937
119864+03
353119864+
03871
119864+03
454119864minus
49202
119864minus48
121119864minus
219000
119864+00
00
119865 2238
119864+09
251119864+
08607
119864+08
873119864+
08965
119864minus43
432119864minus
42161
119864minus217
000119864+
000
0119865 3
295119864+
04302
119864+03
124119864+
03203
119864+03
213119864minus
54566
119864minus54
819119864minus
224000
119864+00
00
119865 4147
119864minus01
425119864minus
02279
119864minus03
628119864minus
03205
119864minus88
918119864minus
88000
119864+00
00
0119865 5
100119864+
10000
119864+00
276119864+
01304
119864+01
478119864minus
24214
119864minus23
478119864minus
112151
119864minus111
00
119865 6821
119864+01
242119864+
00115
119864+01
153119864+
01321
119864minus28
705119864minus
28195
119864minus107
613119864minus
1070
0119865 7
983119864+
04872
119864+03
117119864+
04218
119864+04
00
00
00
119865 8546
119864+03
431119864+
03274
119864+01
812119864+
010
00
00
0119865 9
276119864+
02478
119864+01
255119864+
00794
119864+00
270119864minus
04254
119864minus04
156Eminus0
4775
Eminus0
5282
119864minus05
277119864minus
05119865 10
177119864+
06390
119864+05
180119864+
04347
119864+04
586119864+
01721
119864minus02
583119864+
01152
119864minus01
101E+0
1832
Eminus0
1119865 11
773119864+
02279
119864+01
309119864+
02270
119864+02
00
00
00
119865 12903
119864+02
121119864+
02372
119864+01
601119864+
010
00
00
0119865 13
203119864+
01671
119864minus02
636119864+
00572
119864+00
00
00
00
119865 14108
119864+02
442119864+
00506
119864+01
391119864+
01526
119864minus29
220119864minus
28646
119864minus104
204119864minus
1030
0119865 15
149119864+
03192
119864+02
274119864+
02297
119864+02
414119864+
01979
119864+00
242119864+
01661
119864+00
458Eminus0
2653
Eminus0
2119865 16
930119864+
01152
119864+00
335119864+
01268
119864+01
00
00
00
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
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Applied Computational Intelligence and Soft Computing
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Advances in
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Advances in
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RoboticsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
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2 Scientific Programming
and Artificial Bee Colony (ABC) [7] were introduced by theforaging behavior of ants and bees Cat Swarm Algorithm(CSO) [8] Cuckoo Search Algorithm (CS) [9] and BatAlgorithm (BA) [10] were also proposed by inspiration fromthe intelligence features in behaviors or organisms of swarms
Bat Algorithm proposed by Yang in 2010 is a swarmintelligence algorithm inspired by the echolocation behaviorof bats Echolocation is a type of sonar which is used bybats to detect prey hunt and avoid obstacles With thehelp of echolocation not only can the bats be able todetect the distance of the prey but also they can identifythe shape position and angle of the prey [10] Comparedwith other existing swarm intelligence algorithms BA hasmany advantages such as less control parameters excellentglobal optimization ability and implementation simplicitywhich have shown excellent performances in some classesof optimization problems Due to its simplicity convergencespeed and population feature it has been intensively usedto solve academic and real life problems like multiobjectiveoptimization [11] engineering optimization [12] cluster anal-ysis [13] scheduling problems [14] structure optimization[15] image processing [16] manufacturing designing [17]and various other problems
It has been shown that BA is an excellent and powerfulalgorithm for global optimization problems discrete opti-mization problems and constrained optimization problemsHowever due to the lack of good balance between explorationand exploitation in basic BA the algorithm sometimesfails at finding global optimum and is easily trapped intolocal optima Much efforts have been made to improve theperformance of BA These improvements or modificationscan be divided into two categories
Introduction of New Search Mechanisms for Generating Can-didate Solutions Inspired by DE Xie et al [18] presented amodified approachDLBAwith a new solution updatemecha-nism combining the concept of differential operator and LevyFlight trajectory Li and Zhou [19] embedded the complexvalue encoding mechanism into Bat Algorithm to improvethe diversity and convergence performance Gandomi andYang [20] introduced chaotic maps into the populationinitialization and position update equation and proposeda modified algorithm Bahmani-Firouzi and Azizipanah-Abarghooee [21] proposed four improved velocity updateequations in order to achieve a better balance between theexploration and exploitation Jaddi et al [22] divided theswarm population into two subpopulation groups with eachgroup designed to generate solutions according to differentequations respectively Yilmaz and Kucuksille [23] definedtwo modification structures of the velocity equation inspiredby PSO and then hybridized the modified BA with InvasiveWeed Optimization algorithm The velocity equations usedinformation of the best-so-far solution and the neighborhoodsolutions which can effectively enhance the search ability ofBA
Hybridization of BAwith Other Operators For example Khanand Sahai [24] proposed a hybrid approach involving PSOHS and SA for generating new solutions Wang and Guo [25]
introduced a mutation operator into BA using SA He et al[26] developed a hybrid BA which combines SA and Gaussdistribution Sadeghi et al [27] developed a hybrid BA withlocal search based on PSO Lin et al [28] developed a chaoticBA with Levy Flights and parameters estimated by chaoticmapsMeng et al [29] incorporated the batsrsquo habitat selectionand their self-adaptive compensation for Doppler effect inechoes into the basic BA and proposed a new self-adaptivemodified algorithm NBA
The study is not limited to the above two aspects andmorework can be seen in [30ndash34] and so on
As technology and science develop dramatically fastmore and more complicated optimization problems need tobe solved by advanced numerical methods As an impor-tant heuristic optimization algorithm researches on how toimprove the convergence accuracy and efficiency of BA areof great significance to improve the heuristic optimizationtheory and solve the real world optimization problems
Exploration and exploitation are two critical characteris-tics in the updating process of an algorithm Exploration is theability of the algorithm to find the global optimum solutionsin different areas of the search space while exploitation refersto the capability of the algorithm to find better solutions withprevious experience Researches in the literature show thatfor a swarm intelligence algorithm the exploration capabilityshould be employed first so as to search in the whole spacewhile the exploitation capability should be considered laterby improving the quantity of the solution in the local searchprocess [23]
To achieve a better balance between exploration andexploitation behavior of BA it is highly required to modifythe global search approach to explore the search region anddevelop a local search method to exploit in the nonvisitedregions Accordingly modifications of the search equationand local search strategies for BA are studied in the researchA modified algorithm called OBMLBA is proposed in thisstudy The main difference between OBMLBA and other BAsis the interaction behavior between bats In the proposedapproach the bat explores the search space with the help ofthe best-so-far solution and the neighborhood solutions byusing echolocation which can well balance the explorationand exploitation The new population 119875 is generated by usinga new modified position update equation with frequency119891 defined by sinusoidal function Levy Flight random walkis used to exploit the local search space whereas opposi-tion based learning is used to introduce opposition basedsolutions in population OP Individuals of 119875 and OP aremerged together and the best119873 individuals are used as a newpopulation for the next generation The proposed methodhas been tested on 16 benchmark functions Comparisonwith variants of BA and other state-of-the-art algorithmsdemonstrates the effectiveness and superiority of OBMLBAin solving optimization problems
The paper is organized as follows After the introductionthe basic concept of BA is introduced in Section 2 Theproposed OBMLBA is presented in Section 3 Section 4evaluates the performance of the proposed algorithm bycomparing results with other algorithms Finally conclusionsand future work are discussed in Section 5
Scientific Programming 3
(1) Input the parameters(2) Set iter = 1 Initialize the position and velocity of each individual in the population(3) Evaluate the objective function and related parameter values of each individual(4)While iter le iter max do(5) Generate new solutions using Eq (2)simEq (4)(6) if rand(0 1) gt 119903119894(7) Generate a new local solution around the selected solution 119909lowast using Eq (5)(8) end if(9) if rand(0 1) lt 119860 119894 and 119891(119909119894new) lt 119891(119909119894)(10) update the solution loudness and pulse emission rate by Eq (6)(11) end if(12) end while(13) Output the individual with the smallest objective function value in the population
Algorithm 1 Basic Bat Algorithm
2 Basic Bat Algorithm
The optimization problems considered in this paper aresingle-objective unconstrained continuous problems to beminimized
Bat Algorithm is a heuristic algorithm proposed by Yangin 2010 It is based on the echolocation capability of microbats guiding them on their foraging behavior In BA theposition of a bat represents a possible solution of the givenoptimization problemThe position of the food source foundby the 119894th bat can be expressed as 119909119894 = (1199091198941 1199091198942 119909119894119863)Thefitness of 119909119894 corresponds to the quality of the position the batlocates in
21 Initiation At the beginning bats do not know the loca-tion of food sources They generate a randomly distributedpopulation 119875 of119873 solutions where119873 denotes the number ofbats in the population Each solution can be produced withinthe search space as follows
119909119894119895 = 119909min + rand (0 1) (119909max minus 119909min) (1)
where 119894 = 1 2 119873 119895 = 1 2 119863 119909max and 119909min denotethe upper and lower bounds of the solution in dimension119895 respectively rand(0 1) is a uniformly distributed valuegenerated in the range of [0 1]22 Generation of New Solutions Bats modify the solutionbased on the information of the current location and the bestsource food location Bats navigate by adjusting their flyingdirections using their own and other swarm membersrsquo bestexperience to find the optimum of the problem
At this stage for each food source position 119909119894 a newposition was formulated as follows
119891119894 = 119891min + 120573 (119891max minus 119891min) (2)
V119905119894 = V119905minus1119894 + (119909119905119894 minus 119909lowast)119891119894 (3)
119909119905119894 = 119909119905minus1119894 + V119905119894 (4)
where 119894 represents each bat in the population 119894 = 1 2 119873and 119905 represents the 119905th iteration 119909119905119894 and V119905119894 are the position
and velocity components of the 119894th bat in the population at the119905th iteration 119891119894 denotes the pulse frequency that affects thevelocity of the 119894th bat 119891max and 119891min represent the maximumandminimum of119891119894 120573 is a random number between [0 1] 119909lowastis the best position found by the whole population
23 Local Search Once the new solutions are generated thelocal search is invoked by batsrsquo random walk If the pulseemission rate 119903 of the 119894th bat is smaller than a randomnumberselect 119909119894old from the population and generate a new position119909119894new as follows
119909119894new = 119909119894old + 120576119860119905 (5)
where119909119894old represents a solution chosen in current populationby some mechanism and 120576 is a random vector drawn from auniform distribution 119860119905 is the average loudness of all bats atthe time step 11990524 Solutions Loudness and Pulse Emission Rate UpdatingIf a random number is bigger than the loudness 119860119905 and119891(119909119894new) lt 119891(119909119894) accept the new solution 119909119894new At the sametime the loudness 119860119905 is decreased while its pulse emission 119903119894is increased as follows
119860119905+1119894 = 120572119860119905119894119903119905+1119894 = 1199030119894 (1 minus 119890120574119905)
(6)
where 120572 and 120574 are constants The initial loudness 1198600 andinitial pulse emission rate 1199030 are randomly generated numbersin the range of [1 2] and [0 1] respectively
The pseudocode of Bat Algorithm is presented in Algo-rithm 1
3 Enhanced Bat Algorithm
31 Location Update Equation Based on DE Yilmaz andKucuksille [23] pointed that the update equations of velocityin basic BA provide local search only with the guidance ofthe best-so-far solution whichmakes the BAeasily entrapped
4 Scientific Programming
into a local minimum In order to improve the search abilitythey proposed a modified equation
V119905119894 = 120596V119905minus1119894 + (119909119905119894 minus 119909lowast) 1198911198941205771 + (119909119905119894 minus 119909119905119896) 1198911198941205772 (7)
1205771 + 1205772 = 1 (8)
1205771 = 1 + (120577init minus 1) (itermax minus iter)119899(itermax)119899 (9)
where 119909119905119896 is a solution randomly chosen from the populationand 1205771 and 1205772 are learning factors ranging from 0 to 1 120577initis the initial value of 1205771 itermax is the maximal number ofiterations iter is the number of iterations and 119899 is a nonlinearindex
The second term in the modified search equation is afactor affecting the velocity of the solution with guidanceof the best solution 119909lowast which emphasizes exploitation Thethird term is another factor that affects the velocity withhelp of randomly selected solution 119909119905119896 which emphasizesexploration Effects of the best solution and the kth solutionare adjusted by changing the value of 1205771 Experiments resultsindicate that the modified algorithm can outperform BA inmost test functions
Xie et al [18] proposed an improved BA based on Dif-ferential Evolution operatorThe position updating equationswere defined as follows
119909119905+1119894 = 119909119905best + 1198911119894 (1199091199051199031 minus 1199091199051199032) + 1198912119894 (1199091199051199033 minus 1199091199051199034) (10)
where 119909119905best is the global best location in current population119909119905119903119894 (119894 = 1 2 3 4) is a randomly chosen solution in thepopulation 1198911119894 and 1198912119894 are defined as follows
1198911199051119894 = ((1198911min minus 1198911max) 119905119899119905 + 1198911max)1205731 (11)
1198911199052119894 = ((1198912min minus 1198912max) 119905119899119905 + 1198912max)1205732 (12)
where 1198911max and 1198911min are minimum and maximum fre-quency values of 1198911119894 while 1198912max and 1198912min are minimumandmaximum frequency values of1198912119894 119899119905 is a fixed parameter119905 is the number of iterations 120573 isin [0 1] is a random vectordrawing a uniform distribution The modified algorithm hasshown a better performance than classical BA
Inspired by Yilmaz and Zhoursquos methods we propose amodified location updating equation
119909119905+1119894 = 119909119905119894 + 1198911119894 (119909119905best minus 1199091199051199031) + 1198912119894 (1199091199051199032 minus 1199091199051199033) (13)
where 119909119905119903119894 (119894 = 1 2 3) are individuals randomly selected fromthe population 1198911119894 and 1198912119894 are frequencies
As in mutation operation of DE the frequency 119891 is animportant parameter that affects the convergence perfor-mance of the algorithm The value of 119891 changes according tothe predefined formula In basic BA and DLBA 119891 is definedin (2) (11) and (12) respectively Here we use a sinusoidalformulation [35] permitting certain flexibility in changing
the direction of 119891 The frequencies 1198911119894 and 1198912119894 are defined asfollows
1198911119894 = 12 (sin (2120587 lowast 120596 lowast 119905) lowast iter
itermax+ 1) (14)
1198912119894 = 12 (sin (2120587 lowast 120596 lowast 119905) lowast iter
itermax+ 1) (15)
where 120596 represents frequency of the sinusoidal function
32 Local Search Based on Levy Flight Levy Flight which isbased on the flight behavior of animals and insects has beenvariously studied in literatures [36] It is defined as a kind ofnon-Gauss stochastic process with a randomwalk whose stepsizes are based on Levy stable distribution
There are lots of versions of Levy distribution MantegnaAlgorithm [37] has been tested to be the most efficientmethod to generate Levy distribution values It has beenadopted in many evolution algorithms in order to escapefrom local minimum
When generating a new solution 1199091015840119894 for the 119894th solution 119909119894by performing Levy Flight the new candidate 1199091015840119894 is defined asfollows
1199091015840119894 = 119909119894 + 120572 oplus Levy (120582) (16)
where 120572 is a random step size parameter 120582 is a LevyFlight distribution parameter and oplus denotes entry wisemultiplication
In Mantegna Algorithm the step size 119904 can be defined asfollows
119904 = 120572 oplus Levy (120582) sim 001 119906|V|1120573 (119909119895 minus 119909best) (17)
where 119906 and V are obtained from normal distribution that is
119906 sim 119873 (0 1205902119906) V sim 119873(0 1205902V)
(18)
with
120590119906 = Γ (1 + 120573) sin (1205871205732)120573Γ [(1 + 120573) 2] 2(120573minus1)2
1120573
120590V = 1
(19)
Γ (119911) = int 119905119911minus1119890minus119911119889119911 (20)
The pseudocode of Levy Flight Search is shown inAlgorithm 2
33 Opposition Based Learning Opposition based learningwas proposed by Rahnamayan et al [38] in 2005 It has beensuccessfully used in machine learning and evolutionary com-puting [39] For an optimization problem if the candidatesolutions are near the global optima it is easy to find the idealsolutions with fast convergence speed But if the candidate
Scientific Programming 5
(1) Input the optimization function 119891(119909) and 120573(2) Select the individual 119909119894 which is chosen to be modified by Levy Flight(3) Compute 120590119906 using Eq (19)(4) Compute step size s using Eq (17) and Eq (18)(5) Generate a new candidate solution 1199091015840119894 using Eq (16)(6) Calculate119891(1199091015840)(7) if 119891(1199091015840119894 ) lt 119891(119909119894)(8) 119909119894 = 1199091015840119894 (9) end if
Algorithm 2 Pseudocode of Levy Flight Search
(1) Input original population 119875 population size 119878119873 the dimension of solution 119863 optimization function 119891(119909)(2) Generate opposite population OP as follows(3) for 119894 = 1 119878119873(4) for 119895 = 1 119863(5) 119909119894119895 = 119886119894 + 119887119894 minus 119909119894119895(6) end for(7) end for(8) Merge the original population 119875 and opposite population OP together(9) Calculate the fitness of population 119875OP(10) Choose the top half of population 119875OP as the current population
Algorithm 3 Opposition based optimization
solutions are far away from the global optima it will takemore computational efforts to get the required solutions Theconcept of OBL is to consider the candidate solutions andtheir opposite counterpart solutions simultaneously in anoptimization algorithm Then the greedy selection is appliedbetween them according to their objective function valuesWith the help ofOBL the probability of finding global optimais increased
In basic OBL let 119909 = (1199091 1199092 119909119863) be a119863-dimensionalsolution in the search space with component variables 119909119894 isin[119886119894 119887119894]Then the opposite point = (1 2 119863) is definedby its coordinates 1 2 119863 where
119894 = 119886119894 + 119887119894 minus 119909119894 119894 = 1 2 119863 (21)
OBL can be used not only to initialize the populationinitialization but also to improve the population diversityduring the evolutionary process Once the opposite popula-tion generated it is combinedwith the original population forselection Individuals with fitness ranking in the top half of allcandidate solutions will be selected as the current populationThe pseudocode of OBL is given in Algorithm 3
34 Proposed Modified Bat Algorithm Based on the basic BAand the above considerations we propose amodifiedmethodcalled OBMLBA It is clear that local search with LevyFlight can improve the ability of exploitation Incorporatingwith OBL strategy can enhance the population diversityInspired by DLBA the new location update equation takesadvantage of the information of both the global best solution
and randomly chosen solution near the candidate solutionSo the modified BA can get a good balance between theexploration and exploitation Themain steps of the algorithmare summarized in Algorithm 4
4 Experimental Results and Discussion
In order to evaluate the performance of the proposedalgorithm we select 16 standard benchmark functions toinvestigate the efficiency of the proposed approach Thesefunctions are presented in Table 1The function set comprisesunimodal functions and multimodal functions which canbe used to test the convergence speed and local prematureconvergence problem respectively 1198651sim1198655 are continuousunimodal functions 1198657 is a discontinuous step function1198659 is a noisy quartic function 11986510sim11986516 are multimodalfunctions with high dimension These functions are cate-gorized into low-dimension middle-dimension and high-dimension functions according to 119863 = 15 30 and 60respectively
Related parameters of the basic BADLBA andOBMLBAare presented in Table 2 The maximum number of iterationsis set to be 500 1000 and 2000 in case of119863 = 15 30 and 60
Experiments have been carried out on the test functionswith different dimensions The computational results aresummarized in Tables 3ndash5 in terms of the mean error andstandard deviation of the function error values In additionsome convergence characteristics graphs are shown in Fig-ure 1
6 Scientific Programming
(1) Input the parameters(2) Set iter = 1 Initialize the position and velocity of each individual in the population(3) Evaluate the objective function and related parameter values of each individual(4) While iter le iter max do(5) Apply Levy Flight Local Search strategy using Algorithm 2(6) Generate new solutions using Eq (13)(7) if rand(0 1) gt 119903119894(8) Apply the Opposition based learning using Algorithm 3(9) Generate a new local solution around the selected solution 119909lowast using Eq (5)(10) end if(11) if rand(0 1) lt 119860 119894 and 119891(119909119894new) lt 119891(119909119894)(12) update the solution loudness and pulse emission rate(13) end if(14) end while(15) Output the individual with the smallest objective function value in the population
Algorithm 4 Pseudocode of OBMLBA
Table 1 Benchmark problems
Fun Name Scope 119891min
F1 Sphere [minus100 100] 0F2 Elliptic [minus100 100] 0F3 SumSquare [minus10 10] 0F4 SumPower [minus1 1] 0F5 Schwefel222 [minus10 10] 0F6 Schwefel221 [minus100 100] 0F7 Step [minus100 100] 0F8 Exponential [minus128 128] 0F9 Quartic [minus128 128] 0F10 Rosenbrock [minus5 10] 0F11 Rastrigin [minus512 512] 0F12 Griewank [minus600 600] 0F13 Ackley [minus32 32] 0F14 Alpine [minus10 10] 0F15 Levy [minus10 10] 0F16 Weierstrass [minus05 05] 0
As seen from Tables 3ndash5 OBMLBA can get the globaloptimal on 1198911sim1198914 1198917 1198918 and 11989111sim11989113 with 119863 = 15 30and 60 On 1198916 and 11989114 the precision of solutions obtained byOBMLBA is smaller than the order of 1119864 minus 90 and the orderfor 1119864minus 100 respectively Results indicate that OBMLBA canget solutions extremely close to the global optimal values onmost of the functions
Solutions obtained by BAs are compared with resultsobtained by MLBA and OBMLBA As shown in Tables 3ndash5OBMLBA performs significantly better than basic BA LBAand DLBA on 1198911sim1198916 11989114 and 11989115 All algorithms except forbasic BA and LBA perform well on 1198917 1198918 11989111sim11989113 and 11989116OBMLBA is comparable to DLBA and MLBA on functions1198919 and 11989110
Compared with MLBA OBMLBA provides better per-formance on functions 1198911sim1198916 11989114 and 11989115 For 1198919 11989110 and
Table 2 Parameters setting
Parameters BA DLBA OBMLBAPopulation size119873 40Run time 20Loudness 095Pulse emission 085Factors updating 119860 and 120574 09Minimum of frequency 119891119898119894119899 0Maximum of frequency 119891max 1Factors updating frequency mdash mdash 03
11989115 results obtained by MLBA and OBMLBA are quite closeto each other Both MLBA and OBMLBA get the globaloptimum of 1198917 1198918 11989111sim11989113 and 11989116
From the convergence characteristic graphs it can beobviously observed that OBMLBA is the fastest algorithm inall the considered BAs MLBA is slower than OBMLBA butfaster than DLBA LBA and basic BA OBMLBA exhibits thebest accuracy and achieves the fastest convergence speedTheresults indeed indicate the advantage of themodified positionupdate equation and OBL strategy
On the whole algorithms proposed in this study such asMLBA andOBMLBAwork better than other BAs consideredAnd OBMLBA is more effective than MLBA
In Table 6 OBMLBA is compared with some state-of-the-art algorithms such as ABC PSO OCABC CLPSOand OLPSO-G Results of these algorithms are all deriveddirectly from their corresponding literatures [40] Resultsshow that OBMLBA has outperformed other methods on6 of 7 functions OBMLBA performs worse than ABC andOCABC only on Rosenbrock function
According to the analyses above it can be clearly observedthat OBMLBA provides outstanding performance with fastconvergence speed and high convergence accuracy on mostof the test functions
Scientific Programming 7
Table3Re
sults
ofcomparis
onso
fBAso
n15-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
119119864+
04184
119864+03
260119864+
03253
119864+03
445119864minus
54139
119864minus53
230119864minus
69707
119864minus69
36601
Eminus2
040
119865 2705
119864+07
304119864+
07189
119864+07
193119864+
07102
119864minus42
454119864minus
42396
119864minus67
125119864minus
66347
13Eminus2
020
119865 3810
119864+02
139119864+
02132
119864+02
225119864+
02377
119864minus52
169119864minus
51810
119864minus66
256119864minus
65621
79Eminus2
020
119865 4259
119864minus02
150119864minus
02164
119864minus03
235119864minus
03522
119864minus90
227119864minus
89457
119864minus126
12069119864
minus125
112Eminus3
050
119865 5408
119864+01
724119864+
00108
119864+01
143119864+
01373
119864minus28
145119864minus
27175
119864minus33
552119864minus
33117
Eminus1
04524
41Eminus1
04119865 6
501119864+
01316
119864+00
147119864+
01146
119864+01
304119864minus
27100
119864minus26
250119864minus
35785
119864minus35
668Eminus9
5298
692E
minus94
119865 7111
119864+04
213119864+
03136
119864+03
216119864+
030
00
00
0119865 8
152119864+
00443
119864minus01
126119864minus
01262
119864minus01
00
00
00
119865 9265
119864+00
113119864+
00134
119864minus01
270119864minus
01479
119864minus04
431119864minus
04399
119864minus04
282119864minus
04119
Eminus0
4100
Eminus0
4119865 10
589119864+
04202
119864+04
603119864+
02134
119864+03
135119864+
01210
119864minus01
129E+0
1371
Eminus0
1130
119864+01
340119864minus
01119865 11
131119864+
02816
119864+00
781119864+
01448
119864+01
00
00
00
119865 12106
119864+02
224119864+
01166
119864+01
227119864+
010
00
00
0119865 13
183119864+
01488
119864minus01
107119864+
01565
119864+00
00
00
00
119865 14142
119864+01
104119864+
00529
119864+00
526119864+
00182
119864minus27
792119864minus
27567
119864minus35
179119864minus
34891
Eminus1
06368
23Eminus1
05119865 15
152119864+
02113
119864+01
273119864+
01176
119864+01
747119864minus
01300
119864minus01
278Eminus0
1274
Eminus0
1483
119864minus01
526119864minus
01119865 16
168119864+
01160
119864+00
987119864+
00523
119864+00
00
00
00
8 Scientific Programming
Table4Re
sults
ofcomparis
onso
fBAso
n30-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
405119864+
04280
119864+03
380119864+
03680
119864+03
373119864minus
49167
119864minus48
180119864minus
117568
119864minus117
00
119865 2530
119864+08
115119864+
08954
119864+07
835119864+
07296
119864minus50
125119864minus
49320
119864minus117
101119864minus
1160
0119865 3
515119864+
03797
119864+02
729119864+
02121
119864+03
980119864minus
48438
119864minus47
272119864minus
123651
119864minus123
00
119865 4753
119864minus02
297119864minus
02580
119864minus03
139119864minus
02174
119864minus85
778119864minus
85176
119864minus198
00
0119865 5
233119864+
05428
119864+05
758119864+
01178
119864+02
972119864minus
26433
119864minus25
512119864minus
60162
119864minus59
680Eminus2
100
119865 6683
119864+01
350119864+
00176
119864+01
196119864+
01542
119864minus30
111119864minus
29263
119864minus55
833119864minus
55140
Eminus2
050
119865 7371
119864+04
426119864+
03569
119864+03
814119864+
030
00
00
0119865 8
239119864+
01619
119864+00
123119864+
00262
119864+00
00
00
00
119865 9402
119864+01
691119864+
00225
119864+00
519119864+
00285
119864minus04
283119864minus
04230
119864minus04
120119864minus
04625
Eminus0
5421
Eminus0
5119865 10
385119864+
05651
119864+04
259119864+
04615
119864+04
287119864+
01622
119864minus02
283119864+
01216
119864minus01
282E+0
1248
Eminus0
1119865 11
337119864+
02157
119864+01
213119864+
02734
119864+01
00
00
00
119865 12361
119864+02
339119864+
01374
119864+01
597119864+
010
00
00
0119865 13
196119864+
01271
119864minus01
656119864+
00525
119864+00
00
00
00
119865 14426
119864+01
274119864+
00155
119864+01
142119864+
01126
119864minus27
556119864minus
27291
119864minus60
910119864minus
60330
Eminus2
150
119865 15500
119864+02
897119864+
01696
119864+01
604119864+
01901
119864+00
307119864+
00578
119864+00
282119864+
00517
E+0
0240
E+0
0119865 16
413119864+
01117
119864+00
161119864+
01138
119864+01
00
00
00
Scientific Programming 9
Table5Re
sults
ofcomparis
onso
fBAso
n60
-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
103119864+
05937
119864+03
353119864+
03871
119864+03
454119864minus
49202
119864minus48
121119864minus
219000
119864+00
00
119865 2238
119864+09
251119864+
08607
119864+08
873119864+
08965
119864minus43
432119864minus
42161
119864minus217
000119864+
000
0119865 3
295119864+
04302
119864+03
124119864+
03203
119864+03
213119864minus
54566
119864minus54
819119864minus
224000
119864+00
00
119865 4147
119864minus01
425119864minus
02279
119864minus03
628119864minus
03205
119864minus88
918119864minus
88000
119864+00
00
0119865 5
100119864+
10000
119864+00
276119864+
01304
119864+01
478119864minus
24214
119864minus23
478119864minus
112151
119864minus111
00
119865 6821
119864+01
242119864+
00115
119864+01
153119864+
01321
119864minus28
705119864minus
28195
119864minus107
613119864minus
1070
0119865 7
983119864+
04872
119864+03
117119864+
04218
119864+04
00
00
00
119865 8546
119864+03
431119864+
03274
119864+01
812119864+
010
00
00
0119865 9
276119864+
02478
119864+01
255119864+
00794
119864+00
270119864minus
04254
119864minus04
156Eminus0
4775
Eminus0
5282
119864minus05
277119864minus
05119865 10
177119864+
06390
119864+05
180119864+
04347
119864+04
586119864+
01721
119864minus02
583119864+
01152
119864minus01
101E+0
1832
Eminus0
1119865 11
773119864+
02279
119864+01
309119864+
02270
119864+02
00
00
00
119865 12903
119864+02
121119864+
02372
119864+01
601119864+
010
00
00
0119865 13
203119864+
01671
119864minus02
636119864+
00572
119864+00
00
00
00
119865 14108
119864+02
442119864+
00506
119864+01
391119864+
01526
119864minus29
220119864minus
28646
119864minus104
204119864minus
1030
0119865 15
149119864+
03192
119864+02
274119864+
02297
119864+02
414119864+
01979
119864+00
242119864+
01661
119864+00
458Eminus0
2653
Eminus0
2119865 16
930119864+
01152
119864+00
335119864+
01268
119864+01
00
00
00
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
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Applied Computational Intelligence and Soft Computing
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ArtificialNeural Systems
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Industrial EngineeringJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
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Scientific Programming 3
(1) Input the parameters(2) Set iter = 1 Initialize the position and velocity of each individual in the population(3) Evaluate the objective function and related parameter values of each individual(4)While iter le iter max do(5) Generate new solutions using Eq (2)simEq (4)(6) if rand(0 1) gt 119903119894(7) Generate a new local solution around the selected solution 119909lowast using Eq (5)(8) end if(9) if rand(0 1) lt 119860 119894 and 119891(119909119894new) lt 119891(119909119894)(10) update the solution loudness and pulse emission rate by Eq (6)(11) end if(12) end while(13) Output the individual with the smallest objective function value in the population
Algorithm 1 Basic Bat Algorithm
2 Basic Bat Algorithm
The optimization problems considered in this paper aresingle-objective unconstrained continuous problems to beminimized
Bat Algorithm is a heuristic algorithm proposed by Yangin 2010 It is based on the echolocation capability of microbats guiding them on their foraging behavior In BA theposition of a bat represents a possible solution of the givenoptimization problemThe position of the food source foundby the 119894th bat can be expressed as 119909119894 = (1199091198941 1199091198942 119909119894119863)Thefitness of 119909119894 corresponds to the quality of the position the batlocates in
21 Initiation At the beginning bats do not know the loca-tion of food sources They generate a randomly distributedpopulation 119875 of119873 solutions where119873 denotes the number ofbats in the population Each solution can be produced withinthe search space as follows
119909119894119895 = 119909min + rand (0 1) (119909max minus 119909min) (1)
where 119894 = 1 2 119873 119895 = 1 2 119863 119909max and 119909min denotethe upper and lower bounds of the solution in dimension119895 respectively rand(0 1) is a uniformly distributed valuegenerated in the range of [0 1]22 Generation of New Solutions Bats modify the solutionbased on the information of the current location and the bestsource food location Bats navigate by adjusting their flyingdirections using their own and other swarm membersrsquo bestexperience to find the optimum of the problem
At this stage for each food source position 119909119894 a newposition was formulated as follows
119891119894 = 119891min + 120573 (119891max minus 119891min) (2)
V119905119894 = V119905minus1119894 + (119909119905119894 minus 119909lowast)119891119894 (3)
119909119905119894 = 119909119905minus1119894 + V119905119894 (4)
where 119894 represents each bat in the population 119894 = 1 2 119873and 119905 represents the 119905th iteration 119909119905119894 and V119905119894 are the position
and velocity components of the 119894th bat in the population at the119905th iteration 119891119894 denotes the pulse frequency that affects thevelocity of the 119894th bat 119891max and 119891min represent the maximumandminimum of119891119894 120573 is a random number between [0 1] 119909lowastis the best position found by the whole population
23 Local Search Once the new solutions are generated thelocal search is invoked by batsrsquo random walk If the pulseemission rate 119903 of the 119894th bat is smaller than a randomnumberselect 119909119894old from the population and generate a new position119909119894new as follows
119909119894new = 119909119894old + 120576119860119905 (5)
where119909119894old represents a solution chosen in current populationby some mechanism and 120576 is a random vector drawn from auniform distribution 119860119905 is the average loudness of all bats atthe time step 11990524 Solutions Loudness and Pulse Emission Rate UpdatingIf a random number is bigger than the loudness 119860119905 and119891(119909119894new) lt 119891(119909119894) accept the new solution 119909119894new At the sametime the loudness 119860119905 is decreased while its pulse emission 119903119894is increased as follows
119860119905+1119894 = 120572119860119905119894119903119905+1119894 = 1199030119894 (1 minus 119890120574119905)
(6)
where 120572 and 120574 are constants The initial loudness 1198600 andinitial pulse emission rate 1199030 are randomly generated numbersin the range of [1 2] and [0 1] respectively
The pseudocode of Bat Algorithm is presented in Algo-rithm 1
3 Enhanced Bat Algorithm
31 Location Update Equation Based on DE Yilmaz andKucuksille [23] pointed that the update equations of velocityin basic BA provide local search only with the guidance ofthe best-so-far solution whichmakes the BAeasily entrapped
4 Scientific Programming
into a local minimum In order to improve the search abilitythey proposed a modified equation
V119905119894 = 120596V119905minus1119894 + (119909119905119894 minus 119909lowast) 1198911198941205771 + (119909119905119894 minus 119909119905119896) 1198911198941205772 (7)
1205771 + 1205772 = 1 (8)
1205771 = 1 + (120577init minus 1) (itermax minus iter)119899(itermax)119899 (9)
where 119909119905119896 is a solution randomly chosen from the populationand 1205771 and 1205772 are learning factors ranging from 0 to 1 120577initis the initial value of 1205771 itermax is the maximal number ofiterations iter is the number of iterations and 119899 is a nonlinearindex
The second term in the modified search equation is afactor affecting the velocity of the solution with guidanceof the best solution 119909lowast which emphasizes exploitation Thethird term is another factor that affects the velocity withhelp of randomly selected solution 119909119905119896 which emphasizesexploration Effects of the best solution and the kth solutionare adjusted by changing the value of 1205771 Experiments resultsindicate that the modified algorithm can outperform BA inmost test functions
Xie et al [18] proposed an improved BA based on Dif-ferential Evolution operatorThe position updating equationswere defined as follows
119909119905+1119894 = 119909119905best + 1198911119894 (1199091199051199031 minus 1199091199051199032) + 1198912119894 (1199091199051199033 minus 1199091199051199034) (10)
where 119909119905best is the global best location in current population119909119905119903119894 (119894 = 1 2 3 4) is a randomly chosen solution in thepopulation 1198911119894 and 1198912119894 are defined as follows
1198911199051119894 = ((1198911min minus 1198911max) 119905119899119905 + 1198911max)1205731 (11)
1198911199052119894 = ((1198912min minus 1198912max) 119905119899119905 + 1198912max)1205732 (12)
where 1198911max and 1198911min are minimum and maximum fre-quency values of 1198911119894 while 1198912max and 1198912min are minimumandmaximum frequency values of1198912119894 119899119905 is a fixed parameter119905 is the number of iterations 120573 isin [0 1] is a random vectordrawing a uniform distribution The modified algorithm hasshown a better performance than classical BA
Inspired by Yilmaz and Zhoursquos methods we propose amodified location updating equation
119909119905+1119894 = 119909119905119894 + 1198911119894 (119909119905best minus 1199091199051199031) + 1198912119894 (1199091199051199032 minus 1199091199051199033) (13)
where 119909119905119903119894 (119894 = 1 2 3) are individuals randomly selected fromthe population 1198911119894 and 1198912119894 are frequencies
As in mutation operation of DE the frequency 119891 is animportant parameter that affects the convergence perfor-mance of the algorithm The value of 119891 changes according tothe predefined formula In basic BA and DLBA 119891 is definedin (2) (11) and (12) respectively Here we use a sinusoidalformulation [35] permitting certain flexibility in changing
the direction of 119891 The frequencies 1198911119894 and 1198912119894 are defined asfollows
1198911119894 = 12 (sin (2120587 lowast 120596 lowast 119905) lowast iter
itermax+ 1) (14)
1198912119894 = 12 (sin (2120587 lowast 120596 lowast 119905) lowast iter
itermax+ 1) (15)
where 120596 represents frequency of the sinusoidal function
32 Local Search Based on Levy Flight Levy Flight which isbased on the flight behavior of animals and insects has beenvariously studied in literatures [36] It is defined as a kind ofnon-Gauss stochastic process with a randomwalk whose stepsizes are based on Levy stable distribution
There are lots of versions of Levy distribution MantegnaAlgorithm [37] has been tested to be the most efficientmethod to generate Levy distribution values It has beenadopted in many evolution algorithms in order to escapefrom local minimum
When generating a new solution 1199091015840119894 for the 119894th solution 119909119894by performing Levy Flight the new candidate 1199091015840119894 is defined asfollows
1199091015840119894 = 119909119894 + 120572 oplus Levy (120582) (16)
where 120572 is a random step size parameter 120582 is a LevyFlight distribution parameter and oplus denotes entry wisemultiplication
In Mantegna Algorithm the step size 119904 can be defined asfollows
119904 = 120572 oplus Levy (120582) sim 001 119906|V|1120573 (119909119895 minus 119909best) (17)
where 119906 and V are obtained from normal distribution that is
119906 sim 119873 (0 1205902119906) V sim 119873(0 1205902V)
(18)
with
120590119906 = Γ (1 + 120573) sin (1205871205732)120573Γ [(1 + 120573) 2] 2(120573minus1)2
1120573
120590V = 1
(19)
Γ (119911) = int 119905119911minus1119890minus119911119889119911 (20)
The pseudocode of Levy Flight Search is shown inAlgorithm 2
33 Opposition Based Learning Opposition based learningwas proposed by Rahnamayan et al [38] in 2005 It has beensuccessfully used in machine learning and evolutionary com-puting [39] For an optimization problem if the candidatesolutions are near the global optima it is easy to find the idealsolutions with fast convergence speed But if the candidate
Scientific Programming 5
(1) Input the optimization function 119891(119909) and 120573(2) Select the individual 119909119894 which is chosen to be modified by Levy Flight(3) Compute 120590119906 using Eq (19)(4) Compute step size s using Eq (17) and Eq (18)(5) Generate a new candidate solution 1199091015840119894 using Eq (16)(6) Calculate119891(1199091015840)(7) if 119891(1199091015840119894 ) lt 119891(119909119894)(8) 119909119894 = 1199091015840119894 (9) end if
Algorithm 2 Pseudocode of Levy Flight Search
(1) Input original population 119875 population size 119878119873 the dimension of solution 119863 optimization function 119891(119909)(2) Generate opposite population OP as follows(3) for 119894 = 1 119878119873(4) for 119895 = 1 119863(5) 119909119894119895 = 119886119894 + 119887119894 minus 119909119894119895(6) end for(7) end for(8) Merge the original population 119875 and opposite population OP together(9) Calculate the fitness of population 119875OP(10) Choose the top half of population 119875OP as the current population
Algorithm 3 Opposition based optimization
solutions are far away from the global optima it will takemore computational efforts to get the required solutions Theconcept of OBL is to consider the candidate solutions andtheir opposite counterpart solutions simultaneously in anoptimization algorithm Then the greedy selection is appliedbetween them according to their objective function valuesWith the help ofOBL the probability of finding global optimais increased
In basic OBL let 119909 = (1199091 1199092 119909119863) be a119863-dimensionalsolution in the search space with component variables 119909119894 isin[119886119894 119887119894]Then the opposite point = (1 2 119863) is definedby its coordinates 1 2 119863 where
119894 = 119886119894 + 119887119894 minus 119909119894 119894 = 1 2 119863 (21)
OBL can be used not only to initialize the populationinitialization but also to improve the population diversityduring the evolutionary process Once the opposite popula-tion generated it is combinedwith the original population forselection Individuals with fitness ranking in the top half of allcandidate solutions will be selected as the current populationThe pseudocode of OBL is given in Algorithm 3
34 Proposed Modified Bat Algorithm Based on the basic BAand the above considerations we propose amodifiedmethodcalled OBMLBA It is clear that local search with LevyFlight can improve the ability of exploitation Incorporatingwith OBL strategy can enhance the population diversityInspired by DLBA the new location update equation takesadvantage of the information of both the global best solution
and randomly chosen solution near the candidate solutionSo the modified BA can get a good balance between theexploration and exploitation Themain steps of the algorithmare summarized in Algorithm 4
4 Experimental Results and Discussion
In order to evaluate the performance of the proposedalgorithm we select 16 standard benchmark functions toinvestigate the efficiency of the proposed approach Thesefunctions are presented in Table 1The function set comprisesunimodal functions and multimodal functions which canbe used to test the convergence speed and local prematureconvergence problem respectively 1198651sim1198655 are continuousunimodal functions 1198657 is a discontinuous step function1198659 is a noisy quartic function 11986510sim11986516 are multimodalfunctions with high dimension These functions are cate-gorized into low-dimension middle-dimension and high-dimension functions according to 119863 = 15 30 and 60respectively
Related parameters of the basic BADLBA andOBMLBAare presented in Table 2 The maximum number of iterationsis set to be 500 1000 and 2000 in case of119863 = 15 30 and 60
Experiments have been carried out on the test functionswith different dimensions The computational results aresummarized in Tables 3ndash5 in terms of the mean error andstandard deviation of the function error values In additionsome convergence characteristics graphs are shown in Fig-ure 1
6 Scientific Programming
(1) Input the parameters(2) Set iter = 1 Initialize the position and velocity of each individual in the population(3) Evaluate the objective function and related parameter values of each individual(4) While iter le iter max do(5) Apply Levy Flight Local Search strategy using Algorithm 2(6) Generate new solutions using Eq (13)(7) if rand(0 1) gt 119903119894(8) Apply the Opposition based learning using Algorithm 3(9) Generate a new local solution around the selected solution 119909lowast using Eq (5)(10) end if(11) if rand(0 1) lt 119860 119894 and 119891(119909119894new) lt 119891(119909119894)(12) update the solution loudness and pulse emission rate(13) end if(14) end while(15) Output the individual with the smallest objective function value in the population
Algorithm 4 Pseudocode of OBMLBA
Table 1 Benchmark problems
Fun Name Scope 119891min
F1 Sphere [minus100 100] 0F2 Elliptic [minus100 100] 0F3 SumSquare [minus10 10] 0F4 SumPower [minus1 1] 0F5 Schwefel222 [minus10 10] 0F6 Schwefel221 [minus100 100] 0F7 Step [minus100 100] 0F8 Exponential [minus128 128] 0F9 Quartic [minus128 128] 0F10 Rosenbrock [minus5 10] 0F11 Rastrigin [minus512 512] 0F12 Griewank [minus600 600] 0F13 Ackley [minus32 32] 0F14 Alpine [minus10 10] 0F15 Levy [minus10 10] 0F16 Weierstrass [minus05 05] 0
As seen from Tables 3ndash5 OBMLBA can get the globaloptimal on 1198911sim1198914 1198917 1198918 and 11989111sim11989113 with 119863 = 15 30and 60 On 1198916 and 11989114 the precision of solutions obtained byOBMLBA is smaller than the order of 1119864 minus 90 and the orderfor 1119864minus 100 respectively Results indicate that OBMLBA canget solutions extremely close to the global optimal values onmost of the functions
Solutions obtained by BAs are compared with resultsobtained by MLBA and OBMLBA As shown in Tables 3ndash5OBMLBA performs significantly better than basic BA LBAand DLBA on 1198911sim1198916 11989114 and 11989115 All algorithms except forbasic BA and LBA perform well on 1198917 1198918 11989111sim11989113 and 11989116OBMLBA is comparable to DLBA and MLBA on functions1198919 and 11989110
Compared with MLBA OBMLBA provides better per-formance on functions 1198911sim1198916 11989114 and 11989115 For 1198919 11989110 and
Table 2 Parameters setting
Parameters BA DLBA OBMLBAPopulation size119873 40Run time 20Loudness 095Pulse emission 085Factors updating 119860 and 120574 09Minimum of frequency 119891119898119894119899 0Maximum of frequency 119891max 1Factors updating frequency mdash mdash 03
11989115 results obtained by MLBA and OBMLBA are quite closeto each other Both MLBA and OBMLBA get the globaloptimum of 1198917 1198918 11989111sim11989113 and 11989116
From the convergence characteristic graphs it can beobviously observed that OBMLBA is the fastest algorithm inall the considered BAs MLBA is slower than OBMLBA butfaster than DLBA LBA and basic BA OBMLBA exhibits thebest accuracy and achieves the fastest convergence speedTheresults indeed indicate the advantage of themodified positionupdate equation and OBL strategy
On the whole algorithms proposed in this study such asMLBA andOBMLBAwork better than other BAs consideredAnd OBMLBA is more effective than MLBA
In Table 6 OBMLBA is compared with some state-of-the-art algorithms such as ABC PSO OCABC CLPSOand OLPSO-G Results of these algorithms are all deriveddirectly from their corresponding literatures [40] Resultsshow that OBMLBA has outperformed other methods on6 of 7 functions OBMLBA performs worse than ABC andOCABC only on Rosenbrock function
According to the analyses above it can be clearly observedthat OBMLBA provides outstanding performance with fastconvergence speed and high convergence accuracy on mostof the test functions
Scientific Programming 7
Table3Re
sults
ofcomparis
onso
fBAso
n15-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
119119864+
04184
119864+03
260119864+
03253
119864+03
445119864minus
54139
119864minus53
230119864minus
69707
119864minus69
36601
Eminus2
040
119865 2705
119864+07
304119864+
07189
119864+07
193119864+
07102
119864minus42
454119864minus
42396
119864minus67
125119864minus
66347
13Eminus2
020
119865 3810
119864+02
139119864+
02132
119864+02
225119864+
02377
119864minus52
169119864minus
51810
119864minus66
256119864minus
65621
79Eminus2
020
119865 4259
119864minus02
150119864minus
02164
119864minus03
235119864minus
03522
119864minus90
227119864minus
89457
119864minus126
12069119864
minus125
112Eminus3
050
119865 5408
119864+01
724119864+
00108
119864+01
143119864+
01373
119864minus28
145119864minus
27175
119864minus33
552119864minus
33117
Eminus1
04524
41Eminus1
04119865 6
501119864+
01316
119864+00
147119864+
01146
119864+01
304119864minus
27100
119864minus26
250119864minus
35785
119864minus35
668Eminus9
5298
692E
minus94
119865 7111
119864+04
213119864+
03136
119864+03
216119864+
030
00
00
0119865 8
152119864+
00443
119864minus01
126119864minus
01262
119864minus01
00
00
00
119865 9265
119864+00
113119864+
00134
119864minus01
270119864minus
01479
119864minus04
431119864minus
04399
119864minus04
282119864minus
04119
Eminus0
4100
Eminus0
4119865 10
589119864+
04202
119864+04
603119864+
02134
119864+03
135119864+
01210
119864minus01
129E+0
1371
Eminus0
1130
119864+01
340119864minus
01119865 11
131119864+
02816
119864+00
781119864+
01448
119864+01
00
00
00
119865 12106
119864+02
224119864+
01166
119864+01
227119864+
010
00
00
0119865 13
183119864+
01488
119864minus01
107119864+
01565
119864+00
00
00
00
119865 14142
119864+01
104119864+
00529
119864+00
526119864+
00182
119864minus27
792119864minus
27567
119864minus35
179119864minus
34891
Eminus1
06368
23Eminus1
05119865 15
152119864+
02113
119864+01
273119864+
01176
119864+01
747119864minus
01300
119864minus01
278Eminus0
1274
Eminus0
1483
119864minus01
526119864minus
01119865 16
168119864+
01160
119864+00
987119864+
00523
119864+00
00
00
00
8 Scientific Programming
Table4Re
sults
ofcomparis
onso
fBAso
n30-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
405119864+
04280
119864+03
380119864+
03680
119864+03
373119864minus
49167
119864minus48
180119864minus
117568
119864minus117
00
119865 2530
119864+08
115119864+
08954
119864+07
835119864+
07296
119864minus50
125119864minus
49320
119864minus117
101119864minus
1160
0119865 3
515119864+
03797
119864+02
729119864+
02121
119864+03
980119864minus
48438
119864minus47
272119864minus
123651
119864minus123
00
119865 4753
119864minus02
297119864minus
02580
119864minus03
139119864minus
02174
119864minus85
778119864minus
85176
119864minus198
00
0119865 5
233119864+
05428
119864+05
758119864+
01178
119864+02
972119864minus
26433
119864minus25
512119864minus
60162
119864minus59
680Eminus2
100
119865 6683
119864+01
350119864+
00176
119864+01
196119864+
01542
119864minus30
111119864minus
29263
119864minus55
833119864minus
55140
Eminus2
050
119865 7371
119864+04
426119864+
03569
119864+03
814119864+
030
00
00
0119865 8
239119864+
01619
119864+00
123119864+
00262
119864+00
00
00
00
119865 9402
119864+01
691119864+
00225
119864+00
519119864+
00285
119864minus04
283119864minus
04230
119864minus04
120119864minus
04625
Eminus0
5421
Eminus0
5119865 10
385119864+
05651
119864+04
259119864+
04615
119864+04
287119864+
01622
119864minus02
283119864+
01216
119864minus01
282E+0
1248
Eminus0
1119865 11
337119864+
02157
119864+01
213119864+
02734
119864+01
00
00
00
119865 12361
119864+02
339119864+
01374
119864+01
597119864+
010
00
00
0119865 13
196119864+
01271
119864minus01
656119864+
00525
119864+00
00
00
00
119865 14426
119864+01
274119864+
00155
119864+01
142119864+
01126
119864minus27
556119864minus
27291
119864minus60
910119864minus
60330
Eminus2
150
119865 15500
119864+02
897119864+
01696
119864+01
604119864+
01901
119864+00
307119864+
00578
119864+00
282119864+
00517
E+0
0240
E+0
0119865 16
413119864+
01117
119864+00
161119864+
01138
119864+01
00
00
00
Scientific Programming 9
Table5Re
sults
ofcomparis
onso
fBAso
n60
-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
103119864+
05937
119864+03
353119864+
03871
119864+03
454119864minus
49202
119864minus48
121119864minus
219000
119864+00
00
119865 2238
119864+09
251119864+
08607
119864+08
873119864+
08965
119864minus43
432119864minus
42161
119864minus217
000119864+
000
0119865 3
295119864+
04302
119864+03
124119864+
03203
119864+03
213119864minus
54566
119864minus54
819119864minus
224000
119864+00
00
119865 4147
119864minus01
425119864minus
02279
119864minus03
628119864minus
03205
119864minus88
918119864minus
88000
119864+00
00
0119865 5
100119864+
10000
119864+00
276119864+
01304
119864+01
478119864minus
24214
119864minus23
478119864minus
112151
119864minus111
00
119865 6821
119864+01
242119864+
00115
119864+01
153119864+
01321
119864minus28
705119864minus
28195
119864minus107
613119864minus
1070
0119865 7
983119864+
04872
119864+03
117119864+
04218
119864+04
00
00
00
119865 8546
119864+03
431119864+
03274
119864+01
812119864+
010
00
00
0119865 9
276119864+
02478
119864+01
255119864+
00794
119864+00
270119864minus
04254
119864minus04
156Eminus0
4775
Eminus0
5282
119864minus05
277119864minus
05119865 10
177119864+
06390
119864+05
180119864+
04347
119864+04
586119864+
01721
119864minus02
583119864+
01152
119864minus01
101E+0
1832
Eminus0
1119865 11
773119864+
02279
119864+01
309119864+
02270
119864+02
00
00
00
119865 12903
119864+02
121119864+
02372
119864+01
601119864+
010
00
00
0119865 13
203119864+
01671
119864minus02
636119864+
00572
119864+00
00
00
00
119865 14108
119864+02
442119864+
00506
119864+01
391119864+
01526
119864minus29
220119864minus
28646
119864minus104
204119864minus
1030
0119865 15
149119864+
03192
119864+02
274119864+
02297
119864+02
414119864+
01979
119864+00
242119864+
01661
119864+00
458Eminus0
2653
Eminus0
2119865 16
930119864+
01152
119864+00
335119864+
01268
119864+01
00
00
00
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
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Distributed Sensor Networks
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Volume 2014
International Journal of
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Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
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RoboticsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
4 Scientific Programming
into a local minimum In order to improve the search abilitythey proposed a modified equation
V119905119894 = 120596V119905minus1119894 + (119909119905119894 minus 119909lowast) 1198911198941205771 + (119909119905119894 minus 119909119905119896) 1198911198941205772 (7)
1205771 + 1205772 = 1 (8)
1205771 = 1 + (120577init minus 1) (itermax minus iter)119899(itermax)119899 (9)
where 119909119905119896 is a solution randomly chosen from the populationand 1205771 and 1205772 are learning factors ranging from 0 to 1 120577initis the initial value of 1205771 itermax is the maximal number ofiterations iter is the number of iterations and 119899 is a nonlinearindex
The second term in the modified search equation is afactor affecting the velocity of the solution with guidanceof the best solution 119909lowast which emphasizes exploitation Thethird term is another factor that affects the velocity withhelp of randomly selected solution 119909119905119896 which emphasizesexploration Effects of the best solution and the kth solutionare adjusted by changing the value of 1205771 Experiments resultsindicate that the modified algorithm can outperform BA inmost test functions
Xie et al [18] proposed an improved BA based on Dif-ferential Evolution operatorThe position updating equationswere defined as follows
119909119905+1119894 = 119909119905best + 1198911119894 (1199091199051199031 minus 1199091199051199032) + 1198912119894 (1199091199051199033 minus 1199091199051199034) (10)
where 119909119905best is the global best location in current population119909119905119903119894 (119894 = 1 2 3 4) is a randomly chosen solution in thepopulation 1198911119894 and 1198912119894 are defined as follows
1198911199051119894 = ((1198911min minus 1198911max) 119905119899119905 + 1198911max)1205731 (11)
1198911199052119894 = ((1198912min minus 1198912max) 119905119899119905 + 1198912max)1205732 (12)
where 1198911max and 1198911min are minimum and maximum fre-quency values of 1198911119894 while 1198912max and 1198912min are minimumandmaximum frequency values of1198912119894 119899119905 is a fixed parameter119905 is the number of iterations 120573 isin [0 1] is a random vectordrawing a uniform distribution The modified algorithm hasshown a better performance than classical BA
Inspired by Yilmaz and Zhoursquos methods we propose amodified location updating equation
119909119905+1119894 = 119909119905119894 + 1198911119894 (119909119905best minus 1199091199051199031) + 1198912119894 (1199091199051199032 minus 1199091199051199033) (13)
where 119909119905119903119894 (119894 = 1 2 3) are individuals randomly selected fromthe population 1198911119894 and 1198912119894 are frequencies
As in mutation operation of DE the frequency 119891 is animportant parameter that affects the convergence perfor-mance of the algorithm The value of 119891 changes according tothe predefined formula In basic BA and DLBA 119891 is definedin (2) (11) and (12) respectively Here we use a sinusoidalformulation [35] permitting certain flexibility in changing
the direction of 119891 The frequencies 1198911119894 and 1198912119894 are defined asfollows
1198911119894 = 12 (sin (2120587 lowast 120596 lowast 119905) lowast iter
itermax+ 1) (14)
1198912119894 = 12 (sin (2120587 lowast 120596 lowast 119905) lowast iter
itermax+ 1) (15)
where 120596 represents frequency of the sinusoidal function
32 Local Search Based on Levy Flight Levy Flight which isbased on the flight behavior of animals and insects has beenvariously studied in literatures [36] It is defined as a kind ofnon-Gauss stochastic process with a randomwalk whose stepsizes are based on Levy stable distribution
There are lots of versions of Levy distribution MantegnaAlgorithm [37] has been tested to be the most efficientmethod to generate Levy distribution values It has beenadopted in many evolution algorithms in order to escapefrom local minimum
When generating a new solution 1199091015840119894 for the 119894th solution 119909119894by performing Levy Flight the new candidate 1199091015840119894 is defined asfollows
1199091015840119894 = 119909119894 + 120572 oplus Levy (120582) (16)
where 120572 is a random step size parameter 120582 is a LevyFlight distribution parameter and oplus denotes entry wisemultiplication
In Mantegna Algorithm the step size 119904 can be defined asfollows
119904 = 120572 oplus Levy (120582) sim 001 119906|V|1120573 (119909119895 minus 119909best) (17)
where 119906 and V are obtained from normal distribution that is
119906 sim 119873 (0 1205902119906) V sim 119873(0 1205902V)
(18)
with
120590119906 = Γ (1 + 120573) sin (1205871205732)120573Γ [(1 + 120573) 2] 2(120573minus1)2
1120573
120590V = 1
(19)
Γ (119911) = int 119905119911minus1119890minus119911119889119911 (20)
The pseudocode of Levy Flight Search is shown inAlgorithm 2
33 Opposition Based Learning Opposition based learningwas proposed by Rahnamayan et al [38] in 2005 It has beensuccessfully used in machine learning and evolutionary com-puting [39] For an optimization problem if the candidatesolutions are near the global optima it is easy to find the idealsolutions with fast convergence speed But if the candidate
Scientific Programming 5
(1) Input the optimization function 119891(119909) and 120573(2) Select the individual 119909119894 which is chosen to be modified by Levy Flight(3) Compute 120590119906 using Eq (19)(4) Compute step size s using Eq (17) and Eq (18)(5) Generate a new candidate solution 1199091015840119894 using Eq (16)(6) Calculate119891(1199091015840)(7) if 119891(1199091015840119894 ) lt 119891(119909119894)(8) 119909119894 = 1199091015840119894 (9) end if
Algorithm 2 Pseudocode of Levy Flight Search
(1) Input original population 119875 population size 119878119873 the dimension of solution 119863 optimization function 119891(119909)(2) Generate opposite population OP as follows(3) for 119894 = 1 119878119873(4) for 119895 = 1 119863(5) 119909119894119895 = 119886119894 + 119887119894 minus 119909119894119895(6) end for(7) end for(8) Merge the original population 119875 and opposite population OP together(9) Calculate the fitness of population 119875OP(10) Choose the top half of population 119875OP as the current population
Algorithm 3 Opposition based optimization
solutions are far away from the global optima it will takemore computational efforts to get the required solutions Theconcept of OBL is to consider the candidate solutions andtheir opposite counterpart solutions simultaneously in anoptimization algorithm Then the greedy selection is appliedbetween them according to their objective function valuesWith the help ofOBL the probability of finding global optimais increased
In basic OBL let 119909 = (1199091 1199092 119909119863) be a119863-dimensionalsolution in the search space with component variables 119909119894 isin[119886119894 119887119894]Then the opposite point = (1 2 119863) is definedby its coordinates 1 2 119863 where
119894 = 119886119894 + 119887119894 minus 119909119894 119894 = 1 2 119863 (21)
OBL can be used not only to initialize the populationinitialization but also to improve the population diversityduring the evolutionary process Once the opposite popula-tion generated it is combinedwith the original population forselection Individuals with fitness ranking in the top half of allcandidate solutions will be selected as the current populationThe pseudocode of OBL is given in Algorithm 3
34 Proposed Modified Bat Algorithm Based on the basic BAand the above considerations we propose amodifiedmethodcalled OBMLBA It is clear that local search with LevyFlight can improve the ability of exploitation Incorporatingwith OBL strategy can enhance the population diversityInspired by DLBA the new location update equation takesadvantage of the information of both the global best solution
and randomly chosen solution near the candidate solutionSo the modified BA can get a good balance between theexploration and exploitation Themain steps of the algorithmare summarized in Algorithm 4
4 Experimental Results and Discussion
In order to evaluate the performance of the proposedalgorithm we select 16 standard benchmark functions toinvestigate the efficiency of the proposed approach Thesefunctions are presented in Table 1The function set comprisesunimodal functions and multimodal functions which canbe used to test the convergence speed and local prematureconvergence problem respectively 1198651sim1198655 are continuousunimodal functions 1198657 is a discontinuous step function1198659 is a noisy quartic function 11986510sim11986516 are multimodalfunctions with high dimension These functions are cate-gorized into low-dimension middle-dimension and high-dimension functions according to 119863 = 15 30 and 60respectively
Related parameters of the basic BADLBA andOBMLBAare presented in Table 2 The maximum number of iterationsis set to be 500 1000 and 2000 in case of119863 = 15 30 and 60
Experiments have been carried out on the test functionswith different dimensions The computational results aresummarized in Tables 3ndash5 in terms of the mean error andstandard deviation of the function error values In additionsome convergence characteristics graphs are shown in Fig-ure 1
6 Scientific Programming
(1) Input the parameters(2) Set iter = 1 Initialize the position and velocity of each individual in the population(3) Evaluate the objective function and related parameter values of each individual(4) While iter le iter max do(5) Apply Levy Flight Local Search strategy using Algorithm 2(6) Generate new solutions using Eq (13)(7) if rand(0 1) gt 119903119894(8) Apply the Opposition based learning using Algorithm 3(9) Generate a new local solution around the selected solution 119909lowast using Eq (5)(10) end if(11) if rand(0 1) lt 119860 119894 and 119891(119909119894new) lt 119891(119909119894)(12) update the solution loudness and pulse emission rate(13) end if(14) end while(15) Output the individual with the smallest objective function value in the population
Algorithm 4 Pseudocode of OBMLBA
Table 1 Benchmark problems
Fun Name Scope 119891min
F1 Sphere [minus100 100] 0F2 Elliptic [minus100 100] 0F3 SumSquare [minus10 10] 0F4 SumPower [minus1 1] 0F5 Schwefel222 [minus10 10] 0F6 Schwefel221 [minus100 100] 0F7 Step [minus100 100] 0F8 Exponential [minus128 128] 0F9 Quartic [minus128 128] 0F10 Rosenbrock [minus5 10] 0F11 Rastrigin [minus512 512] 0F12 Griewank [minus600 600] 0F13 Ackley [minus32 32] 0F14 Alpine [minus10 10] 0F15 Levy [minus10 10] 0F16 Weierstrass [minus05 05] 0
As seen from Tables 3ndash5 OBMLBA can get the globaloptimal on 1198911sim1198914 1198917 1198918 and 11989111sim11989113 with 119863 = 15 30and 60 On 1198916 and 11989114 the precision of solutions obtained byOBMLBA is smaller than the order of 1119864 minus 90 and the orderfor 1119864minus 100 respectively Results indicate that OBMLBA canget solutions extremely close to the global optimal values onmost of the functions
Solutions obtained by BAs are compared with resultsobtained by MLBA and OBMLBA As shown in Tables 3ndash5OBMLBA performs significantly better than basic BA LBAand DLBA on 1198911sim1198916 11989114 and 11989115 All algorithms except forbasic BA and LBA perform well on 1198917 1198918 11989111sim11989113 and 11989116OBMLBA is comparable to DLBA and MLBA on functions1198919 and 11989110
Compared with MLBA OBMLBA provides better per-formance on functions 1198911sim1198916 11989114 and 11989115 For 1198919 11989110 and
Table 2 Parameters setting
Parameters BA DLBA OBMLBAPopulation size119873 40Run time 20Loudness 095Pulse emission 085Factors updating 119860 and 120574 09Minimum of frequency 119891119898119894119899 0Maximum of frequency 119891max 1Factors updating frequency mdash mdash 03
11989115 results obtained by MLBA and OBMLBA are quite closeto each other Both MLBA and OBMLBA get the globaloptimum of 1198917 1198918 11989111sim11989113 and 11989116
From the convergence characteristic graphs it can beobviously observed that OBMLBA is the fastest algorithm inall the considered BAs MLBA is slower than OBMLBA butfaster than DLBA LBA and basic BA OBMLBA exhibits thebest accuracy and achieves the fastest convergence speedTheresults indeed indicate the advantage of themodified positionupdate equation and OBL strategy
On the whole algorithms proposed in this study such asMLBA andOBMLBAwork better than other BAs consideredAnd OBMLBA is more effective than MLBA
In Table 6 OBMLBA is compared with some state-of-the-art algorithms such as ABC PSO OCABC CLPSOand OLPSO-G Results of these algorithms are all deriveddirectly from their corresponding literatures [40] Resultsshow that OBMLBA has outperformed other methods on6 of 7 functions OBMLBA performs worse than ABC andOCABC only on Rosenbrock function
According to the analyses above it can be clearly observedthat OBMLBA provides outstanding performance with fastconvergence speed and high convergence accuracy on mostof the test functions
Scientific Programming 7
Table3Re
sults
ofcomparis
onso
fBAso
n15-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
119119864+
04184
119864+03
260119864+
03253
119864+03
445119864minus
54139
119864minus53
230119864minus
69707
119864minus69
36601
Eminus2
040
119865 2705
119864+07
304119864+
07189
119864+07
193119864+
07102
119864minus42
454119864minus
42396
119864minus67
125119864minus
66347
13Eminus2
020
119865 3810
119864+02
139119864+
02132
119864+02
225119864+
02377
119864minus52
169119864minus
51810
119864minus66
256119864minus
65621
79Eminus2
020
119865 4259
119864minus02
150119864minus
02164
119864minus03
235119864minus
03522
119864minus90
227119864minus
89457
119864minus126
12069119864
minus125
112Eminus3
050
119865 5408
119864+01
724119864+
00108
119864+01
143119864+
01373
119864minus28
145119864minus
27175
119864minus33
552119864minus
33117
Eminus1
04524
41Eminus1
04119865 6
501119864+
01316
119864+00
147119864+
01146
119864+01
304119864minus
27100
119864minus26
250119864minus
35785
119864minus35
668Eminus9
5298
692E
minus94
119865 7111
119864+04
213119864+
03136
119864+03
216119864+
030
00
00
0119865 8
152119864+
00443
119864minus01
126119864minus
01262
119864minus01
00
00
00
119865 9265
119864+00
113119864+
00134
119864minus01
270119864minus
01479
119864minus04
431119864minus
04399
119864minus04
282119864minus
04119
Eminus0
4100
Eminus0
4119865 10
589119864+
04202
119864+04
603119864+
02134
119864+03
135119864+
01210
119864minus01
129E+0
1371
Eminus0
1130
119864+01
340119864minus
01119865 11
131119864+
02816
119864+00
781119864+
01448
119864+01
00
00
00
119865 12106
119864+02
224119864+
01166
119864+01
227119864+
010
00
00
0119865 13
183119864+
01488
119864minus01
107119864+
01565
119864+00
00
00
00
119865 14142
119864+01
104119864+
00529
119864+00
526119864+
00182
119864minus27
792119864minus
27567
119864minus35
179119864minus
34891
Eminus1
06368
23Eminus1
05119865 15
152119864+
02113
119864+01
273119864+
01176
119864+01
747119864minus
01300
119864minus01
278Eminus0
1274
Eminus0
1483
119864minus01
526119864minus
01119865 16
168119864+
01160
119864+00
987119864+
00523
119864+00
00
00
00
8 Scientific Programming
Table4Re
sults
ofcomparis
onso
fBAso
n30-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
405119864+
04280
119864+03
380119864+
03680
119864+03
373119864minus
49167
119864minus48
180119864minus
117568
119864minus117
00
119865 2530
119864+08
115119864+
08954
119864+07
835119864+
07296
119864minus50
125119864minus
49320
119864minus117
101119864minus
1160
0119865 3
515119864+
03797
119864+02
729119864+
02121
119864+03
980119864minus
48438
119864minus47
272119864minus
123651
119864minus123
00
119865 4753
119864minus02
297119864minus
02580
119864minus03
139119864minus
02174
119864minus85
778119864minus
85176
119864minus198
00
0119865 5
233119864+
05428
119864+05
758119864+
01178
119864+02
972119864minus
26433
119864minus25
512119864minus
60162
119864minus59
680Eminus2
100
119865 6683
119864+01
350119864+
00176
119864+01
196119864+
01542
119864minus30
111119864minus
29263
119864minus55
833119864minus
55140
Eminus2
050
119865 7371
119864+04
426119864+
03569
119864+03
814119864+
030
00
00
0119865 8
239119864+
01619
119864+00
123119864+
00262
119864+00
00
00
00
119865 9402
119864+01
691119864+
00225
119864+00
519119864+
00285
119864minus04
283119864minus
04230
119864minus04
120119864minus
04625
Eminus0
5421
Eminus0
5119865 10
385119864+
05651
119864+04
259119864+
04615
119864+04
287119864+
01622
119864minus02
283119864+
01216
119864minus01
282E+0
1248
Eminus0
1119865 11
337119864+
02157
119864+01
213119864+
02734
119864+01
00
00
00
119865 12361
119864+02
339119864+
01374
119864+01
597119864+
010
00
00
0119865 13
196119864+
01271
119864minus01
656119864+
00525
119864+00
00
00
00
119865 14426
119864+01
274119864+
00155
119864+01
142119864+
01126
119864minus27
556119864minus
27291
119864minus60
910119864minus
60330
Eminus2
150
119865 15500
119864+02
897119864+
01696
119864+01
604119864+
01901
119864+00
307119864+
00578
119864+00
282119864+
00517
E+0
0240
E+0
0119865 16
413119864+
01117
119864+00
161119864+
01138
119864+01
00
00
00
Scientific Programming 9
Table5Re
sults
ofcomparis
onso
fBAso
n60
-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
103119864+
05937
119864+03
353119864+
03871
119864+03
454119864minus
49202
119864minus48
121119864minus
219000
119864+00
00
119865 2238
119864+09
251119864+
08607
119864+08
873119864+
08965
119864minus43
432119864minus
42161
119864minus217
000119864+
000
0119865 3
295119864+
04302
119864+03
124119864+
03203
119864+03
213119864minus
54566
119864minus54
819119864minus
224000
119864+00
00
119865 4147
119864minus01
425119864minus
02279
119864minus03
628119864minus
03205
119864minus88
918119864minus
88000
119864+00
00
0119865 5
100119864+
10000
119864+00
276119864+
01304
119864+01
478119864minus
24214
119864minus23
478119864minus
112151
119864minus111
00
119865 6821
119864+01
242119864+
00115
119864+01
153119864+
01321
119864minus28
705119864minus
28195
119864minus107
613119864minus
1070
0119865 7
983119864+
04872
119864+03
117119864+
04218
119864+04
00
00
00
119865 8546
119864+03
431119864+
03274
119864+01
812119864+
010
00
00
0119865 9
276119864+
02478
119864+01
255119864+
00794
119864+00
270119864minus
04254
119864minus04
156Eminus0
4775
Eminus0
5282
119864minus05
277119864minus
05119865 10
177119864+
06390
119864+05
180119864+
04347
119864+04
586119864+
01721
119864minus02
583119864+
01152
119864minus01
101E+0
1832
Eminus0
1119865 11
773119864+
02279
119864+01
309119864+
02270
119864+02
00
00
00
119865 12903
119864+02
121119864+
02372
119864+01
601119864+
010
00
00
0119865 13
203119864+
01671
119864minus02
636119864+
00572
119864+00
00
00
00
119865 14108
119864+02
442119864+
00506
119864+01
391119864+
01526
119864minus29
220119864minus
28646
119864minus104
204119864minus
1030
0119865 15
149119864+
03192
119864+02
274119864+
02297
119864+02
414119864+
01979
119864+00
242119864+
01661
119864+00
458Eminus0
2653
Eminus0
2119865 16
930119864+
01152
119864+00
335119864+
01268
119864+01
00
00
00
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
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Applied Computational Intelligence and Soft Computing
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Human-ComputerInteraction
Advances in
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Scientific Programming 5
(1) Input the optimization function 119891(119909) and 120573(2) Select the individual 119909119894 which is chosen to be modified by Levy Flight(3) Compute 120590119906 using Eq (19)(4) Compute step size s using Eq (17) and Eq (18)(5) Generate a new candidate solution 1199091015840119894 using Eq (16)(6) Calculate119891(1199091015840)(7) if 119891(1199091015840119894 ) lt 119891(119909119894)(8) 119909119894 = 1199091015840119894 (9) end if
Algorithm 2 Pseudocode of Levy Flight Search
(1) Input original population 119875 population size 119878119873 the dimension of solution 119863 optimization function 119891(119909)(2) Generate opposite population OP as follows(3) for 119894 = 1 119878119873(4) for 119895 = 1 119863(5) 119909119894119895 = 119886119894 + 119887119894 minus 119909119894119895(6) end for(7) end for(8) Merge the original population 119875 and opposite population OP together(9) Calculate the fitness of population 119875OP(10) Choose the top half of population 119875OP as the current population
Algorithm 3 Opposition based optimization
solutions are far away from the global optima it will takemore computational efforts to get the required solutions Theconcept of OBL is to consider the candidate solutions andtheir opposite counterpart solutions simultaneously in anoptimization algorithm Then the greedy selection is appliedbetween them according to their objective function valuesWith the help ofOBL the probability of finding global optimais increased
In basic OBL let 119909 = (1199091 1199092 119909119863) be a119863-dimensionalsolution in the search space with component variables 119909119894 isin[119886119894 119887119894]Then the opposite point = (1 2 119863) is definedby its coordinates 1 2 119863 where
119894 = 119886119894 + 119887119894 minus 119909119894 119894 = 1 2 119863 (21)
OBL can be used not only to initialize the populationinitialization but also to improve the population diversityduring the evolutionary process Once the opposite popula-tion generated it is combinedwith the original population forselection Individuals with fitness ranking in the top half of allcandidate solutions will be selected as the current populationThe pseudocode of OBL is given in Algorithm 3
34 Proposed Modified Bat Algorithm Based on the basic BAand the above considerations we propose amodifiedmethodcalled OBMLBA It is clear that local search with LevyFlight can improve the ability of exploitation Incorporatingwith OBL strategy can enhance the population diversityInspired by DLBA the new location update equation takesadvantage of the information of both the global best solution
and randomly chosen solution near the candidate solutionSo the modified BA can get a good balance between theexploration and exploitation Themain steps of the algorithmare summarized in Algorithm 4
4 Experimental Results and Discussion
In order to evaluate the performance of the proposedalgorithm we select 16 standard benchmark functions toinvestigate the efficiency of the proposed approach Thesefunctions are presented in Table 1The function set comprisesunimodal functions and multimodal functions which canbe used to test the convergence speed and local prematureconvergence problem respectively 1198651sim1198655 are continuousunimodal functions 1198657 is a discontinuous step function1198659 is a noisy quartic function 11986510sim11986516 are multimodalfunctions with high dimension These functions are cate-gorized into low-dimension middle-dimension and high-dimension functions according to 119863 = 15 30 and 60respectively
Related parameters of the basic BADLBA andOBMLBAare presented in Table 2 The maximum number of iterationsis set to be 500 1000 and 2000 in case of119863 = 15 30 and 60
Experiments have been carried out on the test functionswith different dimensions The computational results aresummarized in Tables 3ndash5 in terms of the mean error andstandard deviation of the function error values In additionsome convergence characteristics graphs are shown in Fig-ure 1
6 Scientific Programming
(1) Input the parameters(2) Set iter = 1 Initialize the position and velocity of each individual in the population(3) Evaluate the objective function and related parameter values of each individual(4) While iter le iter max do(5) Apply Levy Flight Local Search strategy using Algorithm 2(6) Generate new solutions using Eq (13)(7) if rand(0 1) gt 119903119894(8) Apply the Opposition based learning using Algorithm 3(9) Generate a new local solution around the selected solution 119909lowast using Eq (5)(10) end if(11) if rand(0 1) lt 119860 119894 and 119891(119909119894new) lt 119891(119909119894)(12) update the solution loudness and pulse emission rate(13) end if(14) end while(15) Output the individual with the smallest objective function value in the population
Algorithm 4 Pseudocode of OBMLBA
Table 1 Benchmark problems
Fun Name Scope 119891min
F1 Sphere [minus100 100] 0F2 Elliptic [minus100 100] 0F3 SumSquare [minus10 10] 0F4 SumPower [minus1 1] 0F5 Schwefel222 [minus10 10] 0F6 Schwefel221 [minus100 100] 0F7 Step [minus100 100] 0F8 Exponential [minus128 128] 0F9 Quartic [minus128 128] 0F10 Rosenbrock [minus5 10] 0F11 Rastrigin [minus512 512] 0F12 Griewank [minus600 600] 0F13 Ackley [minus32 32] 0F14 Alpine [minus10 10] 0F15 Levy [minus10 10] 0F16 Weierstrass [minus05 05] 0
As seen from Tables 3ndash5 OBMLBA can get the globaloptimal on 1198911sim1198914 1198917 1198918 and 11989111sim11989113 with 119863 = 15 30and 60 On 1198916 and 11989114 the precision of solutions obtained byOBMLBA is smaller than the order of 1119864 minus 90 and the orderfor 1119864minus 100 respectively Results indicate that OBMLBA canget solutions extremely close to the global optimal values onmost of the functions
Solutions obtained by BAs are compared with resultsobtained by MLBA and OBMLBA As shown in Tables 3ndash5OBMLBA performs significantly better than basic BA LBAand DLBA on 1198911sim1198916 11989114 and 11989115 All algorithms except forbasic BA and LBA perform well on 1198917 1198918 11989111sim11989113 and 11989116OBMLBA is comparable to DLBA and MLBA on functions1198919 and 11989110
Compared with MLBA OBMLBA provides better per-formance on functions 1198911sim1198916 11989114 and 11989115 For 1198919 11989110 and
Table 2 Parameters setting
Parameters BA DLBA OBMLBAPopulation size119873 40Run time 20Loudness 095Pulse emission 085Factors updating 119860 and 120574 09Minimum of frequency 119891119898119894119899 0Maximum of frequency 119891max 1Factors updating frequency mdash mdash 03
11989115 results obtained by MLBA and OBMLBA are quite closeto each other Both MLBA and OBMLBA get the globaloptimum of 1198917 1198918 11989111sim11989113 and 11989116
From the convergence characteristic graphs it can beobviously observed that OBMLBA is the fastest algorithm inall the considered BAs MLBA is slower than OBMLBA butfaster than DLBA LBA and basic BA OBMLBA exhibits thebest accuracy and achieves the fastest convergence speedTheresults indeed indicate the advantage of themodified positionupdate equation and OBL strategy
On the whole algorithms proposed in this study such asMLBA andOBMLBAwork better than other BAs consideredAnd OBMLBA is more effective than MLBA
In Table 6 OBMLBA is compared with some state-of-the-art algorithms such as ABC PSO OCABC CLPSOand OLPSO-G Results of these algorithms are all deriveddirectly from their corresponding literatures [40] Resultsshow that OBMLBA has outperformed other methods on6 of 7 functions OBMLBA performs worse than ABC andOCABC only on Rosenbrock function
According to the analyses above it can be clearly observedthat OBMLBA provides outstanding performance with fastconvergence speed and high convergence accuracy on mostof the test functions
Scientific Programming 7
Table3Re
sults
ofcomparis
onso
fBAso
n15-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
119119864+
04184
119864+03
260119864+
03253
119864+03
445119864minus
54139
119864minus53
230119864minus
69707
119864minus69
36601
Eminus2
040
119865 2705
119864+07
304119864+
07189
119864+07
193119864+
07102
119864minus42
454119864minus
42396
119864minus67
125119864minus
66347
13Eminus2
020
119865 3810
119864+02
139119864+
02132
119864+02
225119864+
02377
119864minus52
169119864minus
51810
119864minus66
256119864minus
65621
79Eminus2
020
119865 4259
119864minus02
150119864minus
02164
119864minus03
235119864minus
03522
119864minus90
227119864minus
89457
119864minus126
12069119864
minus125
112Eminus3
050
119865 5408
119864+01
724119864+
00108
119864+01
143119864+
01373
119864minus28
145119864minus
27175
119864minus33
552119864minus
33117
Eminus1
04524
41Eminus1
04119865 6
501119864+
01316
119864+00
147119864+
01146
119864+01
304119864minus
27100
119864minus26
250119864minus
35785
119864minus35
668Eminus9
5298
692E
minus94
119865 7111
119864+04
213119864+
03136
119864+03
216119864+
030
00
00
0119865 8
152119864+
00443
119864minus01
126119864minus
01262
119864minus01
00
00
00
119865 9265
119864+00
113119864+
00134
119864minus01
270119864minus
01479
119864minus04
431119864minus
04399
119864minus04
282119864minus
04119
Eminus0
4100
Eminus0
4119865 10
589119864+
04202
119864+04
603119864+
02134
119864+03
135119864+
01210
119864minus01
129E+0
1371
Eminus0
1130
119864+01
340119864minus
01119865 11
131119864+
02816
119864+00
781119864+
01448
119864+01
00
00
00
119865 12106
119864+02
224119864+
01166
119864+01
227119864+
010
00
00
0119865 13
183119864+
01488
119864minus01
107119864+
01565
119864+00
00
00
00
119865 14142
119864+01
104119864+
00529
119864+00
526119864+
00182
119864minus27
792119864minus
27567
119864minus35
179119864minus
34891
Eminus1
06368
23Eminus1
05119865 15
152119864+
02113
119864+01
273119864+
01176
119864+01
747119864minus
01300
119864minus01
278Eminus0
1274
Eminus0
1483
119864minus01
526119864minus
01119865 16
168119864+
01160
119864+00
987119864+
00523
119864+00
00
00
00
8 Scientific Programming
Table4Re
sults
ofcomparis
onso
fBAso
n30-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
405119864+
04280
119864+03
380119864+
03680
119864+03
373119864minus
49167
119864minus48
180119864minus
117568
119864minus117
00
119865 2530
119864+08
115119864+
08954
119864+07
835119864+
07296
119864minus50
125119864minus
49320
119864minus117
101119864minus
1160
0119865 3
515119864+
03797
119864+02
729119864+
02121
119864+03
980119864minus
48438
119864minus47
272119864minus
123651
119864minus123
00
119865 4753
119864minus02
297119864minus
02580
119864minus03
139119864minus
02174
119864minus85
778119864minus
85176
119864minus198
00
0119865 5
233119864+
05428
119864+05
758119864+
01178
119864+02
972119864minus
26433
119864minus25
512119864minus
60162
119864minus59
680Eminus2
100
119865 6683
119864+01
350119864+
00176
119864+01
196119864+
01542
119864minus30
111119864minus
29263
119864minus55
833119864minus
55140
Eminus2
050
119865 7371
119864+04
426119864+
03569
119864+03
814119864+
030
00
00
0119865 8
239119864+
01619
119864+00
123119864+
00262
119864+00
00
00
00
119865 9402
119864+01
691119864+
00225
119864+00
519119864+
00285
119864minus04
283119864minus
04230
119864minus04
120119864minus
04625
Eminus0
5421
Eminus0
5119865 10
385119864+
05651
119864+04
259119864+
04615
119864+04
287119864+
01622
119864minus02
283119864+
01216
119864minus01
282E+0
1248
Eminus0
1119865 11
337119864+
02157
119864+01
213119864+
02734
119864+01
00
00
00
119865 12361
119864+02
339119864+
01374
119864+01
597119864+
010
00
00
0119865 13
196119864+
01271
119864minus01
656119864+
00525
119864+00
00
00
00
119865 14426
119864+01
274119864+
00155
119864+01
142119864+
01126
119864minus27
556119864minus
27291
119864minus60
910119864minus
60330
Eminus2
150
119865 15500
119864+02
897119864+
01696
119864+01
604119864+
01901
119864+00
307119864+
00578
119864+00
282119864+
00517
E+0
0240
E+0
0119865 16
413119864+
01117
119864+00
161119864+
01138
119864+01
00
00
00
Scientific Programming 9
Table5Re
sults
ofcomparis
onso
fBAso
n60
-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
103119864+
05937
119864+03
353119864+
03871
119864+03
454119864minus
49202
119864minus48
121119864minus
219000
119864+00
00
119865 2238
119864+09
251119864+
08607
119864+08
873119864+
08965
119864minus43
432119864minus
42161
119864minus217
000119864+
000
0119865 3
295119864+
04302
119864+03
124119864+
03203
119864+03
213119864minus
54566
119864minus54
819119864minus
224000
119864+00
00
119865 4147
119864minus01
425119864minus
02279
119864minus03
628119864minus
03205
119864minus88
918119864minus
88000
119864+00
00
0119865 5
100119864+
10000
119864+00
276119864+
01304
119864+01
478119864minus
24214
119864minus23
478119864minus
112151
119864minus111
00
119865 6821
119864+01
242119864+
00115
119864+01
153119864+
01321
119864minus28
705119864minus
28195
119864minus107
613119864minus
1070
0119865 7
983119864+
04872
119864+03
117119864+
04218
119864+04
00
00
00
119865 8546
119864+03
431119864+
03274
119864+01
812119864+
010
00
00
0119865 9
276119864+
02478
119864+01
255119864+
00794
119864+00
270119864minus
04254
119864minus04
156Eminus0
4775
Eminus0
5282
119864minus05
277119864minus
05119865 10
177119864+
06390
119864+05
180119864+
04347
119864+04
586119864+
01721
119864minus02
583119864+
01152
119864minus01
101E+0
1832
Eminus0
1119865 11
773119864+
02279
119864+01
309119864+
02270
119864+02
00
00
00
119865 12903
119864+02
121119864+
02372
119864+01
601119864+
010
00
00
0119865 13
203119864+
01671
119864minus02
636119864+
00572
119864+00
00
00
00
119865 14108
119864+02
442119864+
00506
119864+01
391119864+
01526
119864minus29
220119864minus
28646
119864minus104
204119864minus
1030
0119865 15
149119864+
03192
119864+02
274119864+
02297
119864+02
414119864+
01979
119864+00
242119864+
01661
119864+00
458Eminus0
2653
Eminus0
2119865 16
930119864+
01152
119864+00
335119864+
01268
119864+01
00
00
00
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
6 Scientific Programming
(1) Input the parameters(2) Set iter = 1 Initialize the position and velocity of each individual in the population(3) Evaluate the objective function and related parameter values of each individual(4) While iter le iter max do(5) Apply Levy Flight Local Search strategy using Algorithm 2(6) Generate new solutions using Eq (13)(7) if rand(0 1) gt 119903119894(8) Apply the Opposition based learning using Algorithm 3(9) Generate a new local solution around the selected solution 119909lowast using Eq (5)(10) end if(11) if rand(0 1) lt 119860 119894 and 119891(119909119894new) lt 119891(119909119894)(12) update the solution loudness and pulse emission rate(13) end if(14) end while(15) Output the individual with the smallest objective function value in the population
Algorithm 4 Pseudocode of OBMLBA
Table 1 Benchmark problems
Fun Name Scope 119891min
F1 Sphere [minus100 100] 0F2 Elliptic [minus100 100] 0F3 SumSquare [minus10 10] 0F4 SumPower [minus1 1] 0F5 Schwefel222 [minus10 10] 0F6 Schwefel221 [minus100 100] 0F7 Step [minus100 100] 0F8 Exponential [minus128 128] 0F9 Quartic [minus128 128] 0F10 Rosenbrock [minus5 10] 0F11 Rastrigin [minus512 512] 0F12 Griewank [minus600 600] 0F13 Ackley [minus32 32] 0F14 Alpine [minus10 10] 0F15 Levy [minus10 10] 0F16 Weierstrass [minus05 05] 0
As seen from Tables 3ndash5 OBMLBA can get the globaloptimal on 1198911sim1198914 1198917 1198918 and 11989111sim11989113 with 119863 = 15 30and 60 On 1198916 and 11989114 the precision of solutions obtained byOBMLBA is smaller than the order of 1119864 minus 90 and the orderfor 1119864minus 100 respectively Results indicate that OBMLBA canget solutions extremely close to the global optimal values onmost of the functions
Solutions obtained by BAs are compared with resultsobtained by MLBA and OBMLBA As shown in Tables 3ndash5OBMLBA performs significantly better than basic BA LBAand DLBA on 1198911sim1198916 11989114 and 11989115 All algorithms except forbasic BA and LBA perform well on 1198917 1198918 11989111sim11989113 and 11989116OBMLBA is comparable to DLBA and MLBA on functions1198919 and 11989110
Compared with MLBA OBMLBA provides better per-formance on functions 1198911sim1198916 11989114 and 11989115 For 1198919 11989110 and
Table 2 Parameters setting
Parameters BA DLBA OBMLBAPopulation size119873 40Run time 20Loudness 095Pulse emission 085Factors updating 119860 and 120574 09Minimum of frequency 119891119898119894119899 0Maximum of frequency 119891max 1Factors updating frequency mdash mdash 03
11989115 results obtained by MLBA and OBMLBA are quite closeto each other Both MLBA and OBMLBA get the globaloptimum of 1198917 1198918 11989111sim11989113 and 11989116
From the convergence characteristic graphs it can beobviously observed that OBMLBA is the fastest algorithm inall the considered BAs MLBA is slower than OBMLBA butfaster than DLBA LBA and basic BA OBMLBA exhibits thebest accuracy and achieves the fastest convergence speedTheresults indeed indicate the advantage of themodified positionupdate equation and OBL strategy
On the whole algorithms proposed in this study such asMLBA andOBMLBAwork better than other BAs consideredAnd OBMLBA is more effective than MLBA
In Table 6 OBMLBA is compared with some state-of-the-art algorithms such as ABC PSO OCABC CLPSOand OLPSO-G Results of these algorithms are all deriveddirectly from their corresponding literatures [40] Resultsshow that OBMLBA has outperformed other methods on6 of 7 functions OBMLBA performs worse than ABC andOCABC only on Rosenbrock function
According to the analyses above it can be clearly observedthat OBMLBA provides outstanding performance with fastconvergence speed and high convergence accuracy on mostof the test functions
Scientific Programming 7
Table3Re
sults
ofcomparis
onso
fBAso
n15-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
119119864+
04184
119864+03
260119864+
03253
119864+03
445119864minus
54139
119864minus53
230119864minus
69707
119864minus69
36601
Eminus2
040
119865 2705
119864+07
304119864+
07189
119864+07
193119864+
07102
119864minus42
454119864minus
42396
119864minus67
125119864minus
66347
13Eminus2
020
119865 3810
119864+02
139119864+
02132
119864+02
225119864+
02377
119864minus52
169119864minus
51810
119864minus66
256119864minus
65621
79Eminus2
020
119865 4259
119864minus02
150119864minus
02164
119864minus03
235119864minus
03522
119864minus90
227119864minus
89457
119864minus126
12069119864
minus125
112Eminus3
050
119865 5408
119864+01
724119864+
00108
119864+01
143119864+
01373
119864minus28
145119864minus
27175
119864minus33
552119864minus
33117
Eminus1
04524
41Eminus1
04119865 6
501119864+
01316
119864+00
147119864+
01146
119864+01
304119864minus
27100
119864minus26
250119864minus
35785
119864minus35
668Eminus9
5298
692E
minus94
119865 7111
119864+04
213119864+
03136
119864+03
216119864+
030
00
00
0119865 8
152119864+
00443
119864minus01
126119864minus
01262
119864minus01
00
00
00
119865 9265
119864+00
113119864+
00134
119864minus01
270119864minus
01479
119864minus04
431119864minus
04399
119864minus04
282119864minus
04119
Eminus0
4100
Eminus0
4119865 10
589119864+
04202
119864+04
603119864+
02134
119864+03
135119864+
01210
119864minus01
129E+0
1371
Eminus0
1130
119864+01
340119864minus
01119865 11
131119864+
02816
119864+00
781119864+
01448
119864+01
00
00
00
119865 12106
119864+02
224119864+
01166
119864+01
227119864+
010
00
00
0119865 13
183119864+
01488
119864minus01
107119864+
01565
119864+00
00
00
00
119865 14142
119864+01
104119864+
00529
119864+00
526119864+
00182
119864minus27
792119864minus
27567
119864minus35
179119864minus
34891
Eminus1
06368
23Eminus1
05119865 15
152119864+
02113
119864+01
273119864+
01176
119864+01
747119864minus
01300
119864minus01
278Eminus0
1274
Eminus0
1483
119864minus01
526119864minus
01119865 16
168119864+
01160
119864+00
987119864+
00523
119864+00
00
00
00
8 Scientific Programming
Table4Re
sults
ofcomparis
onso
fBAso
n30-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
405119864+
04280
119864+03
380119864+
03680
119864+03
373119864minus
49167
119864minus48
180119864minus
117568
119864minus117
00
119865 2530
119864+08
115119864+
08954
119864+07
835119864+
07296
119864minus50
125119864minus
49320
119864minus117
101119864minus
1160
0119865 3
515119864+
03797
119864+02
729119864+
02121
119864+03
980119864minus
48438
119864minus47
272119864minus
123651
119864minus123
00
119865 4753
119864minus02
297119864minus
02580
119864minus03
139119864minus
02174
119864minus85
778119864minus
85176
119864minus198
00
0119865 5
233119864+
05428
119864+05
758119864+
01178
119864+02
972119864minus
26433
119864minus25
512119864minus
60162
119864minus59
680Eminus2
100
119865 6683
119864+01
350119864+
00176
119864+01
196119864+
01542
119864minus30
111119864minus
29263
119864minus55
833119864minus
55140
Eminus2
050
119865 7371
119864+04
426119864+
03569
119864+03
814119864+
030
00
00
0119865 8
239119864+
01619
119864+00
123119864+
00262
119864+00
00
00
00
119865 9402
119864+01
691119864+
00225
119864+00
519119864+
00285
119864minus04
283119864minus
04230
119864minus04
120119864minus
04625
Eminus0
5421
Eminus0
5119865 10
385119864+
05651
119864+04
259119864+
04615
119864+04
287119864+
01622
119864minus02
283119864+
01216
119864minus01
282E+0
1248
Eminus0
1119865 11
337119864+
02157
119864+01
213119864+
02734
119864+01
00
00
00
119865 12361
119864+02
339119864+
01374
119864+01
597119864+
010
00
00
0119865 13
196119864+
01271
119864minus01
656119864+
00525
119864+00
00
00
00
119865 14426
119864+01
274119864+
00155
119864+01
142119864+
01126
119864minus27
556119864minus
27291
119864minus60
910119864minus
60330
Eminus2
150
119865 15500
119864+02
897119864+
01696
119864+01
604119864+
01901
119864+00
307119864+
00578
119864+00
282119864+
00517
E+0
0240
E+0
0119865 16
413119864+
01117
119864+00
161119864+
01138
119864+01
00
00
00
Scientific Programming 9
Table5Re
sults
ofcomparis
onso
fBAso
n60
-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
103119864+
05937
119864+03
353119864+
03871
119864+03
454119864minus
49202
119864minus48
121119864minus
219000
119864+00
00
119865 2238
119864+09
251119864+
08607
119864+08
873119864+
08965
119864minus43
432119864minus
42161
119864minus217
000119864+
000
0119865 3
295119864+
04302
119864+03
124119864+
03203
119864+03
213119864minus
54566
119864minus54
819119864minus
224000
119864+00
00
119865 4147
119864minus01
425119864minus
02279
119864minus03
628119864minus
03205
119864minus88
918119864minus
88000
119864+00
00
0119865 5
100119864+
10000
119864+00
276119864+
01304
119864+01
478119864minus
24214
119864minus23
478119864minus
112151
119864minus111
00
119865 6821
119864+01
242119864+
00115
119864+01
153119864+
01321
119864minus28
705119864minus
28195
119864minus107
613119864minus
1070
0119865 7
983119864+
04872
119864+03
117119864+
04218
119864+04
00
00
00
119865 8546
119864+03
431119864+
03274
119864+01
812119864+
010
00
00
0119865 9
276119864+
02478
119864+01
255119864+
00794
119864+00
270119864minus
04254
119864minus04
156Eminus0
4775
Eminus0
5282
119864minus05
277119864minus
05119865 10
177119864+
06390
119864+05
180119864+
04347
119864+04
586119864+
01721
119864minus02
583119864+
01152
119864minus01
101E+0
1832
Eminus0
1119865 11
773119864+
02279
119864+01
309119864+
02270
119864+02
00
00
00
119865 12903
119864+02
121119864+
02372
119864+01
601119864+
010
00
00
0119865 13
203119864+
01671
119864minus02
636119864+
00572
119864+00
00
00
00
119865 14108
119864+02
442119864+
00506
119864+01
391119864+
01526
119864minus29
220119864minus
28646
119864minus104
204119864minus
1030
0119865 15
149119864+
03192
119864+02
274119864+
02297
119864+02
414119864+
01979
119864+00
242119864+
01661
119864+00
458Eminus0
2653
Eminus0
2119865 16
930119864+
01152
119864+00
335119864+
01268
119864+01
00
00
00
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
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RoboticsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Scientific Programming 7
Table3Re
sults
ofcomparis
onso
fBAso
n15-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
119119864+
04184
119864+03
260119864+
03253
119864+03
445119864minus
54139
119864minus53
230119864minus
69707
119864minus69
36601
Eminus2
040
119865 2705
119864+07
304119864+
07189
119864+07
193119864+
07102
119864minus42
454119864minus
42396
119864minus67
125119864minus
66347
13Eminus2
020
119865 3810
119864+02
139119864+
02132
119864+02
225119864+
02377
119864minus52
169119864minus
51810
119864minus66
256119864minus
65621
79Eminus2
020
119865 4259
119864minus02
150119864minus
02164
119864minus03
235119864minus
03522
119864minus90
227119864minus
89457
119864minus126
12069119864
minus125
112Eminus3
050
119865 5408
119864+01
724119864+
00108
119864+01
143119864+
01373
119864minus28
145119864minus
27175
119864minus33
552119864minus
33117
Eminus1
04524
41Eminus1
04119865 6
501119864+
01316
119864+00
147119864+
01146
119864+01
304119864minus
27100
119864minus26
250119864minus
35785
119864minus35
668Eminus9
5298
692E
minus94
119865 7111
119864+04
213119864+
03136
119864+03
216119864+
030
00
00
0119865 8
152119864+
00443
119864minus01
126119864minus
01262
119864minus01
00
00
00
119865 9265
119864+00
113119864+
00134
119864minus01
270119864minus
01479
119864minus04
431119864minus
04399
119864minus04
282119864minus
04119
Eminus0
4100
Eminus0
4119865 10
589119864+
04202
119864+04
603119864+
02134
119864+03
135119864+
01210
119864minus01
129E+0
1371
Eminus0
1130
119864+01
340119864minus
01119865 11
131119864+
02816
119864+00
781119864+
01448
119864+01
00
00
00
119865 12106
119864+02
224119864+
01166
119864+01
227119864+
010
00
00
0119865 13
183119864+
01488
119864minus01
107119864+
01565
119864+00
00
00
00
119865 14142
119864+01
104119864+
00529
119864+00
526119864+
00182
119864minus27
792119864minus
27567
119864minus35
179119864minus
34891
Eminus1
06368
23Eminus1
05119865 15
152119864+
02113
119864+01
273119864+
01176
119864+01
747119864minus
01300
119864minus01
278Eminus0
1274
Eminus0
1483
119864minus01
526119864minus
01119865 16
168119864+
01160
119864+00
987119864+
00523
119864+00
00
00
00
8 Scientific Programming
Table4Re
sults
ofcomparis
onso
fBAso
n30-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
405119864+
04280
119864+03
380119864+
03680
119864+03
373119864minus
49167
119864minus48
180119864minus
117568
119864minus117
00
119865 2530
119864+08
115119864+
08954
119864+07
835119864+
07296
119864minus50
125119864minus
49320
119864minus117
101119864minus
1160
0119865 3
515119864+
03797
119864+02
729119864+
02121
119864+03
980119864minus
48438
119864minus47
272119864minus
123651
119864minus123
00
119865 4753
119864minus02
297119864minus
02580
119864minus03
139119864minus
02174
119864minus85
778119864minus
85176
119864minus198
00
0119865 5
233119864+
05428
119864+05
758119864+
01178
119864+02
972119864minus
26433
119864minus25
512119864minus
60162
119864minus59
680Eminus2
100
119865 6683
119864+01
350119864+
00176
119864+01
196119864+
01542
119864minus30
111119864minus
29263
119864minus55
833119864minus
55140
Eminus2
050
119865 7371
119864+04
426119864+
03569
119864+03
814119864+
030
00
00
0119865 8
239119864+
01619
119864+00
123119864+
00262
119864+00
00
00
00
119865 9402
119864+01
691119864+
00225
119864+00
519119864+
00285
119864minus04
283119864minus
04230
119864minus04
120119864minus
04625
Eminus0
5421
Eminus0
5119865 10
385119864+
05651
119864+04
259119864+
04615
119864+04
287119864+
01622
119864minus02
283119864+
01216
119864minus01
282E+0
1248
Eminus0
1119865 11
337119864+
02157
119864+01
213119864+
02734
119864+01
00
00
00
119865 12361
119864+02
339119864+
01374
119864+01
597119864+
010
00
00
0119865 13
196119864+
01271
119864minus01
656119864+
00525
119864+00
00
00
00
119865 14426
119864+01
274119864+
00155
119864+01
142119864+
01126
119864minus27
556119864minus
27291
119864minus60
910119864minus
60330
Eminus2
150
119865 15500
119864+02
897119864+
01696
119864+01
604119864+
01901
119864+00
307119864+
00578
119864+00
282119864+
00517
E+0
0240
E+0
0119865 16
413119864+
01117
119864+00
161119864+
01138
119864+01
00
00
00
Scientific Programming 9
Table5Re
sults
ofcomparis
onso
fBAso
n60
-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
103119864+
05937
119864+03
353119864+
03871
119864+03
454119864minus
49202
119864minus48
121119864minus
219000
119864+00
00
119865 2238
119864+09
251119864+
08607
119864+08
873119864+
08965
119864minus43
432119864minus
42161
119864minus217
000119864+
000
0119865 3
295119864+
04302
119864+03
124119864+
03203
119864+03
213119864minus
54566
119864minus54
819119864minus
224000
119864+00
00
119865 4147
119864minus01
425119864minus
02279
119864minus03
628119864minus
03205
119864minus88
918119864minus
88000
119864+00
00
0119865 5
100119864+
10000
119864+00
276119864+
01304
119864+01
478119864minus
24214
119864minus23
478119864minus
112151
119864minus111
00
119865 6821
119864+01
242119864+
00115
119864+01
153119864+
01321
119864minus28
705119864minus
28195
119864minus107
613119864minus
1070
0119865 7
983119864+
04872
119864+03
117119864+
04218
119864+04
00
00
00
119865 8546
119864+03
431119864+
03274
119864+01
812119864+
010
00
00
0119865 9
276119864+
02478
119864+01
255119864+
00794
119864+00
270119864minus
04254
119864minus04
156Eminus0
4775
Eminus0
5282
119864minus05
277119864minus
05119865 10
177119864+
06390
119864+05
180119864+
04347
119864+04
586119864+
01721
119864minus02
583119864+
01152
119864minus01
101E+0
1832
Eminus0
1119865 11
773119864+
02279
119864+01
309119864+
02270
119864+02
00
00
00
119865 12903
119864+02
121119864+
02372
119864+01
601119864+
010
00
00
0119865 13
203119864+
01671
119864minus02
636119864+
00572
119864+00
00
00
00
119865 14108
119864+02
442119864+
00506
119864+01
391119864+
01526
119864minus29
220119864minus
28646
119864minus104
204119864minus
1030
0119865 15
149119864+
03192
119864+02
274119864+
02297
119864+02
414119864+
01979
119864+00
242119864+
01661
119864+00
458Eminus0
2653
Eminus0
2119865 16
930119864+
01152
119864+00
335119864+
01268
119864+01
00
00
00
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
8 Scientific Programming
Table4Re
sults
ofcomparis
onso
fBAso
n30-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
405119864+
04280
119864+03
380119864+
03680
119864+03
373119864minus
49167
119864minus48
180119864minus
117568
119864minus117
00
119865 2530
119864+08
115119864+
08954
119864+07
835119864+
07296
119864minus50
125119864minus
49320
119864minus117
101119864minus
1160
0119865 3
515119864+
03797
119864+02
729119864+
02121
119864+03
980119864minus
48438
119864minus47
272119864minus
123651
119864minus123
00
119865 4753
119864minus02
297119864minus
02580
119864minus03
139119864minus
02174
119864minus85
778119864minus
85176
119864minus198
00
0119865 5
233119864+
05428
119864+05
758119864+
01178
119864+02
972119864minus
26433
119864minus25
512119864minus
60162
119864minus59
680Eminus2
100
119865 6683
119864+01
350119864+
00176
119864+01
196119864+
01542
119864minus30
111119864minus
29263
119864minus55
833119864minus
55140
Eminus2
050
119865 7371
119864+04
426119864+
03569
119864+03
814119864+
030
00
00
0119865 8
239119864+
01619
119864+00
123119864+
00262
119864+00
00
00
00
119865 9402
119864+01
691119864+
00225
119864+00
519119864+
00285
119864minus04
283119864minus
04230
119864minus04
120119864minus
04625
Eminus0
5421
Eminus0
5119865 10
385119864+
05651
119864+04
259119864+
04615
119864+04
287119864+
01622
119864minus02
283119864+
01216
119864minus01
282E+0
1248
Eminus0
1119865 11
337119864+
02157
119864+01
213119864+
02734
119864+01
00
00
00
119865 12361
119864+02
339119864+
01374
119864+01
597119864+
010
00
00
0119865 13
196119864+
01271
119864minus01
656119864+
00525
119864+00
00
00
00
119865 14426
119864+01
274119864+
00155
119864+01
142119864+
01126
119864minus27
556119864minus
27291
119864minus60
910119864minus
60330
Eminus2
150
119865 15500
119864+02
897119864+
01696
119864+01
604119864+
01901
119864+00
307119864+
00578
119864+00
282119864+
00517
E+0
0240
E+0
0119865 16
413119864+
01117
119864+00
161119864+
01138
119864+01
00
00
00
Scientific Programming 9
Table5Re
sults
ofcomparis
onso
fBAso
n60
-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
103119864+
05937
119864+03
353119864+
03871
119864+03
454119864minus
49202
119864minus48
121119864minus
219000
119864+00
00
119865 2238
119864+09
251119864+
08607
119864+08
873119864+
08965
119864minus43
432119864minus
42161
119864minus217
000119864+
000
0119865 3
295119864+
04302
119864+03
124119864+
03203
119864+03
213119864minus
54566
119864minus54
819119864minus
224000
119864+00
00
119865 4147
119864minus01
425119864minus
02279
119864minus03
628119864minus
03205
119864minus88
918119864minus
88000
119864+00
00
0119865 5
100119864+
10000
119864+00
276119864+
01304
119864+01
478119864minus
24214
119864minus23
478119864minus
112151
119864minus111
00
119865 6821
119864+01
242119864+
00115
119864+01
153119864+
01321
119864minus28
705119864minus
28195
119864minus107
613119864minus
1070
0119865 7
983119864+
04872
119864+03
117119864+
04218
119864+04
00
00
00
119865 8546
119864+03
431119864+
03274
119864+01
812119864+
010
00
00
0119865 9
276119864+
02478
119864+01
255119864+
00794
119864+00
270119864minus
04254
119864minus04
156Eminus0
4775
Eminus0
5282
119864minus05
277119864minus
05119865 10
177119864+
06390
119864+05
180119864+
04347
119864+04
586119864+
01721
119864minus02
583119864+
01152
119864minus01
101E+0
1832
Eminus0
1119865 11
773119864+
02279
119864+01
309119864+
02270
119864+02
00
00
00
119865 12903
119864+02
121119864+
02372
119864+01
601119864+
010
00
00
0119865 13
203119864+
01671
119864minus02
636119864+
00572
119864+00
00
00
00
119865 14108
119864+02
442119864+
00506
119864+01
391119864+
01526
119864minus29
220119864minus
28646
119864minus104
204119864minus
1030
0119865 15
149119864+
03192
119864+02
274119864+
02297
119864+02
414119864+
01979
119864+00
242119864+
01661
119864+00
458Eminus0
2653
Eminus0
2119865 16
930119864+
01152
119864+00
335119864+
01268
119864+01
00
00
00
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Scientific Programming 9
Table5Re
sults
ofcomparis
onso
fBAso
n60
-dim
ensio
nbenchm
arkfunctio
ns
Fun
BALB
ADLB
AMLB
AOBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SD119865 1
103119864+
05937
119864+03
353119864+
03871
119864+03
454119864minus
49202
119864minus48
121119864minus
219000
119864+00
00
119865 2238
119864+09
251119864+
08607
119864+08
873119864+
08965
119864minus43
432119864minus
42161
119864minus217
000119864+
000
0119865 3
295119864+
04302
119864+03
124119864+
03203
119864+03
213119864minus
54566
119864minus54
819119864minus
224000
119864+00
00
119865 4147
119864minus01
425119864minus
02279
119864minus03
628119864minus
03205
119864minus88
918119864minus
88000
119864+00
00
0119865 5
100119864+
10000
119864+00
276119864+
01304
119864+01
478119864minus
24214
119864minus23
478119864minus
112151
119864minus111
00
119865 6821
119864+01
242119864+
00115
119864+01
153119864+
01321
119864minus28
705119864minus
28195
119864minus107
613119864minus
1070
0119865 7
983119864+
04872
119864+03
117119864+
04218
119864+04
00
00
00
119865 8546
119864+03
431119864+
03274
119864+01
812119864+
010
00
00
0119865 9
276119864+
02478
119864+01
255119864+
00794
119864+00
270119864minus
04254
119864minus04
156Eminus0
4775
Eminus0
5282
119864minus05
277119864minus
05119865 10
177119864+
06390
119864+05
180119864+
04347
119864+04
586119864+
01721
119864minus02
583119864+
01152
119864minus01
101E+0
1832
Eminus0
1119865 11
773119864+
02279
119864+01
309119864+
02270
119864+02
00
00
00
119865 12903
119864+02
121119864+
02372
119864+01
601119864+
010
00
00
0119865 13
203119864+
01671
119864minus02
636119864+
00572
119864+00
00
00
00
119865 14108
119864+02
442119864+
00506
119864+01
391119864+
01526
119864minus29
220119864minus
28646
119864minus104
204119864minus
1030
0119865 15
149119864+
03192
119864+02
274119864+
02297
119864+02
414119864+
01979
119864+00
242119864+
01661
119864+00
458Eminus0
2653
Eminus0
2119865 16
930119864+
01152
119864+00
335119864+
01268
119864+01
00
00
00
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
10 Scientific Programming
Table6Com
paris
onso
fthe
state-of-the-artalgorithm
son30-dim
ensio
nbenchm
arkfunctio
ns
Fun
ABC
PSO
OCA
BCCL
PSO
OLP
SO-G
OBM
LBA
Mean
SDMean
SDMean
SDMean
SDMean
SDMean
SDSphere
680119864minus
14874
119864minus14
334119864minus
14539
119864minus14
405119864minus
41697
119864minus41
158119864minus
12770
119864minus13
412119864minus
54634
119864minus54
00
Schw
eel222
458119864minus
08198
119864minus08
170119864minus
10139
119864minus10
913119864minus
22798
119864minus22
251119864minus
08584
119864minus09
985119864minus
30101
119864minus29
680Eminus2
100
Step
00
00
00
00
00
00
Rosenb
rock
168Eminus0
1204
Eminus0
1280
119864+01
217119864+
01439
119864minus01
726119864minus
01113
119864+01
850119864+
01215
119864+01
299119864+
01282
119864+01
248119864minus
01Ra
strigin
290119864minus
07864
119864minus07
357119864+
01689
119864+00
00
909119864minus
05125
119864minus04
107119864+
00992
119864minus01
00
Grie
wank
169119864minus
05488
119864minus05
153119864minus
03432
119864minus03
00
902119864minus
09857
119864minus09
483119864minus
03863
119864minus03
00
Ackley
109119864minus
06596
119864minus07
820119864minus
08673
119864minus08
383119864minus
15983
119864minus16
366119864minus
07757
119864minus08
798119864minus
15203
119864minus15
00
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Scientific Programming 11
BALBADLBA
MLBAOBMLBA
200 400 600 800 10000Generation
Alpine function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
BALBADLBA
MLBAOBMLBA
Levy function with D = 30
100
101
102
103
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Quartic function with D = 30
10minus6
10minus4
10minus2
100
102
104
Fitn
ess
200 400 600 800 10000Generation
BALBADLBA
MLBAOBMLBA
Schwefel221 function with D = 30
10minus250
10minus200
10minus150
10minus100
10minus50
100
1050
Fitn
ess
200 400 600 800 10000Generation
Figure 1 Convergence performance of BAs on the test functions
5 Conclusions
Bat Algorithm a metaheuristic algorithm proposed recentlyhas been successfully used to solve different types of opti-mization problems However it also has some inefficiency onexploration and exploitation capabilities
In this study a new variant of Bat Algorithm calledOBMLBA is presentedThe exploration and exploitation abil-ities of BA are balanced and enhanced in the search processby combining the information of the best-so-far solution andthe neighborhood solutions into the search equations At thesame time the proposed equation uses sinusoidal function
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
12 Scientific Programming
to define the bat pulse frequency 119891 allowing flexibilityin changing the direction of 119891 Furthermore Levy Flightrandom walk is taken into consideration to escape fromlocal optima The concept of opposition based learning isintroduced to the algorithm to improve the diversity andconvergence capability The proposed approach has beensuccessfully used on 16 benchmark functions Results andcomparisons with the basic BA DLBA and other state-of-the-art algorithms show that OBMLBA has outperformedconsidered approaches in most of the functions Howeverthese experiments are just small scale optimization problemsThe performance of the proposed algorithm on large scaleoptimization problems and real world optimization problemsshould be deeply investigatedThe future study of thismethodmay focus on investigating themathematical foundations andapplications of the proposed algorithm
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National NatureScience Foundation of China under Grant 61402534 bythe Shandong Provincial Natural Science FoundationChina under Grant ZR2014FQ002 and by the FundamentalResearch Funds for the Central Universities under Grant16CX02010A
References
[1] S Rao Engineering OptimizationTheory and Practice NewAgeInternational 1996
[2] X Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010
[3] K S Tang K FMan S Kwong andQHe ldquoGenetic algorithmsand their applicationsrdquo IEEE Signal Processing Magazine vol 13no 6 pp 22ndash37 1996
[4] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[5] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 Perth Australia December1995
[6] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004
[7] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005
[8] S-C Chu P-W Tsai and J-S Pan ldquoCat swarm optimizationrdquoin PRICAI 2006 Trends in Artificial Intelligence Q Yang and GWebb Eds vol 4099 of Lecture Notes in Computer Science pp854ndash858 Springer Berlin Germany 2006
[9] X-S Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and Biologically
Inspired Computing (NABIC rsquo09) pp 210ndash214 CoimbatoreIndia December 2009
[10] X-S Yang ldquoA new metaheuristic bat-inspired algorithmrdquo inNature Inspired Cooperative Strategies for Optimization pp 65ndash74 Springer 2010
[11] T C Bora L D S Coelho and L Lebensztajn ldquoBat-inspiredoptimization approach for the brushless DC wheel motorproblemrdquo IEEE Transactions on Magnetics vol 48 no 2 pp947ndash950 2012
[12] M R Sathya and M M T Ansari ldquoLoad frequency controlusing Bat inspired algorithm based dual mode gain schedulingof PI controllers for interconnected power systemrdquo Interna-tional Journal of Electrical Power amp Energy Systems vol 64 pp365ndash374 2015
[13] S Mishra K Shaw and D Mishra ldquoA new meta-heuristic batinspired classification approach for microarray datardquo ProcediaTechnology vol 4 pp 802ndash806 2012
[14] P Musikapun and P Pongcharoen ldquoSolving multi-stage multi-machine multi-product scheduling problem using bat algo-rithmrdquo in Proceedings of the 2nd International Conference onManagement and Artificial Intelligence (IPEDR rsquo12) vol 35 pp98ndash102 2012
[15] OHasancebi T Teke andO Pekcan ldquoA bat-inspired algorithmfor structural optimizationrdquo Computers and Structures vol 128pp 77ndash90 2013
[16] J Zhang and G Wang ldquoImage matching using a bat algorithmwith mutationrdquo Applied Mechanics and Materials vol 203 pp88ndash93 2012
[17] E S Ali ldquoOptimization of Power System Stabilizers using BATsearch algorithmrdquo International Journal of Electrical Power andEnergy Systems vol 61 pp 683ndash690 2014
[18] J Xie Y Zhou and H Chen ldquoA novel bat algorithm based ondifferential operator and Levy flights trajectoryrdquoComputationalIntelligence and Neuroscience vol 2013 Article ID 453812 13pages 2013
[19] L L Li andY Q Zhou ldquoA novel complex-valued bat algorithmrdquoNeural Computing andApplications vol 25 no 6 pp 1369ndash13812014
[20] AH Gandomi andX-S Yang ldquoChaotic bat algorithmrdquo Journalof Computational Science vol 5 no 2 pp 224ndash232 2014
[21] B Bahmani-Firouzi and R Azizipanah-Abarghooee ldquoOptimalsizing of battery energy storage for micro-grid operation man-agement using a new improved bat algorithmrdquo InternationalJournal of Electrical Power and Energy Systems vol 56 pp 42ndash54 2014
[22] N S Jaddi S Abdullah and A R Hamdan ldquoMulti-populationcooperative bat algorithm-based optimization of artificial neu-ral networkmodelrdquo Information Sciences vol 294 pp 628ndash6442015
[23] S Yilmaz and E U Kucuksille ldquoA new modification approachon bat algorithm for solving optimization problemsrdquo AppliedSoft Computing vol 28 pp 259ndash275 2015
[24] K Khan and A Sahai ldquoA comparison of BA GA PSOBP and LM for training feed forward neural networks in e-learning contextrdquo International Journal of Intelligent Systemsand Applications vol 4 no 7 pp 23ndash29 2012
[25] G Wang and L Guo ldquoA novel hybrid bat algorithm withharmony search for global numerical optimizationrdquo Journal ofApplied Mathematics vol 2013 Article ID 696491 21 pages2013
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Scientific Programming 13
[26] X-S He W-J Ding and X-S Yang ldquoBat algorithm basedon simulated annealing and gaussian perturbationsrdquo NeuralComputing and Applications vol 25 no 2 pp 459ndash468 2014
[27] J Sadeghi S M Mousavi S T A Niaki and S SadeghildquoOptimizing a bi-objective inventory model of a three-echelonsupply chain using a tuned hybrid bat algorithmrdquo Transporta-tion Research Part E Logistics and Transportation Review vol70 no 1 pp 274ndash292 2014
[28] J-H Lin C-W Chou C-H Yang and H-L Tsai ldquoA chaoticlevy flight bat algorithm for parameter estimation in nonlineardynamic biological systemsrdquo Journal of Computing and Infor-mation Technology vol 2 no 2 pp 56ndash63 2012
[29] X-B Meng X Z Gao Y Liu and H Zhang ldquoA novel batalgorithm with habitat selection and Doppler effect in echoesfor optimizationrdquo Expert Systems with Applications vol 42 no17-18 pp 6350ndash6364 2015
[30] X Cai W Li L Wang Q Kang Q Wu and X HuangldquoBat algorithm with Gaussian walk for directing orbits ofchaotic systemsrdquo International Journal of Computing Scienceand Mathematics vol 5 no 2 pp 198ndash208 2014
[31] S Yilmaz and E U Kucuksille ldquoImproved Bat Algorithm(IBA) on continuous optimization problemsrdquo Lecture Notes onSoftware Engineering vol 1 no 3 pp 279ndash283 2013
[32] A M Taha and A Y C Tang ldquoBat algorithm for roughset attribute reductionrdquo Journal of Theoretical and AppliedInformation Technology vol 51 no 1 pp 1ndash8 2013
[33] P-W Tsai J-S Pan B-Y Liao M-J Tsai and V Istanda ldquoBatalgorithm inspired algorithm for solving numerical optimiza-tion problemsrdquo Applied Mechanics and Materials vol 148-149pp 134ndash137 2012
[34] I Pavlyukevich ldquoLevy flights non-local search and simulatedannealingrdquo Journal of Computational Physics vol 226 no 2 pp1830ndash1844 2007
[35] A Draa S Bouzoubia and I Boukhalfa ldquoA sinusoidal differ-ential evolution algorithm for numerical optimisationrdquo AppliedSoft Computing Journal vol 27 pp 99ndash126 2015
[36] X-S Yang and S Deb ldquoEagle strategy using Levy walk andfirefly algorithms for stochastic optimizationrdquo Studies in Com-putational Intelligence vol 284 pp 101ndash111 2010
[37] R N Mantegna ldquoFast accurate algorithm for numerical sim-ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994
[38] S Rahnamayan H R Tizhoosh and M M A SalamaldquoOpposition-based differential evolution for optimization ofnoisy problemsrdquo in Proceedings of the 2006 IEEE Congress onEvolutionary Computation (CEC rsquo06) pp 1865ndash1872 VancouverCanada July 2006
[39] H R Tizhoosh ldquoOpposition-based learning a new schemefor machine intelligencerdquo in Proceedings of the InternationalConference on Computational Intelligence for Modelling Controland Automation pp 695ndash701 2005
[40] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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