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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International Journal of

ReconfigurableComputing

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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

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International Journal of

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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

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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Distributed Sensor Networks

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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

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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 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

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

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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Advances in

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Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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httpwwwhindawicom Volume 2014

Advances in

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International Journal of

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ArtificialNeural Systems

Advances in

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RoboticsJournal 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

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

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

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

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 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