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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 526315 9 pageshttpdxdoiorg1011552013526315

Research ArticleA Simple and Efficient Artificial Bee Colony Algorithm

Yunfeng Xu12 Ping Fan3 and Ling Yuan1

1 School of Computer Science and Technology Huazhong University of Science and Technology Wuhan 430074 China2Department of Information Security Engineering Chinese Peoplersquos Public Security University Beijing 100038 China3 School of Computer Science Hubei University of Science and Technology Xianning 437100 China

Correspondence should be addressed to Ping Fan bjfan208126com

Received 7 September 2012 Revised 30 November 2012 Accepted 30 December 2012

Academic Editor Rui Mu

Copyright copy 2013 Yunfeng Xu 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

Artificial bee colony (ABC) is a new population-based stochastic algorithm which has shown good search abilities on manyoptimization problems However the original ABC shows slow convergence speed during the search process In order to enhancethe performance of ABC this paper proposes a new artificial bee colony (NABC) algorithm which modifies the search patternof both employed and onlooker bees A solution pool is constructed by storing some best solutions of the current swarm Newcandidate solutions are generated by searching the neighborhoodof solutions randomly chosen from the solution pool Experimentsare conducted on a set of twelve benchmark functions Simulation results show that our approach is significantly better or at leastcomparable to the original ABC and seven other stochastic algorithms

1 Introduction

Optimization problems arise in many application areas suchas engineering economy and management Effective andefficient optimization algorithms are always required to tackleincreasingly complex real-world optimization problems Inthe past several years some swarm intelligence algorithmsinspired by the social behaviors of birds fish or insects havebeen proposed to solve optimization problems such as par-ticle swarm optimization (PSO) [1] ant colony optimization(ACO) [2] artificial bee colony (ABC) [3] and firefly algo-rithm (FA) [4] A recent study has shown that ABC performssignificantly better or at least comparable to other swarmintelligence algorithms [5]

ABC is a new swarm intelligence algorithm proposedby Karaboga in 2005 which is inspired by the behavior ofhoney bees [3] Since the development of ABC it has beenapplied to solve different kinds of problems [6] Similar toother stochastic algorithms ABC also faces up some chal-lenging problems For example ABC shows slow convergencespeed during the search process Due to the special searchpattern of bees a new candidate solution is generated byupdating a random dimension vector of its parent solution

Therefore the offspring (new candidate solution) is similarto its parent and the convergence speed becomes slowMoreover ABC easily falls into local minima when handlingcomplex multimodal problems The search pattern of bees isgood at exploration but poor at exploitation [7] However agood optimization algorithm should balance exploration andexploitation during the search process

To improve the performance of ABC this paper proposesa new search pattern for both employed and onlooker beesIn the new approach some best solutions are utilized toaccelerate the convergence speed In addition a solution poolis constructed by storing the best 100119901 solutions in thecurrent swarm with 119901 isin (0 1] The best solution used in thesearch pattern is randomly selected from the solution poolThis is helpful to balance the exploration and exploitationExperiments are conducted on twelve benchmark functionsSimulation results show that our approach outperforms theoriginal ABC and several other stochastic algorithms

The rest of the paper is organized as follows In Section 2the original ABC algorithm is presented Section 3 gives abrief overview of related work Section 4 describes theproposed approach In Section 5 experimental studies arepresented Finally the work is concluded in Section 6

2 Mathematical Problems in Engineering

2 Artificial Bee Colony

Artificial bee colony (ABC) algorithm is a recently proposedoptimization technique which simulates the intelligent for-aging behavior of honey bees A set of honey bees is calledswarm which can successfully accomplish tasks throughsocial cooperation In the ABC algorithm there are threetypes of bees employed bees onlooker bees and scout beesThe employed bees search food around the food source intheirmemorymeanwhile they share the information of thesefood sources to the onlooker bees The onlooker bees tend toselect good food sources from those found by the employedbees The food source that has higher quality (fitness) willhave a large chance to be selected by the onlooker bees thanthe one of lower quality The scout bees are translated froma few employed bees which abandon their food sources andsearch new ones [8]

In the ABC algorithm the first half of the swarm con-sists of employed bees and the second half constitutes theonlooker beesThe number of employed bees or the onlookerbees is equal to the number of solutions in the swarm [3]

The ABC generates a randomly distributed initial popu-lation of SN solutions (food sources) where SN denotes theswarm size Let 119883

119894= 1199091198941 1199091198942 119909

119894119863 represent the 119894th

solution in the swarm where 119863 is the dimension size Eachemployed bee119883

119894generates a new candidate solution119881

119894in the

neighborhood of its present position as follows119907119894119895

= 119909119894119895

+ 120601119894119895

sdot (119909119894119895

minus 119909119896119895

) (1)

where 119883119896

is a randomly selected candidate solution(119894 = 119896) 119895 is a random dimension index selected from the set1 2 119863 and 120601

119894119895is a random number within [minus1 1]

Once the new candidate solution 119881119894is generated a greedy

selection is used If the fitness value of 119881119894is better than that

of its parent 119883119894 then update 119883

119894with 119881

119894 otherwise keep 119883

119894

unchangeableAfter all employed bees complete the search process they

share the information of their food sources with the onlookerbees through waggle dances An onlooker bee evaluates thenectar information taken from all employed bees and choosesa food source with a probability related to its nectar amountThis probabilistic selection is really a roulette wheel selectionmechanism which is described as follows

119901119894=

fit119894

sumSN119895=1

fit119895

(2)

where fit119894is the fitness value of the 119894th solution in the swarm

As seen the better the solution 119894 the higher the probability ofthe 119894th food source selected

If a position cannot be improved over a predefinednumber (called limit) of cycles then the food source isabandoned Assume that the abandoned source is 119883

119894 then

the scout bee discovers a new food source to be replaced with119883119894as follows

119909119894119895

= 119897119887119895+ rand (0 1) sdot (119906119887119895 minus 119897119887

119895) (3)

where rand(0 1) is a random number within [0 1] basedon a normal distribution and 119897119887 119906119887 are lower and upperboundaries of the119895th dimension respectively

3 Related Work

Since the development of ABC it has attracted muchattention for its excellent characteristics In the last decadedifferent versions of ABCs have been applied to variousproblems In this section we present a brief review of theseABC algorithms

Karaboga and Akay [5] presented a comparative studyof ABC A large set of benchmark functions are tested inthe experiments Results show that the ABC is better thanor similar to those of other population-based algorithmswith the advantage of employing fewer control parametersInspired by differential evolution (DE) algorithm Gao andLiu [7 9] proposed two improved versions of ABC In [7] anew search pattern called ABCbest1 is utilized to acceleratethe convergence speed In [9] ABCbest1 and anothersearch pattern called ABCrand1 are employed Moreover aparameter 119902 is intruded to control the frequency of these twopatterns Zhu and Kwong [8] utilized the search informationof the global best solution (119892best) to guide the search ofABC Reported results show that the new approach achievesbetter results than the original ABC algorithm Akay andKaraboga [10] proposed a modified ABC algorithm in whichtwo new search patterns frequency and magnitude of theperturbation are employed to improve the convergence rateResults show that the original ABC algorithm can efficientlysolve basic and simple functions while the modified ABCalgorithm obtains promising results on hybrid and complexfunctions when compared to some state-of-the-art algo-rithms Banharnsakun et al [11] modified the search patternof the onlooker bees in which the best feasible solutionsfound so far are shared globally among the entire swarmTherefore the new candidate solutions are similar to thecurrent best solution Kang et al [12] proposed a Rosen-brock ABC (RABC) algorithmwhich combines Rosenbrockrsquosrotational direction method with the original ABC Thereare two alternative phases of RABC the exploration phaserealized by ABC and the exploitation phased completed bythe Rosenbrock method Wu et al [13] combined harmonysearch (HS) and the ABC algorithm to construct a hybridalgorithm Comparison results show that the hybrid algo-rithm outperforms ABC HS and other heuristic algorithmsLi et al [14] proposed an improved ABC algorithm called I-ABC in which the best-so-far solution inertia weight andacceleration coefficients are introduced to modify the searchprocess Moreover a hybrid ABC algorithm (PS-ABC) basedon 119892best-guided ABC (GABC) [8] and I-ABC is proposedResults show that PS-ABC converges faster than I-ABC andABC

Karaboga and Ozturk [15] used ABC algorithm for dataclustering Experiments are conducted on thirteen typicaltest data sets from UCL Machine Learning Repository Theperformance of ABC is compared with PSO and other nineclassification techniques Simulation results demonstrate thatthe ABC algorithm can efficiently solve data clusteringZhang et al [16] also used ABC algorithm for clusteringThree data sets are tested The performance of ABC iscompared with genetic algorithm simulated annealing tabusearch ACO and K-NM-PSO Results demonstrate the

Mathematical Problems in Engineering 3

effectiveness of ABC on clustering Karaboga and Ozturk[17] applied ABC to solve fuzzy clustering Three data setsincluding cancer diabetes and heart chosen from UCIdatabase are tested Results indicate that the performance ofABC is successful in fuzzy clustering

TheABCalgorithm is usually used to solve unconstrainedoptimization problems In [18] Karaboga and Akay investi-gated the performance of ABC on constrained optimizationproblems In order to handle constraints Debrsquos rules con-sisting of three simple heuristic rules are employed Mezura-Montes and Velez-Koeppel [19] proposed an elitist ABCalgorithm for constrained real-parameter optimization inwhich the operators used by different types of bees are mod-ified Additionally a dynamic tolerance control mechanismfor equality constraints is utilized to facilitate the approachto the feasible region of the search space Yeh and Hsieh[20] proposed a penalty-guided ABC algorithm to solvereliability redundancy allocation problems Sabat et al [21]presented an application of ABC to extract the small signalequivalent circuitmodel parameters of GaASmetal-extendedsemiconductor field effect transistor (MESFT) device Theperformance comparison shows that ABC is better than PSO

It is known that the ABC algorithm is good at solvingoptimization problems over continuous search space Fordiscrete optimization problems it is a big challenge for theABC algorithm Li et al [22] used a hybrid Pareto-based ABCalgorithm to solve flexible job shop-scheduling problems Inthe new algorithm each food sources is represented by twovectors that is the machine assignment and the operationscheduling Moreover an external Pareto archive set is uti-lized to record nondominated solutions In [23] Kashan et aldesigned a new ABC algorithm called DisABC to optimizebinary structured problems Szeto et al [24] proposed anenhanced ABC algorithm to solve capacitated vehicle routingproblem The performance of the new approach is testedon two sets of standard benchmark instances Simulationresults show that the new algorithm outperforms the originalABC and several other existing algorithms Pan et al [25]presented a discrete ABC algorithm hybridized with a variantof iterated greedy algorithm to solve a permutation flow shop-scheduling problem with the total flow time criterion

4 Proposed Approach

Differential evolution (DE) has shown excellent search abili-ties on many optimization problems Like other population-based stochastic algorithms DE also starts with an initialpopulation with randomly generated candidate solutionsAfter initialization DE repeats three operations mutationcrossover and selection Among these operations mutationoperation is very important The mutation scheme highlyinfluences the performance of DEThere are several differentmutation schemes such as DErand1 DErand2 DEbest1and DEbest2 [26]

The property of amutation scheme determines the searchbehavior of individuals in the population For DErand1it results in good exploration but slow convergence speedFor DEbest1 it obtains fast convergence speed but poor

exploration The DErand1 and DEbest1 are described asfollows

119907119894119895

= 1199091199031119895

+ 119865 sdot (1199091199032119895

minus 1199091199033119895

)

119907119894119895

= 119909best119895 + 119865 sdot (1199091199031119895

minus 1199091199032119895

) (4)

where11988311990311198831199032 and119883

1199033are three randomly selected individ-

uals from the current population 119894 = 1199031 = 1199032 = 1199033 119883best is thebest individual found so far and the parameter 119865 is known asthe scale factor which is usually set to 05

As seen the search pattern of employed and onlookerbees is similar to themutation schemes ofDE It is known thatthe ABC algorithm is good at exploration but it shows slowconvergence speed By combining theDEbest1 and the ABCalgorithm it may accelerate the convergence speed of ABCHowever this hybridization is not a new idea In [7] Gaoand Liu embedded DErand1 and DEbest1 into the ABCalgorithm To balance the exploration and exploitation a newparameter 119902 is introduced Results reported in [7] show thatthe parameter 119902 is problem oriented and an empirical value119902 = 07 is used

In this paper we propose a newABC (calledNABC) algo-rithm by employing a modified DEbest1 strategy NABCdiffers from other hybrid algorithms [7 9] which combineABC and DE Although the global best individual usedin DEbest1 can accelerate the convergence speed by theattraction it may result in attracting too fast It meansthat new solutions move to the global best solution veryquickly To tackle this problem a solution pool is constructedby storing the best 100p solutions in the current swarmwith 119901 isin (0 1] The idea is inspired by an adaptive DEalgorithm (JADE) [27] It shares in commonwith the conceptof belief space of cultural algorithm (CA) [28] Both of themutilize some successful solutions stored in solution pool orsituational knowledge to guide other individuals But theupdating rule of the solution pool or situational knowledge isdifferentThenewABCbest1 strategy is described as follows

119907119894119895

= 119909119901

best119895 + 120601119894119895

sdot (1199091199031119895

minus 1199091199032119895

) (5)

where 119909119901best119895 is randomly chosen from the solution pool1198831199031

1198831199032are two randomly selected candidate solutions from the

current swarm 119894 = 1199031 = 1199032 119895 is a random dimension indexselected from the set 1 2 119863 and 120601

119894119895is a random

number within [minus1 1] Empirical studies show that a goodchoice of the parameter119901 should be set between 008 and 015In this paper 119901 is set to 01 for all experiments

According to the new search pattern described in (5)new candidate solutions are generated around some bestsolutionsThis is helpful to accelerate the convergence speedFor the existing ABCbest1 strategy proposed in [7] it onlysearches the neighborhood of the global best solution In ourapproach bees can search the neighborhood of different bestsolutions This can help avoid fast attraction

Themain steps of our new approachNABC algorithm arelisted as follows

Step 1 Randomly initialize the swarm

Step 2 Update the solution pool


Table 1 Benchmark functions used in the experiments

Name Function Range OptSphere 119891

1= sum119863

119894=11199092

119894[minus100 100] 0

Schwefel 222 1198912= sum119863

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + prod

119863

119894=1119909119894

[minus10 10] 0

Schwefel 12 1198913= sum119863

119894=1(sum119894

119895=1119909119895)2 [minus100 100] 0

Schwefel 221 1198914= max

119894(1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863) [minus100 100] 0

Rosenbrock 1198915= sum119863minus1

119894=1[100(119909

119894+1minus 1199092

119894)2+ (119909119894minus 1)2] [minus30 30] 0

Step 1198916= sum119863

119894=1(lfloor119909119894+ 05rfloor)

2 [minus100 100] 0

Quartic with noise 1198917= sum119863

119894=11198941199094

119894+ rand[0 1) [minus128 128] 0

Schwefel 226 1198918= sum119863

119894=1minus119909119894sin (radic|119909

119894|) [minus500 500] minus125695

Rastrigin 1198919= sum119863

119894=1[1199092

119894minus 10 cos(2120587119909

119894) + 10] [minus512 512] 0

Ackley 11989110

= minus20 exp(minus02radic1

119863sum119863

119894=11199092119894) minus exp(

1

119863sum119863

119894=1cos (2120587119909

119894)) + 20 + 119890 [minus32 32] 0

Griewank 11989111

=1

4000sum119863

119894=11199092

119894minus prod119863

119894=1cos(

119909119894

radic119894) + 1 [minus600 600] 0

Penalizedf 12 =

120587

11986310sin2

(3120587119910119894) + sum119863minus1

119894=1(119910119894minus 1)2[1 + 10sin2

(120587119910119894+1

)]

+(119910119863minus 1)2 + sum119863

119894=1119906(119909119894 5 100 4)

[minus50 50] 0

Step 3 For each employed bee generate a new candidatesolution119881

119894according to (5) Evaluate the fitness of119881

119894and use

a greed selection to choose a better one between119883119894and 119881

119894as

the new119883119894

Step 4 Each onlooker bee calculates 119901119894according to (2)

Step 5 Generate a new 119881119894according to (5) based on 119901

119894and

the current solution 119883119894(food source) Evaluate the fitness of

119881119894and use a greed selection to choose a better one between

119883119894and 119881

119894as the new119883

119894

Step 6 The scout bee determines the abandoned 119883119894 if exists

and update it by (3)

Step 7 Update the best solution found so far and cycle =cycle + 1

Step 8 If the number of cycles reaches to the maximumvalue MCN then stop the algorithm and output the resultsotherwise go to Step 2

Compared to the original ABC algorithm our approachNABC does not add extra operations except for the con-struction of the solution pool However this operation doesnot add the computational complexity Both NABC and theoriginal ABC have the same computational complexity

5 Experimental Study

51 Test Functions In order to verify the performance ofNABC experiments are conducted on a set of twelve bench-mark functions These functions were early used in [29]

According to their properties they are divided into twoclasses unimodal functions (119891

1minus 1198917) and multimodal func-

tions (1198918minus11989112) All functions areminimization problems and

their dimensional size is 30 Table 1 presents the descriptionsof these functions where Opt is the global optimum

The experiments are performed on the same computerwith Intel (R) Core (TM)2 Duo CPU T6400 (200GHz) and2GB RAM Our algorithm is implemented using C++ andcomplied with Microsoft Visual C++ 60 under theWindowsXP (SP3)

52 Comparison of NABC with ABC In order to investigatethe effectiveness of our new search pattern this sectionpresents a comparison of NABC with the original ABCalgorithm In the experiments both NABC and ABC usethe same parameter settings The population size SN limitand maximum number of cycles (MSN) are set to 100 100and 1000 respectively The parameter 119901 is set to 01 basedon empirical studies All results reported in this section areaveraged over 30 independent runs

Table 2 presents the computational results of ABC andNABC on the twelve functions where ldquoMeanrdquo indicates themean function value and ldquoStd Devrdquo represents the standarddeviation The best results between ABC and NABC areshown in bold From the results it can be seen that NABCachieves better results than ABC on all test functions exceptfor 1198916 On this function both the two algorithms can find the

global optimum For 1198918and 119891

9 NABC can successfully find

the global optimum while ABC converges to near-optimalsolutions It demonstrates that the new search pattern usedin NABC is helpful to improve the accuracy of solutions

In order to compare the convergence speed of NABC andABC Figure 1 lists the convergence processes of them on


Table 2 Results achieved by the ABC algorithm and NABC

Functions ABC NABCMean Std Dev Mean Std Dev

1198911

375119890 minus 10 273119890 minus 10 475e minus 16 386e minus 161198912


123119890 + 04 239119890 + 03 990e + 03 167e + 031198914

392119890 + 01 152119890 + 01 145e + 01 432e + 001198915

283119890 + 00 179119890 + 00 450e minus 02 238e minus 021198916

000e + 00 000e + 00 000e + 00 000e + 001198917


minus123324 157119890 + 02 minus125695 123e minus 101198919

290119890 minus 09 531119890 minus 09 000e + 00 000e + 0011989110




some representative functions As seen NABC shows fasterconvergence speed than ABC It confirms that the new searchpattern can accelerate the convergence speed

53 Comparison of NABC with Other Algorithms To furtherverify the performance of NABC this section comparesNABC with other population-based algorithms includingsome recently proposed ABC algorithms

531 Comparison of NABC with Evolution Strategies Thissection focuses on the comparison of the NABC algorithmwith Evolution Strategies (ES)The versions of the ES includeclassical evolution strategies (CES) [30] fast evolution strate-gies (FES) [30] covariance matrix adaptation evolutionstrategies (CMA-ES) [31] and evolutionary strategies learnedwith automatic termination (ESLAT) [32]

Theparameter settings of CES FES CMA-ES andESLATcan be found in [32] For NABC the population size and themaximum number of fitness evaluations are set to 20 and100000 (it means that the MSN is 2500) respectively Theparameter limit is set to 600 [5] The parameter 119901 is set to01 based on empirical studies All algorithms are conductedon 50 runs for each test function

Table 3 presents the comparison results of CES FESCMA-ES ESLAT and NABC Results of CES FES CMA-ES and ESLAT were taken from Table 20 in [5] Amongthese algorithms the best results are shown in bold Thelast column of Table 3 reports the statistical significancelevel of the difference of the means of NABC and the bestalgorithm among the four evolution strategies Note that hereldquo+rdquo represents the 119905 value of 49 degrees of freedom which issignificant at a 005 level of significance by two-tailed testldquosdotrdquo indicates the difference of means which is not statisticallysignificant and ldquoNArdquomeans not applicable covering cases forwhich the two algorithms achieve the same accuracy results[33]

From the results it can be seen that NABC outperformsCES and FES on eight functions while CES and FES achievebetter results on three For 119891

6 CES FES and NABC find

the global optimum while ESLAT and CMA-ES fail to solveit NABC performs better than ESLAT on ten functionswhile ESLAT outperforms NABC for the rest of the twofunctions CMA-ES achieves better results than NABC onthree functions while NABC performs better for the restof the nine functions The comparison results show thatthe evolutionary strategies perform better than NABC onunimodal functions such as 119891

1minus 1198914 NABC outperforms the

evolutionary strategies on all multimodal functions (1198918minus11989112)

532 Comparison of NABC with Other Improved ABC Algo-rithms In this section we present a comparison of NABCwith three recently proposed ABC algorithms The involvedalgorithms are listed as follows

(1) 119892best-guided ABC algorithm (GABC) [8](2) Improved ABC algorithm (I-ABC) [14](3) Hybridization of GABC and I-ABC (PS-ABC) [14](4) Our approach NABC

In the experiments the population size SN is set to 40and limit equals 200 The maximum number of cycles is setas 1000 Other parameter settings of GABC I-ABC and PS-ABC can be found in [14] The parameter 119901 used in NABCis set to 01 based on empirical studies All algorithms areconducted 30 times for each test function and the meanfunction values are reported

Table 4 presents the comparison results of NABC withthree other ABC algorithms Results of GABC I-ABC andPS-ABC were taken from Tables 4 and 5 in [14] The bestresults among the four algorithms are shown in bold Fromthe results NABC outperforms GABC on all test functionsexcept for 119891

2 On this function GABC is slightly better than

NABC I-ABC achieves better results than NABC on fivefunctions while NABC performs better on six functionsFor the rest of 119891

9 I-ABC PS-ABC and NABC can find the

global optimum PS-ABC obtains better results than NABCon six functions while NABC outperforms PS-ABC on fivefunctions Both I-ABC and PS-ABC achieve significantlybetter results on three unimodal functions such as119891

11198912 and


Table 3 Comparison of NABC with evolution strategies

Functions CES FES ESLAT CMA-ES NABC SignificanceMean Mean Mean Mean Mean

1198911

170e minus 26 250119890 minus 04 200119890 minus 17 970119890 minus 23 288119890 minus 16 1198912


338119890 + 02 140119890 minus 03 610119890 minus 06 710e minus 23 686119890 + 03 1198914

241119890 + 00 550119890 minus 03 780119890 minus 01 540e minus 12 430119890 minus 01 1198915

277119890 + 01 333119890 + 01 193119890 + 00 400119890 minus 01 262e minus 01 +1198916

000e + 00 000e + 00 200119890 minus 02 144119890 + 00 000e + 00 NA1198917

470119890 minus 02 120e minus 02 390119890 minus 01 230119890 minus 01 138119890 minus 02 +1198918

minus8000 minus125564 minus2300 minus76371 minus125695 +1198919

134119890 + 01 160119890 minus 01 465119890 + 00 518119890 + 01 000e + 00 +11989110

600119890 minus 13 120119890 minus 02 180119890 minus 08 690119890 minus 12 325e minus 14 +11989111


146119890 + 00 280119890 minus 06 150119890 minus 12 120119890 minus 04 252e minus 16 +

Table 4 Comparison of NABC with other ABC algorithms

Functions GABC I-ABC PS-ABC NABC SignificanceMean Mean Mean Mean

1198911

626119890 minus 16 000e + 00 000e + 00 543119890 minus 16 1198912

936119890 minus 16 000e + 00 000e + 00 624119890 minus 15 1198913

109119890 + 04 143119890 + 04 611e + 03 844119890 + 03 1198914

126119890 + 01 127119890 minus 197 000e + 00 579119890 + 00 1198915

748119890 + 00 264119890 + 01 159119890 + 00 145e minus 01 +1198916

249119890 minus 09 384119890 minus 10 572119890 minus 16 000e + 00 +1198917

156119890 minus 01 196119890 minus 02 215119890 minus 02 172e minus 02 +1198918

minus124073 minus1225103 minus125642 minus125695 +1198919

331119890 minus 02 000e + 00 000e + 00 000e + 00 NA11989110

778119890 minus 10 888e minus 16 888e minus 16 107119890 minus 13 11989111

696119890 minus 04 000e + 00 000e + 00 111119890 minus 16 11989112

585119890 minus 16 711119890 minus 12 553119890 minus 16 467e minus 16 +

1198914 On these functions they can find the global optimum

while GABC and NABC only find near-optimal solutionsexcept for 119891

4 For 119891

4 both GABC and NABC fall into local

minima I-ABC and PS-ABC successfully find the globaloptimum on 119891

11 while GABC and NABC fail For function

11989110 I-ABC and PS-ABC are slightly better than NABC For

other two multimodal functions 1198918and 11989112 NABC performs

better than other three ABC algorithms Compared to I-ABCand PS-ABC our approach NABC is simpler and easier toimplement

6 Conclusions

Artificial bee colony is a new optimization technique whichhas shown to be competitive to other population-basedstochastic algorithms However ABC and other stochasticalgorithms suffer from the same problems For example theconvergence speed of ABC is typically slower than PSO andDE Moreover the ABC algorithm easily gets stuck when

handling complex multimodal problems The main reasonis that the search pattern of both employed and onlookerbees is good at exploration but poor at exploitation In orderto balance the exploration and exploitation of ABC thispaper proposes a new ABC variant (NABC) It is known thatDEbest1 mutation scheme is good at exploitation Basedon DEbest1 a new search pattern called ABCbest1 withsolution pool is proposed Our approach differs from otherimproved ABC algorithms by hybridization of DEbest1 andABC

To verify the performance of our approach a set oftwelve benchmark functions are used in the experimentsComparison of NABC with ABC demonstrates that ournew search pattern can effectively accelerate the convergencespeed and improve the accuracy of solutions Another com-parison demonstrates that NABC is significantly better or atleast comparable to other stochastic algorithms Compared toother improvedABC algorithms our approach is simpler andeasier to implement


151050

0 200 400 600 800 1000

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)

Number of cycles (1198911)

(a)

151050

0 200 400 600 800 1000

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)

5minus


(b)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

9

10

11

12


(c)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

minus2000

minus4000

minus6000

minus8000

minus10000

minus12000

0


(d)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(e)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(f)

0 200 400 600 800 1000

1050

minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)

minus40

ABCNABC


(g)

0 200 400 600 800 1000

1510

20

50

minus5

minus10

minus15

minus20

minus25

minus30

minus35Meanbestfunctio

nvalue(

log)

minus40

ABCNABC


(h)

Figure 1 The convergence processes of ABC and NABC on some functions


Acknowledgments

This work was supported by the National Natural ScienceFund of Hubei Province under Grant 2012FFB00901 theScience and Technology Research Project of Xianning Cityunder Grant XNKJ-1203 Doctoral Start Fund of Hubei Uni-versity of Science and Technology under Grant BK1204 theTeaching Research Project of Hubei University of Science andTechnology under Grant 2012X016B Application InnovationProject of the Ministry of Public Security under Grant2005yycxhbst117 The Key Research Project of Science andTechnology of Hubei Province under Grant 2007AA301C33Key Lab of InformationNetwork Security ofMinistry of Pub-lic Security under Grant C09602 and Application InnovativeProject of Public Security of Hubei Province under Granthbst2009sjkycx014

References

[1] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[2] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics B vol 26 no 1 pp 29ndash41 1996

[3] D Karaboga ldquoAn idea based on honey bee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[4] X S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-Inspired Com-puting vol 2 no 2 pp 78ndash84 2010

[5] D Karaboga and B Akay ldquoA comparative study of artificial Beecolony algorithmrdquo Applied Mathematics and Computation vol214 no 1 pp 108ndash132 2009

[6] D Karaboga and B Akay ldquoA survey algorithms simulating beeswarm intelligencerdquo Artificial Intelligence Review vol 31 no 1ndash4 pp 61ndash85 2009

[7] W F Gao and S Y Liu ldquoA modified artificial bee colonyalgorithmrdquo Computers and Operations Research vol 39 no 3pp 687ndash697 2012

[8] G Zhu and S Kwong ldquoGbest-guided artificial bee colonyalgorithm for numerical function optimizationrdquo Applied Math-ematics and Computation vol 217 no 7 pp 3166ndash3173 2010

[9] W Gao and S Liu ldquoImproved artificial bee colony algorithm forglobal optimizationrdquo Information Processing Letters vol 111 no17 pp 871ndash882 2011

[10] B Akay and D Karaboga ldquoA modified Artificial Bee Colonyalgorithm for real-parameter optimizationrdquo Information Sci-ences vol 192 pp 120ndash142 2012

[11] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in Artificial Bee Colony algorithmrdquoApplied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[12] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[13] B Wu C Qian W Ni and S Fan ldquoHybrid harmony searchand artificial bee colony algorithm for global optimizationproblemsrdquoComputers ampMathematics with Applications vol 64no 8 pp 2621ndash2634 2012

[14] G Li P Niu and X Xiao ldquoDevelopment and investigation ofefficient artificial bee colony algorithm for numerical function

optimizationrdquo Applied Soft Computing vol 12 no 1 pp 320ndash332 2012

[15] D Karaboga and C Ozturk ldquoA novel clustering approachArtificial BeeColony (ABC) algorithmrdquoApplied SoftComputingJournal vol 11 no 1 pp 652ndash657 2011

[16] C Zhang D Ouyang and J Ning ldquoAn artificial bee colonyapproach for clusteringrdquo Expert Systems with Applications vol37 no 7 pp 4761ndash4767 2010

[17] D Karaboga and C Ozturk ldquoFuzzy clustering with artificial beecolony algorithmrdquo Scientific Research and Essays vol 5 no 14pp 1899ndash1902 2010

[18] D Karaboga and B Akay ldquoA modified Artificial Bee Colony(ABC) algorithm for constrained optimization problemsrdquoApplied Soft Computing Journal vol 11 no 3 pp 3021ndash30312011

[19] E Mezura-Montes and R E Velez-Koeppel ldquoElitist artificialbee colony for constrained real-parameter optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 1ndash8 2010

[20] W-C Yeh and T-J Hsieh ldquoSolving reliability redundancyallocation problems using an artificial bee colony algorithmrdquoComputers amp Operations Research vol 38 no 11 pp 1465ndash14732011

[21] S L Sabat S K Udgata and A Abraham ldquoArtificial beecolony algorithm for small signal model parameter extractionof MESFETrdquo Engineering Applications of Artificial Intelligencevol 23 no 5 pp 689ndash694 2010

[22] J Q Li Q K Pan S X Xie and S Wang ldquoA hybridartificial bee colony algorithm for flexible job shop schedulingproblemsrdquo International Journal of Computers Communicationsand Control vol 6 no 2 pp 286ndash296 2011

[23] M H Kashan N Nahavandi and A H Kashan ldquoDisABC anew artificial bee colony algorithm for binary optimizationrdquoApplied Soft Computing vol 12 no 1 pp 342ndash352 2012

[24] W Y Szeto Y Wu and S C Ho ldquoAn artificial bee colony algo-rithm for the capacitated vehicle routing problemrdquo EuropeanJournal of Operational Research vol 215 no 1 pp 126ndash135 2011

[25] Q-K Pan M F Tasgetiren P N Suganthan and T J ChualdquoA discrete artificial bee colony algorithm for the lot-streamingflow shop scheduling problemrdquo Information Sciences vol 181no 12 pp 2455ndash2468 2011

[26] 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

[27] J Zhang and A C Sanderson ldquoJADE adaptive differentialevolution with optional external archiverdquo IEEE Transactions onEvolutionary Computation vol 13 no 5 pp 945ndash958 2009

[28] R G Reynolds ldquoAn introduction to cultural algorithmsrdquo inProceedings of the 3rd Annual Conference on EvolutionaryProgramming pp 131ndash139 1994

[29] X Yao Y Liu and G Lin ldquoEvolutionary programming madefasterrdquo IEEE Transactions on Evolutionary Computation vol 3no 2 pp 82ndash102 1999

[30] X Yao and Y Liu ldquoFast evolution strategiesrdquo Control andCybernetics vol 26 no 3 pp 467ndash496 1997

[31] N Hansen and A Ostermeier ldquoAdapting arbitrary normalmutation distributions in evolution strategies the covariancematrix adaptationrdquo in Proceedings of the IEEE InternationalConference on Evolutionary Computation (ICEC rsquo96) pp 312ndash317 May 1996


[32] A Hedar andM Fukushima ldquoEvolution strategies learned withautomatic termination criteriardquo in Proceedings of the Conferenceon Soft Computing and Intelligent Systems and the InternationalSymposium on Advanced Intelligent Systems pp 1ndash9 TokyoJapan 2006

[33] S Das A Abraham U K Chakraborty and A KonarldquoDifferential evolution using a neighborhood-based mutationoperatorrdquo IEEE Transactions on Evolutionary Computation vol13 no 3 pp 526ndash553 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Artificial Bee Colony

Artificial bee colony (ABC) algorithm is a recently proposedoptimization technique which simulates the intelligent for-aging behavior of honey bees A set of honey bees is calledswarm which can successfully accomplish tasks throughsocial cooperation In the ABC algorithm there are threetypes of bees employed bees onlooker bees and scout beesThe employed bees search food around the food source intheirmemorymeanwhile they share the information of thesefood sources to the onlooker bees The onlooker bees tend toselect good food sources from those found by the employedbees The food source that has higher quality (fitness) willhave a large chance to be selected by the onlooker bees thanthe one of lower quality The scout bees are translated froma few employed bees which abandon their food sources andsearch new ones [8]

In the ABC algorithm the first half of the swarm con-sists of employed bees and the second half constitutes theonlooker beesThe number of employed bees or the onlookerbees is equal to the number of solutions in the swarm [3]

The ABC generates a randomly distributed initial popu-lation of SN solutions (food sources) where SN denotes theswarm size Let 119883

119894= 1199091198941 1199091198942 119909

119894119863 represent the 119894th

solution in the swarm where 119863 is the dimension size Eachemployed bee119883

119894generates a new candidate solution119881

119894in the

neighborhood of its present position as follows119907119894119895

= 119909119894119895

+ 120601119894119895

sdot (119909119894119895

minus 119909119896119895

) (1)

where 119883119896

is a randomly selected candidate solution(119894 = 119896) 119895 is a random dimension index selected from the set1 2 119863 and 120601

119894119895is a random number within [minus1 1]

Once the new candidate solution 119881119894is generated a greedy

selection is used If the fitness value of 119881119894is better than that

of its parent 119883119894 then update 119883

119894with 119881

119894 otherwise keep 119883

119894

unchangeableAfter all employed bees complete the search process they

share the information of their food sources with the onlookerbees through waggle dances An onlooker bee evaluates thenectar information taken from all employed bees and choosesa food source with a probability related to its nectar amountThis probabilistic selection is really a roulette wheel selectionmechanism which is described as follows

119901119894=

fit119894

sumSN119895=1

fit119895

(2)

where fit119894is the fitness value of the 119894th solution in the swarm

As seen the better the solution 119894 the higher the probability ofthe 119894th food source selected

If a position cannot be improved over a predefinednumber (called limit) of cycles then the food source isabandoned Assume that the abandoned source is 119883

119894 then

the scout bee discovers a new food source to be replaced with119883119894as follows

119909119894119895

= 119897119887119895+ rand (0 1) sdot (119906119887119895 minus 119897119887

119895) (3)

where rand(0 1) is a random number within [0 1] basedon a normal distribution and 119897119887 119906119887 are lower and upperboundaries of the119895th dimension respectively

3 Related Work

Since the development of ABC it has attracted muchattention for its excellent characteristics In the last decadedifferent versions of ABCs have been applied to variousproblems In this section we present a brief review of theseABC algorithms

Karaboga and Akay [5] presented a comparative studyof ABC A large set of benchmark functions are tested inthe experiments Results show that the ABC is better thanor similar to those of other population-based algorithmswith the advantage of employing fewer control parametersInspired by differential evolution (DE) algorithm Gao andLiu [7 9] proposed two improved versions of ABC In [7] anew search pattern called ABCbest1 is utilized to acceleratethe convergence speed In [9] ABCbest1 and anothersearch pattern called ABCrand1 are employed Moreover aparameter 119902 is intruded to control the frequency of these twopatterns Zhu and Kwong [8] utilized the search informationof the global best solution (119892best) to guide the search ofABC Reported results show that the new approach achievesbetter results than the original ABC algorithm Akay andKaraboga [10] proposed a modified ABC algorithm in whichtwo new search patterns frequency and magnitude of theperturbation are employed to improve the convergence rateResults show that the original ABC algorithm can efficientlysolve basic and simple functions while the modified ABCalgorithm obtains promising results on hybrid and complexfunctions when compared to some state-of-the-art algo-rithms Banharnsakun et al [11] modified the search patternof the onlooker bees in which the best feasible solutionsfound so far are shared globally among the entire swarmTherefore the new candidate solutions are similar to thecurrent best solution Kang et al [12] proposed a Rosen-brock ABC (RABC) algorithmwhich combines Rosenbrockrsquosrotational direction method with the original ABC Thereare two alternative phases of RABC the exploration phaserealized by ABC and the exploitation phased completed bythe Rosenbrock method Wu et al [13] combined harmonysearch (HS) and the ABC algorithm to construct a hybridalgorithm Comparison results show that the hybrid algo-rithm outperforms ABC HS and other heuristic algorithmsLi et al [14] proposed an improved ABC algorithm called I-ABC in which the best-so-far solution inertia weight andacceleration coefficients are introduced to modify the searchprocess Moreover a hybrid ABC algorithm (PS-ABC) basedon 119892best-guided ABC (GABC) [8] and I-ABC is proposedResults show that PS-ABC converges faster than I-ABC andABC

Karaboga and Ozturk [15] used ABC algorithm for dataclustering Experiments are conducted on thirteen typicaltest data sets from UCL Machine Learning Repository Theperformance of ABC is compared with PSO and other nineclassification techniques Simulation results demonstrate thatthe ABC algorithm can efficiently solve data clusteringZhang et al [16] also used ABC algorithm for clusteringThree data sets are tested The performance of ABC iscompared with genetic algorithm simulated annealing tabusearch ACO and K-NM-PSO Results demonstrate the





4 Proposed Approach




119907119894119895

= 1199091199031119895

+ 119865 sdot (1199091199032119895

minus 1199091199033119895

)

119907119894119895

= 119909best119895 + 119865 sdot (1199091199031119895

minus 1199091199032119895

) (4)

where11988311990311198831199032 and119883





119907119894119895

= 119909119901

best119895 + 120601119894119895

sdot (1199091199031119895

minus 1199091199032119895

) (5)




119894119895is a random









1= sum119863

119894=11199092

119894[minus100 100] 0

Schwefel 222 1198912= sum119863

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + prod

119863

119894=1119909119894

[minus10 10] 0

Schwefel 12 1198913= sum119863

119894=1(sum119894

119895=1119909119895)2 [minus100 100] 0


119894(1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863) [minus100 100] 0


119894=1[100(119909

119894+1minus 1199092

119894)2+ (119909119894minus 1)2] [minus30 30] 0

Step 1198916= sum119863

119894=1(lfloor119909119894+ 05rfloor)

2 [minus100 100] 0


119894=11198941199094

119894+ rand[0 1) [minus128 128] 0

Schwefel 226 1198918= sum119863

119894=1minus119909119894sin (radic|119909

119894|) [minus500 500] minus125695


119894=1[1199092

119894minus 10 cos(2120587119909

119894) + 10] [minus512 512] 0

Ackley 11989110


119863sum119863

119894=11199092119894) minus exp(

1

119863sum119863

119894=1cos (2120587119909

119894)) + 20 + 119890 [minus32 32] 0

Griewank 11989111

=1

4000sum119863

119894=11199092


119894=1cos(

119909119894

radic119894) + 1 [minus600 600] 0

Penalizedf 12 =

120587

11986310sin2

(3120587119910119894) + sum119863minus1

119894=1(119910119894minus 1)2[1 + 10sin2

(120587119910119894+1

)]

+(119910119863minus 1)2 + sum119863

119894=1119906(119909119894 5 100 4)

[minus50 50] 0



119894and use


119894as

the new119883119894



119894and



119883119894and 119881

119894as the new119883

119894






















1198911



123119890 + 04 239119890 + 03 990e + 03 167e + 031198914

392119890 + 01 152119890 + 01 145e + 01 432e + 001198915

283119890 + 00 179119890 + 00 450e minus 02 238e minus 021198916

000e + 00 000e + 00 000e + 00 000e + 001198917



290119890 minus 09 531119890 minus 09 000e + 00 000e + 0011989110






















11198912 and




1198911





277119890 + 01 333119890 + 01 193119890 + 00 400119890 minus 01 262e minus 01 +1198916

000e + 00 000e + 00 200119890 minus 02 144119890 + 00 000e + 00 NA1198917



134119890 + 01 160119890 minus 01 465119890 + 00 518119890 + 01 000e + 00 +11989110






1198911

626119890 minus 16 000e + 00 000e + 00 543119890 minus 16 1198912

936119890 minus 16 000e + 00 000e + 00 624119890 minus 15 1198913

109119890 + 04 143119890 + 04 611e + 03 844119890 + 03 1198914

126119890 + 01 127119890 minus 197 000e + 00 579119890 + 00 1198915

748119890 + 00 264119890 + 01 159119890 + 00 145e minus 01 +1198916




331119890 minus 02 000e + 00 000e + 00 000e + 00 NA11989110


696119890 minus 04 000e + 00 000e + 00 111119890 minus 16 11989112




4 For 119891







6 Conclusions





151050

0 200 400 600 800 1000

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)


(a)

151050

0 200 400 600 800 1000

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)

5minus


(b)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

9

10

11

12


(c)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

minus2000

minus4000

minus6000

minus8000

minus10000

minus12000

0


(d)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(e)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(f)

0 200 400 600 800 1000

1050

minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)

minus40

ABCNABC


(g)

0 200 400 600 800 1000

1510

20

50

minus5

minus10

minus15

minus20

minus25

minus30


nvalue(

log)

minus40

ABCNABC


(h)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













4 Proposed Approach




119907119894119895

= 1199091199031119895

+ 119865 sdot (1199091199032119895

minus 1199091199033119895

)

119907119894119895

= 119909best119895 + 119865 sdot (1199091199031119895

minus 1199091199032119895

) (4)

where11988311990311198831199032 and119883





119907119894119895

= 119909119901

best119895 + 120601119894119895

sdot (1199091199031119895

minus 1199091199032119895

) (5)




119894119895is a random









1= sum119863

119894=11199092

119894[minus100 100] 0

Schwefel 222 1198912= sum119863

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + prod

119863

119894=1119909119894

[minus10 10] 0

Schwefel 12 1198913= sum119863

119894=1(sum119894

119895=1119909119895)2 [minus100 100] 0


119894(1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863) [minus100 100] 0


119894=1[100(119909

119894+1minus 1199092

119894)2+ (119909119894minus 1)2] [minus30 30] 0

Step 1198916= sum119863

119894=1(lfloor119909119894+ 05rfloor)

2 [minus100 100] 0


119894=11198941199094

119894+ rand[0 1) [minus128 128] 0

Schwefel 226 1198918= sum119863

119894=1minus119909119894sin (radic|119909

119894|) [minus500 500] minus125695


119894=1[1199092

119894minus 10 cos(2120587119909

119894) + 10] [minus512 512] 0

Ackley 11989110


119863sum119863

119894=11199092119894) minus exp(

1

119863sum119863

119894=1cos (2120587119909

119894)) + 20 + 119890 [minus32 32] 0

Griewank 11989111

=1

4000sum119863

119894=11199092


119894=1cos(

119909119894

radic119894) + 1 [minus600 600] 0

Penalizedf 12 =

120587

11986310sin2

(3120587119910119894) + sum119863minus1

119894=1(119910119894minus 1)2[1 + 10sin2

(120587119910119894+1

)]

+(119910119863minus 1)2 + sum119863

119894=1119906(119909119894 5 100 4)

[minus50 50] 0



119894and use


119894as

the new119883119894



119894and



119883119894and 119881

119894as the new119883

119894






















1198911



123119890 + 04 239119890 + 03 990e + 03 167e + 031198914

392119890 + 01 152119890 + 01 145e + 01 432e + 001198915

283119890 + 00 179119890 + 00 450e minus 02 238e minus 021198916

000e + 00 000e + 00 000e + 00 000e + 001198917



290119890 minus 09 531119890 minus 09 000e + 00 000e + 0011989110






















11198912 and




1198911





277119890 + 01 333119890 + 01 193119890 + 00 400119890 minus 01 262e minus 01 +1198916

000e + 00 000e + 00 200119890 minus 02 144119890 + 00 000e + 00 NA1198917



134119890 + 01 160119890 minus 01 465119890 + 00 518119890 + 01 000e + 00 +11989110






1198911

626119890 minus 16 000e + 00 000e + 00 543119890 minus 16 1198912

936119890 minus 16 000e + 00 000e + 00 624119890 minus 15 1198913

109119890 + 04 143119890 + 04 611e + 03 844119890 + 03 1198914

126119890 + 01 127119890 minus 197 000e + 00 579119890 + 00 1198915

748119890 + 00 264119890 + 01 159119890 + 00 145e minus 01 +1198916




331119890 minus 02 000e + 00 000e + 00 000e + 00 NA11989110


696119890 minus 04 000e + 00 000e + 00 111119890 minus 16 11989112




4 For 119891







6 Conclusions





151050

0 200 400 600 800 1000

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)


(a)

151050

0 200 400 600 800 1000

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)

5minus


(b)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

9

10

11

12


(c)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

minus2000

minus4000

minus6000

minus8000

minus10000

minus12000

0


(d)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(e)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(f)

0 200 400 600 800 1000

1050

minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)

minus40

ABCNABC


(g)

0 200 400 600 800 1000

1510

20

50

minus5

minus10

minus15

minus20

minus25

minus30


nvalue(

log)

minus40

ABCNABC


(h)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1= sum119863

119894=11199092

119894[minus100 100] 0

Schwefel 222 1198912= sum119863

119894=1

10038161003816100381610038161199091198941003816100381610038161003816 + prod

119863

119894=1119909119894

[minus10 10] 0

Schwefel 12 1198913= sum119863

119894=1(sum119894

119895=1119909119895)2 [minus100 100] 0


119894(1003816100381610038161003816119909119894

1003816100381610038161003816 1 le 119894 le 119863) [minus100 100] 0


119894=1[100(119909

119894+1minus 1199092

119894)2+ (119909119894minus 1)2] [minus30 30] 0

Step 1198916= sum119863

119894=1(lfloor119909119894+ 05rfloor)

2 [minus100 100] 0


119894=11198941199094

119894+ rand[0 1) [minus128 128] 0

Schwefel 226 1198918= sum119863

119894=1minus119909119894sin (radic|119909

119894|) [minus500 500] minus125695


119894=1[1199092

119894minus 10 cos(2120587119909

119894) + 10] [minus512 512] 0

Ackley 11989110


119863sum119863

119894=11199092119894) minus exp(

1

119863sum119863

119894=1cos (2120587119909

119894)) + 20 + 119890 [minus32 32] 0

Griewank 11989111

=1

4000sum119863

119894=11199092


119894=1cos(

119909119894

radic119894) + 1 [minus600 600] 0

Penalizedf 12 =

120587

11986310sin2

(3120587119910119894) + sum119863minus1

119894=1(119910119894minus 1)2[1 + 10sin2

(120587119910119894+1

)]

+(119910119863minus 1)2 + sum119863

119894=1119906(119909119894 5 100 4)

[minus50 50] 0



119894and use


119894as

the new119883119894



119894and



119883119894and 119881

119894as the new119883

119894






















1198911



123119890 + 04 239119890 + 03 990e + 03 167e + 031198914

392119890 + 01 152119890 + 01 145e + 01 432e + 001198915

283119890 + 00 179119890 + 00 450e minus 02 238e minus 021198916

000e + 00 000e + 00 000e + 00 000e + 001198917



290119890 minus 09 531119890 minus 09 000e + 00 000e + 0011989110






















11198912 and




1198911





277119890 + 01 333119890 + 01 193119890 + 00 400119890 minus 01 262e minus 01 +1198916

000e + 00 000e + 00 200119890 minus 02 144119890 + 00 000e + 00 NA1198917



134119890 + 01 160119890 minus 01 465119890 + 00 518119890 + 01 000e + 00 +11989110






1198911

626119890 minus 16 000e + 00 000e + 00 543119890 minus 16 1198912

936119890 minus 16 000e + 00 000e + 00 624119890 minus 15 1198913

109119890 + 04 143119890 + 04 611e + 03 844119890 + 03 1198914

126119890 + 01 127119890 minus 197 000e + 00 579119890 + 00 1198915

748119890 + 00 264119890 + 01 159119890 + 00 145e minus 01 +1198916




331119890 minus 02 000e + 00 000e + 00 000e + 00 NA11989110


696119890 minus 04 000e + 00 000e + 00 111119890 minus 16 11989112




4 For 119891







6 Conclusions





151050

0 200 400 600 800 1000

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)


(a)

151050

0 200 400 600 800 1000

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)

5minus


(b)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

9

10

11

12


(c)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

minus2000

minus4000

minus6000

minus8000

minus10000

minus12000

0


(d)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(e)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(f)

0 200 400 600 800 1000

1050

minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)

minus40

ABCNABC


(g)

0 200 400 600 800 1000

1510

20

50

minus5

minus10

minus15

minus20

minus25

minus30


nvalue(

log)

minus40

ABCNABC


(h)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198911



123119890 + 04 239119890 + 03 990e + 03 167e + 031198914

392119890 + 01 152119890 + 01 145e + 01 432e + 001198915

283119890 + 00 179119890 + 00 450e minus 02 238e minus 021198916

000e + 00 000e + 00 000e + 00 000e + 001198917



290119890 minus 09 531119890 minus 09 000e + 00 000e + 0011989110






















11198912 and




1198911





277119890 + 01 333119890 + 01 193119890 + 00 400119890 minus 01 262e minus 01 +1198916

000e + 00 000e + 00 200119890 minus 02 144119890 + 00 000e + 00 NA1198917



134119890 + 01 160119890 minus 01 465119890 + 00 518119890 + 01 000e + 00 +11989110






1198911

626119890 minus 16 000e + 00 000e + 00 543119890 minus 16 1198912

936119890 minus 16 000e + 00 000e + 00 624119890 minus 15 1198913

109119890 + 04 143119890 + 04 611e + 03 844119890 + 03 1198914

126119890 + 01 127119890 minus 197 000e + 00 579119890 + 00 1198915

748119890 + 00 264119890 + 01 159119890 + 00 145e minus 01 +1198916




331119890 minus 02 000e + 00 000e + 00 000e + 00 NA11989110


696119890 minus 04 000e + 00 000e + 00 111119890 minus 16 11989112




4 For 119891







6 Conclusions





151050

0 200 400 600 800 1000

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)


(a)

151050

0 200 400 600 800 1000

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)

5minus


(b)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

9

10

11

12


(c)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

minus2000

minus4000

minus6000

minus8000

minus10000

minus12000

0


(d)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(e)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(f)

0 200 400 600 800 1000

1050

minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)

minus40

ABCNABC


(g)

0 200 400 600 800 1000

1510

20

50

minus5

minus10

minus15

minus20

minus25

minus30


nvalue(

log)

minus40

ABCNABC


(h)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198911





277119890 + 01 333119890 + 01 193119890 + 00 400119890 minus 01 262e minus 01 +1198916

000e + 00 000e + 00 200119890 minus 02 144119890 + 00 000e + 00 NA1198917



134119890 + 01 160119890 minus 01 465119890 + 00 518119890 + 01 000e + 00 +11989110






1198911

626119890 minus 16 000e + 00 000e + 00 543119890 minus 16 1198912

936119890 minus 16 000e + 00 000e + 00 624119890 minus 15 1198913

109119890 + 04 143119890 + 04 611e + 03 844119890 + 03 1198914

126119890 + 01 127119890 minus 197 000e + 00 579119890 + 00 1198915

748119890 + 00 264119890 + 01 159119890 + 00 145e minus 01 +1198916




331119890 minus 02 000e + 00 000e + 00 000e + 00 NA11989110


696119890 minus 04 000e + 00 000e + 00 111119890 minus 16 11989112




4 For 119891







6 Conclusions





151050

0 200 400 600 800 1000

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)


(a)

151050

0 200 400 600 800 1000

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)

5minus


(b)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

9

10

11

12


(c)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

minus2000

minus4000

minus6000

minus8000

minus10000

minus12000

0


(d)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(e)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(f)

0 200 400 600 800 1000

1050

minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)

minus40

ABCNABC


(g)

0 200 400 600 800 1000

1510

20

50

minus5

minus10

minus15

minus20

minus25

minus30


nvalue(

log)

minus40

ABCNABC


(h)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










151050

0 200 400 600 800 1000

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)


(a)

151050

0 200 400 600 800 1000

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Meanbestfunctio

nvalue(

log)

5minus


(b)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

9

10

11

12


(c)

0 200 400 600 800 1000

Meanbestfunctio

nvalue(

log)

minus2000

minus4000

minus6000

minus8000

minus10000

minus12000

0


(d)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(e)

0 200 400 600 800 1000

1050minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)


(f)

0 200 400 600 800 1000

1050

minus5

minus10

minus15

minus20

minus25

minus30

minus35

Meanbestfunctio

nvalue(

log)

minus40

ABCNABC


(g)

0 200 400 600 800 1000

1510

20

50

minus5

minus10

minus15

minus20

minus25

minus30


nvalue(

log)

minus40

ABCNABC


(h)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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