Research ArticleEnhancing Artificial Bee Colony Algorithm withSelf-Adaptive Searching Strategy and Artificial ImmuneNetwork Operators for Global Optimization

Tinggui Chen1 and Renbin Xiao2

1 College of Computer Science amp Information Engineering Zhejiang Gongshang University Zhejiang ProvinceHangzhou 310018 China

2 Institute of Systems Engineering Huazhong University of Science and Technology Hubei Province Wuhan 430074 China

Correspondence should be addressed to Renbin Xiao rbxiaohusteducn

Received 10 November 2013 Accepted 30 December 2013 Published 18 February 2014

Academic Editors Z Cui and X Yang

Copyright copy 2014 T Chen and R XiaoThis 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) algorithm inspired by the intelligent foraging behavior of honey bees was proposed by KarabogaIt has been shown to be superior to some conventional intelligent algorithms such as genetic algorithm (GA) artificial colonyoptimization (ACO) and particle swarm optimization (PSO) However the ABC still has some limitations For example ABC caneasily get trapped in the local optimum when handing in functions that have a narrow curving valley a high eccentric ellipse orcomplex multimodal functions As a result we proposed an enhanced ABC algorithm called EABC by introducing self-adaptivesearching strategy and artificial immune network operators to improve the exploitation and exploration The simulation resultstested on a suite of unimodal or multimodal benchmark functions illustrate that the EABC algorithm outperforms ACO PSO andthe basic ABC in most of the experiments

1 Introduction

With the rapid development of communication technologycomputer technology and network technology humans putforward higher demand for efficient intelligent technolo-gies However in view of the complexity constraint andnonlinearity of practical issues searching for all kinds ofemerging intelligent computing technologies for solving largeand complex problems has been paid attention by more andmore scholars

As one of typical intelligent computing approachesswarm intelligence that combines biology and social basedheuristics has become a research interest to many researchscientists of related fields in recent years It is based on thecollective behavior of social insects flock of birds or schoolsof fish The key components of swarm intelligence are self-organization and division of labor In a self-organizationsystem each of the covered unitsmay respond to local stimuliindividually and act together to accomplish a global task

via division of labor without a centralized supervision Theentire system can adapt to internal and external changes effi-ciently [1 2] Particle swarm optimization (PSO) algorithmintroduced by Hsieh et al in 2008 [3] can be thought of asa typical swarm whose individual agents are birds and hasbeen widely used in all kinds of combination optimizationproblems [4ndash7] What is more other algorithms such asant colony optimization (ACO) [8 9] and artificial immunenetwork (aiNet) [10 11] can also be considered as subfields ofswarm intelligence

Nowadays an artificial bee colony (ABC) algorithminspired by the intelligent foraging behavior of honey beeswas proposed by Karaboga [12] Due to its simplicity and easeof implementation the ABC algorithm has captured muchattention and has been widely applied to solve many practicaloptimization problems such as supply chain management[13] and scheduling optimization [14] In addition a setof well-known numerical comparisons have demonstratedthat the performance of ABC algorithm is competitive to

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 438260 12 pageshttpdxdoiorg1011552014438260

2 The Scientific World Journal

other intelligent ones including genetic algorithm (GA)PSO differential evolution (DS) and evolution strategy (ES)although it uses fewer control parameters [15ndash17]

However similar to other intelligent algorithms the ABCstill has some limitations For example the convergence speedof ABC is slow because of its stochastic nature What ismore ABC can easily get trapped in the local optimumwhenhanding in functions that have a narrow curving valley a higheccentric ellipse or complex multimodal functions [18] Allthese insufficiencies prevent the further applications of theABC algorithm

Therefore in this work some modifications to the stan-dard ABC algorithm are introduced for global optimizationof numerical functions Firstly ABC algorithm is extendedby employing the chaotic systems and the diversity-basedmethod when producing the initial population Next self-adaptive searching strategy is incorporated into the employedbee search process to improve the exploitation In additionthe enhanced algorithm retains the main steps of ABCand incorporates an aiNet-based search technique whereaiNet algorithm has powerful multimodal searching abilityas well as good stabilization due to its negative selection andnetwork compression operators Therefore ABC and aiNethave complementary advantages and a hybrid of the twomaybe a possible strategy to improve the performances of ABC

The remainder of this paper is organized as follows InSection 2 the works related to the ABC algorithm are sum-marized In Section 3 the basic ABC algorithm is describedSection 4 describes the modified ABC algorithm combinedwith self-adaptive searching strategy and artificial immunenetwork operators In Section 5 the testing of the proposedalgorithm through 15 benchmark functions problems iscarried out and the simulation results are compared Finallyconclusions and future works are provided in Section 6

2 Previous Works on the ABC Algorithm

TheABC algorithm imitated the foraging behavior of honey-bee andwas first applied to numerical optimization problemsHowever due to its weaknesses mentioned in Section 1 someresearchers proposedmany improved strategies For exampleAlatas [19] used different chaotic maps to generate sequencessubstituting random numbers for different parameters ofABC when producing initial population Moreover Gao andLiu [18 20] also employed both the chaotic systems andopposition-based learning methods to enhance the globalconvergence In these two literature works authors alsodeveloped an improved solution search equation which wasbased on that the bee searched only for the best solutionof the previous iteration to improve the exploitation Theexperiments derived from a set of 28 benchmark functionsdemonstrated that the performance of thismethodwas betterthan the other methods Unlike these studies mentionedabove inspired by PSO Zhu andKwong [21] proposedGbest-guided ABC algorithm by incorporating the information ofthe global best solution into the solution search equationto improve the exploitation Banharnsakun et al [22] pre-sented a best-so-far method for solution updates in the ABC

algorithm and the searching method based on a dynamicadjustment of search range depending on the iteration wasalso introduced for scout bees The test results showed thatthe proposed method was able to produce higher qualitysolutions with faster convergence than either the originalABC or the current state-of-the-art ABC-based algorithm

Besides numerical optimization the ABC algorithm hasbeen widely used to solve large-scale problems and engineer-ing design optimization Some representative applications areintroduced as follows Kang et al [23] used a hybrid ABCalgorithm which combines Nelder-Mead simplex methodwith ABC algorithm for structural inverse analysis problemsand its performance outperforms other heuristic meth-ods Singh [24] applied the ABC algorithm for the leaf-constrained minimum spanning tree (LCMST) problem andcompared the approach against GA ACO and tabu searchIn the literature [24] it was reported that the proposedalgorithm was superior to the other methods in termsof solution qualities and computational time Zhang et al[25] developed the ABC clustering algorithm to optimallypartition 119873 objectives into 119870 cluster and Debrsquos rules wereused to direct the search direction of each candidate Panet al [26] used the discrete ABC algorithm to solve the lot-streaming flow shop scheduling problem with the criterionof total weighted earliness and tardiness penalties under boththe idling and no-idling cases Samanta and Chakraborty[27] employed ABC algorithm to search out the optimalcombinations of different operating parameters for threewidely used nontraditionalmachining (NTM) processes thatis electrochemical machining electrochemical dischargemachining and electrochemical micromachining processesAlejandro et al [28] used the ABC algorithm in order tofind the optimal distribution of material with the aim ofestablishing a standard time for this duty by examininghow this was applied in a local manufacturing plant Thesimulation results showed that using this approach might beconvenient to set the standard times in the selected companyAll these researches illustrated that the ABC algorithm haspowerful ability to solve much more complex engineeringproblems

3 The Original Artificial BeeColony Algorithm

The artificial bee colony has been inspired by the intelli-gent behavior of real honey bees The honey bees in thisalgorithm are categorized into three groups employed beesonlooker bees and scout bees The first half of the colonyconsists of employed bees and the other half includes theonlookers Each solution in the search space consists of a setof optimization parameters which represent a food sourcepopulation The number of employed bees is equal to thenumber of food sources around the hive In other words forevery food source there is only one employed bee What ismore onlooker bees wait in the hive and decide on a foodsource to exploit based on the information shared by theemployed bees Scout bees are translated froma few employedbees whose food source has been exhausted by the bees

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Similar to the other swarm intelligence algorithms ABCis an iterative process The units of the original ABC algo-rithm can be explained as follows

31 The Initial Population of Solutions The initial populationof solutions is filled with SN number of randomly generated119863-dimensional real-valued vectors (ie food sources) Eachfood source is generated as follows

119909119895

119894 = 119909119895

min + rand (0 1) (119909119895max minus 119909

119895

min) (1)

where 119894 = 1 2 SN 119895 = 1 2 119863 119909119895min and 119909119895max arethe lower and upper bounds for the dimension119895 respectivelyThese food sources are randomly assigned to SN number ofemployed bees and their fitness is evaluated

After initialization the population of the food sourceis subjected to repeat cycle of the search processes of theemployed bees the onlooker bees and the scout bees

32The Search Phase of Employed Bees In this phase in orderto produce a candidate food position from the old one theABC uses the following equation

V119895119894 = 119909119895

119894 + 120593119895

119894 (119909119895

119894 minus 119909119895

119896) (2)

where 119895 isin 1 2 119863 and 119896 isin 1 2 SN are randomlychosen indexes Although 119896 is determined randomly it hasto be different from 119894 120593119894119895 is a random number in the range[minus1 1] Equation (2) denotes that within the neighborhoodof every food source site represented by 119909119894 a food source V119894 isdetermined by changing one parameter of 119909119894

Once V119894 is obtained it will be evaluated and comparedto 119909119894 A greedy selection is applied between 119909119894 and V119894then the better one is selected depending on fitness valuesrepresenting the nectar amount of the food sources at 119909119894 andV119894 If the fitness of V119894 is equal to or better than that of 119909119894 V119894will replace 119909119894 and become a newmember of the populationsotherwise 119909119894 is retained

33 The Selection Phase of Onlooker Bees In this phase eachonlooker bee selects one of the food sources depending on thefitness value obtained from the employed bees The fitness-based probability selection scheme may be a roulette wheelranking based stochastic universal sampling tournamentselection or another selection scheme In original ABCroulette wheel selection scheme is employed described as anequation below

119875119894 =fit (119909119894)

sumSN119898=1 fit (119909119898)

(3)

where fit(119909119894) is the fitness value of solution 119894 Obviously thehigher the fit(119909119894) is the more probability is that the 119894th foodsource is selected After the food source is selected onlookerbees will go to the selected food source and produce a newcandidate position in the neighborhood of the selected foodsource by using (2)

34 Scout Bee Phase In a cycle after all employed andonlooker bees complete their searches the ABC algorithmchecks if there is any exhausted source to be abandoned Ifa position cannot be improved further through a predeter-mined number of cycles then that food source is assumedto be abandoned The scouts can accidentally discover richentirely unknown food sourcesThis operation can be definedas in (4) shown as follows This process helps avoid subopti-mal solutions The value of predetermined number of cyclesis called ldquolimitrdquo for abandoning a food source which is animportant control parameter of ABC algorithm

119909119894 = 119909min + rand (0 1) (119909max minus 119909min) (4)

where 119909min and 119909max are the lower and upper bounds ofvariable 119909119894

35Main Steps of the Original Artificial Colony Bee AlgorithmBased on the above explanation there are three controlparameters used in the original ABC the number of thefood sources which is equal to the number of employed bees(SN) the value of limit and the maximum cycle number(MEN) Detailed pseudocode of the ABC algorithm is givenin Algorithm 1 [15]

4 Enhancing Artificial Bee Algorithm withArtificial Immune Network

The original version of ABC algorithm is very efficient formultidimensional basic functions However the convergencerate of the algorithm is poor when working with somecomplex multimodal functions and composite functionsFurthermore due to its poor exploration process the ABCalgorithm easily gets trapped in a local optimum In orderto improve these limitations existing in the ABC algorithmsome modifications inspired by the artificial immune net-work (ai-Net) algorithm so as to accelerate the convergencerate have been introduced in the search process of theoriginal ABC algorithm In addition an improved searchmechanism based on the self-adaptive strategy as well asa novel generation method of the initial population is alsoproposed

41 Generation of the Initial Population One of the modifi-cations in the ABC algorithm is generating effective initialpopulation which can affect the convergence rate and thequality of the final solution Generally random initializationis the most adopted approach to generate initial populationwhich often makes solutions concentrated in a local areaChaotic sequences derived from a chaotic map have beenproven easy and fast to store there is no need for storage oflong sequences Recently chaotic sequences have been usedinstead of random sequences and shown somewhat goodresults in many applications Therefore chaotic maps areintroduced in ABC to improve the global convergence byescaping the local solutions in [19] Meanwhile in order toincrease the population diversity similar individuals shouldbe gotten rid of The main principle is to compare theaffinity inspired by ai-Net algorithm between two different

4 The Scientific World Journal

(1) Generate the initial population 119909119894 (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (2)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Calculate the probability values 119875119894 for the solution (119909119894) by (3)(7) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(8) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(9) Memorize the best solution so far(10) Cycle = cycle +1(11) Until cycle = MEN

Algorithm 1 Pseudocode of main body of ABC algorithm

individuals As a result this work proposes a novel initial-ization approach which uses chaotic systems and affinity-based compression method to produce initial populationHere according to the literature [19] sinus map is selectedand its equation is defined as follows

119888119898119899+1 = 23(119888119898119899)2 sin(120587119888119898

119899)

119888119898119899 = 0 1 2 119873

(5)

where 119899 is the iteration counter and119873 is the maximum num-ber of chaotic iterations Furthermore the affinity equationbetween two different individuals is defined as Euclideandistance shown in (6)

affinity (119909119894 119909119896) = radic119863

sum

119895=1

(119909119895

119894 minus 119909119895

119896)2

(119894 = 119896 119895 isin (1 2 119863)) gt 120585

(6)

where 120585 is a threshold value defined in advance so as to controlthe difference between two individuals 119863 is the number ofoptimization parameters Based on these operators we pro-pose the following algorithm to generate initial populationand its corresponding pseudocode is given in Algorithm 2

42 An Improved Search Mechanism Based on the Self-Adaptive Strategy As mentioned above the original ABCalgorithm is good at exploration but poor at exploitationdue to two reasons On the one hand in (2) the coefficient120593119894119895 is a uniform random number in [minus1 1] and 119909119896119895 is a

random individual in the population therefore the searchprocess of solutions illustrated by (2) is random enoughfor exploration [21] On the other hand a greedy selectionmechanism is employed between the old and candidatesolutions which may easily make solutions get trapped in

a local optimal What is more the slower convergence rateof the algorithm is another limitation when working withsome complex composite functions As a result we introducea self-adaptive strategy to improve its search process and thedetailed explanations are as follows

Firstly we can see from (2) that there is only one differentelement between the candidate solution and the old one (iethe 119895th element) This search strategy may be efficient inearlier iterations However when the solution approaches toa local optimum its search efficiency becomes poor in lateriterations To handle this limitation similar to [29 30] weintroduce a parameter 119871 to control the difference betweenthe candidate solution and the old one where how to choosethe value of the parameter 119871 is very important Generally thehigher the value of 119871 is the more information is brought intothe candidate solution In the literature [18] the parameter119871 is a fixed constant and chosen according to simulationexperiments Different from this approach mentioned abovethis paper proposes a self-adaptive adjustable strategy todetermine the parameter 119871 and the corresponding searchequation is given below

V119897119894 = 119909119897119894 + 119888119898

119897119894 (119909119897119894 minus 119909119897119896) 119897 = 1 2 119871 (7)

119871 = 1 +

1003817100381710038171003817100381710038171003817119863(

iteration2 timesMEN

)

1003817100381710038171003817100381710038171003817 (8)

where 119888119898119897119894 is a chaotic map defined by (5)119863 is the number ofoptimization parameters A symbol sdot denotes a roundingoperator We can see from (8) that onlooker bees will searchbetter solutions in only one direction in the first iterationand they will search in the whole space with the increasein the value of 119871 when the solutions are closely to the localoptimum in later iterations However we limit the value of 119871nomore than 1+1198632This is because with the higher valueof 119871 the solution generated by the search equations (7)-(8)

The Scientific World Journal 5

Set the population size SN the number of randomly generated individuals SN119903 ≫ SN the maximum number of chaoticiterations119873 gt 200 the number of optimization parameters D and the individual counter 119894 = 1 119895 = 1- -Chaotic systems- -for 119894 = 1 to SN119903 do

for 119895 = 1 to D doRandomly initialize the first chaotic variable cm0119895 isin (0 1) and set iteration counter 119899 = 0for 119899 = 1 to N do

cm119899+1119895 = 23(cm119899119895)2 sin(120587cm

119899119895)

end for119909119895

119894 = 119909119895

min + cm119895119899(119909119895max minus 119909

119895

min)

end forend for- - Affinity-based compression method- -Set the threshold value 120585 and the individual counter 119894 = 1 119895 = 1for 119894 = 1 to (SN119903 minus 1) do

for 119896 = 119894 + 1 to SN119903 do

if affinity(119909119894 119909119896) = radicsum119863

119895=1(119909119895

119894 minus 119909119895

119896)2lt 120585

SN119903 = SN119903 minus 1 delete the individual 119896 in the populationend if

end forend forSelecting SN fittest individuals form set (119883(SN119903)) as initial population

Algorithm 2 Modified initialization step of ABC algorithm

is more likely random search operator As a result (7)-(8) willdynamically adjust the position of onlooker bees by allowingthem to explore with a wider search space in later iterationsAs the number of the iterations increases the correspondingsearch space of onlooker bees will also increase

The second modification in the ABC algorithm lies inthe selection probability of onlooker bees associated with thefood source In the original ABC algorithm the fitness valuesobtained from the employed bees are adopted to determinethe selection probabilities of onlooker bees Nevertheless thefitness comparisons among different individuals only reflectthe qualitative information In this work based on the ideaof the fitness evolution we introduce an environment factor120578119894 corresponding to every food source so as to evaluate itsexploitation potential In other words the parameter 120578119894 isused to evaluate quantitatively the environment situationsof exploitation for every food source As the number ofiterations increases the higher the value of 120578119894 the betterexploitation environment it corresponds to At this momentmore onlooker bees will follow the corresponding employedone to its food source position with higher nectar amount inorder to accelerate exploitation efficiency Conversely if thevalue of 120578119894 is lower its corresponding solution lies in theworseexploitation environment which means that it is difficult tofind out a better solution around the old food source andless onlooker bees will be attracted by the employed oneAs a result how to define 120578119894 needs explain Generally theparameter 120578119894 is associated with both a fitness change amountΔ119891 and a count accumulator 119862 where Δ119891 denotes the fitnessdifference associated with the same food source betweentwo adjacent generations which reflects the exploitation

potential of the corresponding food source position Besidesa count accumulator119862will be explained in the following textThe equations below reflect the fitness change amount Δ119891between two adjacent generations

Δ119891119905 (119909119894) =

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(9)

Δ1198911015840119905 (119909119894) = exp [Δ119891119905 (119909119894)] = log

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(10)

where the parameter 119905 denotes the number of iterations andthe symbol | | means absolute value sign As can be seenfrom (9) the value range of Δ119891 is between 0 and 1 Whenthe value of Δ119891 is higher it means the corresponding foodsource has a higher exploitation potential and is largerlypossible to find out a better solution and vice versa Howeverat later iterations the value of Δ119891 will be very small andneed appropriate amplification As a result we adopt powerfunction to amplifyΔ119891 here and take the number 119890 as its baseDue to this reason (9) is substituted for (10)

In addition if Δ119891 is less than a given small value inadvance (ie Δ119891 le Δ1198910 and Δ1198910 is the threshold of Δ119891) theimproved ABC algorithm may trigger a count accumulatorcalled Counter (represented by 119862 in this paper) which is usedto record how many times the quality of the solution hasnot improved (it corresponds to the number of iterations inthe algorithm) The following equation is used to express thecount accumulator119862 at the 119905th iteration Generally the larger

6 The Scientific World Journal

the value of 119862 the less exploitation potential that the foodsource position corresponds to

119862 (119905) =

119862 (119905 minus 1) + 119879 (119905) =

119905

sum

119894=119904

119879 (119894) 119879 (119905) = 1

0 119879 (119905) = 0

(11)

where119879 represents a pulse signalWhenΔ119891 gt 0 orΔ119891 ge Δ1198910119879 = 0 conversely when Δ119891 = 0 or Δ119891 le Δ1198910 119879 = 1 It meansthat

119879 (119905) = 1 (Δ119891 (119905) = 0) or (Δ119891 (119905) le Δ119891(119905)0) 0 (Δ119891 (119905) gt 0) or (Δ119891 (119905) ge Δ119891(119905)0)

(12)

Note that according to the performance requirement ofspecific problems the maximum times of the stagnation ofglobal extrema 119862max should be defined in advance whichmeans if a minimum of a function has not been updated forcontinuous 119862max iterations the current exploitation area hasfew potentials to find out the better solution and computationresources should be redistributed Generally we define thatthe 119862max is equal to or not less than 5 In addition 119862(119905)is normalized for simplifying the problem and 1198621015840(119905) can beobtained which is expressed as follows

1198621015840(119905) =

sum119905119894=119904 119879 (119894) + 1

119862max + 1 (13)

On the basis of the definitions of the fitness changeamount Δ119891and the count accumulator 119862 the environmentfactor 120578 is defined as follows

120578 (119905) =Δ1198911015840(119905)

1198621015840 (119905) (14)

Submitting (10) and (13) into (14) we can achieve thefollowing equation

120578 (119905) =(119862max + 1) sdot exp

1003816100381610038161003816(fit119905 (119909119894) minus fit119905minus1 (119909119894)) fit119905minus1 (119909119894)1003816100381610038161003816

sum119905119894=119904 119879 (119894) + 1

(15)

In accordance with the expression of the environmentfactor 120578(119909119894) corresponding to every food source the selectionprobability of onlooker bees associated with the food sourcecan be substituted with the following equation

119875119894 =120578119905 (119909119894)

sumSN119898=1 120578119905 (119909119898)

(16)

43 Enhancing Convergence Efficiency with Artificial ImmuneNetwork Operators In the basic ABC system artificial beesfly around in the search space Some (like employed andonlooker bees) choose food source depending on the experi-ence of themselves and their nest mates and then adjust theirpositions but others (like scouts) fly and choose the foodsources randomly without using experience If the nectar

amount of a new source is higher than that of the previousone in their memory they memorize the new food sourceposition and forget the previous one Thus the ABC systemcombines local search methods carried out by employedand onlooker bees with global search methods managedby Karaboga and Basturk [15] However unlike the ABCsystem the concept of artificial immune system (AIS) wasoriginated by observing how the defense mechanism ofnatural immune system protects against attacks by antigensThere are numerous AIS algorithms developed for a varietyof applications where artificial immune network (aiNet forshort) is a typical one and its algorithms and models areoriginally proposed to perform information compressionand data clustering based on artificial immune system (AIS)theory [31] Immune network-based algorithms are similarto clonal selection algorithms in that they both measurethe goodness of antibodies by affinities and both methodsinclude a series of steps for selecting cloning and mutatingantibodiesThemajor difference is that the immune network-based algorithms are represented bynetwork graph structures[32] Compared with other ones the immune network-basedalgorithms employ extra procedures of antibody pruning andsuppressing This allows the models to generate a smallerless-redundant population of antibody representatives whichis desirable for solving multimodal function optimizationComparing ABC optimization with ai-Net algorithm wecan see that the advantages of ABC optimization lie in itsneighborhood search method according to the profitabilityof food sources However ai-Net algorithm adopts fixedclonal individuals to perform local search which has certainblindness In addition due to introducing network com-pression negative selection and other operators ai-Net canmaintain the diversity of the population and reduce thepossibility of being trapped into a local minimum Unlike theai-Net algorithm ABC optimization maintains populationdiversity through random search of scout bees which hasobvious limitation Based on the analysis mentioned aboveif network compression and negative selection deriving fromai-Net algorithm are introduced into ABC optimization thisimproved one may have a powerful and efficient multimodalsearching ability as well as good stabilization The detailedprocess is as follows

Different employed bee individual corresponds to differ-ent food source position In order to eliminate redundantand similar food sources negative selection and networkcompression are used to compare with the similarities amongvarious individuals The Euclidean distance of two employedbee individuals119883119894 and119883119896 is adopted as shown in (17)

119889 (119883119894 119883119896) = radic

119863

sum

119895=1

(119883119895

119894 minus 119883119895

119896)2

(119894 = 119896) (17)

In order to simplify the problem the affinity concept isintroduced which is obtained by the following equation usingthe normalization method

119860 (119883119894 119883119896) =1

1 + 119889 (119883119894 119883119896) (18)

The Scientific World Journal 7

(1) Generate the initial population 119909119894 based on chaotic maps and affinity strategy (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (7)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Adopt negative selection and network compression to eliminate redundant and similar food sources by using (18)(7) Randomly generate the same number of new individuals(8) Calculate the probability values 119875119894 for the solution (119909119894) by (16)(9) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(10) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(11) Memorize the best solution so far(12) Cycle = cycle + 1(13) Until cycle = MEN

Algorithm 3 Pseudocode of main body of the enhanced ABC algorithm

where the value range of 119860(119883119894119883119896) is between 0 and 1 Thesmaller the value of 119860(119883119894119883119896) is the larger the value of119860(119883119894119883119896) is which means that two different employed beeindividuals have a higher similarity Specially when119860(119883119894119883119896)equals 1 these two ones are identical According to negativeselection and network compression operators redundant andsimilar food sources should be eliminated We predefine athreshold value 120576 so as to realize wipe-off of redundantindividuals It also means when 119860(119883119894119883119896) is equal to or greatthan 120576 we think these two ones are identical and only one canbe retained and other should bewiped off Repeat this processuntil the affinity of any two individuals in a population isless than 120576 In doing so the population size may be reducedNevertheless in order to keep the population size unchangedthe same number of new individuals need generating ran-domlyThrough negative selection and network compressionoperators the exploitation efficiencywill be improved and thecorresponding convergence rate of the algorithm will also beaccelerated

44 Main Steps of the Enhanced Artificial Bee Colony Algo-rithm Based on the above analysis three main improve-ments including novel generation of initial population self-adaptive searching strategy and redundant individual com-pression operator are presented and the detailed pseudo-codeis given in Algorithm 3

5 Experimental Studies on FunctionOptimization Problems

51 Benchmark Functions and Parameter Settings In thissection numerical experiment is used to test the performanceof the enhanced ABC (shorthand for EABC) proposed in this

paper Summarized in Table 1 are the 15 scalable benchmarkfunctions 1198911 sim 11989110 are continuous unimodal functions11989111 sim 11989115 are multimodal functions and the number oftheir local minima increases exponentially with the problemdimension

In order to testify the performance of different intelligentalgorithms we compare the EABC with the standard ACOPSO and ABC In all simulations the population size ofACO PSO ABC and EABC is 50 The maximum numberof function evaluations (FE) is set to 5000 The thresholdvalue of the affinity 120576 is 09 Other related parameter valuesof ACO PSO and ABC are referred in the literature [17]All experiment results reported are obtained based on 30independent runsThe experiment results are the best worstmean and standard deviation of the statistical experimentaldata

52 Simulation Results The performance on the solutionaccuracy of EABC is compared with that of ACO PSO andABC Table 2 shows the optimization of the 15 benchmarkfunctions obtained in the 30 independent runs by eachalgorithm and some interesting results can be found inTable 2

Firstly almost all algorithms have identical performanceon most of unimodal functions 1198911 to 1198914 1198917 1198918 and 11989114However on other functions these four algorithms showdifferent performance especially for multimodal ones suchas 11989111 11989112 11989113 and 11989115 Fox example on function 11989115the best values obtained by ACO PSO ABC and EABCare minus10296 minus699347 minus125669 and minus1525687 respectivelyIt means that EABC can be efficiently applied for solvingmultimodal and multidimensional function optimizationproblems due to its abundant operators such as clonal

8 The Scientific World Journal

Table 1 Benchmark functions used in experiments

Number Function Dimension Property Range Min

1 1198911(119909) =

119863

sum

119894=1

1199092

119894 30 Unimodal [minus100 100] 0

2 1198912(119909) =

119863

sum

119894=1

1198941199092

119894 30 Unimodal [minus10 10] 0

3 1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 5 Unimodal [minus512 512] 0

4 1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 30 Unimodal [minus100 100] 0

5 1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 30 Unimodal [minus128 128] 0

6 1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 30 Unimodal [minus100 100] 0

7 1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) 2 Unimodal [minus100 100] minus1

8 1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 2 Unimodal [minus10 10] 0

9 1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 30 Unimodal [minus10 10] 0

10 11989110 (119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 30 Unimodal [minus30 30] 0

11 11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 30 Multimodal [minus512 512] 0

12 11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 30 Multimodal [minus600 600] 0

13 11989113 (119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 10 Multimodal [minus50 50] 0

14 11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

2 Multimodal [minus100 100] 0

15 11989115 (119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817) 30 Multimodal [minus500 500] minus125695

selection and negative selection which outperforms ACOPSO and ABC In addition on the one hand on basicunimodal functions both the basic ABC and EABC havealmost identical solving performance on the other handon the multimodal functions these two ones display hugedifference

Secondly the EABC algorithm can find optimal or closer-to-optimal solutions on the complex multimodal functions11989111 11989112 11989113 and 11989114 Although the result of multi-dimensionfunction 11989115 is slightly far from the known global optimumthe EABC is superior to the other algorithms all the sameAt the same time for almost all benchmark functionsstandard deviations of the EABC obtained from the statisticalexperimental data are no greater than those of others expectfor 11989110 In addition the differences of EABC between thebest andworst solutions for these 15 benchmark functions are

relatively smaller than those of others in the 30 independentsimulation runs All thesemean that the EABC algorithm hasbetter robustness than others It is also clear that EABC canwork better in almost all cases and gets better performancethan ACO PSO and ABC

Summarizing the statementsmentioned above the EABCcan prevent bees from being trapped into the local minimumaccelerate convergence process search with more efficiencyand improve exploitation abilities for basic ABC

53 Analysis and Discussion In this section the effectsof each modification on the performance of EABC arediscussed First of all corresponding to three modificationswe named the basic ABC with the proposed initializationas IABC the one with the proposed self-adaptive searchingstrategy as SABC and the one with the proposed immune

The Scientific World Journal 9

Table 2 Benchmark functions used in experiments for testing the performances of EABC ACO PSO and ABC

Function number Min ACO PSO ABC EABC

1198911(119909) =

119863

sum

119894=1

1199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198912(119909) =

119863

sum

119894=1

1198941199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 0

Best 0 0 0 0Worst 000289 000321 000543 000422Mean 000136 000116 000300 000196SD 000219 000276 000387 000208

1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 0

Best 0 0 0 0Worst 000246 000305 000110 000400Mean 000180 000156 00066 000210SD 000039 000058 000092 000037

1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) minus1

Best minus1 minus1 minus1 minus1Worst minus1 minus1 minus1 minus1Mean minus1 minus1 minus1 minus1SD 0 0 0 0

1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 0

Best 066667 06667 0 0Worst 066667 06667 0 0Mean 066667 06667 0 0SD 000001 000001 0 0

11989110(119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 0

Best 87513 105433 196788 91578Worst 324215 246711 542333 269874Mean 182039 150886 331227 173558SD 50361 241702 1541443 114774

11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 0

Best 526677 435774 0 0Worst 532331 441131 0 0Mean 529226 439771 0 0SD 45649 117286 0 0

11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 0

Best 001470 0017112 0008531 0Worst 001499 0017989 0017356 0Mean 001479 0017391 0011447 0SD 000296 0020808 0001223 0

10 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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httpwwwhindawicom Volume 2014

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Advances in

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 The Scientific World Journal

other intelligent ones including genetic algorithm (GA)PSO differential evolution (DS) and evolution strategy (ES)although it uses fewer control parameters [15ndash17]

However similar to other intelligent algorithms the ABCstill has some limitations For example the convergence speedof ABC is slow because of its stochastic nature What ismore ABC can easily get trapped in the local optimumwhenhanding in functions that have a narrow curving valley a higheccentric ellipse or complex multimodal functions [18] Allthese insufficiencies prevent the further applications of theABC algorithm

Therefore in this work some modifications to the stan-dard ABC algorithm are introduced for global optimizationof numerical functions Firstly ABC algorithm is extendedby employing the chaotic systems and the diversity-basedmethod when producing the initial population Next self-adaptive searching strategy is incorporated into the employedbee search process to improve the exploitation In additionthe enhanced algorithm retains the main steps of ABCand incorporates an aiNet-based search technique whereaiNet algorithm has powerful multimodal searching abilityas well as good stabilization due to its negative selection andnetwork compression operators Therefore ABC and aiNethave complementary advantages and a hybrid of the twomaybe a possible strategy to improve the performances of ABC

The remainder of this paper is organized as follows InSection 2 the works related to the ABC algorithm are sum-marized In Section 3 the basic ABC algorithm is describedSection 4 describes the modified ABC algorithm combinedwith self-adaptive searching strategy and artificial immunenetwork operators In Section 5 the testing of the proposedalgorithm through 15 benchmark functions problems iscarried out and the simulation results are compared Finallyconclusions and future works are provided in Section 6

2 Previous Works on the ABC Algorithm

TheABC algorithm imitated the foraging behavior of honey-bee andwas first applied to numerical optimization problemsHowever due to its weaknesses mentioned in Section 1 someresearchers proposedmany improved strategies For exampleAlatas [19] used different chaotic maps to generate sequencessubstituting random numbers for different parameters ofABC when producing initial population Moreover Gao andLiu [18 20] also employed both the chaotic systems andopposition-based learning methods to enhance the globalconvergence In these two literature works authors alsodeveloped an improved solution search equation which wasbased on that the bee searched only for the best solutionof the previous iteration to improve the exploitation Theexperiments derived from a set of 28 benchmark functionsdemonstrated that the performance of thismethodwas betterthan the other methods Unlike these studies mentionedabove inspired by PSO Zhu andKwong [21] proposedGbest-guided ABC algorithm by incorporating the information ofthe global best solution into the solution search equationto improve the exploitation Banharnsakun et al [22] pre-sented a best-so-far method for solution updates in the ABC

algorithm and the searching method based on a dynamicadjustment of search range depending on the iteration wasalso introduced for scout bees The test results showed thatthe proposed method was able to produce higher qualitysolutions with faster convergence than either the originalABC or the current state-of-the-art ABC-based algorithm

Besides numerical optimization the ABC algorithm hasbeen widely used to solve large-scale problems and engineer-ing design optimization Some representative applications areintroduced as follows Kang et al [23] used a hybrid ABCalgorithm which combines Nelder-Mead simplex methodwith ABC algorithm for structural inverse analysis problemsand its performance outperforms other heuristic meth-ods Singh [24] applied the ABC algorithm for the leaf-constrained minimum spanning tree (LCMST) problem andcompared the approach against GA ACO and tabu searchIn the literature [24] it was reported that the proposedalgorithm was superior to the other methods in termsof solution qualities and computational time Zhang et al[25] developed the ABC clustering algorithm to optimallypartition 119873 objectives into 119870 cluster and Debrsquos rules wereused to direct the search direction of each candidate Panet al [26] used the discrete ABC algorithm to solve the lot-streaming flow shop scheduling problem with the criterionof total weighted earliness and tardiness penalties under boththe idling and no-idling cases Samanta and Chakraborty[27] employed ABC algorithm to search out the optimalcombinations of different operating parameters for threewidely used nontraditionalmachining (NTM) processes thatis electrochemical machining electrochemical dischargemachining and electrochemical micromachining processesAlejandro et al [28] used the ABC algorithm in order tofind the optimal distribution of material with the aim ofestablishing a standard time for this duty by examininghow this was applied in a local manufacturing plant Thesimulation results showed that using this approach might beconvenient to set the standard times in the selected companyAll these researches illustrated that the ABC algorithm haspowerful ability to solve much more complex engineeringproblems

3 The Original Artificial BeeColony Algorithm

The artificial bee colony has been inspired by the intelli-gent behavior of real honey bees The honey bees in thisalgorithm are categorized into three groups employed beesonlooker bees and scout bees The first half of the colonyconsists of employed bees and the other half includes theonlookers Each solution in the search space consists of a setof optimization parameters which represent a food sourcepopulation The number of employed bees is equal to thenumber of food sources around the hive In other words forevery food source there is only one employed bee What ismore onlooker bees wait in the hive and decide on a foodsource to exploit based on the information shared by theemployed bees Scout bees are translated froma few employedbees whose food source has been exhausted by the bees

The Scientific World Journal 3

Similar to the other swarm intelligence algorithms ABCis an iterative process The units of the original ABC algo-rithm can be explained as follows

31 The Initial Population of Solutions The initial populationof solutions is filled with SN number of randomly generated119863-dimensional real-valued vectors (ie food sources) Eachfood source is generated as follows

119909119895

119894 = 119909119895

min + rand (0 1) (119909119895max minus 119909

119895

min) (1)

where 119894 = 1 2 SN 119895 = 1 2 119863 119909119895min and 119909119895max arethe lower and upper bounds for the dimension119895 respectivelyThese food sources are randomly assigned to SN number ofemployed bees and their fitness is evaluated

After initialization the population of the food sourceis subjected to repeat cycle of the search processes of theemployed bees the onlooker bees and the scout bees

32The Search Phase of Employed Bees In this phase in orderto produce a candidate food position from the old one theABC uses the following equation

V119895119894 = 119909119895

119894 + 120593119895

119894 (119909119895

119894 minus 119909119895

119896) (2)

where 119895 isin 1 2 119863 and 119896 isin 1 2 SN are randomlychosen indexes Although 119896 is determined randomly it hasto be different from 119894 120593119894119895 is a random number in the range[minus1 1] Equation (2) denotes that within the neighborhoodof every food source site represented by 119909119894 a food source V119894 isdetermined by changing one parameter of 119909119894

Once V119894 is obtained it will be evaluated and comparedto 119909119894 A greedy selection is applied between 119909119894 and V119894then the better one is selected depending on fitness valuesrepresenting the nectar amount of the food sources at 119909119894 andV119894 If the fitness of V119894 is equal to or better than that of 119909119894 V119894will replace 119909119894 and become a newmember of the populationsotherwise 119909119894 is retained

33 The Selection Phase of Onlooker Bees In this phase eachonlooker bee selects one of the food sources depending on thefitness value obtained from the employed bees The fitness-based probability selection scheme may be a roulette wheelranking based stochastic universal sampling tournamentselection or another selection scheme In original ABCroulette wheel selection scheme is employed described as anequation below

119875119894 =fit (119909119894)

sumSN119898=1 fit (119909119898)

(3)

where fit(119909119894) is the fitness value of solution 119894 Obviously thehigher the fit(119909119894) is the more probability is that the 119894th foodsource is selected After the food source is selected onlookerbees will go to the selected food source and produce a newcandidate position in the neighborhood of the selected foodsource by using (2)

34 Scout Bee Phase In a cycle after all employed andonlooker bees complete their searches the ABC algorithmchecks if there is any exhausted source to be abandoned Ifa position cannot be improved further through a predeter-mined number of cycles then that food source is assumedto be abandoned The scouts can accidentally discover richentirely unknown food sourcesThis operation can be definedas in (4) shown as follows This process helps avoid subopti-mal solutions The value of predetermined number of cyclesis called ldquolimitrdquo for abandoning a food source which is animportant control parameter of ABC algorithm

119909119894 = 119909min + rand (0 1) (119909max minus 119909min) (4)

where 119909min and 119909max are the lower and upper bounds ofvariable 119909119894

35Main Steps of the Original Artificial Colony Bee AlgorithmBased on the above explanation there are three controlparameters used in the original ABC the number of thefood sources which is equal to the number of employed bees(SN) the value of limit and the maximum cycle number(MEN) Detailed pseudocode of the ABC algorithm is givenin Algorithm 1 [15]

4 Enhancing Artificial Bee Algorithm withArtificial Immune Network

The original version of ABC algorithm is very efficient formultidimensional basic functions However the convergencerate of the algorithm is poor when working with somecomplex multimodal functions and composite functionsFurthermore due to its poor exploration process the ABCalgorithm easily gets trapped in a local optimum In orderto improve these limitations existing in the ABC algorithmsome modifications inspired by the artificial immune net-work (ai-Net) algorithm so as to accelerate the convergencerate have been introduced in the search process of theoriginal ABC algorithm In addition an improved searchmechanism based on the self-adaptive strategy as well asa novel generation method of the initial population is alsoproposed

41 Generation of the Initial Population One of the modifi-cations in the ABC algorithm is generating effective initialpopulation which can affect the convergence rate and thequality of the final solution Generally random initializationis the most adopted approach to generate initial populationwhich often makes solutions concentrated in a local areaChaotic sequences derived from a chaotic map have beenproven easy and fast to store there is no need for storage oflong sequences Recently chaotic sequences have been usedinstead of random sequences and shown somewhat goodresults in many applications Therefore chaotic maps areintroduced in ABC to improve the global convergence byescaping the local solutions in [19] Meanwhile in order toincrease the population diversity similar individuals shouldbe gotten rid of The main principle is to compare theaffinity inspired by ai-Net algorithm between two different

4 The Scientific World Journal

(1) Generate the initial population 119909119894 (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (2)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Calculate the probability values 119875119894 for the solution (119909119894) by (3)(7) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(8) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(9) Memorize the best solution so far(10) Cycle = cycle +1(11) Until cycle = MEN

Algorithm 1 Pseudocode of main body of ABC algorithm

individuals As a result this work proposes a novel initial-ization approach which uses chaotic systems and affinity-based compression method to produce initial populationHere according to the literature [19] sinus map is selectedand its equation is defined as follows

119888119898119899+1 = 23(119888119898119899)2 sin(120587119888119898

119899)

119888119898119899 = 0 1 2 119873

(5)

where 119899 is the iteration counter and119873 is the maximum num-ber of chaotic iterations Furthermore the affinity equationbetween two different individuals is defined as Euclideandistance shown in (6)

affinity (119909119894 119909119896) = radic119863

sum

119895=1

(119909119895

119894 minus 119909119895

119896)2

(119894 = 119896 119895 isin (1 2 119863)) gt 120585

(6)

where 120585 is a threshold value defined in advance so as to controlthe difference between two individuals 119863 is the number ofoptimization parameters Based on these operators we pro-pose the following algorithm to generate initial populationand its corresponding pseudocode is given in Algorithm 2

42 An Improved Search Mechanism Based on the Self-Adaptive Strategy As mentioned above the original ABCalgorithm is good at exploration but poor at exploitationdue to two reasons On the one hand in (2) the coefficient120593119894119895 is a uniform random number in [minus1 1] and 119909119896119895 is a

random individual in the population therefore the searchprocess of solutions illustrated by (2) is random enoughfor exploration [21] On the other hand a greedy selectionmechanism is employed between the old and candidatesolutions which may easily make solutions get trapped in

a local optimal What is more the slower convergence rateof the algorithm is another limitation when working withsome complex composite functions As a result we introducea self-adaptive strategy to improve its search process and thedetailed explanations are as follows

Firstly we can see from (2) that there is only one differentelement between the candidate solution and the old one (iethe 119895th element) This search strategy may be efficient inearlier iterations However when the solution approaches toa local optimum its search efficiency becomes poor in lateriterations To handle this limitation similar to [29 30] weintroduce a parameter 119871 to control the difference betweenthe candidate solution and the old one where how to choosethe value of the parameter 119871 is very important Generally thehigher the value of 119871 is the more information is brought intothe candidate solution In the literature [18] the parameter119871 is a fixed constant and chosen according to simulationexperiments Different from this approach mentioned abovethis paper proposes a self-adaptive adjustable strategy todetermine the parameter 119871 and the corresponding searchequation is given below

V119897119894 = 119909119897119894 + 119888119898

119897119894 (119909119897119894 minus 119909119897119896) 119897 = 1 2 119871 (7)

119871 = 1 +

1003817100381710038171003817100381710038171003817119863(

iteration2 timesMEN

)

1003817100381710038171003817100381710038171003817 (8)

where 119888119898119897119894 is a chaotic map defined by (5)119863 is the number ofoptimization parameters A symbol sdot denotes a roundingoperator We can see from (8) that onlooker bees will searchbetter solutions in only one direction in the first iterationand they will search in the whole space with the increasein the value of 119871 when the solutions are closely to the localoptimum in later iterations However we limit the value of 119871nomore than 1+1198632This is because with the higher valueof 119871 the solution generated by the search equations (7)-(8)

The Scientific World Journal 5

Set the population size SN the number of randomly generated individuals SN119903 ≫ SN the maximum number of chaoticiterations119873 gt 200 the number of optimization parameters D and the individual counter 119894 = 1 119895 = 1- -Chaotic systems- -for 119894 = 1 to SN119903 do

for 119895 = 1 to D doRandomly initialize the first chaotic variable cm0119895 isin (0 1) and set iteration counter 119899 = 0for 119899 = 1 to N do

cm119899+1119895 = 23(cm119899119895)2 sin(120587cm

119899119895)

end for119909119895

119894 = 119909119895

min + cm119895119899(119909119895max minus 119909

119895

min)

end forend for- - Affinity-based compression method- -Set the threshold value 120585 and the individual counter 119894 = 1 119895 = 1for 119894 = 1 to (SN119903 minus 1) do

for 119896 = 119894 + 1 to SN119903 do

if affinity(119909119894 119909119896) = radicsum119863

119895=1(119909119895

119894 minus 119909119895

119896)2lt 120585

SN119903 = SN119903 minus 1 delete the individual 119896 in the populationend if

end forend forSelecting SN fittest individuals form set (119883(SN119903)) as initial population

Algorithm 2 Modified initialization step of ABC algorithm

is more likely random search operator As a result (7)-(8) willdynamically adjust the position of onlooker bees by allowingthem to explore with a wider search space in later iterationsAs the number of the iterations increases the correspondingsearch space of onlooker bees will also increase

The second modification in the ABC algorithm lies inthe selection probability of onlooker bees associated with thefood source In the original ABC algorithm the fitness valuesobtained from the employed bees are adopted to determinethe selection probabilities of onlooker bees Nevertheless thefitness comparisons among different individuals only reflectthe qualitative information In this work based on the ideaof the fitness evolution we introduce an environment factor120578119894 corresponding to every food source so as to evaluate itsexploitation potential In other words the parameter 120578119894 isused to evaluate quantitatively the environment situationsof exploitation for every food source As the number ofiterations increases the higher the value of 120578119894 the betterexploitation environment it corresponds to At this momentmore onlooker bees will follow the corresponding employedone to its food source position with higher nectar amount inorder to accelerate exploitation efficiency Conversely if thevalue of 120578119894 is lower its corresponding solution lies in theworseexploitation environment which means that it is difficult tofind out a better solution around the old food source andless onlooker bees will be attracted by the employed oneAs a result how to define 120578119894 needs explain Generally theparameter 120578119894 is associated with both a fitness change amountΔ119891 and a count accumulator 119862 where Δ119891 denotes the fitnessdifference associated with the same food source betweentwo adjacent generations which reflects the exploitation

potential of the corresponding food source position Besidesa count accumulator119862will be explained in the following textThe equations below reflect the fitness change amount Δ119891between two adjacent generations

Δ119891119905 (119909119894) =

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(9)

Δ1198911015840119905 (119909119894) = exp [Δ119891119905 (119909119894)] = log

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(10)

where the parameter 119905 denotes the number of iterations andthe symbol | | means absolute value sign As can be seenfrom (9) the value range of Δ119891 is between 0 and 1 Whenthe value of Δ119891 is higher it means the corresponding foodsource has a higher exploitation potential and is largerlypossible to find out a better solution and vice versa Howeverat later iterations the value of Δ119891 will be very small andneed appropriate amplification As a result we adopt powerfunction to amplifyΔ119891 here and take the number 119890 as its baseDue to this reason (9) is substituted for (10)

In addition if Δ119891 is less than a given small value inadvance (ie Δ119891 le Δ1198910 and Δ1198910 is the threshold of Δ119891) theimproved ABC algorithm may trigger a count accumulatorcalled Counter (represented by 119862 in this paper) which is usedto record how many times the quality of the solution hasnot improved (it corresponds to the number of iterations inthe algorithm) The following equation is used to express thecount accumulator119862 at the 119905th iteration Generally the larger

6 The Scientific World Journal

the value of 119862 the less exploitation potential that the foodsource position corresponds to

119862 (119905) =

119862 (119905 minus 1) + 119879 (119905) =

119905

sum

119894=119904

119879 (119894) 119879 (119905) = 1

0 119879 (119905) = 0

(11)

where119879 represents a pulse signalWhenΔ119891 gt 0 orΔ119891 ge Δ1198910119879 = 0 conversely when Δ119891 = 0 or Δ119891 le Δ1198910 119879 = 1 It meansthat

119879 (119905) = 1 (Δ119891 (119905) = 0) or (Δ119891 (119905) le Δ119891(119905)0) 0 (Δ119891 (119905) gt 0) or (Δ119891 (119905) ge Δ119891(119905)0)

(12)

Note that according to the performance requirement ofspecific problems the maximum times of the stagnation ofglobal extrema 119862max should be defined in advance whichmeans if a minimum of a function has not been updated forcontinuous 119862max iterations the current exploitation area hasfew potentials to find out the better solution and computationresources should be redistributed Generally we define thatthe 119862max is equal to or not less than 5 In addition 119862(119905)is normalized for simplifying the problem and 1198621015840(119905) can beobtained which is expressed as follows

1198621015840(119905) =

sum119905119894=119904 119879 (119894) + 1

119862max + 1 (13)

On the basis of the definitions of the fitness changeamount Δ119891and the count accumulator 119862 the environmentfactor 120578 is defined as follows

120578 (119905) =Δ1198911015840(119905)

1198621015840 (119905) (14)

Submitting (10) and (13) into (14) we can achieve thefollowing equation

120578 (119905) =(119862max + 1) sdot exp

1003816100381610038161003816(fit119905 (119909119894) minus fit119905minus1 (119909119894)) fit119905minus1 (119909119894)1003816100381610038161003816

sum119905119894=119904 119879 (119894) + 1

(15)

In accordance with the expression of the environmentfactor 120578(119909119894) corresponding to every food source the selectionprobability of onlooker bees associated with the food sourcecan be substituted with the following equation

119875119894 =120578119905 (119909119894)

sumSN119898=1 120578119905 (119909119898)

(16)

43 Enhancing Convergence Efficiency with Artificial ImmuneNetwork Operators In the basic ABC system artificial beesfly around in the search space Some (like employed andonlooker bees) choose food source depending on the experi-ence of themselves and their nest mates and then adjust theirpositions but others (like scouts) fly and choose the foodsources randomly without using experience If the nectar

amount of a new source is higher than that of the previousone in their memory they memorize the new food sourceposition and forget the previous one Thus the ABC systemcombines local search methods carried out by employedand onlooker bees with global search methods managedby Karaboga and Basturk [15] However unlike the ABCsystem the concept of artificial immune system (AIS) wasoriginated by observing how the defense mechanism ofnatural immune system protects against attacks by antigensThere are numerous AIS algorithms developed for a varietyof applications where artificial immune network (aiNet forshort) is a typical one and its algorithms and models areoriginally proposed to perform information compressionand data clustering based on artificial immune system (AIS)theory [31] Immune network-based algorithms are similarto clonal selection algorithms in that they both measurethe goodness of antibodies by affinities and both methodsinclude a series of steps for selecting cloning and mutatingantibodiesThemajor difference is that the immune network-based algorithms are represented bynetwork graph structures[32] Compared with other ones the immune network-basedalgorithms employ extra procedures of antibody pruning andsuppressing This allows the models to generate a smallerless-redundant population of antibody representatives whichis desirable for solving multimodal function optimizationComparing ABC optimization with ai-Net algorithm wecan see that the advantages of ABC optimization lie in itsneighborhood search method according to the profitabilityof food sources However ai-Net algorithm adopts fixedclonal individuals to perform local search which has certainblindness In addition due to introducing network com-pression negative selection and other operators ai-Net canmaintain the diversity of the population and reduce thepossibility of being trapped into a local minimum Unlike theai-Net algorithm ABC optimization maintains populationdiversity through random search of scout bees which hasobvious limitation Based on the analysis mentioned aboveif network compression and negative selection deriving fromai-Net algorithm are introduced into ABC optimization thisimproved one may have a powerful and efficient multimodalsearching ability as well as good stabilization The detailedprocess is as follows

Different employed bee individual corresponds to differ-ent food source position In order to eliminate redundantand similar food sources negative selection and networkcompression are used to compare with the similarities amongvarious individuals The Euclidean distance of two employedbee individuals119883119894 and119883119896 is adopted as shown in (17)

119889 (119883119894 119883119896) = radic

119863

sum

119895=1

(119883119895

119894 minus 119883119895

119896)2

(119894 = 119896) (17)

In order to simplify the problem the affinity concept isintroduced which is obtained by the following equation usingthe normalization method

119860 (119883119894 119883119896) =1

1 + 119889 (119883119894 119883119896) (18)

The Scientific World Journal 7

(1) Generate the initial population 119909119894 based on chaotic maps and affinity strategy (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (7)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Adopt negative selection and network compression to eliminate redundant and similar food sources by using (18)(7) Randomly generate the same number of new individuals(8) Calculate the probability values 119875119894 for the solution (119909119894) by (16)(9) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(10) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(11) Memorize the best solution so far(12) Cycle = cycle + 1(13) Until cycle = MEN

Algorithm 3 Pseudocode of main body of the enhanced ABC algorithm

where the value range of 119860(119883119894119883119896) is between 0 and 1 Thesmaller the value of 119860(119883119894119883119896) is the larger the value of119860(119883119894119883119896) is which means that two different employed beeindividuals have a higher similarity Specially when119860(119883119894119883119896)equals 1 these two ones are identical According to negativeselection and network compression operators redundant andsimilar food sources should be eliminated We predefine athreshold value 120576 so as to realize wipe-off of redundantindividuals It also means when 119860(119883119894119883119896) is equal to or greatthan 120576 we think these two ones are identical and only one canbe retained and other should bewiped off Repeat this processuntil the affinity of any two individuals in a population isless than 120576 In doing so the population size may be reducedNevertheless in order to keep the population size unchangedthe same number of new individuals need generating ran-domlyThrough negative selection and network compressionoperators the exploitation efficiencywill be improved and thecorresponding convergence rate of the algorithm will also beaccelerated

44 Main Steps of the Enhanced Artificial Bee Colony Algo-rithm Based on the above analysis three main improve-ments including novel generation of initial population self-adaptive searching strategy and redundant individual com-pression operator are presented and the detailed pseudo-codeis given in Algorithm 3

5 Experimental Studies on FunctionOptimization Problems

51 Benchmark Functions and Parameter Settings In thissection numerical experiment is used to test the performanceof the enhanced ABC (shorthand for EABC) proposed in this

paper Summarized in Table 1 are the 15 scalable benchmarkfunctions 1198911 sim 11989110 are continuous unimodal functions11989111 sim 11989115 are multimodal functions and the number oftheir local minima increases exponentially with the problemdimension

In order to testify the performance of different intelligentalgorithms we compare the EABC with the standard ACOPSO and ABC In all simulations the population size ofACO PSO ABC and EABC is 50 The maximum numberof function evaluations (FE) is set to 5000 The thresholdvalue of the affinity 120576 is 09 Other related parameter valuesof ACO PSO and ABC are referred in the literature [17]All experiment results reported are obtained based on 30independent runsThe experiment results are the best worstmean and standard deviation of the statistical experimentaldata

52 Simulation Results The performance on the solutionaccuracy of EABC is compared with that of ACO PSO andABC Table 2 shows the optimization of the 15 benchmarkfunctions obtained in the 30 independent runs by eachalgorithm and some interesting results can be found inTable 2

Firstly almost all algorithms have identical performanceon most of unimodal functions 1198911 to 1198914 1198917 1198918 and 11989114However on other functions these four algorithms showdifferent performance especially for multimodal ones suchas 11989111 11989112 11989113 and 11989115 Fox example on function 11989115the best values obtained by ACO PSO ABC and EABCare minus10296 minus699347 minus125669 and minus1525687 respectivelyIt means that EABC can be efficiently applied for solvingmultimodal and multidimensional function optimizationproblems due to its abundant operators such as clonal

8 The Scientific World Journal

Table 1 Benchmark functions used in experiments

Number Function Dimension Property Range Min

1 1198911(119909) =

119863

sum

119894=1

1199092

119894 30 Unimodal [minus100 100] 0

2 1198912(119909) =

119863

sum

119894=1

1198941199092

119894 30 Unimodal [minus10 10] 0

3 1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 5 Unimodal [minus512 512] 0

4 1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 30 Unimodal [minus100 100] 0

5 1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 30 Unimodal [minus128 128] 0

6 1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 30 Unimodal [minus100 100] 0

7 1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) 2 Unimodal [minus100 100] minus1

8 1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 2 Unimodal [minus10 10] 0

9 1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 30 Unimodal [minus10 10] 0

10 11989110 (119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 30 Unimodal [minus30 30] 0

11 11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 30 Multimodal [minus512 512] 0

12 11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 30 Multimodal [minus600 600] 0

13 11989113 (119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 10 Multimodal [minus50 50] 0

14 11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

2 Multimodal [minus100 100] 0

15 11989115 (119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817) 30 Multimodal [minus500 500] minus125695

selection and negative selection which outperforms ACOPSO and ABC In addition on the one hand on basicunimodal functions both the basic ABC and EABC havealmost identical solving performance on the other handon the multimodal functions these two ones display hugedifference

Secondly the EABC algorithm can find optimal or closer-to-optimal solutions on the complex multimodal functions11989111 11989112 11989113 and 11989114 Although the result of multi-dimensionfunction 11989115 is slightly far from the known global optimumthe EABC is superior to the other algorithms all the sameAt the same time for almost all benchmark functionsstandard deviations of the EABC obtained from the statisticalexperimental data are no greater than those of others expectfor 11989110 In addition the differences of EABC between thebest andworst solutions for these 15 benchmark functions are

relatively smaller than those of others in the 30 independentsimulation runs All thesemean that the EABC algorithm hasbetter robustness than others It is also clear that EABC canwork better in almost all cases and gets better performancethan ACO PSO and ABC

Summarizing the statementsmentioned above the EABCcan prevent bees from being trapped into the local minimumaccelerate convergence process search with more efficiencyand improve exploitation abilities for basic ABC

53 Analysis and Discussion In this section the effectsof each modification on the performance of EABC arediscussed First of all corresponding to three modificationswe named the basic ABC with the proposed initializationas IABC the one with the proposed self-adaptive searchingstrategy as SABC and the one with the proposed immune

The Scientific World Journal 9

Table 2 Benchmark functions used in experiments for testing the performances of EABC ACO PSO and ABC

Function number Min ACO PSO ABC EABC

1198911(119909) =

119863

sum

119894=1

1199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198912(119909) =

119863

sum

119894=1

1198941199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 0

Best 0 0 0 0Worst 000289 000321 000543 000422Mean 000136 000116 000300 000196SD 000219 000276 000387 000208

1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 0

Best 0 0 0 0Worst 000246 000305 000110 000400Mean 000180 000156 00066 000210SD 000039 000058 000092 000037

1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) minus1

Best minus1 minus1 minus1 minus1Worst minus1 minus1 minus1 minus1Mean minus1 minus1 minus1 minus1SD 0 0 0 0

1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 0

Best 066667 06667 0 0Worst 066667 06667 0 0Mean 066667 06667 0 0SD 000001 000001 0 0

11989110(119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 0

Best 87513 105433 196788 91578Worst 324215 246711 542333 269874Mean 182039 150886 331227 173558SD 50361 241702 1541443 114774

11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 0

Best 526677 435774 0 0Worst 532331 441131 0 0Mean 529226 439771 0 0SD 45649 117286 0 0

11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 0

Best 001470 0017112 0008531 0Worst 001499 0017989 0017356 0Mean 001479 0017391 0011447 0SD 000296 0020808 0001223 0

10 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

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The Scientific World Journal 3

Similar to the other swarm intelligence algorithms ABCis an iterative process The units of the original ABC algo-rithm can be explained as follows

31 The Initial Population of Solutions The initial populationof solutions is filled with SN number of randomly generated119863-dimensional real-valued vectors (ie food sources) Eachfood source is generated as follows

119909119895

119894 = 119909119895

min + rand (0 1) (119909119895max minus 119909

119895

min) (1)

where 119894 = 1 2 SN 119895 = 1 2 119863 119909119895min and 119909119895max arethe lower and upper bounds for the dimension119895 respectivelyThese food sources are randomly assigned to SN number ofemployed bees and their fitness is evaluated

After initialization the population of the food sourceis subjected to repeat cycle of the search processes of theemployed bees the onlooker bees and the scout bees

32The Search Phase of Employed Bees In this phase in orderto produce a candidate food position from the old one theABC uses the following equation

V119895119894 = 119909119895

119894 + 120593119895

119894 (119909119895

119894 minus 119909119895

119896) (2)

where 119895 isin 1 2 119863 and 119896 isin 1 2 SN are randomlychosen indexes Although 119896 is determined randomly it hasto be different from 119894 120593119894119895 is a random number in the range[minus1 1] Equation (2) denotes that within the neighborhoodof every food source site represented by 119909119894 a food source V119894 isdetermined by changing one parameter of 119909119894

Once V119894 is obtained it will be evaluated and comparedto 119909119894 A greedy selection is applied between 119909119894 and V119894then the better one is selected depending on fitness valuesrepresenting the nectar amount of the food sources at 119909119894 andV119894 If the fitness of V119894 is equal to or better than that of 119909119894 V119894will replace 119909119894 and become a newmember of the populationsotherwise 119909119894 is retained

33 The Selection Phase of Onlooker Bees In this phase eachonlooker bee selects one of the food sources depending on thefitness value obtained from the employed bees The fitness-based probability selection scheme may be a roulette wheelranking based stochastic universal sampling tournamentselection or another selection scheme In original ABCroulette wheel selection scheme is employed described as anequation below

119875119894 =fit (119909119894)

sumSN119898=1 fit (119909119898)

(3)

where fit(119909119894) is the fitness value of solution 119894 Obviously thehigher the fit(119909119894) is the more probability is that the 119894th foodsource is selected After the food source is selected onlookerbees will go to the selected food source and produce a newcandidate position in the neighborhood of the selected foodsource by using (2)

34 Scout Bee Phase In a cycle after all employed andonlooker bees complete their searches the ABC algorithmchecks if there is any exhausted source to be abandoned Ifa position cannot be improved further through a predeter-mined number of cycles then that food source is assumedto be abandoned The scouts can accidentally discover richentirely unknown food sourcesThis operation can be definedas in (4) shown as follows This process helps avoid subopti-mal solutions The value of predetermined number of cyclesis called ldquolimitrdquo for abandoning a food source which is animportant control parameter of ABC algorithm

119909119894 = 119909min + rand (0 1) (119909max minus 119909min) (4)

where 119909min and 119909max are the lower and upper bounds ofvariable 119909119894

35Main Steps of the Original Artificial Colony Bee AlgorithmBased on the above explanation there are three controlparameters used in the original ABC the number of thefood sources which is equal to the number of employed bees(SN) the value of limit and the maximum cycle number(MEN) Detailed pseudocode of the ABC algorithm is givenin Algorithm 1 [15]

4 Enhancing Artificial Bee Algorithm withArtificial Immune Network

The original version of ABC algorithm is very efficient formultidimensional basic functions However the convergencerate of the algorithm is poor when working with somecomplex multimodal functions and composite functionsFurthermore due to its poor exploration process the ABCalgorithm easily gets trapped in a local optimum In orderto improve these limitations existing in the ABC algorithmsome modifications inspired by the artificial immune net-work (ai-Net) algorithm so as to accelerate the convergencerate have been introduced in the search process of theoriginal ABC algorithm In addition an improved searchmechanism based on the self-adaptive strategy as well asa novel generation method of the initial population is alsoproposed

41 Generation of the Initial Population One of the modifi-cations in the ABC algorithm is generating effective initialpopulation which can affect the convergence rate and thequality of the final solution Generally random initializationis the most adopted approach to generate initial populationwhich often makes solutions concentrated in a local areaChaotic sequences derived from a chaotic map have beenproven easy and fast to store there is no need for storage oflong sequences Recently chaotic sequences have been usedinstead of random sequences and shown somewhat goodresults in many applications Therefore chaotic maps areintroduced in ABC to improve the global convergence byescaping the local solutions in [19] Meanwhile in order toincrease the population diversity similar individuals shouldbe gotten rid of The main principle is to compare theaffinity inspired by ai-Net algorithm between two different

4 The Scientific World Journal

(1) Generate the initial population 119909119894 (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (2)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Calculate the probability values 119875119894 for the solution (119909119894) by (3)(7) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(8) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(9) Memorize the best solution so far(10) Cycle = cycle +1(11) Until cycle = MEN

Algorithm 1 Pseudocode of main body of ABC algorithm

individuals As a result this work proposes a novel initial-ization approach which uses chaotic systems and affinity-based compression method to produce initial populationHere according to the literature [19] sinus map is selectedand its equation is defined as follows

119888119898119899+1 = 23(119888119898119899)2 sin(120587119888119898

119899)

119888119898119899 = 0 1 2 119873

(5)

where 119899 is the iteration counter and119873 is the maximum num-ber of chaotic iterations Furthermore the affinity equationbetween two different individuals is defined as Euclideandistance shown in (6)

affinity (119909119894 119909119896) = radic119863

sum

119895=1

(119909119895

119894 minus 119909119895

119896)2

(119894 = 119896 119895 isin (1 2 119863)) gt 120585

(6)

where 120585 is a threshold value defined in advance so as to controlthe difference between two individuals 119863 is the number ofoptimization parameters Based on these operators we pro-pose the following algorithm to generate initial populationand its corresponding pseudocode is given in Algorithm 2

42 An Improved Search Mechanism Based on the Self-Adaptive Strategy As mentioned above the original ABCalgorithm is good at exploration but poor at exploitationdue to two reasons On the one hand in (2) the coefficient120593119894119895 is a uniform random number in [minus1 1] and 119909119896119895 is a

random individual in the population therefore the searchprocess of solutions illustrated by (2) is random enoughfor exploration [21] On the other hand a greedy selectionmechanism is employed between the old and candidatesolutions which may easily make solutions get trapped in

a local optimal What is more the slower convergence rateof the algorithm is another limitation when working withsome complex composite functions As a result we introducea self-adaptive strategy to improve its search process and thedetailed explanations are as follows

Firstly we can see from (2) that there is only one differentelement between the candidate solution and the old one (iethe 119895th element) This search strategy may be efficient inearlier iterations However when the solution approaches toa local optimum its search efficiency becomes poor in lateriterations To handle this limitation similar to [29 30] weintroduce a parameter 119871 to control the difference betweenthe candidate solution and the old one where how to choosethe value of the parameter 119871 is very important Generally thehigher the value of 119871 is the more information is brought intothe candidate solution In the literature [18] the parameter119871 is a fixed constant and chosen according to simulationexperiments Different from this approach mentioned abovethis paper proposes a self-adaptive adjustable strategy todetermine the parameter 119871 and the corresponding searchequation is given below

V119897119894 = 119909119897119894 + 119888119898

119897119894 (119909119897119894 minus 119909119897119896) 119897 = 1 2 119871 (7)

119871 = 1 +

1003817100381710038171003817100381710038171003817119863(

iteration2 timesMEN

)

1003817100381710038171003817100381710038171003817 (8)

where 119888119898119897119894 is a chaotic map defined by (5)119863 is the number ofoptimization parameters A symbol sdot denotes a roundingoperator We can see from (8) that onlooker bees will searchbetter solutions in only one direction in the first iterationand they will search in the whole space with the increasein the value of 119871 when the solutions are closely to the localoptimum in later iterations However we limit the value of 119871nomore than 1+1198632This is because with the higher valueof 119871 the solution generated by the search equations (7)-(8)

The Scientific World Journal 5

Set the population size SN the number of randomly generated individuals SN119903 ≫ SN the maximum number of chaoticiterations119873 gt 200 the number of optimization parameters D and the individual counter 119894 = 1 119895 = 1- -Chaotic systems- -for 119894 = 1 to SN119903 do

for 119895 = 1 to D doRandomly initialize the first chaotic variable cm0119895 isin (0 1) and set iteration counter 119899 = 0for 119899 = 1 to N do

cm119899+1119895 = 23(cm119899119895)2 sin(120587cm

119899119895)

end for119909119895

119894 = 119909119895

min + cm119895119899(119909119895max minus 119909

119895

min)

end forend for- - Affinity-based compression method- -Set the threshold value 120585 and the individual counter 119894 = 1 119895 = 1for 119894 = 1 to (SN119903 minus 1) do

for 119896 = 119894 + 1 to SN119903 do

if affinity(119909119894 119909119896) = radicsum119863

119895=1(119909119895

119894 minus 119909119895

119896)2lt 120585

SN119903 = SN119903 minus 1 delete the individual 119896 in the populationend if

end forend forSelecting SN fittest individuals form set (119883(SN119903)) as initial population

Algorithm 2 Modified initialization step of ABC algorithm

is more likely random search operator As a result (7)-(8) willdynamically adjust the position of onlooker bees by allowingthem to explore with a wider search space in later iterationsAs the number of the iterations increases the correspondingsearch space of onlooker bees will also increase

The second modification in the ABC algorithm lies inthe selection probability of onlooker bees associated with thefood source In the original ABC algorithm the fitness valuesobtained from the employed bees are adopted to determinethe selection probabilities of onlooker bees Nevertheless thefitness comparisons among different individuals only reflectthe qualitative information In this work based on the ideaof the fitness evolution we introduce an environment factor120578119894 corresponding to every food source so as to evaluate itsexploitation potential In other words the parameter 120578119894 isused to evaluate quantitatively the environment situationsof exploitation for every food source As the number ofiterations increases the higher the value of 120578119894 the betterexploitation environment it corresponds to At this momentmore onlooker bees will follow the corresponding employedone to its food source position with higher nectar amount inorder to accelerate exploitation efficiency Conversely if thevalue of 120578119894 is lower its corresponding solution lies in theworseexploitation environment which means that it is difficult tofind out a better solution around the old food source andless onlooker bees will be attracted by the employed oneAs a result how to define 120578119894 needs explain Generally theparameter 120578119894 is associated with both a fitness change amountΔ119891 and a count accumulator 119862 where Δ119891 denotes the fitnessdifference associated with the same food source betweentwo adjacent generations which reflects the exploitation

potential of the corresponding food source position Besidesa count accumulator119862will be explained in the following textThe equations below reflect the fitness change amount Δ119891between two adjacent generations

Δ119891119905 (119909119894) =

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(9)

Δ1198911015840119905 (119909119894) = exp [Δ119891119905 (119909119894)] = log

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(10)

where the parameter 119905 denotes the number of iterations andthe symbol | | means absolute value sign As can be seenfrom (9) the value range of Δ119891 is between 0 and 1 Whenthe value of Δ119891 is higher it means the corresponding foodsource has a higher exploitation potential and is largerlypossible to find out a better solution and vice versa Howeverat later iterations the value of Δ119891 will be very small andneed appropriate amplification As a result we adopt powerfunction to amplifyΔ119891 here and take the number 119890 as its baseDue to this reason (9) is substituted for (10)

In addition if Δ119891 is less than a given small value inadvance (ie Δ119891 le Δ1198910 and Δ1198910 is the threshold of Δ119891) theimproved ABC algorithm may trigger a count accumulatorcalled Counter (represented by 119862 in this paper) which is usedto record how many times the quality of the solution hasnot improved (it corresponds to the number of iterations inthe algorithm) The following equation is used to express thecount accumulator119862 at the 119905th iteration Generally the larger

6 The Scientific World Journal

the value of 119862 the less exploitation potential that the foodsource position corresponds to

119862 (119905) =

119862 (119905 minus 1) + 119879 (119905) =

119905

sum

119894=119904

119879 (119894) 119879 (119905) = 1

0 119879 (119905) = 0

(11)

where119879 represents a pulse signalWhenΔ119891 gt 0 orΔ119891 ge Δ1198910119879 = 0 conversely when Δ119891 = 0 or Δ119891 le Δ1198910 119879 = 1 It meansthat

119879 (119905) = 1 (Δ119891 (119905) = 0) or (Δ119891 (119905) le Δ119891(119905)0) 0 (Δ119891 (119905) gt 0) or (Δ119891 (119905) ge Δ119891(119905)0)

(12)

Note that according to the performance requirement ofspecific problems the maximum times of the stagnation ofglobal extrema 119862max should be defined in advance whichmeans if a minimum of a function has not been updated forcontinuous 119862max iterations the current exploitation area hasfew potentials to find out the better solution and computationresources should be redistributed Generally we define thatthe 119862max is equal to or not less than 5 In addition 119862(119905)is normalized for simplifying the problem and 1198621015840(119905) can beobtained which is expressed as follows

1198621015840(119905) =

sum119905119894=119904 119879 (119894) + 1

119862max + 1 (13)

On the basis of the definitions of the fitness changeamount Δ119891and the count accumulator 119862 the environmentfactor 120578 is defined as follows

120578 (119905) =Δ1198911015840(119905)

1198621015840 (119905) (14)

Submitting (10) and (13) into (14) we can achieve thefollowing equation

120578 (119905) =(119862max + 1) sdot exp

1003816100381610038161003816(fit119905 (119909119894) minus fit119905minus1 (119909119894)) fit119905minus1 (119909119894)1003816100381610038161003816

sum119905119894=119904 119879 (119894) + 1

(15)

In accordance with the expression of the environmentfactor 120578(119909119894) corresponding to every food source the selectionprobability of onlooker bees associated with the food sourcecan be substituted with the following equation

119875119894 =120578119905 (119909119894)

sumSN119898=1 120578119905 (119909119898)

(16)

43 Enhancing Convergence Efficiency with Artificial ImmuneNetwork Operators In the basic ABC system artificial beesfly around in the search space Some (like employed andonlooker bees) choose food source depending on the experi-ence of themselves and their nest mates and then adjust theirpositions but others (like scouts) fly and choose the foodsources randomly without using experience If the nectar

amount of a new source is higher than that of the previousone in their memory they memorize the new food sourceposition and forget the previous one Thus the ABC systemcombines local search methods carried out by employedand onlooker bees with global search methods managedby Karaboga and Basturk [15] However unlike the ABCsystem the concept of artificial immune system (AIS) wasoriginated by observing how the defense mechanism ofnatural immune system protects against attacks by antigensThere are numerous AIS algorithms developed for a varietyof applications where artificial immune network (aiNet forshort) is a typical one and its algorithms and models areoriginally proposed to perform information compressionand data clustering based on artificial immune system (AIS)theory [31] Immune network-based algorithms are similarto clonal selection algorithms in that they both measurethe goodness of antibodies by affinities and both methodsinclude a series of steps for selecting cloning and mutatingantibodiesThemajor difference is that the immune network-based algorithms are represented bynetwork graph structures[32] Compared with other ones the immune network-basedalgorithms employ extra procedures of antibody pruning andsuppressing This allows the models to generate a smallerless-redundant population of antibody representatives whichis desirable for solving multimodal function optimizationComparing ABC optimization with ai-Net algorithm wecan see that the advantages of ABC optimization lie in itsneighborhood search method according to the profitabilityof food sources However ai-Net algorithm adopts fixedclonal individuals to perform local search which has certainblindness In addition due to introducing network com-pression negative selection and other operators ai-Net canmaintain the diversity of the population and reduce thepossibility of being trapped into a local minimum Unlike theai-Net algorithm ABC optimization maintains populationdiversity through random search of scout bees which hasobvious limitation Based on the analysis mentioned aboveif network compression and negative selection deriving fromai-Net algorithm are introduced into ABC optimization thisimproved one may have a powerful and efficient multimodalsearching ability as well as good stabilization The detailedprocess is as follows

Different employed bee individual corresponds to differ-ent food source position In order to eliminate redundantand similar food sources negative selection and networkcompression are used to compare with the similarities amongvarious individuals The Euclidean distance of two employedbee individuals119883119894 and119883119896 is adopted as shown in (17)

119889 (119883119894 119883119896) = radic

119863

sum

119895=1

(119883119895

119894 minus 119883119895

119896)2

(119894 = 119896) (17)

In order to simplify the problem the affinity concept isintroduced which is obtained by the following equation usingthe normalization method

119860 (119883119894 119883119896) =1

1 + 119889 (119883119894 119883119896) (18)

The Scientific World Journal 7

(1) Generate the initial population 119909119894 based on chaotic maps and affinity strategy (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (7)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Adopt negative selection and network compression to eliminate redundant and similar food sources by using (18)(7) Randomly generate the same number of new individuals(8) Calculate the probability values 119875119894 for the solution (119909119894) by (16)(9) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(10) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(11) Memorize the best solution so far(12) Cycle = cycle + 1(13) Until cycle = MEN

Algorithm 3 Pseudocode of main body of the enhanced ABC algorithm

where the value range of 119860(119883119894119883119896) is between 0 and 1 Thesmaller the value of 119860(119883119894119883119896) is the larger the value of119860(119883119894119883119896) is which means that two different employed beeindividuals have a higher similarity Specially when119860(119883119894119883119896)equals 1 these two ones are identical According to negativeselection and network compression operators redundant andsimilar food sources should be eliminated We predefine athreshold value 120576 so as to realize wipe-off of redundantindividuals It also means when 119860(119883119894119883119896) is equal to or greatthan 120576 we think these two ones are identical and only one canbe retained and other should bewiped off Repeat this processuntil the affinity of any two individuals in a population isless than 120576 In doing so the population size may be reducedNevertheless in order to keep the population size unchangedthe same number of new individuals need generating ran-domlyThrough negative selection and network compressionoperators the exploitation efficiencywill be improved and thecorresponding convergence rate of the algorithm will also beaccelerated

44 Main Steps of the Enhanced Artificial Bee Colony Algo-rithm Based on the above analysis three main improve-ments including novel generation of initial population self-adaptive searching strategy and redundant individual com-pression operator are presented and the detailed pseudo-codeis given in Algorithm 3

5 Experimental Studies on FunctionOptimization Problems

51 Benchmark Functions and Parameter Settings In thissection numerical experiment is used to test the performanceof the enhanced ABC (shorthand for EABC) proposed in this

paper Summarized in Table 1 are the 15 scalable benchmarkfunctions 1198911 sim 11989110 are continuous unimodal functions11989111 sim 11989115 are multimodal functions and the number oftheir local minima increases exponentially with the problemdimension

In order to testify the performance of different intelligentalgorithms we compare the EABC with the standard ACOPSO and ABC In all simulations the population size ofACO PSO ABC and EABC is 50 The maximum numberof function evaluations (FE) is set to 5000 The thresholdvalue of the affinity 120576 is 09 Other related parameter valuesof ACO PSO and ABC are referred in the literature [17]All experiment results reported are obtained based on 30independent runsThe experiment results are the best worstmean and standard deviation of the statistical experimentaldata

52 Simulation Results The performance on the solutionaccuracy of EABC is compared with that of ACO PSO andABC Table 2 shows the optimization of the 15 benchmarkfunctions obtained in the 30 independent runs by eachalgorithm and some interesting results can be found inTable 2

Firstly almost all algorithms have identical performanceon most of unimodal functions 1198911 to 1198914 1198917 1198918 and 11989114However on other functions these four algorithms showdifferent performance especially for multimodal ones suchas 11989111 11989112 11989113 and 11989115 Fox example on function 11989115the best values obtained by ACO PSO ABC and EABCare minus10296 minus699347 minus125669 and minus1525687 respectivelyIt means that EABC can be efficiently applied for solvingmultimodal and multidimensional function optimizationproblems due to its abundant operators such as clonal

8 The Scientific World Journal

Table 1 Benchmark functions used in experiments

Number Function Dimension Property Range Min

1 1198911(119909) =

119863

sum

119894=1

1199092

119894 30 Unimodal [minus100 100] 0

2 1198912(119909) =

119863

sum

119894=1

1198941199092

119894 30 Unimodal [minus10 10] 0

3 1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 5 Unimodal [minus512 512] 0

4 1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 30 Unimodal [minus100 100] 0

5 1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 30 Unimodal [minus128 128] 0

6 1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 30 Unimodal [minus100 100] 0

7 1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) 2 Unimodal [minus100 100] minus1

8 1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 2 Unimodal [minus10 10] 0

9 1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 30 Unimodal [minus10 10] 0

10 11989110 (119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 30 Unimodal [minus30 30] 0

11 11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 30 Multimodal [minus512 512] 0

12 11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 30 Multimodal [minus600 600] 0

13 11989113 (119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 10 Multimodal [minus50 50] 0

14 11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

2 Multimodal [minus100 100] 0

15 11989115 (119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817) 30 Multimodal [minus500 500] minus125695

selection and negative selection which outperforms ACOPSO and ABC In addition on the one hand on basicunimodal functions both the basic ABC and EABC havealmost identical solving performance on the other handon the multimodal functions these two ones display hugedifference

Secondly the EABC algorithm can find optimal or closer-to-optimal solutions on the complex multimodal functions11989111 11989112 11989113 and 11989114 Although the result of multi-dimensionfunction 11989115 is slightly far from the known global optimumthe EABC is superior to the other algorithms all the sameAt the same time for almost all benchmark functionsstandard deviations of the EABC obtained from the statisticalexperimental data are no greater than those of others expectfor 11989110 In addition the differences of EABC between thebest andworst solutions for these 15 benchmark functions are

relatively smaller than those of others in the 30 independentsimulation runs All thesemean that the EABC algorithm hasbetter robustness than others It is also clear that EABC canwork better in almost all cases and gets better performancethan ACO PSO and ABC

Summarizing the statementsmentioned above the EABCcan prevent bees from being trapped into the local minimumaccelerate convergence process search with more efficiencyand improve exploitation abilities for basic ABC

53 Analysis and Discussion In this section the effectsof each modification on the performance of EABC arediscussed First of all corresponding to three modificationswe named the basic ABC with the proposed initializationas IABC the one with the proposed self-adaptive searchingstrategy as SABC and the one with the proposed immune

The Scientific World Journal 9

Table 2 Benchmark functions used in experiments for testing the performances of EABC ACO PSO and ABC

Function number Min ACO PSO ABC EABC

1198911(119909) =

119863

sum

119894=1

1199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198912(119909) =

119863

sum

119894=1

1198941199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 0

Best 0 0 0 0Worst 000289 000321 000543 000422Mean 000136 000116 000300 000196SD 000219 000276 000387 000208

1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 0

Best 0 0 0 0Worst 000246 000305 000110 000400Mean 000180 000156 00066 000210SD 000039 000058 000092 000037

1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) minus1

Best minus1 minus1 minus1 minus1Worst minus1 minus1 minus1 minus1Mean minus1 minus1 minus1 minus1SD 0 0 0 0

1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 0

Best 066667 06667 0 0Worst 066667 06667 0 0Mean 066667 06667 0 0SD 000001 000001 0 0

11989110(119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 0

Best 87513 105433 196788 91578Worst 324215 246711 542333 269874Mean 182039 150886 331227 173558SD 50361 241702 1541443 114774

11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 0

Best 526677 435774 0 0Worst 532331 441131 0 0Mean 529226 439771 0 0SD 45649 117286 0 0

11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 0

Best 001470 0017112 0008531 0Worst 001499 0017989 0017356 0Mean 001479 0017391 0011447 0SD 000296 0020808 0001223 0

10 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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ReconfigurableComputing

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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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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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 The Scientific World Journal

(1) Generate the initial population 119909119894 (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (2)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Calculate the probability values 119875119894 for the solution (119909119894) by (3)(7) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(8) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(9) Memorize the best solution so far(10) Cycle = cycle +1(11) Until cycle = MEN

Algorithm 1 Pseudocode of main body of ABC algorithm

individuals As a result this work proposes a novel initial-ization approach which uses chaotic systems and affinity-based compression method to produce initial populationHere according to the literature [19] sinus map is selectedand its equation is defined as follows

119888119898119899+1 = 23(119888119898119899)2 sin(120587119888119898

119899)

119888119898119899 = 0 1 2 119873

(5)

where 119899 is the iteration counter and119873 is the maximum num-ber of chaotic iterations Furthermore the affinity equationbetween two different individuals is defined as Euclideandistance shown in (6)

affinity (119909119894 119909119896) = radic119863

sum

119895=1

(119909119895

119894 minus 119909119895

119896)2

(119894 = 119896 119895 isin (1 2 119863)) gt 120585

(6)

where 120585 is a threshold value defined in advance so as to controlthe difference between two individuals 119863 is the number ofoptimization parameters Based on these operators we pro-pose the following algorithm to generate initial populationand its corresponding pseudocode is given in Algorithm 2

42 An Improved Search Mechanism Based on the Self-Adaptive Strategy As mentioned above the original ABCalgorithm is good at exploration but poor at exploitationdue to two reasons On the one hand in (2) the coefficient120593119894119895 is a uniform random number in [minus1 1] and 119909119896119895 is a

random individual in the population therefore the searchprocess of solutions illustrated by (2) is random enoughfor exploration [21] On the other hand a greedy selectionmechanism is employed between the old and candidatesolutions which may easily make solutions get trapped in

a local optimal What is more the slower convergence rateof the algorithm is another limitation when working withsome complex composite functions As a result we introducea self-adaptive strategy to improve its search process and thedetailed explanations are as follows

Firstly we can see from (2) that there is only one differentelement between the candidate solution and the old one (iethe 119895th element) This search strategy may be efficient inearlier iterations However when the solution approaches toa local optimum its search efficiency becomes poor in lateriterations To handle this limitation similar to [29 30] weintroduce a parameter 119871 to control the difference betweenthe candidate solution and the old one where how to choosethe value of the parameter 119871 is very important Generally thehigher the value of 119871 is the more information is brought intothe candidate solution In the literature [18] the parameter119871 is a fixed constant and chosen according to simulationexperiments Different from this approach mentioned abovethis paper proposes a self-adaptive adjustable strategy todetermine the parameter 119871 and the corresponding searchequation is given below

V119897119894 = 119909119897119894 + 119888119898

119897119894 (119909119897119894 minus 119909119897119896) 119897 = 1 2 119871 (7)

119871 = 1 +

1003817100381710038171003817100381710038171003817119863(

iteration2 timesMEN

)

1003817100381710038171003817100381710038171003817 (8)

where 119888119898119897119894 is a chaotic map defined by (5)119863 is the number ofoptimization parameters A symbol sdot denotes a roundingoperator We can see from (8) that onlooker bees will searchbetter solutions in only one direction in the first iterationand they will search in the whole space with the increasein the value of 119871 when the solutions are closely to the localoptimum in later iterations However we limit the value of 119871nomore than 1+1198632This is because with the higher valueof 119871 the solution generated by the search equations (7)-(8)

The Scientific World Journal 5

Set the population size SN the number of randomly generated individuals SN119903 ≫ SN the maximum number of chaoticiterations119873 gt 200 the number of optimization parameters D and the individual counter 119894 = 1 119895 = 1- -Chaotic systems- -for 119894 = 1 to SN119903 do

for 119895 = 1 to D doRandomly initialize the first chaotic variable cm0119895 isin (0 1) and set iteration counter 119899 = 0for 119899 = 1 to N do

cm119899+1119895 = 23(cm119899119895)2 sin(120587cm

119899119895)

end for119909119895

119894 = 119909119895

min + cm119895119899(119909119895max minus 119909

119895

min)

end forend for- - Affinity-based compression method- -Set the threshold value 120585 and the individual counter 119894 = 1 119895 = 1for 119894 = 1 to (SN119903 minus 1) do

for 119896 = 119894 + 1 to SN119903 do

if affinity(119909119894 119909119896) = radicsum119863

119895=1(119909119895

119894 minus 119909119895

119896)2lt 120585

SN119903 = SN119903 minus 1 delete the individual 119896 in the populationend if

end forend forSelecting SN fittest individuals form set (119883(SN119903)) as initial population

Algorithm 2 Modified initialization step of ABC algorithm

is more likely random search operator As a result (7)-(8) willdynamically adjust the position of onlooker bees by allowingthem to explore with a wider search space in later iterationsAs the number of the iterations increases the correspondingsearch space of onlooker bees will also increase

The second modification in the ABC algorithm lies inthe selection probability of onlooker bees associated with thefood source In the original ABC algorithm the fitness valuesobtained from the employed bees are adopted to determinethe selection probabilities of onlooker bees Nevertheless thefitness comparisons among different individuals only reflectthe qualitative information In this work based on the ideaof the fitness evolution we introduce an environment factor120578119894 corresponding to every food source so as to evaluate itsexploitation potential In other words the parameter 120578119894 isused to evaluate quantitatively the environment situationsof exploitation for every food source As the number ofiterations increases the higher the value of 120578119894 the betterexploitation environment it corresponds to At this momentmore onlooker bees will follow the corresponding employedone to its food source position with higher nectar amount inorder to accelerate exploitation efficiency Conversely if thevalue of 120578119894 is lower its corresponding solution lies in theworseexploitation environment which means that it is difficult tofind out a better solution around the old food source andless onlooker bees will be attracted by the employed oneAs a result how to define 120578119894 needs explain Generally theparameter 120578119894 is associated with both a fitness change amountΔ119891 and a count accumulator 119862 where Δ119891 denotes the fitnessdifference associated with the same food source betweentwo adjacent generations which reflects the exploitation

potential of the corresponding food source position Besidesa count accumulator119862will be explained in the following textThe equations below reflect the fitness change amount Δ119891between two adjacent generations

Δ119891119905 (119909119894) =

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(9)

Δ1198911015840119905 (119909119894) = exp [Δ119891119905 (119909119894)] = log

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(10)

where the parameter 119905 denotes the number of iterations andthe symbol | | means absolute value sign As can be seenfrom (9) the value range of Δ119891 is between 0 and 1 Whenthe value of Δ119891 is higher it means the corresponding foodsource has a higher exploitation potential and is largerlypossible to find out a better solution and vice versa Howeverat later iterations the value of Δ119891 will be very small andneed appropriate amplification As a result we adopt powerfunction to amplifyΔ119891 here and take the number 119890 as its baseDue to this reason (9) is substituted for (10)

In addition if Δ119891 is less than a given small value inadvance (ie Δ119891 le Δ1198910 and Δ1198910 is the threshold of Δ119891) theimproved ABC algorithm may trigger a count accumulatorcalled Counter (represented by 119862 in this paper) which is usedto record how many times the quality of the solution hasnot improved (it corresponds to the number of iterations inthe algorithm) The following equation is used to express thecount accumulator119862 at the 119905th iteration Generally the larger

6 The Scientific World Journal

the value of 119862 the less exploitation potential that the foodsource position corresponds to

119862 (119905) =

119862 (119905 minus 1) + 119879 (119905) =

119905

sum

119894=119904

119879 (119894) 119879 (119905) = 1

0 119879 (119905) = 0

(11)

where119879 represents a pulse signalWhenΔ119891 gt 0 orΔ119891 ge Δ1198910119879 = 0 conversely when Δ119891 = 0 or Δ119891 le Δ1198910 119879 = 1 It meansthat

119879 (119905) = 1 (Δ119891 (119905) = 0) or (Δ119891 (119905) le Δ119891(119905)0) 0 (Δ119891 (119905) gt 0) or (Δ119891 (119905) ge Δ119891(119905)0)

(12)

Note that according to the performance requirement ofspecific problems the maximum times of the stagnation ofglobal extrema 119862max should be defined in advance whichmeans if a minimum of a function has not been updated forcontinuous 119862max iterations the current exploitation area hasfew potentials to find out the better solution and computationresources should be redistributed Generally we define thatthe 119862max is equal to or not less than 5 In addition 119862(119905)is normalized for simplifying the problem and 1198621015840(119905) can beobtained which is expressed as follows

1198621015840(119905) =

sum119905119894=119904 119879 (119894) + 1

119862max + 1 (13)

On the basis of the definitions of the fitness changeamount Δ119891and the count accumulator 119862 the environmentfactor 120578 is defined as follows

120578 (119905) =Δ1198911015840(119905)

1198621015840 (119905) (14)

Submitting (10) and (13) into (14) we can achieve thefollowing equation

120578 (119905) =(119862max + 1) sdot exp

1003816100381610038161003816(fit119905 (119909119894) minus fit119905minus1 (119909119894)) fit119905minus1 (119909119894)1003816100381610038161003816

sum119905119894=119904 119879 (119894) + 1

(15)

In accordance with the expression of the environmentfactor 120578(119909119894) corresponding to every food source the selectionprobability of onlooker bees associated with the food sourcecan be substituted with the following equation

119875119894 =120578119905 (119909119894)

sumSN119898=1 120578119905 (119909119898)

(16)

43 Enhancing Convergence Efficiency with Artificial ImmuneNetwork Operators In the basic ABC system artificial beesfly around in the search space Some (like employed andonlooker bees) choose food source depending on the experi-ence of themselves and their nest mates and then adjust theirpositions but others (like scouts) fly and choose the foodsources randomly without using experience If the nectar

amount of a new source is higher than that of the previousone in their memory they memorize the new food sourceposition and forget the previous one Thus the ABC systemcombines local search methods carried out by employedand onlooker bees with global search methods managedby Karaboga and Basturk [15] However unlike the ABCsystem the concept of artificial immune system (AIS) wasoriginated by observing how the defense mechanism ofnatural immune system protects against attacks by antigensThere are numerous AIS algorithms developed for a varietyof applications where artificial immune network (aiNet forshort) is a typical one and its algorithms and models areoriginally proposed to perform information compressionand data clustering based on artificial immune system (AIS)theory [31] Immune network-based algorithms are similarto clonal selection algorithms in that they both measurethe goodness of antibodies by affinities and both methodsinclude a series of steps for selecting cloning and mutatingantibodiesThemajor difference is that the immune network-based algorithms are represented bynetwork graph structures[32] Compared with other ones the immune network-basedalgorithms employ extra procedures of antibody pruning andsuppressing This allows the models to generate a smallerless-redundant population of antibody representatives whichis desirable for solving multimodal function optimizationComparing ABC optimization with ai-Net algorithm wecan see that the advantages of ABC optimization lie in itsneighborhood search method according to the profitabilityof food sources However ai-Net algorithm adopts fixedclonal individuals to perform local search which has certainblindness In addition due to introducing network com-pression negative selection and other operators ai-Net canmaintain the diversity of the population and reduce thepossibility of being trapped into a local minimum Unlike theai-Net algorithm ABC optimization maintains populationdiversity through random search of scout bees which hasobvious limitation Based on the analysis mentioned aboveif network compression and negative selection deriving fromai-Net algorithm are introduced into ABC optimization thisimproved one may have a powerful and efficient multimodalsearching ability as well as good stabilization The detailedprocess is as follows

Different employed bee individual corresponds to differ-ent food source position In order to eliminate redundantand similar food sources negative selection and networkcompression are used to compare with the similarities amongvarious individuals The Euclidean distance of two employedbee individuals119883119894 and119883119896 is adopted as shown in (17)

119889 (119883119894 119883119896) = radic

119863

sum

119895=1

(119883119895

119894 minus 119883119895

119896)2

(119894 = 119896) (17)

In order to simplify the problem the affinity concept isintroduced which is obtained by the following equation usingthe normalization method

119860 (119883119894 119883119896) =1

1 + 119889 (119883119894 119883119896) (18)

The Scientific World Journal 7

(1) Generate the initial population 119909119894 based on chaotic maps and affinity strategy (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (7)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Adopt negative selection and network compression to eliminate redundant and similar food sources by using (18)(7) Randomly generate the same number of new individuals(8) Calculate the probability values 119875119894 for the solution (119909119894) by (16)(9) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(10) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(11) Memorize the best solution so far(12) Cycle = cycle + 1(13) Until cycle = MEN

Algorithm 3 Pseudocode of main body of the enhanced ABC algorithm

where the value range of 119860(119883119894119883119896) is between 0 and 1 Thesmaller the value of 119860(119883119894119883119896) is the larger the value of119860(119883119894119883119896) is which means that two different employed beeindividuals have a higher similarity Specially when119860(119883119894119883119896)equals 1 these two ones are identical According to negativeselection and network compression operators redundant andsimilar food sources should be eliminated We predefine athreshold value 120576 so as to realize wipe-off of redundantindividuals It also means when 119860(119883119894119883119896) is equal to or greatthan 120576 we think these two ones are identical and only one canbe retained and other should bewiped off Repeat this processuntil the affinity of any two individuals in a population isless than 120576 In doing so the population size may be reducedNevertheless in order to keep the population size unchangedthe same number of new individuals need generating ran-domlyThrough negative selection and network compressionoperators the exploitation efficiencywill be improved and thecorresponding convergence rate of the algorithm will also beaccelerated

44 Main Steps of the Enhanced Artificial Bee Colony Algo-rithm Based on the above analysis three main improve-ments including novel generation of initial population self-adaptive searching strategy and redundant individual com-pression operator are presented and the detailed pseudo-codeis given in Algorithm 3

5 Experimental Studies on FunctionOptimization Problems

51 Benchmark Functions and Parameter Settings In thissection numerical experiment is used to test the performanceof the enhanced ABC (shorthand for EABC) proposed in this

paper Summarized in Table 1 are the 15 scalable benchmarkfunctions 1198911 sim 11989110 are continuous unimodal functions11989111 sim 11989115 are multimodal functions and the number oftheir local minima increases exponentially with the problemdimension

In order to testify the performance of different intelligentalgorithms we compare the EABC with the standard ACOPSO and ABC In all simulations the population size ofACO PSO ABC and EABC is 50 The maximum numberof function evaluations (FE) is set to 5000 The thresholdvalue of the affinity 120576 is 09 Other related parameter valuesof ACO PSO and ABC are referred in the literature [17]All experiment results reported are obtained based on 30independent runsThe experiment results are the best worstmean and standard deviation of the statistical experimentaldata

52 Simulation Results The performance on the solutionaccuracy of EABC is compared with that of ACO PSO andABC Table 2 shows the optimization of the 15 benchmarkfunctions obtained in the 30 independent runs by eachalgorithm and some interesting results can be found inTable 2

Firstly almost all algorithms have identical performanceon most of unimodal functions 1198911 to 1198914 1198917 1198918 and 11989114However on other functions these four algorithms showdifferent performance especially for multimodal ones suchas 11989111 11989112 11989113 and 11989115 Fox example on function 11989115the best values obtained by ACO PSO ABC and EABCare minus10296 minus699347 minus125669 and minus1525687 respectivelyIt means that EABC can be efficiently applied for solvingmultimodal and multidimensional function optimizationproblems due to its abundant operators such as clonal

8 The Scientific World Journal

Table 1 Benchmark functions used in experiments

Number Function Dimension Property Range Min

1 1198911(119909) =

119863

sum

119894=1

1199092

119894 30 Unimodal [minus100 100] 0

2 1198912(119909) =

119863

sum

119894=1

1198941199092

119894 30 Unimodal [minus10 10] 0

3 1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 5 Unimodal [minus512 512] 0

4 1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 30 Unimodal [minus100 100] 0

5 1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 30 Unimodal [minus128 128] 0

6 1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 30 Unimodal [minus100 100] 0

7 1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) 2 Unimodal [minus100 100] minus1

8 1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 2 Unimodal [minus10 10] 0

9 1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 30 Unimodal [minus10 10] 0

10 11989110 (119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 30 Unimodal [minus30 30] 0

11 11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 30 Multimodal [minus512 512] 0

12 11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 30 Multimodal [minus600 600] 0

13 11989113 (119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 10 Multimodal [minus50 50] 0

14 11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

2 Multimodal [minus100 100] 0

15 11989115 (119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817) 30 Multimodal [minus500 500] minus125695

selection and negative selection which outperforms ACOPSO and ABC In addition on the one hand on basicunimodal functions both the basic ABC and EABC havealmost identical solving performance on the other handon the multimodal functions these two ones display hugedifference

Secondly the EABC algorithm can find optimal or closer-to-optimal solutions on the complex multimodal functions11989111 11989112 11989113 and 11989114 Although the result of multi-dimensionfunction 11989115 is slightly far from the known global optimumthe EABC is superior to the other algorithms all the sameAt the same time for almost all benchmark functionsstandard deviations of the EABC obtained from the statisticalexperimental data are no greater than those of others expectfor 11989110 In addition the differences of EABC between thebest andworst solutions for these 15 benchmark functions are

relatively smaller than those of others in the 30 independentsimulation runs All thesemean that the EABC algorithm hasbetter robustness than others It is also clear that EABC canwork better in almost all cases and gets better performancethan ACO PSO and ABC

Summarizing the statementsmentioned above the EABCcan prevent bees from being trapped into the local minimumaccelerate convergence process search with more efficiencyand improve exploitation abilities for basic ABC

53 Analysis and Discussion In this section the effectsof each modification on the performance of EABC arediscussed First of all corresponding to three modificationswe named the basic ABC with the proposed initializationas IABC the one with the proposed self-adaptive searchingstrategy as SABC and the one with the proposed immune

The Scientific World Journal 9

Table 2 Benchmark functions used in experiments for testing the performances of EABC ACO PSO and ABC

Function number Min ACO PSO ABC EABC

1198911(119909) =

119863

sum

119894=1

1199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198912(119909) =

119863

sum

119894=1

1198941199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 0

Best 0 0 0 0Worst 000289 000321 000543 000422Mean 000136 000116 000300 000196SD 000219 000276 000387 000208

1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 0

Best 0 0 0 0Worst 000246 000305 000110 000400Mean 000180 000156 00066 000210SD 000039 000058 000092 000037

1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) minus1

Best minus1 minus1 minus1 minus1Worst minus1 minus1 minus1 minus1Mean minus1 minus1 minus1 minus1SD 0 0 0 0

1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 0

Best 066667 06667 0 0Worst 066667 06667 0 0Mean 066667 06667 0 0SD 000001 000001 0 0

11989110(119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 0

Best 87513 105433 196788 91578Worst 324215 246711 542333 269874Mean 182039 150886 331227 173558SD 50361 241702 1541443 114774

11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 0

Best 526677 435774 0 0Worst 532331 441131 0 0Mean 529226 439771 0 0SD 45649 117286 0 0

11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 0

Best 001470 0017112 0008531 0Worst 001499 0017989 0017356 0Mean 001479 0017391 0011447 0SD 000296 0020808 0001223 0

10 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

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The Scientific World Journal 5

Set the population size SN the number of randomly generated individuals SN119903 ≫ SN the maximum number of chaoticiterations119873 gt 200 the number of optimization parameters D and the individual counter 119894 = 1 119895 = 1- -Chaotic systems- -for 119894 = 1 to SN119903 do

for 119895 = 1 to D doRandomly initialize the first chaotic variable cm0119895 isin (0 1) and set iteration counter 119899 = 0for 119899 = 1 to N do

cm119899+1119895 = 23(cm119899119895)2 sin(120587cm

119899119895)

end for119909119895

119894 = 119909119895

min + cm119895119899(119909119895max minus 119909

119895

min)

end forend for- - Affinity-based compression method- -Set the threshold value 120585 and the individual counter 119894 = 1 119895 = 1for 119894 = 1 to (SN119903 minus 1) do

for 119896 = 119894 + 1 to SN119903 do

if affinity(119909119894 119909119896) = radicsum119863

119895=1(119909119895

119894 minus 119909119895

119896)2lt 120585

SN119903 = SN119903 minus 1 delete the individual 119896 in the populationend if

end forend forSelecting SN fittest individuals form set (119883(SN119903)) as initial population

Algorithm 2 Modified initialization step of ABC algorithm

is more likely random search operator As a result (7)-(8) willdynamically adjust the position of onlooker bees by allowingthem to explore with a wider search space in later iterationsAs the number of the iterations increases the correspondingsearch space of onlooker bees will also increase

The second modification in the ABC algorithm lies inthe selection probability of onlooker bees associated with thefood source In the original ABC algorithm the fitness valuesobtained from the employed bees are adopted to determinethe selection probabilities of onlooker bees Nevertheless thefitness comparisons among different individuals only reflectthe qualitative information In this work based on the ideaof the fitness evolution we introduce an environment factor120578119894 corresponding to every food source so as to evaluate itsexploitation potential In other words the parameter 120578119894 isused to evaluate quantitatively the environment situationsof exploitation for every food source As the number ofiterations increases the higher the value of 120578119894 the betterexploitation environment it corresponds to At this momentmore onlooker bees will follow the corresponding employedone to its food source position with higher nectar amount inorder to accelerate exploitation efficiency Conversely if thevalue of 120578119894 is lower its corresponding solution lies in theworseexploitation environment which means that it is difficult tofind out a better solution around the old food source andless onlooker bees will be attracted by the employed oneAs a result how to define 120578119894 needs explain Generally theparameter 120578119894 is associated with both a fitness change amountΔ119891 and a count accumulator 119862 where Δ119891 denotes the fitnessdifference associated with the same food source betweentwo adjacent generations which reflects the exploitation

potential of the corresponding food source position Besidesa count accumulator119862will be explained in the following textThe equations below reflect the fitness change amount Δ119891between two adjacent generations

Δ119891119905 (119909119894) =

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(9)

Δ1198911015840119905 (119909119894) = exp [Δ119891119905 (119909119894)] = log

100381610038161003816100381610038161003816100381610038161003816

fit119905 (119909119894) minus fit119905minus1 (119909119894)fit119905minus1 (119909119894)

100381610038161003816100381610038161003816100381610038161003816

(10)

where the parameter 119905 denotes the number of iterations andthe symbol | | means absolute value sign As can be seenfrom (9) the value range of Δ119891 is between 0 and 1 Whenthe value of Δ119891 is higher it means the corresponding foodsource has a higher exploitation potential and is largerlypossible to find out a better solution and vice versa Howeverat later iterations the value of Δ119891 will be very small andneed appropriate amplification As a result we adopt powerfunction to amplifyΔ119891 here and take the number 119890 as its baseDue to this reason (9) is substituted for (10)

In addition if Δ119891 is less than a given small value inadvance (ie Δ119891 le Δ1198910 and Δ1198910 is the threshold of Δ119891) theimproved ABC algorithm may trigger a count accumulatorcalled Counter (represented by 119862 in this paper) which is usedto record how many times the quality of the solution hasnot improved (it corresponds to the number of iterations inthe algorithm) The following equation is used to express thecount accumulator119862 at the 119905th iteration Generally the larger

6 The Scientific World Journal

the value of 119862 the less exploitation potential that the foodsource position corresponds to

119862 (119905) =

119862 (119905 minus 1) + 119879 (119905) =

119905

sum

119894=119904

119879 (119894) 119879 (119905) = 1

0 119879 (119905) = 0

(11)

where119879 represents a pulse signalWhenΔ119891 gt 0 orΔ119891 ge Δ1198910119879 = 0 conversely when Δ119891 = 0 or Δ119891 le Δ1198910 119879 = 1 It meansthat

119879 (119905) = 1 (Δ119891 (119905) = 0) or (Δ119891 (119905) le Δ119891(119905)0) 0 (Δ119891 (119905) gt 0) or (Δ119891 (119905) ge Δ119891(119905)0)

(12)

Note that according to the performance requirement ofspecific problems the maximum times of the stagnation ofglobal extrema 119862max should be defined in advance whichmeans if a minimum of a function has not been updated forcontinuous 119862max iterations the current exploitation area hasfew potentials to find out the better solution and computationresources should be redistributed Generally we define thatthe 119862max is equal to or not less than 5 In addition 119862(119905)is normalized for simplifying the problem and 1198621015840(119905) can beobtained which is expressed as follows

1198621015840(119905) =

sum119905119894=119904 119879 (119894) + 1

119862max + 1 (13)

On the basis of the definitions of the fitness changeamount Δ119891and the count accumulator 119862 the environmentfactor 120578 is defined as follows

120578 (119905) =Δ1198911015840(119905)

1198621015840 (119905) (14)

Submitting (10) and (13) into (14) we can achieve thefollowing equation

120578 (119905) =(119862max + 1) sdot exp

1003816100381610038161003816(fit119905 (119909119894) minus fit119905minus1 (119909119894)) fit119905minus1 (119909119894)1003816100381610038161003816

sum119905119894=119904 119879 (119894) + 1

(15)

In accordance with the expression of the environmentfactor 120578(119909119894) corresponding to every food source the selectionprobability of onlooker bees associated with the food sourcecan be substituted with the following equation

119875119894 =120578119905 (119909119894)

sumSN119898=1 120578119905 (119909119898)

(16)

43 Enhancing Convergence Efficiency with Artificial ImmuneNetwork Operators In the basic ABC system artificial beesfly around in the search space Some (like employed andonlooker bees) choose food source depending on the experi-ence of themselves and their nest mates and then adjust theirpositions but others (like scouts) fly and choose the foodsources randomly without using experience If the nectar

amount of a new source is higher than that of the previousone in their memory they memorize the new food sourceposition and forget the previous one Thus the ABC systemcombines local search methods carried out by employedand onlooker bees with global search methods managedby Karaboga and Basturk [15] However unlike the ABCsystem the concept of artificial immune system (AIS) wasoriginated by observing how the defense mechanism ofnatural immune system protects against attacks by antigensThere are numerous AIS algorithms developed for a varietyof applications where artificial immune network (aiNet forshort) is a typical one and its algorithms and models areoriginally proposed to perform information compressionand data clustering based on artificial immune system (AIS)theory [31] Immune network-based algorithms are similarto clonal selection algorithms in that they both measurethe goodness of antibodies by affinities and both methodsinclude a series of steps for selecting cloning and mutatingantibodiesThemajor difference is that the immune network-based algorithms are represented bynetwork graph structures[32] Compared with other ones the immune network-basedalgorithms employ extra procedures of antibody pruning andsuppressing This allows the models to generate a smallerless-redundant population of antibody representatives whichis desirable for solving multimodal function optimizationComparing ABC optimization with ai-Net algorithm wecan see that the advantages of ABC optimization lie in itsneighborhood search method according to the profitabilityof food sources However ai-Net algorithm adopts fixedclonal individuals to perform local search which has certainblindness In addition due to introducing network com-pression negative selection and other operators ai-Net canmaintain the diversity of the population and reduce thepossibility of being trapped into a local minimum Unlike theai-Net algorithm ABC optimization maintains populationdiversity through random search of scout bees which hasobvious limitation Based on the analysis mentioned aboveif network compression and negative selection deriving fromai-Net algorithm are introduced into ABC optimization thisimproved one may have a powerful and efficient multimodalsearching ability as well as good stabilization The detailedprocess is as follows

Different employed bee individual corresponds to differ-ent food source position In order to eliminate redundantand similar food sources negative selection and networkcompression are used to compare with the similarities amongvarious individuals The Euclidean distance of two employedbee individuals119883119894 and119883119896 is adopted as shown in (17)

119889 (119883119894 119883119896) = radic

119863

sum

119895=1

(119883119895

119894 minus 119883119895

119896)2

(119894 = 119896) (17)

In order to simplify the problem the affinity concept isintroduced which is obtained by the following equation usingthe normalization method

119860 (119883119894 119883119896) =1

1 + 119889 (119883119894 119883119896) (18)

The Scientific World Journal 7

(1) Generate the initial population 119909119894 based on chaotic maps and affinity strategy (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (7)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Adopt negative selection and network compression to eliminate redundant and similar food sources by using (18)(7) Randomly generate the same number of new individuals(8) Calculate the probability values 119875119894 for the solution (119909119894) by (16)(9) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(10) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(11) Memorize the best solution so far(12) Cycle = cycle + 1(13) Until cycle = MEN

Algorithm 3 Pseudocode of main body of the enhanced ABC algorithm

where the value range of 119860(119883119894119883119896) is between 0 and 1 Thesmaller the value of 119860(119883119894119883119896) is the larger the value of119860(119883119894119883119896) is which means that two different employed beeindividuals have a higher similarity Specially when119860(119883119894119883119896)equals 1 these two ones are identical According to negativeselection and network compression operators redundant andsimilar food sources should be eliminated We predefine athreshold value 120576 so as to realize wipe-off of redundantindividuals It also means when 119860(119883119894119883119896) is equal to or greatthan 120576 we think these two ones are identical and only one canbe retained and other should bewiped off Repeat this processuntil the affinity of any two individuals in a population isless than 120576 In doing so the population size may be reducedNevertheless in order to keep the population size unchangedthe same number of new individuals need generating ran-domlyThrough negative selection and network compressionoperators the exploitation efficiencywill be improved and thecorresponding convergence rate of the algorithm will also beaccelerated

44 Main Steps of the Enhanced Artificial Bee Colony Algo-rithm Based on the above analysis three main improve-ments including novel generation of initial population self-adaptive searching strategy and redundant individual com-pression operator are presented and the detailed pseudo-codeis given in Algorithm 3

5 Experimental Studies on FunctionOptimization Problems

51 Benchmark Functions and Parameter Settings In thissection numerical experiment is used to test the performanceof the enhanced ABC (shorthand for EABC) proposed in this

paper Summarized in Table 1 are the 15 scalable benchmarkfunctions 1198911 sim 11989110 are continuous unimodal functions11989111 sim 11989115 are multimodal functions and the number oftheir local minima increases exponentially with the problemdimension

In order to testify the performance of different intelligentalgorithms we compare the EABC with the standard ACOPSO and ABC In all simulations the population size ofACO PSO ABC and EABC is 50 The maximum numberof function evaluations (FE) is set to 5000 The thresholdvalue of the affinity 120576 is 09 Other related parameter valuesof ACO PSO and ABC are referred in the literature [17]All experiment results reported are obtained based on 30independent runsThe experiment results are the best worstmean and standard deviation of the statistical experimentaldata

52 Simulation Results The performance on the solutionaccuracy of EABC is compared with that of ACO PSO andABC Table 2 shows the optimization of the 15 benchmarkfunctions obtained in the 30 independent runs by eachalgorithm and some interesting results can be found inTable 2

Firstly almost all algorithms have identical performanceon most of unimodal functions 1198911 to 1198914 1198917 1198918 and 11989114However on other functions these four algorithms showdifferent performance especially for multimodal ones suchas 11989111 11989112 11989113 and 11989115 Fox example on function 11989115the best values obtained by ACO PSO ABC and EABCare minus10296 minus699347 minus125669 and minus1525687 respectivelyIt means that EABC can be efficiently applied for solvingmultimodal and multidimensional function optimizationproblems due to its abundant operators such as clonal

8 The Scientific World Journal

Table 1 Benchmark functions used in experiments

Number Function Dimension Property Range Min

1 1198911(119909) =

119863

sum

119894=1

1199092

119894 30 Unimodal [minus100 100] 0

2 1198912(119909) =

119863

sum

119894=1

1198941199092

119894 30 Unimodal [minus10 10] 0

3 1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 5 Unimodal [minus512 512] 0

4 1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 30 Unimodal [minus100 100] 0

5 1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 30 Unimodal [minus128 128] 0

6 1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 30 Unimodal [minus100 100] 0

7 1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) 2 Unimodal [minus100 100] minus1

8 1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 2 Unimodal [minus10 10] 0

9 1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 30 Unimodal [minus10 10] 0

10 11989110 (119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 30 Unimodal [minus30 30] 0

11 11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 30 Multimodal [minus512 512] 0

12 11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 30 Multimodal [minus600 600] 0

13 11989113 (119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 10 Multimodal [minus50 50] 0

14 11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

2 Multimodal [minus100 100] 0

15 11989115 (119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817) 30 Multimodal [minus500 500] minus125695

selection and negative selection which outperforms ACOPSO and ABC In addition on the one hand on basicunimodal functions both the basic ABC and EABC havealmost identical solving performance on the other handon the multimodal functions these two ones display hugedifference

Secondly the EABC algorithm can find optimal or closer-to-optimal solutions on the complex multimodal functions11989111 11989112 11989113 and 11989114 Although the result of multi-dimensionfunction 11989115 is slightly far from the known global optimumthe EABC is superior to the other algorithms all the sameAt the same time for almost all benchmark functionsstandard deviations of the EABC obtained from the statisticalexperimental data are no greater than those of others expectfor 11989110 In addition the differences of EABC between thebest andworst solutions for these 15 benchmark functions are

relatively smaller than those of others in the 30 independentsimulation runs All thesemean that the EABC algorithm hasbetter robustness than others It is also clear that EABC canwork better in almost all cases and gets better performancethan ACO PSO and ABC

Summarizing the statementsmentioned above the EABCcan prevent bees from being trapped into the local minimumaccelerate convergence process search with more efficiencyand improve exploitation abilities for basic ABC

53 Analysis and Discussion In this section the effectsof each modification on the performance of EABC arediscussed First of all corresponding to three modificationswe named the basic ABC with the proposed initializationas IABC the one with the proposed self-adaptive searchingstrategy as SABC and the one with the proposed immune

The Scientific World Journal 9

Table 2 Benchmark functions used in experiments for testing the performances of EABC ACO PSO and ABC

Function number Min ACO PSO ABC EABC

1198911(119909) =

119863

sum

119894=1

1199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198912(119909) =

119863

sum

119894=1

1198941199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 0

Best 0 0 0 0Worst 000289 000321 000543 000422Mean 000136 000116 000300 000196SD 000219 000276 000387 000208

1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 0

Best 0 0 0 0Worst 000246 000305 000110 000400Mean 000180 000156 00066 000210SD 000039 000058 000092 000037

1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) minus1

Best minus1 minus1 minus1 minus1Worst minus1 minus1 minus1 minus1Mean minus1 minus1 minus1 minus1SD 0 0 0 0

1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 0

Best 066667 06667 0 0Worst 066667 06667 0 0Mean 066667 06667 0 0SD 000001 000001 0 0

11989110(119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 0

Best 87513 105433 196788 91578Worst 324215 246711 542333 269874Mean 182039 150886 331227 173558SD 50361 241702 1541443 114774

11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 0

Best 526677 435774 0 0Worst 532331 441131 0 0Mean 529226 439771 0 0SD 45649 117286 0 0

11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 0

Best 001470 0017112 0008531 0Worst 001499 0017989 0017356 0Mean 001479 0017391 0011447 0SD 000296 0020808 0001223 0

10 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 The Scientific World Journal

the value of 119862 the less exploitation potential that the foodsource position corresponds to

119862 (119905) =

119862 (119905 minus 1) + 119879 (119905) =

119905

sum

119894=119904

119879 (119894) 119879 (119905) = 1

0 119879 (119905) = 0

(11)

where119879 represents a pulse signalWhenΔ119891 gt 0 orΔ119891 ge Δ1198910119879 = 0 conversely when Δ119891 = 0 or Δ119891 le Δ1198910 119879 = 1 It meansthat

119879 (119905) = 1 (Δ119891 (119905) = 0) or (Δ119891 (119905) le Δ119891(119905)0) 0 (Δ119891 (119905) gt 0) or (Δ119891 (119905) ge Δ119891(119905)0)

(12)

Note that according to the performance requirement ofspecific problems the maximum times of the stagnation ofglobal extrema 119862max should be defined in advance whichmeans if a minimum of a function has not been updated forcontinuous 119862max iterations the current exploitation area hasfew potentials to find out the better solution and computationresources should be redistributed Generally we define thatthe 119862max is equal to or not less than 5 In addition 119862(119905)is normalized for simplifying the problem and 1198621015840(119905) can beobtained which is expressed as follows

1198621015840(119905) =

sum119905119894=119904 119879 (119894) + 1

119862max + 1 (13)

On the basis of the definitions of the fitness changeamount Δ119891and the count accumulator 119862 the environmentfactor 120578 is defined as follows

120578 (119905) =Δ1198911015840(119905)

1198621015840 (119905) (14)

Submitting (10) and (13) into (14) we can achieve thefollowing equation

120578 (119905) =(119862max + 1) sdot exp

1003816100381610038161003816(fit119905 (119909119894) minus fit119905minus1 (119909119894)) fit119905minus1 (119909119894)1003816100381610038161003816

sum119905119894=119904 119879 (119894) + 1

(15)

In accordance with the expression of the environmentfactor 120578(119909119894) corresponding to every food source the selectionprobability of onlooker bees associated with the food sourcecan be substituted with the following equation

119875119894 =120578119905 (119909119894)

sumSN119898=1 120578119905 (119909119898)

(16)

43 Enhancing Convergence Efficiency with Artificial ImmuneNetwork Operators In the basic ABC system artificial beesfly around in the search space Some (like employed andonlooker bees) choose food source depending on the experi-ence of themselves and their nest mates and then adjust theirpositions but others (like scouts) fly and choose the foodsources randomly without using experience If the nectar

amount of a new source is higher than that of the previousone in their memory they memorize the new food sourceposition and forget the previous one Thus the ABC systemcombines local search methods carried out by employedand onlooker bees with global search methods managedby Karaboga and Basturk [15] However unlike the ABCsystem the concept of artificial immune system (AIS) wasoriginated by observing how the defense mechanism ofnatural immune system protects against attacks by antigensThere are numerous AIS algorithms developed for a varietyof applications where artificial immune network (aiNet forshort) is a typical one and its algorithms and models areoriginally proposed to perform information compressionand data clustering based on artificial immune system (AIS)theory [31] Immune network-based algorithms are similarto clonal selection algorithms in that they both measurethe goodness of antibodies by affinities and both methodsinclude a series of steps for selecting cloning and mutatingantibodiesThemajor difference is that the immune network-based algorithms are represented bynetwork graph structures[32] Compared with other ones the immune network-basedalgorithms employ extra procedures of antibody pruning andsuppressing This allows the models to generate a smallerless-redundant population of antibody representatives whichis desirable for solving multimodal function optimizationComparing ABC optimization with ai-Net algorithm wecan see that the advantages of ABC optimization lie in itsneighborhood search method according to the profitabilityof food sources However ai-Net algorithm adopts fixedclonal individuals to perform local search which has certainblindness In addition due to introducing network com-pression negative selection and other operators ai-Net canmaintain the diversity of the population and reduce thepossibility of being trapped into a local minimum Unlike theai-Net algorithm ABC optimization maintains populationdiversity through random search of scout bees which hasobvious limitation Based on the analysis mentioned aboveif network compression and negative selection deriving fromai-Net algorithm are introduced into ABC optimization thisimproved one may have a powerful and efficient multimodalsearching ability as well as good stabilization The detailedprocess is as follows

Different employed bee individual corresponds to differ-ent food source position In order to eliminate redundantand similar food sources negative selection and networkcompression are used to compare with the similarities amongvarious individuals The Euclidean distance of two employedbee individuals119883119894 and119883119896 is adopted as shown in (17)

119889 (119883119894 119883119896) = radic

119863

sum

119895=1

(119883119895

119894 minus 119883119895

119896)2

(119894 = 119896) (17)

In order to simplify the problem the affinity concept isintroduced which is obtained by the following equation usingthe normalization method

119860 (119883119894 119883119896) =1

1 + 119889 (119883119894 119883119896) (18)

The Scientific World Journal 7

(1) Generate the initial population 119909119894 based on chaotic maps and affinity strategy (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (7)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Adopt negative selection and network compression to eliminate redundant and similar food sources by using (18)(7) Randomly generate the same number of new individuals(8) Calculate the probability values 119875119894 for the solution (119909119894) by (16)(9) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(10) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(11) Memorize the best solution so far(12) Cycle = cycle + 1(13) Until cycle = MEN

Algorithm 3 Pseudocode of main body of the enhanced ABC algorithm

where the value range of 119860(119883119894119883119896) is between 0 and 1 Thesmaller the value of 119860(119883119894119883119896) is the larger the value of119860(119883119894119883119896) is which means that two different employed beeindividuals have a higher similarity Specially when119860(119883119894119883119896)equals 1 these two ones are identical According to negativeselection and network compression operators redundant andsimilar food sources should be eliminated We predefine athreshold value 120576 so as to realize wipe-off of redundantindividuals It also means when 119860(119883119894119883119896) is equal to or greatthan 120576 we think these two ones are identical and only one canbe retained and other should bewiped off Repeat this processuntil the affinity of any two individuals in a population isless than 120576 In doing so the population size may be reducedNevertheless in order to keep the population size unchangedthe same number of new individuals need generating ran-domlyThrough negative selection and network compressionoperators the exploitation efficiencywill be improved and thecorresponding convergence rate of the algorithm will also beaccelerated

44 Main Steps of the Enhanced Artificial Bee Colony Algo-rithm Based on the above analysis three main improve-ments including novel generation of initial population self-adaptive searching strategy and redundant individual com-pression operator are presented and the detailed pseudo-codeis given in Algorithm 3

5 Experimental Studies on FunctionOptimization Problems

51 Benchmark Functions and Parameter Settings In thissection numerical experiment is used to test the performanceof the enhanced ABC (shorthand for EABC) proposed in this

paper Summarized in Table 1 are the 15 scalable benchmarkfunctions 1198911 sim 11989110 are continuous unimodal functions11989111 sim 11989115 are multimodal functions and the number oftheir local minima increases exponentially with the problemdimension

In order to testify the performance of different intelligentalgorithms we compare the EABC with the standard ACOPSO and ABC In all simulations the population size ofACO PSO ABC and EABC is 50 The maximum numberof function evaluations (FE) is set to 5000 The thresholdvalue of the affinity 120576 is 09 Other related parameter valuesof ACO PSO and ABC are referred in the literature [17]All experiment results reported are obtained based on 30independent runsThe experiment results are the best worstmean and standard deviation of the statistical experimentaldata

52 Simulation Results The performance on the solutionaccuracy of EABC is compared with that of ACO PSO andABC Table 2 shows the optimization of the 15 benchmarkfunctions obtained in the 30 independent runs by eachalgorithm and some interesting results can be found inTable 2

Firstly almost all algorithms have identical performanceon most of unimodal functions 1198911 to 1198914 1198917 1198918 and 11989114However on other functions these four algorithms showdifferent performance especially for multimodal ones suchas 11989111 11989112 11989113 and 11989115 Fox example on function 11989115the best values obtained by ACO PSO ABC and EABCare minus10296 minus699347 minus125669 and minus1525687 respectivelyIt means that EABC can be efficiently applied for solvingmultimodal and multidimensional function optimizationproblems due to its abundant operators such as clonal

8 The Scientific World Journal

Table 1 Benchmark functions used in experiments

Number Function Dimension Property Range Min

1 1198911(119909) =

119863

sum

119894=1

1199092

119894 30 Unimodal [minus100 100] 0

2 1198912(119909) =

119863

sum

119894=1

1198941199092

119894 30 Unimodal [minus10 10] 0

3 1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 5 Unimodal [minus512 512] 0

4 1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 30 Unimodal [minus100 100] 0

5 1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 30 Unimodal [minus128 128] 0

6 1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 30 Unimodal [minus100 100] 0

7 1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) 2 Unimodal [minus100 100] minus1

8 1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 2 Unimodal [minus10 10] 0

9 1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 30 Unimodal [minus10 10] 0

10 11989110 (119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 30 Unimodal [minus30 30] 0

11 11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 30 Multimodal [minus512 512] 0

12 11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 30 Multimodal [minus600 600] 0

13 11989113 (119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 10 Multimodal [minus50 50] 0

14 11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

2 Multimodal [minus100 100] 0

15 11989115 (119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817) 30 Multimodal [minus500 500] minus125695

selection and negative selection which outperforms ACOPSO and ABC In addition on the one hand on basicunimodal functions both the basic ABC and EABC havealmost identical solving performance on the other handon the multimodal functions these two ones display hugedifference

Secondly the EABC algorithm can find optimal or closer-to-optimal solutions on the complex multimodal functions11989111 11989112 11989113 and 11989114 Although the result of multi-dimensionfunction 11989115 is slightly far from the known global optimumthe EABC is superior to the other algorithms all the sameAt the same time for almost all benchmark functionsstandard deviations of the EABC obtained from the statisticalexperimental data are no greater than those of others expectfor 11989110 In addition the differences of EABC between thebest andworst solutions for these 15 benchmark functions are

relatively smaller than those of others in the 30 independentsimulation runs All thesemean that the EABC algorithm hasbetter robustness than others It is also clear that EABC canwork better in almost all cases and gets better performancethan ACO PSO and ABC

Summarizing the statementsmentioned above the EABCcan prevent bees from being trapped into the local minimumaccelerate convergence process search with more efficiencyand improve exploitation abilities for basic ABC

53 Analysis and Discussion In this section the effectsof each modification on the performance of EABC arediscussed First of all corresponding to three modificationswe named the basic ABC with the proposed initializationas IABC the one with the proposed self-adaptive searchingstrategy as SABC and the one with the proposed immune

The Scientific World Journal 9

Table 2 Benchmark functions used in experiments for testing the performances of EABC ACO PSO and ABC

Function number Min ACO PSO ABC EABC

1198911(119909) =

119863

sum

119894=1

1199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198912(119909) =

119863

sum

119894=1

1198941199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 0

Best 0 0 0 0Worst 000289 000321 000543 000422Mean 000136 000116 000300 000196SD 000219 000276 000387 000208

1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 0

Best 0 0 0 0Worst 000246 000305 000110 000400Mean 000180 000156 00066 000210SD 000039 000058 000092 000037

1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) minus1

Best minus1 minus1 minus1 minus1Worst minus1 minus1 minus1 minus1Mean minus1 minus1 minus1 minus1SD 0 0 0 0

1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 0

Best 066667 06667 0 0Worst 066667 06667 0 0Mean 066667 06667 0 0SD 000001 000001 0 0

11989110(119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 0

Best 87513 105433 196788 91578Worst 324215 246711 542333 269874Mean 182039 150886 331227 173558SD 50361 241702 1541443 114774

11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 0

Best 526677 435774 0 0Worst 532331 441131 0 0Mean 529226 439771 0 0SD 45649 117286 0 0

11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 0

Best 001470 0017112 0008531 0Worst 001499 0017989 0017356 0Mean 001479 0017391 0011447 0SD 000296 0020808 0001223 0

10 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 7

(1) Generate the initial population 119909119894 based on chaotic maps and affinity strategy (119894 = 1 2 SN)(2) Evaluate the fitness (fit(119909119894)) of the population(3) Set cycle to 1(4) Repeat(5) For each employed bee

Produce new solution V119894 by using (7)Calculate its fitness value fit(V119894)Apply greedy selection process

(6) Adopt negative selection and network compression to eliminate redundant and similar food sources by using (18)(7) Randomly generate the same number of new individuals(8) Calculate the probability values 119875119894 for the solution (119909119894) by (16)(9) For each onlooker bee

Select a solution 119909119894 depending on 119875119894Produce new solution V119895Calculate its fitness value fit(V119895)Apply greedy selection process

(10) If there is an abandoned solution for the scoutthen replace it with a new solution which will be randomly produced by (4)

(11) Memorize the best solution so far(12) Cycle = cycle + 1(13) Until cycle = MEN

Algorithm 3 Pseudocode of main body of the enhanced ABC algorithm

where the value range of 119860(119883119894119883119896) is between 0 and 1 Thesmaller the value of 119860(119883119894119883119896) is the larger the value of119860(119883119894119883119896) is which means that two different employed beeindividuals have a higher similarity Specially when119860(119883119894119883119896)equals 1 these two ones are identical According to negativeselection and network compression operators redundant andsimilar food sources should be eliminated We predefine athreshold value 120576 so as to realize wipe-off of redundantindividuals It also means when 119860(119883119894119883119896) is equal to or greatthan 120576 we think these two ones are identical and only one canbe retained and other should bewiped off Repeat this processuntil the affinity of any two individuals in a population isless than 120576 In doing so the population size may be reducedNevertheless in order to keep the population size unchangedthe same number of new individuals need generating ran-domlyThrough negative selection and network compressionoperators the exploitation efficiencywill be improved and thecorresponding convergence rate of the algorithm will also beaccelerated

44 Main Steps of the Enhanced Artificial Bee Colony Algo-rithm Based on the above analysis three main improve-ments including novel generation of initial population self-adaptive searching strategy and redundant individual com-pression operator are presented and the detailed pseudo-codeis given in Algorithm 3

5 Experimental Studies on FunctionOptimization Problems

51 Benchmark Functions and Parameter Settings In thissection numerical experiment is used to test the performanceof the enhanced ABC (shorthand for EABC) proposed in this

paper Summarized in Table 1 are the 15 scalable benchmarkfunctions 1198911 sim 11989110 are continuous unimodal functions11989111 sim 11989115 are multimodal functions and the number oftheir local minima increases exponentially with the problemdimension

In order to testify the performance of different intelligentalgorithms we compare the EABC with the standard ACOPSO and ABC In all simulations the population size ofACO PSO ABC and EABC is 50 The maximum numberof function evaluations (FE) is set to 5000 The thresholdvalue of the affinity 120576 is 09 Other related parameter valuesof ACO PSO and ABC are referred in the literature [17]All experiment results reported are obtained based on 30independent runsThe experiment results are the best worstmean and standard deviation of the statistical experimentaldata

52 Simulation Results The performance on the solutionaccuracy of EABC is compared with that of ACO PSO andABC Table 2 shows the optimization of the 15 benchmarkfunctions obtained in the 30 independent runs by eachalgorithm and some interesting results can be found inTable 2

Firstly almost all algorithms have identical performanceon most of unimodal functions 1198911 to 1198914 1198917 1198918 and 11989114However on other functions these four algorithms showdifferent performance especially for multimodal ones suchas 11989111 11989112 11989113 and 11989115 Fox example on function 11989115the best values obtained by ACO PSO ABC and EABCare minus10296 minus699347 minus125669 and minus1525687 respectivelyIt means that EABC can be efficiently applied for solvingmultimodal and multidimensional function optimizationproblems due to its abundant operators such as clonal

8 The Scientific World Journal

Table 1 Benchmark functions used in experiments

Number Function Dimension Property Range Min

1 1198911(119909) =

119863

sum

119894=1

1199092

119894 30 Unimodal [minus100 100] 0

2 1198912(119909) =

119863

sum

119894=1

1198941199092

119894 30 Unimodal [minus10 10] 0

3 1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 5 Unimodal [minus512 512] 0

4 1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 30 Unimodal [minus100 100] 0

5 1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 30 Unimodal [minus128 128] 0

6 1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 30 Unimodal [minus100 100] 0

7 1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) 2 Unimodal [minus100 100] minus1

8 1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 2 Unimodal [minus10 10] 0

9 1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 30 Unimodal [minus10 10] 0

10 11989110 (119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 30 Unimodal [minus30 30] 0

11 11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 30 Multimodal [minus512 512] 0

12 11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 30 Multimodal [minus600 600] 0

13 11989113 (119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 10 Multimodal [minus50 50] 0

14 11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

2 Multimodal [minus100 100] 0

15 11989115 (119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817) 30 Multimodal [minus500 500] minus125695

selection and negative selection which outperforms ACOPSO and ABC In addition on the one hand on basicunimodal functions both the basic ABC and EABC havealmost identical solving performance on the other handon the multimodal functions these two ones display hugedifference

Secondly the EABC algorithm can find optimal or closer-to-optimal solutions on the complex multimodal functions11989111 11989112 11989113 and 11989114 Although the result of multi-dimensionfunction 11989115 is slightly far from the known global optimumthe EABC is superior to the other algorithms all the sameAt the same time for almost all benchmark functionsstandard deviations of the EABC obtained from the statisticalexperimental data are no greater than those of others expectfor 11989110 In addition the differences of EABC between thebest andworst solutions for these 15 benchmark functions are

relatively smaller than those of others in the 30 independentsimulation runs All thesemean that the EABC algorithm hasbetter robustness than others It is also clear that EABC canwork better in almost all cases and gets better performancethan ACO PSO and ABC

Summarizing the statementsmentioned above the EABCcan prevent bees from being trapped into the local minimumaccelerate convergence process search with more efficiencyand improve exploitation abilities for basic ABC

53 Analysis and Discussion In this section the effectsof each modification on the performance of EABC arediscussed First of all corresponding to three modificationswe named the basic ABC with the proposed initializationas IABC the one with the proposed self-adaptive searchingstrategy as SABC and the one with the proposed immune

The Scientific World Journal 9

Table 2 Benchmark functions used in experiments for testing the performances of EABC ACO PSO and ABC

Function number Min ACO PSO ABC EABC

1198911(119909) =

119863

sum

119894=1

1199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198912(119909) =

119863

sum

119894=1

1198941199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 0

Best 0 0 0 0Worst 000289 000321 000543 000422Mean 000136 000116 000300 000196SD 000219 000276 000387 000208

1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 0

Best 0 0 0 0Worst 000246 000305 000110 000400Mean 000180 000156 00066 000210SD 000039 000058 000092 000037

1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) minus1

Best minus1 minus1 minus1 minus1Worst minus1 minus1 minus1 minus1Mean minus1 minus1 minus1 minus1SD 0 0 0 0

1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 0

Best 066667 06667 0 0Worst 066667 06667 0 0Mean 066667 06667 0 0SD 000001 000001 0 0

11989110(119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 0

Best 87513 105433 196788 91578Worst 324215 246711 542333 269874Mean 182039 150886 331227 173558SD 50361 241702 1541443 114774

11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 0

Best 526677 435774 0 0Worst 532331 441131 0 0Mean 529226 439771 0 0SD 45649 117286 0 0

11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 0

Best 001470 0017112 0008531 0Worst 001499 0017989 0017356 0Mean 001479 0017391 0011447 0SD 000296 0020808 0001223 0

10 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom

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ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

Biomedical Imaging

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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

8 The Scientific World Journal

Table 1 Benchmark functions used in experiments

Number Function Dimension Property Range Min

1 1198911(119909) =

119863

sum

119894=1

1199092

119894 30 Unimodal [minus100 100] 0

2 1198912(119909) =

119863

sum

119894=1

1198941199092

119894 30 Unimodal [minus10 10] 0

3 1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 5 Unimodal [minus512 512] 0

4 1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 30 Unimodal [minus100 100] 0

5 1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 30 Unimodal [minus128 128] 0

6 1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 30 Unimodal [minus100 100] 0

7 1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) 2 Unimodal [minus100 100] minus1

8 1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 2 Unimodal [minus10 10] 0

9 1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 30 Unimodal [minus10 10] 0

10 11989110 (119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 30 Unimodal [minus30 30] 0

11 11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 30 Multimodal [minus512 512] 0

12 11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 30 Multimodal [minus600 600] 0

13 11989113 (119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 10 Multimodal [minus50 50] 0

14 11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

2 Multimodal [minus100 100] 0

15 11989115 (119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817) 30 Multimodal [minus500 500] minus125695

selection and negative selection which outperforms ACOPSO and ABC In addition on the one hand on basicunimodal functions both the basic ABC and EABC havealmost identical solving performance on the other handon the multimodal functions these two ones display hugedifference

Secondly the EABC algorithm can find optimal or closer-to-optimal solutions on the complex multimodal functions11989111 11989112 11989113 and 11989114 Although the result of multi-dimensionfunction 11989115 is slightly far from the known global optimumthe EABC is superior to the other algorithms all the sameAt the same time for almost all benchmark functionsstandard deviations of the EABC obtained from the statisticalexperimental data are no greater than those of others expectfor 11989110 In addition the differences of EABC between thebest andworst solutions for these 15 benchmark functions are

relatively smaller than those of others in the 30 independentsimulation runs All thesemean that the EABC algorithm hasbetter robustness than others It is also clear that EABC canwork better in almost all cases and gets better performancethan ACO PSO and ABC

Summarizing the statementsmentioned above the EABCcan prevent bees from being trapped into the local minimumaccelerate convergence process search with more efficiencyand improve exploitation abilities for basic ABC

53 Analysis and Discussion In this section the effectsof each modification on the performance of EABC arediscussed First of all corresponding to three modificationswe named the basic ABC with the proposed initializationas IABC the one with the proposed self-adaptive searchingstrategy as SABC and the one with the proposed immune

The Scientific World Journal 9

Table 2 Benchmark functions used in experiments for testing the performances of EABC ACO PSO and ABC

Function number Min ACO PSO ABC EABC

1198911(119909) =

119863

sum

119894=1

1199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198912(119909) =

119863

sum

119894=1

1198941199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 0

Best 0 0 0 0Worst 000289 000321 000543 000422Mean 000136 000116 000300 000196SD 000219 000276 000387 000208

1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 0

Best 0 0 0 0Worst 000246 000305 000110 000400Mean 000180 000156 00066 000210SD 000039 000058 000092 000037

1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) minus1

Best minus1 minus1 minus1 minus1Worst minus1 minus1 minus1 minus1Mean minus1 minus1 minus1 minus1SD 0 0 0 0

1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 0

Best 066667 06667 0 0Worst 066667 06667 0 0Mean 066667 06667 0 0SD 000001 000001 0 0

11989110(119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 0

Best 87513 105433 196788 91578Worst 324215 246711 542333 269874Mean 182039 150886 331227 173558SD 50361 241702 1541443 114774

11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 0

Best 526677 435774 0 0Worst 532331 441131 0 0Mean 529226 439771 0 0SD 45649 117286 0 0

11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 0

Best 001470 0017112 0008531 0Worst 001499 0017989 0017356 0Mean 001479 0017391 0011447 0SD 000296 0020808 0001223 0

10 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

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

The Scientific World Journal 9

Table 2 Benchmark functions used in experiments for testing the performances of EABC ACO PSO and ABC

Function number Min ACO PSO ABC EABC

1198911(119909) =

119863

sum

119894=1

1199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198912(119909) =

119863

sum

119894=1

1198941199092

1198940

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198913(119909) = 25 +

119863

sum

119894=1

lfloor119909119894rfloor 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198914(119909) =

119863

sum

119894=1

(lfloor119909119894 + 05rfloor)2 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198915(119909) =

119863

sum

119894=1

1198941199094

119894 + random(0 1) 0

Best 0 0 0 0Worst 000289 000321 000543 000422Mean 000136 000116 000300 000196SD 000219 000276 000387 000208

1198916(119909) = max119894 10038171003817100381710038171199091198941003817100381710038171003817 1 le 119894 le 119863 0

Best 0 0 0 0Worst 000246 000305 000110 000400Mean 000180 000156 00066 000210SD 000039 000058 000092 000037

1198917(119909) = minus

119863

prod

119894=1

cos (119909119894) exp(minus119863

sum

119894=1

(119909119894 minus 120587)2) minus1

Best minus1 minus1 minus1 minus1Worst minus1 minus1 minus1 minus1Mean minus1 minus1 minus1 minus1SD 0 0 0 0

1198918(119909) = 026

119863

sum

119894=1

1199092

119894 minus 048

119863

prod

119894=1

119909119894 0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

1198919(119909) = (1199091 minus 1)2+

119863

sum

119894=2

119894(21199092

119894 minus 119909119894minus1)2 0

Best 066667 06667 0 0Worst 066667 06667 0 0Mean 066667 06667 0 0SD 000001 000001 0 0

11989110(119909) =

119863

sum

119894=1

100(119909119894+1 minus 1199092

119894 )2+ (1 minus 119909119894)

2 0

Best 87513 105433 196788 91578Worst 324215 246711 542333 269874Mean 182039 150886 331227 173558SD 50361 241702 1541443 114774

11989111(119909) =

119863

sum

119894=1

(119909119894 minus 10 cos (2120587119909119894) + 10) 0

Best 526677 435774 0 0Worst 532331 441131 0 0Mean 529226 439771 0 0SD 45649 117286 0 0

11989112(119909) =1

4000(

119863

sum

119894=1

(119909119894 minus 100)2) minus (

119863

prod

119894=1

cos(119909119894 minus 100

radic119894

)) + 1 0

Best 001470 0017112 0008531 0Worst 001499 0017989 0017356 0Mean 001479 0017391 0011447 0SD 000296 0020808 0001223 0

10 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

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 The Scientific World Journal

Table 2 Continued

Function number Min ACO PSO ABC EABC

11989113(119909) = 10 times 119863 +

119863

sum

119894=1

(1199092

119894 minus 10 cos (2120587119909119894)) 0

Best 456781 2144755 334788 0Worst 877993 3955741 1091447 0Mean 585411 263991 559331 0SD 131142 1556380 1004216 0

11989114(119909) = 05 +

sin2 (radicsum119863119894=1 1199092119894 ) minus 05

(1 + 0001 (sum119863

119894=1 1199092119894 ))2

0

Best 0 0 0 0Worst 0 0 0 0Mean 0 0 0 0SD 0 0 0 0

11989115(119909) =

119863

sum

119894=1

minus 119909119894 sin(radic10038171003817100381710038171199091198941003817100381710038171003817)

minus125695

Best minus10296 minus699347 minus115669 minus125687Worst minus10237 minus688333 minus114988 minus125143Mean minus10266 minus690912 minus115441 minus125511SD 521849 4579577 1254471 1013217

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

FE

Fitness

ABCIABCSABC

OABCEABC

(a) Function 11989113 with119863 = 10

ABCIABCSABC

OABCEABC

0 1000 2000 3000 4000 5000

minus12000

minus10000

minus8000

minus6000

minus4000

minus2000

0

FE

Fitness

(b) Function 11989115 with119863 = 30

Figure 1 Convergence speed of the different ABCs on the two test functions (11989113 11989115)

operators as OABC We compare the convergence speed ofthese different ABCs through two complex high-dimensionmultimodal functions 11989113 and 11989115 in order to find thecontributions of three modifications in EABC to improvethe performance of the algorithm respectively The corre-sponding simulation results are shown in Figure 1 We cansee from Figure 1 that IABC SABC and OABC outperformthe basic ABC which means that the three modificationmeasures mentioned in Section 4 have positive effect onthe convergence speed of the algorithm In addition SABCand OABC are obviously superior to IABC which impliesthat searching strategy and immune operators play moreimportant roles than that of initialization However it isdifficult to compare the contributions between searching

strategy and immune operators on the two test functions thereasons may be that the characteristic of test functions willalso affect the problem-solving efficiency of the algorithm

6 Conclusion

In this paper we have proposed an enhanced artificial beecolony algorithm called EABC through introducing self-adaptive searching strategy and artificial immune networkoperators Subsequently a suite of unimodal or multimodalbenchmark functions are used to testify the performance ofthe proposed algorithmThe simulation results illustrate thatthe EABC algorithm outperforms ACO PSO and the basicABC

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

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

The Scientific World Journal 11

The future work includes the studies on how to applyEABC to more complex discrete dynamic optimizationproblems including product design optimization problemdynamic project scheduling problem and data clusteringproblem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71071141 and 71171089)the Specialized Research Fund for the Doctoral Program ofHigher Education of China (Grant nos 20130142110051 and20103326120001) and Humanity and Sociology Foundationof Ministry of Education of China (Grant no 11YJC630019)

References

[1] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA

[2] X S Yang Z H Cui R B Xiao A H Gandomi and MKaramanoglu Swarm Intelligence and Bio-Inspired Computa-tion Theory and Applications Elsevier Waltham Mass USA2013

[3] S T Hsieh T Y Sun C L Lin and C C Liu ldquoEffectivelearning rate adjustment of blind source separation based onan improved particle swarm optimizerrdquo IEEE Transactions onEvolutionary Computation vol 12 no 2 pp 242ndash251 2008

[4] Z H Cui and X J Cai ldquoIntegral particle swarm optimizationwith dispersed accelerator informationrdquo Fundamenta Informat-icae vol 95 no 4 pp 427ndash447 2009

[5] Z H Cui X J Cai J C Zeng and Y F Yin ldquoPID-controlledparticle swarm optimizationrdquo Journal of Multiple-Valued Logicand Soft Computing vol 16 no 6 pp 585ndash610 2010

[6] C Priya and P Lakshmi ldquoParticle swarm optimisation appliedto real time control of spherical tank systemrdquo InternationalJournal of Bio-Inspired Computation vol 4 no 4 pp 206ndash2162012

[7] A H Gandomi G J Yun X S Yang and S Talatahari ldquoChaos-enhanced accelerated particle swarm optimizationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 18no 2 pp 327ndash340 2013

[8] K M Salama and A A Freitas ldquoLearning Bayesian networkclassifiers using ant colony optimizationrdquo Swarm Intelligencevol 7 no 2-3 pp 229ndash254 2013

[9] H Ahangarikiasari M R Saraji and M Torabi ldquoInvestigationof code complexity of an innovative algorithm based onACO inweighted graph traversing and compare it to traditional ACOand Bellman-Fordrdquo Journal of Bioinformatics and IntelligentControl vol 2 no 1 pp 73ndash78 2013

[10] P B Cao and R B Xiao ldquoAssembly planning using a novelimmune approachrdquo International Journal of AdvancedManufac-turing Technology vol 31 no 7-8 pp 770ndash782 2007

[11] M Bateni A Baraani andA Ghorbani ldquoAlert correlation usingartificial immune recognition systemrdquo International Journal ofBio-Inspired Computation vol 4 no 3 pp 181ndash195 2012

[12] D Karaboga ldquoAn idea on honeybee swarm for numericaloptimizationrdquo Tech Rep TR06 ErciyesUniversity EngineeringFaculty Computer Engineering Department 2005

[13] T Chen and C Ju ldquoA novel artificial bee colony algorithmfor solving the supply chain network design under disruptionscenariosrdquo International Journal of Computer Applications inTechnology vol 47 no 2-3 pp 289ndash296 2013

[14] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[15] D Karaboga and B Basturk ldquoA powerful and efficient algo-rithm for numerical function optimization artificial bee colony(ABC) algorithmrdquo Journal of Global Optimization vol 39 no 3pp 459ndash471 2007

[16] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

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

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

[19] B Alatas ldquoChaotic bee colony algorithms for global numericaloptimizationrdquo Expert Systems with Applications vol 37 no 8pp 5682ndash5687 2010

[20] W Gao and S Liu ldquoA modified artificial bee colony algorithmrdquoComputers and Operations Research vol 39 no 3 pp 687ndash6972012

[21] 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

[22] A Banharnsakun T Achalakul and B Sirinaovakul ldquoThe best-so-far selection in artificial bee colony algorithmrdquo Applied SoftComputing Journal vol 11 no 2 pp 2888ndash2901 2011

[23] F Kang J Li and Q Xu ldquoStructural inverse analysis byhybrid simplex artificial bee colony algorithmsrdquo Computers andStructures vol 87 no 13-14 pp 861ndash870 2009

[24] A Singh ldquoAn artificial bee colony algorithm for the leaf-constrained minimum spanning tree problemrdquo Applied SoftComputing Journal vol 9 no 2 pp 625ndash631 2009

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

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

[27] S Samanta and S Chakraborty ldquoParametric optimization ofsome non-traditional machining processes using artificial beecolony algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 24 no 6 pp 946ndash957 2011

[28] A Alejandro L G Jorge I R Manuel and M Aide ldquoOpti-mization of the material flow in a manufacturing plant byuse of artificial bee colony algorithmrdquo Expert Systems WithApplications vol 40 no 12 pp 4785ndash4790 2013

12 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

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 The Scientific World Journal

[29] S Sundar A Singh and A Rossi ldquoAn artificial bee colonyalgorithm for the 0-1 multidimensional knapsack problemrdquo inProceedings of the 3rd International Conference on Contempo-rary Computing vol 94 of Communications in Computer andInformation Science pp 141ndash151 2010

[30] S Sundar and A Singh ldquoA hybrid heuristic for the set coveringproblemrdquoOperational Research vol 12 no 3 pp 345ndash365 2012

[31] Y Liu and R Xiao ldquoOptimal synthesis of mechanisms for pathgeneration using refined numerical representation basedmodelandAIS based searchingmethodrdquo Journal ofMechanical Designvol 127 no 4 pp 688ndash691 2005

[32] B Gong J Im and G Mountrakis ldquoAn artificial immunenetwork approach to multi-sensor land useland cover classifi-cationrdquo Remote Sensing of Environment vol 115 no 2 pp 600ndash614 2011

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