Research ArticleA Novel Hybrid Clonal Selection Algorithm withCombinatorial Recombination and Modified HypermutationOperators for Global Optimization

Weiwei Zhang Jingjing Lin Honglei Jing and Qiuwen Zhang

School of Computer and Communication Engineering Zhengzhou University of Light Industry Zhengzhou 450000 China

Correspondence should be addressed to Weiwei Zhang anqikeli163com

Received 12 May 2016 Accepted 31 July 2016

Academic Editor Jens Christian Claussen

Copyright copy 2016 Weiwei Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Artificial immune system is one of the most recently introduced intelligence methods which was inspired by biological immunesystem Most immune system inspired algorithms are based on the clonal selection principle known as clonal selection algorithms(CSAs) When coping with complex optimization problems with the characteristics of multimodality high dimension rotationand composition the traditional CSAs often suffer from the premature convergence and unsatisfied accuracy To address theseconcerning issues a recombination operator inspired by the biological combinatorial recombination is proposed at first Therecombination operator could generate the promising candidate solution to enhance search ability of the CSA by fusing theinformation from random chosen parents Furthermore a modified hypermutation operator is introduced to construct morepromising and efficient candidate solutions A set of 16 common used benchmark functions are adopted to test the effectivenessand efficiency of the recombination and hypermutation operators The comparisons with classic CSA CSA with recombinationoperator (RCSA) and CSA with recombination and modified hypermutation operator (RHCSA) demonstrate that the proposedalgorithm significantly improves the performance of classic CSAMoreover comparison with the state-of-the-art algorithms showsthat the proposed algorithm is quite competitive

1 Introduction

Optimization techniques play a very important role in engi-neering design commercial manufacture financial marketinformation science and related areas An optimizationproblem can be expressed as

Optimize 119891 (119883)

subject to 119883 isin Ω(1)

where 119883 = [1199091 1199092 119909

119863] is a D-dimensional vector of

decision variables in the feasible region Ω The optimizationcould be either a minimization problem or a maximizationproblem Traditional optimization algorithms often fail todeal with multimodal nonconvex nondifferentiable prob-lems since most of them rely on the gradient information[1] In the past few decades inspired by nature peoplehave developed many optimization computation methodsto solve the complicated optimization problems including

genetic algorithm (GA) differential evolution (DE) particleswarm optimization (PSO) artificial immune system (AIS)and some new nature-inspired algorithms [2ndash7]

Among them artificial immune system (AIS) is a newlyemerging computational paradigm inspired by the funda-mentals of immune system It abstracts the structure andfunction of the biological immune system and exploits theapplications to solve computational problems In the area ofoptimization numerical comparisons demonstrated that theperformance of AIS is competitive compared to that of theother nature-inspired algorithms [8] Among them clonalselection algorithm is one of well-known AIS and has beenapplied to solve many practical optimization problems [9ndash11] The clonal selection algorithm is inspired from behaviorof B cells in secreting antibody to bind with the invadingantigen In view of the good performance of clonal selectionalgorithms (CSAs) in computational optimization area [12]in this paper our work concentrates on CSA

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2016 Article ID 6204728 14 pageshttpdxdoiorg10115520166204728

2 Computational Intelligence and Neuroscience

CSAs have attracted a lot of attention since they weredeveloped [13ndash15] However with the increase of the com-plexity of the optimization problems the deficiency of CSA isgradually exposed One of the main shortages is that hyper-mutation is the main operator to modify the construction ofthe candidate solution in the traditional CSAs and generallyboth global search and local search are based on adjustingthe step size of the hypermutation It would be barelyunsatisfactory on balancing the diversity and convergence Insome improved versions new randomly generated solutionsare introduced by receptor editing or other mechanisms toincrease diversity However the mechanism of randomlyintroduced solutions along with the step-size controlledhypermutation leads to a blind optima searching processThereafter premature convergence and diversity loss happenwhen dealing with the high-dimensional multimodal opti-mization problems

In view of the above analysis we are motivated toexplore the undeveloped potential of clonal selection theoryin this paper Inspired by the immune response of B cells acombinatorial recombination operator is introduced to sharethe responsibility with hypermutation In the proposedalgorithm recombination operator is proposed to enhancethe search ability of the CSA Moreover the hypermutationoperator is modified to generate more promising candidatesolutions

The rest of this paper is organized as follows In Section 2the general framework of CSA algorithm is presentedand the research works on improved CSAs in the fieldof numerical optimization are reviewed In Section 3 theRHCSA algorithm is proposed based on the new introducedrecombination and the modified hypermutation operatorsSection 4 presents and discusses the experimental resultsFinally the conclusion is drawn in Section 5

2 CSA Framework

The AIS has become popular from the late 1990s Severalbooks journals and conference papers have been publishedin the past few years de Castro and Timmis [16] depictthe main models in AIS such as clonal selection immunenetworks and negative selection theories More detailedreview of AIS and their applications can be found in [15 17ndash19] Here we will give a brief literature review of the contem-porary research efforts in clonal selection based algorithms inhandling the numerical optimization

21 Clonal Selection Algorithms (CSAs) The main idea ofclonal selection theory lies in the phenomenon where B cellreacts to invaded antigen through modifying the receptorcalled antibody The general one named CLOGNALG [20]is one of the representatives for clonal selection algorithmsThere are mainly three operations involved which arecloning hypermutation and selection

The general framework of clonal selection algorithm foroptimization is presented as followsFramework of CSAStep 1 (initialization) Randomly initialize antibody popula-tion

Step 2 (evaluation) Evaluate the objective values of theantibody population as their fitness

Step 3 (cloning) Generate copies of the antibodies

Step 4 (hypermutation) Mutate all the generated copies

Step 5 (selection) Select the one with highest fitness tosurvive

Step 6 Repeat Steps 2ndash5 until a termination criterion is met

Themain features of the CSA framework are (i) the clonenumber usually proportionally to the affinity of antibodywith respect to the antigens (ii) the mutation rate normallyinversely proportional to the affinity and (iii) the absence ofrecombination operators (such as crossover in GAs) Thosecharacteristics expose deficiencies when facing the high-dimensional nonconvex multimodal and multiobjectiveoptimization problems First it is obvious that the cloningoperator consumes high computational resource Secondhypermutation is insufficient to bear the burden of balancingboth the global search and the local search which will lead tothe premature convergence and unsatisfied accuracy Thirdthe lack of consideration on interaction among individualsin the population may lead to the search missing globalawareness that is overly searching one or some areas of thesearch space while leaving the others unvisited

22 Improved CSAs To overcome the abovementionedshortcomings a bunch of improved algorithms based onCSAare proposed The research works can be briefly classified asthree categories

221 Modification of the Operators or Introduction of NewOperators within the CSA Framework Cutello et al intro-duced a real-coded clonal selection algorithm for globaloptimization involving the cloning operator inversely pro-portional hypermutation and aging operator [21] Three ver-sions of somatic contiguous hypermutation operator are ana-lyzed and proven to have better performance than standardbit mutation to some types of optimization problem [22]Khilwani et al [23] proposed a fast clonal algorithm (FCA)which designs a parallel mutation operator comprising Gaus-sian and Cauchy mutation strategies Lu and Yang [24] intro-duce the Cauchy mutation for the improved CSA (IMCSA)Two versions of immune algorithm namedOPT-IMMALG01and OPT-IMMALG combined with the clonal operator Mhypermutation operator aging operator and (120583+120582) selectionoperator based on binary-code and real-code representationrespectively are discussed The experimental results approvethe effectiveness of the algorithms in handing the high-dimensional global numerical optimization problem [12]Randomized clonal expansion strategy is proposed to solvehigh-dimensional global optimization problem in [25]

222 Combine CSA with Immune Network Theory Immunenetwork theory considers that the immune cells have rela-tions with each other and hereafter the cells molecules and

Computational Intelligence and Neuroscience 3

other related substances construct a network CAS combingwith immune network theory is more suitable for handlingmultimodal function optimization

The opt-AINet algorithm is an earlier version of CSAwith artificial immune networks which is proposed tosolve multimodal continuous function optimization [26]Uniform cloning operator affinity-based Gaussian mutationand similarity-based suppressor are proposed To enhancethe parameter adaptation an improved adaptive ArtificialImmune Network called IA-AIS is proposed where affinity-based cloning operator controlled affinity-based Gaussianmutation and dynamic suppressor are introduced [27] Byimitating the social behaviors of animals social leaningmechanism is introduced to the immune network an algo-rithm called AINet-SL is proposed The population is sepa-rated into elitist swarm (ES) and the common swarm (CS)Nonlinear affinity-based cloning self-learningmutation andsocial learning mutation are employed [28] Based on theaffinity measure the candidate solution space is divided intothe elitist space the common space and the poor space anddifferent hypermutation strategies are applied respectivelyin MPAINet [29] Concentration-Based Artificial ImmuneNetwork where the concentration of antibody is introducedto stimulate and maintain the diversity of the populationis applied to continuous optimization [30] combinatorialoptimization [31] and multiobjective optimization [32]

223 Hybrid CSA CSA is also hybrid with PSO DE andother evolutionary strategiesHill-climbing local search oper-ator is combined with immune algorithm in [33] Differentialimmune clonal selection algorithm (DICSA) is put forwardby Gong et al [34] where the differential mutation anddifferential crossover operators are introduced Orthogonalinitialization and neighborhood orthogonal cloning operatorare proposed in an orthogonal immune algorithm (OIA) [35]CSA with nondominated neighborhood selection strategy(NNIA) [36] was proposed for multiobjective optimizationShang et al [37] proposed an immune clonal algorithm(NICA) for multiobjective optimization problems whichmakes improvements on four aspects in comparison with thetraditional clonal selection computing model

Baldwin effect is introduced into CSA and formulatedthe Baldwinian clonal selection algorithm (BCSA) whichguides the evolution of each antibody by the differentialinformation of other antibodies in the population [38] Gonget al [39] substitute the mutation in CSA with the localsearch technique in Lamarckian Clonal Selection Algorithm(LCSA) and adopt recombination operator and tournamentselection operator for numerical optimization An enhancedCSA hybrid learning CSA (HLCSA) [40] is proposed byintroducing two learning mechanisms Baldwinian learningand orthogonal learning

Althoughmany improvements on immune inspired algo-rithm have been realized the abovementioned shortcomingof artificial immune algorithm such as premature conver-gence high computational cost and unsatisfied accuracy thatcan negatively affect the application of immune algorithmremains a problem The search ability of immune algorithmis limited when dealing with high-dimensional complex

optimization with different characteristics In this paper wepropose an improved CSA by introducing a recombinationoperator and modifying the hypermutation operator to copewith the complex optimization problems with the character-istic high-dimensional nonconvex rotated and composedoptimization problems

3 CSA with Combinatorial Recombinationand Modified Hypermutation

31 Combinatorial Recombination In immunology the pres-ence of both recombination and somatic mutation takes theresponsibility for the diversification of antibody genes Therecombination of the immunoglobulin gene segments is thefirst stepwhen the cells are first exposed to antigen In the pastfew decades recombination is mentioned more as crossoverin GA However as depicted in [41] the recombination ofimmunoglobulin genes involved in the production of anti-bodies differs from the recombination (crossover) of parentalgenes in sexual reproduction In the former nucleotides canbe inserted and deleted randomly from the recombined genesegments while in the latter the genetic mixture is generatedfrom parental chromosomes

The basic unit of antibody molecule has a Y-shapestructure which contains two identical light chains andtwo identical heavy chains as shown in Figure 1(a) Variableregions located at the tips of the Y are primarily responsiblefor antigen recognition Within these variable regions somepolypeptide segments show exceptional variability The anti-body molecules can be synthesized by an individual lyingin the way of encoding the amino acid sequences of thevariable domains into DNA chains as well as the randomselection and recombination of gene segments as shownin Figure 1(b) During the development of B cell the genesegments in the libraries are combined and rearranged at thelevel of the DNA With the recombination of gene segmentsrecombination creates a population of cells that varywidely intheir specificity In this way few immune cells are compatiblewith various antigens After altering the base of antibody themutations fine-tune the lymphocyte receptor to better matchthe antigen [41 42]

In the perspective of optimization recombination func-tions as the coarse-grained exploration while hypermutationworks the same way as fine-grained exploitation Inspired bythis a combinatorial recombination operator is proposed asfollows

To avoid the truncation error and the complexity ofthe coding our algorithm is coded in real number andeach dimension of a solution is viewed as a gene segmentThe whole population forms the gene fragments libraryAccording to the recombination in immunology any orderlyrearrangement of gene segments would generate a new Bcell As presented in Figure 2 the recombination could be (a)between two individuals as crossover or (b) among severalindividuals as the combination of randomly selected genesegments With the help of normalization the combinationof gene segments could be in the specific order such asin Figure 2(a) or can be randomly arranged as shown inFigure 2(b) in the computational respective

4 Computational Intelligence and Neuroscience

Variable region onheavy chain

Antigen bindingsites

Variable regionon light chain

Constant regionon light chain

Constant regionon heavy chain

middot middot middotmiddot middot middot

(a)

Rearranged DNA

Gene library

(b)

Figure 1 Structure of antibody (a) and recombination (b) in the variable region

k

k

Xi1

Xi1

Xim

Xim

Xin

Xin

XiD

XiD

Xj1

Xj1

Xjm

Xjm

Xjn

Xjn

XjD

XjD

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

(a)

Xa1 Xa

m Xam Xa

D

Xb1 Xb

m Xbm Xb

D

Xd1 Xd

m Xdm Xd

D

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot

Xa1 Xa

2 Xb3 Xt

D

Xa2 Xa

3 Xd1 Xt

K

middot middot middot middot middot middot middot middot middot

(b)

Figure 2 The way of rearrangement of gene segments (a) between two individuals (b) among several individuals

The former one is similar to the SBX recombinationin GA [42] where the crossover of parents could swapgene segments on one site or many sites To better simulatethe arrangement of gene segments line recombination DEinspired recombination [43] and intelligent recombination[44] are proposed There would be a lot of ways to do thecombinations with optional joined gene segments With therandomness of the arrangements and combination diversityis introduced However as is known to all too much intro-duced diversity would be harmful for the performance Basedon the experimental experience our work focuses on theway to process recombination between two parents A newcombinatorial recombination operator combining with linerecombination is proposed as follows

Randomly choose two individuals from the populationdenoted by119883

119860and119883

119861 and then randomly choose119898 dimen-

sions119898 isin [1119863] fromeach of themwhere dimensions index

could be recorded as vectors119881119860and119881

119861 respectivelyThe new

individuals are generated by

1198831015840119881119860

119860= 120572119883119881119860

119860+ (1 minus 120572)119883

119881119861

119861

1198831015840119881119861

119861= 120572119883119881119861

119861+ (1 minus 120572)119883

119881119860

119860

(2)

where 120572 is a randomly produced number between 0 and 1 Itshould be noted that the range of each dimension of decisionvariable should be normalized at first The new proposedrecombination operator is represented in Figure 3

As shown in Figure 3 two new individuals are generatedthrough the combinational recombination In the example119898 equals 3 It needs to be known that 119894

1could be different

from 1198951 the same as in 119894

2and 1198952and 1198943and 1198953as long as the

normalization has been doneThen the fitness of the new generated individuals is

evaluated Together with the original individuals two with

Computational Intelligence and Neuroscience 5

XA X1A X

i1A X

i2A X

i3A XD

A VA = i1 i2 i3

XB X1B X

j1B X

j2B X

j3B XD

BVB = j1 j2 j3

Recombination

X998400i1A = 120572X

i1A + (1 minus 120572)X

j1B X

998400j1B = 120572X

j1B + (1 minus 120572)X

i1A

X998400i2A = 120572X

i2A + (1 minus 120572)X

j2B X

998400j2B = 120572X

j2B + (1 minus 120572)X

i2A

X998400i3A = 120572X

i3A + (1 minus 120572)X

j3B X

998400j3B = 120572X

j3B + (1 minus 120572)X

i3A

X998400A

XDB

X1A XD

A

X998400B X1

B

X998400i1A

X998400j1B

X998400i2A

X998400j2B

X998400i3A

X998400j3B

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

Figure 3 Recombination process

the higher fitness will survive and the other two individualsare deleted that is choose two individuals with high fitnessfrom the set 119883

119860 119883119861 1198831015840119860 1198831015840119861 Instead of comparing with the

whole population and reserving the elite a better diversitywill be maintained this way

32 Modified Hypermutation Hypermutation operatorbrings diversity for the population by introducingperturbation for each clone Although there are severalways to implement this operator [22] inversely proportionalstrategy is always the main basis The concept of theoperator proposed in [12] is adopted in this research workwhere each candidate solution is subject to M mutationswithout explicitly using a mutation probability The inverselyproportional law is used to determine the number of themutationsM

120572 = exp (minus120588119891lowast (119883119894))

119872 = lfloor(120572 times 119899) + 1rfloor (3)

where 119891lowast(119883119894) isin [0 1] is the normalized fitness of 119883

119894 120588

is the decay constant which determines the shape of themutation rate and lfloorsdotrfloor returns the lower bound integer Then119872mutation is performed on each candidate solution

1198831015840119895

119894=

119883119895

1199031+ 120582 (119883

119895

1199031minus 119883119895

1199032) if 119895 isin rand 119872(119899)

119883119895

119894otherwise

(4)

119883119894(119895) is the 119895th dimension of the 119894th individual rand

119872(119899) isin 1 119899 is randomly chosen 119872 indexes withoutrepetition and 120582 is a random number in the range of[minus1 1] 1199031 1199032 isin 1 2 119873 are randomly selected numbershereafter the amplitude of the hypermutation is controlledautomatically by the difference of randomly selected indi-viduals in the population The mutation equation (4) could

be considered as the variant of differential evolution andis introduced by [45] to modify the original updating foodsource of artificial bee colony algorithm (ABC) where itis improved to benefit for enhancing the search ability ofABC In our proposed algorithm the equation is combinedwith the hypermutation inversely proportionalM strategy asdepicted above The M strategy controls the direction thatis along how many dimensions while the equation controlsthe distance of the mutated clones with their parents Withcombination of both the amplitude of the hypermutation isautomatically controlled according to the distribution of thepopulation

33 Framework of the Proposed Algorithm Combined withthe proposed combinatorial recombination and hypermuta-tion operator the framework of the proposed algorithm is asfollows

Step 1 (initialization) Randomly initialize a population119860119887 of119873 individuals where 119873 denotes the size of the initial pop-ulation Each initial solution 119883

119894= 119883119894(1) 119883

119894(2) 119883

119894(119863)

is produced randomly within the range of boundaries of thedecision space

119883119895

119894= 119883119895

min + rand (0 1) (119883119895

max minus 119883119895

min) (5)

where 119894 = 1 2 119873 119895 = 1 2 119863 where 119863 is thedimension of the decision variables and 119883119895min and 119883119895max arethe lower and upper bounds for the dimension 119895 respectively

Step 2 (evaluation) Evaluate the objective value of eachsolution as its fitness

Step 3 (recombination) Randomly choose two individu-als from the population and implement the combinatorial

6 Computational Intelligence and Neuroscience

recombination as described in Section 31The recombinationrate is set to119873

119903

Step 4 (clonal selection) (i) Cloning each individual 119883119894

generates119873119888copies 1198831

119894 1198832119894 119883

119873119888

119894 where119873

119888is the clone

number which is a user defined constant(ii) Hypermutation each clone 119883119895

119894 119895 = 1 119873

119888 goes

through the hypermutation as described in Section 32 andgenerates the hypermutated clones1198831015840119895

119894

(iii) Selection select the individual with highest fitnessamong119883

119894and hypermutated clones 11988310158401

119894 11988310158402119894 119883

1015840119873119888

119894

Step 5 If stopping condition is not met go to Step 3Otherwise output the best one of the feasible group

It could be observed that instead of being proportional tothe fitness the clone number is a constant which not onlysaves computational source but also avoids individuals beingconcentrated in some decision area and leaving the othersunvisited when handling the multimodal problems

4 Experimental Studies on FunctionOptimization Problems

In this section experiments are conducted to evaluate theperformance of RHCSA by using 16 commonly used globaloptimization benchmark problems These functions containthe characteristics of being unimodal as 119891

1sim1198912 unrotated

multimodal as 1198913sim1198918 rotated multimodal as 119891

9sim11989114 and

composite functions as 11989115sim11989116 Table 1 gives the expression

of the benchmark function The detailed characteristics ofthese functions can be found in [1 46]

The introduced parameters are analyzed at first and thenthe proposed algorithm is compared with traditional CSAin the benchmark functions Finally comparisons betweenthe proposed algorithm and the state-of-the-art evolutionarycomputing models are represented Some discussions on theperformance analysis of RHCSA are also included It needs tobe noted that the same setup for the experiments is used forall the involved peers algorithms that is the comparison of allthe algorithms is under the same configuration of experimentsetup in this paper

41 Experimental Parameters Settings The experiments wereconducted on 16 benchmark functions The maximum num-ber of function evaluations (mFES) is set as 10000lowastD for10-D and 30-D situations respectively Each test is run 30independent times

There are quite few parameters introduced in the pro-posed algorithm For a fair comparison among CSAs theyare tested using the same setting of the parameters that isthe population size119873 is set to 30 and clone number119873

119888is set

to 4 as in [40] Furthermore there are two new parametersintroduced One is119898 which controls the direction of combi-natorial recombination and the other is recombination rateTables 2 and 3 give the experimental results of varying 119898 inthe 16 benchmark functions with 10-D and 30-D For conve-nience the experiment sets 4 proportions to the dimension asthe value to119898 which are [lceil(110)119863rceil lceil(15)119863rceil lceil(13)119863rceil119863]

The recombination operator could introduce diversityto the population through fusing information between ran-domly chosen parents When 119898 is small the generatedoffspring is much similar to one of the chosen parents andonly little dimension obtains information from both parentsWhen119898 is large the generated offspring tend to be the fusionof chosen parents That is to say m controls the offspringbeing much like one of the chosen parents or the fusion ofboth of them From Tables 2 and 3 we can find that the dif-ference among the varying setup ofm is quite small In amorecareful observation lceil(15)119863rceil and lceil(13)119863rceil are more appro-priate to compare with the other situations for both D = 10and D = 30 and lceil(13)119863rceil is even better It could be observedthat experimental results are not sensitive to the parameter119898 and a slightly larger119898may be beneficial for the algorithmIn view of the observation m = lceil(13)119863rceil is chosen in ourexperiments

There is another parameter named recombination rate119873119903 which controls the rate of recombination process which

will be implemented The higher the 119873119903 the more recom-

bination processes that will be executed In the early stagethe population is uniformly distributed in the search spaceand recombination of randomly chosen individuals will bringdiversity to the population As is known to all diversity isbeneficial for enhancing the search ability of the algorithmbut too much diversity will slow the convergence Tables4 and 5 represent the experimental results of varying therecombination rate

FromTables 4 and 5 it could be observed that RHCSAhasa quite robust performance with the varying recombinationrate By comparison 119873

119903which is equal to 07 achieves the

best performance and is adopted in the paper

42 Comparisons with Classic CSA The experiments areimplemented to check if the proposed operators are beneficialfor the performance of the algorithm Firstly recombinationoperator is added to the original CSA denoted as RCSA andthen the modified hypermutation is introduced as RHCSA

Table 6 shows the statistical results of CLONALG RCSAand RHCSA in optimizing the 16 test problems with D =10 based on 30 independent runs including the mean andstandard deviation From the table we can observe thatfor all these test instances the introduced recombinationoperator could obviously improve the performance of theCSA The reason is that the recombination operator couldbring diversity to enhance the search ability and avoidbeing trapped into the local optima The experiments resultson multimodal function 119891

3sim1198918could present the ability

of introduced recombination operator With the modifiedhypermutation operator the performance of the algorithmobtains a further improvement This is because both recom-bination and hypermutation adopt the difference of chosenindividuals to generate the candidate solution in the proposedalgorithm In the early stage the population is distributed inthe search space and the difference is relatively large whichis beneficial for global search and then with the iterationsgoing on the population converges to the optima and thedifference between individuals becomes smaller and smallerthat is the search process turns to be a local search It can

Computational Intelligence and Neuroscience 7

Table 1 Benchmark functions used in our experimental study

Name Test function D S 119891min

Sphere function 1198911(119909) =

119863

sum119894=1

1199092119894 1030 [minus100 100] 0

Rosenbrockrsquos function1198912 (119909) =119863minus1

sum119894=1

(100 (1199092119894minus 119909119894+1)2

+ (119909119894minus 1)2

) 1030 [minus2048 2048] 0

Ackleyrsquos function 1198913(119909) = minus20 exp(minus02radic 1

119863

119863

sum119894=1

1199092119894) minus exp( 1

119863

119863

sum119894=1

(2120587119909119894)) + 20 + 119890 1030 [minus32768 32768] 0

Griewanksrsquos function 1198914(119909) =

119863

sum119894=1

1199092119894

4000minus119863

prod119894=1

cos119909119894

radic119894+ 1 1030 [minus600 600] 0

Weierstrass function1198915(119909) =

119863

sum119894=1

119896maxsum119896=1

[119886119896 cos (2120587119887119896 (119909119894+ 05))] minus 119863

119896maxsum119896=1

119886119896 cos (2120587119887119896 sdot 05)1030 [minus05 05] 0

119886 = 05 119887 = 3 119896max = 20

Rastriginrsquos function 1198916(119909) =

119863

sum119894=1

1199092119894minus 10 cos (2120587119909

119894) + 10 1030 [minus512 512] 0

NoncontRas

1198917(119909) =

119863

sum119894=1

1199102119894minus 10 cos(2120587119910

119894) + 10

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt 05

round (2119909119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge 05

119894 = 1 2 119863

1030 [minus512 512] 0

Schwefelrsquos function 1198918(119909) = 4189829119863 minus

119863

sum119894=1

119909119894sin (1003816100381610038161003816119909119894

100381610038161003816100381605

) 1030 [minus500 500] 0

RotAckleyrsquos function 1198919(119909) = 119891

3(119910) 119910 = 119872 lowast 119909 1030 [minus32768 32768] 0

RotGriewanksrsquosfunction 119891

10(119909) = 119891

4(119910) 119910 = 119872 lowast 119909 1030 [minus600 600] 0

RotWeierstrassfunction 119891

11(119909) = 119891

5(119910) 119910 = 119872 lowast 119909 1030 [minus05 05] 0

RotRastriginrsquosfunction 119891

12(119909) = 119891

6(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

Rotnoncon Rasfunction 119891

13(119909) = 119891

7(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

RotSchwefelrsquosfunction

11989114(119909) = 4189829119863 minus

119863

sum119894=1

119885119894

119885119894=

119910119894sin (1003816100381610038161003816119910119894

100381610038161003816100381605

)10038161003816100381610038161199101198941003816100381610038161003816 le 500

0001 (10038161003816100381610038161199101198941003816100381610038161003816 minus 500)

2 10038161003816100381610038161199101198941003816100381610038161003816 ge 500

119894 = 1 2 119863

119910 = 119872 lowast (119909 minus 42096) + 42096

1030 [minus500 500] 0

Composition 1 11989115= 1198621198651 1030 [minus5 5] 0

Composition 2 11989116= 1198621198652 1030 [minus5 5] 0

be observed that RCSA surpass CSA and be surpassed byRHCSA at all the tested functions It can be concluded thatthe proposed recombination and hypermutation operatorsare effective improving the ability of the CSA

Table 7 shows the experimental results of the algorithmswhen 119863 = 30 Similar results are represented whichindicates that RHCSA is able to handle the high-dimensionaloptimization problems as well

43 Comparisons with the State-of-the-Art Algorithms Tocompare RHCSA with the state-of-the-art algorithms exper-imental results of seven representative evolutionary algo-rithms are listed in Tables 8 and 9 These algorithms areBaldwinian clonal selection algorithm (BCSA) [38] hybridlearning clonal selection algorithm (HLCSA) [40] orthogo-nal crossover based differential evolution (OXDE) [47] self-adaptive differential evolution (SaDE) [48] global and local

8 Computational Intelligence and Neuroscience

Table 2 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 10

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

13521119890 minus 180 plusmn 00000119890 minus 000 153784119890 minus 166 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 22773119890 minus 078 plusmn 49085119890 minus 078

1198912

86187119890 minus 004 plusmn 320378119890 minus 004 81377119890 minus 004 plusmn 30271119890 minus 004 89516119890 minus 004 plusmn 32039119890 minus 004 28971e minus 004 plusmn 36648e minus 004

1198913

52831119890 minus 015 plusmn 42156119890 minus 015 17763119890 minus 015 plusmn 17763119890 minus 015 88817e minus 016 plusmn 00000e minus 000 20724119890 minus 015 plusmn 20511119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

40175119890 minus 015 plusmn 13376119890 minus 015 30923e minus 015 plusmn 14969e minus 015 39292119890 minus 015 plusmn 11420119890 minus 015 42912119890 minus 015 plusmn 10827119890 minus 015

11989110

93401119890 minus 002 plusmn 15480119890 minus 002 18195119890 minus 002 plusmn 10245119890 minus 002 16937e minus 002 plusmn 16091e minus 002 55463119890 minus 002 plusmn 26635119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

24401119890 + 000 plusmn 10935119890 + 000 20556119890 + 000 plusmn 11601119890 + 000 15954e + 000 plusmn 15954e + 000 16638119890 + 000 plusmn 17026119890 + 000

11989113

81024119890 + 000 plusmn 56678119890 + 000 41385119890 + 000 plusmn 21123119890 + 000 40000e + 000 plusmn 21213e + 000 41773119890 + 000 plusmn 37530119890 + 000

11989114

75454119890 minus 008 plusmn 62091119890 minus 007 68712e minus 008 plusmn 70921e minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 92565119890 minus 008 plusmn 86635119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

90023119890 + 000 plusmn 78901119890 + 000 64971119890 + 000 plusmn 79297119890 + 000 52613e + 000 plusmn 50573e + 000 86977119890 + 000 plusmn 74630119890 + 000

Table 3 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 30

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

31129119890 minus 114 plusmn 44031119890 minus 114 82698119890 minus 110 plusmn 00000119890 minus 000 16994e minus 194 plusmn 00000e minus 000 66159119890 minus 108 plusmn 00000119890 minus 000

1198912

19281119890 minus 007 plusmn 89376119890 minus 007 74820119890 minus 007 plusmn 30968119890 minus 007 26862119890 minus 007 plusmn 60065119890 minus 007 74820e minus 008 plusmn 30968e minus 008

1198913

44415119890 minus 016 plusmn 25108119890 minus 016 89382119890 minus 016 plusmn 81571119890 minus 016 88817e minus 016 plusmn 00000e minus 000 21438119890 minus 015 plusmn 26596119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

32039119890 minus 015 plusmn 18147119890 minus 015 26532e minus 015 plusmn 13037e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 43241119890 minus 015 plusmn 29542119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

51900119890 + 001 plusmn 43580119890 + 001 41625119890 + 001 plusmn 91218119890 + 000 26201e + 001 plusmn 83454e + 000 29186119890 + 001 plusmn 67990119890 + 001

11989113

65345119890 + 001 plusmn 34082119890 + 001 51253119890 + 001 plusmn 23126119890 + 001 45502e + 001 plusmn 10609e + 001 48247119890 + 001 plusmn 10493119890 + 001

11989114

30358119890 minus 003 plusmn 38244119890 minus 003 42769119890 minus 003 plusmn 25513119890 minus 003 24186e minus 003 plusmn 53084e minus 003 34354119890 minus 003 plusmn 41538119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

94154119890 minus 002 plusmn 71725119890 minus 002 88757119890 minus 002 plusmn 14384119890 minus 002 88501e minus 002 plusmn 15560e minus 001 96583119890 minus 002 plusmn 17236119890 minus 001

Computational Intelligence and Neuroscience 9

Table 4 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 10

Function 01 05 07 1

1198911

38119119890 minus 119 plusmn 38510119890 minus 119 40163119890 minus 178 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 30122119890 minus 178 plusmn 00000119890 minus 000

1198912

29186119890 minus 004 plusmn 11966119890 minus 004 31115119890 minus 004 plusmn 105044119890 minus 004 89516119890 minus 004 plusmn 320393119890 minus 004 22544e minus 004 plusmn 15452e minus 004

1198913

19741119890 minus 015 plusmn 19767119890 minus 015 14807119890 minus 015 plusmn 19052119890 minus 015 88817e minus 016 plusmn 00000e minus 000 14135119890 minus 015 plusmn 19876119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

56305119890 minus 015 plusmn 13377119890 minus 015 79474119890 minus 015 plusmn 25291119890 minus 015 39292e minus 015 plusmn 11420e minus 015 41535119890 minus 015 plusmn 12103119890 minus 015

11989110

37179119890 minus 002 plusmn 18338119890 minus 002 31530e minus 002 plusmn 17649e minus 002 38363119890 minus 002 plusmn 377691119890 minus 002 37971119890 minus 002 plusmn 37825119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

23153119890 + 000 plusmn 13464119890 + 000 23031119890 + 000 plusmn 13465119890 + 000 15954119890 + 000 plusmn 15954119890 + 000 13685e + 000 plusmn 14679e + 000

11989113

40817119890 + 000 plusmn 55791119890 + 000 38531e + 000 plusmn 22514e + 000 40000119890 + 000 plusmn 21213119890 + 000 48073119890 + 000 plusmn 34243119890 + 000

11989114

55773e minus 008 plusmn 36818e minus 007 69679119890 minus 008 plusmn 59121119890 minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 70125119890 minus 008 plusmn 10013119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

86735119890 + 000 plusmn 83012119890 + 000 62013119890 + 000 plusmn 42016119890 + 000 52613e + 000 plusmn 50573e + 000 56012119890 + 000 plusmn 71232119890 + 000

Table 5 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 30

Function 01 05 07 1

1198911

16848119890 minus 121 plusmn 00000119890 minus 000 11232119890 minus 150 plusmn 00000119890 minus 000 16994119890 minus 194 plusmn 00000119890 minus 000 84243e minus 198 plusmn 00000e minus 000

1198912

33269119890 minus 007 plusmn 81537119890 minus 007 33493119890 minus 007 plusmn 31302119890 minus 007 26862e minus 007 plusmn 60065e minus 007 73295119890 minus 007 plusmn 30183119890 minus 007

1198913

96645119890 minus 016 plusmn 25121119890 minus 016 89312119890 minus 016 plusmn 28421119890 minus 016 88817e minus 016 plusmn 00000e minus 000 26645119890 minus 015 plusmn 27580119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

25600119890 minus 015 plusmn 22178119890 minus 015 24424e minus 015 plusmn 12352e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 25461119890 minus 015 plusmn 22735119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

46654119890 + 001 plusmn 41276119890 + 001 27521119890 + 001 plusmn 10387119890 + 000 26201e + 001 plusmn 83454e + 000 28012119890 + 001 plusmn 12293119890 + 001

11989113

51231119890 + 001 plusmn 13024119890 + 001 45613119890 + 001 plusmn 13450119890 + 001 45502e + 001 plusmn 10609e + 001 47234119890 + 001 plusmn 11249119890 + 001

11989114

26782119890 minus 003 plusmn 32933119890 minus 003 28334119890 minus 003 plusmn 62013119890 minus 003 24186e minus 003 plusmn 53084e minus 003 30187119890 minus 003 plusmn 87612119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

87011119890 minus 002 plusmn 56745119890 minus 002 87329119890 minus 002 plusmn 92581119890 minus 002 88501119890 minus 002 plusmn 15560119890 minus 001 60112e minus 002 plusmn 25832e minus 002

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Computational Intelligence and Neuroscience

CSAs have attracted a lot of attention since they weredeveloped [13ndash15] However with the increase of the com-plexity of the optimization problems the deficiency of CSA isgradually exposed One of the main shortages is that hyper-mutation is the main operator to modify the construction ofthe candidate solution in the traditional CSAs and generallyboth global search and local search are based on adjustingthe step size of the hypermutation It would be barelyunsatisfactory on balancing the diversity and convergence Insome improved versions new randomly generated solutionsare introduced by receptor editing or other mechanisms toincrease diversity However the mechanism of randomlyintroduced solutions along with the step-size controlledhypermutation leads to a blind optima searching processThereafter premature convergence and diversity loss happenwhen dealing with the high-dimensional multimodal opti-mization problems

In view of the above analysis we are motivated toexplore the undeveloped potential of clonal selection theoryin this paper Inspired by the immune response of B cells acombinatorial recombination operator is introduced to sharethe responsibility with hypermutation In the proposedalgorithm recombination operator is proposed to enhancethe search ability of the CSA Moreover the hypermutationoperator is modified to generate more promising candidatesolutions

The rest of this paper is organized as follows In Section 2the general framework of CSA algorithm is presentedand the research works on improved CSAs in the fieldof numerical optimization are reviewed In Section 3 theRHCSA algorithm is proposed based on the new introducedrecombination and the modified hypermutation operatorsSection 4 presents and discusses the experimental resultsFinally the conclusion is drawn in Section 5

2 CSA Framework

The AIS has become popular from the late 1990s Severalbooks journals and conference papers have been publishedin the past few years de Castro and Timmis [16] depictthe main models in AIS such as clonal selection immunenetworks and negative selection theories More detailedreview of AIS and their applications can be found in [15 17ndash19] Here we will give a brief literature review of the contem-porary research efforts in clonal selection based algorithms inhandling the numerical optimization

21 Clonal Selection Algorithms (CSAs) The main idea ofclonal selection theory lies in the phenomenon where B cellreacts to invaded antigen through modifying the receptorcalled antibody The general one named CLOGNALG [20]is one of the representatives for clonal selection algorithmsThere are mainly three operations involved which arecloning hypermutation and selection

The general framework of clonal selection algorithm foroptimization is presented as followsFramework of CSAStep 1 (initialization) Randomly initialize antibody popula-tion

Step 2 (evaluation) Evaluate the objective values of theantibody population as their fitness

Step 3 (cloning) Generate copies of the antibodies

Step 4 (hypermutation) Mutate all the generated copies

Step 5 (selection) Select the one with highest fitness tosurvive

Step 6 Repeat Steps 2ndash5 until a termination criterion is met

Themain features of the CSA framework are (i) the clonenumber usually proportionally to the affinity of antibodywith respect to the antigens (ii) the mutation rate normallyinversely proportional to the affinity and (iii) the absence ofrecombination operators (such as crossover in GAs) Thosecharacteristics expose deficiencies when facing the high-dimensional nonconvex multimodal and multiobjectiveoptimization problems First it is obvious that the cloningoperator consumes high computational resource Secondhypermutation is insufficient to bear the burden of balancingboth the global search and the local search which will lead tothe premature convergence and unsatisfied accuracy Thirdthe lack of consideration on interaction among individualsin the population may lead to the search missing globalawareness that is overly searching one or some areas of thesearch space while leaving the others unvisited

22 Improved CSAs To overcome the abovementionedshortcomings a bunch of improved algorithms based onCSAare proposed The research works can be briefly classified asthree categories

221 Modification of the Operators or Introduction of NewOperators within the CSA Framework Cutello et al intro-duced a real-coded clonal selection algorithm for globaloptimization involving the cloning operator inversely pro-portional hypermutation and aging operator [21] Three ver-sions of somatic contiguous hypermutation operator are ana-lyzed and proven to have better performance than standardbit mutation to some types of optimization problem [22]Khilwani et al [23] proposed a fast clonal algorithm (FCA)which designs a parallel mutation operator comprising Gaus-sian and Cauchy mutation strategies Lu and Yang [24] intro-duce the Cauchy mutation for the improved CSA (IMCSA)Two versions of immune algorithm namedOPT-IMMALG01and OPT-IMMALG combined with the clonal operator Mhypermutation operator aging operator and (120583+120582) selectionoperator based on binary-code and real-code representationrespectively are discussed The experimental results approvethe effectiveness of the algorithms in handing the high-dimensional global numerical optimization problem [12]Randomized clonal expansion strategy is proposed to solvehigh-dimensional global optimization problem in [25]

222 Combine CSA with Immune Network Theory Immunenetwork theory considers that the immune cells have rela-tions with each other and hereafter the cells molecules and

Computational Intelligence and Neuroscience 3

other related substances construct a network CAS combingwith immune network theory is more suitable for handlingmultimodal function optimization

The opt-AINet algorithm is an earlier version of CSAwith artificial immune networks which is proposed tosolve multimodal continuous function optimization [26]Uniform cloning operator affinity-based Gaussian mutationand similarity-based suppressor are proposed To enhancethe parameter adaptation an improved adaptive ArtificialImmune Network called IA-AIS is proposed where affinity-based cloning operator controlled affinity-based Gaussianmutation and dynamic suppressor are introduced [27] Byimitating the social behaviors of animals social leaningmechanism is introduced to the immune network an algo-rithm called AINet-SL is proposed The population is sepa-rated into elitist swarm (ES) and the common swarm (CS)Nonlinear affinity-based cloning self-learningmutation andsocial learning mutation are employed [28] Based on theaffinity measure the candidate solution space is divided intothe elitist space the common space and the poor space anddifferent hypermutation strategies are applied respectivelyin MPAINet [29] Concentration-Based Artificial ImmuneNetwork where the concentration of antibody is introducedto stimulate and maintain the diversity of the populationis applied to continuous optimization [30] combinatorialoptimization [31] and multiobjective optimization [32]

223 Hybrid CSA CSA is also hybrid with PSO DE andother evolutionary strategiesHill-climbing local search oper-ator is combined with immune algorithm in [33] Differentialimmune clonal selection algorithm (DICSA) is put forwardby Gong et al [34] where the differential mutation anddifferential crossover operators are introduced Orthogonalinitialization and neighborhood orthogonal cloning operatorare proposed in an orthogonal immune algorithm (OIA) [35]CSA with nondominated neighborhood selection strategy(NNIA) [36] was proposed for multiobjective optimizationShang et al [37] proposed an immune clonal algorithm(NICA) for multiobjective optimization problems whichmakes improvements on four aspects in comparison with thetraditional clonal selection computing model

Baldwin effect is introduced into CSA and formulatedthe Baldwinian clonal selection algorithm (BCSA) whichguides the evolution of each antibody by the differentialinformation of other antibodies in the population [38] Gonget al [39] substitute the mutation in CSA with the localsearch technique in Lamarckian Clonal Selection Algorithm(LCSA) and adopt recombination operator and tournamentselection operator for numerical optimization An enhancedCSA hybrid learning CSA (HLCSA) [40] is proposed byintroducing two learning mechanisms Baldwinian learningand orthogonal learning

Althoughmany improvements on immune inspired algo-rithm have been realized the abovementioned shortcomingof artificial immune algorithm such as premature conver-gence high computational cost and unsatisfied accuracy thatcan negatively affect the application of immune algorithmremains a problem The search ability of immune algorithmis limited when dealing with high-dimensional complex

optimization with different characteristics In this paper wepropose an improved CSA by introducing a recombinationoperator and modifying the hypermutation operator to copewith the complex optimization problems with the character-istic high-dimensional nonconvex rotated and composedoptimization problems

3 CSA with Combinatorial Recombinationand Modified Hypermutation

31 Combinatorial Recombination In immunology the pres-ence of both recombination and somatic mutation takes theresponsibility for the diversification of antibody genes Therecombination of the immunoglobulin gene segments is thefirst stepwhen the cells are first exposed to antigen In the pastfew decades recombination is mentioned more as crossoverin GA However as depicted in [41] the recombination ofimmunoglobulin genes involved in the production of anti-bodies differs from the recombination (crossover) of parentalgenes in sexual reproduction In the former nucleotides canbe inserted and deleted randomly from the recombined genesegments while in the latter the genetic mixture is generatedfrom parental chromosomes

The basic unit of antibody molecule has a Y-shapestructure which contains two identical light chains andtwo identical heavy chains as shown in Figure 1(a) Variableregions located at the tips of the Y are primarily responsiblefor antigen recognition Within these variable regions somepolypeptide segments show exceptional variability The anti-body molecules can be synthesized by an individual lyingin the way of encoding the amino acid sequences of thevariable domains into DNA chains as well as the randomselection and recombination of gene segments as shownin Figure 1(b) During the development of B cell the genesegments in the libraries are combined and rearranged at thelevel of the DNA With the recombination of gene segmentsrecombination creates a population of cells that varywidely intheir specificity In this way few immune cells are compatiblewith various antigens After altering the base of antibody themutations fine-tune the lymphocyte receptor to better matchthe antigen [41 42]

In the perspective of optimization recombination func-tions as the coarse-grained exploration while hypermutationworks the same way as fine-grained exploitation Inspired bythis a combinatorial recombination operator is proposed asfollows

To avoid the truncation error and the complexity ofthe coding our algorithm is coded in real number andeach dimension of a solution is viewed as a gene segmentThe whole population forms the gene fragments libraryAccording to the recombination in immunology any orderlyrearrangement of gene segments would generate a new Bcell As presented in Figure 2 the recombination could be (a)between two individuals as crossover or (b) among severalindividuals as the combination of randomly selected genesegments With the help of normalization the combinationof gene segments could be in the specific order such asin Figure 2(a) or can be randomly arranged as shown inFigure 2(b) in the computational respective

4 Computational Intelligence and Neuroscience

Variable region onheavy chain

Antigen bindingsites

Variable regionon light chain

Constant regionon light chain

Constant regionon heavy chain

middot middot middotmiddot middot middot

(a)

Rearranged DNA

Gene library

(b)

Figure 1 Structure of antibody (a) and recombination (b) in the variable region

k

k

Xi1

Xi1

Xim

Xim

Xin

Xin

XiD

XiD

Xj1

Xj1

Xjm

Xjm

Xjn

Xjn

XjD

XjD

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

(a)

Xa1 Xa

m Xam Xa

D

Xb1 Xb

m Xbm Xb

D

Xd1 Xd

m Xdm Xd

D

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot

Xa1 Xa

2 Xb3 Xt

D

Xa2 Xa

3 Xd1 Xt

K

middot middot middot middot middot middot middot middot middot

(b)

Figure 2 The way of rearrangement of gene segments (a) between two individuals (b) among several individuals

The former one is similar to the SBX recombinationin GA [42] where the crossover of parents could swapgene segments on one site or many sites To better simulatethe arrangement of gene segments line recombination DEinspired recombination [43] and intelligent recombination[44] are proposed There would be a lot of ways to do thecombinations with optional joined gene segments With therandomness of the arrangements and combination diversityis introduced However as is known to all too much intro-duced diversity would be harmful for the performance Basedon the experimental experience our work focuses on theway to process recombination between two parents A newcombinatorial recombination operator combining with linerecombination is proposed as follows

Randomly choose two individuals from the populationdenoted by119883

119860and119883

119861 and then randomly choose119898 dimen-

sions119898 isin [1119863] fromeach of themwhere dimensions index

could be recorded as vectors119881119860and119881

119861 respectivelyThe new

individuals are generated by

1198831015840119881119860

119860= 120572119883119881119860

119860+ (1 minus 120572)119883

119881119861

119861

1198831015840119881119861

119861= 120572119883119881119861

119861+ (1 minus 120572)119883

119881119860

119860

(2)

where 120572 is a randomly produced number between 0 and 1 Itshould be noted that the range of each dimension of decisionvariable should be normalized at first The new proposedrecombination operator is represented in Figure 3

As shown in Figure 3 two new individuals are generatedthrough the combinational recombination In the example119898 equals 3 It needs to be known that 119894

1could be different

from 1198951 the same as in 119894

2and 1198952and 1198943and 1198953as long as the

normalization has been doneThen the fitness of the new generated individuals is

evaluated Together with the original individuals two with

Computational Intelligence and Neuroscience 5

XA X1A X

i1A X

i2A X

i3A XD

A VA = i1 i2 i3

XB X1B X

j1B X

j2B X

j3B XD

BVB = j1 j2 j3

Recombination

X998400i1A = 120572X

i1A + (1 minus 120572)X

j1B X

998400j1B = 120572X

j1B + (1 minus 120572)X

i1A

X998400i2A = 120572X

i2A + (1 minus 120572)X

j2B X

998400j2B = 120572X

j2B + (1 minus 120572)X

i2A

X998400i3A = 120572X

i3A + (1 minus 120572)X

j3B X

998400j3B = 120572X

j3B + (1 minus 120572)X

i3A

X998400A

XDB

X1A XD

A

X998400B X1

B

X998400i1A

X998400j1B

X998400i2A

X998400j2B

X998400i3A

X998400j3B

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

Figure 3 Recombination process

the higher fitness will survive and the other two individualsare deleted that is choose two individuals with high fitnessfrom the set 119883

119860 119883119861 1198831015840119860 1198831015840119861 Instead of comparing with the

whole population and reserving the elite a better diversitywill be maintained this way

32 Modified Hypermutation Hypermutation operatorbrings diversity for the population by introducingperturbation for each clone Although there are severalways to implement this operator [22] inversely proportionalstrategy is always the main basis The concept of theoperator proposed in [12] is adopted in this research workwhere each candidate solution is subject to M mutationswithout explicitly using a mutation probability The inverselyproportional law is used to determine the number of themutationsM

120572 = exp (minus120588119891lowast (119883119894))

119872 = lfloor(120572 times 119899) + 1rfloor (3)

where 119891lowast(119883119894) isin [0 1] is the normalized fitness of 119883

119894 120588

is the decay constant which determines the shape of themutation rate and lfloorsdotrfloor returns the lower bound integer Then119872mutation is performed on each candidate solution

1198831015840119895

119894=

119883119895

1199031+ 120582 (119883

119895

1199031minus 119883119895

1199032) if 119895 isin rand 119872(119899)

119883119895

119894otherwise

(4)

119883119894(119895) is the 119895th dimension of the 119894th individual rand

119872(119899) isin 1 119899 is randomly chosen 119872 indexes withoutrepetition and 120582 is a random number in the range of[minus1 1] 1199031 1199032 isin 1 2 119873 are randomly selected numbershereafter the amplitude of the hypermutation is controlledautomatically by the difference of randomly selected indi-viduals in the population The mutation equation (4) could

be considered as the variant of differential evolution andis introduced by [45] to modify the original updating foodsource of artificial bee colony algorithm (ABC) where itis improved to benefit for enhancing the search ability ofABC In our proposed algorithm the equation is combinedwith the hypermutation inversely proportionalM strategy asdepicted above The M strategy controls the direction thatis along how many dimensions while the equation controlsthe distance of the mutated clones with their parents Withcombination of both the amplitude of the hypermutation isautomatically controlled according to the distribution of thepopulation

33 Framework of the Proposed Algorithm Combined withthe proposed combinatorial recombination and hypermuta-tion operator the framework of the proposed algorithm is asfollows

Step 1 (initialization) Randomly initialize a population119860119887 of119873 individuals where 119873 denotes the size of the initial pop-ulation Each initial solution 119883

119894= 119883119894(1) 119883

119894(2) 119883

119894(119863)

is produced randomly within the range of boundaries of thedecision space

119883119895

119894= 119883119895

min + rand (0 1) (119883119895

max minus 119883119895

min) (5)

where 119894 = 1 2 119873 119895 = 1 2 119863 where 119863 is thedimension of the decision variables and 119883119895min and 119883119895max arethe lower and upper bounds for the dimension 119895 respectively

Step 2 (evaluation) Evaluate the objective value of eachsolution as its fitness

Step 3 (recombination) Randomly choose two individu-als from the population and implement the combinatorial

6 Computational Intelligence and Neuroscience

recombination as described in Section 31The recombinationrate is set to119873

119903

Step 4 (clonal selection) (i) Cloning each individual 119883119894

generates119873119888copies 1198831

119894 1198832119894 119883

119873119888

119894 where119873

119888is the clone

number which is a user defined constant(ii) Hypermutation each clone 119883119895

119894 119895 = 1 119873

119888 goes

through the hypermutation as described in Section 32 andgenerates the hypermutated clones1198831015840119895

119894

(iii) Selection select the individual with highest fitnessamong119883

119894and hypermutated clones 11988310158401

119894 11988310158402119894 119883

1015840119873119888

119894

Step 5 If stopping condition is not met go to Step 3Otherwise output the best one of the feasible group

It could be observed that instead of being proportional tothe fitness the clone number is a constant which not onlysaves computational source but also avoids individuals beingconcentrated in some decision area and leaving the othersunvisited when handling the multimodal problems

4 Experimental Studies on FunctionOptimization Problems

In this section experiments are conducted to evaluate theperformance of RHCSA by using 16 commonly used globaloptimization benchmark problems These functions containthe characteristics of being unimodal as 119891

1sim1198912 unrotated

multimodal as 1198913sim1198918 rotated multimodal as 119891

9sim11989114 and

composite functions as 11989115sim11989116 Table 1 gives the expression

of the benchmark function The detailed characteristics ofthese functions can be found in [1 46]

The introduced parameters are analyzed at first and thenthe proposed algorithm is compared with traditional CSAin the benchmark functions Finally comparisons betweenthe proposed algorithm and the state-of-the-art evolutionarycomputing models are represented Some discussions on theperformance analysis of RHCSA are also included It needs tobe noted that the same setup for the experiments is used forall the involved peers algorithms that is the comparison of allthe algorithms is under the same configuration of experimentsetup in this paper

41 Experimental Parameters Settings The experiments wereconducted on 16 benchmark functions The maximum num-ber of function evaluations (mFES) is set as 10000lowastD for10-D and 30-D situations respectively Each test is run 30independent times

There are quite few parameters introduced in the pro-posed algorithm For a fair comparison among CSAs theyare tested using the same setting of the parameters that isthe population size119873 is set to 30 and clone number119873

119888is set

to 4 as in [40] Furthermore there are two new parametersintroduced One is119898 which controls the direction of combi-natorial recombination and the other is recombination rateTables 2 and 3 give the experimental results of varying 119898 inthe 16 benchmark functions with 10-D and 30-D For conve-nience the experiment sets 4 proportions to the dimension asthe value to119898 which are [lceil(110)119863rceil lceil(15)119863rceil lceil(13)119863rceil119863]

The recombination operator could introduce diversityto the population through fusing information between ran-domly chosen parents When 119898 is small the generatedoffspring is much similar to one of the chosen parents andonly little dimension obtains information from both parentsWhen119898 is large the generated offspring tend to be the fusionof chosen parents That is to say m controls the offspringbeing much like one of the chosen parents or the fusion ofboth of them From Tables 2 and 3 we can find that the dif-ference among the varying setup ofm is quite small In amorecareful observation lceil(15)119863rceil and lceil(13)119863rceil are more appro-priate to compare with the other situations for both D = 10and D = 30 and lceil(13)119863rceil is even better It could be observedthat experimental results are not sensitive to the parameter119898 and a slightly larger119898may be beneficial for the algorithmIn view of the observation m = lceil(13)119863rceil is chosen in ourexperiments

There is another parameter named recombination rate119873119903 which controls the rate of recombination process which

will be implemented The higher the 119873119903 the more recom-

bination processes that will be executed In the early stagethe population is uniformly distributed in the search spaceand recombination of randomly chosen individuals will bringdiversity to the population As is known to all diversity isbeneficial for enhancing the search ability of the algorithmbut too much diversity will slow the convergence Tables4 and 5 represent the experimental results of varying therecombination rate

FromTables 4 and 5 it could be observed that RHCSAhasa quite robust performance with the varying recombinationrate By comparison 119873

119903which is equal to 07 achieves the

best performance and is adopted in the paper

42 Comparisons with Classic CSA The experiments areimplemented to check if the proposed operators are beneficialfor the performance of the algorithm Firstly recombinationoperator is added to the original CSA denoted as RCSA andthen the modified hypermutation is introduced as RHCSA

Table 6 shows the statistical results of CLONALG RCSAand RHCSA in optimizing the 16 test problems with D =10 based on 30 independent runs including the mean andstandard deviation From the table we can observe thatfor all these test instances the introduced recombinationoperator could obviously improve the performance of theCSA The reason is that the recombination operator couldbring diversity to enhance the search ability and avoidbeing trapped into the local optima The experiments resultson multimodal function 119891

3sim1198918could present the ability

of introduced recombination operator With the modifiedhypermutation operator the performance of the algorithmobtains a further improvement This is because both recom-bination and hypermutation adopt the difference of chosenindividuals to generate the candidate solution in the proposedalgorithm In the early stage the population is distributed inthe search space and the difference is relatively large whichis beneficial for global search and then with the iterationsgoing on the population converges to the optima and thedifference between individuals becomes smaller and smallerthat is the search process turns to be a local search It can

Computational Intelligence and Neuroscience 7

Table 1 Benchmark functions used in our experimental study

Name Test function D S 119891min

Sphere function 1198911(119909) =

119863

sum119894=1

1199092119894 1030 [minus100 100] 0

Rosenbrockrsquos function1198912 (119909) =119863minus1

sum119894=1

(100 (1199092119894minus 119909119894+1)2

+ (119909119894minus 1)2

) 1030 [minus2048 2048] 0

Ackleyrsquos function 1198913(119909) = minus20 exp(minus02radic 1

119863

119863

sum119894=1

1199092119894) minus exp( 1

119863

119863

sum119894=1

(2120587119909119894)) + 20 + 119890 1030 [minus32768 32768] 0

Griewanksrsquos function 1198914(119909) =

119863

sum119894=1

1199092119894

4000minus119863

prod119894=1

cos119909119894

radic119894+ 1 1030 [minus600 600] 0

Weierstrass function1198915(119909) =

119863

sum119894=1

119896maxsum119896=1

[119886119896 cos (2120587119887119896 (119909119894+ 05))] minus 119863

119896maxsum119896=1

119886119896 cos (2120587119887119896 sdot 05)1030 [minus05 05] 0

119886 = 05 119887 = 3 119896max = 20

Rastriginrsquos function 1198916(119909) =

119863

sum119894=1

1199092119894minus 10 cos (2120587119909

119894) + 10 1030 [minus512 512] 0

NoncontRas

1198917(119909) =

119863

sum119894=1

1199102119894minus 10 cos(2120587119910

119894) + 10

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt 05

round (2119909119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge 05

119894 = 1 2 119863

1030 [minus512 512] 0

Schwefelrsquos function 1198918(119909) = 4189829119863 minus

119863

sum119894=1

119909119894sin (1003816100381610038161003816119909119894

100381610038161003816100381605

) 1030 [minus500 500] 0

RotAckleyrsquos function 1198919(119909) = 119891

3(119910) 119910 = 119872 lowast 119909 1030 [minus32768 32768] 0

RotGriewanksrsquosfunction 119891

10(119909) = 119891

4(119910) 119910 = 119872 lowast 119909 1030 [minus600 600] 0

RotWeierstrassfunction 119891

11(119909) = 119891

5(119910) 119910 = 119872 lowast 119909 1030 [minus05 05] 0

RotRastriginrsquosfunction 119891

12(119909) = 119891

6(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

Rotnoncon Rasfunction 119891

13(119909) = 119891

7(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

RotSchwefelrsquosfunction

11989114(119909) = 4189829119863 minus

119863

sum119894=1

119885119894

119885119894=

119910119894sin (1003816100381610038161003816119910119894

100381610038161003816100381605

)10038161003816100381610038161199101198941003816100381610038161003816 le 500

0001 (10038161003816100381610038161199101198941003816100381610038161003816 minus 500)

2 10038161003816100381610038161199101198941003816100381610038161003816 ge 500

119894 = 1 2 119863

119910 = 119872 lowast (119909 minus 42096) + 42096

1030 [minus500 500] 0

Composition 1 11989115= 1198621198651 1030 [minus5 5] 0

Composition 2 11989116= 1198621198652 1030 [minus5 5] 0

be observed that RCSA surpass CSA and be surpassed byRHCSA at all the tested functions It can be concluded thatthe proposed recombination and hypermutation operatorsare effective improving the ability of the CSA

Table 7 shows the experimental results of the algorithmswhen 119863 = 30 Similar results are represented whichindicates that RHCSA is able to handle the high-dimensionaloptimization problems as well

43 Comparisons with the State-of-the-Art Algorithms Tocompare RHCSA with the state-of-the-art algorithms exper-imental results of seven representative evolutionary algo-rithms are listed in Tables 8 and 9 These algorithms areBaldwinian clonal selection algorithm (BCSA) [38] hybridlearning clonal selection algorithm (HLCSA) [40] orthogo-nal crossover based differential evolution (OXDE) [47] self-adaptive differential evolution (SaDE) [48] global and local

8 Computational Intelligence and Neuroscience

Table 2 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 10

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

13521119890 minus 180 plusmn 00000119890 minus 000 153784119890 minus 166 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 22773119890 minus 078 plusmn 49085119890 minus 078

1198912

86187119890 minus 004 plusmn 320378119890 minus 004 81377119890 minus 004 plusmn 30271119890 minus 004 89516119890 minus 004 plusmn 32039119890 minus 004 28971e minus 004 plusmn 36648e minus 004

1198913

52831119890 minus 015 plusmn 42156119890 minus 015 17763119890 minus 015 plusmn 17763119890 minus 015 88817e minus 016 plusmn 00000e minus 000 20724119890 minus 015 plusmn 20511119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

40175119890 minus 015 plusmn 13376119890 minus 015 30923e minus 015 plusmn 14969e minus 015 39292119890 minus 015 plusmn 11420119890 minus 015 42912119890 minus 015 plusmn 10827119890 minus 015

11989110

93401119890 minus 002 plusmn 15480119890 minus 002 18195119890 minus 002 plusmn 10245119890 minus 002 16937e minus 002 plusmn 16091e minus 002 55463119890 minus 002 plusmn 26635119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

24401119890 + 000 plusmn 10935119890 + 000 20556119890 + 000 plusmn 11601119890 + 000 15954e + 000 plusmn 15954e + 000 16638119890 + 000 plusmn 17026119890 + 000

11989113

81024119890 + 000 plusmn 56678119890 + 000 41385119890 + 000 plusmn 21123119890 + 000 40000e + 000 plusmn 21213e + 000 41773119890 + 000 plusmn 37530119890 + 000

11989114

75454119890 minus 008 plusmn 62091119890 minus 007 68712e minus 008 plusmn 70921e minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 92565119890 minus 008 plusmn 86635119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

90023119890 + 000 plusmn 78901119890 + 000 64971119890 + 000 plusmn 79297119890 + 000 52613e + 000 plusmn 50573e + 000 86977119890 + 000 plusmn 74630119890 + 000

Table 3 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 30

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

31129119890 minus 114 plusmn 44031119890 minus 114 82698119890 minus 110 plusmn 00000119890 minus 000 16994e minus 194 plusmn 00000e minus 000 66159119890 minus 108 plusmn 00000119890 minus 000

1198912

19281119890 minus 007 plusmn 89376119890 minus 007 74820119890 minus 007 plusmn 30968119890 minus 007 26862119890 minus 007 plusmn 60065119890 minus 007 74820e minus 008 plusmn 30968e minus 008

1198913

44415119890 minus 016 plusmn 25108119890 minus 016 89382119890 minus 016 plusmn 81571119890 minus 016 88817e minus 016 plusmn 00000e minus 000 21438119890 minus 015 plusmn 26596119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

32039119890 minus 015 plusmn 18147119890 minus 015 26532e minus 015 plusmn 13037e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 43241119890 minus 015 plusmn 29542119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

51900119890 + 001 plusmn 43580119890 + 001 41625119890 + 001 plusmn 91218119890 + 000 26201e + 001 plusmn 83454e + 000 29186119890 + 001 plusmn 67990119890 + 001

11989113

65345119890 + 001 plusmn 34082119890 + 001 51253119890 + 001 plusmn 23126119890 + 001 45502e + 001 plusmn 10609e + 001 48247119890 + 001 plusmn 10493119890 + 001

11989114

30358119890 minus 003 plusmn 38244119890 minus 003 42769119890 minus 003 plusmn 25513119890 minus 003 24186e minus 003 plusmn 53084e minus 003 34354119890 minus 003 plusmn 41538119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

94154119890 minus 002 plusmn 71725119890 minus 002 88757119890 minus 002 plusmn 14384119890 minus 002 88501e minus 002 plusmn 15560e minus 001 96583119890 minus 002 plusmn 17236119890 minus 001

Computational Intelligence and Neuroscience 9

Table 4 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 10

Function 01 05 07 1

1198911

38119119890 minus 119 plusmn 38510119890 minus 119 40163119890 minus 178 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 30122119890 minus 178 plusmn 00000119890 minus 000

1198912

29186119890 minus 004 plusmn 11966119890 minus 004 31115119890 minus 004 plusmn 105044119890 minus 004 89516119890 minus 004 plusmn 320393119890 minus 004 22544e minus 004 plusmn 15452e minus 004

1198913

19741119890 minus 015 plusmn 19767119890 minus 015 14807119890 minus 015 plusmn 19052119890 minus 015 88817e minus 016 plusmn 00000e minus 000 14135119890 minus 015 plusmn 19876119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

56305119890 minus 015 plusmn 13377119890 minus 015 79474119890 minus 015 plusmn 25291119890 minus 015 39292e minus 015 plusmn 11420e minus 015 41535119890 minus 015 plusmn 12103119890 minus 015

11989110

37179119890 minus 002 plusmn 18338119890 minus 002 31530e minus 002 plusmn 17649e minus 002 38363119890 minus 002 plusmn 377691119890 minus 002 37971119890 minus 002 plusmn 37825119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

23153119890 + 000 plusmn 13464119890 + 000 23031119890 + 000 plusmn 13465119890 + 000 15954119890 + 000 plusmn 15954119890 + 000 13685e + 000 plusmn 14679e + 000

11989113

40817119890 + 000 plusmn 55791119890 + 000 38531e + 000 plusmn 22514e + 000 40000119890 + 000 plusmn 21213119890 + 000 48073119890 + 000 plusmn 34243119890 + 000

11989114

55773e minus 008 plusmn 36818e minus 007 69679119890 minus 008 plusmn 59121119890 minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 70125119890 minus 008 plusmn 10013119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

86735119890 + 000 plusmn 83012119890 + 000 62013119890 + 000 plusmn 42016119890 + 000 52613e + 000 plusmn 50573e + 000 56012119890 + 000 plusmn 71232119890 + 000

Table 5 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 30

Function 01 05 07 1

1198911

16848119890 minus 121 plusmn 00000119890 minus 000 11232119890 minus 150 plusmn 00000119890 minus 000 16994119890 minus 194 plusmn 00000119890 minus 000 84243e minus 198 plusmn 00000e minus 000

1198912

33269119890 minus 007 plusmn 81537119890 minus 007 33493119890 minus 007 plusmn 31302119890 minus 007 26862e minus 007 plusmn 60065e minus 007 73295119890 minus 007 plusmn 30183119890 minus 007

1198913

96645119890 minus 016 plusmn 25121119890 minus 016 89312119890 minus 016 plusmn 28421119890 minus 016 88817e minus 016 plusmn 00000e minus 000 26645119890 minus 015 plusmn 27580119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

25600119890 minus 015 plusmn 22178119890 minus 015 24424e minus 015 plusmn 12352e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 25461119890 minus 015 plusmn 22735119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

46654119890 + 001 plusmn 41276119890 + 001 27521119890 + 001 plusmn 10387119890 + 000 26201e + 001 plusmn 83454e + 000 28012119890 + 001 plusmn 12293119890 + 001

11989113

51231119890 + 001 plusmn 13024119890 + 001 45613119890 + 001 plusmn 13450119890 + 001 45502e + 001 plusmn 10609e + 001 47234119890 + 001 plusmn 11249119890 + 001

11989114

26782119890 minus 003 plusmn 32933119890 minus 003 28334119890 minus 003 plusmn 62013119890 minus 003 24186e minus 003 plusmn 53084e minus 003 30187119890 minus 003 plusmn 87612119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

87011119890 minus 002 plusmn 56745119890 minus 002 87329119890 minus 002 plusmn 92581119890 minus 002 88501119890 minus 002 plusmn 15560119890 minus 001 60112e minus 002 plusmn 25832e minus 002

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

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Applied Computational Intelligence and Soft Computing

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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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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 3

other related substances construct a network CAS combingwith immune network theory is more suitable for handlingmultimodal function optimization

The opt-AINet algorithm is an earlier version of CSAwith artificial immune networks which is proposed tosolve multimodal continuous function optimization [26]Uniform cloning operator affinity-based Gaussian mutationand similarity-based suppressor are proposed To enhancethe parameter adaptation an improved adaptive ArtificialImmune Network called IA-AIS is proposed where affinity-based cloning operator controlled affinity-based Gaussianmutation and dynamic suppressor are introduced [27] Byimitating the social behaviors of animals social leaningmechanism is introduced to the immune network an algo-rithm called AINet-SL is proposed The population is sepa-rated into elitist swarm (ES) and the common swarm (CS)Nonlinear affinity-based cloning self-learningmutation andsocial learning mutation are employed [28] Based on theaffinity measure the candidate solution space is divided intothe elitist space the common space and the poor space anddifferent hypermutation strategies are applied respectivelyin MPAINet [29] Concentration-Based Artificial ImmuneNetwork where the concentration of antibody is introducedto stimulate and maintain the diversity of the populationis applied to continuous optimization [30] combinatorialoptimization [31] and multiobjective optimization [32]

223 Hybrid CSA CSA is also hybrid with PSO DE andother evolutionary strategiesHill-climbing local search oper-ator is combined with immune algorithm in [33] Differentialimmune clonal selection algorithm (DICSA) is put forwardby Gong et al [34] where the differential mutation anddifferential crossover operators are introduced Orthogonalinitialization and neighborhood orthogonal cloning operatorare proposed in an orthogonal immune algorithm (OIA) [35]CSA with nondominated neighborhood selection strategy(NNIA) [36] was proposed for multiobjective optimizationShang et al [37] proposed an immune clonal algorithm(NICA) for multiobjective optimization problems whichmakes improvements on four aspects in comparison with thetraditional clonal selection computing model

Baldwin effect is introduced into CSA and formulatedthe Baldwinian clonal selection algorithm (BCSA) whichguides the evolution of each antibody by the differentialinformation of other antibodies in the population [38] Gonget al [39] substitute the mutation in CSA with the localsearch technique in Lamarckian Clonal Selection Algorithm(LCSA) and adopt recombination operator and tournamentselection operator for numerical optimization An enhancedCSA hybrid learning CSA (HLCSA) [40] is proposed byintroducing two learning mechanisms Baldwinian learningand orthogonal learning

Althoughmany improvements on immune inspired algo-rithm have been realized the abovementioned shortcomingof artificial immune algorithm such as premature conver-gence high computational cost and unsatisfied accuracy thatcan negatively affect the application of immune algorithmremains a problem The search ability of immune algorithmis limited when dealing with high-dimensional complex

optimization with different characteristics In this paper wepropose an improved CSA by introducing a recombinationoperator and modifying the hypermutation operator to copewith the complex optimization problems with the character-istic high-dimensional nonconvex rotated and composedoptimization problems

3 CSA with Combinatorial Recombinationand Modified Hypermutation

31 Combinatorial Recombination In immunology the pres-ence of both recombination and somatic mutation takes theresponsibility for the diversification of antibody genes Therecombination of the immunoglobulin gene segments is thefirst stepwhen the cells are first exposed to antigen In the pastfew decades recombination is mentioned more as crossoverin GA However as depicted in [41] the recombination ofimmunoglobulin genes involved in the production of anti-bodies differs from the recombination (crossover) of parentalgenes in sexual reproduction In the former nucleotides canbe inserted and deleted randomly from the recombined genesegments while in the latter the genetic mixture is generatedfrom parental chromosomes

The basic unit of antibody molecule has a Y-shapestructure which contains two identical light chains andtwo identical heavy chains as shown in Figure 1(a) Variableregions located at the tips of the Y are primarily responsiblefor antigen recognition Within these variable regions somepolypeptide segments show exceptional variability The anti-body molecules can be synthesized by an individual lyingin the way of encoding the amino acid sequences of thevariable domains into DNA chains as well as the randomselection and recombination of gene segments as shownin Figure 1(b) During the development of B cell the genesegments in the libraries are combined and rearranged at thelevel of the DNA With the recombination of gene segmentsrecombination creates a population of cells that varywidely intheir specificity In this way few immune cells are compatiblewith various antigens After altering the base of antibody themutations fine-tune the lymphocyte receptor to better matchthe antigen [41 42]

In the perspective of optimization recombination func-tions as the coarse-grained exploration while hypermutationworks the same way as fine-grained exploitation Inspired bythis a combinatorial recombination operator is proposed asfollows

To avoid the truncation error and the complexity ofthe coding our algorithm is coded in real number andeach dimension of a solution is viewed as a gene segmentThe whole population forms the gene fragments libraryAccording to the recombination in immunology any orderlyrearrangement of gene segments would generate a new Bcell As presented in Figure 2 the recombination could be (a)between two individuals as crossover or (b) among severalindividuals as the combination of randomly selected genesegments With the help of normalization the combinationof gene segments could be in the specific order such asin Figure 2(a) or can be randomly arranged as shown inFigure 2(b) in the computational respective

4 Computational Intelligence and Neuroscience

Variable region onheavy chain

Antigen bindingsites

Variable regionon light chain

Constant regionon light chain

Constant regionon heavy chain

middot middot middotmiddot middot middot

(a)

Rearranged DNA

Gene library

(b)

Figure 1 Structure of antibody (a) and recombination (b) in the variable region

k

k

Xi1

Xi1

Xim

Xim

Xin

Xin

XiD

XiD

Xj1

Xj1

Xjm

Xjm

Xjn

Xjn

XjD

XjD

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

(a)

Xa1 Xa

m Xam Xa

D

Xb1 Xb

m Xbm Xb

D

Xd1 Xd

m Xdm Xd

D

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot

Xa1 Xa

2 Xb3 Xt

D

Xa2 Xa

3 Xd1 Xt

K

middot middot middot middot middot middot middot middot middot

(b)

Figure 2 The way of rearrangement of gene segments (a) between two individuals (b) among several individuals

The former one is similar to the SBX recombinationin GA [42] where the crossover of parents could swapgene segments on one site or many sites To better simulatethe arrangement of gene segments line recombination DEinspired recombination [43] and intelligent recombination[44] are proposed There would be a lot of ways to do thecombinations with optional joined gene segments With therandomness of the arrangements and combination diversityis introduced However as is known to all too much intro-duced diversity would be harmful for the performance Basedon the experimental experience our work focuses on theway to process recombination between two parents A newcombinatorial recombination operator combining with linerecombination is proposed as follows

Randomly choose two individuals from the populationdenoted by119883

119860and119883

119861 and then randomly choose119898 dimen-

sions119898 isin [1119863] fromeach of themwhere dimensions index

could be recorded as vectors119881119860and119881

119861 respectivelyThe new

individuals are generated by

1198831015840119881119860

119860= 120572119883119881119860

119860+ (1 minus 120572)119883

119881119861

119861

1198831015840119881119861

119861= 120572119883119881119861

119861+ (1 minus 120572)119883

119881119860

119860

(2)

where 120572 is a randomly produced number between 0 and 1 Itshould be noted that the range of each dimension of decisionvariable should be normalized at first The new proposedrecombination operator is represented in Figure 3

As shown in Figure 3 two new individuals are generatedthrough the combinational recombination In the example119898 equals 3 It needs to be known that 119894

1could be different

from 1198951 the same as in 119894

2and 1198952and 1198943and 1198953as long as the

normalization has been doneThen the fitness of the new generated individuals is

evaluated Together with the original individuals two with

Computational Intelligence and Neuroscience 5

XA X1A X

i1A X

i2A X

i3A XD

A VA = i1 i2 i3

XB X1B X

j1B X

j2B X

j3B XD

BVB = j1 j2 j3

Recombination

X998400i1A = 120572X

i1A + (1 minus 120572)X

j1B X

998400j1B = 120572X

j1B + (1 minus 120572)X

i1A

X998400i2A = 120572X

i2A + (1 minus 120572)X

j2B X

998400j2B = 120572X

j2B + (1 minus 120572)X

i2A

X998400i3A = 120572X

i3A + (1 minus 120572)X

j3B X

998400j3B = 120572X

j3B + (1 minus 120572)X

i3A

X998400A

XDB

X1A XD

A

X998400B X1

B

X998400i1A

X998400j1B

X998400i2A

X998400j2B

X998400i3A

X998400j3B

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

Figure 3 Recombination process

the higher fitness will survive and the other two individualsare deleted that is choose two individuals with high fitnessfrom the set 119883

119860 119883119861 1198831015840119860 1198831015840119861 Instead of comparing with the

whole population and reserving the elite a better diversitywill be maintained this way

32 Modified Hypermutation Hypermutation operatorbrings diversity for the population by introducingperturbation for each clone Although there are severalways to implement this operator [22] inversely proportionalstrategy is always the main basis The concept of theoperator proposed in [12] is adopted in this research workwhere each candidate solution is subject to M mutationswithout explicitly using a mutation probability The inverselyproportional law is used to determine the number of themutationsM

120572 = exp (minus120588119891lowast (119883119894))

119872 = lfloor(120572 times 119899) + 1rfloor (3)

where 119891lowast(119883119894) isin [0 1] is the normalized fitness of 119883

119894 120588

is the decay constant which determines the shape of themutation rate and lfloorsdotrfloor returns the lower bound integer Then119872mutation is performed on each candidate solution

1198831015840119895

119894=

119883119895

1199031+ 120582 (119883

119895

1199031minus 119883119895

1199032) if 119895 isin rand 119872(119899)

119883119895

119894otherwise

(4)

119883119894(119895) is the 119895th dimension of the 119894th individual rand

119872(119899) isin 1 119899 is randomly chosen 119872 indexes withoutrepetition and 120582 is a random number in the range of[minus1 1] 1199031 1199032 isin 1 2 119873 are randomly selected numbershereafter the amplitude of the hypermutation is controlledautomatically by the difference of randomly selected indi-viduals in the population The mutation equation (4) could

be considered as the variant of differential evolution andis introduced by [45] to modify the original updating foodsource of artificial bee colony algorithm (ABC) where itis improved to benefit for enhancing the search ability ofABC In our proposed algorithm the equation is combinedwith the hypermutation inversely proportionalM strategy asdepicted above The M strategy controls the direction thatis along how many dimensions while the equation controlsthe distance of the mutated clones with their parents Withcombination of both the amplitude of the hypermutation isautomatically controlled according to the distribution of thepopulation

33 Framework of the Proposed Algorithm Combined withthe proposed combinatorial recombination and hypermuta-tion operator the framework of the proposed algorithm is asfollows

Step 1 (initialization) Randomly initialize a population119860119887 of119873 individuals where 119873 denotes the size of the initial pop-ulation Each initial solution 119883

119894= 119883119894(1) 119883

119894(2) 119883

119894(119863)

is produced randomly within the range of boundaries of thedecision space

119883119895

119894= 119883119895

min + rand (0 1) (119883119895

max minus 119883119895

min) (5)

where 119894 = 1 2 119873 119895 = 1 2 119863 where 119863 is thedimension of the decision variables and 119883119895min and 119883119895max arethe lower and upper bounds for the dimension 119895 respectively

Step 2 (evaluation) Evaluate the objective value of eachsolution as its fitness

Step 3 (recombination) Randomly choose two individu-als from the population and implement the combinatorial

6 Computational Intelligence and Neuroscience

recombination as described in Section 31The recombinationrate is set to119873

119903

Step 4 (clonal selection) (i) Cloning each individual 119883119894

generates119873119888copies 1198831

119894 1198832119894 119883

119873119888

119894 where119873

119888is the clone

number which is a user defined constant(ii) Hypermutation each clone 119883119895

119894 119895 = 1 119873

119888 goes

through the hypermutation as described in Section 32 andgenerates the hypermutated clones1198831015840119895

119894

(iii) Selection select the individual with highest fitnessamong119883

119894and hypermutated clones 11988310158401

119894 11988310158402119894 119883

1015840119873119888

119894

Step 5 If stopping condition is not met go to Step 3Otherwise output the best one of the feasible group

It could be observed that instead of being proportional tothe fitness the clone number is a constant which not onlysaves computational source but also avoids individuals beingconcentrated in some decision area and leaving the othersunvisited when handling the multimodal problems

4 Experimental Studies on FunctionOptimization Problems

In this section experiments are conducted to evaluate theperformance of RHCSA by using 16 commonly used globaloptimization benchmark problems These functions containthe characteristics of being unimodal as 119891

1sim1198912 unrotated

multimodal as 1198913sim1198918 rotated multimodal as 119891

9sim11989114 and

composite functions as 11989115sim11989116 Table 1 gives the expression

of the benchmark function The detailed characteristics ofthese functions can be found in [1 46]

The introduced parameters are analyzed at first and thenthe proposed algorithm is compared with traditional CSAin the benchmark functions Finally comparisons betweenthe proposed algorithm and the state-of-the-art evolutionarycomputing models are represented Some discussions on theperformance analysis of RHCSA are also included It needs tobe noted that the same setup for the experiments is used forall the involved peers algorithms that is the comparison of allthe algorithms is under the same configuration of experimentsetup in this paper

41 Experimental Parameters Settings The experiments wereconducted on 16 benchmark functions The maximum num-ber of function evaluations (mFES) is set as 10000lowastD for10-D and 30-D situations respectively Each test is run 30independent times

There are quite few parameters introduced in the pro-posed algorithm For a fair comparison among CSAs theyare tested using the same setting of the parameters that isthe population size119873 is set to 30 and clone number119873

119888is set

to 4 as in [40] Furthermore there are two new parametersintroduced One is119898 which controls the direction of combi-natorial recombination and the other is recombination rateTables 2 and 3 give the experimental results of varying 119898 inthe 16 benchmark functions with 10-D and 30-D For conve-nience the experiment sets 4 proportions to the dimension asthe value to119898 which are [lceil(110)119863rceil lceil(15)119863rceil lceil(13)119863rceil119863]

The recombination operator could introduce diversityto the population through fusing information between ran-domly chosen parents When 119898 is small the generatedoffspring is much similar to one of the chosen parents andonly little dimension obtains information from both parentsWhen119898 is large the generated offspring tend to be the fusionof chosen parents That is to say m controls the offspringbeing much like one of the chosen parents or the fusion ofboth of them From Tables 2 and 3 we can find that the dif-ference among the varying setup ofm is quite small In amorecareful observation lceil(15)119863rceil and lceil(13)119863rceil are more appro-priate to compare with the other situations for both D = 10and D = 30 and lceil(13)119863rceil is even better It could be observedthat experimental results are not sensitive to the parameter119898 and a slightly larger119898may be beneficial for the algorithmIn view of the observation m = lceil(13)119863rceil is chosen in ourexperiments

There is another parameter named recombination rate119873119903 which controls the rate of recombination process which

will be implemented The higher the 119873119903 the more recom-

bination processes that will be executed In the early stagethe population is uniformly distributed in the search spaceand recombination of randomly chosen individuals will bringdiversity to the population As is known to all diversity isbeneficial for enhancing the search ability of the algorithmbut too much diversity will slow the convergence Tables4 and 5 represent the experimental results of varying therecombination rate

FromTables 4 and 5 it could be observed that RHCSAhasa quite robust performance with the varying recombinationrate By comparison 119873

119903which is equal to 07 achieves the

best performance and is adopted in the paper

42 Comparisons with Classic CSA The experiments areimplemented to check if the proposed operators are beneficialfor the performance of the algorithm Firstly recombinationoperator is added to the original CSA denoted as RCSA andthen the modified hypermutation is introduced as RHCSA

Table 6 shows the statistical results of CLONALG RCSAand RHCSA in optimizing the 16 test problems with D =10 based on 30 independent runs including the mean andstandard deviation From the table we can observe thatfor all these test instances the introduced recombinationoperator could obviously improve the performance of theCSA The reason is that the recombination operator couldbring diversity to enhance the search ability and avoidbeing trapped into the local optima The experiments resultson multimodal function 119891

3sim1198918could present the ability

of introduced recombination operator With the modifiedhypermutation operator the performance of the algorithmobtains a further improvement This is because both recom-bination and hypermutation adopt the difference of chosenindividuals to generate the candidate solution in the proposedalgorithm In the early stage the population is distributed inthe search space and the difference is relatively large whichis beneficial for global search and then with the iterationsgoing on the population converges to the optima and thedifference between individuals becomes smaller and smallerthat is the search process turns to be a local search It can

Computational Intelligence and Neuroscience 7

Table 1 Benchmark functions used in our experimental study

Name Test function D S 119891min

Sphere function 1198911(119909) =

119863

sum119894=1

1199092119894 1030 [minus100 100] 0

Rosenbrockrsquos function1198912 (119909) =119863minus1

sum119894=1

(100 (1199092119894minus 119909119894+1)2

+ (119909119894minus 1)2

) 1030 [minus2048 2048] 0

Ackleyrsquos function 1198913(119909) = minus20 exp(minus02radic 1

119863

119863

sum119894=1

1199092119894) minus exp( 1

119863

119863

sum119894=1

(2120587119909119894)) + 20 + 119890 1030 [minus32768 32768] 0

Griewanksrsquos function 1198914(119909) =

119863

sum119894=1

1199092119894

4000minus119863

prod119894=1

cos119909119894

radic119894+ 1 1030 [minus600 600] 0

Weierstrass function1198915(119909) =

119863

sum119894=1

119896maxsum119896=1

[119886119896 cos (2120587119887119896 (119909119894+ 05))] minus 119863

119896maxsum119896=1

119886119896 cos (2120587119887119896 sdot 05)1030 [minus05 05] 0

119886 = 05 119887 = 3 119896max = 20

Rastriginrsquos function 1198916(119909) =

119863

sum119894=1

1199092119894minus 10 cos (2120587119909

119894) + 10 1030 [minus512 512] 0

NoncontRas

1198917(119909) =

119863

sum119894=1

1199102119894minus 10 cos(2120587119910

119894) + 10

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt 05

round (2119909119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge 05

119894 = 1 2 119863

1030 [minus512 512] 0

Schwefelrsquos function 1198918(119909) = 4189829119863 minus

119863

sum119894=1

119909119894sin (1003816100381610038161003816119909119894

100381610038161003816100381605

) 1030 [minus500 500] 0

RotAckleyrsquos function 1198919(119909) = 119891

3(119910) 119910 = 119872 lowast 119909 1030 [minus32768 32768] 0

RotGriewanksrsquosfunction 119891

10(119909) = 119891

4(119910) 119910 = 119872 lowast 119909 1030 [minus600 600] 0

RotWeierstrassfunction 119891

11(119909) = 119891

5(119910) 119910 = 119872 lowast 119909 1030 [minus05 05] 0

RotRastriginrsquosfunction 119891

12(119909) = 119891

6(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

Rotnoncon Rasfunction 119891

13(119909) = 119891

7(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

RotSchwefelrsquosfunction

11989114(119909) = 4189829119863 minus

119863

sum119894=1

119885119894

119885119894=

119910119894sin (1003816100381610038161003816119910119894

100381610038161003816100381605

)10038161003816100381610038161199101198941003816100381610038161003816 le 500

0001 (10038161003816100381610038161199101198941003816100381610038161003816 minus 500)

2 10038161003816100381610038161199101198941003816100381610038161003816 ge 500

119894 = 1 2 119863

119910 = 119872 lowast (119909 minus 42096) + 42096

1030 [minus500 500] 0

Composition 1 11989115= 1198621198651 1030 [minus5 5] 0

Composition 2 11989116= 1198621198652 1030 [minus5 5] 0

be observed that RCSA surpass CSA and be surpassed byRHCSA at all the tested functions It can be concluded thatthe proposed recombination and hypermutation operatorsare effective improving the ability of the CSA

Table 7 shows the experimental results of the algorithmswhen 119863 = 30 Similar results are represented whichindicates that RHCSA is able to handle the high-dimensionaloptimization problems as well

43 Comparisons with the State-of-the-Art Algorithms Tocompare RHCSA with the state-of-the-art algorithms exper-imental results of seven representative evolutionary algo-rithms are listed in Tables 8 and 9 These algorithms areBaldwinian clonal selection algorithm (BCSA) [38] hybridlearning clonal selection algorithm (HLCSA) [40] orthogo-nal crossover based differential evolution (OXDE) [47] self-adaptive differential evolution (SaDE) [48] global and local

8 Computational Intelligence and Neuroscience

Table 2 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 10

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

13521119890 minus 180 plusmn 00000119890 minus 000 153784119890 minus 166 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 22773119890 minus 078 plusmn 49085119890 minus 078

1198912

86187119890 minus 004 plusmn 320378119890 minus 004 81377119890 minus 004 plusmn 30271119890 minus 004 89516119890 minus 004 plusmn 32039119890 minus 004 28971e minus 004 plusmn 36648e minus 004

1198913

52831119890 minus 015 plusmn 42156119890 minus 015 17763119890 minus 015 plusmn 17763119890 minus 015 88817e minus 016 plusmn 00000e minus 000 20724119890 minus 015 plusmn 20511119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

40175119890 minus 015 plusmn 13376119890 minus 015 30923e minus 015 plusmn 14969e minus 015 39292119890 minus 015 plusmn 11420119890 minus 015 42912119890 minus 015 plusmn 10827119890 minus 015

11989110

93401119890 minus 002 plusmn 15480119890 minus 002 18195119890 minus 002 plusmn 10245119890 minus 002 16937e minus 002 plusmn 16091e minus 002 55463119890 minus 002 plusmn 26635119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

24401119890 + 000 plusmn 10935119890 + 000 20556119890 + 000 plusmn 11601119890 + 000 15954e + 000 plusmn 15954e + 000 16638119890 + 000 plusmn 17026119890 + 000

11989113

81024119890 + 000 plusmn 56678119890 + 000 41385119890 + 000 plusmn 21123119890 + 000 40000e + 000 plusmn 21213e + 000 41773119890 + 000 plusmn 37530119890 + 000

11989114

75454119890 minus 008 plusmn 62091119890 minus 007 68712e minus 008 plusmn 70921e minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 92565119890 minus 008 plusmn 86635119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

90023119890 + 000 plusmn 78901119890 + 000 64971119890 + 000 plusmn 79297119890 + 000 52613e + 000 plusmn 50573e + 000 86977119890 + 000 plusmn 74630119890 + 000

Table 3 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 30

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

31129119890 minus 114 plusmn 44031119890 minus 114 82698119890 minus 110 plusmn 00000119890 minus 000 16994e minus 194 plusmn 00000e minus 000 66159119890 minus 108 plusmn 00000119890 minus 000

1198912

19281119890 minus 007 plusmn 89376119890 minus 007 74820119890 minus 007 plusmn 30968119890 minus 007 26862119890 minus 007 plusmn 60065119890 minus 007 74820e minus 008 plusmn 30968e minus 008

1198913

44415119890 minus 016 plusmn 25108119890 minus 016 89382119890 minus 016 plusmn 81571119890 minus 016 88817e minus 016 plusmn 00000e minus 000 21438119890 minus 015 plusmn 26596119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

32039119890 minus 015 plusmn 18147119890 minus 015 26532e minus 015 plusmn 13037e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 43241119890 minus 015 plusmn 29542119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

51900119890 + 001 plusmn 43580119890 + 001 41625119890 + 001 plusmn 91218119890 + 000 26201e + 001 plusmn 83454e + 000 29186119890 + 001 plusmn 67990119890 + 001

11989113

65345119890 + 001 plusmn 34082119890 + 001 51253119890 + 001 plusmn 23126119890 + 001 45502e + 001 plusmn 10609e + 001 48247119890 + 001 plusmn 10493119890 + 001

11989114

30358119890 minus 003 plusmn 38244119890 minus 003 42769119890 minus 003 plusmn 25513119890 minus 003 24186e minus 003 plusmn 53084e minus 003 34354119890 minus 003 plusmn 41538119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

94154119890 minus 002 plusmn 71725119890 minus 002 88757119890 minus 002 plusmn 14384119890 minus 002 88501e minus 002 plusmn 15560e minus 001 96583119890 minus 002 plusmn 17236119890 minus 001

Computational Intelligence and Neuroscience 9

Table 4 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 10

Function 01 05 07 1

1198911

38119119890 minus 119 plusmn 38510119890 minus 119 40163119890 minus 178 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 30122119890 minus 178 plusmn 00000119890 minus 000

1198912

29186119890 minus 004 plusmn 11966119890 minus 004 31115119890 minus 004 plusmn 105044119890 minus 004 89516119890 minus 004 plusmn 320393119890 minus 004 22544e minus 004 plusmn 15452e minus 004

1198913

19741119890 minus 015 plusmn 19767119890 minus 015 14807119890 minus 015 plusmn 19052119890 minus 015 88817e minus 016 plusmn 00000e minus 000 14135119890 minus 015 plusmn 19876119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

56305119890 minus 015 plusmn 13377119890 minus 015 79474119890 minus 015 plusmn 25291119890 minus 015 39292e minus 015 plusmn 11420e minus 015 41535119890 minus 015 plusmn 12103119890 minus 015

11989110

37179119890 minus 002 plusmn 18338119890 minus 002 31530e minus 002 plusmn 17649e minus 002 38363119890 minus 002 plusmn 377691119890 minus 002 37971119890 minus 002 plusmn 37825119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

23153119890 + 000 plusmn 13464119890 + 000 23031119890 + 000 plusmn 13465119890 + 000 15954119890 + 000 plusmn 15954119890 + 000 13685e + 000 plusmn 14679e + 000

11989113

40817119890 + 000 plusmn 55791119890 + 000 38531e + 000 plusmn 22514e + 000 40000119890 + 000 plusmn 21213119890 + 000 48073119890 + 000 plusmn 34243119890 + 000

11989114

55773e minus 008 plusmn 36818e minus 007 69679119890 minus 008 plusmn 59121119890 minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 70125119890 minus 008 plusmn 10013119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

86735119890 + 000 plusmn 83012119890 + 000 62013119890 + 000 plusmn 42016119890 + 000 52613e + 000 plusmn 50573e + 000 56012119890 + 000 plusmn 71232119890 + 000

Table 5 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 30

Function 01 05 07 1

1198911

16848119890 minus 121 plusmn 00000119890 minus 000 11232119890 minus 150 plusmn 00000119890 minus 000 16994119890 minus 194 plusmn 00000119890 minus 000 84243e minus 198 plusmn 00000e minus 000

1198912

33269119890 minus 007 plusmn 81537119890 minus 007 33493119890 minus 007 plusmn 31302119890 minus 007 26862e minus 007 plusmn 60065e minus 007 73295119890 minus 007 plusmn 30183119890 minus 007

1198913

96645119890 minus 016 plusmn 25121119890 minus 016 89312119890 minus 016 plusmn 28421119890 minus 016 88817e minus 016 plusmn 00000e minus 000 26645119890 minus 015 plusmn 27580119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

25600119890 minus 015 plusmn 22178119890 minus 015 24424e minus 015 plusmn 12352e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 25461119890 minus 015 plusmn 22735119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

46654119890 + 001 plusmn 41276119890 + 001 27521119890 + 001 plusmn 10387119890 + 000 26201e + 001 plusmn 83454e + 000 28012119890 + 001 plusmn 12293119890 + 001

11989113

51231119890 + 001 plusmn 13024119890 + 001 45613119890 + 001 plusmn 13450119890 + 001 45502e + 001 plusmn 10609e + 001 47234119890 + 001 plusmn 11249119890 + 001

11989114

26782119890 minus 003 plusmn 32933119890 minus 003 28334119890 minus 003 plusmn 62013119890 minus 003 24186e minus 003 plusmn 53084e minus 003 30187119890 minus 003 plusmn 87612119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

87011119890 minus 002 plusmn 56745119890 minus 002 87329119890 minus 002 plusmn 92581119890 minus 002 88501119890 minus 002 plusmn 15560119890 minus 001 60112e minus 002 plusmn 25832e minus 002

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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International Journal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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4 Computational Intelligence and Neuroscience

Variable region onheavy chain

Antigen bindingsites

Variable regionon light chain

Constant regionon light chain

Constant regionon heavy chain

middot middot middotmiddot middot middot

(a)

Rearranged DNA

Gene library

(b)

Figure 1 Structure of antibody (a) and recombination (b) in the variable region

k

k

Xi1

Xi1

Xim

Xim

Xin

Xin

XiD

XiD

Xj1

Xj1

Xjm

Xjm

Xjn

Xjn

XjD

XjD

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middotmiddot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

(a)

Xa1 Xa

m Xam Xa

D

Xb1 Xb

m Xbm Xb

D

Xd1 Xd

m Xdm Xd

D

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot

Xa1 Xa

2 Xb3 Xt

D

Xa2 Xa

3 Xd1 Xt

K

middot middot middot middot middot middot middot middot middot

(b)

Figure 2 The way of rearrangement of gene segments (a) between two individuals (b) among several individuals

The former one is similar to the SBX recombinationin GA [42] where the crossover of parents could swapgene segments on one site or many sites To better simulatethe arrangement of gene segments line recombination DEinspired recombination [43] and intelligent recombination[44] are proposed There would be a lot of ways to do thecombinations with optional joined gene segments With therandomness of the arrangements and combination diversityis introduced However as is known to all too much intro-duced diversity would be harmful for the performance Basedon the experimental experience our work focuses on theway to process recombination between two parents A newcombinatorial recombination operator combining with linerecombination is proposed as follows

Randomly choose two individuals from the populationdenoted by119883

119860and119883

119861 and then randomly choose119898 dimen-

sions119898 isin [1119863] fromeach of themwhere dimensions index

could be recorded as vectors119881119860and119881

119861 respectivelyThe new

individuals are generated by

1198831015840119881119860

119860= 120572119883119881119860

119860+ (1 minus 120572)119883

119881119861

119861

1198831015840119881119861

119861= 120572119883119881119861

119861+ (1 minus 120572)119883

119881119860

119860

(2)

where 120572 is a randomly produced number between 0 and 1 Itshould be noted that the range of each dimension of decisionvariable should be normalized at first The new proposedrecombination operator is represented in Figure 3

As shown in Figure 3 two new individuals are generatedthrough the combinational recombination In the example119898 equals 3 It needs to be known that 119894

1could be different

from 1198951 the same as in 119894

2and 1198952and 1198943and 1198953as long as the

normalization has been doneThen the fitness of the new generated individuals is

evaluated Together with the original individuals two with

Computational Intelligence and Neuroscience 5

XA X1A X

i1A X

i2A X

i3A XD

A VA = i1 i2 i3

XB X1B X

j1B X

j2B X

j3B XD

BVB = j1 j2 j3

Recombination

X998400i1A = 120572X

i1A + (1 minus 120572)X

j1B X

998400j1B = 120572X

j1B + (1 minus 120572)X

i1A

X998400i2A = 120572X

i2A + (1 minus 120572)X

j2B X

998400j2B = 120572X

j2B + (1 minus 120572)X

i2A

X998400i3A = 120572X

i3A + (1 minus 120572)X

j3B X

998400j3B = 120572X

j3B + (1 minus 120572)X

i3A

X998400A

XDB

X1A XD

A

X998400B X1

B

X998400i1A

X998400j1B

X998400i2A

X998400j2B

X998400i3A

X998400j3B

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

Figure 3 Recombination process

the higher fitness will survive and the other two individualsare deleted that is choose two individuals with high fitnessfrom the set 119883

119860 119883119861 1198831015840119860 1198831015840119861 Instead of comparing with the

whole population and reserving the elite a better diversitywill be maintained this way

32 Modified Hypermutation Hypermutation operatorbrings diversity for the population by introducingperturbation for each clone Although there are severalways to implement this operator [22] inversely proportionalstrategy is always the main basis The concept of theoperator proposed in [12] is adopted in this research workwhere each candidate solution is subject to M mutationswithout explicitly using a mutation probability The inverselyproportional law is used to determine the number of themutationsM

120572 = exp (minus120588119891lowast (119883119894))

119872 = lfloor(120572 times 119899) + 1rfloor (3)

where 119891lowast(119883119894) isin [0 1] is the normalized fitness of 119883

119894 120588

is the decay constant which determines the shape of themutation rate and lfloorsdotrfloor returns the lower bound integer Then119872mutation is performed on each candidate solution

1198831015840119895

119894=

119883119895

1199031+ 120582 (119883

119895

1199031minus 119883119895

1199032) if 119895 isin rand 119872(119899)

119883119895

119894otherwise

(4)

119883119894(119895) is the 119895th dimension of the 119894th individual rand

119872(119899) isin 1 119899 is randomly chosen 119872 indexes withoutrepetition and 120582 is a random number in the range of[minus1 1] 1199031 1199032 isin 1 2 119873 are randomly selected numbershereafter the amplitude of the hypermutation is controlledautomatically by the difference of randomly selected indi-viduals in the population The mutation equation (4) could

be considered as the variant of differential evolution andis introduced by [45] to modify the original updating foodsource of artificial bee colony algorithm (ABC) where itis improved to benefit for enhancing the search ability ofABC In our proposed algorithm the equation is combinedwith the hypermutation inversely proportionalM strategy asdepicted above The M strategy controls the direction thatis along how many dimensions while the equation controlsthe distance of the mutated clones with their parents Withcombination of both the amplitude of the hypermutation isautomatically controlled according to the distribution of thepopulation

33 Framework of the Proposed Algorithm Combined withthe proposed combinatorial recombination and hypermuta-tion operator the framework of the proposed algorithm is asfollows

Step 1 (initialization) Randomly initialize a population119860119887 of119873 individuals where 119873 denotes the size of the initial pop-ulation Each initial solution 119883

119894= 119883119894(1) 119883

119894(2) 119883

119894(119863)

is produced randomly within the range of boundaries of thedecision space

119883119895

119894= 119883119895

min + rand (0 1) (119883119895

max minus 119883119895

min) (5)

where 119894 = 1 2 119873 119895 = 1 2 119863 where 119863 is thedimension of the decision variables and 119883119895min and 119883119895max arethe lower and upper bounds for the dimension 119895 respectively

Step 2 (evaluation) Evaluate the objective value of eachsolution as its fitness

Step 3 (recombination) Randomly choose two individu-als from the population and implement the combinatorial

6 Computational Intelligence and Neuroscience

recombination as described in Section 31The recombinationrate is set to119873

119903

Step 4 (clonal selection) (i) Cloning each individual 119883119894

generates119873119888copies 1198831

119894 1198832119894 119883

119873119888

119894 where119873

119888is the clone

number which is a user defined constant(ii) Hypermutation each clone 119883119895

119894 119895 = 1 119873

119888 goes

through the hypermutation as described in Section 32 andgenerates the hypermutated clones1198831015840119895

119894

(iii) Selection select the individual with highest fitnessamong119883

119894and hypermutated clones 11988310158401

119894 11988310158402119894 119883

1015840119873119888

119894

Step 5 If stopping condition is not met go to Step 3Otherwise output the best one of the feasible group

It could be observed that instead of being proportional tothe fitness the clone number is a constant which not onlysaves computational source but also avoids individuals beingconcentrated in some decision area and leaving the othersunvisited when handling the multimodal problems

4 Experimental Studies on FunctionOptimization Problems

In this section experiments are conducted to evaluate theperformance of RHCSA by using 16 commonly used globaloptimization benchmark problems These functions containthe characteristics of being unimodal as 119891

1sim1198912 unrotated

multimodal as 1198913sim1198918 rotated multimodal as 119891

9sim11989114 and

composite functions as 11989115sim11989116 Table 1 gives the expression

of the benchmark function The detailed characteristics ofthese functions can be found in [1 46]

The introduced parameters are analyzed at first and thenthe proposed algorithm is compared with traditional CSAin the benchmark functions Finally comparisons betweenthe proposed algorithm and the state-of-the-art evolutionarycomputing models are represented Some discussions on theperformance analysis of RHCSA are also included It needs tobe noted that the same setup for the experiments is used forall the involved peers algorithms that is the comparison of allthe algorithms is under the same configuration of experimentsetup in this paper

41 Experimental Parameters Settings The experiments wereconducted on 16 benchmark functions The maximum num-ber of function evaluations (mFES) is set as 10000lowastD for10-D and 30-D situations respectively Each test is run 30independent times

There are quite few parameters introduced in the pro-posed algorithm For a fair comparison among CSAs theyare tested using the same setting of the parameters that isthe population size119873 is set to 30 and clone number119873

119888is set

to 4 as in [40] Furthermore there are two new parametersintroduced One is119898 which controls the direction of combi-natorial recombination and the other is recombination rateTables 2 and 3 give the experimental results of varying 119898 inthe 16 benchmark functions with 10-D and 30-D For conve-nience the experiment sets 4 proportions to the dimension asthe value to119898 which are [lceil(110)119863rceil lceil(15)119863rceil lceil(13)119863rceil119863]

The recombination operator could introduce diversityto the population through fusing information between ran-domly chosen parents When 119898 is small the generatedoffspring is much similar to one of the chosen parents andonly little dimension obtains information from both parentsWhen119898 is large the generated offspring tend to be the fusionof chosen parents That is to say m controls the offspringbeing much like one of the chosen parents or the fusion ofboth of them From Tables 2 and 3 we can find that the dif-ference among the varying setup ofm is quite small In amorecareful observation lceil(15)119863rceil and lceil(13)119863rceil are more appro-priate to compare with the other situations for both D = 10and D = 30 and lceil(13)119863rceil is even better It could be observedthat experimental results are not sensitive to the parameter119898 and a slightly larger119898may be beneficial for the algorithmIn view of the observation m = lceil(13)119863rceil is chosen in ourexperiments

There is another parameter named recombination rate119873119903 which controls the rate of recombination process which

will be implemented The higher the 119873119903 the more recom-

bination processes that will be executed In the early stagethe population is uniformly distributed in the search spaceand recombination of randomly chosen individuals will bringdiversity to the population As is known to all diversity isbeneficial for enhancing the search ability of the algorithmbut too much diversity will slow the convergence Tables4 and 5 represent the experimental results of varying therecombination rate

FromTables 4 and 5 it could be observed that RHCSAhasa quite robust performance with the varying recombinationrate By comparison 119873

119903which is equal to 07 achieves the

best performance and is adopted in the paper

42 Comparisons with Classic CSA The experiments areimplemented to check if the proposed operators are beneficialfor the performance of the algorithm Firstly recombinationoperator is added to the original CSA denoted as RCSA andthen the modified hypermutation is introduced as RHCSA

Table 6 shows the statistical results of CLONALG RCSAand RHCSA in optimizing the 16 test problems with D =10 based on 30 independent runs including the mean andstandard deviation From the table we can observe thatfor all these test instances the introduced recombinationoperator could obviously improve the performance of theCSA The reason is that the recombination operator couldbring diversity to enhance the search ability and avoidbeing trapped into the local optima The experiments resultson multimodal function 119891

3sim1198918could present the ability

of introduced recombination operator With the modifiedhypermutation operator the performance of the algorithmobtains a further improvement This is because both recom-bination and hypermutation adopt the difference of chosenindividuals to generate the candidate solution in the proposedalgorithm In the early stage the population is distributed inthe search space and the difference is relatively large whichis beneficial for global search and then with the iterationsgoing on the population converges to the optima and thedifference between individuals becomes smaller and smallerthat is the search process turns to be a local search It can

Computational Intelligence and Neuroscience 7

Table 1 Benchmark functions used in our experimental study

Name Test function D S 119891min

Sphere function 1198911(119909) =

119863

sum119894=1

1199092119894 1030 [minus100 100] 0

Rosenbrockrsquos function1198912 (119909) =119863minus1

sum119894=1

(100 (1199092119894minus 119909119894+1)2

+ (119909119894minus 1)2

) 1030 [minus2048 2048] 0

Ackleyrsquos function 1198913(119909) = minus20 exp(minus02radic 1

119863

119863

sum119894=1

1199092119894) minus exp( 1

119863

119863

sum119894=1

(2120587119909119894)) + 20 + 119890 1030 [minus32768 32768] 0

Griewanksrsquos function 1198914(119909) =

119863

sum119894=1

1199092119894

4000minus119863

prod119894=1

cos119909119894

radic119894+ 1 1030 [minus600 600] 0

Weierstrass function1198915(119909) =

119863

sum119894=1

119896maxsum119896=1

[119886119896 cos (2120587119887119896 (119909119894+ 05))] minus 119863

119896maxsum119896=1

119886119896 cos (2120587119887119896 sdot 05)1030 [minus05 05] 0

119886 = 05 119887 = 3 119896max = 20

Rastriginrsquos function 1198916(119909) =

119863

sum119894=1

1199092119894minus 10 cos (2120587119909

119894) + 10 1030 [minus512 512] 0

NoncontRas

1198917(119909) =

119863

sum119894=1

1199102119894minus 10 cos(2120587119910

119894) + 10

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt 05

round (2119909119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge 05

119894 = 1 2 119863

1030 [minus512 512] 0

Schwefelrsquos function 1198918(119909) = 4189829119863 minus

119863

sum119894=1

119909119894sin (1003816100381610038161003816119909119894

100381610038161003816100381605

) 1030 [minus500 500] 0

RotAckleyrsquos function 1198919(119909) = 119891

3(119910) 119910 = 119872 lowast 119909 1030 [minus32768 32768] 0

RotGriewanksrsquosfunction 119891

10(119909) = 119891

4(119910) 119910 = 119872 lowast 119909 1030 [minus600 600] 0

RotWeierstrassfunction 119891

11(119909) = 119891

5(119910) 119910 = 119872 lowast 119909 1030 [minus05 05] 0

RotRastriginrsquosfunction 119891

12(119909) = 119891

6(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

Rotnoncon Rasfunction 119891

13(119909) = 119891

7(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

RotSchwefelrsquosfunction

11989114(119909) = 4189829119863 minus

119863

sum119894=1

119885119894

119885119894=

119910119894sin (1003816100381610038161003816119910119894

100381610038161003816100381605

)10038161003816100381610038161199101198941003816100381610038161003816 le 500

0001 (10038161003816100381610038161199101198941003816100381610038161003816 minus 500)

2 10038161003816100381610038161199101198941003816100381610038161003816 ge 500

119894 = 1 2 119863

119910 = 119872 lowast (119909 minus 42096) + 42096

1030 [minus500 500] 0

Composition 1 11989115= 1198621198651 1030 [minus5 5] 0

Composition 2 11989116= 1198621198652 1030 [minus5 5] 0

be observed that RCSA surpass CSA and be surpassed byRHCSA at all the tested functions It can be concluded thatthe proposed recombination and hypermutation operatorsare effective improving the ability of the CSA

Table 7 shows the experimental results of the algorithmswhen 119863 = 30 Similar results are represented whichindicates that RHCSA is able to handle the high-dimensionaloptimization problems as well

43 Comparisons with the State-of-the-Art Algorithms Tocompare RHCSA with the state-of-the-art algorithms exper-imental results of seven representative evolutionary algo-rithms are listed in Tables 8 and 9 These algorithms areBaldwinian clonal selection algorithm (BCSA) [38] hybridlearning clonal selection algorithm (HLCSA) [40] orthogo-nal crossover based differential evolution (OXDE) [47] self-adaptive differential evolution (SaDE) [48] global and local

8 Computational Intelligence and Neuroscience

Table 2 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 10

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

13521119890 minus 180 plusmn 00000119890 minus 000 153784119890 minus 166 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 22773119890 minus 078 plusmn 49085119890 minus 078

1198912

86187119890 minus 004 plusmn 320378119890 minus 004 81377119890 minus 004 plusmn 30271119890 minus 004 89516119890 minus 004 plusmn 32039119890 minus 004 28971e minus 004 plusmn 36648e minus 004

1198913

52831119890 minus 015 plusmn 42156119890 minus 015 17763119890 minus 015 plusmn 17763119890 minus 015 88817e minus 016 plusmn 00000e minus 000 20724119890 minus 015 plusmn 20511119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

40175119890 minus 015 plusmn 13376119890 minus 015 30923e minus 015 plusmn 14969e minus 015 39292119890 minus 015 plusmn 11420119890 minus 015 42912119890 minus 015 plusmn 10827119890 minus 015

11989110

93401119890 minus 002 plusmn 15480119890 minus 002 18195119890 minus 002 plusmn 10245119890 minus 002 16937e minus 002 plusmn 16091e minus 002 55463119890 minus 002 plusmn 26635119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

24401119890 + 000 plusmn 10935119890 + 000 20556119890 + 000 plusmn 11601119890 + 000 15954e + 000 plusmn 15954e + 000 16638119890 + 000 plusmn 17026119890 + 000

11989113

81024119890 + 000 plusmn 56678119890 + 000 41385119890 + 000 plusmn 21123119890 + 000 40000e + 000 plusmn 21213e + 000 41773119890 + 000 plusmn 37530119890 + 000

11989114

75454119890 minus 008 plusmn 62091119890 minus 007 68712e minus 008 plusmn 70921e minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 92565119890 minus 008 plusmn 86635119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

90023119890 + 000 plusmn 78901119890 + 000 64971119890 + 000 plusmn 79297119890 + 000 52613e + 000 plusmn 50573e + 000 86977119890 + 000 plusmn 74630119890 + 000

Table 3 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 30

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

31129119890 minus 114 plusmn 44031119890 minus 114 82698119890 minus 110 plusmn 00000119890 minus 000 16994e minus 194 plusmn 00000e minus 000 66159119890 minus 108 plusmn 00000119890 minus 000

1198912

19281119890 minus 007 plusmn 89376119890 minus 007 74820119890 minus 007 plusmn 30968119890 minus 007 26862119890 minus 007 plusmn 60065119890 minus 007 74820e minus 008 plusmn 30968e minus 008

1198913

44415119890 minus 016 plusmn 25108119890 minus 016 89382119890 minus 016 plusmn 81571119890 minus 016 88817e minus 016 plusmn 00000e minus 000 21438119890 minus 015 plusmn 26596119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

32039119890 minus 015 plusmn 18147119890 minus 015 26532e minus 015 plusmn 13037e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 43241119890 minus 015 plusmn 29542119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

51900119890 + 001 plusmn 43580119890 + 001 41625119890 + 001 plusmn 91218119890 + 000 26201e + 001 plusmn 83454e + 000 29186119890 + 001 plusmn 67990119890 + 001

11989113

65345119890 + 001 plusmn 34082119890 + 001 51253119890 + 001 plusmn 23126119890 + 001 45502e + 001 plusmn 10609e + 001 48247119890 + 001 plusmn 10493119890 + 001

11989114

30358119890 minus 003 plusmn 38244119890 minus 003 42769119890 minus 003 plusmn 25513119890 minus 003 24186e minus 003 plusmn 53084e minus 003 34354119890 minus 003 plusmn 41538119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

94154119890 minus 002 plusmn 71725119890 minus 002 88757119890 minus 002 plusmn 14384119890 minus 002 88501e minus 002 plusmn 15560e minus 001 96583119890 minus 002 plusmn 17236119890 minus 001

Computational Intelligence and Neuroscience 9

Table 4 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 10

Function 01 05 07 1

1198911

38119119890 minus 119 plusmn 38510119890 minus 119 40163119890 minus 178 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 30122119890 minus 178 plusmn 00000119890 minus 000

1198912

29186119890 minus 004 plusmn 11966119890 minus 004 31115119890 minus 004 plusmn 105044119890 minus 004 89516119890 minus 004 plusmn 320393119890 minus 004 22544e minus 004 plusmn 15452e minus 004

1198913

19741119890 minus 015 plusmn 19767119890 minus 015 14807119890 minus 015 plusmn 19052119890 minus 015 88817e minus 016 plusmn 00000e minus 000 14135119890 minus 015 plusmn 19876119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

56305119890 minus 015 plusmn 13377119890 minus 015 79474119890 minus 015 plusmn 25291119890 minus 015 39292e minus 015 plusmn 11420e minus 015 41535119890 minus 015 plusmn 12103119890 minus 015

11989110

37179119890 minus 002 plusmn 18338119890 minus 002 31530e minus 002 plusmn 17649e minus 002 38363119890 minus 002 plusmn 377691119890 minus 002 37971119890 minus 002 plusmn 37825119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

23153119890 + 000 plusmn 13464119890 + 000 23031119890 + 000 plusmn 13465119890 + 000 15954119890 + 000 plusmn 15954119890 + 000 13685e + 000 plusmn 14679e + 000

11989113

40817119890 + 000 plusmn 55791119890 + 000 38531e + 000 plusmn 22514e + 000 40000119890 + 000 plusmn 21213119890 + 000 48073119890 + 000 plusmn 34243119890 + 000

11989114

55773e minus 008 plusmn 36818e minus 007 69679119890 minus 008 plusmn 59121119890 minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 70125119890 minus 008 plusmn 10013119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

86735119890 + 000 plusmn 83012119890 + 000 62013119890 + 000 plusmn 42016119890 + 000 52613e + 000 plusmn 50573e + 000 56012119890 + 000 plusmn 71232119890 + 000

Table 5 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 30

Function 01 05 07 1

1198911

16848119890 minus 121 plusmn 00000119890 minus 000 11232119890 minus 150 plusmn 00000119890 minus 000 16994119890 minus 194 plusmn 00000119890 minus 000 84243e minus 198 plusmn 00000e minus 000

1198912

33269119890 minus 007 plusmn 81537119890 minus 007 33493119890 minus 007 plusmn 31302119890 minus 007 26862e minus 007 plusmn 60065e minus 007 73295119890 minus 007 plusmn 30183119890 minus 007

1198913

96645119890 minus 016 plusmn 25121119890 minus 016 89312119890 minus 016 plusmn 28421119890 minus 016 88817e minus 016 plusmn 00000e minus 000 26645119890 minus 015 plusmn 27580119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

25600119890 minus 015 plusmn 22178119890 minus 015 24424e minus 015 plusmn 12352e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 25461119890 minus 015 plusmn 22735119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

46654119890 + 001 plusmn 41276119890 + 001 27521119890 + 001 plusmn 10387119890 + 000 26201e + 001 plusmn 83454e + 000 28012119890 + 001 plusmn 12293119890 + 001

11989113

51231119890 + 001 plusmn 13024119890 + 001 45613119890 + 001 plusmn 13450119890 + 001 45502e + 001 plusmn 10609e + 001 47234119890 + 001 plusmn 11249119890 + 001

11989114

26782119890 minus 003 plusmn 32933119890 minus 003 28334119890 minus 003 plusmn 62013119890 minus 003 24186e minus 003 plusmn 53084e minus 003 30187119890 minus 003 plusmn 87612119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

87011119890 minus 002 plusmn 56745119890 minus 002 87329119890 minus 002 plusmn 92581119890 minus 002 88501119890 minus 002 plusmn 15560119890 minus 001 60112e minus 002 plusmn 25832e minus 002

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

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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

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Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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

Computational Intelligence and Neuroscience 5

XA X1A X

i1A X

i2A X

i3A XD

A VA = i1 i2 i3

XB X1B X

j1B X

j2B X

j3B XD

BVB = j1 j2 j3

Recombination

X998400i1A = 120572X

i1A + (1 minus 120572)X

j1B X

998400j1B = 120572X

j1B + (1 minus 120572)X

i1A

X998400i2A = 120572X

i2A + (1 minus 120572)X

j2B X

998400j2B = 120572X

j2B + (1 minus 120572)X

i2A

X998400i3A = 120572X

i3A + (1 minus 120572)X

j3B X

998400j3B = 120572X

j3B + (1 minus 120572)X

i3A

X998400A

XDB

X1A XD

A

X998400B X1

B

X998400i1A

X998400j1B

X998400i2A

X998400j2B

X998400i3A

X998400j3B

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot middot middot middot middot middot middot

Figure 3 Recombination process

the higher fitness will survive and the other two individualsare deleted that is choose two individuals with high fitnessfrom the set 119883

119860 119883119861 1198831015840119860 1198831015840119861 Instead of comparing with the

whole population and reserving the elite a better diversitywill be maintained this way

32 Modified Hypermutation Hypermutation operatorbrings diversity for the population by introducingperturbation for each clone Although there are severalways to implement this operator [22] inversely proportionalstrategy is always the main basis The concept of theoperator proposed in [12] is adopted in this research workwhere each candidate solution is subject to M mutationswithout explicitly using a mutation probability The inverselyproportional law is used to determine the number of themutationsM

120572 = exp (minus120588119891lowast (119883119894))

119872 = lfloor(120572 times 119899) + 1rfloor (3)

where 119891lowast(119883119894) isin [0 1] is the normalized fitness of 119883

119894 120588

is the decay constant which determines the shape of themutation rate and lfloorsdotrfloor returns the lower bound integer Then119872mutation is performed on each candidate solution

1198831015840119895

119894=

119883119895

1199031+ 120582 (119883

119895

1199031minus 119883119895

1199032) if 119895 isin rand 119872(119899)

119883119895

119894otherwise

(4)

119883119894(119895) is the 119895th dimension of the 119894th individual rand

119872(119899) isin 1 119899 is randomly chosen 119872 indexes withoutrepetition and 120582 is a random number in the range of[minus1 1] 1199031 1199032 isin 1 2 119873 are randomly selected numbershereafter the amplitude of the hypermutation is controlledautomatically by the difference of randomly selected indi-viduals in the population The mutation equation (4) could

be considered as the variant of differential evolution andis introduced by [45] to modify the original updating foodsource of artificial bee colony algorithm (ABC) where itis improved to benefit for enhancing the search ability ofABC In our proposed algorithm the equation is combinedwith the hypermutation inversely proportionalM strategy asdepicted above The M strategy controls the direction thatis along how many dimensions while the equation controlsthe distance of the mutated clones with their parents Withcombination of both the amplitude of the hypermutation isautomatically controlled according to the distribution of thepopulation

33 Framework of the Proposed Algorithm Combined withthe proposed combinatorial recombination and hypermuta-tion operator the framework of the proposed algorithm is asfollows

Step 1 (initialization) Randomly initialize a population119860119887 of119873 individuals where 119873 denotes the size of the initial pop-ulation Each initial solution 119883

119894= 119883119894(1) 119883

119894(2) 119883

119894(119863)

is produced randomly within the range of boundaries of thedecision space

119883119895

119894= 119883119895

min + rand (0 1) (119883119895

max minus 119883119895

min) (5)

where 119894 = 1 2 119873 119895 = 1 2 119863 where 119863 is thedimension of the decision variables and 119883119895min and 119883119895max arethe lower and upper bounds for the dimension 119895 respectively

Step 2 (evaluation) Evaluate the objective value of eachsolution as its fitness

Step 3 (recombination) Randomly choose two individu-als from the population and implement the combinatorial

6 Computational Intelligence and Neuroscience

recombination as described in Section 31The recombinationrate is set to119873

119903

Step 4 (clonal selection) (i) Cloning each individual 119883119894

generates119873119888copies 1198831

119894 1198832119894 119883

119873119888

119894 where119873

119888is the clone

number which is a user defined constant(ii) Hypermutation each clone 119883119895

119894 119895 = 1 119873

119888 goes

through the hypermutation as described in Section 32 andgenerates the hypermutated clones1198831015840119895

119894

(iii) Selection select the individual with highest fitnessamong119883

119894and hypermutated clones 11988310158401

119894 11988310158402119894 119883

1015840119873119888

119894

Step 5 If stopping condition is not met go to Step 3Otherwise output the best one of the feasible group

It could be observed that instead of being proportional tothe fitness the clone number is a constant which not onlysaves computational source but also avoids individuals beingconcentrated in some decision area and leaving the othersunvisited when handling the multimodal problems

4 Experimental Studies on FunctionOptimization Problems

In this section experiments are conducted to evaluate theperformance of RHCSA by using 16 commonly used globaloptimization benchmark problems These functions containthe characteristics of being unimodal as 119891

1sim1198912 unrotated

multimodal as 1198913sim1198918 rotated multimodal as 119891

9sim11989114 and

composite functions as 11989115sim11989116 Table 1 gives the expression

of the benchmark function The detailed characteristics ofthese functions can be found in [1 46]

The introduced parameters are analyzed at first and thenthe proposed algorithm is compared with traditional CSAin the benchmark functions Finally comparisons betweenthe proposed algorithm and the state-of-the-art evolutionarycomputing models are represented Some discussions on theperformance analysis of RHCSA are also included It needs tobe noted that the same setup for the experiments is used forall the involved peers algorithms that is the comparison of allthe algorithms is under the same configuration of experimentsetup in this paper

41 Experimental Parameters Settings The experiments wereconducted on 16 benchmark functions The maximum num-ber of function evaluations (mFES) is set as 10000lowastD for10-D and 30-D situations respectively Each test is run 30independent times

There are quite few parameters introduced in the pro-posed algorithm For a fair comparison among CSAs theyare tested using the same setting of the parameters that isthe population size119873 is set to 30 and clone number119873

119888is set

to 4 as in [40] Furthermore there are two new parametersintroduced One is119898 which controls the direction of combi-natorial recombination and the other is recombination rateTables 2 and 3 give the experimental results of varying 119898 inthe 16 benchmark functions with 10-D and 30-D For conve-nience the experiment sets 4 proportions to the dimension asthe value to119898 which are [lceil(110)119863rceil lceil(15)119863rceil lceil(13)119863rceil119863]

The recombination operator could introduce diversityto the population through fusing information between ran-domly chosen parents When 119898 is small the generatedoffspring is much similar to one of the chosen parents andonly little dimension obtains information from both parentsWhen119898 is large the generated offspring tend to be the fusionof chosen parents That is to say m controls the offspringbeing much like one of the chosen parents or the fusion ofboth of them From Tables 2 and 3 we can find that the dif-ference among the varying setup ofm is quite small In amorecareful observation lceil(15)119863rceil and lceil(13)119863rceil are more appro-priate to compare with the other situations for both D = 10and D = 30 and lceil(13)119863rceil is even better It could be observedthat experimental results are not sensitive to the parameter119898 and a slightly larger119898may be beneficial for the algorithmIn view of the observation m = lceil(13)119863rceil is chosen in ourexperiments

There is another parameter named recombination rate119873119903 which controls the rate of recombination process which

will be implemented The higher the 119873119903 the more recom-

bination processes that will be executed In the early stagethe population is uniformly distributed in the search spaceand recombination of randomly chosen individuals will bringdiversity to the population As is known to all diversity isbeneficial for enhancing the search ability of the algorithmbut too much diversity will slow the convergence Tables4 and 5 represent the experimental results of varying therecombination rate

FromTables 4 and 5 it could be observed that RHCSAhasa quite robust performance with the varying recombinationrate By comparison 119873

119903which is equal to 07 achieves the

best performance and is adopted in the paper

42 Comparisons with Classic CSA The experiments areimplemented to check if the proposed operators are beneficialfor the performance of the algorithm Firstly recombinationoperator is added to the original CSA denoted as RCSA andthen the modified hypermutation is introduced as RHCSA

Table 6 shows the statistical results of CLONALG RCSAand RHCSA in optimizing the 16 test problems with D =10 based on 30 independent runs including the mean andstandard deviation From the table we can observe thatfor all these test instances the introduced recombinationoperator could obviously improve the performance of theCSA The reason is that the recombination operator couldbring diversity to enhance the search ability and avoidbeing trapped into the local optima The experiments resultson multimodal function 119891

3sim1198918could present the ability

of introduced recombination operator With the modifiedhypermutation operator the performance of the algorithmobtains a further improvement This is because both recom-bination and hypermutation adopt the difference of chosenindividuals to generate the candidate solution in the proposedalgorithm In the early stage the population is distributed inthe search space and the difference is relatively large whichis beneficial for global search and then with the iterationsgoing on the population converges to the optima and thedifference between individuals becomes smaller and smallerthat is the search process turns to be a local search It can

Computational Intelligence and Neuroscience 7

Table 1 Benchmark functions used in our experimental study

Name Test function D S 119891min

Sphere function 1198911(119909) =

119863

sum119894=1

1199092119894 1030 [minus100 100] 0

Rosenbrockrsquos function1198912 (119909) =119863minus1

sum119894=1

(100 (1199092119894minus 119909119894+1)2

+ (119909119894minus 1)2

) 1030 [minus2048 2048] 0

Ackleyrsquos function 1198913(119909) = minus20 exp(minus02radic 1

119863

119863

sum119894=1

1199092119894) minus exp( 1

119863

119863

sum119894=1

(2120587119909119894)) + 20 + 119890 1030 [minus32768 32768] 0

Griewanksrsquos function 1198914(119909) =

119863

sum119894=1

1199092119894

4000minus119863

prod119894=1

cos119909119894

radic119894+ 1 1030 [minus600 600] 0

Weierstrass function1198915(119909) =

119863

sum119894=1

119896maxsum119896=1

[119886119896 cos (2120587119887119896 (119909119894+ 05))] minus 119863

119896maxsum119896=1

119886119896 cos (2120587119887119896 sdot 05)1030 [minus05 05] 0

119886 = 05 119887 = 3 119896max = 20

Rastriginrsquos function 1198916(119909) =

119863

sum119894=1

1199092119894minus 10 cos (2120587119909

119894) + 10 1030 [minus512 512] 0

NoncontRas

1198917(119909) =

119863

sum119894=1

1199102119894minus 10 cos(2120587119910

119894) + 10

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt 05

round (2119909119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge 05

119894 = 1 2 119863

1030 [minus512 512] 0

Schwefelrsquos function 1198918(119909) = 4189829119863 minus

119863

sum119894=1

119909119894sin (1003816100381610038161003816119909119894

100381610038161003816100381605

) 1030 [minus500 500] 0

RotAckleyrsquos function 1198919(119909) = 119891

3(119910) 119910 = 119872 lowast 119909 1030 [minus32768 32768] 0

RotGriewanksrsquosfunction 119891

10(119909) = 119891

4(119910) 119910 = 119872 lowast 119909 1030 [minus600 600] 0

RotWeierstrassfunction 119891

11(119909) = 119891

5(119910) 119910 = 119872 lowast 119909 1030 [minus05 05] 0

RotRastriginrsquosfunction 119891

12(119909) = 119891

6(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

Rotnoncon Rasfunction 119891

13(119909) = 119891

7(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

RotSchwefelrsquosfunction

11989114(119909) = 4189829119863 minus

119863

sum119894=1

119885119894

119885119894=

119910119894sin (1003816100381610038161003816119910119894

100381610038161003816100381605

)10038161003816100381610038161199101198941003816100381610038161003816 le 500

0001 (10038161003816100381610038161199101198941003816100381610038161003816 minus 500)

2 10038161003816100381610038161199101198941003816100381610038161003816 ge 500

119894 = 1 2 119863

119910 = 119872 lowast (119909 minus 42096) + 42096

1030 [minus500 500] 0

Composition 1 11989115= 1198621198651 1030 [minus5 5] 0

Composition 2 11989116= 1198621198652 1030 [minus5 5] 0

be observed that RCSA surpass CSA and be surpassed byRHCSA at all the tested functions It can be concluded thatthe proposed recombination and hypermutation operatorsare effective improving the ability of the CSA

Table 7 shows the experimental results of the algorithmswhen 119863 = 30 Similar results are represented whichindicates that RHCSA is able to handle the high-dimensionaloptimization problems as well

43 Comparisons with the State-of-the-Art Algorithms Tocompare RHCSA with the state-of-the-art algorithms exper-imental results of seven representative evolutionary algo-rithms are listed in Tables 8 and 9 These algorithms areBaldwinian clonal selection algorithm (BCSA) [38] hybridlearning clonal selection algorithm (HLCSA) [40] orthogo-nal crossover based differential evolution (OXDE) [47] self-adaptive differential evolution (SaDE) [48] global and local

8 Computational Intelligence and Neuroscience

Table 2 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 10

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

13521119890 minus 180 plusmn 00000119890 minus 000 153784119890 minus 166 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 22773119890 minus 078 plusmn 49085119890 minus 078

1198912

86187119890 minus 004 plusmn 320378119890 minus 004 81377119890 minus 004 plusmn 30271119890 minus 004 89516119890 minus 004 plusmn 32039119890 minus 004 28971e minus 004 plusmn 36648e minus 004

1198913

52831119890 minus 015 plusmn 42156119890 minus 015 17763119890 minus 015 plusmn 17763119890 minus 015 88817e minus 016 plusmn 00000e minus 000 20724119890 minus 015 plusmn 20511119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

40175119890 minus 015 plusmn 13376119890 minus 015 30923e minus 015 plusmn 14969e minus 015 39292119890 minus 015 plusmn 11420119890 minus 015 42912119890 minus 015 plusmn 10827119890 minus 015

11989110

93401119890 minus 002 plusmn 15480119890 minus 002 18195119890 minus 002 plusmn 10245119890 minus 002 16937e minus 002 plusmn 16091e minus 002 55463119890 minus 002 plusmn 26635119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

24401119890 + 000 plusmn 10935119890 + 000 20556119890 + 000 plusmn 11601119890 + 000 15954e + 000 plusmn 15954e + 000 16638119890 + 000 plusmn 17026119890 + 000

11989113

81024119890 + 000 plusmn 56678119890 + 000 41385119890 + 000 plusmn 21123119890 + 000 40000e + 000 plusmn 21213e + 000 41773119890 + 000 plusmn 37530119890 + 000

11989114

75454119890 minus 008 plusmn 62091119890 minus 007 68712e minus 008 plusmn 70921e minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 92565119890 minus 008 plusmn 86635119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

90023119890 + 000 plusmn 78901119890 + 000 64971119890 + 000 plusmn 79297119890 + 000 52613e + 000 plusmn 50573e + 000 86977119890 + 000 plusmn 74630119890 + 000

Table 3 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 30

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

31129119890 minus 114 plusmn 44031119890 minus 114 82698119890 minus 110 plusmn 00000119890 minus 000 16994e minus 194 plusmn 00000e minus 000 66159119890 minus 108 plusmn 00000119890 minus 000

1198912

19281119890 minus 007 plusmn 89376119890 minus 007 74820119890 minus 007 plusmn 30968119890 minus 007 26862119890 minus 007 plusmn 60065119890 minus 007 74820e minus 008 plusmn 30968e minus 008

1198913

44415119890 minus 016 plusmn 25108119890 minus 016 89382119890 minus 016 plusmn 81571119890 minus 016 88817e minus 016 plusmn 00000e minus 000 21438119890 minus 015 plusmn 26596119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

32039119890 minus 015 plusmn 18147119890 minus 015 26532e minus 015 plusmn 13037e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 43241119890 minus 015 plusmn 29542119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

51900119890 + 001 plusmn 43580119890 + 001 41625119890 + 001 plusmn 91218119890 + 000 26201e + 001 plusmn 83454e + 000 29186119890 + 001 plusmn 67990119890 + 001

11989113

65345119890 + 001 plusmn 34082119890 + 001 51253119890 + 001 plusmn 23126119890 + 001 45502e + 001 plusmn 10609e + 001 48247119890 + 001 plusmn 10493119890 + 001

11989114

30358119890 minus 003 plusmn 38244119890 minus 003 42769119890 minus 003 plusmn 25513119890 minus 003 24186e minus 003 plusmn 53084e minus 003 34354119890 minus 003 plusmn 41538119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

94154119890 minus 002 plusmn 71725119890 minus 002 88757119890 minus 002 plusmn 14384119890 minus 002 88501e minus 002 plusmn 15560e minus 001 96583119890 minus 002 plusmn 17236119890 minus 001

Computational Intelligence and Neuroscience 9

Table 4 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 10

Function 01 05 07 1

1198911

38119119890 minus 119 plusmn 38510119890 minus 119 40163119890 minus 178 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 30122119890 minus 178 plusmn 00000119890 minus 000

1198912

29186119890 minus 004 plusmn 11966119890 minus 004 31115119890 minus 004 plusmn 105044119890 minus 004 89516119890 minus 004 plusmn 320393119890 minus 004 22544e minus 004 plusmn 15452e minus 004

1198913

19741119890 minus 015 plusmn 19767119890 minus 015 14807119890 minus 015 plusmn 19052119890 minus 015 88817e minus 016 plusmn 00000e minus 000 14135119890 minus 015 plusmn 19876119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

56305119890 minus 015 plusmn 13377119890 minus 015 79474119890 minus 015 plusmn 25291119890 minus 015 39292e minus 015 plusmn 11420e minus 015 41535119890 minus 015 plusmn 12103119890 minus 015

11989110

37179119890 minus 002 plusmn 18338119890 minus 002 31530e minus 002 plusmn 17649e minus 002 38363119890 minus 002 plusmn 377691119890 minus 002 37971119890 minus 002 plusmn 37825119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

23153119890 + 000 plusmn 13464119890 + 000 23031119890 + 000 plusmn 13465119890 + 000 15954119890 + 000 plusmn 15954119890 + 000 13685e + 000 plusmn 14679e + 000

11989113

40817119890 + 000 plusmn 55791119890 + 000 38531e + 000 plusmn 22514e + 000 40000119890 + 000 plusmn 21213119890 + 000 48073119890 + 000 plusmn 34243119890 + 000

11989114

55773e minus 008 plusmn 36818e minus 007 69679119890 minus 008 plusmn 59121119890 minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 70125119890 minus 008 plusmn 10013119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

86735119890 + 000 plusmn 83012119890 + 000 62013119890 + 000 plusmn 42016119890 + 000 52613e + 000 plusmn 50573e + 000 56012119890 + 000 plusmn 71232119890 + 000

Table 5 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 30

Function 01 05 07 1

1198911

16848119890 minus 121 plusmn 00000119890 minus 000 11232119890 minus 150 plusmn 00000119890 minus 000 16994119890 minus 194 plusmn 00000119890 minus 000 84243e minus 198 plusmn 00000e minus 000

1198912

33269119890 minus 007 plusmn 81537119890 minus 007 33493119890 minus 007 plusmn 31302119890 minus 007 26862e minus 007 plusmn 60065e minus 007 73295119890 minus 007 plusmn 30183119890 minus 007

1198913

96645119890 minus 016 plusmn 25121119890 minus 016 89312119890 minus 016 plusmn 28421119890 minus 016 88817e minus 016 plusmn 00000e minus 000 26645119890 minus 015 plusmn 27580119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

25600119890 minus 015 plusmn 22178119890 minus 015 24424e minus 015 plusmn 12352e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 25461119890 minus 015 plusmn 22735119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

46654119890 + 001 plusmn 41276119890 + 001 27521119890 + 001 plusmn 10387119890 + 000 26201e + 001 plusmn 83454e + 000 28012119890 + 001 plusmn 12293119890 + 001

11989113

51231119890 + 001 plusmn 13024119890 + 001 45613119890 + 001 plusmn 13450119890 + 001 45502e + 001 plusmn 10609e + 001 47234119890 + 001 plusmn 11249119890 + 001

11989114

26782119890 minus 003 plusmn 32933119890 minus 003 28334119890 minus 003 plusmn 62013119890 minus 003 24186e minus 003 plusmn 53084e minus 003 30187119890 minus 003 plusmn 87612119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

87011119890 minus 002 plusmn 56745119890 minus 002 87329119890 minus 002 plusmn 92581119890 minus 002 88501119890 minus 002 plusmn 15560119890 minus 001 60112e minus 002 plusmn 25832e minus 002

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Computational Intelligence and Neuroscience

recombination as described in Section 31The recombinationrate is set to119873

119903

Step 4 (clonal selection) (i) Cloning each individual 119883119894

generates119873119888copies 1198831

119894 1198832119894 119883

119873119888

119894 where119873

119888is the clone

number which is a user defined constant(ii) Hypermutation each clone 119883119895

119894 119895 = 1 119873

119888 goes

through the hypermutation as described in Section 32 andgenerates the hypermutated clones1198831015840119895

119894

(iii) Selection select the individual with highest fitnessamong119883

119894and hypermutated clones 11988310158401

119894 11988310158402119894 119883

1015840119873119888

119894

Step 5 If stopping condition is not met go to Step 3Otherwise output the best one of the feasible group

It could be observed that instead of being proportional tothe fitness the clone number is a constant which not onlysaves computational source but also avoids individuals beingconcentrated in some decision area and leaving the othersunvisited when handling the multimodal problems

4 Experimental Studies on FunctionOptimization Problems

In this section experiments are conducted to evaluate theperformance of RHCSA by using 16 commonly used globaloptimization benchmark problems These functions containthe characteristics of being unimodal as 119891

1sim1198912 unrotated

multimodal as 1198913sim1198918 rotated multimodal as 119891

9sim11989114 and

composite functions as 11989115sim11989116 Table 1 gives the expression

of the benchmark function The detailed characteristics ofthese functions can be found in [1 46]

The introduced parameters are analyzed at first and thenthe proposed algorithm is compared with traditional CSAin the benchmark functions Finally comparisons betweenthe proposed algorithm and the state-of-the-art evolutionarycomputing models are represented Some discussions on theperformance analysis of RHCSA are also included It needs tobe noted that the same setup for the experiments is used forall the involved peers algorithms that is the comparison of allthe algorithms is under the same configuration of experimentsetup in this paper

41 Experimental Parameters Settings The experiments wereconducted on 16 benchmark functions The maximum num-ber of function evaluations (mFES) is set as 10000lowastD for10-D and 30-D situations respectively Each test is run 30independent times

There are quite few parameters introduced in the pro-posed algorithm For a fair comparison among CSAs theyare tested using the same setting of the parameters that isthe population size119873 is set to 30 and clone number119873

119888is set

to 4 as in [40] Furthermore there are two new parametersintroduced One is119898 which controls the direction of combi-natorial recombination and the other is recombination rateTables 2 and 3 give the experimental results of varying 119898 inthe 16 benchmark functions with 10-D and 30-D For conve-nience the experiment sets 4 proportions to the dimension asthe value to119898 which are [lceil(110)119863rceil lceil(15)119863rceil lceil(13)119863rceil119863]

The recombination operator could introduce diversityto the population through fusing information between ran-domly chosen parents When 119898 is small the generatedoffspring is much similar to one of the chosen parents andonly little dimension obtains information from both parentsWhen119898 is large the generated offspring tend to be the fusionof chosen parents That is to say m controls the offspringbeing much like one of the chosen parents or the fusion ofboth of them From Tables 2 and 3 we can find that the dif-ference among the varying setup ofm is quite small In amorecareful observation lceil(15)119863rceil and lceil(13)119863rceil are more appro-priate to compare with the other situations for both D = 10and D = 30 and lceil(13)119863rceil is even better It could be observedthat experimental results are not sensitive to the parameter119898 and a slightly larger119898may be beneficial for the algorithmIn view of the observation m = lceil(13)119863rceil is chosen in ourexperiments

There is another parameter named recombination rate119873119903 which controls the rate of recombination process which

will be implemented The higher the 119873119903 the more recom-

bination processes that will be executed In the early stagethe population is uniformly distributed in the search spaceand recombination of randomly chosen individuals will bringdiversity to the population As is known to all diversity isbeneficial for enhancing the search ability of the algorithmbut too much diversity will slow the convergence Tables4 and 5 represent the experimental results of varying therecombination rate

FromTables 4 and 5 it could be observed that RHCSAhasa quite robust performance with the varying recombinationrate By comparison 119873

119903which is equal to 07 achieves the

best performance and is adopted in the paper

42 Comparisons with Classic CSA The experiments areimplemented to check if the proposed operators are beneficialfor the performance of the algorithm Firstly recombinationoperator is added to the original CSA denoted as RCSA andthen the modified hypermutation is introduced as RHCSA

Table 6 shows the statistical results of CLONALG RCSAand RHCSA in optimizing the 16 test problems with D =10 based on 30 independent runs including the mean andstandard deviation From the table we can observe thatfor all these test instances the introduced recombinationoperator could obviously improve the performance of theCSA The reason is that the recombination operator couldbring diversity to enhance the search ability and avoidbeing trapped into the local optima The experiments resultson multimodal function 119891

3sim1198918could present the ability

of introduced recombination operator With the modifiedhypermutation operator the performance of the algorithmobtains a further improvement This is because both recom-bination and hypermutation adopt the difference of chosenindividuals to generate the candidate solution in the proposedalgorithm In the early stage the population is distributed inthe search space and the difference is relatively large whichis beneficial for global search and then with the iterationsgoing on the population converges to the optima and thedifference between individuals becomes smaller and smallerthat is the search process turns to be a local search It can

Computational Intelligence and Neuroscience 7

Table 1 Benchmark functions used in our experimental study

Name Test function D S 119891min

Sphere function 1198911(119909) =

119863

sum119894=1

1199092119894 1030 [minus100 100] 0

Rosenbrockrsquos function1198912 (119909) =119863minus1

sum119894=1

(100 (1199092119894minus 119909119894+1)2

+ (119909119894minus 1)2

) 1030 [minus2048 2048] 0

Ackleyrsquos function 1198913(119909) = minus20 exp(minus02radic 1

119863

119863

sum119894=1

1199092119894) minus exp( 1

119863

119863

sum119894=1

(2120587119909119894)) + 20 + 119890 1030 [minus32768 32768] 0

Griewanksrsquos function 1198914(119909) =

119863

sum119894=1

1199092119894

4000minus119863

prod119894=1

cos119909119894

radic119894+ 1 1030 [minus600 600] 0

Weierstrass function1198915(119909) =

119863

sum119894=1

119896maxsum119896=1

[119886119896 cos (2120587119887119896 (119909119894+ 05))] minus 119863

119896maxsum119896=1

119886119896 cos (2120587119887119896 sdot 05)1030 [minus05 05] 0

119886 = 05 119887 = 3 119896max = 20

Rastriginrsquos function 1198916(119909) =

119863

sum119894=1

1199092119894minus 10 cos (2120587119909

119894) + 10 1030 [minus512 512] 0

NoncontRas

1198917(119909) =

119863

sum119894=1

1199102119894minus 10 cos(2120587119910

119894) + 10

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt 05

round (2119909119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge 05

119894 = 1 2 119863

1030 [minus512 512] 0

Schwefelrsquos function 1198918(119909) = 4189829119863 minus

119863

sum119894=1

119909119894sin (1003816100381610038161003816119909119894

100381610038161003816100381605

) 1030 [minus500 500] 0

RotAckleyrsquos function 1198919(119909) = 119891

3(119910) 119910 = 119872 lowast 119909 1030 [minus32768 32768] 0

RotGriewanksrsquosfunction 119891

10(119909) = 119891

4(119910) 119910 = 119872 lowast 119909 1030 [minus600 600] 0

RotWeierstrassfunction 119891

11(119909) = 119891

5(119910) 119910 = 119872 lowast 119909 1030 [minus05 05] 0

RotRastriginrsquosfunction 119891

12(119909) = 119891

6(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

Rotnoncon Rasfunction 119891

13(119909) = 119891

7(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

RotSchwefelrsquosfunction

11989114(119909) = 4189829119863 minus

119863

sum119894=1

119885119894

119885119894=

119910119894sin (1003816100381610038161003816119910119894

100381610038161003816100381605

)10038161003816100381610038161199101198941003816100381610038161003816 le 500

0001 (10038161003816100381610038161199101198941003816100381610038161003816 minus 500)

2 10038161003816100381610038161199101198941003816100381610038161003816 ge 500

119894 = 1 2 119863

119910 = 119872 lowast (119909 minus 42096) + 42096

1030 [minus500 500] 0

Composition 1 11989115= 1198621198651 1030 [minus5 5] 0

Composition 2 11989116= 1198621198652 1030 [minus5 5] 0

be observed that RCSA surpass CSA and be surpassed byRHCSA at all the tested functions It can be concluded thatthe proposed recombination and hypermutation operatorsare effective improving the ability of the CSA

Table 7 shows the experimental results of the algorithmswhen 119863 = 30 Similar results are represented whichindicates that RHCSA is able to handle the high-dimensionaloptimization problems as well

43 Comparisons with the State-of-the-Art Algorithms Tocompare RHCSA with the state-of-the-art algorithms exper-imental results of seven representative evolutionary algo-rithms are listed in Tables 8 and 9 These algorithms areBaldwinian clonal selection algorithm (BCSA) [38] hybridlearning clonal selection algorithm (HLCSA) [40] orthogo-nal crossover based differential evolution (OXDE) [47] self-adaptive differential evolution (SaDE) [48] global and local

8 Computational Intelligence and Neuroscience

Table 2 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 10

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

13521119890 minus 180 plusmn 00000119890 minus 000 153784119890 minus 166 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 22773119890 minus 078 plusmn 49085119890 minus 078

1198912

86187119890 minus 004 plusmn 320378119890 minus 004 81377119890 minus 004 plusmn 30271119890 minus 004 89516119890 minus 004 plusmn 32039119890 minus 004 28971e minus 004 plusmn 36648e minus 004

1198913

52831119890 minus 015 plusmn 42156119890 minus 015 17763119890 minus 015 plusmn 17763119890 minus 015 88817e minus 016 plusmn 00000e minus 000 20724119890 minus 015 plusmn 20511119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

40175119890 minus 015 plusmn 13376119890 minus 015 30923e minus 015 plusmn 14969e minus 015 39292119890 minus 015 plusmn 11420119890 minus 015 42912119890 minus 015 plusmn 10827119890 minus 015

11989110

93401119890 minus 002 plusmn 15480119890 minus 002 18195119890 minus 002 plusmn 10245119890 minus 002 16937e minus 002 plusmn 16091e minus 002 55463119890 minus 002 plusmn 26635119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

24401119890 + 000 plusmn 10935119890 + 000 20556119890 + 000 plusmn 11601119890 + 000 15954e + 000 plusmn 15954e + 000 16638119890 + 000 plusmn 17026119890 + 000

11989113

81024119890 + 000 plusmn 56678119890 + 000 41385119890 + 000 plusmn 21123119890 + 000 40000e + 000 plusmn 21213e + 000 41773119890 + 000 plusmn 37530119890 + 000

11989114

75454119890 minus 008 plusmn 62091119890 minus 007 68712e minus 008 plusmn 70921e minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 92565119890 minus 008 plusmn 86635119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

90023119890 + 000 plusmn 78901119890 + 000 64971119890 + 000 plusmn 79297119890 + 000 52613e + 000 plusmn 50573e + 000 86977119890 + 000 plusmn 74630119890 + 000

Table 3 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 30

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

31129119890 minus 114 plusmn 44031119890 minus 114 82698119890 minus 110 plusmn 00000119890 minus 000 16994e minus 194 plusmn 00000e minus 000 66159119890 minus 108 plusmn 00000119890 minus 000

1198912

19281119890 minus 007 plusmn 89376119890 minus 007 74820119890 minus 007 plusmn 30968119890 minus 007 26862119890 minus 007 plusmn 60065119890 minus 007 74820e minus 008 plusmn 30968e minus 008

1198913

44415119890 minus 016 plusmn 25108119890 minus 016 89382119890 minus 016 plusmn 81571119890 minus 016 88817e minus 016 plusmn 00000e minus 000 21438119890 minus 015 plusmn 26596119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

32039119890 minus 015 plusmn 18147119890 minus 015 26532e minus 015 plusmn 13037e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 43241119890 minus 015 plusmn 29542119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

51900119890 + 001 plusmn 43580119890 + 001 41625119890 + 001 plusmn 91218119890 + 000 26201e + 001 plusmn 83454e + 000 29186119890 + 001 plusmn 67990119890 + 001

11989113

65345119890 + 001 plusmn 34082119890 + 001 51253119890 + 001 plusmn 23126119890 + 001 45502e + 001 plusmn 10609e + 001 48247119890 + 001 plusmn 10493119890 + 001

11989114

30358119890 minus 003 plusmn 38244119890 minus 003 42769119890 minus 003 plusmn 25513119890 minus 003 24186e minus 003 plusmn 53084e minus 003 34354119890 minus 003 plusmn 41538119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

94154119890 minus 002 plusmn 71725119890 minus 002 88757119890 minus 002 plusmn 14384119890 minus 002 88501e minus 002 plusmn 15560e minus 001 96583119890 minus 002 plusmn 17236119890 minus 001

Computational Intelligence and Neuroscience 9

Table 4 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 10

Function 01 05 07 1

1198911

38119119890 minus 119 plusmn 38510119890 minus 119 40163119890 minus 178 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 30122119890 minus 178 plusmn 00000119890 minus 000

1198912

29186119890 minus 004 plusmn 11966119890 minus 004 31115119890 minus 004 plusmn 105044119890 minus 004 89516119890 minus 004 plusmn 320393119890 minus 004 22544e minus 004 plusmn 15452e minus 004

1198913

19741119890 minus 015 plusmn 19767119890 minus 015 14807119890 minus 015 plusmn 19052119890 minus 015 88817e minus 016 plusmn 00000e minus 000 14135119890 minus 015 plusmn 19876119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

56305119890 minus 015 plusmn 13377119890 minus 015 79474119890 minus 015 plusmn 25291119890 minus 015 39292e minus 015 plusmn 11420e minus 015 41535119890 minus 015 plusmn 12103119890 minus 015

11989110

37179119890 minus 002 plusmn 18338119890 minus 002 31530e minus 002 plusmn 17649e minus 002 38363119890 minus 002 plusmn 377691119890 minus 002 37971119890 minus 002 plusmn 37825119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

23153119890 + 000 plusmn 13464119890 + 000 23031119890 + 000 plusmn 13465119890 + 000 15954119890 + 000 plusmn 15954119890 + 000 13685e + 000 plusmn 14679e + 000

11989113

40817119890 + 000 plusmn 55791119890 + 000 38531e + 000 plusmn 22514e + 000 40000119890 + 000 plusmn 21213119890 + 000 48073119890 + 000 plusmn 34243119890 + 000

11989114

55773e minus 008 plusmn 36818e minus 007 69679119890 minus 008 plusmn 59121119890 minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 70125119890 minus 008 plusmn 10013119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

86735119890 + 000 plusmn 83012119890 + 000 62013119890 + 000 plusmn 42016119890 + 000 52613e + 000 plusmn 50573e + 000 56012119890 + 000 plusmn 71232119890 + 000

Table 5 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 30

Function 01 05 07 1

1198911

16848119890 minus 121 plusmn 00000119890 minus 000 11232119890 minus 150 plusmn 00000119890 minus 000 16994119890 minus 194 plusmn 00000119890 minus 000 84243e minus 198 plusmn 00000e minus 000

1198912

33269119890 minus 007 plusmn 81537119890 minus 007 33493119890 minus 007 plusmn 31302119890 minus 007 26862e minus 007 plusmn 60065e minus 007 73295119890 minus 007 plusmn 30183119890 minus 007

1198913

96645119890 minus 016 plusmn 25121119890 minus 016 89312119890 minus 016 plusmn 28421119890 minus 016 88817e minus 016 plusmn 00000e minus 000 26645119890 minus 015 plusmn 27580119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

25600119890 minus 015 plusmn 22178119890 minus 015 24424e minus 015 plusmn 12352e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 25461119890 minus 015 plusmn 22735119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

46654119890 + 001 plusmn 41276119890 + 001 27521119890 + 001 plusmn 10387119890 + 000 26201e + 001 plusmn 83454e + 000 28012119890 + 001 plusmn 12293119890 + 001

11989113

51231119890 + 001 plusmn 13024119890 + 001 45613119890 + 001 plusmn 13450119890 + 001 45502e + 001 plusmn 10609e + 001 47234119890 + 001 plusmn 11249119890 + 001

11989114

26782119890 minus 003 plusmn 32933119890 minus 003 28334119890 minus 003 plusmn 62013119890 minus 003 24186e minus 003 plusmn 53084e minus 003 30187119890 minus 003 plusmn 87612119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

87011119890 minus 002 plusmn 56745119890 minus 002 87329119890 minus 002 plusmn 92581119890 minus 002 88501119890 minus 002 plusmn 15560119890 minus 001 60112e minus 002 plusmn 25832e minus 002

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

Submit your manuscripts athttpwwwhindawicom

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International Journal of

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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

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 7

Table 1 Benchmark functions used in our experimental study

Name Test function D S 119891min

Sphere function 1198911(119909) =

119863

sum119894=1

1199092119894 1030 [minus100 100] 0

Rosenbrockrsquos function1198912 (119909) =119863minus1

sum119894=1

(100 (1199092119894minus 119909119894+1)2

+ (119909119894minus 1)2

) 1030 [minus2048 2048] 0

Ackleyrsquos function 1198913(119909) = minus20 exp(minus02radic 1

119863

119863

sum119894=1

1199092119894) minus exp( 1

119863

119863

sum119894=1

(2120587119909119894)) + 20 + 119890 1030 [minus32768 32768] 0

Griewanksrsquos function 1198914(119909) =

119863

sum119894=1

1199092119894

4000minus119863

prod119894=1

cos119909119894

radic119894+ 1 1030 [minus600 600] 0

Weierstrass function1198915(119909) =

119863

sum119894=1

119896maxsum119896=1

[119886119896 cos (2120587119887119896 (119909119894+ 05))] minus 119863

119896maxsum119896=1

119886119896 cos (2120587119887119896 sdot 05)1030 [minus05 05] 0

119886 = 05 119887 = 3 119896max = 20

Rastriginrsquos function 1198916(119909) =

119863

sum119894=1

1199092119894minus 10 cos (2120587119909

119894) + 10 1030 [minus512 512] 0

NoncontRas

1198917(119909) =

119863

sum119894=1

1199102119894minus 10 cos(2120587119910

119894) + 10

119910119894=

119909119894

10038161003816100381610038161199091198941003816100381610038161003816 lt 05

round (2119909119894)

2

10038161003816100381610038161199091198941003816100381610038161003816 ge 05

119894 = 1 2 119863

1030 [minus512 512] 0

Schwefelrsquos function 1198918(119909) = 4189829119863 minus

119863

sum119894=1

119909119894sin (1003816100381610038161003816119909119894

100381610038161003816100381605

) 1030 [minus500 500] 0

RotAckleyrsquos function 1198919(119909) = 119891

3(119910) 119910 = 119872 lowast 119909 1030 [minus32768 32768] 0

RotGriewanksrsquosfunction 119891

10(119909) = 119891

4(119910) 119910 = 119872 lowast 119909 1030 [minus600 600] 0

RotWeierstrassfunction 119891

11(119909) = 119891

5(119910) 119910 = 119872 lowast 119909 1030 [minus05 05] 0

RotRastriginrsquosfunction 119891

12(119909) = 119891

6(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

Rotnoncon Rasfunction 119891

13(119909) = 119891

7(119910) 119910 = 119872 lowast 119909 1030 [minus512 512] 0

RotSchwefelrsquosfunction

11989114(119909) = 4189829119863 minus

119863

sum119894=1

119885119894

119885119894=

119910119894sin (1003816100381610038161003816119910119894

100381610038161003816100381605

)10038161003816100381610038161199101198941003816100381610038161003816 le 500

0001 (10038161003816100381610038161199101198941003816100381610038161003816 minus 500)

2 10038161003816100381610038161199101198941003816100381610038161003816 ge 500

119894 = 1 2 119863

119910 = 119872 lowast (119909 minus 42096) + 42096

1030 [minus500 500] 0

Composition 1 11989115= 1198621198651 1030 [minus5 5] 0

Composition 2 11989116= 1198621198652 1030 [minus5 5] 0

be observed that RCSA surpass CSA and be surpassed byRHCSA at all the tested functions It can be concluded thatthe proposed recombination and hypermutation operatorsare effective improving the ability of the CSA

Table 7 shows the experimental results of the algorithmswhen 119863 = 30 Similar results are represented whichindicates that RHCSA is able to handle the high-dimensionaloptimization problems as well

43 Comparisons with the State-of-the-Art Algorithms Tocompare RHCSA with the state-of-the-art algorithms exper-imental results of seven representative evolutionary algo-rithms are listed in Tables 8 and 9 These algorithms areBaldwinian clonal selection algorithm (BCSA) [38] hybridlearning clonal selection algorithm (HLCSA) [40] orthogo-nal crossover based differential evolution (OXDE) [47] self-adaptive differential evolution (SaDE) [48] global and local

8 Computational Intelligence and Neuroscience

Table 2 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 10

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

13521119890 minus 180 plusmn 00000119890 minus 000 153784119890 minus 166 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 22773119890 minus 078 plusmn 49085119890 minus 078

1198912

86187119890 minus 004 plusmn 320378119890 minus 004 81377119890 minus 004 plusmn 30271119890 minus 004 89516119890 minus 004 plusmn 32039119890 minus 004 28971e minus 004 plusmn 36648e minus 004

1198913

52831119890 minus 015 plusmn 42156119890 minus 015 17763119890 minus 015 plusmn 17763119890 minus 015 88817e minus 016 plusmn 00000e minus 000 20724119890 minus 015 plusmn 20511119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

40175119890 minus 015 plusmn 13376119890 minus 015 30923e minus 015 plusmn 14969e minus 015 39292119890 minus 015 plusmn 11420119890 minus 015 42912119890 minus 015 plusmn 10827119890 minus 015

11989110

93401119890 minus 002 plusmn 15480119890 minus 002 18195119890 minus 002 plusmn 10245119890 minus 002 16937e minus 002 plusmn 16091e minus 002 55463119890 minus 002 plusmn 26635119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

24401119890 + 000 plusmn 10935119890 + 000 20556119890 + 000 plusmn 11601119890 + 000 15954e + 000 plusmn 15954e + 000 16638119890 + 000 plusmn 17026119890 + 000

11989113

81024119890 + 000 plusmn 56678119890 + 000 41385119890 + 000 plusmn 21123119890 + 000 40000e + 000 plusmn 21213e + 000 41773119890 + 000 plusmn 37530119890 + 000

11989114

75454119890 minus 008 plusmn 62091119890 minus 007 68712e minus 008 plusmn 70921e minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 92565119890 minus 008 plusmn 86635119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

90023119890 + 000 plusmn 78901119890 + 000 64971119890 + 000 plusmn 79297119890 + 000 52613e + 000 plusmn 50573e + 000 86977119890 + 000 plusmn 74630119890 + 000

Table 3 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 30

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

31129119890 minus 114 plusmn 44031119890 minus 114 82698119890 minus 110 plusmn 00000119890 minus 000 16994e minus 194 plusmn 00000e minus 000 66159119890 minus 108 plusmn 00000119890 minus 000

1198912

19281119890 minus 007 plusmn 89376119890 minus 007 74820119890 minus 007 plusmn 30968119890 minus 007 26862119890 minus 007 plusmn 60065119890 minus 007 74820e minus 008 plusmn 30968e minus 008

1198913

44415119890 minus 016 plusmn 25108119890 minus 016 89382119890 minus 016 plusmn 81571119890 minus 016 88817e minus 016 plusmn 00000e minus 000 21438119890 minus 015 plusmn 26596119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

32039119890 minus 015 plusmn 18147119890 minus 015 26532e minus 015 plusmn 13037e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 43241119890 minus 015 plusmn 29542119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

51900119890 + 001 plusmn 43580119890 + 001 41625119890 + 001 plusmn 91218119890 + 000 26201e + 001 plusmn 83454e + 000 29186119890 + 001 plusmn 67990119890 + 001

11989113

65345119890 + 001 plusmn 34082119890 + 001 51253119890 + 001 plusmn 23126119890 + 001 45502e + 001 plusmn 10609e + 001 48247119890 + 001 plusmn 10493119890 + 001

11989114

30358119890 minus 003 plusmn 38244119890 minus 003 42769119890 minus 003 plusmn 25513119890 minus 003 24186e minus 003 plusmn 53084e minus 003 34354119890 minus 003 plusmn 41538119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

94154119890 minus 002 plusmn 71725119890 minus 002 88757119890 minus 002 plusmn 14384119890 minus 002 88501e minus 002 plusmn 15560e minus 001 96583119890 minus 002 plusmn 17236119890 minus 001

Computational Intelligence and Neuroscience 9

Table 4 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 10

Function 01 05 07 1

1198911

38119119890 minus 119 plusmn 38510119890 minus 119 40163119890 minus 178 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 30122119890 minus 178 plusmn 00000119890 minus 000

1198912

29186119890 minus 004 plusmn 11966119890 minus 004 31115119890 minus 004 plusmn 105044119890 minus 004 89516119890 minus 004 plusmn 320393119890 minus 004 22544e minus 004 plusmn 15452e minus 004

1198913

19741119890 minus 015 plusmn 19767119890 minus 015 14807119890 minus 015 plusmn 19052119890 minus 015 88817e minus 016 plusmn 00000e minus 000 14135119890 minus 015 plusmn 19876119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

56305119890 minus 015 plusmn 13377119890 minus 015 79474119890 minus 015 plusmn 25291119890 minus 015 39292e minus 015 plusmn 11420e minus 015 41535119890 minus 015 plusmn 12103119890 minus 015

11989110

37179119890 minus 002 plusmn 18338119890 minus 002 31530e minus 002 plusmn 17649e minus 002 38363119890 minus 002 plusmn 377691119890 minus 002 37971119890 minus 002 plusmn 37825119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

23153119890 + 000 plusmn 13464119890 + 000 23031119890 + 000 plusmn 13465119890 + 000 15954119890 + 000 plusmn 15954119890 + 000 13685e + 000 plusmn 14679e + 000

11989113

40817119890 + 000 plusmn 55791119890 + 000 38531e + 000 plusmn 22514e + 000 40000119890 + 000 plusmn 21213119890 + 000 48073119890 + 000 plusmn 34243119890 + 000

11989114

55773e minus 008 plusmn 36818e minus 007 69679119890 minus 008 plusmn 59121119890 minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 70125119890 minus 008 plusmn 10013119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

86735119890 + 000 plusmn 83012119890 + 000 62013119890 + 000 plusmn 42016119890 + 000 52613e + 000 plusmn 50573e + 000 56012119890 + 000 plusmn 71232119890 + 000

Table 5 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 30

Function 01 05 07 1

1198911

16848119890 minus 121 plusmn 00000119890 minus 000 11232119890 minus 150 plusmn 00000119890 minus 000 16994119890 minus 194 plusmn 00000119890 minus 000 84243e minus 198 plusmn 00000e minus 000

1198912

33269119890 minus 007 plusmn 81537119890 minus 007 33493119890 minus 007 plusmn 31302119890 minus 007 26862e minus 007 plusmn 60065e minus 007 73295119890 minus 007 plusmn 30183119890 minus 007

1198913

96645119890 minus 016 plusmn 25121119890 minus 016 89312119890 minus 016 plusmn 28421119890 minus 016 88817e minus 016 plusmn 00000e minus 000 26645119890 minus 015 plusmn 27580119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

25600119890 minus 015 plusmn 22178119890 minus 015 24424e minus 015 plusmn 12352e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 25461119890 minus 015 plusmn 22735119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

46654119890 + 001 plusmn 41276119890 + 001 27521119890 + 001 plusmn 10387119890 + 000 26201e + 001 plusmn 83454e + 000 28012119890 + 001 plusmn 12293119890 + 001

11989113

51231119890 + 001 plusmn 13024119890 + 001 45613119890 + 001 plusmn 13450119890 + 001 45502e + 001 plusmn 10609e + 001 47234119890 + 001 plusmn 11249119890 + 001

11989114

26782119890 minus 003 plusmn 32933119890 minus 003 28334119890 minus 003 plusmn 62013119890 minus 003 24186e minus 003 plusmn 53084e minus 003 30187119890 minus 003 plusmn 87612119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

87011119890 minus 002 plusmn 56745119890 minus 002 87329119890 minus 002 plusmn 92581119890 minus 002 88501119890 minus 002 plusmn 15560119890 minus 001 60112e minus 002 plusmn 25832e minus 002

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

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8 Computational Intelligence and Neuroscience

Table 2 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 10

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

13521119890 minus 180 plusmn 00000119890 minus 000 153784119890 minus 166 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 22773119890 minus 078 plusmn 49085119890 minus 078

1198912

86187119890 minus 004 plusmn 320378119890 minus 004 81377119890 minus 004 plusmn 30271119890 minus 004 89516119890 minus 004 plusmn 32039119890 minus 004 28971e minus 004 plusmn 36648e minus 004

1198913

52831119890 minus 015 plusmn 42156119890 minus 015 17763119890 minus 015 plusmn 17763119890 minus 015 88817e minus 016 plusmn 00000e minus 000 20724119890 minus 015 plusmn 20511119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

40175119890 minus 015 plusmn 13376119890 minus 015 30923e minus 015 plusmn 14969e minus 015 39292119890 minus 015 plusmn 11420119890 minus 015 42912119890 minus 015 plusmn 10827119890 minus 015

11989110

93401119890 minus 002 plusmn 15480119890 minus 002 18195119890 minus 002 plusmn 10245119890 minus 002 16937e minus 002 plusmn 16091e minus 002 55463119890 minus 002 plusmn 26635119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

24401119890 + 000 plusmn 10935119890 + 000 20556119890 + 000 plusmn 11601119890 + 000 15954e + 000 plusmn 15954e + 000 16638119890 + 000 plusmn 17026119890 + 000

11989113

81024119890 + 000 plusmn 56678119890 + 000 41385119890 + 000 plusmn 21123119890 + 000 40000e + 000 plusmn 21213e + 000 41773119890 + 000 plusmn 37530119890 + 000

11989114

75454119890 minus 008 plusmn 62091119890 minus 007 68712e minus 008 plusmn 70921e minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 92565119890 minus 008 plusmn 86635119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

90023119890 + 000 plusmn 78901119890 + 000 64971119890 + 000 plusmn 79297119890 + 000 52613e + 000 plusmn 50573e + 000 86977119890 + 000 plusmn 74630119890 + 000

Table 3 Results (mean plusmn std) of RHCSA with varying sampling points of combinatorial recombination in 16 benchmark functions with119863 = 30

Function 119898 = lceil(110)119863rceil 119898 = lceil(15)119863rceil 119898 = lceil(13)119863rceil 119898 = 119863

1198911

31129119890 minus 114 plusmn 44031119890 minus 114 82698119890 minus 110 plusmn 00000119890 minus 000 16994e minus 194 plusmn 00000e minus 000 66159119890 minus 108 plusmn 00000119890 minus 000

1198912

19281119890 minus 007 plusmn 89376119890 minus 007 74820119890 minus 007 plusmn 30968119890 minus 007 26862119890 minus 007 plusmn 60065119890 minus 007 74820e minus 008 plusmn 30968e minus 008

1198913

44415119890 minus 016 plusmn 25108119890 minus 016 89382119890 minus 016 plusmn 81571119890 minus 016 88817e minus 016 plusmn 00000e minus 000 21438119890 minus 015 plusmn 26596119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

32039119890 minus 015 plusmn 18147119890 minus 015 26532e minus 015 plusmn 13037e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 43241119890 minus 015 plusmn 29542119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

51900119890 + 001 plusmn 43580119890 + 001 41625119890 + 001 plusmn 91218119890 + 000 26201e + 001 plusmn 83454e + 000 29186119890 + 001 plusmn 67990119890 + 001

11989113

65345119890 + 001 plusmn 34082119890 + 001 51253119890 + 001 plusmn 23126119890 + 001 45502e + 001 plusmn 10609e + 001 48247119890 + 001 plusmn 10493119890 + 001

11989114

30358119890 minus 003 plusmn 38244119890 minus 003 42769119890 minus 003 plusmn 25513119890 minus 003 24186e minus 003 plusmn 53084e minus 003 34354119890 minus 003 plusmn 41538119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

94154119890 minus 002 plusmn 71725119890 minus 002 88757119890 minus 002 plusmn 14384119890 minus 002 88501e minus 002 plusmn 15560e minus 001 96583119890 minus 002 plusmn 17236119890 minus 001

Computational Intelligence and Neuroscience 9

Table 4 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 10

Function 01 05 07 1

1198911

38119119890 minus 119 plusmn 38510119890 minus 119 40163119890 minus 178 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 30122119890 minus 178 plusmn 00000119890 minus 000

1198912

29186119890 minus 004 plusmn 11966119890 minus 004 31115119890 minus 004 plusmn 105044119890 minus 004 89516119890 minus 004 plusmn 320393119890 minus 004 22544e minus 004 plusmn 15452e minus 004

1198913

19741119890 minus 015 plusmn 19767119890 minus 015 14807119890 minus 015 plusmn 19052119890 minus 015 88817e minus 016 plusmn 00000e minus 000 14135119890 minus 015 plusmn 19876119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

56305119890 minus 015 plusmn 13377119890 minus 015 79474119890 minus 015 plusmn 25291119890 minus 015 39292e minus 015 plusmn 11420e minus 015 41535119890 minus 015 plusmn 12103119890 minus 015

11989110

37179119890 minus 002 plusmn 18338119890 minus 002 31530e minus 002 plusmn 17649e minus 002 38363119890 minus 002 plusmn 377691119890 minus 002 37971119890 minus 002 plusmn 37825119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

23153119890 + 000 plusmn 13464119890 + 000 23031119890 + 000 plusmn 13465119890 + 000 15954119890 + 000 plusmn 15954119890 + 000 13685e + 000 plusmn 14679e + 000

11989113

40817119890 + 000 plusmn 55791119890 + 000 38531e + 000 plusmn 22514e + 000 40000119890 + 000 plusmn 21213119890 + 000 48073119890 + 000 plusmn 34243119890 + 000

11989114

55773e minus 008 plusmn 36818e minus 007 69679119890 minus 008 plusmn 59121119890 minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 70125119890 minus 008 plusmn 10013119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

86735119890 + 000 plusmn 83012119890 + 000 62013119890 + 000 plusmn 42016119890 + 000 52613e + 000 plusmn 50573e + 000 56012119890 + 000 plusmn 71232119890 + 000

Table 5 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 30

Function 01 05 07 1

1198911

16848119890 minus 121 plusmn 00000119890 minus 000 11232119890 minus 150 plusmn 00000119890 minus 000 16994119890 minus 194 plusmn 00000119890 minus 000 84243e minus 198 plusmn 00000e minus 000

1198912

33269119890 minus 007 plusmn 81537119890 minus 007 33493119890 minus 007 plusmn 31302119890 minus 007 26862e minus 007 plusmn 60065e minus 007 73295119890 minus 007 plusmn 30183119890 minus 007

1198913

96645119890 minus 016 plusmn 25121119890 minus 016 89312119890 minus 016 plusmn 28421119890 minus 016 88817e minus 016 plusmn 00000e minus 000 26645119890 minus 015 plusmn 27580119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

25600119890 minus 015 plusmn 22178119890 minus 015 24424e minus 015 plusmn 12352e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 25461119890 minus 015 plusmn 22735119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

46654119890 + 001 plusmn 41276119890 + 001 27521119890 + 001 plusmn 10387119890 + 000 26201e + 001 plusmn 83454e + 000 28012119890 + 001 plusmn 12293119890 + 001

11989113

51231119890 + 001 plusmn 13024119890 + 001 45613119890 + 001 plusmn 13450119890 + 001 45502e + 001 plusmn 10609e + 001 47234119890 + 001 plusmn 11249119890 + 001

11989114

26782119890 minus 003 plusmn 32933119890 minus 003 28334119890 minus 003 plusmn 62013119890 minus 003 24186e minus 003 plusmn 53084e minus 003 30187119890 minus 003 plusmn 87612119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

87011119890 minus 002 plusmn 56745119890 minus 002 87329119890 minus 002 plusmn 92581119890 minus 002 88501119890 minus 002 plusmn 15560119890 minus 001 60112e minus 002 plusmn 25832e minus 002

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

Submit your manuscripts athttpwwwhindawicom

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International Journal of

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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

Computational Intelligence and Neuroscience 9

Table 4 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 10

Function 01 05 07 1

1198911

38119119890 minus 119 plusmn 38510119890 minus 119 40163119890 minus 178 plusmn 00000119890 minus 000 72585e minus 195 plusmn 00000e minus 000 30122119890 minus 178 plusmn 00000119890 minus 000

1198912

29186119890 minus 004 plusmn 11966119890 minus 004 31115119890 minus 004 plusmn 105044119890 minus 004 89516119890 minus 004 plusmn 320393119890 minus 004 22544e minus 004 plusmn 15452e minus 004

1198913

19741119890 minus 015 plusmn 19767119890 minus 015 14807119890 minus 015 plusmn 19052119890 minus 015 88817e minus 016 plusmn 00000e minus 000 14135119890 minus 015 plusmn 19876119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

56305119890 minus 015 plusmn 13377119890 minus 015 79474119890 minus 015 plusmn 25291119890 minus 015 39292e minus 015 plusmn 11420e minus 015 41535119890 minus 015 plusmn 12103119890 minus 015

11989110

37179119890 minus 002 plusmn 18338119890 minus 002 31530e minus 002 plusmn 17649e minus 002 38363119890 minus 002 plusmn 377691119890 minus 002 37971119890 minus 002 plusmn 37825119890 minus 002

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

23153119890 + 000 plusmn 13464119890 + 000 23031119890 + 000 plusmn 13465119890 + 000 15954119890 + 000 plusmn 15954119890 + 000 13685e + 000 plusmn 14679e + 000

11989113

40817119890 + 000 plusmn 55791119890 + 000 38531e + 000 plusmn 22514e + 000 40000119890 + 000 plusmn 21213119890 + 000 48073119890 + 000 plusmn 34243119890 + 000

11989114

55773e minus 008 plusmn 36818e minus 007 69679119890 minus 008 plusmn 59121119890 minus 007 98137119890 minus 008 plusmn 77851119890 minus 007 70125119890 minus 008 plusmn 10013119890 minus 007

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

86735119890 + 000 plusmn 83012119890 + 000 62013119890 + 000 plusmn 42016119890 + 000 52613e + 000 plusmn 50573e + 000 56012119890 + 000 plusmn 71232119890 + 000

Table 5 Results (mean plusmn std) of RHCSA with varying recombination rate in 16 benchmark functions with 119863 = 30

Function 01 05 07 1

1198911

16848119890 minus 121 plusmn 00000119890 minus 000 11232119890 minus 150 plusmn 00000119890 minus 000 16994119890 minus 194 plusmn 00000119890 minus 000 84243e minus 198 plusmn 00000e minus 000

1198912

33269119890 minus 007 plusmn 81537119890 minus 007 33493119890 minus 007 plusmn 31302119890 minus 007 26862e minus 007 plusmn 60065e minus 007 73295119890 minus 007 plusmn 30183119890 minus 007

1198913

96645119890 minus 016 plusmn 25121119890 minus 016 89312119890 minus 016 plusmn 28421119890 minus 016 88817e minus 016 plusmn 00000e minus 000 26645119890 minus 015 plusmn 27580119890 minus 015

1198914

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198915

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198916

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198917

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198918

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

1198919

25600119890 minus 015 plusmn 22178119890 minus 015 24424e minus 015 plusmn 12352e minus 015 39092119890 minus 015 plusmn 19459119890 minus 015 25461119890 minus 015 plusmn 22735119890 minus 015

11989110

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989111

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989112

46654119890 + 001 plusmn 41276119890 + 001 27521119890 + 001 plusmn 10387119890 + 000 26201e + 001 plusmn 83454e + 000 28012119890 + 001 plusmn 12293119890 + 001

11989113

51231119890 + 001 plusmn 13024119890 + 001 45613119890 + 001 plusmn 13450119890 + 001 45502e + 001 plusmn 10609e + 001 47234119890 + 001 plusmn 11249119890 + 001

11989114

26782119890 minus 003 plusmn 32933119890 minus 003 28334119890 minus 003 plusmn 62013119890 minus 003 24186e minus 003 plusmn 53084e minus 003 30187119890 minus 003 plusmn 87612119890 minus 003

11989115

0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

11989116

87011119890 minus 002 plusmn 56745119890 minus 002 87329119890 minus 002 plusmn 92581119890 minus 002 88501119890 minus 002 plusmn 15560119890 minus 001 60112e minus 002 plusmn 25832e minus 002

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Applied Computational Intelligence and Soft Computing

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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ArtificialNeural Systems

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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

10 Computational Intelligence and Neuroscience

Table 6 Results of traditional clonal selection algorithm and RHCSA when119863 = 10

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

50414119890 minus 008 44098119890 minus 008 69076119890 minus 056 95731119890 minus 055 72585e minus 195 0

1198912

55057119890 + 000 27914119890 + 000 45314119890 minus 003 47750119890 minus 002 89516e minus 004 32039119890 minus 004

1198913

28532119890 minus 003 85712119890 minus 003 23976119890 minus 015 57314119890 minus 015 88817e minus 016 0

1198914

26092119890 minus 002 19092119890 minus 002 0 0 0 0

1198915

12569119890 minus 002 44216119890 minus 003 0 0 0 0

1198916

84141119890 + 000 24848119890 + 000 0 0 0 0

1198917

57681119890 + 000 13571119890 + 000 0 0 0 0

1198918

36436119890 + 002 97648119890 + 001 0 0 0 0

1198919

10619119890 + 000 79728119890 minus 001 13592119890 minus 008 10975119890 minus 008 39292e minus 015 11420119890 minus 015

11989110

42974119890 minus 001 10264119890 minus 001 25301119890 minus 001 11356119890 minus 001 16937e minus 002 16091119890 minus 002

11989111

60837119890 + 000 15496119890 + 000 52403119890 minus 008 12560119890 minus 008 0 0

11989112

42193119890 + 001 54173119890 + 000 36021119890 + 001 12541119890 + 001 27006e + 000 15954119890 + 000

11989113

43672119890 + 001 55570119890 + 000 23702119890 + 001 25091119890 + 001 40000e + 000 21213119890 + 000

11989114

19305119890 + 003 23358119890 + 002 27201119890 minus 003 61034119890 minus 003 98137e minus 008 77851119890 minus 007

11989115

45354119890 + 001 32450119890 + 001 10276119890 minus 033 18734119890 minus 033 0 0

11989116

51701119890 + 001 17347119890 + 001 52613119890 + 000 50573119890 + 000 52613e + 000 50573119890 + 000

Table 7 Results of traditional clonal selection algorithm and RHCSA when119863 = 30

ALGs Clonal selection algorithm (CSA) CSA with recombination RHCSAMean Std Mean Std Mean Std

1198911

64595119890 minus 002 29504119890 minus 002 14668119890 minus 105 30015119890 minus 105 16994e minus 194 0

1198912

27748119890 + 001 21866119890 + 000 18524119890 minus 001 26531119890 minus 001 26862e minus 007 60065119890 minus 007

1198913

31589119890 minus 001 18716119890 minus 001 19187e minus 017 38302119890 minus 017 88817119890 minus 016 0

1198914

16746119890 minus 001 43211119890 minus 002 0 0 0 0

1198915

36752119890 + 001 46907119890 + 000 0 0 0 0

1198916

36752119890 + 001 46907119890 + 000 0 0 0 0

1198917

23935119890 + 001 25226119890 + 000 0 0 0 0

1198918

16279119890 + 003 17813119890 + 002 0 0 0 0

1198919

38700119890 + 000 32556119890 minus 001 18245119890 minus 005 13675119890 minus 005 39092e minus 015 19459119890 minus 015

11989110

83333119890 minus 001 75256119890 minus 002 29176119890 minus 028 13613119890 minus 028 0 0

11989111

33702119890 + 001 27162119890 + 000 11680119890 minus 030 19130119890 minus 030 0 0

11989112

25929119890 + 002 14658119890 + 001 43898119890 + 001 20666119890 + 000 26201e + 001 83454119890 + 000

11989113

25762119890 + 002 21376119890 + 001 54715119890 + 001 17261119890 + 001 45502e + 001 10609119890 + 001

11989114

87875119890 + 003 32136119890 + 002 25296119890 minus 002 16225119890 minus 002 24186e minus 003 53084119890 minus 003

11989115

44892119890 + 001 95942119890 + 000 33097119890 minus 033 16468119890 minus 033 0 0

11989116

39889119890 + 001 43023119890 + 000 25177119890 minus 001 11168119890 minus 001 88501e minus 002 15560119890 minus 001

real-coded genetic algorithm (GL-25) [49] and comprehen-sive learning particle swarm optimization (CLPSO) [1]

Table 8 presents the mean values and standard deviationsof the seven algorithms on the 16 test functions with119863 = 10where the best results are shown in bold face First of allRHCSA performs the best on the 6 unrotated multimodalfunctions 119891

3sim1198918 It could exactly locate the global optima in

7 functions 1198914sim1198918 11989111 and 119891

15 In particular it is observed

that RHCSA surpasses all the other algorithms on functions1198913and11989112with great superiorityMoreover RHCSAHLCSA

BCSA SaDE and CLPSO can obtain the global minima0 on functions 119891

5 1198916 1198917 and 119891

8 RHCSA HLCSA BCSA

SaDE and OXDE get the global optimum of function 11989111

and RHCSA HLCSA SaDE OXDE and GL-25 find theoptimum on function 119891

15 Though RHCSA cannot get the

best results on functions 1198911 1198919 11989110 11989113 and 119891

16 it still gets

the comparable solutions with respect to the best resultsobtained by the other algorithms To the composite problems11989115 RHCSA reaches the global best as HLCSA OXDE SaDE

and GL-25 To the function 11989116 RHCSA surpasses the other

algorithms except for HLCSASimilar results can be observed from Table 9 when 119863

equals 30 RHCSA can consistently obtain good results whichare even slightly better than those on 10-119863 problems RHCSA

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

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

Computational Intelligence and Neuroscience 11

Table 8 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 10

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 72585119890 minus 195 plusmn 0000119890 minus 000 89516119890 minus 004 plusmn 32039119890 minus 004 88817e minus 016 plusmn 0000e minus 000 0 plusmn 0HLCSA 42228119890 minus 053 plusmn 99892119890 minus 053 39087e minus 028 plusmn 66496e minus 028 25757119890 minus 015 plusmn 48648119890 minus 016 0 plusmn 0BCSA 11694119890 minus 037 plusmn 13065119890 minus 037 18521119890 minus 001 plusmn 72808119890 minus 001 26645119890 minus 015 plusmn 00000119890 minus 000 14146119890 minus 001 plusmn 13744119890 minus 001

OXDE 45059119890 minus 056 plusmn 74966119890 minus 056 10265119890 minus 026 plusmn 20786119890 minus 026 20724119890 minus 015 plusmn 13467119890 minus 015 99330119890 minus 004 plusmn 10673119890 minus 002

SaDE 14451119890 minus 176 plusmn 00000119890 minus 000 20249119890 + 000 plusmn 74832119890 minus 001 50330119890 minus 015 plusmn 11109119890 minus 015 18074119890 minus 003 plusmn 38251119890 minus 003

GL-25 10771e minus 321 plusmn 00000e minus 000 20956119890 + 000 plusmn 63579119890 minus 001 27830119890 minus 015 plusmn 14703119890 minus 015 12134119890 minus 002 plusmn 10199119890 minus 002

CLPSO 18154119890 minus 041 plusmn 30360119890 minus 041 21490119890 + 000 plusmn 12450119890 + 000 39672119890 minus 015 plusmn 17413119890 minus 015 74577119890 minus 006 plusmn 21864119890 minus 005

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 0 plusmn 0 66331119890 minus 002 plusmn 25243119890 minus 001 56667119890 minus 001 plusmn 72793119890 minus 001 0 plusmn 0SaDE 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0GL-25 73315119890 minus 007 plusmn 22405119890 minus 006 19633119890 + 000 plusmn 11774119890 + 000 56336119890 + 000 plusmn 12724119890 + 000 28952119890 + 002 plusmn 19959119890 + 002

CLPSO 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39292119890 minus 015 plusmn 11420119890 minus 015 16937119890 minus 002 plusmn 16091119890 minus 002 0 plusmn 0 15954e + 000 plusmn 15954e + 000HLCSA 35527119890 minus 015 plusmn 00000119890 minus 000 28802119890 minus 002 plusmn 19129119890 minus 002 0 plusmn 0 42783119890 + 000 plusmn 20095119890 + 000

BCSA 35527119890 minus 015 plusmn 00000119890 minus 000 32715119890 minus 001 plusmn 17104119890 minus 001 0 plusmn 0 62881119890 + 001 plusmn 16334119890 + 001

OXDE 31974119890 minus 015 plusmn 10840119890 minus 015 49045119890 minus 001 plusmn 27411119890 minus 002 0 plusmn 0 37808119890 + 000 plusmn 19094119890 + 000

SaDE 94739e minus 016 plusmn 15979e minus 015 13704119890 minus 002 plusmn 16048119890 minus 002 0 plusmn 0 39135119890 + 000 plusmn 14295119890 + 000

GL-25 35527119890 minus 015 plusmn 00000119890 minus 000 10545e minus 002 plusmn 10858e minus 002 21771119890 minus 004 plusmn 47010119890 minus 004 33619119890 + 000 plusmn 22861119890 + 000

CLPSO 38606119890 minus 014 plusmn 58665119890 minus 014 28592119890 minus 002 plusmn 15307119890 minus 002 14403119890 minus 002 plusmn 12235119890 minus 002 41634119890 + 000 plusmn 90362119890 minus 001

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 40000119890 + 000 plusmn 21213119890 + 000 98137119890 minus 008 plusmn 77851119890 minus 007 0 plusmn 0 52613119890 + 000 plusmn 50573119890 + 000

HLCSA 42442119890 + 000 plusmn 26469119890 + 000 0 plusmn 0 0 plusmn 0 46177e minus 001 plusmn 81247e minus 001BCSA 65935119890 + 001 plusmn 10618119890 + 001 26391119890 + 002 plusmn 79682119890 + 001 43387119890 minus 031 plusmn 65797119890 minus 031 87291119890 + 000 plusmn 17669119890 + 001

OXDE 30956119890 + 000 plusmn 11245119890 + 000 15792119890 + 001 plusmn 67669119890 + 001 0 plusmn 0 10047119890 + 001 plusmn 30498119890 + 001

SaDE 39534119890 + 000 plusmn 19500119890 + 000 20681119890 + 002 plusmn 86302119890 + 001 0 plusmn 0 18019119890 + 001 plusmn 37308119890 + 001

GL-25 73657119890 + 000 plusmn 22262119890 + 000 52254119890 + 002 plusmn 17963119890 + 002 0 plusmn 0 90000119890 + 001 plusmn 30513119890 + 001

CLPSO 20254e + 000 plusmn 10621e + 000 31281119890 + 002 plusmn 15723119890 + 002 23104119890 minus 002 plusmn 65636119890 minus 002 60233119890 + 000 plusmn 40698119890 + 000

achieves the best performance on functions 1198913 1198914 1198915 1198916 1198917

1198918 11989110 11989111 11989115 and 119891

16compared with the other algorithms

EvenRHCSA cannot find the best results on functions119891911989112

and 11989113 its obtained results are very close to the best results

obtained by the other algorithms For example the best resulton function 119891

12(16549119890 + 001 plusmn 44609119890 + 000) is found

by OXDE while a comparable result obtained by RHCSA is26201119890 + 001 plusmn 83454119890 + 000 RHCSA greatly improves theresults on functions 119891

3 11989114 and 119891

16 It obtains comparatively

good results on functions 1198919 11989112 and 119891

13 It can be observed

that the scalability of RHCSA is pretty well when dealing withhigh-dimensional optimization problems

5 ConclusionThis paper presents a new clonal selection based algorithm inwhich combinatorial recombination and hypermutation areintroduced to enhance the search ability The proposed algo-rithm is tested on 16 commonly used benchmark functionswith unimodal multimodal rotation and composite charac-teristics Firstly the CSAs with and without recombination

12 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

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 Computational Intelligence and Neuroscience

Table 9 Results (mean plusmn std) of RHCSA and other state-of-the-art evolutionary algorithms when119863 = 30

(a)

ALGs 1198911

1198912

1198913

1198914

RHCSA 16994119890 minus 194 plusmn 00000119890 minus 000 74820119890 minus 001 plusmn 30968119890 minus 001 88817e minus 016 plusmn 88817e minus 016 0 plusmn 0HLCSA 71289119890 minus 066 plusmn 20169119890 minus 065 11617e minus 015 plusmn 55448e minus 015 26645119890 minus 015 plusmn 00000119890 minus 000 0 plusmn 0BCSA 29665119890 minus 025 plusmn 84182119890 minus 025 19018119890 + 001 plusmn 26925119890 + 000 15614119890 minus 013 plusmn 39345119890 minus 013 0 plusmn 0OXDE 48545119890 minus 059 plusmn 12064119890 minus 058 26577119890 minus 001 plusmn 10114119890 + 000 26645119890 minus 001 plusmn 00000119890 minus 000 28730119890 minus 003 plusmn 56727119890 minus 003

SaDE 91236119890 minus 150 plusmn 44538119890 minus 149 21973119890 + 001 plusmn 10132119890 + 000 77383119890 minus 001 plusmn 60009119890 minus 001 11999119890 minus 002 plusmn 19462119890 minus 002

GL-25 53539e minus 228 plusmn 00000e minus 000 20832119890 + 001 plusmn 86842119890 minus 001 84969119890 minus 014 plusmn 17664119890 minus 013 97959119890 minus 015 plusmn 32264119890 minus 014

CLPSO 19761119890 minus 029 plusmn 15041119890 minus 029 17605119890 + 001 plusmn 36364119890 + 000 18415119890 minus 014 plusmn 30495119890 minus 015 11102119890 minus 016 plusmn 32467119890 minus 016

(b)

ALGs 1198915

1198916

1198917

1198918

RHCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0HLCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0BCSA 0 plusmn 0 0 plusmn 0 0 plusmn 0 0 plusmn 0OXDE 16214119890 minus 003 plusmn 65408119890 minus 003 94189119890 minus 000 plusmn 20859119890 minus 000 15100119890 + 001 plusmn 29868119890 + 000 39479119890 + 000 plusmn 21624119890 + 001

SaDE 95195119890 minus 002 plusmn 15798119890 minus 001 86230119890 minus 001 plusmn 89502119890 minus 001 63333119890 minus 001 plusmn 76489119890 minus 001 39479119890 + 001 plusmn 64747119890 + 001

GL-25 71724 minus 119890004 plusmn 48027119890 minus 004 23030119890 + 001 plusmn 84952119890 + 000 39096119890 + 001 plusmn 22071119890 + 001 35030119890 + 003 plusmn 68004119890 + 000

CLPSO 0 plusmn 0 0 plusmn 0 87634119890 minus 015 plusmn 13333119890 minus 014 0 plusmn 0

(c)

ALGs 1198919

11989110

11989111

11989112

RHCSA 39092119890 minus 015 plusmn 19459119890 minus 015 0 plusmn 0 0 plusmn 0 26201119890 + 001 plusmn 83454119890 + 000

HLCSA 35527e minus 015 plusmn 00000e minus 000 0 plusmn 0 0 plusmn 0 25471119890 + 001 plusmn 67663119890 + 000

BCSA 13086119890 minus 013 plusmn 34341119890 minus 013 12273119890 minus 002 plusmn 22389119890 minus 002 18516119890 minus 014 plusmn 48488119890 minus 014 32115119890 + 001 plusmn 90135119890 + 000

OXDE 35527e minus 015 plusmn 00000e minus 000 15612119890 minus 003 plusmn 32032119890 minus 003 14210119890 minus 001 plusmn 23765119890 minus 001 16549e + 001 plusmn 44609e + 000SaDE 11708119890 + 000 plusmn 65356119890 minus 001 13096119890 minus 002 plusmn 26787119890 minus 002 20504119890 + 000 plusmn 90887119890 minus 001 25050119890 + 001 plusmn 66415119890 + 000

GL-25 11416119890 minus 013 plusmn 16841119890 minus 013 42040119890 minus 015 plusmn 55414119890 minus 015 56795119890 minus 003 plusmn 26720119890 minus 003 29464119890 + 001 plusmn 22594119890 + 001

CLPSO 14501119890 minus 007 plusmn 70645119890 minus 007 44152119890 minus 007 plusmn 74331119890 minus 007 17977119890 + 000 plusmn 61941119890 minus 001 46287119890 + 001 plusmn 57149119890 + 000

(d)

ALGs 11989113

11989114

11989115

11989116

RHCSA 45502119890 + 001 plusmn 10609119890 + 001 24186e minus 003 plusmn 53084e minus 003 0 plusmn 0 88501e minus 002 plusmn 15560e minus 001HLCSA 47609119890 + 001 plusmn 15104119890 + 001 10663119890 + 003 plusmn 52886119890 + 002 0 plusmn 0 32212119890 + 000 plusmn 10133119890 + 000

BCSA 26912119890 + 001 plusmn 12152119890 + 001 26533119890 + 003 plusmn 53339119890 + 002 20259119890 minus 023 plusmn 63626119890 minus 023 18053119890 + 001 plusmn 18282119890 + 001

OXDE 17959e + 001 plusmn 51559e + 000 43428119890 + 001 plusmn 88341119890 + 001 33333119890 + 000 plusmn 18257119890 + 001 25992119890 + 000 plusmn 56332119890 minus 001

SaDE 22788119890 + 001 plusmn 66879119890 + 000 24742119890 + 003 plusmn 59013119890 + 002 11833119890 minus 031 plusmn 21659119890 minus 031 10208119890 + 001 plusmn 17851119890 + 001

GL-25 96862119890 + 001 plusmn 40914119890 + 001 32335119890 + 003 plusmn 59871119890 + 002 27878119890 minus 028 plusmn 11207119890 minus 027 52187119890 + 001 plusmn 21654119890 + 001

CLPSO 40333119890 + 001 plusmn 75039119890 + 000 26321119890 + 003 plusmn 33553119890 + 002 82952119890 minus 005 plusmn 33295119890 minus 004 79983119890 + 000 plusmn 16728119890 + 000

operator is compared The experimental results indicate thatthe proposed recombination operator is able to improve theperformance of the CSA And then the modified hyper-mutation makes further improvement Finally the proposedalgorithm is compared with the state-of-the-art algorithmsThe experimental results show the competitiveness of theRHCSA

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 61403349 no 61402422 no 61501405

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

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

Computational Intelligence and Neuroscience 13

no 61401404 and no 61302118) Science and TechnologyResearch Key Project of Basic Research Projects in Edu-cation Department of Henan Province (no 15A520033 no14B520066 and no 17B510011) and Doctoral Foundation ofZhengzhou University of Light Industry (no 2013BSJJ044)and in part by the Program for Science and TechnologyInnovation Talents in Universities of Henan Province undergrant no 17HASTIT022

References

[1] J J Liang A K Qin P N Suganthan and S BaskarldquoComprehensive learning particle swarm optimizer for globaloptimization of multimodal functionsrdquo IEEE Transactions onEvolutionary Computation vol 10 no 3 pp 281ndash295 2006

[2] A R Jordehi ldquoEnhanced leader PSO (ELPSO) a new PSOvariant for solving global optimisation problemsrdquo Applied SoftComputing vol 26 pp 401ndash417 2015

[3] A R Jordehi ldquoSeeker optimisation (human group optimi-sation) algorithm with chaosrdquo Journal of Experimental andTheoretical Artificial Intelligence vol 27 no 6 pp 753ndash762 2015

[4] X-S Yang and A H Gandomi ldquoBat algorithm a novelapproach for global engineering optimizationrdquo EngineeringComputations vol 29 no 5 pp 464ndash483 2012

[5] A R Jordehi ldquoBrainstorm optimisation algorithm (BSOA) anefficient algorithm for finding optimal location and setting ofFACTS devices in electric power systemsrdquo International Journalof Electrical Power amp Energy Systems vol 69 pp 48ndash57 2015

[6] A A Heidari R A Abbaspour and A R Jordehi ldquoAn efficientchaotic water cycle algorithm for optimization tasksrdquo NeuralComputing amp Applications 2015

[7] I Boussaıd J Lepagnot and P Siarry ldquoA survey on optimizationmetaheuristicsrdquo Information Sciences vol 237 pp 82ndash117 2013

[8] E Hart C McEwan J Timmis and A Hone ldquoAdvances inartificial immune systemsrdquo Evolutionary Intelligence vol 4 no2 pp 67ndash68 2011

[9] Y Xu Y Jiang and W Deng ldquoAn novel immune geneticalgorithm and its application in WWTPrdquo in Proceedings ofthe Chinese Automation Congress (CAC rsquo15) pp 802ndash808November 2015

[10] R Daoudi K Djemal and A Benyettou ldquoImproving cellsrecognition by local database categorization in ArtificialImmune System algorithmrdquo in Proceedings of the IEEE Interna-tional Conference on Evolving and Adaptive Intelligent Systems(EAIS rsquo15) pp 1ndash6 Douai France December 2015

[11] Z Lu G Pei B Liu and Z Liu ldquoHardware implementationof negative selection algorithm for malware detectionrdquo inProceedings of the IEEE International Conference on ElectronDevices and Solid-State Circuits (EDSSC rsquo15) pp 301ndash304 IEEESingapore June 2015

[12] M Pavone G Narzisi and G Nicosia ldquoClonal selection animmunological algorithm for global optimization over contin-uous spacesrdquo Journal of Global Optimization vol 53 no 4 pp769ndash808 2012

[13] R Liu L Jiao X Zhang and Y Li ldquoGene transposon basedclone selection algorithm for automatic clusteringrdquo InformationSciences vol 204 no 20 pp 1ndash22 2012

[14] L D S Batista F G Guimaraes and J A Ramirez ldquoA dis-tributed clonal selection algorithm for optimization in electro-magneticsrdquo IEEE Transactions on Magnetics vol 45 no 3 pp1598ndash1601 2009

[15] J Zheng Y Chen andW Zhang ldquoA survey of artificial immuneapplicationsrdquo Artificial Intelligence Review vol 34 no 1 pp 19ndash34 2010

[16] L N de Castro and J Timmis Artificial Immune Systems ANew Computational Intelligence Approach Springer LondonUK 2002

[17] N Cruz-Cortes ldquoHandling constraints in global optimizationusing artificial immune systems a surveyrdquo in Constraint-Handling in Evolutionary Optimization pp 237ndash262 SpringerBerlin Germany 2009

[18] E Mezura-Montes and C A Coello Coello ldquoConstraint-handling in nature-inspired numerical optimization pastpresent and futurerdquo Swarm and Evolutionary Computation vol1 no 4 pp 173ndash194 2011

[19] D Dasgupta S Yu and F Nino ldquoRecent advances in artificialimmune systems models and applicationsrdquo Applied Soft Com-puting Journal vol 11 no 2 pp 1574ndash1587 2011

[20] L N de Castro and F J von Zuben ldquoLearning and optimizationusing the clonal selection principlerdquo IEEE Transactions onEvolutionary Computation vol 6 no 3 pp 239ndash251 2002

[21] V Cutello G Nicosia andM Pavone ldquoReal coded clonal selec-tion algorithm for unconstrained global optimization usinga hybrid inversely proportional hypermutation operatorrdquo inProceedings of the ACM Symposium on Applied Computing (SACrsquo06) pp 950ndash954 ACM Dijon France April 2006

[22] T Jansen and C Zarges ldquoAnalyzing different variants ofimmune inspired somatic contiguous hypermutationsrdquo Theo-retical Computer Science vol 412 no 6 pp 517ndash533 2011

[23] N Khilwani A Prakash R Shankar and M K Tiwari ldquoFastclonal algorithmrdquo Engineering Applications of Artificial Intelli-gence vol 21 no 1 pp 106ndash128 2008

[24] H Lu and J Yang ldquoAn improved clonal selection algorithmfor job shop schedulingrdquo in Proceedings of the InternationalSymposium on Intelligent Ubiquitous Computing and Education(IUCE rsquo09) pp 34ndash37 Chengdu China May 2009

[25] X Liu L Shi R Chen and H Chen ldquoA novel clonal selectionalgorithm for global optimization problemsrdquo in Proceedingsof the International Conference on Information Engineeringand Computer Science (ICIECS rsquo09) pp 1ndash4 Wuhan ChinaDecember 2009

[26] L N De Castro and J Timmis ldquoAn artificial immune networkfor multimodal function optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo02) pp 699ndash704IEEE Honolulu Hawaii USA May 2002

[27] Z Li Y Zhang and H-Z Tan ldquoIA-AIS an improved adaptiveartificial immune system applied to complex optimizationproblemsrdquoApplied SoftComputing vol 11 no 8 pp 4692ndash47002011

[28] Z Li C He and H-Z Tan ldquoAINet-SL artificial immune net-work with social learning and its application in FIR designingrdquoApplied Soft Computing Journal vol 13 no 12 pp 4557ndash45692013

[29] Z Li J Li and J Zhou ldquoAn improved artificial immune networkfor multimodal function optimizationrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp766ndash771 IEEE Changsha China June 2014

[30] G P Coelho and F J V Zuben ldquoA concentration-based artificialimmune network for continuous optimizationrdquo in Proceedingsof the IEEECongress on EvolutionaryComputation pp 1ndash8 2010

[31] G P Coelho F O De Franca and F J V Zuben ldquoA con-centration-based artificial immune network for combinatorial

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

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

14 Computational Intelligence and Neuroscience

optimizationrdquo in Proceedings of the IEEE Congress of Evolution-ary Computation (CEC rsquo11) pp 1242ndash1249 New Orleans LaUSA June 2011

[32] G P Coelho and F J V Zuben ldquoA concentration-basedartificial immune network for multi-objective optimizationrdquo inProceedings of the IEEE Congress on Evolutionary Computationpp 343ndash357 2011

[33] A R Yildiz ldquoAn effective hybrid immune-hill climbing opti-mization approach for solving design and manufacturing opti-mization problems in industryrdquo Journal of Materials ProcessingTechnology vol 209 no 6 pp 2773ndash2780 2009

[34] M Gong L Zhang L Jiao and W Ma ldquoDifferential immuneclonal selection algorithmrdquo in Proceedings of the InternationalSymposium on Intelligent Signal Processing and CommunicationsSystems (ISPACS rsquo07) pp 666ndash669 Xiamen China December2007

[35] M Gong L Jiao F Liu and W Ma ldquoImmune algorithm withorthogonal design based initialization cloning and selectionfor global optimizationrdquo Knowledge amp Information Systems vol25 no 3 pp 523ndash549 2010

[36] M Gong L Jiao H Du and L Bo ldquoMulti-objective immunealgorithmwith non-dominated neighbor-based selectionrdquo Evo-lutionary Computation vol 16 no 2 pp 225ndash255 2008

[37] R Shang L Jiao F Liu and W Ma ldquoA novel immune clonalalgorithm for MO problemsrdquo IEEE Transactions on Evolution-ary Computation vol 16 no 1 pp 35ndash50 2012

[38] M Gong L Jiao and L Zhang ldquoBaldwinian learning in clonalselection algorithm for optimizationrdquo Information Sciences vol180 no 8 pp 1218ndash1236 2010

[39] M Gong L Jiao J Yang and F Liu ldquoLamarckian learningin clonal selection algorithm for numerical optimizationrdquoInternational Journal on Artificial Intelligence Tools vol 19 no1 pp 19ndash37 2010

[40] Y Peng and B Lu ldquoHybrid learning clonal selection algorithmrdquoInformation Sciences vol 296 pp 128ndash146 2015

[41] L N De Castro and F J V Zuben ldquoArtificial immune systemspart Imdashbasic theory and applicationsrdquo Tech Rep 210 Univer-sidade Estadual de Campinas 1999

[42] H Bersini ldquoThe immune and the chemical crossoverrdquo IEEETransactions on Evolutionary Computation vol 6 no 3 pp306ndash313 2002

[43] Y Qi Z HouM Yin H Sun and J Huang ldquoAn immunemulti-objective optimization algorithm with differential evolutioninspired recombinationrdquo Applied Soft Computing Journal vol29 pp 395ndash410 2015

[44] R Liu C L Ma F He W P Ma and L C Jiao ldquoReferencedirection based immune clone algorithm for many-objectiveoptimizationrdquo Frontiers in Computer Science vol 8 no 4 pp642ndash655 2014

[45] W-F Gao S-Y Liu and L-L Huang ldquoA novel artificialbee colony algorithm based on modified search equation andorthogonal learningrdquo IEEE Transactions on Cybernetics vol 43no 3 pp 1011ndash1024 2013

[46] J J Liang P N Suganthan and K Deb ldquoNovel compositiontest functions for numerical global optimizationrdquo in Proceedingsof the IEEE Swarm Intelligence Symposium (SIS rsquo05) pp 68ndash75Pasadena Calif USA June 2005

[47] YWang Z Cai and Q Zhang ldquoEnhancing the search ability ofdifferential evolution through orthogonal crossoverrdquo Informa-tion Sciences vol 185 no 1 pp 153ndash177 2012

[48] A K Qin V L Huang and P N Suganthan ldquoDifferential evo-lution algorithm with strategy adaptation for global numericaloptimizationrdquo IEEE Transactions on Evolutionary Computationvol 13 no 2 pp 398ndash417 2009

[49] C Garcıa-Martınez M Lozano F Herrera D Molina and AM Sanchez ldquoGlobal and local real-coded genetic algorithmsbased on parent-centric crossover operatorsrdquo European Journalof Operational Research vol 185 no 3 pp 1088ndash1113 2008

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