A comparative review of approaches to prevent premature convergence in GA

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Applied Soft Computing

jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a s o c

Review article

A comparative review of approaches to prevent premature convergence in GA

Hari Mohan Pandey

^a,∗

, Ankit Chaudhary

^b

, Deepti Mehrotra

^c

aDepartmentofComputerScience&Engineering,AmityUniversity,Noida,UttarPradesh,India

bDepartmentofComputerScience,MUM,IA,USA

cAmitySchoolofEngineering&Technology,AmityUniversity,Noida,UttarPradesh,India

a r t i c l e i n f o

Articlehistory:

Received31March2013

Receivedinrevisedform11August2014 Accepted11August2014

Availableonline6September2014 Keywords:

Evolutionaryalgorithms Geneticalgorithm Markovchain Prematureconvergence Schematheory Statisticalmechanics

a b s t r a c t

ThispapersurveysstrategiesappliedtoavoidprematureconvergenceinGeneticAlgorithms(GAs).

GeneticAlgorithmbelongstothesetofnatureinspiredalgorithms.TheapplicationsofGAcoverwide domainssuchasoptimization,patternrecognition,learning,scheduling,economics,bioinformatics,etc.

FitnessfunctionisthemeasureofGA,distributedrandomlyinthepopulation.Typically,theparticu- larvalueforeachgenestartdominatingasthesearchevolves.Duringtheevolutionarysearch,ﬁtness decreasesasthepopulationconverges,thisleadstotheproblemsoftheprematureconvergenceand slowﬁnishing.Inthispaper,adetailedandcomprehensivesurveyofdifferentapproachesimplemented topreventprematureconvergencewiththeirstrengthsandweaknessesispresented.Thispaperalso discussesthedetailsaboutGA,factorsaffectingtheperformanceduringthesearchforglobaloptimaand briefdetailsaboutthetheoreticalframeworkofGeneticalgorithm.Thesurveyedresearchisorganized inasystematicorder.Adetailedsummaryandanalysisofreviewedliteraturearegivenforthequick review.Acomparisonofreviewedliteraturehasbeenmadebasedondifferentparameters.Theunder- lyingmotivationforthispaperistoidentifymethodsthatallowthedevelopmentofnewstrategiesto preventprematureconvergenceandtheeffectiveutilizationofgeneticalgorithmsinthedifferentarea ofresearch.

Contents

1. Introduction... 1048

2. FactorsaffectingGAs... 1049

3. Theoreticalframework ... 1049

3.1. Schematheory ... 1049

3.2. Markovchainstheory... 1050

3.3. Statisticalmechanics... 1050

4. Approachesforpreventingprematureconvergence... 1051

4.1. Crowdingmethod... 1051

4.1.1. Strengthsandweaknesses... 1051

4.2. Incestpreventionalgorithm... 1051

4.3. Scheduledsharingapproach... 1052

4.4. MigrationmodelandnCUBEbasedapproach... 1053

4.5. Cooperationbasedapproach... 1053

∗Correspondingauthor.Tel.:+919810625304.

E-mailaddress:[email protected](H.M.Pandey).

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4.6. Syntacticanalysisofconvergence... 1053

4.7. Pygmyalgorithm... 1054

4.8. Adaptiveprobabilitybasedapproach... 1054

4.9. Socialdisastertechnique... 1055

4.10. Theislandmodelgeneticalgorithm... 1055

4.11. Shiftingbalancetheoryindynamicenvironment... 1056

4.12. Randomoffspringgenerationapproach... 1058

4.13. Chaosoperatorbasedapproach... 1059

4.14. Self-adaptiveselectionpressuresteeringapproach(SASEGASA)... 1060

4.15. Multi-combinativestrategy ... 1060

4.16. GAusingself-organizingmaps... 1061

4.17. Age-layeredpopulationstructure(ALPS)approach... 1062

4.18. Strengthsandweaknesses... 1064

4.19. Numberstructuringapproach... 1064

4.20. Hybridparticleswarmoptimization(PSO)andGA... 1064

4.21. Selectivemutationbasedapproach... 1065

4.22. Elitematingpool(EMP)approach... 1065

4.23. Dynamicapplicationofreproductionoperator... 1066

4.24. HybridstrategyusingEMPAandDARO... 1067

4.25. Frequencycrossoverwithninedifferentmutations... 1068

5. Summaryandanalysesofreviewedliteratures... 1068

6. Conclusions... 1075

References... 1075

1. Introduction

GeneticAlgorithms (GAs)are search and optimization algo- rithmsbasedonnaturalselectionandgenetics.GeneticAlgorithms areoneofthemostpopularalgorithmsinthecategoriesofevo- lutionary algorithms. The basicprinciples of GAs were initially developedbyHolland[1]andfurthercarriedbyDeJongandGold- berg.GoldbergandMichalewiczhavegiventhedetailedoverview andimplementationofGAsinvariousﬁelds[2,3].

GAoperatesonapopulationofsolutionsrepresentedbysome encoding.DuringtheimplementationprocessofGAseverysolu- tionorindividualisassignedaﬁtnesswhichisthemeasureofGAs.

Theﬁtnessofeachindividualisdirectlyrelatedtotheobjective functionforoptimizationproblem[4].Then,usingthereproduc- tion,crossoverandmutationoperatorstheindividualpopulation canbemodiﬁedtoanewpopulation[5].

In GAs, the searching is iterativelyguided by the ﬁtness of thecurrent parentgeneration.Whenever, we applyGAs for an optimizationproblem,itrunsoverthousandsofindividual,each representsa solution.Theobtainedsolutionsareevaluated and recombinedtogetoffspring.Ithasbeenprovenin[1–3,6]thatthe previousgenerationsdetailsareonlyimplicitlyandpartiallypre- servedinthecurrentgeneration.Hence,theregenerationcanbe hardtomanagebecauseofthefollowingreasons:

(a)In regeneration an individual population appears that had alreadybeenseeninthesearchspace.

(b)Geneticdriftcanaffectthesearchforoptimizationinawrong direction.

(c)Asaresult,abigpartofthesearchspaceisleftunexploredfor theglobaloptimum.

GAshasgainedpopularitybecauseitcanbeappliedinawide rangeof problems,includingmultimodalfunction optimization, machinelearning,patternrecognition,imageprocessing,natural languageprocessing,grammarinference[4,5,8]andmanymore.

But the behaviors of GAs have not been foundas adaptive as expected.Acommonproblemisthatofprematureconvergence [7].

Theprimaryinterestofthispaperistoshowthedetailedsur- veyofdifferentapproachesproposedandappliedforsolvingthe prematureconvergenceprobleminGAs.

Section2offersadetaileddescriptionofdifferentfactorsaffect- ingGAsearch.Inthissectionwediscusseddiversityofpopulation and selective pressure. The aim of Section 3 is toprovide the theoreticalframework tounderstandthegeneticmechanism of evolutionarysearch process. Thissectionbrieﬂy describesvari- oustheories suchasSchematheory, Markovchain theory,and Statistical mechanism. Thissection also discusses and presents

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the important literatures to understand the genetic dynamics throughMarkovchainandstatisticalmechanics.Themajorparts ofthispapercontainedinSection4.Thissectiongivesthedetailed andcomprehensivesurveyonapproachesofhandlingpremature convergencewithinGAs. In addition,strengthsandweaknesses have been provided for each approach. The surveyed research works have been organized according to the publication. Sur- veyedresearcheshavebeensummarizedinSection5usingvarious parameters.TheconclusionsaredrawninSection6.Lastly,refer- encesaregivenwhichcoverstheimportantliteraturesinthearea ofGA.

2. FactorsaffectingGAs

A genetic algorithm is a method of optimization based on Darwin’snaturalselectiontheory.Darwin’stheorywasbasedon survivaloftheﬁttesttoreﬁneasetofsolutioninaniterativeman- ner[2].Darwin’stheorysaysthatevolutionoccursifandonlyif threeconditionsmet[9](seeFig.1).TheconditionsshowninFig.1 arenecessary;itmeansthatifduetoanyreasonanyonecondition doesnotmeetintheregenerationprocess,naturalselectionwill ceaseatthatgeneration.

GAstartswithasetofsolutionwhichisrepresentedbychro- mosome.Byusinganappropriateselectiontechniqueasolution fromonepopulationispickeddependingontheﬁtnessandused toformanewoffspring.ThisprocessrepeateduntilGAreachedto thethresholdorthemaximumnumberofgenerations.Fig.2shows theﬂowchartoftheStandardGeneticAlgorithms(SGA).

The performance of GA largely depends on the crossover and mutationoperations. Hence,adapting thesuitable valueof crossoverandmutationisvery importantbecauseitguidesthe searchprocessandmaintainsthediversityinthepopulation[19].

Asdiscussed,GAisthepopulationbasedsearchmethod.There- fore, selection of the population is the critical step because if

Struggle for Existence

Variations Inheritance of the

variations Darwin's Theory

Fig.1.NecessaryandsufﬁcientconditionsofDarwin’stheory.

Start

Initial Population Generator Fitness = Fitness P (G) Crossover Mutation

Reproduction

Replacement to incorporate new individual population Selection Process (Natural

Selection)

Termination?

End

Fig.2.Flowchartofstandardgeneticalgorithm[8].

Population Diversity

Search Space

Fitness Function

Selection Pressure Number of

Individuals Problem Difficulty

Initial Population

Fig.3. FactorsaffectingGA’sbehavior.

populationhasnotbeenselectedinanintelligentmannerthenit willbedifﬁculttogettheappropriateanswertotheproblem.Typ- ically,selectiondoneattwoplaces,oneisinthebeginningknown asInitialPopulationselectionorgenerationofrandominitialpopu- lationandthesecondoneistheselectionofpopulationforthenext generation.Hence,beforeapplyingtheselectionorgenerationof populationonehastotakesomefactorsintoaccountasshown inFig.3.Inthispaper,weareconsideringtwoimportantfactors, namelyPopulationDiversityandSelectionPressure.

Inordertoreachtotheglobaloptimaandexplorethesearch spaceadequately,maintainingpopulationdiversityisveryimpor- tant.Also,ithadbeenexplainedclearlythatdegreeofpopulation diversity is one of the main causes of premature convergence [27,28].Itisnecessarytoselectthebestsolutionofthecurrent generationtodirecttheGAtoreachtotheglobaloptimum.Theten- dencytoselectthebestmemberofthecurrentgenerationisknown asselectivepressure.Selectionpressureisalsoakeyfactorthat playsanimportantroleinmaintaininggeneticdiversity.Aproper balancingisrequiredbetweengeneticdiversityandselectionpres- suretodirectGAsearchtoconvergeinatimeeffectivemannerand achieveglobaloptima.Highselectivepressurereducesthegenetic diversity;henceinthissituationprematureconvergencecanoccur whilethelittleselectivepressureprohibitsGAtoconvergetoan optimuminreasonabletime.

3. Theoreticalframework

This sectionprovides anoverview of different theoriespre- sented to address theoretical concerns related to evolutionary theory. Thereare various tools and methods available suchas Schematheory,Markovchaintheory,Dimensionalanalysis,Order statistics, Quantitative genetics, Orthogonal functional analysis, Quadraticaldynamicalsystems,andStatisticalphysics.Wehave concentratedourstudyonthreemethods,namelySchematheory sinceitisconsideredasthefundamentaltheory,Markovchainthe- orysinceitishelpfulinsettingtheboundariesforconvergenceand manymorefeaturesandStatisticalmechanicstheory.

3.1. Schematheory

Holland[136]presentedSchematheoryanditwasacceptedas fundamentaltheorytounderstandGA.Itwaswidelyuseduntilthe early1990’s.Itwasclaimedthatinapopulationof‘n’chromosomes, GAprocessO(n³)schemataintoeachgeneration.Thispropertyis knownasintrinsic/implicitparallelism[1].Aschemaisasimilarity templateusedtodescribeasubsetofstringdisplayssimilarityat certainstringposition.Itcanbeformedbyternaryalphabet{0,1,∗}, where‘*’representsanotationsymbolshowsthedescriptionofall possiblesimilaritiesamongstringofaparticularlengthandalpha- bet[137].Lengthandorderofschemawereusedtounderstand

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theworkingofschematheory.Lengthcanbedefinedasthedis- tancebetweenfirstandlastdefinedpositionintheschemawhile theorderofschemaisthenumberofdefinedpositions.TheGA modeledinschematheoryknownascanonicalGA,whichworks onbinarystrings.

Althoughtheschematheorywasconsideredas fundamental theory,butitwascriticizedbymanyresearchers.Someofthem arepointedoutas:

•ItisnotsuitabletoexplainthedynamicalorlimitbehaviorofGA.

•Implicitlyseparationofproblemtosomeextent.

Theignoranceoftheseassumptionsleadstothe“Buildingblock hypothesis”whichclearlyexplainsthebehaviorofGA[2].Inaddi- tion,Beryer[139]presentedanalternativemethodtoshowhow GAoperates.Reevesand Rowe[138]presentedthedetailing of criticismofschematheoryintheirbook“GeneticAlgorithms:Prin- ciplesandPerspectives(AGuidetoGAtheory).Onemorereason ofreplacementofschematheoryisthesuitabilityofMarkovchain theoryintheﬁeldofevolutionaryalgorithms(EAs).

3.2. Markovchainstheory

ThetermEvolutionaryAlgorithm (EA) standsfora family of stochasticproblemsolverbasedonprinciplesthatcanbefoundin biologicalevolution[140].Astochasticalgorithmcanbeconsidered asarandomsequenceofeventsE={X1,X2,X3,...}occurringin time,whereeach{Xt}representsarandomvariableandthepossi- blevaluesthatthesevariablescantakearecalledthestatesofthe system[138].AMarkovchainisarandomsequenceofvariables X₀,X₁,X₂...satisﬁestheMarkovproperty,i.e.thesimplestkind ofdependencewithinastochasticprocessiswhenthedistribution oftimetdependsonlyonwhathappenedatthetimet−1[138].

Instatisticalterms,thisisknownas“memorylessproperty”andcan bedeﬁnedas[142]:

P(X_n₊₁=x|X_n=x_n,...,|X₀=x₀)=P(X_n₊₁=x|X_n=x_n)

Rudolph[141]presentedthematrixpropertyforMarkovchain theoryasgivenbelow:

AsquarematrixP=(p_ij):N×Nissaidtobe a.Nonnegative(P≥0),ifp_ij≥0∀^i,^j∈{1,...,N} b.Positive(P>0),ifp_ij>0∀^i,^j∈{1,...,N}

AnonnegativesquarematrixP=(p_ij):N×Nissaidtobe c.Primitive,if∃ⁿ∈N:Pⁿispositive

d.Reducible,ifPcanbebroughtintotheform(withsquarematrices CandT)

C 0 R T

Byapplyingthesamepermutationstorowsandcolumns.

e.Irreducible,ifit’snotreducible f.Stochastic,if

N j=

p_ij=1∀ⁱ∈{1,...,N}

AstochasticmatrixP=(p_ij):N×Nissaidtobe g.Stable,ifithasidenticalrows

h.Columnsallowableifithasatleastonepositiveentryineach column.

MatricesthatdescribethetransitionprobabilitiesinaMarkov chainarestochasticmatricesandwillbereferredtoasa“transi- tionmatrix”[142].Inthenextparagraph,wehaveprovidedabrief

reviewofMarkovchaintheorywhichshowstheapplicabilityof thismodel.

Horn[160]usedMarkovchainstoanalyzethestochasticeffects ofthenichingoperatorofanichedGA.ItwasprovedthataMarkov chainisaneffectivetooltostudythedynamicnatureofanichedGA.

Theeffectof“AbsorbingMarkovchain”and“ErgodicMarkovchain”

wasshowntoestimatetheconvergenceofanichedGA[160].Eiben etal.[154]studiedclassicalGAandsimulatedannealing(SA)and presentedAbstractGeneticAlgorithm(AGA)whichgeneralizeand unifiesGAandSA.Forthebetterclarificationpurpose,similarity anddissimilaritybetweenclassicalGAandAGAwaspresented.In addition,Markovchainmodelwasusedtoshowtheevolutionof AGA.Leungetal.[158]discussedtheabilityoftheGAsearchby applyingzeromutationprobabilityandtherelationshipsbetween prematureconvergence and effectof GA parameters werepre- sented.UsingMarkovchain,thereasonsofchoosingzeromutation probabilitywhenanalyzingprematureconvergencewasdemon- strated.JunandXinghuo[156]showedtheoreticalanalysistoshow theconditionfortheconvergenceofEA.Thenecessaryandsuffi- cientconditionsfortheconvergenceofEAtotheglobaloptimum werederivedwiththehelpofMarkovchaintheorytodescribethe limitingbehavior.Inaddition,theupperandlowerboundofthe convergenceraterequiredfortheEAwasexplored.StarkandSpall [155]presentedcomputablerateofconvergenceforGAviaMarkov chainanalysis.Huang[145]extendedtheworkpresentedin[146]

byapplyingMarkovmodeldevelopedbyNixandVose[144].Reeves andRowe[138]presentedtheapplicabilityofMarkovchainthe- orytodescribethelimitingdistributionofsimpleGAinwhichthey explainedtheeffectofselection,mutationandcrossover.Inaddi- tion,modelingGAusesMarkovchaintheorywaspresentedusing suitableexample.Auger[143]studiedtheconvergenceproperty ofself-adaptiveevolutionarystrategiesviaphi-irreducibleMarkov chain. Basically,he used recurrentMarkov chainsto provethe existenceofcriticalsizeofpopulation,whichwascalculatedusing differentparameters of algorithms and determines theconver- gence rate of thealgorithm. Braak [148]presented Differential EvolutionMarkovChain(DE-MC) which combinestheessential featuresofdifferentialevolution(DE)presentedin[149–151]for globaloptimizationoverrealparameterspacewithMarkovchain MonteCarlo(MCMC)[152,153]tosolveanimportantproblemof MCMCinrealparameterspaces,namelythatofchoosinganappro- priatescaleandorientationforjumpingdistribution.Clark[142]

appliedthistheoryformodelingstochasticoptimizationalgorithm.

HéderandBitó[157]implementeda GAtoprovidespecial site diversitytosimultaneouslyoptimizedownlinkanduplinkthesig- naltointerferenceandnoiseratio(SINR)valueinbroadbandﬁxed wireless(BFWA)accesssystem.Fortheconvergenceanalysis,i.e.

todeterminethemaximumnumberofiterationstheyusedMarkov chainswhereeverystaterepresentsapossiblepopulationcontain- ingdifferentterminalstation(TS)andbasestation(BS)assignment setasindividuals.Suzuki[147]suggestedlargedeviationprinciple approachforGAusingMarkovchains.Polietal.[159]presented exactschematheoryandMarkovchainmodelGeneticProgram- ming(GP)andvariablelengthGAforthehomologouscrossover.In theabsenceofmutationtherelationshipbetweenapproximateand exactschematheoryfordifferentformsofhomologouscrossover wasshown.

AlthoughMarkovchainmodelwasimplementedforconver- gence analysis. But it was found limited by the complexity of calculations and it doesnot showthepredictive descriptionof geneticdynamics[140,161].

3.3. Statisticalmechanics

Itisabranchofmathematicalphysics.Itcanbeusedtostudy thestateofuncertaintyofamechanicalsystemusingprobability

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theory which works on the average behavior of the system [161,162]. Prügel et al. [163] and Shapiro [164] described the GAusingstatisticalmechanics.Shapiro[164]showedstatistical formulation of GA to studythe dynamics of GA searching and explained how statisticalmechanics theoryaddresses thelimi- tationsofSchemaand Markovchain theory.Twosourceswere identiﬁedinstudyingthedynamicsofevolutionaryapproachesas givenbelow[164]:

•Thedynamicsarestochastic–selectioninvolvessamplingand mutation and crossover involve probabilistic changes to the strings.

•Thedynamicsactsonthespaceofpossiblepopulations,aspace ofastronomicaldimensionality

Itwassaidthatbyderivingadeterministicsetofequations, onecandescribe theevolution which mustbesufﬁcientlysim- pleto allow for sometype of study. Thereare many previous works available which proves the applicability of this theory [138,165,161,166–168].

4. Approachesforpreventingprematureconvergence

This section gives the detailed explanation of the different approachesgivenbyresearchersformanagingtheriskofprema- tureconvergence inGAs. We have reviewedvarious literatures andpresentingtwenty-fourapproacheswiththeirstrengthsand weakness.

4.1. Crowdingmethod

CrowdingwasintroducedbyDeJong[38]toeliminatethemost similarindividualwheneveranewoneentersinasubpopulation.

Theaimofpresentingthistechniquewastomaintainthepopula- tiondiversityandtoﬁxtheprematureconvergence.Itwasapplied inthesurvival solutionstepofgeneticalgorithms.Theprimary objectiveof introducingit inthe survivalselection is todecide whichindividualamongthoseinthecurrentpopulationandtheir offspringindividualswillpasstothenextgeneration.

Mahfoud[12]presented a modiﬁed versionof theDe Jong’s strategy ofreplacing offspring of themostsimilar parents.The modiﬁedversionofcrowdingwasfoundeffectiveinmaintaining populationdiversityandpreventingprematureconvergence.This approachworksasfollows:

Let, P_i=Next generation population, p_i=First individual, c_i=Secondindividual(child),d(x,y)=distancebetweentwoindi- vidualsxandy

Then,if((d(p₁,c₁)+d(p₂,c₂))<(d(p₁,c₂)+d(p₂,c₁)) P1←Winnerbetweenp1andc1

P2←Winnerbetweenp2andc2

Else

P₁←Winnerbetweenp₁andc₂ P2←Winnerbetweenp2andc1

Three variations of crowding approach were presented, namelyStandardCrowding[38],DeterministicCrowding[12]and RestrictedTournament Selection (RTS) [84]. Standard crowding was found to be limited in multimodal function optimization.

Deterministiccrowding(competition betweenchildrenandpar- entsofidenticalniches)approachwasproposedbyMahfoud[12]

toaddressthelimitations[17]ofthestandardcrowdingapproach suggestedbyDeJong[38].Theresultingorderofcomplexityof deterministiccrowding was O(N). RTS adapts standard tourna- mentselectionformultimodalfunctionoptimization.IncaseofRTS twoelementsfromthepopulationwaspickedtoapplyreproduc- tionoperatorsandthenarandomsampleofCF(crowdingfactor)

individualwastakenfromthepopulationasinstandardcrowding.

ThisprocedurewasrepeatedN/2times.Theorderofcomplexityof RTSisO(CF.N)whichvariesfromO(N)toO(N²)dependingonthe valueofCF[85].

4.1.1. Strengthsandweaknesses

Thismethodwasfoundeffectiveforproblemsofalllevelsof difficulties,especiallyformultimodalfunctionoptimization.Using this,onecansolveproblemsthataremuchmoredifficultthanthose solvablebytraditionalbasedhill-climbing.Itusesarandomsample ofCFindividualwhereCFisthecrowdingfactorratherthanfitness proportionateselectiontoresolvetheselectionpressure.Thereis noneedofsharingfunctionwhichisthekeyelementofthesharing approachproposedbyHolland[1]andfurtherexpandedbyGold- bergandRichardson[16].Selectionofgoodsharingfunctionneeds a-priorknowledgeoftheobjectivefunctioncharacteristics[74].

Theproblemwiththisapproachisthatitmayloseloweroptima whichmaylieonacrossoverpathtohigheroptima.Maintaining diversityinthepopulationwasthemaingoalofthisapproachand algorithmwassuccessfultosomeextent,butstochasticerrorsby thelowvalueofcrowdingfactoraffecttheperformance.Mainte- nanceofbitwisediversityistheproblem. Itdoesnotmodelthe methodbywhichapopulationarrivesastablemixtureofspecies, butinsteadstrivestomaintainthediversityofthepre-existingmix- ture[75].Handlingreplacementerrorswerethemainconcernwith thistechnique.Inaddition,writingreplacementrulesarenotan easytask

4.2. Incestpreventionalgorithm

ToavoidprematureconvergenceofGAtwoapproacheswere introduced,namelythedynamicapplicationofcrossoverandmuta- tion operator and the population partial re-initialization [31].

Theprimary objectiveof applyingthecrossoveroperationis to exchangecertaininformationbetweentwoparentchromosomes atthesametimeonemustkeepinmindthatimportantinforma- tionshouldnotbelost.Hence,theselectionofmatingpairisavery importantstepinremovingtheinsufﬁcienciesofGAs.

Nicoar˘a[31]and EshelmanandSchaffer[32] usedHamming distanceasameasureforselectingthematingpair.Theideaisthat ifHammingdistanceisaboveathresholdthenselectstheparent chromosomeformatingotherwiseno.Beforestarting,thethresh- oldissettoahighvalueandduringrunningofGAifnopairwas chosenwiththeinitializedthreshold,thevaluecanbedecreased.

Adetailedformula(Eq.(1))wasgivenforcrossoveroperator and mutation operators (Eq. (2)). Eq. (2) is the modiﬁed for- mula as given in [33]. Results of [31] were compared against three other genetic algorithms, namely NSGA-II [34], Canonic geneticalgorithm and Elitistgeneticalgorithm. For thesimula- tionawell-knownjobshopschedulingproblemwasimplemented andperformancesofNSGA-DAR(Non-dominatedSortingGenetic Algorithm-DynamicApplicationofgeneticoperatorsandpopula- tionpartialre-initialization)weretestedontheft10testinstance andtheobtainedresultswerefoundencouraging.

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⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

1,ifDdominatesP1andP2 max

1−k1∗t G ,0.5

,ifDdominatesoneparent

andnodominancerelationwithotherparents max

0.5−k2∗t G ,0

,ifDisdominatedbyatleastoneparent

0.5,ifnodominancerelationexistbetweenDandP1,P2 0,otherwise

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Shimodaira [78] employed theconcept of incest prevention andproposedanewgeneticalgorithmDiversityControl-oriented

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GeneticAlgorithm(DCGA).Incestpreventionwasappliedinsur- vivalselectionandthisincreasedthediversitybygivingindividuals withfatherHammingdistancefromthebestsolutionahigherprob- abilityofsurvival[77].Matsui[79]incorporatedtwonewselections scheme,namelycorrelativetournamentselection(CTS)forrepro- ductionandcorrelativefamilybasedselection(CFBS)forsurvival.

CTSchoosethematewithhigherﬁtnessandHammingdistance fromasetofcandidates.

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⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

1, ifM(s)dominatess

0,ifsdominatesM(s)orM(s)isnotvalid max

1−k3∗t G ,0.5

,ifnodominancerelation existsbetweensandM(s)

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AsimilarapproachwasproposedbyFernandesandRosa[80]

calledasnegativeassertivematingintoGAwithvaryingpopu- lation size(GAVaPS). They showedthat GAVaPS mimic natural species’behavior byselectingparents, accordingtoparenthood orphenotypesimilarity.Astrategyofchoosingdifferentcrossover methodswasproposedin[81].Itwassuggestedthatonecanchoose theappropriatecrossovermechanismtoavoidprematureconver- genceonthebasisofthecouple’sHammingdistance.Incontrastto incestprevention,threenewmethodsofselectionofmatingpairs forGAweresuggested[82].Itwastriedtomimicinbreedingin naturewhichrestrictsindividualstomatewithoneseparatedby aminimalHammingdistance.Experimentalresultsshowedthat theproposedapproachcouldenhanceexploitation,reducethedis- ruptionofschemaandperformbetterthanaconventionalgenetic algorithm(CGA)andincestpreventionincertainproblems.

Thisapproachworkseffectivelywhencombinedwithmultiple crossoversonmultipleparents. Therefore, itwasgiven a name Multiple Crossover on Multiple Parents withIncest Prevention (MCMPIP)[76]. It was foundeffective for multimodal function optimization.In addition,it avoidspremature convergence and improvesthesolutionattheexpenseofconvergencespeed[77].

Theapproachsuggestedin[31]ishelpfulintwofold.First,dynamic applicationofgeneticoperatorstakestheadvantagesoftheiden- tiﬁcationoftheoperatorstopromotethebeneﬁcialeffectsofall theavailableoperatorsduringallevolution.Second,thepopulation partialre-initializationachievesthemaingoalonlyinthecritical situation,whentheriskofprematureconvergenceisbigenough whichsavesthecomputationalcost.

Thedrawbacksofthisapproachare:itsufferswiththeproblem ofanincreaseincomputationcostforcalculatingtheindividual’s Hammingdistance.ComparisonofindividualHammingdistance withthresholdincreasesthecomputationalcost.

Toovercome the problem(additional cost requires in com- putingtheindividual’sHammingdistance)ofincestprevention, CraighurstandMartine[83]consideredthefamilytreetodisal- lowincestbyadheringtotheancestrybasedincesttree.Thelevel ofincestlawhelpsinallowingordisallowingthematingamong familymembers.Experimentalresultsproveitseffectivenesson solutionqualityanddiversitymaintenance.Thisapproachshowed theslowconvergencerateandwassensitivetothemutationrate whileincestpreventionyieldedasatisfactoryoutcome.

4.3. Scheduledsharingapproach

Palmer,andStephen[15]presentedanatureinspiredapproach knownas ScheduledSharing topreventpremature convergence byapplyinganextensioncalledassharingtostandardGA.This approachhandlesprematureconvergencebyrequiringorganisms withinanichetocompetewiththeirneighbors.Initially,onepicks thenichesizeaslargeaspossibletoencouragetheorganismto

spreadacrossthesearchspace.Thereafter,exponentiallydecays andallowgradualclusteringoftheorganism.Thisapproachisthe extensiontothestandardsharingalgorithm[16,17].Theapproach givenin[15]over[16,17]isthemodificationoffixednichesharing tohandlemoredifficultandrealistictestfunction.

Palmer and Stephen [15] not only worked on sharingalgo- rithm,butalsotheyworkedonsampledsharing,whichremoves O(n²) communication betweenprocessors by standard sharing.

Experimentalresultsprovethatscheduledsharingcandelaythe convergenceaslongasrequirewhich continuesthesearchpro- cessefﬁciently.Tocalculatetheconvergenceofapopulationsearch spacecanbedividedintosubspaceorbuckets.Suppose,xirepresent thepercentageofthepopulationofithbucket,thenthepercentage ofthetotalpopulationcanbecalculatedas:

PercentageofTotalPopulation=

ix_i² (3)

Eq.(3)isanimportantmeasuretodecidetheconvergenceofa population,highervaluerepresentsthatthepopulationismore concentratedin oneor afew bucketswhile thelowermeasure showsthatpopulationsaremoredistributedacrossthebuckets.

Twoclasses oflandscapeswereused, namelyEasyand Difﬁcult landscapesgeneratedbytheweightsW_iofthesub-functions.Easy landscapesarethosehaveasmallnumberoflocaloptimawithwide basinsofattractionwhiledifﬁcultlandscapeshavejustoppositeof theeasylandscapes.ExperimentalresultsprovethatbothSched- uledSharingandNon-Sharingalgorithmsperformedbetterthanthe RandomSearchalgorithmatallpopulationsizesonbothclassesof landscapesandpreventprematureconvergenceofageneticalgo- rithm[15].

Thisapproachwasfoundeffectiveonlargepopulationgenetic optimizationofruggedtwodimensionallandscapes.Itishelpful inimprovingofflineperformancebyrequiringorganismwithina nichetocompetewiththeirneighbors.Itwasshownthatitallows theapplicationofsharingforcomplexproblemswherethereisno definablebestnichesize,withoutviolatingblackboxprinciplein thecalculationofnichesize.Thisapproachtakestheadvantages ofsimulatedannealingincalculatingthenichesizebyapplyingan exponentialdecreaseintheschedule.Itreducesthecomputational costbyallowinggradualclusteringoforganisminareasofgreatest interest.ItprovidesflexiblecontroloftheconvergenceofaGA.

Thisapproachallowsthefeedbacktodeﬁneaproblemdependent sharingschedule.

Althoughthistechniqueoutperformedoverrandomsearchand standardsharingtechniques.Butstilltherearesomelimitations.

•Thekeyconcernofthisapproachisﬁndingtheappropriatesched- uleforapplyingsharing.

•It requires prior knowledge to writethe appropriate sharing functionwhichisadifﬁculttaskbecauseofarealoptimization problem, noinformation aboutthesearch spaceanddistance betweentheoptimaisgenerallyavailable.

•Itwasfoundthat thisapproach improvestheperformance of standardsharingapproachbyincorporatingtheconceptofclus- teranalysisanddynamicniching.Butstillthecomputationaltime toobtaintheﬁtnessofindividualsdominatesthecomputational costofcomparisons.

•Itwasacceptedthattomaketheschedulesharingmoreeffective, adifﬁcultydetectorisneededwhichmightbeusedtodecideupon theschedule[15].

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4.4. MigrationmodelandnCUBEbasedapproach

Balakrishnan[24]presentedaMigrationModeltomaintainthe diversityandvariationinthegeneticpropertyofindividualpopu- lationwhichkeepstheGAsearchprocessalive.Thebehaviorof migration model is very similarto anordinary communication model,whichallowsindividualstomigratefromonepopulationto another.Therefore,migrationmodelhelpsinﬁndingglobalsolu- tionandpreventprematureconvergence.FeaturesofnCUBEmodel wereusedtodesign themigrationmodelbecauseithasseveral reasonsaslistedin[24].

ThreedifferentscenariosnamelyBestIndividual,MeanIndivid- ualandRandomIndividualweresuggestedtomigrateanindividual fromonepopulationtoanother.Itwasobservedthatonceanindi- vidualreachedtotargetpopulationthemigratedpopulationgets welcomedinadifferentways,suchasWarm,NeutralandColdwel- come.Coldwelcomewasusedforthesimulationin[24].

Twoexperimentswereconductedtoensuretheeffectiveness.

Theﬁrstexperimentteststhesuitabilityofmigrationofindividual basedondifferentmigrationstrategieswhilethesecondexperi- mentgivestheanswertothescalabilityissueofthesystem[24].

Thesimulationresultsofboththeexperimentswerefoundeffec- tivetopreventprematureconvergence.Inaddition,itimprovesthe performanceofGAtosolvethecomplexoptimizationproblem.

The present approach was powered by maintaining inde- pendentpopulationand allowinginterpopulationmigrationto maintainthediversityandalleviateprematureconvergence.nCube modelwasusedfortheexperimentwhichimprovestheperfor- manceofGAsearch.Inaddition,itwasevidentthatthisapproach isoneofthebestapproachesforcomplexoptimizationproblems.

Butstilltherearesomekeyissuesinapplyingthistechnique.

First,settingthemigrationintervalplayanimportantrolebecause both largeand smallintervals beingdesirable. Ifthe migration intervalistoolargethenitincreasesthepossibilityofpremature convergencewhereasthesmallmigrationinterval increasesthe interactionamongthepopulationwhichresultsinanincreasein thecomputationtime.Second,makingdecisionforanindividualto migrateoutofapopulationisacrucialfactor.Third,howtochoose numberofindependentpopulationsbecausesolutionstatuswillbe affectedifmoreindependentpopulationsareavailable.

4.5. Cooperationbasedapproach

Cooperationbasedgeneticalgorithmtopreventprematurecon- vergencewaspresentedin[18].Thefactwhichwasshownin[18]

isthatthehardnessofaprobleminherentlyrelatedtothewaywe representtheproblem.Therefore,thedifﬁcultyleveloftheproblem canincreaseordecreasebasedontherepresentationtechniques used.Toprovethisfact1’scountingproblem(max1problem)was implementedandthedifﬁcultylevelshowedusingdifferentrep- resentationtechniques.

Theideaofcooperatinggeneticalgorithmsaysthatinsteadof usingonlyonerepresentationapplyvarietiesofGAconcurrently eachoptimizingthesameobjectbutusingitsownrepresentation.

Inotherwords,afteraﬁxednumberofgenerations,eachGApro- videsotherGA’swithitsbestlocaloptimafound.Experimental resultswerefoundeffectiveforsolvingthoseproblemswhichwas eitherdifﬁcultorcannotbesolvablebysequentialGA.

Fonlupetal.[18]claimedthatthevarietiesofrepresentation techniques could be helpful in avoiding premature convergence. Experiments were conducted to prove the claim which showed the effectiveness of the presented approach. Various

representation techniquesarethe keyconcept inremoving the correlationbetweentheneighboringindividualandmaintainsthe diversityofthepopulation.Thisapproachimprovesthespeedwith- outapplyingparallelism.Anexchangeinindividualscreatesnew pathsofresearchandallowsGAtochoosethenewcreatedpaths.

The weaknesseswiththis approach are: (a)selection of the appropriaterepresentationtechniquestomaintainthediversityin thepopulationbecausehowonecansaythatthenewrepresen- tationtechniquewillworkeffectively.(b)Inthepresentedmodel [18]itwasconsideredthatafter20generationseachGA_ibroadcasts toeveryGA_j|j₌_/_iitis10individual,butwhatistheguaranteethat sameassumptionwillworkforanyotherproblem.(c)Dealingwith morethanonerepresentationisnotaneasytask.

4.6. Syntacticanalysisofconvergence

TheaverageHammingdistanceofapopulationwasusedasa syntacticmetrictopreventprematureconvergenceandpredictthe convergencetime[29,30].Theprimarygoalofusingsyntacticmet- ricistogettheprobabilisticboundonthetimeconvergenceofGAs.

Itwasshownthattheanalysisofﬂatfunctionprovidesworst-case timecomplexityofstaticfunctionswhichdevelopsatheoretical basisfortheprematureconvergence.

Itwasobservedthatmutationisoneofthewaysofmaintain- ingdiversityof thepopulation[29].Butthehighmutationrate raisesproblems.Inaddition,italsoincreasestherisktodestroy good schemasasbadones.Theseobservations wereworkedas motivationalfactorstocome-upwithnewapproaches.

Therefore,itwassuggestedthatinplaceofincreasingmutation rates,pickanindividualandadditsbitcomplementtothepopu- lation[29].Theaimofdoingthiswastoensurethateachalleleis presentandthepopulationspansthewholeencodedspaceofthe problem.Thepresenceofcomplementaryindividualsalsomakes aGAmoreresistanttodeception.Dependingontheassumptions about thesearch space, one canpick individual to becomple- mented.Variousapproaches weresuggestedfordoing thissuch asprobabilistic, randomselectionand others.It wasfoundthat randomlyselectingtheindividualtobecomplementedmakesthe leastnumberofassumptionsanditmaybethebeststrategyinthe absenceofotherinformation.Forconductingtheexperimentmin- imumﬁtnessindividualwerereplacedbyeitherthecomplement ofarandomlypickedindividualinthecurrentpopulationorthe complementofthecurrentbestindividual.

Ifl_irepresentthenumberofbitsneededtorepresentvariable iinadeceptiveproblem,thenthefunctionsusedin[29]canbe describedusingEq.(4).

Deceptive(x_i)=

n i=1

x_i if x_i=/0

2^lⁱ⁺² if x_i=0

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For10-bitand20-bitdeceptiveproblemDec1andDec2were usedwhichcanberepresentedas:

Dec1:Deceptive(x²₁,x⁸₂) (5)

Dec2:Deceptive(x²₁,x⁸₂,x₃⁵,x⁵₄) (6) InEqs.(5)and(6)superscriptsdenotethenumberofbitsneeded foreachvariableinDec1andDec2respectively.

Theresultsobtainedexperimentallywerecomparedwithclas- sicalGA(CGA)andaGAwithcompliments(GAC),whichproves thatGACeasilyreachedtotheoptimumwhereasCGAfailstoﬁnd theoptimum.

LouisandGregory[29]acceptedthattheGACdoesnotgiveguar- anteeforanoptimalsolutiontoallproblems,whereasMessyGA [45]canmakemuchstrongerguarantees.Butthedrawbackofthe messygeneticalgorithmisitscomputationalcomplexity.It was

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alsoshownthatGACperformedmuchbetterthantheCGAonAck- ley’sOneMax[46]andnoworseontheDeJongtestsuite[38].

Also,itwasagreedthatGAhardproblemsthatarebothdeceptive andepistaticmayneedmaskedcrossoverorotherlengthinde- pendentrecombinant operatorsto besolved successfully using complements[47].

Atimeconvergencemodelwaspresentedtopredictthetime toconvergence[29].Thestringsimilarityassumptionwasmade todevelopthismodel.Thismodelworkssuccessfullyforunimodal function.Thebehaviorofmultimodalfunction,geneticdriftand otherscanalsobepredictableusingthismodel.Themodelstarts withtheEq.(7)whichcomputesthechangeinHammingaverage pergeneration.

h_t+1=f(ht) (7)

whereht=Hammingaverageingenerationtandht+1=Hamming averageinthenextgeneration.

Twoobservationsweremadeforchoosingthefunctionf(ht).

Theﬁrstobservationsaysthat“theschematheoremalongwiththe similarityassumptionindicatesf(ht)islinear(seeEq.(8))”.

h_t+1=aht+b (8)

Thesecondobservationsays that“without mutationthe ﬁnal Hammingaverageiszero”whichimpliesthatb=0.Byputtingthis valueinEq.(8),wewillget.

h_t+1=aht (9)

ThegeneralsolutionoftheEq.(9)isgivenas:

ht=

l/2 t=0 a^th₀ t>0

(10) ThevalueofacanbeobtainedbykeepingtrackoftheHam- mingaverage,whilerunningageneticalgorithm[29].Itwasshown thatthemodelworks effectivelyand a goodpredictionof time complexityispossible.

Theprimarystrength ofthisapproachisthat itprovidesthe timelimitstotheconvergenceoftheGA.Inaddition,combining ﬁtnesspredictionwithHammingaveragepredictionprovidethe detailsabouthowmuchprogressispossiblewithaGAalongwith boundsonhowmuchremainingworkcanbedone.Syntacticanaly- sismodelformsthebasistoseetheeffectofselectionandmutation onthebehaviorofGA.Itwasevidentfromtheresultthatgoodpre- dictionsoftimecomplexityarepossible,evenfromaroughmodel thatuseseasilycomputablesyntacticinformation.

Theproblemswiththisapproachare:mutationrateaffectsthe performanceoftheGAinthelatergeneration.Inaddition,comput- ingupperboundandlowerboundisnotcosteffectiveprocess.It wasacceptedthatqualitativepredictionmaynotbepossiblewith syntacticinformation[29,30].

4.7. Pygmyalgorithm

Rayan[43]describedanewselectionschemetoreachtoopti- malsolution.Actually,hesuggestedPygmyalgorithmtodealwith multi-objectiveproblems.Thisalgorithmwasoriginallyappliedto theproblemofevolvingminimalsortingnetworks,whichmustnot onlybeabletosortnumbers,butalsoinasfewlessnumberaspos- sible[43].Thisapproachusestwoﬁtnessfunctionsandmaintains twolistsofparents,oneforeachcriterionoftheproblem.Itwas clearlyshownthatifthepopulationusingPygmyalgorithmthen thechancestoconvergetoanoptimalsolutionismorecompared toothertechniquesusesaﬁtnessfunctionwhichisasumofthe assumedcriteria.In[44]Pygmyalgorithmwasappliedforhandling

prematureconvergenceinthecaseofgrammarinduction.Inorder toimplementPygmyalgorithmandtoselecttheresultinggram- marwithminimumnumberofruleprinciplewasusedwhichwas consideredasanotherﬁtnesscriteriontocreateasecondparent list.

Thisapproachusesalesscomputationalcostcomparetothe previousapproachessuchasCrowding[38],Sharing[16],restricted mating,spatial matingand others. It reducestherisk of losing geneticmaterial comparedtothemethodswhich rewardparsi- mony[86]ortrytocombinethetwomeasures[87]runtheriskof individualstradingoffvariouspartsoftheﬁtnessfunction.Pygmy algorithmmaintainstwolistsofﬁtnesswhichremovesthepossibil- itiesofinbreedingbetweenindividualswithsimilarperformance.

In addition,thisapproach avoidsthechancesfor individualsto breedwithcloserelativeslikesiblingsorevenparentsofopposite sex.

Theweaknesswiththisapproachisthatinsomesituationsit isnotwellsuitedformulti-objectiveproblems.Inordertogetthe solutionofmulti-objectiveproblemsanindividualmustsolvetwo ormoresmallerproblemsofthesamenaturetosolvethemain problem.Pygmyalgorithmdoesnotaddresssomeofthesubprob- lemssimplybecauseindividualsinthesamegroupcouldnotmate witheachother.

ToaddresstheaboveissueRyan[43]presentedRacialHarmony.

Itwasevidentin[43]thatbyapplyingracesamongtheindivid- ualinthesamegrouptheperformanceofPygmyalgorithmwas improved. RacialPreferenceFactor (RPF)wasusedasa measure ofprobabilitythat anindividualwillchooseanindividualfrom anotherracewhenselectingamate.

4.8. Adaptiveprobabilitybasedapproach

In many previous works [68,88–90] the idea of adapting crossoverandmutationwasappliedforthebettermentofGAper- formance.Schafferetal.[88]implementedacrossovermechanism wherethecrossoverpointswerechosendependingontheperfor- manceofthegeneratedoffspring.Davis[89]suggestedmechanism ofadaptingprobabilitiesbasedontheperformanceoftheoperators.

Heallottedhigherprobabilitiestotheoperatorswhichcreateand causethegenerationofbetterstrings.Thetechniquepresentedin [89]wasdevelopedinthecontextofasteadystateGA[90].Fogarty [90]showedtheeffectofvaryingmutationrateovergenerations.In hisworkheshowedthattheexponentialdecreaseinmutationrate showsbetterperformanceforasingleapplication.Whitleyetal.

[68]presentedthattheprobabilityofmutationisadynamicvari- ableparameterwhichcanbedeterminedbytheHammingdistance betweentheparentsolutions.Itwasevidentin[68]thatadapt- ingmutationprobabilitydynamicallysigniﬁcantly improvesthe performance.

AdaptiveGeneticAlgorithm(AGA)(SrinivasandPatnaik)isone oftheefﬁcientmethodsforoptimizingthemultimodalfunction [19].Thisapproachdiffersfromtheexistingapproaches [89,90]

ascrossoverandmutationprobabilitywasdeterminedfor each individualasafunction ofitsﬁtness.Whitelyet al.[68]imple- mentedadaptivemutationapproachinthecontextofsteadystate GA,whereasAGAisbasedongenerationalreplacementstrategy.

AGAemploybestsolutionprotectionstrategywhichresolvesthe highleveldisruptionwhilesteadystateGAappliespopulationary elitisminwhichthereisnoneedtoprotectthebestsolution.

Topreventprematureconvergenceduringtheimplementation ofGAaprobabilitybasedapproachwasadoptedwhichvariesthe probabilitiesofcrossoverandmutationdependingontheﬁtness valueofthesolution.Adetaileddesignwasgivenforsuitablychoos- ingthecrossover(Pc)andmutation (Pm)probabilities [19].The

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objectiveofthisapproachwasnotonlytopresentamechanism tochoosecrossoverandmutationprobabilitiesbutalsopresent asuitabletechniquetopreventtheGAfromgettingtrappedata localoptimum.Itwasshownthattheadaptivevalueofcrossover andmutationshouldbefrom0.0to1.0and0.0to0.5respectively.

Itwasproventhatthelowvalueofthecrossoverprobabilityand highvalueofthemutationprobabilityleadtoprematureconver- genceorashiftfromgeneticalgorithmstorandomsearch.These factshelpinpreventingprematureconvergenceoftheGA.

Extensiveexperimentswereconductedtoshowtheeffective- nessof AGA.For theexperimentvarious optimizationfunctions wereusedsuchasDeJong’sfunctionsknownasspikyfunction(f5, f6,f7),Order-3Deceptiveproblem,TravelingSalesmanProblem(TSP) andmanymore.Theoutcomeoftheexperimentwasfoundtobe encouraging.

Themainstrengthofthisapproachisthatitprovidesabasis foradaptingthecrossoverandmutationprobabilitydependingon thefitnessvalueofthesolution,i.e.forlow fitnessvalueassign highvalueofP_c andP_mand viceversa. AGAprotectsbestsolu- tionfromthehighlevelsofdisruptionwhichreceiveszeroPcwith smallamountofPmthisinformationishelpfulinreducingtheextra computationcostofapplyingcrossover.Inthisapproachacriteria weresetas“allsolutionwithafitnessvaluelessthantheaver- agefitnessvalueofthepopulationhavePm=0.5whichshowsthat sub-averagesolutionsarecompletelydisruptedandtotallynew solutionsarecreated”thiscriteriawasalsofoundeffectiveinsav- ingthecomputationalcostandhelpsGAnottogettrappedata localoptimum.

Asdiscussed,thisapproachgives aconcretebasis forselect- ingthecrossoverandmutationprobability.Itwasfoundthatmost oftheindividuals weretrappedatalocaloptimumbecausethe explorationeffectofmutationwaslow.ToaddressthisissueJung presentedaSelectiveMutationbasedapproach(discussedinSection 4.21)[35].

4.9. Socialdisastertechnique

Grefenstette[91]attemptedTravelingSalesmanProblemand suggestedtheconceptofgreedycrossover.Thecrossoveropera- torimplementedwascalled“greedycrossover”becauseitalways preferslocallycheaperpaths.Pickingalocallycheaperpathwas basedontheknowledgeabouttheproblemandshowedpromis- ingresultsthanmoreexpensiveone.Itwasevidentin[91]that crossoverbasedonknowledge,improveconvergence,butitmight derivetolocalminimaandleadstoprematureconvergence.

ToaddresstheaboveissueSocialDisasterTechnique(Kureichick etal.)wassuggestedwhichconsidersthewholepopulation[42].

Kureichicketal.[42]directedtheireffortstoavoidthefollowing tworeasons:

•Immoderatecrossovergreediness

•Lowinﬂuenceofrandomfactorsintendedtomakethepopulation getoutoflocalminima.

Theconceptofsuper-individualwassuggestedwhichtakesover thechargeofthepopulation.Catastrophicoperatorwasusedinthis methodtomaintainthepopulationdiversity.Packingand judg- mentdayoperatorwasusedfortheexperiment.Theworkingof packingoperatoristoapplyseparation,i.e.separatesallthetours havethesamelengthandonlyoneremainsunchanged,whereas incaseofjudgmentdayoperatoronlythebestinstanceremains unchanged.Inaddition,warpoperatorswereusedinpopulation, whichintroducespartialorfullrandomization.

AnexperimentwasconductedontheTravelingSalesmanProb- lem(TSP)usingthebenchmarksgivenbyOliver’s30[93],Eilon’s 50andEilon’s75[92].Itwasshownthatthethirtytownproblem wassolvedeasilywhile50and75townTSPswasfoundharder inthepresentedapproach.Theresultof5and75TSPswascom- paredagainsttheresultsgivenin[92]anditwasobservedthatthe suggestedapproachshowedtheshortertour.

Themethoddescribedabovebroughtsigniﬁcantimprovements, someofthemare:introducednewoperatorsprovidethefacility tousethepopulationpooleffectively.Itissuitableforthesmall populationsizeproblem.Warpoperatorbroughtthepartialorfull randomnesswhichhelpsthepopulationtogetoutofthelocalmin- ima.Decreaseinthegreedinessofthecrossoveroperatorincreases thevariationamongthepopulation.

Theweaknessofthisapproachis:therewasnosufﬁcientinfor- mationisprovidedtotelluptowhatextentgreedinessofcrossover shouldbedecreased.Thisapproachdoesnotﬁtforcomplicated problems,i.e.forlargepopulationsizeproblemsthisapproachcon- sumesmoretimetoconverge.

4.10. Theislandmodelgeneticalgorithm

Vose[94]andWhitleyetal.[95]presentedexactmodelsofan SGAusinginfinitepopulationassumption.Onederivationofthese modelswaspresentedin [95]while itwasextended tolookat finite populationsusingMarkov models[94].It wasfoundthat theweaknessoftheinfinitepopulationmodelwasthatonecould nottakeintoaccountthesamplingbiasintroducedbyusingfinite populations,whereasthefinitepopulationMarkovmodelwastoo expensivetoexecuteverysmallproblems.Therefore,itwassug- gestedthatusingadistributiontakenfromafinitepopulationone canintroducefiniteeffectsintotheinfinitemodel.

Whitleyetal.[69]suggestedanapplicationoftheinﬁnitepop- ulationmodeltoseethebehaviorofmultipleSGArunningparallel.

TheseSGAwasallowedtoexchangeinformationtimetotime.It wasshowninpreviousresearchthatparallelGAyieldbetterper- formancethanserial[70–72].IslandModelwasusedtoimplement GAonbothserialandparallelmachines[65–68].Inanislandmodel eachmachineexecutesaGAandmaintainsitssubpopulationfor directing thesearch.It wasobservedthatanisland modeluses multiplepopulationsforeachmachineinparallelimplementation which helpstopreservegeneticdiversity,since eachislandcan potentiallyfollowadifferentsearchtrajectorythroughthesearch space[69].

In order to implement island model two parameters were introduced,namelymigrationintervalandmigrationsize.Thenum- ber of generations between migrations was represented by a migrationintervalwhilemigrationsizeisthenumberofindivid- ualsinthepopulationtomigrate.Eachmachineintheislandmodel periodicallyexchangeaportionoftheirpopulationusingaprocess knownasmigration.

ThefunctionGwasusedtorepresenttrajectoryoftheinfinite populationSGA,whichworksinavectorp[94].p^t representthe populationatthetimetwhichshowsthesamplingdistributionof afiniteorinfinitepopulation.Thesearchprocessentersintonew generationusingfunctionGasdepictedinEq.(11).

p^t+1=G(p^t) (11)

wherep^t⁺¹representnextgenerationaninfinitelylargepopula- tion. It alsorepresentsthe expectedsampling distributionof a finitepopulation[69].TheconstructiondetailsofGwasgivenby Whitley[96].Itwasobservedthat thebehaviorofthefunction GwasexactlycorrespondtotheSGAprovidedtheinfinitelarge

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Island-1

Island-5 Island-6

Island-3 Island-2

Island-4 migrate most fit 5%

migrate most fit 5%

Delete the least fit 5% immediately after receiving most fit 5%

After X generate apply migration

Fig.4.BasicorganizationofIslandmodelusingsixislands.Forfirstmigrationafter Xgenerationmigratemostfit5%fromIsland-1toIsland-2(immediateneighboron theleft).AssoonasIsland-2receivethemigrateddataimmediatelydeletetheleast fit5%ofthepopulationthesameprocessrepeatedtoeveryotherislandexcluding itself.

population.Thedetailsabouttheﬁniteandinﬁnitepopulationwere coveredinvaryingpopulationsize(smallandlarge)[69].

Whitleyetal.[69]usedfunctionGtodevelopanidealizedIsland GA.Theideawastotalpopulationwasdividedintoanumberof subpopulationscalledas“islands”.Theseislandsareresponsibleto executeaversionofGAoneachsubpopulation.Eachislandman- agesa separatecopyof Gfor SGA using1-pointcrossover.The crossoverratechosenwas0.6.Itwasfoundthatmigrationoccurs betweensubpopulationinarestrictivemannerusingtemporaland spatialconsiderations.

ThemigrationstrategyappliedinIslandmodelisdepictedin Fig.4.Wehaveusedsixislandstodemonstratetheworking.From Fig.4itisclearthatalltheislandsarearrangedinaring.Forthe firstmigrationIsland-1migratemostfit5%afterXgenerationto itsimmediateneighborinthelefti.e.Island-2.AssoonasIsland-2 receivesthemigratedinformationitdeletestheleastfit5%ofits population.ForthesecondmigrationIsland-2migratethemostfit 5%toitsimmediateneighbortoitslefti.e.Island-3afterXgenera- tions.ThismigrationprocessoccurseveryXgenerationuntileach islandhassendonesetofstringstoeveryotherexcludingitself thentheprocessrepeated[69].

Experiments were conductedusing two separableand non- separableproblems.Twoseparableproblemswerechosenbecause theyrepresentexamplesfromawell-knownsetoflinearlysepara- bletestproblemwhilethenon-separableproblemwaschosento demonstratehowtheaddedinteractionbetweenparameterscan affecttheGAperformance.GENITORwasusedasasearchenginefor theexperiment[73].Theexperimentalresultsshownin[69]prove thattheIslandModelgeneticalgorithmsoutperformandmaintain thepopulationdiversity,avoidprematureconvergenceandreach totheglobaloptima.

Island GA helpsto preservegenetic diversity becauseevery islandpotentiallyfollowsadifferentsearchtrajectorythroughthe searchspace[69].Anabstractmodelwaspresentedwhichsupport ittooutperformsinglepopulationmodelsonlinearlyseparable problems.Itissuitableforlargetotalpopulationsizeproblems.

This model drawn the attention of researchers since it offerssomedegreeofindependenceandhenceexploredifferent regions of thesearch space while at thesame time sharing of

SBT Phases

Phase-1: Exploratory Phase Phase-2: Mass Selection Phase-3: Interdeme Selection

Random genetic drift and allow small population to explore

Uses the lucky gene combination of phase-1

Increse in size of more successful gene Increse in rate of migration Excess population shift the allele frequencies to nearby population A continual shifting of control from one adaptive peak to a superior one [97].

Fig.5. Threedistinctphasesofshiftingbalancetheory.

informationallowed.Theconceptofmigrationwasfoundeffec- tivewhichhelpsinmaintainingpopulationdiversityandthekey toresolvingprematureconvergence.

Thelimitationof thisapproach is themigrationinterval, i.e.

afterhowmanygenerationsoneshouldapplymigrationofbest fit individualfromone island toanother.Anotherfactor which was not defined clearly was migration size. For the presented experiments5%bestfitindividualwasmigratedbutwhatisthe guaranteethatitwillbesuitableforalltypesofoptimizationprob- lems. It wasacceptedthat thepresented approach couldshow verycomplexandunexpectedbehaviorwhenappliedtoseparable problem.

4.11. Shiftingbalancetheoryindynamicenvironment

Wright [20] presented Shifting Balance Theory (SBT). The ill behaviorofevolutionarysystemwasobserved,i.e.gettingtrapped atalocaloptimum.Itwasexplainedthatforafitnesslandscape theremightbetwopeaksorevenmore.Inruggedfieldofthischar- acter(verymanypeaks),selectionwilleasilycarrythespeciesto thenearestpeak,buttheremaybeinnumerableotherpeakswhich arehigherbutseparatedby“valleys”.Wineberg[97]showedhe thoughtofevolutionasamechanismbywhichthespeciesmay continuallyfinditswayfromlowertohigherpeaks.Theterm“trial anderror”wassuggestedasexploratorymechanismswhichtake thespeciestothenexthigherpeak,despitetheharshselection pressuresofthevalleyforcingthespeciesbacktotheoriginalhill [98].

HartlandClark[98]presentedagoodsummaryofSBTintheir book“PrinciplesofPopulationGenetics”.Theysuggestedthatsub- divisionofthepopulationintoasetofsmallsemi-isolateddemes increasesthepossibilityfordemestoexplorethefullrangeofthe adaptivetopographyandreachtothehighestfitnessvalueona convolutedadaptive surface.Reductionof largepopulationinto smallerdemesreducesthepossibilityofrandomgeneticdriftof allelefrequenciesandprovidestheflexibilitytoexploretheadap- tivetopography more or less independently. It was foundthat therandomgeneticdrift inany subpopulationleadstotempo- rary reductionin fitness. Thiscouldberectifiedbyselection in a largesize populationwhich leads thesearch process topass througha“valley”ofreducedfitnessandpossiblyendup“climbing”apeakoffitnesshigherthantheoriginal.Itincreasesthesize ifasubpopulationreachedtotheadaptivepeakonthefittedsur- face.Inaddition,migrationofindividualgeneswillbehigherand thereforethefavorablegenecombinationsaregraduallyspreading throughouttheentiresetofsubpopulationsbymeansofinterdeme selection.

HartlandClarkidentifythreedistinctphasesanddemonstrated theroleofeachphase[98]asdepictedinFig.5.

Wright’s Demes was one of the signiﬁcant contributions in justifying the introduction of a novel GA implementation

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particularly for parallel GA. It was observed that both ﬁne- grained and coarse grained model (Island model) is presented whencreatingparallelGA.Thesetechniquesuse“demes”toindi- cateamatingpoolofindividualslocallyclusteredgeographically [97]. We havealready discussed Island model(Section4.10)in whicheachprocessor runsanSGAonitsownsubpopulationor deme.

Cohoonetal.[99]initiatedtheassociationoftheWright’sSBT withIslandmodel style ofparallelGA. Cohoonetal.[100]jus- tifiedtheapplicabilityofislandmodel.Hepresentedtheideaof punctuatedequilibriaforparallelGAusingWright’stheory[20]and EldregeandGould’stheory[101]ofpunctuatedequilibria.Simi- lartoislandmodel,manyresearchersworkedonafinegrained modelusingWright’stheory[102–107].Inthiscase,eachproces- sorisgivenonlyasinglememberofthepopulation.Eachmember ispermittedtomatewithitsnearestneighborsorpossiblywithin asmallneighborhood.Thebasisofthisdesignwasthefactthatfast communicationmaybeachievedamongneighbor.Theneighbors canbearrangedusingvarioustopologies[97].Here,diffusionof geneticmaterialsoccursthroughneighborhoodoverlapwhichwas identifiedasdemes.

Although Wright’s theorywas acceptedand appliedin sev- eralapplications.Butthistheorywasfoundcontroversial.Coyne, BartonandTurelli[108]questionedaboutthevalidityandimpor- tance of SBT. Coyne et al. [112] studied both theory (Fisher’s theoryandWright’stheory)andrejectedtheideathat“Wright’s SBTtheoryhasplayedamajorroleinadaptiveevolution”.Instead Darwin’sandFishertheory[111]“adaptationisnaturalselection actingondifferencesamongindividualswithoutgeneticdrift, pop- ulationsubdivisionanddifferentialmigrationplayingthevitalroles hypothesizedbytheSBT”wasaccepted[112].Peck,EllnerandGould [109]andWadeandGoodnight[110]presentedaframeworkto support theSBT theory and claim that the dismissal of SBT is premature.

AsdiscussedtheSBTtheorywascontroversialstillOppacher and Wineberg [21] presented the modified version of SBT in whichtheoriginalgoalwastoavoiddefectsinherentinitsorig- inalformulationandpreventprematureconvergenceinGAs.The newalgorithmimplementedisknownasShiftingBalanceGenetic Algorithm(SBGA)whichimprovestheperformanceinadynamic environment.Twodifferenttypesof populationgroups, namely CoregroupandColonieswereusedinthemodifiedSBT.Thecore groupisthelargecentralpopulation,whileasmallergroupcalled ascolonies.Themainjobofcoloniesisfindingthesearchspaceof thelandscapesthatthecoregroupisunabletofind.Inthiswayone canpreventprematureconvergence.Todevelopacompletesystem fewmeasureswereconsideredsuchaspopulationdiversity(mea- suredbyEq.(12)),anddistance(measuredbyEq.(13).Distance representsthedistancebetweenasinglechromosomeandpop- ulationingenespace,isamodificationofthediversitymeasure [21].

Diversity(P)= 1 LN(N−1)

N i=1

N j=1

HD(p_i,p_j) (12)

Distance(c,P)= 1 LN

N i=1

HD(c,P_i) (13)

whereLrepresentsthelengthofachromosome,Nisthesizeofthe populationP,p_iistheithchromosomeinthepopulation,HDrepre- sentstheHammingdistancefunction.ThetermHD(c,P_i)showsthe Hammingdistancebetweenasinglechromosomecwhosedistance isbeingmeasuredandpopulationP_i.

Selected using regular fitness fucntion

Selected based on the distance from the core group New Population after reproduction of colony

divide into two parts divide into two parts

Fig.6.Structureofnewpopulationafterreproductionofcolony.

Onemoremeasurewasconsideredknownascontainment(Eq.

(14)).Itwasconsideredasameasureoftheextenttowhichone populationiscontainedwithinanother.

Containment(A,B)= 1 M

M i=1

WithinDistance(a_i,B)

= 1 MN

M i=1

N j=1

dp(a_i,b_j)

(14)

Where,withinDistance(c,P)=_N¹

N i=1

dp(c,p_i)

d_p(a,b)=

1 if Distance(a,P)<Distance(b,P)

0 Otherwise

a_iandb_jistheithandjthmemberofthepopulationAandB, respectively,andM,NrepresentsthepopulationsizeofAandB, respectively.

Table1presentsthestateofaffairsofthecolonyandtherem- edysuggestedtokeepcolonyawayfromthecore.Themeasureof containment(containmentfunction)canbeusedforthispurpose whichcanbecalculatedusingEq.(14).

ItisdepictedinFig.6thatwhenthecolonyisreadyforrepro- ductionthenewpopulationwasdividedintotwohalves.Member’s ofthefirst partwasselectedusing theregular fitnessfunction, whereasthememberofthesecondpartwasselectedbasedonthe distancefromthecoregroupusingthefactthatthegreaterthe distancehigherthefitness.

UnliketheSBTtheoryofimmigrationofcolonymembers,send- ingmembersfromthecoretothecolony,OppacherandWineberg [21]suggestednomigrationfromthecoretothecolony.Although thecolonystillsendsmigrantsbacktothecore.Itwasobservedthat colonymemberswereselectedrandomly,stochasticallyorusing elitesubgroupfor immigrationintothecorebased ontheirﬁt- ness.Addingthecolonymembersintocoreresultedinatemporary increaseinthesizewhichcouldbereducedtonormalsizeduring thereproductionbyexertingselectionpressureonthecore.Asthe migrationofcolonymembersdisruptsthecoregroup(seeTable1), hencetorectifythisissue,time(migrationinterval)wasgivenfor eachcolonytoevolvepotentiallyusefulmembers.

Toensuretheeffectivenessoftheimplementedapproachtwo experimentswereconductedusingtheF2functionfromtheDe Jongtestsuitecomposedwiththeonedimensionalversionofthe GriewangkfunctionknownasF8.Thecombinedfunctionwasused fortheexperimentcalledasF8F2function.Theprimaryaimofthe ﬁrstexperimentwastochecktheprematureconvergencewhile thesecondexperimentwasconductedtoensurethedynamicenvi- ronments.Theexperimentalresultswerefoundsigniﬁcantlybetter andencouragingthanthestandardGAtopreventprematurecon- vergenceandalsoruninadynamicenvironment.

Themainstrengthofthisapproachwastoresolvepremature convergenceanduptoanextendedthisapproachworkssuccess- fully.Thisapproachwasstrengthenedbyprovidingtheconceptof

(12)

Table1

Stateofaffairsofcolonyandtheirremedy.

S.No. Stateofaffairs Remedy

1 Ifcolonyremainscompletelyoutsidethecore. Noaction,Freelyevolveusingregularﬁtnessfunction,allowlookingpotentially interestingareaignoredbythecore.

2 Ifcolonymovesortryingtomoveintoareasearchedby thecoregroup.

Pushedcolonyawayfromthecoregroup.

3 Ifthecolonyisattheoutsideboundaryofthecore Pushcolonygentlyaway

4 Ifthecolonyisalmostentirelycontainedinthecore Applystrongpressuretopushitintonovelterritory

coloniestosearchareasofﬁtnesslandscapesmissedbythecore group.Theﬂowofcolonymemberofthecoregroupafteratime intervalincreasesthediversityofthecoreandhelpsinpreven- tingprematureconvergence.Inaddition,itwasfoundsuitablein adynamicenvironment.InSBGA,ifthepeakshiftsawayfromthe coregroupthecolonyarealreadyinanareaawayfromthecore.

OppacherandWineberg[21]assumednomigrationbetween colonyandcoreandremedialapproachesweresuggestedforthe colonymemberifentersinthecoresearchareastillsomeofthe colonymemberjumpsintothecoregroupwhichmightbeprob- lematicinmanagingthediversity.Thisapproachsufferswiththe problemof migration interval, i.e. when to migratethe colony member.Lastly,itwasacceptedthatthereisnodynamicmodel availableofthedynamicenvironment.

4.12. Randomoffspringgenerationapproach

RochaandJose[22]presentedanewapproachtoremovethe insufficiencyofGA.ThisapproachisknownasRandomOffspring Generation(ROG).BylookingattheflowchartshowninFig.7one caneasilyunderstandtheworkingofROGprocess.Theflowchart clearlydemonstratesthereasonof occurringof prematurecon- vergenceinGA.Theansweris fora largeamountofindividual isusingthesimilargeneticmaterialduringthereproductionpro- cess.Inotherwords,wecansaythatforthecrossoveroperation samegeneticmaterialswereused.Inthiscasereproductionpro- cesswillnotshowanyeffectbecausetheoffspringbredarenothing butsimplycloneoftheirparents.

Therefore,topreventprematureconvergenceinGA,ROGpro- cesscouldbeappliedbeforeapplyingthereproductionoperations.

Theideaisﬁrsttestthesimilarityofgeneticmaterial,ifgenetic materialsfoundsimilarthangeneratetherandomoffspringwhich willcodearandomsolutionfortheproblemotherwiseapplyrepro- ductionoperation.In[22]twodifferentstrategieswereusedinROG namely1-RO(oneoffspringpertwoparents)and2-RO.Incaseof1- RO,onlyoneoffspringwillbegeneratedandotheroneobtained fromtheirparenttoreachtotheresultwhile in2-ROboththe offspringwillberandomlybred.

ROGstrategywasappliedtoaselectedsetofTravelingSalesman Problem(TSP)[23].FortheexperimentTSPproblemwasusedas abenchmark.Thedataobtainedfromtheexperimentswerecom- paredwithotherapproachessuchasAdaptiveMutationRate(AMR) andSocialDisasterTechniques(SDT)onvariouscrossoveropera- tors.Thecomparativeresultsprovethatthetechniquepresentedin [22]isfarbettertopreventprematureconvergencetolocaloptima andhenceimprovetheperformanceofGAs.

Itwasshownexperimentallythatthepresentedapproachyields thebestresultscomparedtoadaptivemutationrate(AMR)(Sec- tion 4.21) and social disaster technique (SDT)(Section 4.9).As observedAMRshowedthesolutionwithasimilardegreeofquality asobtainedbySGAwhereasSDTineitherofitsformdidn’tshow anyimprovementinthesolution.Anotherimportantmeritofthis approachisthesimplenatureinimplementation.

Rocha and Jose [22] accepted that the weakness of this approachisthecomputationaloverheadsdisregardedsincethere is no change in the selection methodology compared to other approachessuchasCrowding[38](Section4.1),Sharing[16]and ScheduledSharing[15](Section4.3).

Start

Initial Population Generator Set Fitness = Fitness P(G)

Crossover Mutation Reproduction

Replacement to incorporate new individual population

Replacement to incorporate new individual population Not Similar

Generate offspring

Genetic Material? Similar

Terminator?

End Random offspring generation process

No

Yes

No