An adaptive multi-population differential artificial bee colony algorithm for many-objective service composition in cloud manufacturing

Zhou, Jiajun and Yao, Xifan and Lin, Yingzi and Chan, Felix T.S. and Li,

Yun (2018) An adaptive multi-population differential artificial bee colony

algorithm for many-objective service composition in cloud

manufacturing. Information Sciences, 456. pp. 50-82. ISSN 0020-0255 ,

http://dx.doi.org/10.1016/j.ins.2018.05.009

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In: Information

Sciences

An

adaptive

multi-population

differential

artificial

bee

colony

algorithm

for

many-objective

service

composition

in

cloud

manufacturing

Jiajun

Zhou

a

_,

_Xifan

_Yao

a,∗

_,

_Yingzi

_Lin

b

_,

_Felix

_T.S.

_Chan

c

_,

_Yun

_Li

d

a School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, Guangdong, China b Intelligent Human Machine Systems Lab, Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA

c Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong d Faculty of Engineering, University of Strathclyde, Glasgow G1 1XQ, UK

a

b

s

t

r

a

c

t

Severalconflictingcriteriamustbeoptimized simultaneouslyduringtheservice

compo-sitionand optimalselection (SCOS)incloud manufacturing,amongwhichtradeoff

opti-mizationregardingthequality ofthecompositeservicesisakeyissueinsuccessful

im-plementationofmanufacturingtasks.Thisstudyimprovestheartificialbeecolony(ABC)

algorithmbyintroducingasynergeticmechanismforfoodsourceperturbation,anew

di-versitymaintenancestrategy,andanovelcomputingresourcesallocationschemeto

han-dlecomplicatedmany-objectiveSCOSproblems.Specifically,differentialevolution(DE)

op-eratorswith distinctsearchbehaviors areintegratedintothe ABCupdating equationto

enhancethelevelofinformationexchangebetweentheforagingbees,andthecontrol

pa-rametersforreproductionoperatorsareadaptedindependently.Meanwhile,ascalarization

basedapproachwithactivediversitypromotionisusedtoenhancetheselectionpressure.

Inthisproposal,multiplesizeadjustablesubpopulationsevolvewithdistinctreproduction

operatorsaccordingtotheutilityofthegeneratingoffspringsothatmorecomputational

resourceswillbeallocatedtothebetterperformingreproductionoperators.Experiments

foraddressingbenchmarktestinstancesandSCOSproblemsindicatethattheproposed

al-gorithmhasacompetitiveperformanceandscalabilitybehaviorcomparedwithcontesting

algorithms.

1. Introduction

Cloudmanufacturing(CMfg)referstoaservice-orientednetworkedmanufacturingmodelinwhichserviceconsumersare enabledtoconfigure,select,andutilizeresourcesondemandsoastocompletecustomizedmanufacturingtasks[9].Adapted fromthecloud computingmodelandInternet ofthings(IoT) intothefield ofmanufacturing,CMfgwasfirstcoinedby Li etal.[17]andisgainingsignificantattentionfromboth academiaandindustry.CMfgaimsatconnectingconsumerswith smallandmedium-sizedenterprisesorindividual engineerstoformtemporary,reconfigurableproductionlines,providing consumerswithcapableandqualifieddesignandmanufacturingservicesinavirtualcommunity.

∗ _{Corresponding author.}

Tofacilitatetheaccessandutilizationof resources inheterogeneousenvironments, manufacturing resources or capac-ities inCMfg are virtualized anddelivered in the formofservices. However, dueto the functional limitationofa single service,the complexityofamanufacturing taskcalls fora compositeserviceinwhich multipleisolated resourcesare or-ganizedinto a set ofinteracting serviceswith morepowerful functions. Onthe other hand, theincreasing diversityand complexityofcustomers’individualizationrequirementshasresultedinthedifficultyforasinglefunctionalserviceto com-pleteauser’stask.Fromtheperspectiveofcostandefficiency,aspecializedserviceforeachtaskmeansrelativelyhighcost andslowresponse.Therefore,assemblingmodularservices intoanaggregatedoneto meetacustomer’srequirementwill strengthenthe flexibilityandagilityforCMfg.Foracomplicatedmanufacturing task,theCMfgsystemdecomposesitinto severalsubtasksandsearchesall qualifiedservicesforeachsubtasktoformacandidateservicepool,albeitwithdifferent QoS(qualityofservice).Then,thesystemselectsoneservicefromeach servicepool togeneratea compositeservicewith thegivenmulti-objectiveoptimization(e.g.,multipleconflictingQoSpropertiesneedtobeoptimizedsimultaneously),and suchaprocessiscalledservicecompositionoptimalselection(SCOS).Obviously,withthemassofservicesavailableonthe pool,SCOSbelongstothesetofNP-hardmulti-objectiveoptimizationproblemswithcharacteristicsofnonlinearobjective functions,combinatorialexplosion,andhigh-dimensionaldecisionspace.

Variousapproaches havebeen proposed for solving such problems (e.g.,[31,33]). Dueto the combinatorial explosion, exactapproachesor exhaustivesearch algorithms forSCOSare computationally expensiveorsufferfromfinding an opti-malsolutionwithina polynomialtime.Toaddressthisissue,scholarshavedevelopedmanyevolutionaryalgorithmssuch asthegeneticalgorithm(GA)[31,33]andparticleswarm optimization(PSO) [34].Inthesestudies,SCOSindifferentkinds ofscenariosforachievingspecific goalsisdiscussed. However,almost allofthem transformthemulti-objectiveproblems (MOPs)intosingleobjectiveproblems(SOPs)by weightedcombination. Thisissomewhat rigidintermsofflexibility.Due totheconflictingnatureofmultipleobjectivesinMOPs,asinglesolution,therefore,isnotavailable.Instead,asetof trade-off solutions,calledaParetooptimalset,isgenerallypreferredpractically.Userscanselectoneofthemaccordingtotheir preference.Therefore,providingmultiplealternativeswithParetooptimalityinasinglerunbyusingmulti-objective evolu-tionaryalgorithms(MOEAs)isbecomingmoredesirable.ComparedwithSOPs,MOPsaremorecomplex,whoseoptimization targetistoachievethebesttradeoff betweenmultipleobjectives,andthealgorithmsmayneedmoreinteractionstoform ahybridoptimizerwithpowerfulsearchingability.

Theartificialbeecolony(ABC)algorithmisafairly newswarm intelligentalgorithminspiredby honeybeeforaging be-havior,andwasfirstdevelopedbyKarabogaandBasturk[14].Withfewerparameters,ABCisrelativelyefficient,robustand successfulinsolvingtherealworldproblems[30].ThebasicversionofABCisveryefficientintacklingSOPs[15].However, its performance is not effectivein MOP andmany-objective problem contexts. Thisis mainly dueto the inefficientfood sourceperturbationinthecaseofahighdimensionalobjectivespace.Toovercomesuch drawbacks,we introduceseveral promisingDE mutationstrategies tostrengthenthe honeybeeforagingbehavior, andadjustthe operatorcontrol parame-tersfor goodoffspring reproduction. Inaddition, the quality-indicator basedfitness assignment approach isemployed to identifyprominentsearchregionsintheABConlookersstage.Furthermore,a well-maintainedexternalarchivetailoredfor many-objectivecontextsisemployedtoguidethecollaborationofsearchoperators.

The rest of this paper is organized as follows. Section 2 reviews the relevant work concerning service composition and multi-objective evolutionary computing. In Section 3, the SCOS problem and its optimization model are described.

Section4explainsthenewlyproposed algorithmanditskeycomponents.InSection5,comparisonexperimentsonawide rangeofbenchmarkproblemsarediscussedandanalyzed.TheproposedalgorithmisthenappliedtotackleSCOSproblems inSection6.Finally,conclusionsaregiveninSection7.

2. Relatedwork

Fromthestandpoint ofserviceaccessibilitiesandresourceallocation efficiency,designingall thepossiblerequired ser-vicesforataskintherealworldisextremely difficultandissomewhat rigidintermsofflexibility,particularlyfora cus-tomer’spersonalizeddemand.Providingsimpleandsinglefundamentalservicesthatcanbe configured,selected,and col-laboratedforcustomizedscenariosmakemoresensesincesuchapproachesaremoreagile,scalableandelastic.Asaresult, howtoselectappropriateservicesfrommultiplecandidateswithdifferentQoSpropertiestogenerateacompositeservice whileguarantyingthebestQoSoftheresultmustbe investigatedandaddressed. Unfortunately,sucha processis charac-terizedwithNP-hard complexitydue tothe exuberant growthofservices offered inthe cloud, andmostof the existing methodsmeetaperformancebottleneckwhenhandlingthecomposition.

CloudcomputingfurtheracceleratestheincreaseofavailableservicesontheInternet,andQoShasdrawnmoreattention duetomanycandidatesmeetingthesamefunctionalrequirement,albeitwithdistinctQoS.Asaresult,itisachallengefor compositionsystemstochooseanoptimalsetofrequiredservicesfromdifferentservicepoolsefficientlywhileguaranteeing theoverallQoS.Compositiontechniquesforcloudservicesfirstappearedin2009[13],andafterwards,substantialacademic effortsweremadeinthisfield.Withthecapabilitiesofhandlingcomplexsituations,gradient-freemechanisms,and obtain-ingnear-optimalsolutionswithin ashortperiodoftime,evolutionary algorithmshaveattractedmuchattentioninrecent years,becomingapopularapproachintheliterature forsolving SCOSproblems.In[33],GAanditsvariantswere usedin partnerselectioninvirtualmanufacturing.In[34],anenhancedPSOwasdevelopedforSCOSinamanufacturinggrid.Huang etal.[11]usedanewchaoscontroloptimalalgorithmwithbetterpopulationdiversitytoaddresstheSCOSproblem,which performedbetter than the chaotic GA. In [32], a parallel chaos optimizernamed FC-PACO-RM wasintroduced to handle

SCOSinCMfg.Lailietal.[16]analyzedthecharacteristicsofresourceservicesintheprivate cloud,andconstructedadual schedulingmodelformanufacturingserviceselection.

Generally,theaforementionedQoS-awareSCOSapproachesshowdistinctcharacteristicsandhavebeenappliedin differ-entscenarios.However,thereisstillroom foradaptabilityandefficiencyimprovementintheseapproaches.Infact,dueto thecomplexityoftheCMfgenvironmentandthelargescale ofavailablecandidateservices,currentmethodsare undesir-ableto someextentandsufferfromsearchingspaceexplosion,difficultiessuch asloss ofdiversityandlocalconvergence associatedwiththeevolutionaryprocess.Combining themeritsofseveralalgorithms isan effectiveway,which,however, oftenleadstothedifficultyindetermininganappropriatecollaborativemanner.Moreover,fromtheaspectoftheobjective dimension,mostexistingapproachestrytocreatea utilityfunctionbyaggregatingmultipleobjectivestogether,which es-sentiallytransformsthemulti-objectiveservicecompositionintosingleobjectiveoptimization.Inpractice,whetherornot theaggregationfunctionscan trulyreflecttheimportanceofdifferentQoSpropertiesremainsdoubtful.Anotherdrawback oftheaggregation-basedmethodconcerns theinherentcharacteristicsofthe aggregationfunction.Forinstance,weighted sumaggregationisincapableoffindingnon-convexpartsofthePF(ParetoFront).Thus, toprovidediversifiedalternatives forusers,apowerfulalgorithmforSCOSinamany-objectivecontextisrequired.

ABC isa stochastic search algorithm based onthe intelligent swarm behavior ofbees, andhasbeen appliedfor solv-ing realwordoptimizationproblems[30,36]having greatcomputational difficulties.Inrecentwork, severalmodifications weremadefortheoriginal ABCaswell ashybridizationwithother meta-heuristicmethods[36,48,49].Examinationof re-liable publications showsthat ABC potentially outperformsother well-known meta-heuristic approachessuch as GAand PSO[15].Nonetheless,limitedeffortshavebeendevotedtoSCOSinCMfg.Indeed,severalexistinghybridalgorithmstryto addressSCOSbuttheyonlyfocusononeobjectivebyusingascalarization-basedapproach[49]ortwo representative ob-jectivesbyreduction[31,48].Inpractice,theobjectivereductionstrategycanbeoflimitedusebecauseitishardtosatisfy customers’manydiversepreferencesinfocusingondifferentQoSaspects,whichofteninvolvefourormoreobjectives.This phenomenongivesrisetoanewterm,formallyknownasmany-objectiveoptimizationproblems(MaOPs).MaOPsposegreat challengestoexistingPareto-basedmulti-objectiveevolutionaryalgorithms(MOEAs)suchasNSGA-II[6]andSPEA2[50],as theirPareto-dominance-basedselectionpressureisseverelydegradedwiththeincreasingnumberofobjectives.Asaresult, Pareto-basedMOEAs sufferfrom slowconvergenceto the PF.On the contrary,MOEA based on decomposition(MOEA/D) dividesacomplexMOPintoanumberofSOPsandsolvestheminacollaborativemanner,whichfacilitatesconvergenceby thescalarization-basedapproach.However,apotentialdrawbackofMOEA/Disthathighconvergencepressuremayleadthe algorithmto convergeonsomenarrow partsofPFwhenthepopulation diversityisnot wellmaintained.Inrecentyears, a numberofmany-objectiveevolutionaryalgorithms (MaOEAs)havebeensuggested. Based ondifferentselection criteria, MaOEAscanberoughlycategorizedintofivetypes:

(1) Modified-Pareto-dominance-basedapproaches:referringtothoseovercomingtheineffectivenessofParetooptimality causedbythecurseofdimensionality,including

θ

-dominance[41],fuzzyParetodominance[10]andgriddominance

[40].

(2) Decomposition-basedapproaches:referringtothoseusingtheframeworkofMOEA/D(e.g.,MOEA/D-DU[42],EFR-RR

[42],RPEA[26] and

θ

-DEA [41])orMOEA/D-M2M[24] withanenhanced replacementstrategyordifferentranking schemes.

(3) Indicator-based approaches: referring to those utilizing various performance indicators like IGD-indicator [35], HV indicator[1],andR2indicator[7]toevaluatethequalityofsolutions,suchasAR-MOEA[35],HypE[1],andMOMBI-II

[7].

(4) Diversity-based approaches: referring to those attempting to alleviate the loss of selection pressure through im-proved diversity management schemes, e.g., SPEA2+SDE with shift-based density estimation [23], NSGA-III with niche-preservationoperator[5],MaOEA-DDFCbasedondirectionaldiversityandfavorableconvergence[4],VaEAwith vectoranglebaseddiversitypromotion[39],SPEA/Rwithreferencedirection-baseddensityestimator[12],and1by1EA withadistributionindicator[25].

(5) Hybridapproaches:referringtothoseadoptinghybridselectionmechanismstoemphasizebothapproximationand di-versity,includingMaOEAsbasedonthecombinationofParetodominance-anddecomposition-basedselection mecha-nisms(e.g.MOEA/DD[20],andDPP[22]),andMaOEAsbasedonthecombinationofParetodominance-and indicator-basedselectionmechanisms(e.g.,P-SEA[8]andTwo-Arch2[37]).

MotivatedbythesuccessofmultipleselectioncriteriaandthemeritofABC,wesuggestanewoptimizer,whoseselection criteria,basedondecomposition,indicatorandparetodominance,aremergedintotheABCframeworkwitha complemen-taryeffect.Tobespecific,decompositionbasedselectionisusedtoenhancetheselectionpressuretowardsthetruePF.The synergyofpareto-anddecomposition-basedapproachesisemployedtomaintaintheexternalarchiveforrobustsearching. TheIndicatorbasedapproachisadoptedtoidentifythepotentialsearchregions.Asforreproduction,multipledifferentDE operatorsareemployedinanadaptivemanner.ThisproposedhybridalgorithmfollowstheidenticalframeworkofABCwhile introducingmultiplereproductionoperatorsandasynergyofdistinctselectioncriteria.Multi-populationswithvaryingsize aredesignedtoadjustcomputationalresourcesbetweenoperatorsautomaticallyasevolutionproceeds.

Fig. 1. Overview of SCOS procedure.

Table 1

Aggregation QoS evaluation.

Structure Time Cost Reliability Availability Reputation

Sequence Q1(seq) = n  i=1 q1 (C S i) Q2(seq) = n  i=1 q2 (C S i) Q3(seq) = n  i=1 q3 (C S i) Q4(seq) = n  i=1 q4 (C S i) Q5(seq) = n  i=1 q5 (C S i) /n Parallel Q1(par) = max ( q 1 (C S i)) Q2(par) = n  i=1 q2(C S i) Q3(par) = n  i=1 q3(C S i) Q4(par) = n  i=1 q4(C S i) Q5(par) = n  i=1 q5(C S i) /n Selective Q1 (sel) = n  i=1 ( q 1(C S i) ∗λi) Q2 (sel) = n  i=1 ( q 2(C S i) ∗λi) Q3 (sel) = n  i=1 ( q 3(C S i) ∗λi) Q4 (sel) = n  i=1 ( q 4(C S i) ∗λi) Q5 (sel) = n  i=1 ( q 5(C S i) ∗λi) Circular Q1 (cir) = θ∗ n  i=1 q1 (C S i) Q2 (cir) = θ∗ n  i=1 q2 (C S i) Q3 (cir) = n  i=1 q3 (C S i) Q4 (cir) = n  i=1 q4 (C S i) Q5 (cir) = n  i=1 q5 (C S i) /n 3. Problemstatement

A complicated manufacturing task in CMfg can be represented by the directed acyclic diagram (DAG) with flow-dependentstructures(e.g.,sequence,parallel,selective,circular),inwhichthenodes,numberedfrom1ton,representthe decomposedsubtasks,andthedirected acyclicarrowsdenotethe dependencyrelationshipsamongthesubtasks. Multiple candidateservices(CSs)aregeneratedaccordingtothefunctionalrequirementsofeachsubtask,i.e., asetofserviceswith identicalfunctionalitybutdifferentnon-functionalpropertiesformsthecorrespondingcloudserviceset(CSS).Formally,all theCSsmeetingtherequirementsoftheithsubtaskST_iaredenotedasCSS_i.TheCSScanbegeneratedbysemanticmatching oftheservicefunctionalitybetweensupplyanddemand.EachCSisassociatedwithseveralQoSparameters.Fig.1presents theprocedureofSCOSalongwithitsarchitecturelayersandtheoptimalselectionschema.Theoretically,acomposite man-ufacturingservice(CMS)consistsofmultipleCSsrespectivelyselectedfromdifferentCSSs.IfthesizeofavailableCSsforST_i

iski,wherei∈{1,2,,n},therewillbeni=1kipossiblecompositions,andtheaimofSCOSistofindaCMSwiththebest

QoSfromallpossibleones.

Inpractice,QoSconsistsofseveralnon-functionalmetricsfocusingondifferentaspects(e.g.,time,cost,reliability, avail-abilityandreputation),soit ishardto achievea solutionwiththebeststate inall metricssimultaneously. Instead,there existsasetoftradeoff options,knownasPareto-optimalsolutions,whereanimprovementofonemetricmayresultin de-teriorationoftheothers.Forinstance,timeandcostcanbeclearlyopposed,andimprovingthetimewouldprobablyimply adeteriorationincost.Takingtheimportanceofmeasurabilityintoconsideration,thispaperfocusesonfiverepresentative QoScriteriashowninTable 1,withthespecificmeaning oftheseQoScriteriaexplained in[11,32].Interestedreadersare referredto[13]formoreQoScriteriaincloud-basedsystems.

GiventheQoScriteriamentionedabove,a QoSsetofcomponentservicescanbe expressedasq={q1,q2,q3,q4,q5},a QoSsetofCMSisdescribedbyQ={Q1,Q2,Q3,Q4,Q5},andtheQoSvaluesofaCMSaredeterminedby theQoSvaluesof itscomponentservicesaswellastheinterconnectionstructureamongcomponentservices.

Fig. 2. Four basic composite structures.

Ingeneral, thereare fourbasiccomposite structures:sequence,parallel,selectiveandcircular,asshowninFig. 2,and detailedformulasforaggregationQoScriteriavaluesofcompositeserviceswithdifferentstructuresarepresentedinTable1. BasedontheformulasinTable1,theparallel,selectiveandcircularstructurescanbetransformedintosequencestructures

[34].NotethatthescaleofaggregatedQoScriteria’svaluescouldbeverydifferent,whichwillseriouslyaffectthe scalariza-tionfunctionindecompositionbasedMOEAs.Toavoidthisissue,alltheaggregatedQoSvaluesarenormalizedinthesame rangeof[0,1].Specifically,theQoScriteriacanbeclassifiedintopositiveandnegativeones.Forpositivecriteria,larger val-uesmeanbetterperformance,whilefornegativeones,smallervaluesmeanbetterperformance.Thenormalizationformulas forpositiveandnegativecriteriaareshowninequationsbelow,respectively.

˜ Qr

(

CMS

)

=

⎧

⎨

⎩

agqr(CS j_i i)  −agmin qr(CSSi)  agmax qr(CSSi)  −agmin qr(CSSi) , ifag



max qr

(

CSSi

)



 =ag



minqr

(

CSSi

)



1, otherwise (1) ˜ Qr

(

CMS

)

=

⎧

⎨

⎩

agmax qr(CSSi)  −agqr(CS j_i i)  agmax qr(CSSi)  −agmin qr(CSSi) , ifag



max qr

(

CSSi

)



 =ag



minqr

(

CSSi

)



1, otherwise (2)

whereoneCSperCSSisselectedtoconstituteaCMS,i.e.,CMS=

{

CSj1 1,· · · ,CS ji i,· · · ,CS jn n

}

,CS ji

idenotestheselectedCSfrom

CSSi,ji∈[1,k]andi=1,2,· · · ,n;qr

(

CS_iji

)

denotesthevalueofrthQoScriteriaforCS_iji;

max

qr

(

CSSi

)

and

min

qr

(

CSSi

)

are,

respec-tively,themaximumandminimumpossiblevaluesofrthQoScriteriaforCSsinCSSi.Theaggregationsymbol(i.e.,ag={



,



,avg,minandmax}) canbeused accordingto thecharacteristicsoftheQoScriteriaandinterconnectionstructures be-tweencomponentservices,asshowninTable1.

When MaOEAs are adopted to tackle SCOS, each QoS criterion corresponds to an objective function. Let x=

{

x1,· · · ,xi,· · ·xn

}

be a CMS,where xirepresents theselectedCS ji

i forSTi. Theobjective functionsforthemaximum

qual-itycriterioncanberewrittenasfollowsinthecontextofminimizationproblems: Minimize F

(

x

)

=

{

f1

(

x

)

,· · · ,fr

(

x

)

,...,fm

(

x

)

}

Subjectto: x∈



(3)

where,

fr

(

x

)

=1− ˜Qr

(

CMS

)

(4)

whereF:



→Rm _{consistsofm objectivefunctions(QoS criteria);R}m _{istheobjective space;andxisthe decisionvector}

consisting ofn variables.



=n

i=1[1,ki]∈Rn is called thedecision space.Such a problemcan be calledmany-objective

SCOSifm≥4.

4. Proposedalgorithm:MS-DABC

4.1. Frameworkoftheproposedalgorithm

TheproposedMulti-populationSelf-adaptionDifferentialABC(MS-DABC)followsthesameschemaasthebasicABC,but involves crucial improvementstrategies inmany-objective contexts, such as foodsource updating, external archive(EXA)

Algorithm 1 General framework of MS-DABC. Initialization()

while termination criterion is not fulfilled do g = g + 1 ;/ ∗_{update generation counter} ∗_/ Update_Parameters(); Send_Employed_Bees(); Send_Onlooker_Bees(); Send_Scout_Bees(); Update_Utility(); end

Output EXA; / ∗_{Return the Pareto front} ∗_/

maintenance, computational resource allocation amongsubpopulations, and potential search region detection in the on-lookerbeephase.Specifically,severalbeecoloniesevolveindependentlywithdifferentreproductionoperatorsandexchange informationperiodically,andthe populationsize foreach colonycan be adjustedbased ontheutility ofthe correspond-ing subpopulation, thereby resulting in a dynamical configuration of the computational efforts. The EXA, maintained by Pareto-dominanceanddecomposition-basedselectionsinasteadystatemanner,isemployed tomeasuretheutilityofthe subpopulations.Themainfeaturesoftheproposedalgorithmaresummarizedasfollows:

(1) Referencevectorguidedmatingrestriction:inamany-objectivespace,arbitrarilyselectedsolutionsforthe reproduc-tionoperation arelikely toproduce an offspring faraway fromits parents.This maycausean unacceptable distri-butionofsolutionsandleadtosevere deteriorationintheconvergenceofthealgorithm.Thus,mating restrictionis introducedsothatmatingparentscanbechosenfromaneighborhood,asmuchaspossible.

(2) The adaptive synergy ofmultiple evolutionary operators: to exploit the meritsof different operators, we combine multipleevolutionaryoperatorsbyusingdynamicconfigurationschema,inspiredbytheideasin[38]and[21]. How-ever,differing fromthe previousensemblemethodconcentratingon SOPs[38]orMOPsusingdecompositionbased techniques(see [21]), in which the fitness improvement can be measured straightforwardly, herethe selection of operatorsisbasedontheircontributionstoEXAmembers.

(3) Indicatorbasedprominentsearch region detection: traditionally,in orderto identifythe potential search region in theonlooker beephase, Paretodominance anddensity informationis employed tocompute thefitness value of a givensolutionforproblemsinvolvingtwoorthreeobjectives(see[6,50]).However,duetothedominanceresistance issuein many-objective space, such information failsto identify the difference between solutions andis naturally unsatisfactoryin fitness assignment. To remedy thisissue, a performance indicator is employed to achieve fitness assignmentinprominentsearchregiondetection.

(4) Hierarchical elite-preserving of the EXA: a new elite-preserving strategy, which combines dominance and decomposition-basedapproaches,isusedtobalancetheconvergenceanddiversityofsolutionsintheEXA.The updat-ingoftheEXAisconductedbasedonnondominatedsorting,thelocalniche-countsofthesubregions,andscalarization functionsinahierarchicalmanner,whereaworstsolutionintermsofdominancecansurviveincaseitisassociated withan isolated subregion. Thisisbeneficial forachievingabalance betweentheconvergenceanddiversity ofthe

EXA.

ThemainprocedureofMS-DABCisshowninAlgorithm1,whichisconstitutedinfourmainparts:Initialization,Update Parameters,BeeForagingBehavior(e.g.,SendEmployedBees,SendOnlookerBeesandSendScoutBees)andUpdateUtility. NotethatthearchivemanagementstrategyisembeddedinthesubpartsofBeeForagingBehaviorinasteadystatemanner, computational resource allocation is achieved in Update Utility, andthe parameter adaptation ofreproduction operators is realized in Update Parameters.To deal with MaOPs with badly scaled PFs, the objective vectors of the solutions are normalizedineachgenerationbyidealandnadirpoints,assuggestedin[39].Inthefollowingsections,thekeycomponents ofMS-DABCaregivenaftertheintroductionofsomepreliminaryknowledgeofABC.

4.2.BasicABCalgorithm

ABC[15]isinspired by thefoodforagingbehavior ofhoneybees.The optimizationprocedureofABC startsbythe ini-tializationsimilarwithothermetaheuristics.Thenthewholebeecolonyisdividedintothreesubgroups,i.e.,theemployed bees,onlookersandscouts,toperformdifferenttasksuntiltheterminationconditionissatisfied.Employedbeestryto ex-plorenewfoodsources by learningfromneighborsandthensharethe nectaramountofthefoodsources withonlooker bees.Givenafoodsourcexi,theemployedbeeitriestofindoutatemporarypositionvibythefollowingequation:

v

_{i j}=xi j+

φ

i j

(

xi j−xk j

)

(5)

wherexijandvij representthejthvariableofxiandvi,respectively;

ϕ

ijisrandomlyselectedinrange[−1,1];i,kandjare

Algorithm 2 Initialization().

1: / ∗_{The setting of control parameters} ∗_/

2: Set generation counter g = 0, utility update period ng , the abandonment criteria limit , set the control 3: parameters of reproduction parameters in subpopulations μC R k = 0 . 5 , μFk = 0 . 5 , πk = 0 , S CR,k = φ,

4: SCR,k = φ, S F,k = φ, and Fesk = 0 for k = 1 , 2 , 3 , 4 ;

5: / ∗_{The initialization of food sources} ∗_/

6: Generate an initial population pop = { X 1 , ·· · X N_{} by random sampling from ;}

7: Randomly partition pop into subpo p k( k = 1 , 2 , 3 , 4 ) with respect to their sizes;

8: Initialize ideal point Z ∗_{= ( Z} ∗ 1 , ·· · Z ∗

m) by setting z ∗i = min { f i( x 1) , ·· · f i( x N)} :

9: EXA ← pop ;

10: / ∗_{Generation of reference ( weight ) vectors} ∗_/ 11: Generate a set of reference vectors ={λ1 , .. . _λN_};

12: Perform the assignment of neighborhood E(i) = { i 1 , ·· · , i T} , where λi1 , ·· ·λiT are the closest

13: vectors to λi based on the Euclidean distance;

respectively.Foraminimizationproblem,thefitnessvalueforvicanbecalculatedby:

fiti=

1/

(

1+fi

)

if fi≥0

1+abs

(

fi

)

iffi<0 (6)

Accordingtothequality(nectaramount)ofthefoodsourcesharedbytheemployedbees,anonlookerprobabilistically choosestovisitafoodsourcebyusingequationbelow:

pi=fiti

N

k=1fitk (7)

wherefit_irepresentsthefitnessoftheithsolution,i∈{1,2,…,N}.Then anewfoodsourceisgeneratedby usingEq.(5)and thebetteroneismemorizedforfurtherexploitationinthenextiterations.

If the quality ofa food source cannot be improved by a predetermined number oftimes named limit, the bee of a correspondingfoodsourcebecomesascoutthatexploresanewfoodsourcerandomly.

4.3. Initialization

The initializationprocedureofMS-DABCisprovided inAlgorithm2,whichcontainsthreemainaspects:thesettingof controlparameters,theinitializationoffoodsources,andthegenerationofreference(weight)vectors.Parametersinvolved inthealgorithmareintroducedinthefollowingsections.TheinitializationoffoodsourcesisthesameastheoriginalABC aforementioned.Toalleviatetheblindnessofparentsmatingforoffspringproduction,thebeecolonyusestheneighborhood conceptofMOEA/D[45]formatingrestriction,eachfoodsourceisassociatedwithasubproblem,andeachsubproblemis predefinedbyauniqueweightvector,whichalsospecifiesasearchsubregionintheobjectivespace.

Tobespecific,asetofevenlyspreadweightvectors

={

λ

1_,…,

_λ

N_{}isgeneratedbyusingasimplex-latticedesignmethod}

suggestedin[45].Inthecaseofobjectivenumberm>5,thetwo-layerweightvectorgenerationmethodproposedin[20]is adoptedforbetteruniformity.Eachweightvector

λ

i₌

₍

_λ

i

1,

λ

i2,...,

λ

im

)

,specifiesauniquesearchregion



i,whichisdefined

as:



i₌

_{

_F

₍

_x

₎

_|

_θ

i

₍

_x

₎

_≤

_θ

j

₍

_x

₎

_,

_∀

_j∈

_{

_{1,· · · ,N}

_}}

₍₈₎

θ

i

₍

_x

₎

₌

_λ

i_,F

₍

_x

₎

_−z∗

=arccos



λ

i_×

₍

_F

₍

_x

₎

_−z∗

₎



λ

i



_

_×

_F

₍

_x

₎

_−z∗



(9) wherei∈{1,...,N},x∈



,and

θ

i_{(x)istheacuteanglebetween}

_λ

i_andF(x)-z∗_,z∗_{istheidealpoint.Thereafter,forasolution} x,theindexofitsassociatedsubregion



k_isdefinedas:

k=argmin

i∈{1,··· ,N}



F

(

x

)

−z∗_,

_λ

i

(10) Byintroducing theneighbor conceptofMOEA/D,each foodsourceisassociated witha weightvector, whichhasbeen assignedwithaneighborhoodbasedontheEuclideandistance.Thereafter,eachfoodsourcecanbeoptimizedbyusingfood sourceschosenfromitsneighborsubregions.

4.4. Foodsourceupdate

Theproposedalgorithmmaintainstwotypesofpopulations,thecurrentpopulationpopandtheEXA.Toupdatepop,the scalaroptimizationsubproblemsofMOEA/Dareemployedtoestimate thequalityofthesolutions.Foreach weightvector inMOEA/D,theresultingproblemiscalledasubproblem,whichcanevaluatethefitnessvalueofasolution.Threepopular

Fig. 3. Illustration of PBI.

aggregationapproaches, the weightedsum, weightedTchebycheff and penalty-based boundary intersection (PBI), can be usedto form subproblems.The PBI approach isadopted inthis paper,due toits advantages inhandling many-objective problems.ThePBIapproachisdefinedas:

Minimize gpbi



x



λ

i_,z∗



=d1+

θ

d2 (11) whered1=





(

F

(

x

)

−z∗

)

T

λ

i







λ

i



_

(12) andd2=











F

(

x

)

−



z∗_−d 1

λ

i



λ

i



_













(13) where

θ

isauser-definedpenaltyfactor;d1 andd2 arethelengthoftheprojectionofvector(F(x)-z∗)ontheweightvector

λ

i_{andtheperpendiculardistancefromF(x)to}

_λ

i_{,respectively.z}∗₌

₍

_z∗

1,· · · ,z∗m

)

istheidealpointanditsithcomponentcan

becomputedby:

z∗

i =min_x

∈fi

(

x

)

(14)

Fig.3presentsasimpleillustrationofthePBIapproach.

Theupdatingprocedureofpopisbasedonlocalmatingandlocalreplacement.Localmatingconsidersselectingparents withmatingrestriction(e.g.,fromneighboringsubregions) toproduceoffspringwhilelocalreplacementmeanseach child solutionreplaces,atmost,acertainnumberofsolutionsinitsneighboringsubregions.Specifically,givenasubproblemi,an offspringsolutionxcwillbegeneratedbyapplyingreproductionoperatorsonsomeselectedparents fromtheneighboring

subregions.Then,xcisusedtoreplaceatmostnrsolutionsinpopifitperformsbetterthanthemintermsoftheirrespective

subproblems’aggregationvalues.

4.5.Followingprobabilityofonlookerbees

Traditionally,thefitnessinvolvedinEq.(7)forMOEAsisevaluatedbasedonParetodominancerelationshipandthe den-sityinformation(suchascrowdingdistance[6]inNSGA-IIandkthnearestdistance[50]inSPEA2),andtheformerisalways preferabletothelatter. Suchmechanismprovidesmoreopportunity forsolutionsbeingthefirstParetofront butpossibly withpoordiversity,soitwillbias thesearchtowardasmallsegmentofPF.Ontheotherhand,Paretodominancefailsto distinguishsolutionsinthehigh-dimensionalobjectivespace[5],andfitnessassignmentbasedpurelyoncrowdingdistance maycauselossofselection pressureforsolutions toconvergetowardthePF, andthe searchwill belargely directionless. Inspiredbythepromisingperformance ofindicator-basedMOEAsinaddressing MaOPs,indicator-basedfitnessassignment isemployedtoguidethefoodsourceselectionforonlookers.

Inprinciple,anyperformanceindicator (e.g.,HV[1]andepsilonmetricIε+[37]) canbeappliedinfitnessevaluation.In thisstudy,we usetheepsilon metricIε+, duetothe highcomputationalcost ofHV.Tobe specific, thefitness ofafood sourcexisestimatedbasedonepsilonmetric:

fit

(

x

)

=

x′∈P\x

subjectto Iε+

(

x′,x

)

=min

 ε

|

fi

(

x′

)

−

ε

≤fi

(

x

)

,i∈

{

1,· · · ,m

}



(16) wherePrepresentsthecurrentpopulation,parameterkisascalingfactorthatissetas0.05inthisstudy.Eq.(15) approxi-matestheHVcontributionsofasolutionregardingtheoptimizationgoalbyestimatingthesumofɛneededtocoverxby theremainingmembersofthepopulationP.Eq.(16)givestheminimumdistanceinwhichsolutionx’ canbetranslatedin onepossibledimensionoftheobjectivespacesuchthatxisweaklydominated.Inessence,fit(x)isadoptedtoestimatethe “lossinquality” ofthewholesolutionsetifxisremovedawayfromP.Notethatthehigherthevalueoffitness,thebetter thequalityofthefoodsource.Toavoida solutionwiththeextremely largestfitnessalwaysbeingselected,hencecausing thepopulationprematurelytoconvergeintoa localoptimum,insteadofusingtherawfitnessvalue,therankingof popu-lationmembersisusedtoassignthefollowingprobabilityofonlookers.Wecansortthepopulationmembersaccordingto fitnessindescendingorder.Theranksofindividualsarethenassignedasfollows:

Ranki=N−i, i=1,2,· · · ,N (17)

whereNisthepopulationsize.Thenthefollowingprobabilitypioftheithfoodsourceisdefinedas:

pi=Ranki/N, i=1,2,· · · ,N (18)

4.6. Synergyofadaptivereproductionoperatorswithcomplementaryeffect

In our proposal,MS-DABC partitions the whole population into four subgroups evolvingwith a distinct reproduction operator.TheadaptionofcontrolparametersfortheseoperatorsisdiscussedindetailinSection4.7.Inaddition, subpopu-lations’sizesaredynamicallyadjustedbasedontheiroperators’performance,withrelateddetailsexplainedinSection4.8.

Regardingtheoperatorselection,severalrecommendationshavebeenprovidedin[3,27,29,43]andtheirsettingsare de-termined by the characteristics ofthe problem. The original DE adopts the “rand/1” mutationoperation, which exhibits strongexplorationabilitybutpoorconvergenceperformance.Forthe“rand/2” mutationoperation,itleadstomore pertur-bation than“rand/1”, asmore target vectorsareselected. The “current-to-pbest/1” mutationoperation was introduced to improvesearchrobustnessbyexploitingthehistoricalrecordofsuccess[43].The“current-to-rand/1” mutationoperationis verysuitableforhandlingrotatedproblemsduetotherotation-likeperturbation[29].Mutationoperationslike“best/1”and “best/2” introduce the global best guided term in the update equation, which may cause premature resultswhensolving problemswithcomplexfitnesslandscapes.Therefore,thesemutationoperationshavenotbeenemployedinMS-DABC.

Basedon theaboveconsideration,fourmutationoperationswithcomplementaryeffects,i.e., “rand/1”,“rand/2”, “rand-to-best/2” and“current-to-pbest/1”are adoptedinthe foursubgroups respectively.Thecombinationofthosemutation op-erationswithbinomial crossoverformsa numberofevolutionary operators.Givena solutionxi=(xi1, xi2,…,xin),the new

candidatesolutionvi=(vi1,vi2,...,vin)isproducedasfollows:

“rand/1/bin”

v

_{i, j=}



xr1 ,j+F

(

xr2 ,j−xr3 ,j

)

, if

(

rand

(

0,1

)

≤CR

)

or

(

j=jrand

)

xi, j, otherwise (19) “rand/2/bin”

v

_{i, j=}



xr₁ ,j+F

(

xr₂ ,j−xr₃ ,j

)

+F

(

xr₄ ,j−xr₅ ,j

)

, if

(

rand

(

0,1

)

≤CR

)

or

(

j=jrand

)

xi, j, otherwise (20) “rand-to-best/2/bin”

v

_{i, j}=



xi, j+F

(

xbest, j−xi, j

)

+F

(

xr₁ ,j−xr₂ ,j

)

+F

(

xr₃ ,j−xr₄ ,j

)

, if

(

rand

(

0,1

)

≤CR

)

or

(

j=jrand

)

xi, j, otherwise

(21) “current-to-pbest/1/bin”

v

_{i, j}=



xi, j+F

(

xpbest,i, j−xi, j+xr1 ,j− ˜xr2 ,j

)

, if

(

rand

(

0,1

)

≤CR

)

or

(

j=jrand

)

xi, j, otherwise

(22) where indices r1, r2, r3, r4, and r5 are random mutually exclusive integers within the range [1, N]; j∈{1, 2, …, n}; F is thescaling factor;CR iscrossoverrate;Letop1,op2,op3,op4 denoteoperators inEqs.(19)–(22),respectively;xbest,j inop3 indicatesthejthelementofxbest,whichisarandomlyselectedglobalbestsolutionfromtheEXA;xpbest,i,j inop4denotesthe

jthelementofxpbest,i,whichrepresentsthepersonalbestofxi,i.e.,anondominatedarchiveofpersonalbest;andx˜r2,jinop4 representsthejthelementofx˜r₂,whichispicked upfromtheunionsetofevolvingpopulationandarchiveset.Notethat

thearchiveforop4 provides informationaboutthe searchdirectionandisdevotedtostrengthenthe diversity,aspointed outby[43].

4.7.Parametersadaptationforreproductionoperators

Variousconflicting conclusions havebeen drawn withregard to the manual parameter tuning ofscaling factor Fand crossover rateCR [27], which were performed in differentproblem contexts and hence lack generality. In fact, different parametersettingsmayberequiredatdifferentstagesofevolutionsoastoachievethebestperformance.Inourempirical studyofself-adaptiveparameterstrategiesproposedin[3,29,43],theonein[43]wasverifiedtobethemostsuitable.Thus, itischosentobetheparameteradaptionapproachinMS-DABC.

Atgenerationg,letCR_i,k bethecrossoverrateofkthreproductionoperatorusedby individualx_i,sampledthroughthe followequation:

CRi,k=randni,k

(

μ

CRk, 0.1

)

(23)

whereCR_i,k is confined inthe range [0, 1],

μ

CR_k representsthe average value of CR_i,k andis set to 0.5in initialization. AssumedthatSCR,k bethesetofsuccessfulCRi,k,then

μ

CRk,atgenerationg,isevaluatedby:

μ

CRk=

(

1−c

)

·

μ

CRk+c·meanA

(

SCR,k

)

(24)

wherecisaconstantparameterwithin[0.05,0.2],meanA(.)denotesthearithmeticmeanfunction.

Similarly,thescalingfactorofthekthreproductionoperatorusedbyxi,i.e.,Fi,kisupdatedaccordingtoCauchy

distribu-tion:

Fi,k=randci,k

(

μ

Fk, 0.1

)

(25)

where

μ

Fkdenotesthelocationparameter,and0.1isthescaleparameter.Fi,k∈[0,1].

LetSF,k be the set of successful CRi,k at generation g, and

μ

Fk is set to 0.5 initially, so atgeneration g, the location

parameter

μ

F_kisupdatedas:

μ

Fk=

(

1−c

)

·

μ

Fk+c·meanL

(

SF,k

)

(26)

wheremean_L(.)denotestheLehmerfunction:

meanL=  F∈SFF 2  F∈SFF (27) Then,thescalingfactorandcrossoverrateareadaptedandsampledfromgraduallyadaptiveprobabilitydistributions.

4.8.Subpopulationsizeadaption

Fromtheperspectiveofresourceconservation,computationaleffortsshouldbe distributedtothesubpopulationsbased ontheir utilities, which areobtained throughthe analysisof search datafrom theprevious evolutionary process. Inthis study,we define andcalculate a utility

π

k _{foreach subpopulationk. During every ng generation, the utility}

_π

k _willbe

updatedandthesizesofsubpopulationsadjustedcorrespondingly.

Indetail, let popbe the whole population, andsubpop1,subpop2, subpop3 andsubpop4 be thesubpopulations ofpop. Clearly,wehave:

pop=k=1,...Ksubpopk (28)

LetthepopsizebeN,andthesizeofsubpopkbeNk,

γ

kistheratiobetweensubpopkandpop,then:

Nk=

γ

k·N, k=1,2,...,K (29)

where

γ

k∈[0,1],

K

k=1

γ

k=1,andKisthetotalnumberofsubpopulations.

TheEXA, whichis detailedinSection 4.9,is employed to reservethe elite solutionsfound andupdate right afterthe generationofanewcandidate.AflagisthenmarkedtoeachmemberofEXAtoshowitisobtainedbywhichsubpopulation. Ifasolutionisfoundbymorethanonesubgroup,theflagcontainsallindexesofthesesubgroups.Letbk

gbethenumberof

EXAmembersprovidedbysubpop_kinthegthgeneration.AtthegenerationG,theutility

π

k_isupdatedas:

π

k₌ G−1  g=G−ng bk g/

Fesk, k=1,2,...,K (30)

whereG isthe currentgeneration,G−1

g=G−ngbkg isthe total numberofEXA membersfound by subpopk within the last ng

generations, and

Fesk is the consumed function evaluations by subpopk within the last ng generations. Then, the ratio

betweensubpopkandpop,i.e.,

γ

k,canbeevaluatedasfollows:

γ

k=

π

k/

_K

k=1

π

k_{, k=1,2,...,K (31)}

andthesizeofsubpopkcanbeevaluated:

Nk=

γ

k·N=



π

k

_K k=1

π

k



·N, k=1,2,...,K (32)

Aminimumsize(i.e.,5)issetforallsubpopulationstoguaranteethateachofthemhasachancetoevolve.Clearly,the highertheutilityforasubgroup,themoreindividualsareassignedtoit,whichmeansthatbetterperformedsubpopulations are allocatedwithmorecomputational resources.With thealgorithmproceeding, the utilityrecalculation andpopulation partitioningoperationare executedperiodically. Atthebeginning,the subpopulationsaresetto thesamesize.Ifthe cur-rentgeneration numberGisa multipleof ng,thencompute

π

k _{foreach subpopulationkduringthe last nggenerations,}

andupdate thesizesof all subpopulations.Inaddition, tomaintainthe diversityofpopulation andfacilitateinformation exchangeamongsubpopulations,themembersofthesubpopulationsarerandomlypickedfromthewholepopulationonce theregroupingoperationistriggered.

4.9. Elite-preservingstrategy

To alleviate the difficulties resulting from different methods, we integrate the merits of Pareto dominance, niche-preservation anddecomposition-based approachesto guidethe elite-preserving process in a hierarchicalmanner. In par-ticular,thebalancebetweentheproximityanddistributionofEXAisachievedbytwodifferentmodules,i.e.,Pareto-based Ranking(PR),andNiche&Decomposition basedSelection(NDS), separately.PRisonlyemployed asa sortingoperatorto rankEXAmembersintodifferentNon-dominationLevels(NDLs),whileawell-designedNDS,inwhichthedensityindicator (i.e.,nichecount)iscombinedwiththedecomposition-basedtechnique,canbeusedtoprunetheinferiorsolutionsofEXA

inamorefine-grainedmanner.

Itshouldbepointedoutthat,insteadofperformingEXAupdatingafterallmembersofthewholeoffspringaregenerated,

EXA isupdated rightafter thegeneration ofa newcandidatein asteady-state manner,so that theelite informationcan be utilizedassoon aspossible.More specifically, ateach generationforeach solution xi,an offspring xc is generatedby

performingareproductionoperatoronseveralsolutionsselectedfromtheneighborsubregionsofx_i.Then,xc iscombined

withEXA to form a intermediate EXA’. Afterwards,one of inferior solutions fromEXA’ is identified andeliminated. The methodin[19]isemployedfortheupdatingofNDLssothatthecomputationalcostisreduced.

Asmentionedearlier,identificationoftheworstsolutionx′_{fromEXA’ismadeinahierarchicalmanner.Atthebeginning,}

theintermediateEXA’,acombinationofEXAandXc,isdividedintoseveralNDLs,i.e.,F1,,Fl .Afterwards,weidentifythe

mostcrowdedsubregionassociatedwithsolutionsinFl,denotedas



xcrowd∈Fl

(

x

)

,wherethecrowdeddegreeofasubregionis

determinedby thenumberofsolutions (nichecount)associatedwithit.Then, thefollowingtwoscenarios areconsidered accordingtothenumberofsolutionsin



crowd

x∈Fl

(

x

)

,i.e.,

|



crowd x∈Fl

(

x

)

|

:

Case1:

|



_xcrowd_∈F

l

(

x

)

|

=1,i.e.,onlyonesolutionexistsinthemostcrowedsubregionassociatedwiththosesolutionsinFl,

(if|Fl|>1,theneachmemberofFlislocatedinanisolatedarea)whichmeansthat



xcrowd∈Fl

(

x

)

islikelytobeanunexploited

subregion.Inthiscase,thesolutionin



_xcrowd_∈F

l

(

x

)

shouldbereservedforthenextroundofexploitation.Instead,weidentify

the most crowdedsubregion associated withsolutions inFl−1, so on andso forth. Once

|



xcrowd∈Fl−k

(

x

)

|

>1(k=0,…,l−1), we

find the worst solution x′ _from



crowd

x∈Fl−k

(

x

)

in terms of the largest aggregation function value of the subproblem, and x ′

is eliminated fromEXA’ .In the extremecase,

|



crowd

x∈Fl−k

(

x

)

|

=1(k=l−1), which meanseach memberofEXA’ isassociated

withanisolatedsubregion.Inthiscase,solutionx′fromFl withtheworstaggregationfunctionvalueofthesubproblemis

eliminatedfromEXA’.AnexampleofthisisillustratedinFig.4(a),wheretheEXA′ _{isdividedinto4NDLs,i.e.,F}

1=

{

x1,x3

}

,

F2=

{

x5,x6

}

,F3=

{

x4

}

,andF4=

{

x2

}

,themostcrowedsubregionassociatedwiththesesolutionsinFlis



4,whichhasonly

onememberx2_,thus



4 _{islikelytobeanunexploitedsubregion,andx}2_{shouldbereserved.Next,thesamegoesforx}4 _in

F3,whichshouldbereserved.Afterwards,subregionsassociatedwiththesesolutionsinF2,i.e.



2and



5,areinvestigated, andapparentlythemorecrowdedoneis



5_{,whichhastwomembersx}3 _andx6_.Finally,x6_{iseliminated sinceithasthe} largestaggregationfunctionvalueinthecorrespondingsubregion.

Case2:

|



_xcrowd_∈F

l

(

x

)

|

>1,i.e.,multiplesolutionsexistin



crowd

x∈Fl

(

x

)

.Theworstsolutionfromwhich,identifiedbythe

high-est aggregation function value of corresponding subproblem, is eliminated from EXA’. Such an example is illustrated in

Fig.4(b),where thesubregionassociatedwithsolutions inFl is



4.Apparently|



4|>1,so x2 isremovedfromEXA’

be-causeit hasthe worstaggregation functionvalue andthere isa better solutionx6 _{forthe samesubproblem interms of} convergence.

Itshouldbepointedoutthat,duringtheidentificationofthemostcrowdedsubregion,theremaybemultiplesubregions havingthesamenichecount.Inthiscase,theonewiththelargestsumofaggregationfunctionvaluesisselected:

c=argmax

j∈S



x∈j

gj

₍

_x

₎

₍₃₃₎

where Sdenotes theindices setof subregionswiththe samecrowded degree,



j _{representsthe subregionofthe}

corre-sponding subproblem j, and gj_{(x) indicates the aggregation function of the subproblem j forthe current solution x. The}

procedureoftheEXAupdateispresentedinAlgorithm3,wheretherecurrencemethodofworstmemberidentificationis giveninAlgorithm4.

Differing fromNSGA-III, where Pareto dominance is considered as a major selection criteria, Pareto based ranking in ourproposed approach isonlyemployed asa sortingoperator insteadofselection criteria,andthe selection isbasedon

Fig. 4. Examples of the worst member elimination in EXA .

Algorithm 3 UPDATEEXA ( EXA , x c ).

Input : external archive EXA; offspring solution x c Output: external archive EXA

EXA′ ← EXA ∪ x c

NDLs = { F 1 , ·· · , F end} : % ∗ Use the method in [19] to update the nondomination level structure of EXA ∗/

l = | NDLs | if|crowd x∈Fl (x)| == 1 then if l! = 1 then l ← l − 1 x ′ ← ID _ worst _{( NDLs, EXA} ′ , l ₎ else // l == 1

x ′ ← identify the worst member of F

end in terms of PBI

end else

Identify the most crowded subregion d

associated with those solutions in Fl

x ′ ← argmaxg d (x), x ∈ _d end

EXA ← EXA ′ \ x ′

thedensity indicator andaggregationfunctionssequentially. In particular,the densityindicator,which isbased on niche counts, concerns the diversity issue; while the selection operation, determined by aggregation function values,concerns the convergence issue. In other words, it is a diversity-first and convergence-second method. For the case in Fig. 4(a), due to the dominance priority strategy of NSGA-III, solution x2 _{is removed without reservation since it is located in} the last NDL. However, in our proposed approach, solution x2 _{is reserved as it is associated with an isolation} subre-gion.Instead, solutionx6 _{is removedduetoits worst aggregation functionvalue in themostcrowded subregion. Inthis} study, a selection priority is provided to the member Fl in caseit is located in an isolated subregion. On the contrary,

since solutions in the former NDL havehigher priority in NSGA-III, x2 _{is eliminated and the population diversity is not} well-maintained.

4.10.TheimplementationprocedureofMS-DABCforSCOSproblems

Basedtheaboveanalysis,thedetailedimplementationofMS-DABCforsolvingMaOPsispresentedinAlgorithm5,where the algorithm starts by initializing the bee colony and setting the parameters, then enters the evolution loop until the maximumnumberofgenerations,maxG,isachieved.At first,thecolony israndomlydivided intoseveralsubgroupswith equalsizes,whichareindependentlyevolvedbyemployingdifferentevolutionaryoperators.TheEXA,whichismaintained byafine-grainedselectionmechanism,isusedtoguidethesearchprocess.Basedonthecontributionsoftheevolutionary operatorsfortheEXAmembers,thesizesofthesubgroupsareadjustedperiodically.Parameterupdatingfortheevolutionary

Algorithm 4 ID _ worst ( NDLs, EXA, l ) .

Input: external archive EXA ; nondomination level structure of EXA : NDLs ; current level : l

Output : the worst member of EXA : x ′

if|crowd x ∈ Fl (x)| == 1 then if l! = 1 then l ← l − 1 x ′ ← ID _ worst _{( NDLs, EXA, l )} else// l == 1

x ′ ← identify the worst member of F

end in terms of PBI

end else

Identify the most crowded subregion d

associated with those solutions in Fl

x ′ ← argmaxg(x), x ∈ _d end

operatorsisgiveninlines5–9,regroupingofsubpopulationisgiveninlines32–39;thefoodsourcepositionupdateaswell asEXAmaintainingstrategyaregiveninAlgorithm6;andotherkeycomponentsareelaboratedbefore.

5. ExperimentsonbenchmarkMOPsandMaOPs

Aseriesof experimentsare carriedout withtheaimoftesting theperformance oftheproposed algorithm insolving MOPsand MaOPs.InSection 5.1, experimentalsettings are briefly describedatfirst, andinSection 5.2,we comparethe proposedalgorithmwithseveralstate-of-the-artMOEAshavingacompetitiveperformance.Then,thescalabilityofproposed algorithmfortacklingmany-objectiveproblemsisinvestigatedinSection5.3.Next,inSection5.4,theaimistoevaluatethe effectivenessofimprovementstrategies forMS-DABC.Finally,experiments inSection 5.5are performedtoinvestigatethe sensitivityofparametersontheproposedalgorithm.

5.1. Experimentalsettings

(1) Test problems: Severaltypesof benchmarkMOPswithvariouscharacteristics anddifferentdifficultiesarechosen in ourexperimentalstudies,including:1) thetri-objectiveDTLZfamilyofscalableinstances:DTLZ1-DTLZ7;and2)the bi-objectiveWFGtoolkitproblems:WFG1-WFG9.Theabovetestinstanceshavevariousfeaturessuchasnonlinearor degenerategeometry,variablelinkage,andmulti-modal,whichposeagreatchallengetoMOEAs.Moredetailsofthe testMOPsaredescribedin[1,2].

(2) Performance metrics: In order toevaluate the performance of MOEAson thetest MOPs, both invertedgenerational distance(IGD)andhyper-volume(HV)metricscanbeusedtomeasuretheconvergenceanddiversityoftheobtained solutionsset.However,thecomputationalcomplexityofHVistoohighforPFswithmanyobjectives.Therefore,only IGD is employed as the performance metric in our experiments. The detailed calculation of IGD metric has been elaboratedin[12,23].AlowvalueofIGDmeansthattheobtainedsolutionsetisveryclosetothetruePF.Inaddition, indicatorSPREAD(△)isemployedtomeasuretheuniformityofthesolutionsetfound,assuggestedin[39].

(3) Parameterssetting:Inourstudy,theproposedalgorithmiscomparedwithseveralstate-of-the-artMOEAs.Parameters settings of the MOEAs are adopted as recommended in the literature, as summarized in Table 2, where n is the variabledimension,pmandpcarethemutationandcrossoverprobabilitiesrespectively,

η

mand

η

carethedistribution

indexes of crossover and mutation operators respectively, T is the neighborhood size of weight vectors, nr is the

maximumnumberofsolutions replacedbyeachnewsolution,and

δ

istheprobabilityofselectingparentsolutions fromneighborhood.ThepopulationsizeNandthemaximumfunctionevaluations(Max_FEs)areadjusted according tothecomplexityofMOPs.Nissetas200,andMax_FEs=50,000.Toallowafaircomparison,thesettingsofT,

δ

,and

nrforMS-DABCarethesameasotheralgorithms,sincethecontrolparametersinvolvedintheevolutionaryoperators

mentionedinSection4.6canbedynamicallyconfigured,andonlyfourparametersneedtobeset,i.e.,N,maxG,limit, andng.Amongthem,Nisa commonparameters andissetthesameasotherpeeralgorithms,maxGisdecidedby

Max_FEsandN,andlimitissetas20nassuggestedby[14].Thus, onlyngneedstobetuned andits valueissetas 30basedonthesensitivityanalysisprovidedinSection5.5.

Forfaircomparison,all thealgorithms takethesameMax_FEs forthe sameinstance.Togetstatisticallysound conclu-sions,thecomparedalgorithmsareexecuted30timesforeachtestinstance,andthemeanandthestandarddeviations(std) ofthemetricvaluesarerecordedforcomparison.Anon-parametrictest(i.e.,Wilcoxon’sranksumtest)ata0.05significance levelisfurtherperformedtoassessthestatisticalsignificanceoftheresultsobtainedbythecomparedMOEAs.

Algorithm 6 Food source update.

Table 2

Parameter settings for the compared algorithms.

Algorithms Parameter settings

NSGA-II pc = 0.9, p m = 1/ n, ηc = 20, ηm = 20 SMPSO C1 ∈ [1.5, 2.5], C 2 ∈ [1.5, 2.5], p m = 1/ n, ηm = 20 SMS-EMOA pc = 0.9, p m = 1/ n, ηc = 20, ηm = 20 HypE pc = 0.9, p m = 1/ n, ηc = 20, ηm = 20 MOEA/D-DE pm = 1/ n, ηm = 20, CR = 1, F = 0.5, δ= 0.9, T = 20, n r = 2 MOEA/D-DRA pm = 1/ n, ηm = 20, CR = 1, F = 0.5, δ= 0.9, T = 0.1 ∗N , n r = 0.01 ∗N ENS-MOEA/D N = 100, CR = 1, F = 0.5, LP = 50, ε= 0.01 MOEA/D-FRRMAB CR = 1, F = 0.5, p m = 1/ n, ηm = 20, T = 20, δ= 0.9, n r = 2, C = 5.0, W = 0.5 ∗N, D = 1.0 MS-DABC limit = 20 n, ng = 30, μCR = 0.5, μF = 0.5, T = 20, δ= 0.9, n r = 2

5.2. ComparisonofMS-DABCwithstate-of-the-artMOEAs

TheMS-DABCiscomparedwithstate-of-the-artMOEAsincludingNSGA-II1_[6],SMPSO2_{[28],SMS-EMOA}2 _[2],HypE3_[1],

MOEA/D-DE4 _{[18], MOEA/D-DRA}4 _{[44], ENS-MOEA/D}5 _{[47], andMOEA/D-FRRMAB}6 _{[21]. Particularly, NSGA-II andSMPSO}

are two well-known Pareto-based MOEAs, HypE andSMS-EMOAare two representative indicator-basedMOEAs, andthe restaredecomposition-basedones.Thereasonswe selectthose variantsofMOEA/D forpeercomparisonare explainedas follows. First, MOEA/D-DE uses extra measures formaintaining population diversity, which is very promising intackling MOPswithcomplicatedPSshapes. Second,MOEA/D-DRAemploys utilitymetrics forsubproblemsto decideonpromising searchregions,whichcanachievedynamicalresourceallocation.Third,ENS-MOEA/DandMOEA/D-FRRMABalsoimplement

1 The code of NSGA-II is from _{https://www.egr.msu.edu/ ∼kdeb/codes.shtml .}

2 The codes of SMPSO and SMS-EMOA are available at _{https://sourceforge.net/projects/jmetal/?source=directory .} 3 The code of HypE is from _{http://www.tik.ee.ethz.ch/pisa .}

4 The codes of MOEA/D-DE and MOEA/D-DRA are from _{http://dces.essex.ac.uk/staff/zhang/webofmoead.htm .} 5 The code of ENS-MOEA/D can be downloaded from _{http://www.ntu.edu.sg/home/epnsugan/ .}

Table 3

IGD-metric comparisons of the algorithms for DTLZ problems.

theco-evolving paradigms (e.g.,the ensemble of multipleparameters andadaptive operator selection)to coordinate the samesearchasours.Therefore,itismeaningfultoconductacomparisonamongthem.

5.2.1. ComparisonsontheDTLZinstances

TheexperimentalresultsofthecomparedMOEAsforDTLZproblemsarelistedinTable3,wherethemeanandstandard deviation(inparentheses) ofthe IGDmetricvaluesfor30 runsarerecorded. Thenumbers next tothe parenthesesrefer totherankingsofalgorithmsforthatproblem,whichareobtainedbasedontheaveragemetricvalue,andthebestresult ishighlightedinboldtype withadarkgraybackground,andthesecond bestishighlightedinlightgraybackground.The symbols“ + ”,“ = ” and“–”,respectivelyindicatethatMS-DABCperformsstatisticallybetterthan,similartoandworsethan thecomparedalgorithmsbasedonWilcoxon’sranksumtest.ThelastrowdenotesthatMS-DABCwinsonwproblems,ties ontproblems,andlosesonlproblems,comparedwithitscorrespondingcompetitor.Forsimplicity,theMOEA/D-FRRMAB isreferred asFRRMABforshort.From Table 3,itcan be seenthat MS-DABChasthe bestIGDmetric value infourcases while ENS-MOEA/Dperforms the bestin two cases. However, the final rankimplies that MOEA/D-FRRMAB achieves the secondbest whileENS-MOEA/Displacedthird.ThisisduetoMOEA/D-FRRMAB producingmorestableandbetterresults onDTLZ2andDTLZ4thanENS-MOEA/D.Anotherpromisingalgorithm,MOEA/D-DRA,beingrankedslightlylowerthan ENS-MOEA/D,achievesfourthplace. Regardingthe comparisonresultsofWilcoxon’s ranksumtest givenin thelast row, MS-DABC is superior to NSGA-II, SMPSO, SMS-EMOA, HypE, MOEA/D-DE, MOEA/D-DRA, ENS-MOEA/D, andMOEA/D-FRRMAB respectivelywith6,6,6,6,6,6,5,and3winsoutof7problems,onlybeatenbySMPSOonDTLZ5,ENS-MOEA/DonDTLZ1 andDTLZ6,andMOEA/D-FRRMABonDTLZ1.Itisalsonotedthatvariants ofMOEA/D suchasMOEA/D-DRA,ENS-MOEA/D andMOEA/D-FRRMABperformbetterthanMOEA/D-DE,especiallyoncomplicatedproblemssuchasDTLZ1andDTLZ3with multimodality,andDTLZ4withbiaslandscapes.Thereasonmaybeduetothesynergeticmechanismofmultiplestrategies inthesevariants,whichissuitablefordifferenttypesofsearchlandscapes.

Fig.5presentstheboxplotsforSPREADmetricvaluesofsolutionsobtainedbythecomparedMOEAsover30independent runs.Table4givestheStatisticsofSPREADcomparisonsofMS-DABCwiththeotherrivals,accordingtotheWilcoxon’srank sumtest. Clearly, theseresultsindicate that MS-DABCperforms significantly better than its competitors on47 out of 56 comparisonsintermsofSPREADmetric.WhileMS-DABCisonlyoutperformedbyitsrivalson2outof56comparisons.To bespecific,MS-DABCisdefeatedbyENS-MOEA/DandMOEA/D-FRRMABonDTLZ1andDTLZ3,respectively.Thesuperiority ofMS-DABCismoreobviousoncomplexproblemswithirregularPFs,e.g.,DTLZ5-7.

Fig. 5. Boxplots for SPREAD comparison of MOEAs on DTLZ problems.

Table 4

Statistics of SPREAD comparisons on DTLZ problems.

NSGA-II SMPSO SMS-EMOA HypE MOEA/D-DE MOEA/D-DRA ENS-MOEA/D FRRMAB

+ 7 5 7 6 7 7 3 5

= 0 2 0 1 0 0 3 1

– 0 0 0 0 0 0 1 1

Fig. 6. Boxplots for SPREAD comparison of MOEAs on WFG problems.

5.2.2. ComparisonsontheWFGinstances

Comparison results of WFG benchmarks on IGD metric are presented in Table 5, from which we can observe that MS-DABC yields 7 out of9 best metric values,while SMPSOand HypE win only 1test case, respectively. Regarding the Wilcoxon’s ranksumtest, comparedwiththeother eightMOEAs, MS-DABCachievessignificant betterresultin61 outof 72 IGD comparisons. In terms ofoverall final rank, MS-DABC is ranked the best, followed by SMS-EMOAand MOEA/D-FRRMAB. Theseobservationsdemonstratethe advantagesofMS-DABCin tacklingWFGproblems.It isinteresting to note thattwo indicatorbasedalgorithms,i.e.,SMS-EMOAandHypE,performwell onWFGinstances,rankedsecondandfourth overall inthiscomparison.It isalso observedthat, by usingsome adaptivestrategies (e.g.,dynamical resourceallocation forMOEA/D-DRA, adaptiveneighborhood size forENS-MOEA/D), the modifiedalgorithms performbetter than their stan-dardversionMOEA/D-DE.MS-DABCandMOEA/D-FRRMABsharesomesimilaritiesinmaintainingdiversitywiththehelpof adaptiveoperatorselection,andtheybothshowpromisingperformanceinaddressingWFGinstances.

Fig.6 illustrates the boxplotsforSPREAD metricvaluesover 30runs obtainedby MS-DABCandits competitors, from whichit canbe observedthatMS-DABCperforms eithercomparablyorremarkablybetterthan itsrivalsonmosttest in-stancesexceptWFG3,whereitisbeatenbySMPSOandSMS-EMOA.TheSPREADmetriccomparisonsofMS-DABCwithits rivals,regardingWilcoxon’sranksumtest,arepresentedinTable6,whichrevealsthatMS-DABCachievessignificantbetter resultsin 67out of72comparisons.Especially,MS-DABC significantlyoutperforms MOEA/D-FRRMABin 8out of9cases.

Table 5

IGD-metric comparisons of the algorithm for WFG problems.

Table 6

Statistics of SPREAD comparisons on WFG problems.

NSGA-II SMPSO SMS-EMOA HypE MOEA/D-DE MOEA/D-DRA ENS-MOEA/D FRRMAB

+ 9 8 8 7 9 9 9 8

= 0 0 0 2 0 0 0 1

– 0 1 1 0 0 0 0 0

Thisimpliesthat,besidesadaptiveoperatorselection,thewellmaintainedEXAandpotentialregiondetectionofonlookers couldbeotherimportantfactorsforthebetterdistributionofsolutionsobtainedbyMS-DABC.

5.3.ThescalabilityofMS-DABCfortacklingMaOPs

Inordertoinvestigatethescalabilityofourproposedalgorithmintacklingproblemswithmanyobjectives,MS-DABCis comparedwithsomestate-of-the-artalgorithmsonDTLZ1toDTLZ4oftheDTLZbenchmarkproblems,whichcanbescaled toan arbitrarynumberof objectivesanddecisionvariables. Tobe morespecific, thenumberofobjectives foreach DTLZ instancevariesfrom5to15,i.e.,m∈{5,8,10,13,15},andthenumberofdecisionvariablesissetasn=m+r−1,wherer