ContentslistsavailableatScienceDirect
Journal
of
Manufacturing
Systems
jo u r n al h om ep age : w w w . e l s e v i e r . c o m / l o c a t e / j m a n s y s
Multi-objective
fuzzy
disassembly
line
balancing
using
a
hybrid
discrete
artificial
bee
colony
algorithm
Can
B.
Kalayci
a,1,
Arif
Hancilar
a,
Askiner
Gungor
a,∗,
Surendra
M.
Gupta
b,2aDepartmentofIndustrialEngineering,PamukkaleUniversity,KinikliKampusu,20070Denizli,Turkey
b334SN,DepartmentofMechanicalandIndustrialEngineering,NortheasternUniversity,360HuntingtonAvenue,Boston,MA02115,USA
a
r
t
i
c
l
e
i
n
f
o
Articlehistory: Received27May2014
Receivedinrevisedform8October2014 Accepted24November2014
Availableonline18December2014 Keywords:
Fuzzydisassemblylinebalancing Combinatorialoptimization Metaheuristics
Artificialbeecolony Variableneighborhoodsearch
a
b
s
t
r
a
c
t
Thispaperpresentsafuzzyextensionofthedisassemblylinebalancingproblem(DLBP)withfuzzy
taskprocessingtimessinceuncertaintyisthemaincharacterofreal-worlddisassemblysystems.The
processingtimesoftasksareformulatedbytriangularfuzzymembershipfunctions.Thebalancemeasure
functionismodifiedaccordingtofuzzycharacteristicsofthedisassemblyline.Ahybriddiscreteartificial
beecolonyalgorithmisproposedtosolvetheproblemwhoseperformanceisstudiedoverawell-known
testproblemtakenfromopenliteratureandoveranewdatasetintroducedinthisstudy.Furthermore,
theinfluenceofthefuzzinessonthecomputationalcomplexityofHDABCisevaluatedandthesolution
qualityoftheproposedalgorithmiscomparedagainstdiscreteandtraditionalversionsoftheartificialbee
colonyalgorithm.Computationalcomparisonsdemonstratethesuperiorityoftheproposedalgorithm.
©2014TheSocietyofManufacturingEngineers.PublishedbyElsevierLtd.Allrightsreserved.
1. Introduction
Atatimewheneconomicandecologicalresourcesarerapidly depleted,productrecoverythataimstominimizetheamountof wastesenttolandfillsisgainingalotofimportance.Comprehensive reviewsofenvironmentallyconsciousmanufacturingandproduct recoverycanbefoundin[1,2].Productrecoveryconsistsofseveral stepsinwhichthefirstcrucialstepisdisassembly[3]. Disassem-blyisamethodicalextractionofvaluableparts[4]fromdiscarded productsthroughaseriesofoperations.Someoftheobjectivesof disassemblyaregivenasfollows[5]:recoveryof valuableparts andsubassemblies,removalofhazardousparts,extractionofparts fromtheremainderoftheproductwhichcanbesenttoinventory forfutureuse,achievingenvironmentallyfriendlymanufacturing standards,retrievalofpartsorsubassembliesofdiscontinued prod-uctstosatisfyasuddendemandfor theseparts,decreasingthe amountof residuetobesent tolandfills.Thus thedesign ofan efficientdisassemblylinehasaconsiderableindustrialand envi-ronmentalimportance.
∗ Correspondingauthor.Tel.:+902582963141;fax:+902582963262. E-mailaddresses:cbkalayci@pau.edu.tr(C.B.Kalayci),arif.hancilar@gmail.com (A.Hancilar),askiner@pau.edu.tr(A.Gungor),gupta@neu.edu(S.M.Gupta).
1 Tel.:+902582963209;fax:+902582963262. 2 Tel.:+16173734846;fax:+16173732921.
DisassemblyLineBalancingProblem(DLBP)isamulti-objective problemasdescribedin[6]andhasbeenmathematicallyproven tobe NP-completein [7] makingthe goalto achieve the opti-malbalancecomputationallyexpensive.NP-completeorNP-hard termsarethewaysof showingthat certainclassesofproblems are not solvable in realistic time [8]. Exponential time com-plexityofexhaustivesearch limitsitsapplicationtolargesized instances althoughitworks wellinobtainingoptimalsolutions forsmallsized instances.Anefficientsearchmethod,therefore, needstobeemployedtoattainan (near)optimalsolution.The waysofapproachingtothiscombinatorialoptimizationproblem intheliteraturecanbedividedintotwocategories; mathemati-calprogrammingtechniques[9–12]andmetaheuristics[7,13–24]. Metaheuristics are more popular since the problem quickly becomesunsolvablewithmathematicalprogrammingtechniques fora practical sizedproblem. See[25]for moreinformation on DLBP.
Disassemblyoperationshaveuniquecharacteristicsandcannot beconsideredasthereverseofassembly operations.Pleasesee
[17]fordetails.Althoughtherearesignificantdifferencesbetween assembly line balancing and disassembly line balancing prob-lems,disassemblylinebalancingliteraturehasbeenbuildingup ontopoftheassembly linebalancing literaturesince thereare alsosomesimilaritiesbetweenthem.Theliteratureoffuzziness inassemblylinebalancingisofprimaryinterestofthisstudy.The fuzzinessintheliteratureofassemblylinebalancingwasfirst men-tionedin[26,27]bysolvingasimplefuzzyassemblylinebalancing http://dx.doi.org/10.1016/j.jmsy.2014.11.015
problemappliedtoasingleproductexample.Afuzzymixedmodel linebalancingproblemwitha fuzzybinarylinearprogramming modelwasconsideredin[28]usingaheuristicsolutionapproach thatdealswithfuzzyprocessingtimes.Fuzzymultiobjective two-sidedassembly linebalancing problemwassolvedusingabees algorithmin[29]withthreefuzzygoals:maximizethework slack-nessindex,minimizethetotalbalancedelayand maximizethe lineefficiency.Geneticalgorithmswereusedtosolvemulti objec-tivefuzzyassemblylinebalancingproblemtype2[30]andfuzzy assemblylinebalancingproblemtypeE[31].
In disassembly,there is a highdegreeof uncertainty in the structureandthequalityofthereturnedproducts.Thestructural unknownsofend-of-life(EOL)productsmaycreateother uncer-tainties,suchasestimationofdisassemblytasktimeofeachpart duetobothmachineandhumanfactors.Tothebestofour knowl-edge,thereisonlyonepreviouslypublishedwork[32]thatdeals withfuzzinessinmixedmodeldisassemblylinebalancing prob-lemusingbinaryfuzzygoalprogramming.Yet,in thisstudy,in ordertodealwithimprecisedata,fuzzynumbersareintroduced torepresenttheprocessingtimeof eachjob.Thus, afuzzy dis-assemblylinebalancingmodelisobtained.Theconsiderationof fuzzinessforthesolutionofdisassemblylinebalancingproblem isofimmenseinterestsincethedataobtainedfrommorerealistic situationsareimpreciseanduncertain.Aimingtofillthisgap,this paperintroducesanewhybriddiscreteartificialbeecolony algo-rithm(HDABC)forsolvingthemultiobjectivefuzzydisassembly linebalancingproblem(FDLBP).Thefuzzyprocessingtimesof dis-assemblytasksarerepresentedbytriangularfuzzymembership functions.Twodifferentevaluation mechanismsareusedinthe proposedalgorithm:lexicographicmethodthatfocusesoneach objective accordingtopredefined prioritiesand fixedweighted methodthattriestooptimizeeachconflictingobjective concur-rentlywiththehopeofconstructinganefficientParetofrontier.The performanceofHDABCiscomparedagainstdiscreteandtraditional versionsof theartificial beecolony algorithm ontwo different cases.Moreover,theinfluenceofthefuzzinessonthe computa-tionalcomplexityofHDABCisevaluated.
2. Fuzzydisassemblylinebalancingmodel 2.1. Notation
asolutioninVNS
BD fuzzybalancedelaytime
BE fuzzybalanceefficiency cmax maximumcycletime
˜c fuzzycycletime
di demandquantityofpartirequested
fs thenectaramount(i.e.,thefitnessvalue)ofafoodsource
(s)intheABCalgorithm
g currentiteration(generation)numberofthealgorithm hi binaryvalue;1ifpartiishazardous,else0.
i partidentification,taskcount(1,...,n) imp non-improvediterationcount
IP set(i,j)ofpartssuchthattaskimustprecedetaskj j partidentification,taskcount(1,...,n)
k workstationcount(1,...,m) LBj lowerboundfordimensionj
m number of workstations required for a given solution sequence
n numberofpartsforremoval(dimensionoftheproblem) Ps theselectionprobabilityofafoodsource(s)
ps populationsizeoftheABCalgorithm PSi ithpartinasolutionsequence
r0,1 auniformlydistributedrealnumberin[0,1]
r−1,1 auniformlydistributedrealnumberin[−1,1] s asolution(bee)count(1,...,ps)
SI smoothnessindex
STj totalfuzzystationtime;thesumofprocessingtimeat
workstationj
ti partremovaltimeofparti
UBj upperboundfordimensionj
xjk binaryvalue;1ifpartjisassignedtostationk,0otherwise.
yij binaryvalue;1iftaskiisexecutedaftertaskj,0otherwise.
2.2. Problemformulation
Assumptionsofmultiobjectivefuzzydisassemblylinebalancing problemaregivenasfollows:thereisonlyonetypeofproductinthe disassemblyline;completedisassemblyisperformed;partremoval timesareknownandassumedtobefuzzyinteger;hazardousparts areknowninadvance;anddemandisalsoknowninadvanceand itsquantityisdeterministic.
Mathematicalformulationofthemulti-objectiveDLBPwasfirst introduced in [7]as a deterministic model.Based on themain conceptsof[7],afuzzyextensionoftheproblemisformulized con-sideringfuzzypartremovaltimesandfuzzycycletimeasfollows:
minf1=m (1) minf2=
m k=1 (c −STk) (2) minf3= n i=1 i×hPSi, hPSi= 1 hazardous 0 otherwise (3) minf4= n i=1 i×dPSi, dPSi∈N,∀
PSi (4) Subjectto: m k=1 xjk=1, j=1,...,n (5)⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
n i=1 ti c⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
≤n (6) m k=1 ST≤c (7) xik≤ m k=1 xjk,∀
(i,j)∈IP (8)Objectivefunctions;(1)aimstominimizethenumberof work-stations,(2)balancesthesmoothnessindex(SI)
whichshowsthe balanceofworkloaddistributedamongthestations,(3) aimsto removehazardouscomponentasearlyaspossiblewhereas(4)aims tohandleahighdemandcomponentasearlyaspossible.Equation(5),guaranteesthatalltasksareassignedtoexactlyone worksta-tion.Constraints;(6)guaranteesthatthenumberofworkstations withaworkloaddoesnotexceedthepermittednumber,(7) guaran-teesthattheworkcontentofworkstationcannotexceedthecycle time,(8)guaranteesthatallthedisassemblyprecedence relation-shipbetweenpartsshouldbesatisfied.
Inadditiontothemainobjectives,weevaluatetheperformance ofdisassemblylineinafuzzyenvironmentusingtwomeasures. Thefirstmeasureiscalledthebalanceefficiency,i.e.,BE
asinEq.(9). Anothermeasureisthebalancedelaytime,i.e.,BD asinEq.(10). Balancedelaytime(BD) showstheunusedcapacityoftheline.This measureisobtainedbysummingtheidletimeofallthestations. BE = m k=1 STk NWS×c (9) BD = m k=1 (c−STk) (10)2.3. Fuzzificationofthedisassemblylinebalancingproblem
Inthisstudy,thepurposeofusingfuzzydataapproachisto representmorerealisticsituations,inotherwords,todealwith uncertainty.Thefuzzinessofdataisrepresentedbytriangularfuzzy numbersasin[30,31].Triangularfuzzynumbersarithmeticis per-formedasfollowswhere˛andˇrepresentfuzzynumbers:
A+B=(˛1+ˇ1,˛2+ˇ2,˛3+ˇ3) A−B =(˛1−ˇ3,˛2−ˇ2,˛3−ˇ1) A×B=(˛1·ˇ1,˛2·ˇ2,˛3·ˇ3) A B=˛ 1 ˇ3, ˛2 ˇ2, ˛3 ˇ1 (11)
Tocompare thefuzzynumbers, thegreatestassociate ordinary numberrankingmethodisusedasfollows:
F(
A)=˛1+2˛2+˛34 (12)
TheobjectiveistodeterminethesetofParetooptimalsolutions basedontherankedvaluesofthefuzzycriteria.
3. Artificialbeecolonyalgorithm
Swarmintelligenceisaresearchbranchofartificialintelligence thataimstobringsolutionstotheproblemsbymodelingthe popu-lationofinteractingagentsorswarmsthatareabletoself-organize. Theagentsareabletoexchangeinformationinordertoshare expe-riences,andtheperformanceoftheoverallswarmemergesfrom thecollectionofthesimpleagents’interactionsandactions[33]. Anantcolony,aflockofbirds,flyingparticles,immunesystemor abeecolonyaretypicalexamplesofaswarmsystem.ArtificialBee Colony(ABC)Algorithmthatwasoriginallyproposedfor contin-uousfunctionoptimizationproblemsby[34]isanoptimization algorithmbasedontheintelligentbehaviorofhoneybeeswarm. ThemainstepsofthebasicABCalgorithmbasedon[35]areshown
inTable1.
Table1
MainstepsofthebasicABCalgorithm. Step Description
1 Initializethepopulation 2 Evaluatethesolutions 3 Employedbeesphase
4 Calculateprobabilitiesforonlookerbees 5 Onlookerbeesphase
6 Scoutbeesphase
7 Memorizethebestsolutionachievedsofar 8 Ifmaximumcyclenumberisnotreached,gotoStep3
9 Stop
Apossibleoptimumsolutiontotheproblemisrepresentedby afoodsourceandthequalityofthesolutionisdefinedbythe nec-taramountfoundinthatfoodsource.Therearethreestepsineach cycleasfollows:releasingtheemployedandonlookerbeesonto thefoodsources,calculatingtheirnectaramountsandsendinga determinednumberofscoutbeesrandomlyontothepossiblefood sources[36].Inotherwords,employed,onlookerandscoutbeesare themainmechanismsoftheABCalgorithmtogenerateasetof fea-siblesolutions,exploitthesesolutionsandexplorenewsolutions, respectively.Ifasolutionisnotimprovedbyapre-setnumberof tri-als,thenemployedbeeleavesthatfoodsourceandtheemployed beebecomesascoutinordertoexplorenewareasofthesearch space.IntheABCalgorithm,onlookersandemployedbeescarry outtheexploitationprocessinthesearchspacewhilethescouts controltheexplorationprocess.Forfurtherdetailsofthebasic algo-rithmthereadersunfamiliarwiththeABCalgorithmarereferred to[34]whichintroducedtheoriginalalgorithm.Basedon[35],the detailedstepsofthebasicABCalgorithmaregivenasfollows:
Atthefirststep,ABCgeneratesarandomlydistributedinitial foodsourcepositions.Theinitialpopulationofsolutionsisfilled withpsnumberofrandomlygeneratedn-dimensionalreal-valued vectorsthatcorrespondstofoodsourcepositions.LetXi{xi1,xi2,...,
xin}representtheithfoodsourceinthepopulationandeachfood
sourceisgeneratedaccordingtoEq.(14)asfollows: xsj=LBj+(UBj−LBj)×r−1,1 for j=1,2,...n
and s=1,2,...ps (14)
Intheemployedbeesphase,eachemployedbeegeneratesanew foodsourceintheneighborhoodofitspresentpositionaccording toequation(15)asfollows:
xnew(j)=xsj+(xsj−xpj)×r−1,1 for p=1,2,...,ps
and j=1,2,...,n (15)
Oncea newsolutionis obtained,it willbeevaluatedand com-paredtotheprevioussolution.Ifthefitnessofthenewsolution isequaltoorbetterthanthatofprevioussolution,newsolution willreplacetheprevioussolutionandbecomeanewmemberof thepopulation;otherwiseprevioussolutionisretained.Thus, a greedyselectionmechanismisemployedbetweentheoldandnew candidatesolutions.
Intheonlookerbeephase,thenectarinformationgatheredfrom employedbeephaseisevaluatedbyanonlookerbeeandafood sourceisthenselectedaccordingtoitsprobabilityvaluecalculated usingthefollowingequation.
ps=
fpss s=1fs(16)
Inshort,aroulettewheelselectionmechanismisappliedtoselecta foodsource.Oncetheonlookerhasselectedthefoodsource,it pro-ducesamodificationusingEq.(15).Asinthecaseoftheemployed beephase,areplacementofthefoodsourcetakesplaceifthe mod-ified foodsourcehasa betteror equalnectar amountthanthe previousone,anditfinallybecomesanewmemberinthe pop-ulation.Ifasolutionisnotimprovedbyapre-setnumberoftrials limit,thefoodsourceisleftout,andthecorrespondingemployed beebecomesascout.Thescoutproducesanewsolutionrandomly accordingtothefollowingequation.
xsj=LBj+(UBj−LBj)×r0,1 for j=1,2,...n
4. TheproposedHDABCalgorithm 4.1. Solutionrepresentation
TheoriginalABCalgorithmisdesignedforcontinuous optimiza-tionproblems.However,standardcontinuousencodingschemeof ABCcannotbedirectlyusedtosolvetheFDLBP.Inordertoapply ABCtoFDLBP,constructingadirectrelationshipbetweenthetask sequenceandthevectorofindividualsinABCisrequired.The per-mutationbasedrepresentationisusedasin[26]toencodeeach solutioninHDABCalgorithm.
4.2. Populationinitializationandsolutionconstructionstrategy InitialpopulationisgeneratedrandomlyfortheHDABC algo-rithm. A solutionconstruction is started withopeningthefirst workstation.A stationorientedassignmentprocedurebased on
[14]isusedtofillnextworkstationswithremainingtasks.Inthis procedure, first, a setof candidatetasks is identified. A taskis insertedintothesetifitsatisfiesthreecriteria:first,ithasnot alreadybeenassignedtooneormoreofthepreviousworkstations, second,itspredecessorshasalreadybeencompletedandthird,its fuzzytasktimeshouldbelessthanorequaltotheremaining avail-ablefuzzytimeoftheworkstationunderconsideration.Then,atask israndomlyselectedfromthecandidatetasksetandassignedto theworkstation.Ifthereisnocandidatetaskforthatworkstation, anewworkstationisopenedwithafuzzycycletime.Thisprocess continuesuntilalltasksareassignedtoanumberofworkstations. 4.3. Neighborhoodstructures
Inthisstudy,anumberofneighborhoodstructures[insert,swap, twopointright,onepointrightandonepointleftoperators]are employed.Eachneighborhoodstructure changestheinitial con-figuration.Insertoperator (N1)randomlyselectsa positionofa taskand insertsit toanotherrandomlyselectedpositioninthe disassemblysequence.Swapoperator(N2)randomlyselectstwo tasksinthelinesequence,swapstheirpositionsandreturnsanew feasiblesolution.Twopointrightoperator(N3)randomlyselects twocutpositionsinthedisassemblysequence,reconstructsanew sub-sequencewithinthesecutpositionswhilenotchangingthe positionsof taskslocatedoutofthetwo selectedcutpositions. Onepointrightoperator(N4)randomlyselectsacutpositionin thedisassemblysequence,reconstructsanewsequencestarting fromthispositionaccordingtothesolutionconstructionstrategy whilekeepingthepositionsoftasksasalreadyassignedbeforethe cutposition.Onepointleftoperator(N5)randomlyselectsacut pointinthedisassemblysequenceandkeepsthepositionsofthe taskswhichareassignedafterthiscutpointwhilereconstructinga newsub-sequencebeforethecutpositionaccordingtothesolution constructionstrategy.
4.4. VNSapproach
Avariableneighborhood(VNS)basedapproach[37]isemployed in order to further exploit the solutions that HDABC cannot improve.VNSusestheideaofsystematicallychangingthe neigh-borhoodsin ordertoimprove thecurrentsolutionand aimsto furtherexplore thesolutionspace,which may notbeexplored byasimplelocalsearchtechnique[38].Shakingandlocalsearch operatorsareusedintheimplementationoftheVNS.The shak-ingoperatordecidesthesearchdirectionoftheVNSfromasetof neighborhoods.Therefore,eachsolutionobtainedbytheshaking operatorisfurtherevaluatedwiththelocalsearchoperatorinorder toexplorenewpromisingneighborhoodsofthecurrentsolution. Ifthereisanimprovement,thentheshakingoperationsreturnsto
thefirstoperator,otherwise,shakingcontinueswiththenext oper-ation.Afterreachingthemaximumnumberofshakingoperations, thesearchprocedurecontinueswiththefirstoperationinthenew iterationuntilVNSiterationlimitisreached.TheVNSprocedureis giveninTable2.
4.5. Evaluationmechanism
The evaluation mechanism is concerned withthe computa-tionoftheobjectivefunctionforeachsolution.Inthisstudy,the solutions are evaluated with a lexicographic methodin which objectivesareevaluatedunderthecontrolofadecisionmakerin a hierarchicalmanneraccordingtopredefinedpriorities. Objec-tivesareconsideredwithahigherpriorityinthefollowingorder: first,second,thirdandfourthobjectives.Therefore,withthis lex-icographicevaluationmethod,thealgorithmtriestooptimizethe first objectiveby ignoring theothers. Then, thealgorithm tries tooptimizesecond,thirdandfourthobjectives,respectively.Itis notpossibletoconstructanefficientParetosetsincelexicographic evaluationmethodmakesthealgorithmignoresomebetter solu-tionsaccordingtootherobjectives.However,thealgorithm will probablycatchthebestsolutionforthefirstobjectivesincethe lexicographicevaluationmethodfocusesonthefirstobjectivewith thehighestpriority.Infact,thesecondobjectiveshouldalsoreach bestvaluesamongotherssincethesecondobjectivebehaves par-alleltothefirstobjective.However,thirdandfourthobjectivesare notlikelytoreachtheirbestvaluessincetheyconflictwiththefirst andsecondobjectives.
Anotherapproachistoassignweightstoeachobjectiveinorder toobtainasingleobjectiveoptimizationproblem.Thus,the algo-rithmcanfocusonasingleobjectiveandanefficientfrontiermaybe found.Ingeneral,therearethreedifferentmethodstoassignthese weightstoeachobjective:fixed,randomandadaptiveweighted methods.Inourpreliminaryexperiments,fixedweightedmethod outperformedothers. Therefore,second evaluationmethodthat wasinvestigatedisfixedweightedmethod.Inthismethod,multi objectiveoptimizationproblemissolvedwithequalweights asso-ciatedwitheachobjective.Thus,anadditionalobjectivefunction isobtainedusingthefollowingequation.
minf5= 4
i=1 wi×fi where 4 i=1 wi=1 (18)4.6. Paretofiltermechanism
Realworldproblemsgenerallyrequiresimultaneous optimiza-tionofmultiple(oftenconflicting)objectives.Itisnotpossibleto findonesolutionthatperfectlysatisfiesallofourneeds.Inthiscase, adecisionmakerhastomakeachoiceregardingwhichobjective hasthehighest priority.Asmentioned in theintroduction sec-tion,theFDLBPisamulti-objectiveoptimizationproblemwhere anumberofobjectivesneedtobeminimizedconcurrently.
Onthecontrarytotheoptimizationproblemwithasingle objec-tive,inamulti-objectiveoptimizationproblem,thereisatrade-off amongobjectivesofasolutionwhichisknownasParetooptimal setwhichcontainsallnon-dominatedsolutions[39].InthisPareto optimalset,asolutionisimprovedintermsofaprioritizedobjective byloweringtheimportanceofatleastoneoftheotherconflicting objectives.Multi-objectiveoptimizationenablesdecisionmakerto selectanoptimalsolutionintheformofParetooptimalsetbasedon hispriorityamongobjectivesofthesolutionforFDLBP.To guaran-teethatthesolutionsobtainedarenon-dominated,asimplePareto filteringapproachisembeddedintoHDABCalgorithm.ThePareto setisiterativelyupdatedtogetclosertoanefficientPareto-optimal front.Oncethealgorithmgeneratesasolutionthatdominatesany
Table2
TheVNSprocedure.
Step Description
1 Takeinitialsolution0fromHDABCalgorithmasaninput
2 Acceptinitialsolution0asthebestsolution1tostartshakingprocedure
3 IfVNSiterationlimitisreached,gotoStep11
4 Applyashakingoperatorrandomlytothecurrentbestsolution1andobtaina2solutionintheneighborhoodofthecurrentbestsolution1
5 Ifsolution2isnotbetterthancurrentbestsolution1,gotoStep7
6 Savesolution2ascurrentbestsolution1andgobacktoStep3
7 Acceptsolution2assolution3forstartinglocalsearchprocedure
8 Applyalocalsearchoperatortothesolution3andobtainasolution4
9 Ifsolution4isbetterthancurrentbestsolution1,save4ascurrentbestsolution1andgotoStep3
10 IfVNSiterationlimitisnotreached,gotoStep8
11 StopVNSprocedureandsendbestfoundsolution1totheHDABCalgorithm
Table3
TheprocedureofHDABC. Step Description
1 Start
2 Initializeinputdata,precedencerelationshipsandthepredefinedparameters(ps,eb,ob,timelimit,ilimit) 3 Generateinitialpopulationaccordingtostationbasedtaskassignmentprocedure
4 EvaluateinitialpopulationandconstructPOPmatrixthatkeepseachsolutionandassociatedfitnessvalues(f1,f2,f3,f4,f5).
5 SetNIvectorwhichholdsthenon-improvediterationnumberforeachmemberofthepopulationtoazerovector 6 ApplyParetofiltermechanismandsavePOSmatrixthatkeepsPareto-optimalsolutionsfrominitialpopulation
8 Savebestsofarsolution,applyParetofiltermechanismandupdatePOSmatrixthatkeepsPareto-optimalsolutionsaccordingtocurrentpopulation 9 Releaseallemployedbeestofoodsourcesasfollows:
9.1 Applyrandomlyselectedneighborhoodstructuretoeachsolutioninthecurrentpopulationandobtainnewsolutions 9.1.1 Ifanimprovementisnotachievedasaresultoftheneighborhoodoperator,gotoStep9.1.3
9.1.2 Savenewsolutionandreplaceoldsolutioninthecurrentpopulation,gotoStep9.1.4 9.1.3 IncreaserelatedpositionofNIvectortosavenon-improvediterationcount 10 Releaseeachonlookerbeeasfollows:
10.1 Applyroulettewheelselectionmechanismandselectafoodsourcefornextonlookerbee. 10.2 Applyrandomlyselectedneighborhoodstructuretotheselectedsolutionandobtainanewsolution 10.3 Ifanimprovementisnotachievedasaresultoftheneighborhoodoperator,gotoStep10.5 10.4 Savenewsolutionandreplacetheselectedoldsolutioninthecurrentpopulation,goto10.6 10.5 IncreaserelatedpositionedmemberofNIvectortosavenon-improvediterationcount 10.6 Ifthereareanyremainingonlookersathand,gotoStep10.1
11 Startscoutbeesphaseasfollows:
11.1 Abooleanstatementischeckedforeachmemberinthepopulationasfollows:
11.1.1 Ifnon-improvediterationcount(imp)islessthanpredefinediterationlimit(ilimit),gotoStep11.1.7 11.1.2 Generatearandomnumber(r)in[0,1]
11.1.3 Ifr≤0.5,gotoStep12.1.5 11.1.4 ApplyVNSalgorithmtothesolution 11.1.5 Generatearandomsolution
11.1.6 Animprovementisachieved,settherelatedpositionedmemberofNIvectortozerovalue 11.1.7 Ifanyuncheckedmemberispresent,gotoStep11.1.1
12 Checktimelimit,ifnotexceededgotoStep8 13 Saveglobalstatisticsandprintfinalresults
14 Stop
solutionintheParetoset,itreplacesthesolution(s)whichwere dominatedintheset.
4.7. HDABCprocedure
TheprocedureofHDABCisgiveninTable3. 5. Numericalresults
TheproposedalgorithmwascodedinC++andtestedonanIntel Xeon2.5GHzprocessorwith8GBRAM. Theperformanceofthe HDABCalgorithm wasevaluatedover two cases; awell-known dataset (25-partmobile phone example)[40] and a newdata set(47-partlaptopexample)introducedinthisstudy. Consider-ingfuzzinessfortheprocessingtimes,fuzzydataarerepresented bytriangularfuzzymembershipfunctions.Thedataforthetwo scenariosaregiveninTables4and5,respectively.
In order to show the effectiveness of the algorithm, fixed weightedand lexicographic evaluation mechanismis tested on DLBPbefore solving FDLBP. The numerical results of proposed algorithmarecomparedwithotherstudiesfromtheliteraturein
Table6.Asaresult,HDABCalgorithmperformedbetterwithina
limitedtime frameof 10s. Afterextended experimentation,the followingsettingsforHDABCalgorithm’scontrolparameterswere foundtobethebest:numberofemployedbees,numberofonlooker beesandnumberoftasksareequaltoeachother,iterationlimit toreleasescoutbeesisequalto100whilevariableneighborhood searchlimitisequalto10.
Inordertounderstandhowfuzzinessinfluencesthe computa-tionalcomplexity,weinvestigatedthebehaviorofthealgorithmin asinglerunfordifferentcases:25-partand47-partDLBPinstances against25-partand47-partFDLBPinstances.Thealgorithmwas runfor 100sand thecumulative numberof solutions obtained bythealgorithmwasobserved.AsdepictedinFig.1,the cumula-tivenumberoffeasiblesolutionsfoundbythealgorithmdecreased whentheproblemsizeincreasedand/orfuzzinesswasconsidered. ItcanbedrawnfromFig.1thatfuzzinessincreasesthecomplexity ofthealgorithmbecauseitreducesthefeasiblenumberofsolutions obtainedinagiventimesincewhenthenumberoffeasible solu-tionsincrease,thetimetoreach(near)optimumsolutionsismore likelytodecrease.AfurtherobservationcanbemadefromFig.1
Table4
Dataofmobilephoneinstance.
Taskno. Description Time Demand Hazardous Predecessor
1 Antenna (1,3,8) 4 Yes 0 2 Battery (1,2,6) 7 Yes 0 3 Antennaguide (2,3,4) 1 No 1,2 4 Bolt(Type1)A (8,10,12) 1 No 0 5 Bolt(Type1)B (8,10,12) 1 No 0 6 Bolt(Type2)1 (12,15,18) 1 No 2 7 Bolt(Type2)2 (12,15,18) 1 No 2 8 Bolt(Type2)3 (12,15,18) 1 No 2 9 Bolt(Type2)4 (12,15,18) 1 No 3 10 Clip (1,2,3) 2 No 4,5 11 Rubberseal (1,2,3) 1 No 10 12 Speaker (1,2,6) 4 Yes 11 13 Whitecable (1,2,3) 1 No 6,7,8,9 14 Red/bluecable (1,2,3) 1 No 6,7,8,9 15 Orangecable (1,2,3) 1 No 6,7,8,9 16 Metaltop (1,2,3) 1 No 6,7,8,9 17 Frontcover (1,2,3) 2 No 13,14 18 Backcover (2,3,4) 2 No 15
19 Circuitboard (10,12,18) 8 Yes 13,14,16,18
20 Plasticscreen (4,5,6) 1 No 17
21 Keyboard (1,2,3) 4 No 17
22 LCD (4,5,6) 6 No 21
23 Subkeyboard (10,12,18) 7 Yes 16,22
24 InternalIC (1,2,3) 1 No 19,23
25 Microphone (1,2,6) 4 Yes 21
Fig.1. ComputationalcomplexitycomparisonofDLBPandFDLBPfor25partand47partinstances.
oncomputationalcomplexityisnotashighascomparedtosmaller sizedproblems.
Afterthecomputationalcomplexity analysis,30 replications, starting each time from a differentnumber seed, onthe prob-leminstanceisperformedwithinatimeframeof100stosolve FDLBPinstances.Thealgorithmisterminatedoneachreplication afterthistimelimitisreached.Thebestsolutionsobtainedusing lexicographicevaluationmechanismarepresentedinTable7.
AtypicalParetooptimalsetobtainedbyHDABCalgorithmwith fixedweightedevaluationmechanismformobilephoneexample andlaptopexampleisgiveninTables8and9,respectively.
Theresultsobtainedshowedthatboth methodshave advan-tages against each other. Fixed weighted method was able to construct a more diverse Pareto frontier while lexicographic
methodwasmoresuccessfultooptimizetheobjectivewiththe highestprioritysincelexicographicmethodmaydisregardsome Paretooptimalsolutionsbecauseofthepredefinedprioritieswith abiasedfocusoncertainobjectives.Asdiscussedearlier,adecision makerisabletocomparethesolutionsandselectthemost desir-ableoneaccordingtohis/herneeds.Forexample,forbothproduct cases,thedecisionmakerchoosessolution1,ifsmoothnessindex andnumberofworkstationismorecriticalforthecompany.On theotherhand,s/hemaychoosesolution2ifremovinghazardous andhighdemandedcomponentsaremoreimportant.
In order to showthe performance of HDABC, we compared HDABC with traditional artificial bee colony (ABC) algorithm and discrete version of artificialbee colony (DABC) algorithm. We tested each algorithm separately on both FDLBP instances.
Table5
Dataoflaptopinstance.
Taskno. Description Time Demand Hazardous Predecessor
1 2boltgroup (14,16,18) 1 No 0
2 4boltgroup (28,32,36) 1 No 1
3 Harddrive (3,4,9) 3 Yes 2
4 Harddrivecover (2,4,6) 1 No 2
5 Battery (3,4,9) 5 Yes 0 6 Batterycover (4,6,8) 1 No 5 7 Notebookfeet (8,10,12) 1 No 0 8 2boltgroup (12,16,20) 1 No 0 9 ZIFconnector (4,5,6) 2 No 8 10 4boltgroup (28,32,36) 1 No 8 11 CDdriver (3,4,9) 4 Yes 9,10 12 CDdrivercover (4,6,8) 1 No 10 13 1boltgroup (6,8,10) 1 No 0 14 RAMcover (1,2,3) 2 No 13 15 3boltgroup (20,24,28) 1 No 0 16 Stripcover (5,7,9) 1 No 15 17 4boltgroup (28,32,36) 1 No 16 18 Keyboard (4,6,8) 2 No 17
19 RAM(bottom) (3,4,9) 7 Yes 14
20 2boltgroup (12,16,20) 1 No 18
21 Modem (3,4,9) 6 Yes 20
22 PCIcard (3,4,9) 7 Yes 18
23 4boltgroup (28,32,36) 1 No 22
24 4boltgroup (28,32,36) 1 No 22
25 Monitor (12,16,20) 6 Yes 23,24
26 11boltgroup (76,88,100) 1 No 19
27 Heatsinkcover (6,8,10) 2 No 26
28 4boltgroup (28,32,36) 1 No 27
29 Heatsink (3,6,11) 4 Yes 28
30 1boltgroup (6,8,10) 1 No 29 31 Processor (3,4,9) 7 Yes 30 32 13boltgroup (98,104,110) 1 No 1,5,9,18 33 Topcover (14,18,22) 2 No 32 34 Cable (2,4,6) 2 No 33 35 1boltgroup (6,8,10) 1 No 33 36 Speaker (7,8,12) 4 Yes 34,35 37 9boltgroup (60,72,84) 1 No 36 38 Systemboard (6,8,10) 1 No 37 39 9boltgroup (60,72,84) 1 No 38 40 Fan (8,10,12) 3 No 39 41 2boltgroup (12,16,20) 1 No 40
42 Audioboard (3,4,9) 3 Yes 41
43 2boltgroup (12,16,20) 1 No 36
44 LEDboard (3,4,9) 4 Yes 43
45 4boltgroup (28,32,36) 1 No 44
46 Touchpad (2,4,6) 3 No 45
47 RAM(top) (3,4,9) 7 Yes 18
Table6
BestresultsforDLBP.
Researchwork f1 f2 f3 f4
MOACO[14] 9 9 – 883
RL[41] 9 9 97 862
HDABCwithfixedweightedevaluationmechanism 9 9 80 824
HDABCwithlexicographicevaluationmechanism 9 9 76 825
Table7
BestresultsofHDABCwithlexicographicevaluationmechanismforFDLBP.
Example f1 f2 f3 f4 BD
SI BEMobilephone 12 14,048 85 919 (0,33,107) (0,12.042,32.109) (0.505,0.810,1) Laptop 10 62,376 336 2840 (0,184,388) (0,61.595,126.317) (0.647,0.823,1)
Table8
ParetooptimalsolutionsfoundbyHDABCwithfixedweightedevaluationmechanismforFDLBPwith25-partmobilephoneinstance.
Solution BD
SI BE f1 f2 f3 f41 (0,33,107) (0,12.12,32.17) (0.50,0.81,1) 12 14,10 86 935
2 (11,48,125) (8.54,19.84,37.32) (0.46,0.75,1) 13 21,39 74 814
Table9
ParetooptimalsolutionsfoundbyHDABCwithfixedweightedevaluationmechanismforFDLBPwith47-partlaptopinstance.
Solution BD
SI BE f1 f2 f3 f41 (0,184,388) (0,62.21,126.55) (0.65,0.82,1) 10 62.74 322 2678
2 (83,288,498) (44.76,103.37,160.85) (0.59,0.75,0.97) 11 103.09 288 2471
Boldentriesshowsbettersolutionsintermsofobjectivefunctionvalues.
Fig.2. Improvementofthesolutionsoveriterationswitheachalgorithm’sbestperformancefor25partFDLBPprobleminstance.
Fig.3. Improvementofthesolutionsoveriterationswitheachalgorithm’sbestperformancefor47partFDLBPprobleminstance.
Figs.2and3demonstratetheimprovementofthesolutionsover
iterationswithinatimelimitof100sfor25-partand47-partFDLBP instances,respectively.Allalgorithmscomparedreachedthesame solutions within the time frame, however, HDABC algorithm achievesthesolutionfasterthantheothers.Figs.4and5depictthe bestsolutionsreachedbyallalgorithmson30samplerunsof 25-partand47-partFDLBPinstances,respectively.InFigs.4and5,it canbeseenthatHDABCwasmorerobustsinceitwasabletoreach bettersolutionsinmanyofsampleruns.For47-partinstance,the performancedifferenceismoreobvious.Fig.6presentsastatistical
analysisfor30samplerunresultsindicatingthatwithin95%of confidenceintervalHDABCismorerobustandefficient.
In order to understand the influence of integrated VNS approach on HDABC algorithm, a comparison between DABC and HDABC was done in terms of number of improvements achieved within 100s of time frame. As can be seen in
Tables 10 and 11, HDABC algorithm was able to achieve 286
(and 287for47-part FDLBPinstance)improvements with inte-grated VNS mechanism while DABC was able to achieve 197 improvements.
Fig.4. 30samplerunsof25partFLDBPinstancewitheachalgorithm.
Fig.5. 30samplerunsof47partFLDBPinstancewitheachalgorithm.
Table10
Numberofimprovementsachievedbyneighborhoodstructuresin100s(25partFDLBPprobleminstance):asamplerunofHDABCalgorithm. VNSlocalsearchstructures
Pure(DABCalone) N1 N2 N3 N4 N5 Sum
Withlocalsearch VNSshakingstructures N1 51 18 6 1 1 0 77 N2 61 24 5 3 2 0 95 N3 14 3 1 3 1 0 22 N4 51 8 4 2 1 0 66 N5 20 4 1 1 0 0 26 Sum 197 57 17 10 5 0 286
Fig.6.Performancecomparisonofmethodswithin95%confidenceintervalforFDLBPprobleminstances.
Table11
Numberofimprovementsachievedbyneighborhoodstructuresin100s(47partFDLBPprobleminstance):asamplerunofHDABCalgorithm. VNSlocalsearchstructures
Pure(DABCalone) N1 N2 N3 N4 N5 Sum
Withlocalsearch VNSshakingstructures N1 47 27 5 3 2 0 84 N2 92 17 4 1 0 0 114 N3 16 7 4 0 1 0 28 N4 20 9 0 1 0 0 30 N5 22 6 1 1 1 0 31 Sum 197 66 14 6 4 0 287 6. Conclusions
Thisstudycontributestotheliteraturebypresentingafuzzy extensionofthedisassemblylinebalancingproblem(DLBP)with fuzzytaskprocessingtimesusingahybriddiscreteartificialbee colony(HDABC)algorithmasasolutionapproachforthefirsttime. Thealgorithmaimstosolvethefuzzydisassemblylinebalancing problemwithobjectivesofminimizingthenumberof worksta-tions,optimizingfuzzysmoothnessindexoftheline,maximizing theearliest removalof hazardous componentsand maximizing theearliestremovalofhighdemandedcomponents.Balancedelay andbalanceefficiencyofthedisassemblylinewasalsomeasured. Theeffectivenessofproposedalgorithmistestedusingamobile phone instancewith 25 parts from theliterature and a laptop instancewith47partsintroducedinthisstudy.Thetotalcost func-tionisformulatedaccordingtolexicographicandfixedweighted mechanism.Inthefixedweightedmethod,thetotalcostfunction iscalculated withequallyweightedsumofmultiple objectives. Asforthelexicographicmethod,itwasassumedthatamanager alreadydecidedtheprioritiesoftheobjectives.Theinfluencesof bothmechanismsontheperformanceoftheproposedHDABCwere examined.Theexaminationtounderstandtheinfluenceof fuzzi-nessonthecomplexityofthealgorithmindicatedthatfuzziness increasesthecomplexitybecauseitreducesthefeasiblenumber ofsolutionsobtainedinagiventimesincewhenthenumberof
feasiblesolutionsincrease,thetimetoreach(near)optimum solu-tionsismorelikelytodecrease.Furtheranalysissuggestedthatthe performanceofHDABCcomparedtotraditionalartificialbeecolony (ABC)algorithmanddiscreteversionofartificialbeecolony(DABC) algorithmwasthebestintermssolutionqualityandrobustness. ThisworkislimitedtosinglemodelFDLBPandconsidersa com-pletedisassemblyofcertainproducts.Asforfurtherstudies,more complexDLBPsshouldbeconsidered.Anideaistoaddressmixed modelorpartialDLBPwithfuzzyjobprocessingtimes.
Acknowledgment
This work is partially supported by the Scientific Research Projects Fund of Pamukkale University under the Grant No. 2010FBE051.
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