Design of exible assembly line to minimize equipment cost JOSEPH BUKCHIN and MICHAL TZUR Department of Industrial Engineering Faculty of Engineering TelAviv University TelAviv Israel Email bukchinen - PDF document

Design of exible assembly line to minimize equipment cost JOSEPH BUKCHIN and MICHAL TZUR Department of Industrial Engineering Faculty of Engineering TelAviv University TelAviv  Israel Email bukchinen
Design of exible assembly line to minimize equipment cost JOSEPH BUKCHIN and MICHAL TZUR Department of Industrial Engineering Faculty of Engineering TelAviv University TelAviv  Israel Email bukchinen

Design of exible assembly line to minimize equipment cost JOSEPH BUKCHIN and MICHAL TZUR Department of Industrial Engineering Faculty of Engineering TelAviv University TelAviv Israel Email bukchinen - Description


tauacil or tzurengtauacil Received April 1997 and accepted July 1999 In this paper we develop an optimal and a heuristic algorithm for the problem of designing a 57519exible assembly line when several equipment alternatives are available The design p ID: 36671 Download Pdf

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Presentation on theme: "Design of exible assembly line to minimize equipment cost JOSEPH BUKCHIN and MICHAL TZUR Department of Industrial Engineering Faculty of Engineering TelAviv University TelAviv Israel Email bukchinen"— Presentation transcript


Designof¯exibleassemblylinetominimizeequipmentcostJOSEPHBUKCHINandMICHALTZURDepartmentofIndustrialEngineering,FacultyofEngineering,Tel-AvivUniversity,Tel-Aviv69978,IsraelE-mail:bukchin@eng.tau.ac.ilortzur@eng.tau.ac.ilReceivedApril1997andacceptedJuly1999Inthispaperwedevelopanoptimalandaheuristicalgorithmfortheproblemofdesigninga¯exibleassemblylinewhenseveralequipmentalternativesareavailable.Thedesignproblemaddressesthequestionsofselectingtheequipmentandassigningtasksto IIETransactions,585±598 SALBisproventobeanNP-Hardproblem[5],andre-sultingfromthisistheconclusionthatourproblemisalsoNP-Hard.Therearerelativelyfewstudiesthataddresstheprob-leminwhichthereismorethanonetypeofequipment.GravesandHolmesRed®eld[6]considerthedesignproblemwithseveralequipmentalternatives,whenmultiproductsareassembledonthesameline.Theyassumeacompleteorderingoftasksofthesameproductandlargesimilaritiesamongdi€erentproducts.Theseassumptionsresultinarelativelysmallnumberofcandidatework-stations(anumberwhichisproportionaltothetotalnumberoftasksandiscloseto,butgreaterthantwo),whichthereforesimpli®estheproblemcon-siderably.Theiralgorithmindeedenumeratesallfeasibleworkstations,selectsthebestequipmentforeach,andthenchoosesthebestsetofworkstations.PreviousworkonthesingleproductdesignproblemwithequipmentselectionincludesGravesandWhitney[7]andGravesandLamar[8],butinbotharticlesthesequenceoftasksisalsoassumedtobe®xed.etal.[9]discussprocessingalternativesinamanualassemblylineasanextensionofSALB.Eachprocessingalternativeisrelatedtoagivensetoftasksi.e.,representsalimitedequipmentselectionwhichmaybetotheexistingequipmentinthestation,andthedecisioniswhethertouseeachsuchalternativeinordertoshortenthetasksduration,atagivencost.Sincethelineismanual,eachtaskmaybeperformedateachsta-tion.TheirsolutionprocedureconsistsofabranchandboundalgorithminwhichaSALBproblemissolvedineverynodeofthebranchandboundtree,thereforethisalgorithmmaybeusedonlyforasmallnumberofpos-sibleprocessingalternatives.RubinovitzandBukchin[10]presentabranchandboundalgorithmfortheproblemofdesigningandbal-ancingaroboticassemblylinewhenseveralrobottypesareavailableandtheobjectiveistominimizethenumberofworkstations.Theirmodelisaspecialcaseofours,inwhichalloftheequipmentalternativeshaveidenticalpurchasingcosts.TsaiandYao[11]proposedaheuristicapproachforthedesignofa¯exibleroboticassemblylinewhichproducesafamilyofproducts.Giventheworktobedoneineachstation,thedemandofeachproductandabudgetconstraint,theheuristicdeterminestherobottypeandnumberofrobotsrequiredineachworkstation.Theirobjectiveistominimizethestandarddeviationofoutputratesofallworkstations,whichistheirmeasure-mentforabalancedline.Theremainderofthepaperisorganizedasfollows:inSection2weintroducethenotationandaformulationoftheproblemandillustrateitwithanexample.InSection3wedeveloptwotypesoflowerbounds,thatareusedlaterinouralgorithm.InSection4wedescribeourexactbranchandboundalgorithmandpresentsomequalita-tiveinsightswithrespecttotheproblem'sparameters,basedonanempiricalstudythatweperformed.Wealsoexaminethequalityofthelowerboundthatwedevelopedfortheproblem.InSection5wediscusshowaheuristicprocedure,basedonthebranchandboundalgorithm,maybedesignedfortheverylargeproblems.Finally,Section6containsourconclusions.2.ProblemformulationInthissectionweintroducethenotationaswellasourpreciseassumptions,andpresentanintegerprogrammingformulationoftheproblem.Basedonthisformulationwedevelop,inthenextsection,lowerboundsfortheproblem.Toillustratethemodel'sassumptionsandhelpthereaderfollowouranalysis,weprovideattheendofthissectionanexampleproblem.Theproblemisde®nedbythefollowingparameters:=durationoftaskwhenperformedbyequipment...=costofequipmenttype...;(Weuseequipmenttypethisshouldcausenoconfusion.)=requiredcycletime;=setofimmediatepredecessorsoftaskThefollowingassumptionsarestatedtoclarifythesettinginwhichtheproblemarises:(i)Thereisagivensetofequipmenttypes,eachtypeisassociatedwithaspeci®ccost.Theequipmentcostisassumedtoincludethepurchasingandoperationalcostofusingtheequipment.(ii)Theprecedencerelationbetweenassemblytasksis(iii)Theassemblytaskscannotbefurthersubdivided.(iv)Thedurationofataskisdeterministic,butde-pendsontheequipmentselectedtoperformthe(v)Ataskcanbeperformedatanystationoftheassemblyline,providedthattheequipmentse-lectedforthisstationiscapableofperformingthetask,andthatprecedencerelationsaresatis®ed.(vi)Thetotaldurationtimeoftasksthatareassignedtoagivenstationshouldnotexceedthepre-determinedcycletime.(vii)Asingleequipmentisassignedtoeachstationontheline.(viii)Asingleproductisassembledontheline.(ix)Materialhandling,loadingandunloadingtimesarenegligibleorincludedinthetasksduration.(x)Setupandtoolchangingtimesarenegligibleorincludedinthetask'sduration.Thedecisionsthathavetobemadeaddresstwoissues:(i)thedesignissue,wheretheequipmenthastobese-BukchinandTzur lectedandassignedtostations;and(ii)theassignmentofalltaskstothestations,suchthattheprecedenceaswellasthecycletimeconstraintsaresatis®ed.Thefollowingtwosetsofbinarydecisionvariablescorrespondtoeachofthesetwoissues,respectively.IntheAppendixwesummarizeallthenotationusedthroughoutthepaper.Wede®neforeveryequipmentandeverystation1ifequipmentisassignedtostation0otherwiseInaddition,wede®neforeverytask,everyequipmentandstationnumber1iftaskisperformedbyequipmentatstation0otherwiseThefollowingistheresultingintegerprogrammingformulationoftheproblem,denotedas(P1):subjecttosubjecttoTheobjectivefunction(1)representsthetotaldesigncosttobeminimized.Notethatthenumberoftasks,,servesasanupperboundforthenumberofstations.Constraintset(2)ensuresthatiftaskisanimmediatepredecessoroftask,thenitcannotbeassignedtoastationwithahigherindexthanthestationtowhichtaskisassigned.Constraintset(3)ensuresthateachtaskisperformedexactlyonce.Constraintset(4)representstherelationshipbetweentheandthevariablesbynotallowinganytasktobeperformedonagivenpieceofequipmentinagivenstation,ifthisequipmentisnotassignedtothatstation.Also,ifagivenpieceofequipmentassignedtoagivenstation,constraintset(4)speci®esthecycletimerequirement.Constraintset(5)representstherequire-mentofatmostonepieceofequipmentatanystationandconstraintsets(6)and(7)de®nethedecisionvari-ablestobebinary.Sincethisisthe®rsttimethatthisproblemhasbeenconsidered,theformulationisnew,al-thoughelementsofithaveappearedpreviouslyinthelit-erature.Theformulationconsistsofvariablesandconstraints,butthemainimportanceofthisformulationistherelaxationresultingfromit,whichen-ablesustoobtaingoodbounds,asexplainedinSection3.AnexampleproblemOurexampleproblemisbasedontheexampleanalyzedinPintoetal.[9].Inparticular,weadoptedtheprece-dencediagramoftheir10tasksproblem,showninFig.1.Inourexample,aproductisassembledonanautomatedassemblyline,usingthreedi€erenttypesofequipment(machines).ThecostofeachequipmenttypeandthetimerequiredtoperformeveryassemblytaskbyeachoftheselectedequipmenttypesareshowninFig.1.Emptyel-ementsinthedurationtableimplythatthetaskcannotbeperformedbytheassociatedequipmenttype.Wecancompareamongdi€erentequipmenttypesalongthreedimensions:cost,speedand¯exibility(num-beroftasksthatcanbeperformedbytheequipment).Whennoequipmenttypeisdominatedbytheothers,atradeo€existsbetweendi€erenttypes,withrespecttoatleasttwooftheabove-mentionedproperties.Forexam-ple:afastand¯exibleequipmenttypeislikelytobemoreexpensive.Inourexample,onecanobservethateachequipmenttypehasanadvantageovertheothersinoneofthethreedimensions:±ahighly¯exibleequipmenttype,namely,apieceofequipmentwhichiscapableofperformingalargenumberofassemblytasks(alltasks,inthis±afastassemblyequipmenttypecharacterizedbyshorttaskduration.±theleastexpensiveassemblyequipment.TheIPformulationofthisexample,basedonformula-tion(P1)presentedinSection2,consistsof330binaryvariablesand61constraints.Wesolvedthisproblembyouralgorithm(de-scribedinSection4),determiningtaskassignmentsandequipmentselection,whileminimizingthetotalequip-mentcost(1),subjecttoacycletimeconstraintof50.Theoptimalcon®gurationwasobtainedin0.05secondsandisshowninFig.2.Theminimalequipmentcostrequiredforacycletimeof50is$360000(threemachinesof$100000eachplusonemachineof$60000).Thetradeo€betweenthedi€erenttypesisdemonstratedintheoptimalsolution,bythefactthatallthreetypesofequipmentareused.Animportantconclusiondrawnfromthisexampleisthataslongasagivenequipmenttypeisnotdominatedbyanothertypealongallthree,itmaybeincludedintheoptimalcon®gura-Designof¯exibleassemblyline 3.LowerboundsInthissectionwedeveloplowerboundsfortheproblem,aswellasforsubproblemsofit.Asweshowbelow,theboundsareobtainedbyrelaxingsomeoftheconstraintsoftheformulation(P1)andsolvingtherelaxedproblem;theboundsareusedinourbranchandboundalgorithmthatwillbediscussedinthenextsection.Considerproblem(P1),andmakethefollowingre-laxationstoit:(i)Eliminatetheprecedenceconstraints(2).(ii)Sumtheconstraintsin(4)overallstations,foreachequipmenttype.Theresultingsetofconstraints,denotedby(4)isthefollowing:Asaresultoftheserelaxations,constraintset(5)isnolongermeaningfulsinceallthestationsarenowconsid-intheformulation.Equation(4)impliesthatitisnotrequiredtokeepthecycletimeconstraintineverystation,onlythecycletimeconstraint,representingacapacityconstraintforeachequipmenttype.Thereforewede®nethefollowingnewdecisionvariables,whichareindependentofthestations:totalnumberoftype1iftaskisperformedbyequipment0otherwiseTherelaxedformulation,denotedas(P2),isnowde-scribedasfollows:subjectto Fig.1.Precedencediagram,tasktimesandequipmentcosts. Fig.2.Optimalcon®gurationoftheexampleproblem(totalcost=360000).BukchinandTzur Here,(8)isequivalentto(1),representingthetotalequipmentcosttobeminimized;(9)replaces(3),ensuringthateachtaskisperformedexactlyonce,and(10)isinfactconstraint(4)discussedabove.Tosimplifytheproblemfurther,werelaxtheconstraintsre-gardingthevariables(12)andobtainproblem(P3).Therefore,problem(P3)isde®nedby(8)±(11)and:andweprovethefollowing:Theorem1.ThefollowingsolutionisoptimalforproblemifECIfmorethanoneindexjachievestheminimumchooseoneofthemBeforeprovingthetheoremformally,letus®rstex-plainitintuitively.Note®rstthateachunitofanequipmenttypemaybeassignedasmuchworkasthecycletime,.Therefore,thenumberofunits(whichmaybefractional)tobepurchasedfromeachequipmenttypeisthesumofthedurationofalltasksassignedtothistype,dividedbythecycletime,resultingin(14).Thismeansthatinordertoperformacertaintask,saybyacertainequipmenttype,say,afractionoftheequipmentneedstobepurchased,whichequalstothefractionofcycletimerequiredtoperformit,i.e.,Thecostofthisfractionofequipmentis:Comparingthecostsofallequipmentalternativesforagiventask,andchoosingthetypewhosecostismini-mal,oneobtainsEquation(13).Wenowprovideamoreformalproof.ProofofTheorem1.Note®rstthatthesolutionde®nedby(13)and(14)isfeasible.Notealsothatgiventothevariables,thesolutiontothevariablesasde®nedby(14)is.Thereforeitremainstoprovethatthesolutionofthevariablesasde®nedby(13)isAssume,bycontradiction,thatthissolutionisnotoptimal,thereforethereexistsavariableintheopti-malsolutions.t.1butThisvariable,associatedwithtask,contributesunitstothevariableandthereforetotheobjectivevalueof(P3).Ifinsteadwechoosefortask1forthatsatis®esthenthecontributiontothevariableisunitsand,byde®nition)totheobjectivevalueof(P3),acontradictiontotheoptimalityoftheformersolution.Wede®ne:isdeterminedby(13)and(14).Corollary1.isalowerboundtothevalueofTheCorollaryistruesince1istheoptimalobjectivevalueofproblem(P3)whichisarelaxationofproblem(P1).Inconclusion,wehaveshownhowtoobtainalowerboundtotheproblem,whichiseasytocompute.Thedeviationofthisboundfromtheoptimalsolutionvalueresultsfromignoringtheprecedenceconstraints,fromconsideringthecycletimerequirementinaggregationtoallstations(i.e.,ataskmaybeperformedinmorethanonestation),andfromtheabilitytouseafractionofapieceofequipment.Inthenextsectionabranchandboundalgo-rithmisdeveloped,whichusestheproposedlowerbound.Intheprocessofsolvingproblem(P1)viathebranchandboundalgorithm,anodeinthebranchandboundtreerepresentsapartialsolution,inwhichsomeofthetaskshavealreadybeenassignedtospeci®cequipmenttypes.Forthisnode,thecalculationofalowerboundisrequired.Thisleadsustoconsideraoftherelaxedproblem(P3),inwhichitisgiventhatasubsetoftheoriginalsetoftasksisperformedbyanalreadyde-terminedsetofequipmenttypes.Inaddition,agivennumberoftimeunits,say,arestillavailableonthelastselectedequipmenttype,saytype.Sincethisequipmenthasalreadybeenpurchased,nocostisassociatedwithtimeunits.Thesubproblemhastodeterminethenumberandtypeofequipmenttobepurchased(atminimalcost)inordertoperformtheoftasks,say,andtoassignthetasksintothenewequipmentwhilesatisfyingtheaggregate(andpossiblyfractional)cycletimeconstraint.Wedenotethissubproblemas(P4)andstateitsexactsubjecttoDesignof¯exibleassemblyline Asdiscussed,thisformulationisidenticalto(P3),ex-ceptthatonlytasksinthesetareconsidered,andtheoriginalconstraint(10)isreplacedby(10a)and(10b).Constraint(10b)isamodi®cationoftheoriginalcon-straint(10)forequipmenttype,whichre¯ectsthetimeunitsonthisequipmenttype.Ideally,allStimeunitsofequipmentshouldbeused,inwhichcasethedesiredvalueforthevariablesmaybefractional.Thereforewedenoteby(P5)asubproblemwhichisarelaxationofproblem(P4),obtainedbyallowingthevariablestobefractional.Thisrelaxationenablesustosolveproblem(P5)tooptimality,providinguswithalowerboundtothevalueofproblem(P4).(Asbecomesclearfromthealgorithmbelow,atmosttwowhichrefertothesametask,willbefractional).Weusethefollowingalgorithm,denotedas(TaskEquipmentSelection),tosolveproblem(P5):Step1.betheequipmenttypeforwhichStep2.bethetaskforwhichStep3.then:set.If,goto(Step5);otherwise,gobackto(Step2).Otherwise:ifStep4.ForeveryStep5.Thebasicideaofthisalgorithmisto®rstassigntaskstofreetimeunitsofequipmenttype.Recallthatwhennofreetimeunitsareavailable(asinproblem(P3)),everytaskisassignedtotheequipmentwhichhastheminimalvalueofwhichwede®nehere(Step(1)ofAlgorithmTES)asequipmenttype.Thisisalsothesolutionforproblem(P5),oncethefreetimeunitsofhavebeenexhausted.Therefore,thetasksthatareassignedtothefreetimeunitsofequipmentarethoseforwhichthe``alternativecost''perunittimeofusageofequipment,de®nedinStep(2)ofthealgo-rithm,ismaximal.TheoptimalityofAlgorithmTESisstatedinthenexttheorem.Theorem2.AlgorithmTESproducestheoptimalsolutionforsubproblemNote®rstthatthesolutionproducedbytheal-gorithmisfeasible.Itisalsoclearthatanoptimalsolu-tionwillnecessarilyuseallfreetimeunitsofequipment.Moreover,oncetheseunitsareusedup,therestoftheproblemisofthetypeofproblem(P3)(onlywithlesstasks),andthereforethesolutionisasde®nedinSteps(1),(4)and(5)ofAlgorithmTES.Itremainstoprovethatthechoiceoftaskstobeassignedtoequipment,asdescribedinSteps2and3,isoptimal.Assumethatthesuggestedsolution(thesolutionpro-ducedbyAlgorithmTES)assignstothefreetimeunitsofthetasksintheset...1andwhere01.Nowassumebycontradictionthatintheoptimalsolutionforsome,i.e.,atleastonetimeunitofthefreeunitsofequipmentisallocatedtoataskwhichisnotin,andconsiderthe®rstsuchunit.(Wediscusshereonlytheusageoftheunitsofequipmenttype,asiftheyaremarked;theassignmentoftaskstoadditionalequipmentofthattypearenotrelevanthere).Asaresult,one(maybeadditional)timeunitofataskin(saytask)hastobeassignedtootherequipment(insteadof);asdiscussedearlier,thebestalternativeistheequipmentidenti®edinStep(1)ofAlgorithmTES.Ifweconsiderthecontributiontotheobjectivevalueofproblem(P5)ofthetimeunitwhoseassignmentdi€ersbetweenthesuggestedsolutionandtheoptimalsolution,weobtainthatinthesuggestedsolutionthecontributionisandintheoptimalsolutionthecontributionis.Byde®nition(Step(2)ofAlgorithmTES),thelatterishigherthantheformer,acontradictiontotheoptimalityofthelattersolution.Wede®ne:isobtainedfromAlgorithmTES.Corollary2.isalowerboundtothevalueof4.ThebranchandboundalgorithmBranchandboundalgorithmshavebeenextensivelyusedforsolvingcomplexcombinatorialproblems,includingassemblylinedesignandbalancingproblems[10,12].Inthisstudy,afrontiersearchbranchandboundalgorithmisdevelopedforminimizingthetotalequipmentcost.Theadvantageofafrontiersearchbranchandboundalgo-BukchinandTzur rithmisthatthenumberofnodesinvestigatedinthebranchandboundtree,isminimal.Inaddition,theuseofsubproblemsandlowerboundsateachnodeofthebranchandboundtree,whicharespeci®ctotheprobleminvestigated,considerablyimprovethee€ectivenessofthealgorithm.Theyweredevelopedintheprevioussection,andtheirusewillbeillustratedinthissection.Throughoutthealgorithm,workstationsareopened(established)sequentially,equipmentisselectedandplacedinthenewlyopenedworkstation,andtaskstobeperformedbytheselectedequipmentareassignedtothisworkstation.Therefore,throughoutthealgorithm,partialsolutionstotheproblemexist,whichdescribepartialassignmentsoftaskstoequipmentandstations.Inaddition,foreachpartialsolutionalowerboundmaybecomputedbasedonthesolutionofsubproblem(P5),asdescribedbelow.Thealgorithmendswhenalltasksareassignedtoequipmentandworkstations,andtheobtainedsolutionvalueisnolargerthanthelowerboundofallpartialsolutions.InSections4.1±4.3wedescribethedetailsofthealgorithm:AnodeinthebranchandboundtreeEachbranchandboundnoderepresentsonepartialsolutionoftheoriginalproblem.Apartialsolutionischaracterizedbyasetoftasks,,whichhavealreadybeenassignedtostations,alongwiththeequipmentse-lectedtoperformthesetasks,i.e.,theequipmentselectedforthesestations.Amongthestationsthatwereusedthusfarinthepartialsolution,thelastopenedstationistheonlyonetowhichtasksmaystillbeassigned.Finally,suchapartialsolutionisassociatedwithanaccumulated,representingthecostofpurchasingtheequipmentdecideduponthusfar.Wede®netheslackofthelastopenedstationatnode,asthedi€erencebetweentherequiredcycletimeandthetimealreadyassignedtothatstationbysomeofthetasksin.Anytask,isatobeassignedtothelaststationopenedifthefollowingconditionshold:(i)Thetaskhasnopredecessors,oritspredecessorsarealreadyassigned.(ii)Thetimetoperformtaskbythealreadyselectedequipmenttype(atthelastopenedstation),,isnolargerthantheremainingslack,Ifthesetofcandidatetasksisnotempty,thestationisde®nedasanopenstation.Otherwise,ifthesetofcan-didatetasksisempty,thestationisde®nedasaThelowerboundThelowerboundwhichiscalculatedforeachnodeofthebranchandboundtree,consistsoftwoelements.The®rstelement,associatedwithdecisions,isthe(exact)costofthealreadyselectedequipmentinthepartialsolutionassociatedwithnode,aknownvaluewhichwedenoted.Thesecondelement,associatedwithsions,isalowerboundonthecostoftheequipmentwhichisyettobeselectedforthesetofyetunassignedisthecomplementofintheoriginalsetoftasks).Thissecondelementiscomputedinoneoftwoways,accordingtowhetherthelastopenedstationisclosedoropen:(i)Ifnoderepresentsaclosedstation,there-mainingdecisionsconcerntheassignmentofthetasksintonewstationsthatneedtobeopened,whoseequipmenttypeshavenotyetbeenchosen.Notethatthisisexactlyproblem(P1)(seeSection2),onlylimitedtothesetoftasksin.Thereforethelowerboundfortheelementassociatedwithfuturecostsofnodeisaclosedstation,whereisobtainedbycalculatingthevalueof(15)tothesetoftasksinSection3).(ii)Ifnoderepresentsanopenstation,therelax-ationwhichisequivalentto1butinadditiontakesintoconsiderationthelastopenedstation,isrepresentedbyaproblemwhichisintheformof(P4).Equipmentin(P4)representstheequip-menttypeofthelastopenedstationinthepartialsolutionofnode,andin(P4)representstheremainingslackofthatstation,.Thereforethelowerboundfortheelementassociatedwithfuturecostsofnodeisanopenstationis,whereisthesolutionof(P5)(therelaxationof(P4)),obtainedbysolvingAlgorithmSummingupthetwoelementsdiscussedaboveofthelowerboundofagivennode,weconcludethatthelowerboundofisaclosedstation,andisanopenstation.Inbothcases,thelowerboundiseasilyStagesofthealgorithmThemainstagesoftheproposedalgorithmareasfol-Stage1Creationofthe®rstlevelofthebranchandbound.Atthisleveleachnodecontainsataskwhichdoesnothaveprecedencerequirements,alongwithanequip-menttypethatiscapableofperformingthistask.Suchanodeisgeneratedforeveryfeasibleequipment±taskStage2Selectionofanodetobeextended.Asdescribedabove,alowerboundoftheoptimalcostiscalculatedforeachnodeofthetree.Theopennode(nodewithoutDesignof¯exibleassemblyline descendants)withthelowestlowerboundisselectedforfurtherextension,representingourchoiceofafrontiersearchalgorithm.Stage3Nodeextension.Eachdescendantoftheextendednodecontainsanassignmentofanewsingletask.Iftheextendednoderepresentsanopenstation,anextensionisperformedforeachcandidatetask.Iftheextendednoderepresentsaclosedstation,anewstationisopened,andtheextensionisperformedforeveryfeasibleequipment±taskcombination.Stage4Eliminationofdominatednodes.Eachtimeastationbecomesclosed,acomparisonbetweenthecurrentnodeandallotheropennodesthatareassociatedwithclosedstations,isperformedinordertoeliminatedomi-natednodes.Thedominatednodecouldbeeitherthenewone,orapreviouslycreatednode.Thedominanceruleisdescribedasfollows:assumethatatnode,asetoftaskshasalreadybeenassigned,withanassociatedequip-mentcost,.Atanothernode,,asetoftasksalreadybeenassigned,withanassociatedequipmentcost.Nodeisdominatedbynode,andthereforecanbeeliminated.Stage5Endcondition.Ifanextendednodecontainsalltasks,anditssolutionvalueisnolargerthanthelowerboundofallopennodes,anoptimalsolutionhasbeenfound.Otherwise,thealgorithmproceedsasinStage2ExperimentalstudyfortheoptimalalgorithmWehavecodedourbranchandboundalgorithmandconductedanexperimentalstudy.Ascanbeconcludedfromtherunningtimereportedinthenextsection,theoptimalbranchandboundalgorithmiscapableofsolv-ingmoderateproblemsizesinareasonableamountoftime,i.e.,problemswithafewdozenoftasksandwith®vetotenequipmenttypes.Thisisonlyanapproxima-tion,sincethevariabilityoftheruntimefordi€erentinstancesofthesamesizeisquitelarge.Thepurposeoftheexperimentalstudypresentedinthissectionistoex-aminetheimpactofvariousproblemparametersonthealgorithmsperformance,andtoinvestigatethee€ective-nessoftheinitiallowerbound(1),measuredbyitsdistancefromtheoptimalsolutionvalue.Wereportonthreeperformancemeasuresinthisstudy:(i)Thesizeofthebranchandboundtree(thetotalnumberofnodesgenerated).(ii)Themaximumnumberofopennodesinthebranchandboundtree.(iii)Therunningtimeofthealgorithm.Infact,thecomplexityofthealgorithm(ameasureofitsrunningtime)isapproximatelythenumberofnodesgen-erated,multipliedbythecomplexityoftheworktobedoneateachnode.Thelatteris,whereisthemaximumnumberofopennodesinthebranchandboundtree,sincecalculating1or2is,andforeachnewnodeitslowerboundhastobeplacedinasortedlistof.Whiletherunningtimeperformancemeasureimpliesthecapabilitiesofthealgorithm,theothertwoperformancemeasureshavetheadvantageofbeingindependentofthecodingeciencyandthecomputertype.Themaximalnumberofnodesopenedsimulta-neouslyduringarunisameasureofthememoryspacerequired(inadditiontoitsimpactonthecomplexityofthealgorithm),seethenextsectionsformoredetails.Weexaminedtheimpactof®veparametersoftheproblemonthe®rsttwoperformancemeasuresmen-tionedabove.Atwolevelfullfactorialexperimentalde-signhasbeenperformed,examiningthesigni®cancelevelofeachfactor.Theparametersandthevaluesthatwereexaminedaredescribednext.Thenumberoftasks.Thenumberoftaskswassetto15and30.Equipmentalternatives.Thenumberofequipmentalternativeswassettothreeand®ve.Variabilityoftaskduration.Thedurationofeverytaskwasgeneratedfromauniformdistribution.Weexaminedadistributionwithasmallvariance,),andadistributionwithahighvari-anceU(0),whereistheexpectedvalueofthetaskduration..The-ratioisameasureforthe¯exibilityincreatingassemblysequences,developedbyMans-oorandYadin[13],andde®nedasfollows:Letanelementofaprecedencematrix,suchthat:1iftaskprecedestask0otherwise,whereisthenum-berofzeroesin,andisthenumberofassemblytasks.The-ratiovalueisthereforebetweenzero,whentherearenoprecedenceconstraintsbetweentasks(anysequenceisfeasible),andone,whenonlyasingleassemblysequenceisfeasible.As-semblytasksareoftencharacterizedbyrelatively-ratios.Hence,precedencediagramswithratiosof0.1and0.4weregeneratedinthisstudy..The-ratioisameasureforthe¯exibilityoftheassemblyequipment,developedbyRub-inovitzandBukchin[10],andde®nedasfollows:anelementinamatrix,representthetimetoperformtaskbyequipmenttype.Iftaskcannotbeperformedbyequipmenttypeissetequaltoin®nity.Letrepresentthenumberofelementssettoin®nity,bethenumberoftasks,bethenumberofequipmenttypes,then-ratio=1.The-ratiovalueisBukchinandTzur thereforebetweenzero,wheneachtaskcanbeperformedbyonlyasingleequipmenttype,andone,wheneachtaskcanbeperformedbyanyoneoftheequipmentalternatives.Thevalueofthe-ratiointhisstudywassetto0.3and0.6.Inadditiontotheseparametersettings,wenotethatthecostofeachequipmenttypewasdeterminedasadecreas-ingfunctionofthevalueof.Thislatterchoiceensuresthatnoequipmenttypemaydominateanyotheralongthetwodimensionsofexpectedtaskdurationandcost.ThethirddesiredpropertyofanequipmenttypewhichwasdiscussedinSection2,namelyits¯exibility,wasgeneratedarbitrarilyaccordingtothespeci®ed-ratio,inordertopreservegeneralityofpossibleequipmentcharacteristics.Thetotalnumberofexperimentsinatwo-level,®ve-factorsfull-factorialexperimentalstudyis232.Wegenerated10instancesforeachexperiment,resultinginatotalof320algorithmruns.Theresultsoftheexperimentsareanalyzedwithre-specttotheimpactthateachfactorhasoneachofthefollowingthreevaluesofinterest:(i)thetotalnumberofnodesvisited;(ii)themaximalnumberofopennodes;and(iii)thedi€erencebetweentheinitiallowerboundandtheoptimalsolutionvalue.TheresultsarepresentedintheformofstandardANOVA(AnalysisofVariance)tables(Tables1±3),includingthevaluesofthemaine€ects,aswellastheirsigni®cancelevels.Followingeachtable,wediscusstheresults,andprovideadditionalinsight.InTable1,thevaluesofthemaine€ectsrepresentthedi€erencesintheaveragenumberofnodesbetweentheexperimentswithhighandlowvalueofeachfactor.Wecanseethatallmaine€ectsarehighlysigni®cant,withverysmall-values.Eventheleastsigni®cantfactor,thedurationvariability,hasa-valueoflessthan2%.Notsurprisinglywediscoverthatthe®rsttwomaine€ectsarepositive,thatis,increasingthenumberoftasksorthenumberofalternativeequipmenttypesleadstoalargerbranchandboundtree.Thetreesizeisalsohighlyandpositivelya€ectedbythevalueofthe-ratio,whichcanbeexplainedbytheincreaseinthenumberofassemblyalternatives(sequences)forhigher-ratiovalues.Asimilarphenomenonoccursforthe-ratio,wherehighvaluesofthismeasuremeanthattherearemanyalter-nativesfortheequipmentassignments,leadingtoalargertree.Thelesspredictableresultisthenegativesignofthedurationvariabilitye€ect.Hereweseethatasmallervariabilityleadstoalargertreesize.Webelievethatthereasonforthisisthatsmallvariabilityamongtasks'durationincreasesthenumberofcandidatetaskstobeassignedateachstageofthebranchandboundproceduresincemoretaskshavesimilarduration.Theresultswithrespecttothemaximumnumberofopennodes,whichisourmeasureforthememoryspacerequired,aresummarizedinTable2.WecanseethatthereisasimilaritybetweentheresultsofTables1and2,andthatthemaine€ectsinbothhavethesamesigns.Thisimpliesthatboththerunningtimeandthememoryrequirementsarea€ectedbythesefactorsinthesameway.Thesimilaritycanalsobenoticedwhenlookingat-valuecolumn,thoughthe-valuesinTable2aregenerallyhigher.Fouroutofthe®vefactorsarehighlysigni®cant,whilethedurationvariabilityfactorhasahigh-value(0.17),andcannotbeidenti®edassigni®cant. Table1.TheimpactoffactorsonthetotalnumberofnodesFactorE€ectSSdfMSFp(1)No.oftasks11240101.07E81101.07E850.737.693E-12(2)No.ofEq.Types7978509.18E71509.18E725.567.433E-07(3)Durationvariability3790114.89E71114.89E75.770.0169359-ratio11217100.66E81100.66E850.528.434E-12-ratio9019650.76E71650.76E732.662.615E-08Error951.69E8314303.09E6TotalSS128.09E9319 Table2.TheimpactoffactorsonthemaximalnumberofopennodesFactorE€ectSSdfMSFp(1)No.oftasks1022834.80E51834.80E536.963.603E-09(2)No.ofEq.types859589.94E51589.94E526.125.676E-07(3)Durationvariability230423.91E41423.91E41.880.1716794-ratio1048878.30E51878.30E538.891.498E-09-ratio929690.18E51690.18E530.566.97E-08Error102.61E7314326.78E4TotalSS132.50E7319Designof¯exibleassemblyline Finally,weexaminedtheimpactofthemainfactorsonthelowerbounde€ectiveness,measuredasthedi€erencebetweentheinitiallowerboundandtheoptimalsolutionvalue.Itisinterestingtodiscover(seeTable3)thesignofeache€ectandtonotethatall®vemaine€ectsarehighlysigni®cant.Weobservethatthegapbetweenthelowerboundandtheoptimalsolutionvalueisanincreasingfunctionofthenumberofequipmenttypesandthedura-tionvariability;itisadecreasingfunctionofthenumberoftasks,aswellasthe-ratioand-ratio.Theaveragegapbetweentheinitiallowerboundandtheoptimalsolutionvaluewas33.9%,whichisreasonableconsideringthere-laxationthatwehavemade.Theminimalgapwasob-tainedforproblemswith30tasks,threeequipmenttypes,smallvariabilityoftaskduration,an-ratioof0.4andan-ratioof0.6,withanaveragegapof14.1%forthesecharacteristics.Theaveragelargestgap,whichwasob-tainedfortheoppositefactorvalues,wasequalto51.7%.Thisinformationisusefulforassessingthedistanceofaheuristicsolution'svaluefromtheoptimalsolution'sval-ue,whenaheuristicalgorithmisemployed.Asuggestedheuristicfortheproblemisdescribedinthenextsection.5.Heuristicalgorithm±descriptionandexperimentsThefrontiersearchbranchandboundalgorithmrequireslargecomputerresourcesinordertosolveverylargeproblems,andthereforeaheuristicisrequiredformostrealworldproblems.Inthissectionwepresentaheuristicprocedurewhosecontrolparametermaybechosenac-cordingtotheproblemsize.Thiscontrolparameterde-termineshowmanynodesofthetreemaybeskipped,andthereforeisresponsiblefortherunningtime,forthememoryrequirements,aswellasforthedistancefromoptimalityoftheresultingsolution.Accordingtotherulesofthefrontiersearchbranchandboundprocedure,thenodewiththesmallestlowerboundisextendedateachiteration.However,someofthesenodeshaveaverysmallprobabilityofeventuallyprovidingtheoptimalsolution,andtheirextensionisessentialonlyforprovingtheoptimalityofthesolution.Intheproposedheuristic,wemodi®edthenodeselectionrule,inordertoavoidtheextensionofsuchnodes.beanopennodeatthetreelevel,withalower.Letbeanotheropennodeatthetreelevel,withalowerbound.De®netobetheinitiallowerboundoftheproblem(beforedoinganyassign-ment).Notethatthelevelsofthetreearenumberedsuchthattherootofthetreeislevel0andthehighestindexlevelislevel,wherealltasksarealreadyassigned.Thenodeselectionruleismodi®edasfollows:Step1.If,and,selectnodeStep2.If,andStep2.1.If XÿLBYNXÿNYK selectnode,otherwise,selectnodeTheparameterinstep2.1istheheuristic'scontrolparameter;theselectionofthevalueofisdiscussedNotethatinStep1above,theusualnodeselectionruleisapplied,whileinStep2thisruleissometimesreversed.AccordingtoStep2,wepreferhighindexedoverlowindexednodesiftheirlowerboundsareonlyslightlylarger.Thereasoningforthisisthatbythetimethelowerindexednodewillbecomeahigherindexednode,itmayaccumulatehighercoststhanthedi€erenceintheirlowerbounds.Theleft-handsideinStep2.1oftheselectionprocessrepresentstheaveragecostperlevelforthelevelsbetweennodes,andthisiscomparedwiththeaveragecostperlevelthatwasaccumulatedalongthebranchthatreachedthehigherindexednode,.Iftheformerissmallerthanthelatter,itmaybeanindicationthatthebranchthatemanatesfromnodehasbetterchancesofprovidingtheoptimalsolution.Thisisweightedbythecontrolparameter,,whichrepresentsthetradeo€betweenthetreesizeandthesolutionsquality.When0,theinequalityinStep2.1neverholds,sothatthenodewiththelowestlowerboundisalwaysselected,andtheoptimalsolutionisachieved.Ontheotherhand,ifisverylarge,nodeswithhighindexedlevelsarealwayspreferableovernodeswithlowindexedlevels,andaheuristicsolutionisquicklyobtained.However,suchasolutionisnotlikelytobeagoodone. Table3.Theimpactoffactorsonthedi€erencebetweentheinitiallowerboundandtheoptimalsolutionvalueFactorE€ectSSdfMSFp(1)No.oftasks0.0530.22110.22158.572.52E-13(2)No.ofEq.types0.0540.22910.22960.538.60E-14(3)Durationvariability0.1090.95510.955252.720.00E+000.0780.48510.485128.184.33E-250.1431.63111.631431.430.00E+00Error1.3033140.0041TotalSS4.824319BukchinandTzur Inordertochooseagoodvaluefor,namely,avalueinwhichtheproblemissolvableinareasonabletimeandthesolutioniscloseenoughtotheoptimum,asensitivityanalysisonthevalueofwasperformed.Weselectedtheeightproblemsthatrequiredthelongesttimetobesolvedbytheoptimalalgorithm,allfromthecategoryof30tasks,®veequipmentalternatives,smallvarianceoftaskduration,high-ratioandhighWeexaminedthoseproblemsfor10di€erentvaluesofbetweenzeroto10.TheresultsarepresentedinTable4,wherewecanseethatforeachandeachoftheeightproblemsthesolutionvalue,thesizeofthebranchandboundtree,themaximalnumberofopennodescreatedduringthealgorithmrunandtheCPUtime.TheCPUtimereportedisinseconds,usingaPentiumII266MHzprocessor.Thememoryrequirementforeachnodeofthebranchandboundtreeisapproximately150Bytes,im-plyingthatthelargestmemoryrequirement,amongtheeightproblems,wasabout2.5MB(forProblem4).Theresultsmarkedwithanasteriskareoptimal.Itisapparentfromthetablethatforeachproblemthereisapointinwhichthetreesize,alongwiththeCPUtime,increasesdramatically,andthenalmostimmediatelytheoptimalsolutionisobtained.Forallproblems,theopti-malsolutionwasobtainedforarelativelysmallnumberofnodes,comparedwiththeoptimalalgorithm.Thestochasticnatureoftheheuristicisalsorecognized,whereinsomecasesalowervalueofcausedanincreaseintheobjectivevalue(seeforexampleProblem1,1and7).Toidentifytherecommendedvalueofthissetofproblems,twographswerecreatedandarepresentedinFig.3(aandb).Figure3(a)showsthedi€erencebetweentheheuristic'sandtheoptimalsolu-tion'svalues,whereeachpointassociatedwithaspeci®cvalueofisanaverageoftheeightresults.Figure3(b)showstheratiobetweentheaveragesizeofthebranchandboundtreefortheheuristicalgorithmandtheav-eragesizerequiredbytheoptimalsolution.Wecanseeaclearsimilarityanddependencybetweenthetwographs,whichcanbedividedintothreeranges.Inthe®rstrangehashighvalues,theheuristicsolutionvalueismuchlargerthantheoptimalsolutionvalue(adi€erenceof32%for10),andtheratiobetweenthetwobranchandboundtreesisverysmall.Whenthedi€erenceinthevaluesbecomesmuchsmaller(8.2%)andtheaveragesizeoftheheuristic'sbranchandboundtreeincreasessigni®cantly.Finally,whenthegraphbecomessharperwithahigherrateofin-crease;atthatpoint,arelativelygoodsolutionisob-tained,withanaveragedi€erenceof1.7%fromtheoptimalsolution,andwithonly5.3%oftheaveragetreesizeoftheoptimalsolution.Beyondthatpoint,whenissmallerthanone,thetreesizeisincreasingdramati-callywhilethesolutionvalueisonlyslightlyimproving.Forthissetofproblems,thevalueof1providesarelativelygoodsolutionwherethesizeofthetreeisrelativelysmall.Whilethe``best''valueofmaybedependentontheparameters'characteristics,weexpectthatingeneralsmallvaluesofwillprovidegoodandfastsolutionsforthemostdicultproblemsthatcannotbesolvedtooptimality.Duetothelargevariabilityinsolutiontime,asolutionmaynotbeobtainedasfastasexpectedforcertainproblems.Inthesecasesourrecommendationisto®rstrunthealgorithmwithalargevalueof,inordertoobtainafastheuristicsolution;thenbydecreasingitsvaluegradually,weexpectthatthesolutionobtainedwillbeimproved,Thisprocessmayberepeatedaslongasasolutionisobtainedinareasonableamountoftime.6.ConclusionsInthispaperweproposedanewmethodforthedesignofa¯exibleassemblylinewhichmayconsistofseveraltypesofassemblyequipment.Thepurposeofthedesignprocessistochoosethetypeofequipmenttoplaceineverystationofthelineandtodeterminetheassign-mentoftaskstoeachequipmenttype,wheretheob-jectiveisminimizingtotalequipmentcost.ThisdesignproblemisNP-hardsinceaspecialcaseofitisthesimpleassemblylinebalancingproblem,whichisknowntobeNP-hard.Wepresentaformulationoftheproblem,basedonwhichwedeveloplowerboundsforboththecompleteandalsoforpartialproblems.Theselowerboundsarethenusedinabranchandboundalgorithm.Ourbranchandboundalgorithmalsousesadominanceruleforcuttingbranchesofthebranchandboundtree,thereforereducingitsrunningtime.Althoughthealgorithmhasanexponentialcomplexity,itiscapableofsolvingproblemsofmoderatesize.Sinceitisadesignproblemwhichhastobesolvedonlyeveryonceinawhileandnotfrequentlyduringoperation,weareabletodevotetoitrelativelylargecomputationalresources.Finally,wedevelopedaheuristicprocedurewhichmaybeusedforlargeproblemsthatcannotbesolvedbytheoptimalalgorithm.Theheuristicisvery¯exibleinde-terminingitsaccuracyononehand,anditscomputa-tionaltimeontheotherhand.Thetrade-o€betweentheaccuracyandthecomputationaltimeiscontrolledbytheheuristicscontrolparameter.Anexperimentalstudydemonstratedthesensitivityoftheaccuracyandthecomputationaltimeoftheheuristicasafunctionofthecontrolparameter,andimpliedonitspreferredvaluefortheexaminedsetofproblems.Wenotethatbysolvingourproblemafewtimes,fordi€erentvaluesofthecycletimeparameter,weareabletoaddressthehigherlevelproblem,inwhichthecycletimeisadecisionvariable.Inparticular,analternativetothedesignofasinglelinewithacycletimeof,isadesignoflines,withacycletimeofeach,thereforeprovidingDesignof¯exibleassemblyline Table4.HeuristicsresultsProblem1Problem2Problem3Problem4KTreeSol.1TreeSol.2TreeSol.3TreeSol.410107750.051900122840.051740111780.05157088530.0514405107700.051780127840.051440115730.05155098580.0514203112720.051750127630.051490137770.051420109510.0513702112690.051640165640.051360158780.05142093390.0515001.5106630.051620240670.0513705631190.171320146610.0514001379800.111280*66903972.25133016241980.44129066636122.581290*0.7127858526.101290182439208.351330103494573.951270*31338153823.781290*0.519359136013.241280*33375163720.431320*134576626.271270*869274263122.8713000.349776379951.411280*42029193026.751320*1994811489.561270*752955558102.931290*09869911365196.091280*55109293339.541320*25965190514.061270*16361816788550.851290* Table4.Problem5Problem6Problem7Problem8KTreeSol.5TreeSol.6TreeSol.7TreeSol.81085490.051880100630.05158075420.051840105700.051900593520.051700105640.05140075420.051840110700.0519203101520.051670105640.05140086400.051630109510.0516902162660.051520124610.05140090400.051640132600.0514901.5167680.051520221820.051400114400.051410460920.111320*166532331.81142047565011.7113401588770.38141037323301.161320*0.72378271411.261370*1440914997.741300*108526514.551350*61242180954.371320*0.568474183055.421370*22654241516.64132039705199532.411350*47148308839.281320*0.370684337259.811370*35454343229.721300*60834322956.851350*67928595584.861320*01032085847128.861370*53617640861.191300*69131559575.351350*16132016414581.391320*BukchinandTzur thesamethroughput.Since,thenumberofseparatelines,isnotlikelytobehigh,itisstillreasonabletosolvetheproblemforeveryresultingvalueofcycletime.[1]Baybars,I.(1986)Asurveyofexactalgorithmforthesimpleassemblylinebalancingproblem.ManagementScience,909±[2]Ghosh,S.andGagnon,R.J.(1989)Acomprehensiveliteraturereviewandanalysisofthedesign,balancingandschedulingofassemblysystems.InternationalJournalofProductionResearch,637±670.[3]Scholl,A.(1999)BalancingandSequencingofAssemblyLines,2ndedition.Physica-Verlag,Heidelberg,NewYork.[4]Sarin,S.C.andErel,E.(1990)Developmentofcostmodelforthesingle-modelstochasticassemblylinebalancingproblem.nationalJournalofProductionResearch,1305±1316.[5]Karp,R.M.(1972)Reducibilityamongcombinatorialproblems,ComplexityofComputerComputation,Miller,R.E.andThatcher,J.W.(eds.),PlenumPress,NewYork,pp.85±103.[6]Graves,S.C.andHolmesRed®eld,C.(1988)Equipmentselectionandtaskassignmentformultiproductassemblysystemdesign.TheInternationalJournalofFlexibleManufacturingSystems[7]Graves,S.C.andWhitney,D.E.(1979)Amathematicalpro-grammingprocedurefortheequipmentselectionandsystemevaluationinprogrammableassembly,inProceedingsoftheEighteenthIEEEConferenceonDecisionandControl,FtLau-derdale,FL.pp.531±536.[8]Graves,S.C.andLamar,B.W.(1983)Anintegerprogrammingprocedureforassemblydesignproblems.OperationsResearch(3),522±545.[9]Pinto,P.A.,Dannenbring,D.G.andKhumawala,B.M.(1983)Assemblylinebalancingwithprocessingalternatives:anappli-ManagementScience,817±830.[10]Rubinovitz,J.andBukchin,J.(1993)RALB±aheuristicalgo-rithmfordesignandbalancingofroboticassemblylines.oftheCIRP,497±500.[11]Tsai,D.M.andYao,M.J.(1993)Aline-balanced-basecapacityplanningprocedureforseries-typeroboticassemblyline.Inter-nationalJournalofProductionResearch,1901±1920.[12]Johnson,J.R.(1988)OptimallybalancinglargeassemblylineswithFABLE.ManagementScience,240.[13]Mansoor,E.M.andYadin,M.(1971)Ontheproblemofas-semblylinebalancing,inDevelopmentsinOperationsResearchAvi-Itzhak,B.(ed),GordonandBreach,NewYork,p.361.Appendix±Summaryofnotation=durationoftaskwhenperformedbyequipment...=costofequipmenttypeC=requiredcycletime;=setofimmediatepredecessorsoftaskDecisionvariables1ifequipmentisassignedtostation0otherwise.1iftaskisperformedbyequipment0otherwise.totalnumberoftype1iftaskisperformedby0otherwise.Lowerbounds1=thesolutionofproblem(P3)andalowerboundforproblem(P1);2=thesolutionofproblem(P5)andalowerboundforproblem(P4).Optimalbranchandboundrelatedvalues=anodethebranchandboundtree;=thecostofpurchasingtheequipmentdecideduponthusfarinthepartialsolutionassociatedwithnode=slackofthelastopenedstationatnode=asetoftaskswhichhavealreadybeenassignedtostationsinthepartialsolutionassociatedwithnode=thecomplementofintheoriginalsetoftasks,i.e.,thesetoftaskswhichhavenotbeenassignedyettostationsinthepartialsolutionassociatedwithnode Fig.3.(a)Acomparisonbetweentheheuristicandtheoptimalsolution,and(b)theratioforthetreesizeoftheheuristicandthatoftheoptimalsolution.Designof¯exibleassemblyline =thevalueof1whenonlythesetoftasks=thevalueof2,giventhesetoftasks=thelowerboundofnodeParametersusedintheexperimentalstudy=theexpectationofthetaskduration;-ratio=2,whereisthenumberofzeroesinthematrixwhoseelementsare:1iftaskprecedestask0otherwise-ratio=1=thenumberofelementsthatequaltoin®nity.Heuristicbranchandboundrelatedvalues=treelevelofnode=lowerboundofnodeinthebranchandbound=theinitiallowerboundoftheproblem;=theheuristic'scontrolparameter.JosephBukchinisamemberoftheDepartmentofIndustrialEngi-neeringatTelAvivUniversity.HereceivedB.Sc.,M.Sc.andD.Sc.degreesinIndustrialEngineeringattheTechnion,Haifa,Israel.Hismainresearchinterestsareintheareasofassemblysystemsdesign,assemblylinebalancing,facilitydesign,designofcellularmanufac-turingsystems,operationalschedulingaswellasworkstationdesignwithrespecttocognitiveandphysicalaspectsofthehumanoperator.MichalTzurreceivedherB.A.fromTelAvivUniversity,Israel,andherM.PhilandPh.D.inManagementSciencefromColumbiaUni-versity.MichaljoinedtheFacultyofIndustrialEngineeringatTelAvivUniversity,Israel,in1994afterspending3yearsattheOperationsandInformationManagementdepartmentattheWhartonSchoolattheUniversityofPennsylvania.Herresearchinterestsareintheareasoflogisticsystems,inventorymanagementcombinedwithforecasthori-zonresults,operationsschedulingandproductionplanning.ContributedbytheManufacturingSystemsControlDepartmentBukchinandTzur

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