Production Planning and Control, Vol.12, N 4, pp , 2001

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1 ACommonCycleApproachtoLotSchedulinginMultistage ManufacturingSystems LAAS-CNRS,7AvenueduColonelRoche,31077ToulouseFRANCE Jean-ClaudeHennet assumptionofconstantdemandratesforalltheend-products.underthecommoncycleapproach, productionplanningprobleminajob-shop.thecyclicnatureoftheproblemisrelatedtothea-priori ACyclicEconomicLot-SizingandSchedulingProblem(CELSP)isformulatedtosolveamultistage Abstract thedemandoverthecommoncyclehorizon.thepapershowsthattheceslpcanbesolvedina eachgenericjobconsistsofproducinganassociatedend-productinthequantityrequiredtomeet decomposedwayandthatitssolutioncaneasilybeimplementedandadjustedtolimitedvariations Multistagemanufacturingsystems,planning,scheduling,optimization. ofdemands. Keywords 1

2 describethemultistagestructureoftheproducts. 1INTRODUCTION Thisstudyisdevotedtotheorganizationofproductioninmanufacturingsystemsproducingfamilies ofproductsthroughseveralprocessingandassemblystages.aninput-outputapproachisusedto usefulforcontrollingandmonitoringthesystem,butintermsofproductionorganization,itmay points,respectivelyrelatedtoproductsandprocesses.descriptionofamanufacturingsysteminterms ofprocessesfocusesonthepropertiesoftheexistingresources,whichareforinstancethemachines, pallets,carriersandworkteams.suchadescriptionisdictatedbytheexistingfacilities.itisvery Productionactivityinamanufacturingsystemcanbedescribedfromtwocomplementaryview- improveeciencyonlymarginally. Then,inthelate1980's,Grubbstromandhisco-authors(GrubbstromandOvrin,1992;GrubbstromandMolinder,1994)haveintroducedlead-timesintheirinput-outputbasedmodels,underthe formofconstantdelays.asimilarapproachhasbeenusedbyhennet(hennet,1998;hennetand elsofproductionandinventorysystemshasbecomeclassicalsincetheworkofveinott(veinott,1969). ysisoftheproductsstructure:primaryproductsaretransformedintoend-productsthroughoneorsev- eralstageswhichcombineandtransformproducts.theuseofinput-outputanalysistoconstructmod- Aproduct-relateddescriptionofamanufacturingsystemisobtainedthroughaninput-outputanal- Barthes,1998)tobuildclosed-loopplanningpolicies. andcostparametersassociatedwiththeresources,itaimsatsolvingthecombinedlot-sizingand hasinputnodes,outputnodesandnocycle.suchadescriptioncanbeveryuseful,inparticularfor re-engineeringandre-designpurposes.thisstudyusesaninput-outputrepresentationofmultistage manufacturingsystems,todeneorre-denejobsandtaskstobeperformed.thenusingtheproperties Graphically,amultistagemanufacturingstructurecanberepresentedbyagozintograph,which schedulingproblem. iterativeresolutionofstandardjob-shopschedulingproblems. ingproblem(elmaghraby,1978)),whichisknowntobenp-hard,andforwhichthesearchforan optimalprocedureisstillanopenproblem(bourlandandyano,1997).however,underthecommoncycleapproach,themultiresourceelsphasbeenreduced(ouennicheandboctor,1998)toan ThisproblemisamultiresourceextensionoftheclassicalELSP(EconomicLotsizingandSchedul- 2

3 introducedin(bomberger,1966)tosolvetheelsp.intheccapproach,eachtaskisperformed exactlyonceineachcycle.somesucientconditionsofoptimalityoftheccassumptionhavebeen establishedin(gallego,1990).inaddition,computationalexperienceshowsthatccschedulesare closetooptimalwhenprocessingtimesarenottoodierentandthattheyoftencomparefavorably TheCommonCycle(CC)approachisthesimplestversionoftheBasicPeriod(BP)assumption, undertheassumptionofconstantdemandrates.then,itisshownthatanormalizedscheduling toheuristicsusedtosolvethemorecomplexproblemsbasedonlessrestrictivebp(basicperiod) problemcanbesolvedrst,independentlyofthevalueofthecommoncycleperiod.theoptimal assumptions. valueoftheperiodisthencomputedbyanexplicitformulaoftheeoqtype. themultiresourceelspundertheccapproach.somebasicpropertiesofthecelspareestablished, TheCyclicEconomicLotsizingandSchedulingProblem(CELSP)studiedinthispaperdescribes 2AMultistageModel problemcanbeeasilyadaptedtolimitedvariationsofthedemand. Intermsofimplementation,thecomputedsolutionofthecombinedlot-sizingandscheduling 2.1TheInput-OutputApproach ABillofMaterialscanbeconstructedtosummarizetheinputandoutputowsofamanufacturing systemoversomeworkinghorizon,t.vectortddenotesthecumulatedexternaloutputvectorand TQthecumulatedproduction/supplyvectoroverT.Then,astaticLeontief-typerepresentationof thesystemovertis: (R S)isnotfullrow-rank,itrepresentsthepossibilitytoobtainthesamecombinationofoutputs whererandsarenonnegativematrices.ristheoutputmatrix(forinternalandexternaltransformations)andsthe(internal)consumptionmatrix.thenetproductionmatrix,(r S)relatesdtoQ. Forgeneralmanufacturingsystems,matricesRandShavethesamerectangularstructure.Ifmatrix (R S)TQ=Td (1) throughdierentactivities.if(r S)isnotfullcolumn-rank,itcorrespondstothepossibilityto substitutesomeproductioninputsbysomeotherones. 3

4 ufacturingactivityhasseveralinputproducts,andproductsofthesametypemaybeusedbyseveral activities.howeverthereisaone-to-onecorrespondencebetweenoutputproductsandactivities:each productionofoneunitofproducticonsumescomponentsj=1;:::;minquantitiesji,forj=1;:::;m. manufacturingactivityhasexactlyoneoutputproduct.accordingtoagivenmanufacturingrecipe, Theproductstructuresstudiedinthispaperarerepresentedbygeneralacyclicgraphs.Eachman- Furthermore,thegraphoftheproductstructurehasnocycle. index,matrixristheidentitymatrix,i,andmatrixsisthenonnegativematrix=((ji)). whichtypicallycharacterizetheeldofapplicationofrequirementsplanning,andparticularlymrp assemblygraphsaresquarematrices.ifactivitiesandassociatedoutputproductshavethesame techniques(seee.g.(baker,1993)).theinputandoutputmatricesrepresentingsuchgeneralized SuchconvergentmanufacturingstructurescanberepresentedbyGozintographs(Vazsonyi,1955), levellforproductswhicharecomponentsofproductsoflevelsstriclylessthanlandofatleastone levell 1product.Levellsisthemaximalnumberofstagesinthemanufacturingprocess. Anexamplewithm=5,n=2isdescribedbythegozintographofFig.1. Productscanthenbepartitionedintolevels(seee.g.(Salomon,1991)).Level0isforendproducts, 1 Level 2 : primary products A B Productsbeingorderedinagreementwiththeincreasingorderoftheirlevel,theinputmatrix Figure1:Anexampleofamultistageproductstructure Level 1 : intermediate product C Level 0 : end products D E

5 isalower-triangularnonnegativematrixwithzerosonthemaindiagonal: Thefollowingresultcanthenbestated: =26400:::0 m1:::m;m : (2) Input-Outputproductionmodel: Any(nonnegative)demandvectorTdcanbesatisedoverTwithinthedomainofvalidityofthe Theorem1 (3).Thenetproductionmatrix,(I )hasallitso-diagonalelementsnonpositive.itmaybe ToprovetheTheorem,itsucestoprovetheexistenceofanonnegativevectorQsolutionofequation Proof (I )TQ=Td; (3) ofanm-matrix(alsocalledaclasskmatrix(kohleretal.,1975))tobeanessentiallynonpositive matrixwithallitsprincipalminorspositive.therefore,matrix(i )isanm-matrix(berman triangular,anditsdiagonalelementsareequalto1.therefore,det(i )=1,andmoregenerally, alltheprincipalminorsofmatrix(i )areequalto1.itisawell-knowncharacteristicproperty calledanessentiallynonpositivematrix(bermanandplemmons,1979).furthermore,itislower- (BermanandPlemmons,1979). ratedcanbefullledusingthenonnegativeproduction/supplyratevectorq,uniquelydenedby: andplemmons,1979).thisimplies,inparticular,thatitsinverse(i ) 1isanonnegativematrix Adirectconsequenceofthispropertyisthat,aslongasthemodelapplies,anynonnegativedemand Q=(I ) 1d (4) 5

6 Nonnegativityofmatrix(I ) 1canbeeasilycheckeddirectly: Thefollowingrelationsareobtained,form>2.: 8<:bi+1;i=i+1;ifori=1;:::m 1 (I ) 1=26410:::0 bm1:::bm;m 11 b bi+k;i=i+k;i+pj=1;:::;k 1i+k;i+jbi+j;ifori=1;:::m 2;k=2;:::;m i: : Fromtheserelations,itcomesbyinductionthatsincematrixisnonnegative,matrix(I ) 1is alsononnegative. mustalsotakeintoaccounttheconstraintsandcostsrelatedtotheuseofparticularresources. ofthismodelinproductionplanningisthereforeveryappealing.however,arealisticproductionplan yieldingvectorqtwithinthetimehorizont,usingtheresourcesavailable.aninput-outputanalysis thusprovidesasimplemodeltodeterminetheproductionrequirementsofalltheproducts.theuse TheLeontiefmodel(4)appliesovertimeperiodTifthereexistsafeasibleproductionschedule Theconsideredproductionplanningproblemisdescribedbythefollowingparameters: 2.2ProductionPlanning products, productsaretheend-products,andthem nremainingproductsareintermediateandprimary -,theinput-outputmatrixofthesystem,withdimensionmm -nnumberofend-products -didemandrateforend-producti(i=1;:::;n);ddenotesthedemandratevector. -mnumberofproducts(primary,intermediateandend-products).byconvention,thenrst -hjunitaryinventoryholdingcostofproductj;j=1;:::;mperunitoftime pacityconstraintsusuallyrequiresaniterativeadjustmentprocess.recently,someattemptshavebeen niques,productionandstoragearenotgloballyoptimized.inaddition,satisfactionofproductionca- -pjproductionrateofproductj;j=1;:::;m -fjsetupcostofproductj;j=1;:::;m. InMaterialRequirementPlanning(MRP)andManufacturingResourcePlanning(MRPII)tech- 6

7 madetobuildmathematicalmodelsoftherequirementplanningmethod(gravesetal.,1998;grubbstromandmolinder,1994;hennetandbarthes,1998).inparticular,thedynamicmodelof(graves etal.,1998)describestheforecastupdatingprocessinarollinghorizonframework,andthemodelin constraints(forinstancethemultilevellotsizingproblem)orintegratethemunderasimpliedform. (HennetandBarthes,1998)representsproductionplanningasatrackingproblem. Onthecontrary,problemswithsmalltimebuckets,suchastheContinuousSetupLotsizingProblem acrucialrole(salomon,1991).modelswithlargetimebucketseitherdonotintegratetheresource timeframework.inthesemodels,thelengthoftheelementaryperiod,calledthetimebucket,plays Mostdeterministiclotsizingmodelsforproductionplanninghavebeenformulatedinthediscrete (CSLP),representmoreaccuratelytheresourceconstraintsoftheschedulingproblem.Butthey rapidlybecomeuntractablewhenthetimehorizonisincreased,becauseofthelargenumberofboolean variablesinvolvedintheirformulation. addressthelotsizingandschedulingissuesthroughtheresolutionofacycliceconomiclotsizing times,whichmaybeconsideredacceptableinsomeplanningproblems,becomesclearlyunrealisticat therealtimelevel.hence,inthecontextofrepetitivemultistageproduction,itisproposedtojointly beintegratedintotheplanningmodeltoguaranteefeasibility.also,theassumptionofconstantlead- Asstressedin(Rouxetal.,1999)foramulti-sitejob-shopproblem,realschedulingdecisionsmust Muckstadt,1994).Input-outputmodelsarecombinedwithcyclicschedulingconstraintstoformulate andschedulingproblem(celsp).themodellingapproachissimilartotheoneusedin(loerchand thelot-sizingdecisionprobleminamultistagemanufacturingsystemunderfairlystabledemands.in addition,thedenitionofthecelsprequirestheknowledgeoftheresourceassignmentpolicy.the consideredmanufacturingcontextisajob-shopwhereeachproductionactivity,denedbyitsoutput product,j,withj2(1;:::;m),mustbecompletedonaparticularresource,r.aresourcemaybe assignedtoseveralactivities. forend-products.thisratherunrealisticassumptionisusedtobuildanominalproductionschedule overanoptimizedproductionperiod.theproblemofadjustmentoftherealscheduletorealdemand 3CyclicLotScheduling ThemultistageCELSPisnowadressedundertheassumptionofconstantdemandrates(di,i=1;:::;n) 7

8 study,theexistenceofsafetystocksisimplicit,andtherealdemandsforproductsmayapplytoall theproductsandmaycoversafetystocksregulationaroundtheirnominallevels,aswellasactual usefulforoptimizingthedistributionofsafetystockswithinamultistageproductionmodel.inthis isconsideredinsection4.inadditiontotheseadjusments,andtolimittheirimportance,theuseof externaldemandvariations. safetystocksisclassicaltocompensatefordemanductuations.input-outputmodelsareparticularly 3.1GenericJobsandGenericTasks whichisk-periodichasbeenprovedusingtimedpetrinets(carlierandchretienne,1988).butdueto theproblemcomplexity,manyauthorshaverestrictedtheclassofinvestigatedschedulestoperiodic(or innitelyoften.forsucharecurrentjob-shopschedulingproblem,theexistenceofanoptimalsolution overaninnitetimehorizoncanbecharacterizedbyasetofgenerictasksthathavetobeperformed Asthedemandratesaresupposedconstant,thecostminimizingsolutionofthelotschedulingproblem 1-periodic)solutions(HanenandMunier,1995).Inparticular,thecommoncycleapproachhasbeen theoperationsmustbeperformedexactlyoncewithintheselectedtimehorizon.theyarecalledthe developedforsimultaneousdeterminationoflotsizesandschedules(ouennicheandboctor,1998). sucestocharacterizetheschedulewithinonetimehorizonhavingthelengthoftheperiod,t.all Underthecommoncycleapproach,theproductionschedulingpolicyisanundenitelylongperiodical repetitionofthesamesequencesoftasksonthedierentmachines.todenesuchaschedule,it generictasks.someschedulingpropertieswillbeestablishedtosimplifythetreatmentofthismodel. therelevantentitiesforprecedenceconstraintsarethegenerictaskswithineachgenericjob,notthe Itthenbecomesnecessarytodeveloptheinput-outputgraphintosubgraphs.Eachsubgraphrepresents agenericjob. amountsofproducts.thisisparticularlyclearforintermediateproductsusedbyseveralendproducts. Whenusingtheinput-outputmodel,oneencountersadicultytoformulateschedulingconstraints: inquantitytdi.thegenerictaskswhichconstitutejobiareassociatedwithallthepathsofthe gozintographhavingproductiastheirterminalnode.thus,byconstruction,thesubgraphofeach producti,theassociatedoperationisproductionofquantityjllitdiofproductj.inthesubgraph jobisapureassemblygraph.forinstance,ifthereisapathoflength2,say(j;l;i)fromproductjto Eachgenericjobi(i=1;:::;n)isassociatedwiththeendproducti(i=1;:::;n),tobeproduced 8

9 othersublotsofthesamecomponent,andpossiblytothesublotsoftheothercomponentsproduced totheothergenerictasksofthesamejobthroughprecedenceconstraints.butitisalsorelatedtothe nentwithinaperiodissplitintosublots.anactivityproducingasublotisagenerictask.itisrelated ofjobj,thereisaprecedenceconstraintbetweenoperation(j;l;i)andoperation(l;i). bythesameresourceifthisisasharedresource. Ifseveralgenericjobsrequirethesametypeofcomponents,thentheproductionlotofthiscompo- tasks1,3,5,7,8. andcomposedoftasks2,4,6,9,10andtheotheroneassociatedwithend-product(e)andcomposedof graphoffig.2.thisdevelopedgraphdescribestwogenericjobs:oneassociatedwithend-product(d) Inagreementwiththisdenitionofgenericjobs,thegozintographofFig.1isdevelopedintothe Product A on Resource 3 Figure2:GraphdevelopingthegraphofFig.1 6 Product B Resource Product C Resource 1 disjunctiveresourcesharingconstraints:resource1isusedtoproduceproductsdande,resource2 constraintsbetweenthesublotsofthesameproduct(productsa,b,c)anddashedzonesrepresenting sentingthegozintoparameters.twotypeofdashedzonesareconsidered;dashedzonesrepresenting Inthedevelopedgraph,arcscorrespondtoprecedenceconstraints,withvaluationsstillrepre- 2 1 Product D Product E isusedtoproduceproductsbandc,andresource3isusedtoproduceproducta. 9

10 tojobiifi=j(s):itisthestartingoperationofapath(j;:::;i)endingatnodeiintheprecedence graphofgenerictasks. Letpdenotethetotalnumberofgenericmanufacturingtasks.Agenerictasks2(1;:::;p)belongs i=sinquantitydit.thedurationofterminaltasksisst,with endproductarethesame. itsassociatedpathreducestonodei.forsimplicity,theindexofaterminaltaskandoftheassociated Thus,atasks2(1;:::;n)isaterminaltask.Suchataskconsistsinproducingtheend-product Thengenerictasksproducingtheend-productsarecalledterminaltasks.Ifsisaterminaltask, isdenedbythepath(j;l;:::;i),thentask(s)isdenedby(l;:::;i).tasksconsistsinproducing thedevelopedprecedencegraph(thegraphoffig.2fortheexample).itisdenoted(s).iftasks Ifs2(n+1;:::;p),thenitisnotaterminaltask,andthereisauniquegenerictaskaftersin s=ds=psfors=1;:::;n: (5) productj=p(s)inquantityqst,withqs=jll0idi: Thedurationofgenerictasks2(n+1;:::;p)issT,with ThemainvariablesoftheCELSParethedurationofthecommoncycleperiodTand,withina 3.2ProblemFormulations=qs=pjforj=P(s): (6) typicaltimeintervalwithdurationt,thegenericperiod[0;t],thestarttimest(s)ofgenerictasks assignedtoproductionofproductj. productj:(j)=fs;p(s)=jg:thecardinalityof(j),whichisthenumberofgenerictasks producingj,isdenoted!j.theresourcer= (s)ofthegenerictasksproducingjistheresource Let(j)denotethesetofgenerictasksofproduction(orsupply)ofthesameintermediate(orprimary) s2(1;:::;p).thesubsequentdatesoftaskssimilartosaret(s)+ktwithk=1;2;:::. rg:thecardinalityoftheset(r)isdenotedr.10 taskstobecompletedonaparticularresource,r,isdenoted(r)anddenedby:(r)=fs; (s)= Resourceconstraintscanbecombinedwithdisjunctiveconstraintsonsublots.Thesetofgeneric

11 3.2.1Theproblemconstraints Constraintsareformulatedoverthegenericperiod. Resourceconstraints PrecedenceconstraintsT(s)+sTT(t)ifsprecedest,s>n Resourcesaresupposedsimpleandunitary.Resourceconstraintsapplytothegenerictasksofthe sameproductandtothegenerictasksofthevariousproductsusingthesameresource. T(s)+sTTfors2(1;:::;n) (7) (8) tointroduceauxiliarybooleanvariablesxskunderthefollowingconvention: Toobtainaconjunctivelinearformulationofsuchdisjunctiveconstraints,aclassicaltechniqueis xsk=8><>:1iftasksisinthekthposition T(s)+sT T(t)T(2 xsk xt;k+1) 8s;t; (s)= (t)=r;k=1;:::;r 1: withr 0otherwise Xk=1xsk=18ssuchthatr= (s) onresourcer= (s) (10) (11) (9) particularbyalargeconstant,usuallydenotedm.suchaformulationwitha"bigm"isoftenused attherighthandsideoftheinequality,couldbereplacedbyanypositivescalarlargerthant,in Notethatinformulation(10)ofdisjunctiveconstraints,variableTwhichappearsmultiplicatively Xs2rxsk=18r2(1;:::;R);k2(1;:::r): forresourceconstraintsinjob-shopschedulingproblems(e.g.in(ouennicheandboctor,1998)).here, (12) No-waitconstraintswithinlots theuseofvariabletinplaceofmin(10)isduetotheinterpretationoftheproblemasacyclic schedulingproblemforwhichtheheightbetweentwoinstancesofatask(denedasin(hanenand Munier,1995))isequalto1forallthegenerictasks,underthecommoncycleapproach. Withinthegenericperiod,sublotsofthesamecomponentcorrespondtoitsuseindierentstages 11

12 evaluatedin(mckoyandp.j.egbelu,1999).inparticular,ithasbeenshownthatthenon-grouping continuouslyprocessedwithinoneaggregatedlot.thesetwostrategieshavebeencomparedand strategymaynotablydecreasecycletimes,butthatthisadvantagehastobebalancedagainstthe fordierentendproducts.twodierentstrategiesmaybeinvestigatedforthesequencingofsublots: theymaybeprocessedseparately,asiftheywererelatedtodierentcomponents,ortheymaybe increaseinthenumberofsetups. Itisthusassumedthatallthesublotsofthesamelotshouldbeproducedinsequencewithout interruption,inordertohaveonlyoneset-upcostfortheconsideredproductwithineachperiod. Thissectionshowshowtheconstraintsofthegroupingstrategymaybeintegratedinthemodel. LetTb(j)denotethestartdateofproductj,possiblydenedasthelatestdatesuchthat: Theconstraintsofnon-idlingwithinlotstaketheform: andlette(j)denotethecompletiondateofproductj,possiblydenedastheearliestdatesuchthat: Te(j)T(s)+sT8s2(j): Tb(j)T(s)8s2(j); (14) (13) 3.2.2TheObjectiveFunctionTe(j) Tb(j)=TX Theobjectiveistominimizethesumofsetupcostsandinventoryholdingcostsperunitoftime.It s2(j)s: (15) canbedecomposedintothreeterms:j=c1+c2+c3 -set-upcostsperunitoftime:c1=(pmj=1fj)=t withc1;c2;c3denedasfollows: -inventoryholdingcostsforendproductsperunitoftime:c2=tk2withk2=pni=1hi(1 di=pi)di=2 -inventoryholdingcostsforintermediateandprimaryproductsperunitoftime: (16) (OuennicheandBoctor,1998). c3=pps=n+1hjqs(t((s)) T(s)+qsT=2pj qst=2pl)withj=p(s)andl=p((s)). Theexpressionsofinventoryholdingcostsperunitoftimearesimilartotheonesobtainedin 12

13 Itisproposedtoreformulatetheproblemunderthechangesofvariables: 3.3ADecomposedResolutionMethod latedasfollows: Thenewvariablesarecallednormalizeddates.Constraints(7),(8),(10),(15)arethenreformu- Z(s)=T(s)=Tfors=1;:::;p;Zb(j)=Tb(j)=T;Ze(j)=Te(j)=Tforj=1;:::;m: Z(s)+sZ(t)ifsbeforet,s2(n+1;:::;p) Z(s)+s1fors2(1;:::;n) (17) Constraints(9),(11),(12)remainunchanged.Thetermc3inJ(16)canbere-writtenintheform: Z(s)+s Z(t)(2 xsk xt;k+1)8s;t; (s)= (t)=r;k=1;:::;r 1: Ze(j) Zb(j)=X s2(j)s: (20) (18) (19) c3=tk3fork3=pps=n+1hjqs(z((s)) Z(s)+qs=2pj qs=2pl),withj=p(s)andl=p((s)). TheCELSPcanbesolvedoptimallyinadecomposedway: -solverstanormalizedschedulingproblemwithvariablesz(s)andxsk ThisnewformulationoftheconstraintsoftheCELSPallowstostatethefollowingTheorem: -computetheoptimalvalueoftbyaneoq-typeformula: Theorem2 -computetheoptimalstarttimesofoptimalproductionlotsbyt(s)=tz(s). Proof T=vut(mXj=1fj)=(k2+k3): (21) variabletdoesnotappearintheconstraintsandissimplyamultiplyingfactorinthecriterionterm thenormalizedcelsp.moreover,theonlytermofcriterionjinvolvingvariablesz(s)isc3,as VariableT>0isnotinvolvedinanyoftheconstraints(17),(18),(19),(20),(9),(11),(12)of 13

14 thefollowinglinearcriteriontobeminimized:c=pps=n+1hjqs(z((s)) Z(s)): notfeasible.asthenextstep,assumingthattheproblemisfeasible,theoptimalvalueoftcanbe c3,theoptimalsolutionofthecelspwithrespecttovariablesz(s)andxskcanbeindependently explicitelyobtainedthroughtheoptimalityconditionsofcriterionjwithrespecttot.itisthen obtainedbysolvingthenormalizedschedulingproblemdenedbytheaforelistedconstraintsandby expressedbytheeoq-typeformula(21)statedinthetheorem. InfeasibilityofthenormalizedschedulingproblemmayhappenonlyifthecompleteCELSPis parameterintheexpressionofdisjunctiveconstraintsconsiderablyimprovesthenumericalrobustness oftheresolutionprocess. problemcaneasilybesolvedbymixedlinearprogramming.thefactofavoidingtheuseofa"bigm" Asforanyjob-shopschedulingproblem,thecurseofdimensionalityimposestouseheuristics Aslongasthenumberofbooleanvariablesxskdoesnotgettoolarge,thenormalizedscheduling more. usedinthiscasebuttheoptimalityofthevalueoftcomputedbyformula(21)isnotguaranteedany numberofjobsgetslarge,asitisgenerallythecaseinpractice.thedecomposedmethodcanstillbe ratherthananoptimalresolutiontechniqueinsolvingthenormalizedschedulingproblemwhenthe Thedicultytousetheresultsoflot-sizingandschedulingoptimisationproblemsinindustrialapplicationshasbeenstressedbymanyauthors(M.Ben-Daya,1999;Brandimarteetal.,1995).Thisis oftheproblemdata.clearly,demanductuationsareanimportantcauseofchangesintheproblem data.inmanyindustries,theseuctuationsarecontrolledthroughamasterproductionschedule (MPS)overarollingtimehorizon.IncreasingthelengthofthefrozenpartoftheMPShasbeen mainlyduetothecombinedeectofthehighsensitivityofthesolutionsandofthehighvariability showntobeacriticalfactortoreducethenervousnessofthemrpsystem(zhaoandlee,1996). areretatedtomachinefailureanddeterioration,qualityinspectionandmaintenance.theimportance ofthesefactorshasbeenshowntobeparticularlyhighinmultistagemanufacturingsystems(m.ben- Apartfromvariationsofdemandratesforendproducts,someofthemainreasonsfordatachanges 4AReactiveScheduler 14

15 changesareconsidered,itispossibletoobtaingoodimplementablesolutionsbyadaptationofthe nominaloptimizedsolution. Daya,1999). 1.Theadaptedsolutionischosenperiodical,withthesameperiodTasinthenominalsolution. However,aslongasthedemandandproductionpatternremainsstationaryandthatlimiteddata thenominalone.thiscanbeachievedbyreplacing,inthenormalizedgenericschedulingproblem, constraints(19)bythesetofprecedenceconstraintsdescribingthenominalsequenceoftaskson resources. 3.Theadaptedsolutionisobtainedbysolvingthemodiednormalizedgenericschedulingproblem 2.Thesequenceoftasksoneachmachineischosenidenticalintheadaptedsolutionandin usingthecurrentdataandtheassumptionsof1and2. 5NumericalEvaluation itonlyinvolvesconjunctiveconstraintsandrealvariables. Themodiednormalizedgenericschedulingproblemismuchsimplerthantheoriginalonebecause stages.intherststage,thecompleteproblemissolvedforthenominalvaluesofdemandand Thepracticalimplementationoftheproposedlot-sizingandschedulingtechniqueproceedsintwo productionrates.thesecondstageisthereal-timeadjustmentofthesolutioninresponsetoreal uctuationsofdemandandproductionrates.suchanadjustmentispossibleonlyifproduction overload.themethodisnowillustratedonthemultistageproductstructureoffig.1andfig.2,under dierentsetsofdata. 5.1Thenominaloptimalschedule capacityovertheperiodisnotexceeded.furtheradjustmentsarerequiredinthecaseofcapacity Foreachproductj,theunitholdingcostishj=0:01. demandratesforintermediateproductsarenull,andthevectorofproductionratesis[ ]. Atthenominalpoint,demandrateforproductEisd1=0:8,demandrateforproductDisd2=0:5, 15

16 chartoffig.3,withvariablesz(q)expressedaspercentagesoftheperiod. TheoptimalnominalsolutionofthenormalizedschedulingproblemisdisplayedthroughtheGantt obtainedusingtheeoq-typeexpression(21)ist'14:0timeunits. Thesumofxedcostsforthe5productshasbeentakenequalto1000.Then,thevalueofT Figure3:Ganttchartforagenericperiod cent.thesolutionsobtainedfortheoptimalandforthesimpliedschedulingproblemarereported Inthedataofthenumericalexample,vectorsdandparenowrandomlyvariedupto10to50per 5.2Thereactiveschedule Resource 3 Resource 2 Resource % 20% 40% 60% 80% 100% time axis ontable1.maximal disturbance level 10% 20% 30% 40% 50% 16 Number of infeasible solutions 0/10 0/10 2/10 3/10 6/10 Average relative suboptimality of the schedule < 10-4 < Average relative difference in the period length 4% 7% 10% 12% 18% Suboptimality gap using the nominal period 0.1% 0.5% 0.9% 1.3% 1.7%

17 imposingthatthesequenceoftasksoneachmachineidenticaltotheoneobtainedforthenominal Twotypesofsimplicationcanbemadetoadjustthescheduletoreal-timeconstraints. Accordingtotheapproachdevelopedintheprecedingsection,therstsimplicationconsistsin Table1 setofdata.suchasimplicationleadstoconsiderablesavingsincomputationaltime.thesimplied thissimplicationismerelymarginal(lessthan1=1000inrelativevalue). schedulingproblemtobesolvedovereachperiodisastandardlinearprogramonlyhavingcontinuous variables.onthecontrary,theoriginalschedulingproblemalsoinvolvesthebooleanvariablesxsk period.fromthecomputationalviewpoints,thesavingsarenegligiblesinceagoodapproximationof denedin(9).onthesmallsizeexampleconsideredhere,thecomputationaltimeisdecreasedbya theoptimalvalueoftheperiodisobtainedbyapplyingtheeoq-typeformula(21)totheproblem factorrangingfrom20to100.itisclearfromtable1thattheaveragecriteriondeteriorationdueto dataandthesolutionofthesimpliedschedulingproblem.table1showsthatfordatavariationsup to50percent,theaveragerelativedierenceintheperiodlengthiscloseto20percent.andthatit Thesecondsimplicationconsistsofusingthenominalperiodinsteadoftheoptimaloneateach inducesasuboptimalitygapofalmost2percent.thus,fromthepureoptimalityviewpoint,itwould beecienttoadjusteachperiodandtheproductionlotsizesaccordingly.however,fromapratical viewpoint,itiscertainlypreferabletomaintaintheproductionperiodatitsnominalvalueaslongas Theproposedreactiveschedulingtechniquehastobeadjustedtodealwiththecaseswhennofeasible thechangesindataremainlowanddonotinvolveaclearstationarytrend. 5.3Adjustmenttoreal-timeconditions solutionexists.thiscasefrequentlyoccursunderimportantvariationsofdemandloadand/orproductionrates.thecomputedfrequenciesofoccurenceofsuchcasesarereportedonthesecondrow oftable1. notfeasible,and,fromthepracticalviewpointthereisnorealrestrictioninassessingfeasibilityonly solutionwhiletheoriginalschedulingproblemisfeasible,thiscasehasneverbeenencounteredinthe experiments.thus,ingeneral,bothproblemscanbesimultaneouslyconsideredaseitherfeasibleor Itisimportanttonotethat,althoughthesimpliedschedulingproblemmayhavenofeasible 17

18 orbacklogs.thecyclicapproachisnotnaturallywell-suitedfordealingwithsuchcases.however, job-shop,eitherbecauseofexcessivedemandratesorlowproductivity.ifthesystemdoesnotpossess extraresources,theonlysolutiontothisproblemistoproducelessthanthedemand,withlosssales fromthesolutionofthesimpliedproblem. theceslpcanbemodiedtotreatthiscase. Theabsenceofasolutiontothegenericschedulingproblemcorrespondstoanoverloadofthe normalizedgenericschedulingproblemisnotfeasible,afeasiblemakespanproblemcanbeformulated. thesimpliedschedulingproblem,exceptforconstraints(18),whicharereplacedby(23): Thenormalizedmakespanvalue,denoted,canbeminimisedunderalltheschedulingconstraintsof portionalityfactorleadstothereductionofallthegenerictaskdurationssbythisfactor.ifthe Itcanbeobservedfromequations(5),(6)thatreducingallthedemandratesbythesamepro- Z(s)+sZ(t)ifsbeforetinthenominalschedule,s2(n+1;:::;p) Z(s)+sfors2(1;:::;n) Minimizesubjectto (22) thereisnowaitingtimealongthispath,thevalueofthemakespancanbelimitedto1(or1 )by Theoptimalvalueofthemakespanisthelengthofthecriticalpathinthegraphoftasks.As Ze(j) Zb(j)=X s2(j)s: (24) (23) dividingallthedemandratedby(resp.by easilysolved.then,accordingtothecompanypolicy,theremainingdemandratevector,(1 )d maycorrespondtolostsalesortobacklogsforthefollowingperiod. thenbere-formulatedunderthereduceddemandrates.thisproblemisclearlyfeasibleandcanbe 1 ).Thenormalizedgenericschedulingproblemcan reliabilityoftheequipmentsandworkteams.suchdisturbancesmustbetakenintoaccountwhen 6Conclusions buildingthelot-sizingandschedulingpolicy,speciallyinthecaseofmultistageproduction,forwhich disturbancesareoftenampliedthroughtheproductstructure. Mostofthemodernmanufacturingcompaniesoperateinvariablecompetitivemarkets,withthe consequenceofhavingtoreacttouctuatingdemands.anotherfactorofdisturbancesistheimperfect 18

19 withtheadaptivityadvantageofadjustingthelengthsoftaskstotherealevolutionofdemandand productivity. Theapproachfollowedinthispapercombinestherepetitivepropertiesofcyclictasksequencing problemintoanormalizedschedulingproblem,whichcanbesolvedrstandindependently,andalot sizingproblem.itisofcurrentpracticetoseparatelytreattheproblemsoflot-sizingandscheduling, constantdemandandproductionrates.thecommoncycleapproachallowstodecomposethecyclic butusuallywiththelot-sizingproblemtreatedrst.here,byfocusingtheanalysisonstoragecosts, Anormalizedlot-sizingandschedulingproblemhasrstbeenformulatedundertheassumptionof forsolvingthehighlydisturbedcases,forwhichthemanufacturingsystemisoverloaded. ithasbeenestablishedthatschedulingdecisionshaveanimportantinuenceonlot-sizingdecisions. thenominalsequencingoftasksonresourcesbutchangingthedatesanddurationsoftasks.the eciencyofthisadjusmentschemehasbeenshownnumerically.amethodhasnallybeenproposed Arapidandecientadjustmentofthenominalscheduletorealdataisobtainedbymaintaining References Baker,K.R.(1993).Requirementsplanning.In:HandbooksinOperationsResearchandManagement Berman,A.andR.J.Plemmons(1979).NonNegativeMatricesintheMathematicalSciences.AcademicPress. Science,vol.4,S.C.Graves,A.H.G.RinnooyKan,P.H.ZipkinEds..North-Holland.pp.571{ 628. Bourland,K.E.andC.A.Yano(1997).Acomparisonofsolutionapproachesforthexed-sequence Bomberger,E.(1966).Adynamicprogrammingapproachtoalotsizeschedulingproblem.ManagementScience,Vol.12,pp Brandimarte,P.,W.UkovitchandA.Villa(1995).Factorylevelaggregatescheduling:Bridgingthe economiclotschedulingproblem.iietransactions,vol.29,pp gapbetweenoptimizedschedulingandrealtimecontrol.in:optimizationmodelsandconcepts inproductionmanagement,p.brandimarteanda.villaeds..gordonandbreachpublishers. pp.187{

20 Carlier,J.andP.Chretienne(1988).Timedpetrinetsschedules.In:AdvancesinPetriNets:Lecture Elmaghraby,S.E.(1978).Theeconomiclotschedulingproblem(elsp):reviewandextensions.ManagementScience,Vol.24,pp Gallego,G.(1990).Anextensiontotheclassofeasyeconomiclotschedulingproblems.IIETransactions,vol.22,pp NotesinComputerScience,No.340.Springer.pp.62{84. Graves,S.C.,D.B.KletterandW.B.Hetzel(1998).Adynamicmodelforrequirementplanningwith Grubbstrom,R.W.andA.Molinder(1994).Materialrequirementplanningemployinginput-output applicationtosupplychainoptimization.operationsresearch,vol.46,suppno.3,pp.s35-s49. Grubbstrom,R.W.andP.Ovrin(1992).Intertemporalgeneralizationoftherelationshipbetween analysisandlaplacetransforms.intl.j.productioneconomics,vol.35,pp Hanen,C.andA.Munier(1995).Cyclicschedulingonparallelprocessors:Anoverview.In:Scheduling materialrequirementplanningandinput-outputanalysis.intl.j.productioneconomics,vol. 26,pp Hennet,J.C.(1998).Afeedbackcontrolapproachtomultistageproductionplanningbasedoninputoutputanalysis.In:LAASReportNo98159,12thInternationalConferenceonInput-Output TheoryanditsApplications,P.Chretienne,E.G.Coman,J.K.Lenstra,Z.LiuEds.JohnWiley andsonsltd.pp.193{226. Hennet,J.C.andI.Barthes(1998).Closed-loopplanningofmulti-levelproductionunderresource Kohler,G.J.,A.B.WhinstonandG.P.Wright(1975).OptimizationoverLeontiefsubstitutionsystems. constraints.in:proceedingsoftheifacsymposiumincom'98,nancy,(france). Techniques,New-York(USA),18-22Mai1998,14p. Loerch,A.G.andJ.A.Muckstadt(1994).Anapproachtoproductionplanningandschedulingin NorthHolland/AmericanElsevier. M.Ben-Daya,A.Rahim(1999).Multi-stagelotsizingmodelswithimperfectprocessesandinspection errors.productionplanningandcontrol,vol.10,pp cyclicallyscheduledmanufacturingsystems.int.journalofproductionresearch,vol32,pp

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