IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING VOL - PDF document

etal.:STOCHASTICMODELINGOFANAUTOMATEDGUIDEDVEHICLESYSTEMWITHONEVEHICLEANDACLOSED-LOOPPATH517SomeoftheadvantagesoftheMarkovchainapproachareasfollows:1)Itiseasytounderstand;thetransitionprobabilitiesarecharacterizedby12simpleconditions.2)Itissimplecon-ceptuallybecausealloneneedsis (where denotestheofthejobinterarrivaltime),whichcanbeeasilycalculatedforanydistributionfortheinterarrivaltimeofjobs.3)Itproducesastate-spacecollapsetherebymakingitfeasibletocomputethelimitingprobabilities.Andnally,4)itisquiteaccurate;thisisreectedbyfavorablecomparisonswithasimulationbenchmark.Themainthemeofthispaperwastodevelopasimple,al-thoughapproximate,Markovchaintohelpstudytheperfor-manceofanAGVsystem(withonevehicleandaclosed-loopoptimizeitscapacity.ThecostandcapacityoftheAGVaretwoparametersthatareinextricablylinkedtoeachother,andthiswasanattempttoquantifyandanalyzethisre-lationship.ExtendingthismodeltodropoffAGVs,multipleAGVs,orsystemsthatdosharesomeofthepropertiesas-sumedhereshouldmakeforexcitingtopicsforfurtherresearch.TheoremA.1:Iftheinterarrivaltimeistriangularlydis-tributedwith ,Properties1and2aresatisProof:Forthetriangulardistribution,for Also,for and Sinceweareinterestedinobtainingthelimitsfor tendingtozero,wedonotconsiderthecasefor .Then Then,itfollowsthat Property1isthussatised.Similarly,forproperty2,wehave applyingL'Hospital'sruletwice CKNOWLEDGMENTTheauthorswouldliketothankallthereviewersfortheirdetailedcomments;inparticular,theyexpressgratitudetothereviewerwhosuggestedanewtitleforthepaperandmadesev-eralothercommentsthatwereusedtoimprovethequalityofthiswork.[1]O.Bakkalbasi,Flowpathnetworkdesignandlayoutconformaterialdeliverysystems,Ph.D.dissertation,GeorgiaInst.Tech-nology,Atlanta,GA,1989.[2]Y.A.BozerandJ.H.Park,NewpartitioningschemesfortandemAGVsystems,ProgressinMaterial-HandlingResearch,1992AnnArbor,MI:Braun-Brunled,1993,pp.317[3]Y.A.BozerandM.M.Srinivasan,TandemAGVsystems:Apar-titioningalgorithmandperformancecomparisonwithconventionalAGVsystems,Eur.J.OperationalRes.,vol.63,pp.173191,1992.[4]P.Chevalier,Y.Pochet,andL.Talbott,Designofa2-stationsautomatedguidedvehiclesystems,QuantitativeApproachestoDistributionLogisticsandSupplyChainManagement,A.Kolse,M.Speranza,andL.W.Eds,Eds.Berlin,Germany:Springer,2002,pp.[5]J.DuanandJ.Simonato,AmericanoptionpricingunderGARCHbyaMarkovchainapproximation,J.EconomicDynamicsContr.,vol.25,no.11,pp.16891718,2001.[6]P.J.Egbelu,Theuseofnon-simulationapproachesinestimatingvehiclerequirementsinanautomatedguidedvehiclebasedtransportMater.Flow,vol.4,pp.1732,1987.[7]P.J.EgbeluandJ.M.A.Tanchoco,Characterizationofautomaticguidedvehicledispatchingrules,Int.J.ProductionRes.,vol.22,no.3,pp.359374,1984.[8]GendreauandM.Etude,Approfondidunmodeledequilibrepouraffectationdespassagersdanslesreseauxdetransportencommun,Ph.D.dissertation,Univ.deMontreal,Montreal,Canada,1984.[9]S.Heragu,FacilitiesDesign.Boston,MA:PWS,1997.[10]M.E.JohnsonandM.L.Brandeau,Stochasticmodelingforauto-matedmaterialhandlingsystemdesignandcontrol,Transportation,vol.30,no.4,pp.330348,1996.[11]P.Koo,J.Jang,andJ.Suh,EstimationofpartwaitingtimeandsizinginAGVsystems,Int.J.FlexibleManuf.Syst.,vol.16,pp.228,2005.[12]H.J.KushnerandP.Dupuis,NumericalMethodsforStochasticCon-trolProblemsinContinuousTime,2nded.NewYork:Springer,2001.[13]A.M.LawandW.D.Kelton,SimulationModelingandAnalysisNewYork:McGraw-Hill,2000.[14]L.Leung,S.K.Khator,andD.Kimbler,AssignmentofAGVswithdifferentvehicletypes,Mater.Flow,vol.4,no.12,pp.6572,1987.[15]B.MahadevanandT.T.Narendran,EstimationofnumberofAGVsforFMS:Ananalyticalmodel,Int.J.ProductionRes.,vol.31,no.7,pp.16551670,1993.[16]W.L.MaxwellandJ.A.Muckstadt,Designofautomatedguidedve-hiclesystems,IIETrans.,vol.14,no.2,pp.114124,1982.[17]W.B.Powell,Iterativealgorithmsforbulkarrival,bulkservice,andnon-Poissonarrivals,TransportationSci.,vol.20,pp.6579,1986.[18]S.M.Ross,IntroductiontoProbabilityModels.SanDiego,CA:Aca-demic,1997.[19]J.J.Solberg,AmathematicalmodelofcomputerizedmanufacturingProc.5thInt.Conf.ProductionRes.,Tokyo,Japan,Aug.[20]M.M.Srinivasan,Y.A.Bozer,andM.Cho,Trip-basedhandlingsys-tems:Throughputcapacityanalysis,IIETrans.,vol.26,no.1,pp.89,1994.[21]Subramaniam,J.S.Stidham,Jr,andC.J.Lautenbacher,Airlineyieldmanagementwithoverbooking,cancellationsandno-shows,Trans-portationSci.,vol.33,no.2,pp.147167,1999.[22]J.M.A.Tanchoco,P.J.Egbelu,andF.Taghaboni,ofthetotalnumberofvehiclesinanAGV-basedmaterialtransportMater.Flow,vol.4,no.12,pp.3351,1987.[23]T.Tansupawuth,OptimizingthecapacityofanAGVusingasto-chasticsimulation,M.S.thesis,ColoradoStateUniv.,Pueblo,2002.[24]T.U.M.I.Wrandeau,Designingasingle-vehicleautomatedguidedvehiclesystemwithmultipleloadcapacity,TransportationSci.,vol.30,no.4,pp.351353,1996. etal.:STOCHASTICMODELINGOFANAUTOMATEDGUIDEDVEHICLESYSTEMWITHONEVEHICLEANDACLOSED-LOOPPATH515TABLEXVIXV(VERAGEUMBEROFAITINGATACHACHINEFORISTRIBUTEDNTERARRIVAL TABLEXVIIROBABILITYHATAGVLEAVESWOORRRIVALS TABLEXVIIIXVII(ROBABILITYHATAGVLEAVESWOOREHINDFORRRIVALS showsthecostvaluesforthedifferentcapacitiesandtheneedforoptimization.Figs.5and6showtheeffectofabiggervalueof onthevalueof forSystems30and31,respectively.Figs.7and8showtheeffectofthesameon forSystems30and31,respectively.Asseenfromthesegraphs,theaccu-racyoftheMarkovmodeldecreasesas increases.Clearly,as decreases,thediscrete-timeapproximationgetsclosertoitscontinuouslimit[see(2)and(4)].TABLEXIXROBABILITYHATAGVLEAVESWOORISTRIBUTEDNTERARRIVAL TABLEXXXX().PROBABILITYHATAGVLEAVESWOOREHINDFORISTRIBUTEDNTERARRIVAL TABLEXXIAPACITY )ANDOPTIMALCOST(C ACHINES =$300 =$550 =$10000 TABLEXXIIAPACITY )ANDOPTIMALCOST(C ACHINES =$300 =$550 =$10000 ThemajorconclusionfromourexperimentsisthattheMarkovchainmodelproducesasolutionwithagoodqualityanddoessoinaveryreasonableamountofcomputertime,whichislessthan2minutesonaPentiumprocessorPCwith2-GHzCPUfrequencyand250-MGRAMsize.Thesimulationmodelincomparisontakesaboutfourtimesasmuchtime. 506IEEETRANSACTIONSONAUTOMATIONSCIENCEANDENGINEERING,VOL.5,NO.3,JULY2008 Fig.1.Schematicviewof-machinesystem.Theperformancemetricsthatweexploreforevaluationandoptimizationofthesystemare:1)averagenumberofjobswaitingateachmachine;2)probabilitythatthenumberofjobswaitingatamachinewhentheAGVdepartsfromthesamemachineexceedsagivenvalue;3)averagelong-runcostofrunningthesystem.III.MARKOVA.SystemStateWeintroduceaMarkovchainmodelforanalyzingthesystem.Lettheset denotetheentiresystemthatcontains (orpick-uppoints).Then Dropoffpoint,Machine1,Machine2 Machine Asubsetof ,called ,canbedenedasfollows: Dropoffpoint,Machine (1)for .Thebehavioroftheabove-describedsystemcanbemodeledwithasequenceofMarkovchainsassociatedwiththefollowing: TheMarkovchainassociatedwith willbereferredtoas thMarkovchain.Becauseeachmachinewillbeanalyzedindependently,weconsider Markovchains,oneforeachma-chine.Also,aniceproperty(Theorem4.2)ofthesystemisthatonecanusetheinvariantdistribution(limitingprobabilities)of thMarkovchaintocomputethesameforthe Markovchain.Thisprovidesforasimplerecursiveschemethatcanbeusedtodeterminesomeimportantsystemperformancemeasuresassociatedwiththeentiresystemfromtheinvariantdistributions.Inordertoconstructadiscrete-timeMarkovchain,i.e.,,wemustdiscretizetime.Welet denoteapositiveandsmallunitoftime.Wethenhavethefollowingde1)DeÞnition3.1:For ,let denotethemax-imumlengthofthetimeintervalduringwhichtheprobabilityoftwoormorearrivalsofjobs,atthe thmachine,islessthan ,i.e., (2)where isthenumberofarrivalsatthe thmachineduringatimeintervaloflength Nextweintroducesomenotation. Maximumnumberofjobsthatcanwaitatthe thmachine. TimespentbytheAGVintravelingfrom tomachine wheremachine0isthedropoffpoint.Thisalsoincludestheloadingtimeonmachine when andtheunloadingtime . Anintegermultipleof suchthat (3) Probabilitythat jobsarriveatthe thmachineinatimeintervaloflength . Numberofstatesinthe thMarkovchain.(StatesaredenedbelowinDenition3.2.) capacityoftheAGV. availablecapacityoftheAGV.(Itisarandomvariableforallmachinesbuttherstforwhichitequals .)Let denotethetimebetweenthe thandthe( arrivalofjobsatmachine .Then,sincejobscontinuallyarriveateachmachine,forany Wewillobservethesystemafterunittime,i.e., ,fol-lowingastandardconventionintheliterature(see[18,p.435]),andafterunittime,anewepochwillbeassumedtohavebegun.Thus,ifonespeci ,thelength(timeduration)ofanyepochinthe thMarkovchainwillequal Thefollowingphenomenawillbetreatedaseventsforthe thMarkovchain:1)Ajobarrivesatamachine;2)theAGVdepartsfromthe thmachine;and3)theAGVarrivesatthe machine.Aneventwillsignalthebeginningofanewepoch.Theprobabilityoftwoormorearrivalsinoneepochwillbeas-sumedtobenegligibleforsmall ,andthedenitionof ensuresthat.Furthermore,wewillassumethatifajobarrivesduringanepoch,wewillmovethateventforwardintimetocoincidewiththeendofthatepoch.Since isasmallquantity,when issmall,thisshouldnotposeseriousproblems.ThisapproximationisnecessarytoensuretheMarkovproperty etal.:STOCHASTICMODELINGOFANAUTOMATEDGUIDEDVEHICLESYSTEMWITHONEVEHICLEANDACLOSED-LOOPPATH509Case4:If , , , , , ,and ,then Case5:If , , , , , ,and ,then ConsiderthescenarioinwhichtheAGVtravelsfromthedropoffpointtomachine duringthe thepoch,arrivesatma- bytheendofthe thepoch,andthebufferatmachine isfull.Then,wehavethefollowing.Case6:If , , , , , ,and ,then ConsiderthescenarioinwhichtheAGVtravelsfrommachine tothedropoffpointduringthe thepoch,thearrivaltothedropoffpointdoesnotoccurbytheendoftheepochandthebufferatmachine isnotfull.Then,ifnojobarrivesduringthe thepoch,wehaveCase7,andifajobdoesarrivewehaveCase8.Case7:If , , , , , ,and ,then Case8:If , , , , , ,and ,then ConsiderthescenarioinwhichtheAGVtravelsfrommachine tothedropoffpointduringthe thepoch,thearrivaltothedropoffpointdoesnotoccurbytheendofthe thepochandthebufferatmachine isfull.Then,wehavethefollowing.Case9:If , , , , , ,and ,then ConsiderthescenarioinwhichtheAGVtravelsfrommachine tothedropoffpointduringthe thepoch,itarrivesatthedropoffpointbytheendoftheepoch,andthebufferatmachine isnotfull.Then,ifnojobarrivesduringthe thepoch,wehaveCase10.Ifajobarrivaloccursduringthe thepoch,wehaveCase11.Case10:If , , , , , ,and ,then Case11:If , , , , , ,and ,then Finally,considerthescenarioinwhichtheAGVtravelsfrom tothedropoffpointduringthe thepoch,itarrivesatthedropoffpointbytheendoftheepoch,andthebufferat isfull.Then,wehavethefollowing.Case12:If , , , , , ,and ,then IV.PEASURESANDPTIMIZATIONFromDenition3.2,thenumberofstatesinthe thMarkovchain,i.e., ,canbecomputedas Thefollowingwell-knownresultallowsustodeterminetheinvariantdistribution(limitingprobabilities)oftheunderlyingMarkovchains.1)Theorem4.1: denotetheone-steptransitionprob-abilitymatrixofaMarkovchain.Ifthematrixisaperiodicandirreducible,andif ,whose thelementisdenotedby ,de-notesacolumnvectorofsize ,thensolvingthefollowingsystemoflinearequationsyieldsthelimitingprobabilitiesoftheMarkovchain: and Fromthedenitionofourtransitionprobabilities,itisnothardtoshowthateachstateispositiverecurrentandaperiodicandthatthereisasinglecommunicatingclassofstates.Then,theaboveresultholdsandwemayuseittocomputethelimitingWenextdeneafunction, ,thatassignsanintegervalueintheset toeachstateinthe thMarkovchain where denotesthestateinthe thMarkovchainatagivenepoch, .Notethat ,theepochindex,issuppressedherefromthenotationof(5)toincreaseclarity.Ourtransitionprobabilitiesforthe thMarkovchainwerecomputedunderthe etal.:STOCHASTICMODELINGOFANAUTOMATEDGUIDEDVEHICLESYSTEMWITHONEVEHICLEANDACLOSED-LOOPPATH511 machinestobeservedbyanAGVwhosecapacityis ,isgivenby (13)where and areconstantsrepresentingtheoperatingcostperunittimeofanAGVhavingunitcapacityandtheholdingcostperaunittimeforonejob,respectively; denotesthexedcostofthevehicleIntheuniformizationprocedure,theprobabilityoftwoormorearrivalsinoneepochisneglected.Thus,afewarrivalsareunaccountedfor.Thiscausesareductioninthevaluesofbothperformancemeasures,i.e., and fortherstmachine.ThiserroraffectsthecapacitydistributionandresultsinareducedavailablecapacityfortheAGVwhenitvisitsthesubsequentmachines.Thiserrorinthecapacitydis-tributionpartlynegatestheerrorintroducedbyuniformization.Hence,atallmachinesbuttherst,wherethereisnoerrorines-timatingthecapacity,theoverallerrorintheperformance-mea-surevaluesislow.Attherstmachine,thecapacityisequaltothemaximumcapacity,andsothisreductiondoesnotoccur,therebyproducingahighererror.Thisisreectedinthenumer-icalresultspresentedlater.Alsonotethatonehastondthelimitingprobabilitiesof differentMarkovchains,onechainassociatedwithaspecimachineinthesystem.However,tondthelimitingprobabilityofthespecicstateinaMarkovchainassociatedwithmachine ,where ,onehastondthelimitingprobabilitiesofthe differentMarkovchainsassociatedwithmachine (seeTheorem4.2),sincetheavailablecapacityoftheAGVcanassumeanyvaluebetweenzeroand whenitarrivesatanymachinebuttherstmachine.(Notethatsincetheavail-ablecapacityoftheAGVis whenitarrivesattherstma-chine,thelimitingprobabilitiesoftheMarkovchainassociatedwiththerstmachinecanbefoundwithouttheuseofTheorem4.2.)Hence,thecomputationalworkassociatedwithanygivenmachinebuttherstconstitutesofthecomputationalworkofcalculatingthelimitingprobabilitiesof MarkovchainsplusthelimitingprobabilitiesoftheMarkovchainassociatedwiththegivenmachineviaTheorem4.2.Therefore,thecom-plexityoftheMarkovchainmodelisarst-orderpolynomialinthenumberofmachines Theadvantageofourcomputa-tionalschemeisthatthestatespacecollapsesforeachMarkovchain,therebymakingthecomputationofthelimitingprobabil-itiesfeasible.6)AGVCapacityOptimization:Weareinterestedinopti-mizingtheAGVscapacitywithrespecttothecostofoperatingthesystem.Theoptimizationproblemconsideredinthispaperistodetermine tominimizetheexpressionin(13).Otherwaysforoptimizationcouldbeconsidereddependingonwhatthemanagerdesires.Oneexampleisto suchthat where and aresetbymanagerialpolicy.OptimizationisperformedbyanexhaustiveenumerationoftheAGVsmaximumcapacityvariable, .Sincethestatespaceofthisdiscreteoptimizationproblem,whichhasasingledecisionvariable,isverysmall,anexhaustiveenumerationisV.COMPUTATIONALESULTSSimulationisbyfarthemostextensivelyusedtoolforperfor-manceevaluationofAGVsystems,becauseitprovidesuswithveryaccurateestimatesofperformancemeasures.Asaresult,itisessentialthatwecompareourresultstothoseobtainedfromasimulationmodel.A.SimulationModelConsideraprobabilityspace ,where denotesthe(universal)setofallpossibleroundtripsoftheAGV, thesigmaeldofsubsetsof ,and denotesaprobabilitymea-sureon .Usingadiscrete-eventsimulator,itispossibletogeneraterandomsamples fromthemeasur-ablespace.Thesamplescanthenbeusedtoestimatevaluesofalltheperformancemeasuresderivedintheprevioussection. denotethenumberofjobswaitingnearthe machineattime inthesimulationsample .Then,fromthestronglawoflargenumbers,withprobability1 Also,withprobability1 (15)where denotesthenumberofoccasionsinwhichthenumberofjobsleftbehindatmachine (i.e.,jobsnotpickedupbytheAGVbecauseitisfull)equalsorexceeds in tothemachineinthesimulationsample B.PerformanceTestsforMarkovModelWeconductednumericalexperimentswithourmodeltode-terminethepracticalityoftheapproachandtobenchmarkitsperformancewithasimulationmodelthatisguaranteedtoper-formwellbutisconsiderablyslower.TheerrorofourmodelwithrespecttothesimulationestimateisdenedbyError(%) where denotestheestimatefromtheMarkovchainmodel denotesthesamefromthesimulationmodel.Wepresentresultsontwo-machineandve-machinesystems,alongwithasimpleexampletoillustrateourmethodology.TablesIandIIdenetheparametersforthesystemswithtwomachines,andTablesXIandXIIdenetheparametersforthesystemswithvemachinesstudied.InTableI, notestherateofarrivalofjobsatmachine .Weusedtheexpo-nentialandgammatomodelthedistributionoftheinterarrivaltimeofjobs.TheperformancemetricsofthesystemswithtwomachinesandvemachinesareshowninTablesIIIVIIIandTablesXIIIXX,respectively.Thesetablesalsoshowthesimu-lationestimatesandthecorrespondingerrorvalueswhicharecalculatedasdenedabove.Convergencewasachievedwith etal.:STOCHASTICMODELINGOFANAUTOMATEDGUIDEDVEHICLESYSTEMWITHONEVEHICLEANDACLOSED-LOOPPATH505theexpensesincurredinbuyingahighercapacityvehicle.Re-ducinginventoryleadstoreducedinventory-holdingcostsandcongestion.ThecostofanAGVmostcertainlydependsonitscapacity.AttachingatrailertoanAGVincreasesitscapacity,andhencevariable-capacityAGVsareseeninmanyreal-worldsystems.Althoughattachingatrailercanconstrainitsmove-mentsinsomeways,buyinganewAGVisaconsiderablymoreexpensiveproposition.BecauseofthecomplexityinastochasticAGVsystem,an-alyticalmodelsforperformanceevaluationarenotcommonlyfoundintheliterature.Thereare,ofcourse,importantexcep-tions,someofwhichwediscussnext.MaxwellandMuckstadt[16]presentedananalyticaldeterministicmodeltondthemin-imumnumberofAGVsrequiredinagivensystem,whichwasextendedbyLeungetal.[14]toconsideradditionalvehicletypes.Egbelu[6]describedfouranalyticalmodelsforasystemsimilartothatofMaxwellandMuckstadt[16].Tanchocoetal.[22]presentamodelbasedontheCAN-Qsoftware(Computer-izedAnalysisofNetworkofQueues,seeSolberg[19])forde-terminingthenumberofAGVsneeded.CAN-Qusessophisti-catedqueuingtheoryconcepts,butitsblack-boxnaturehasper-hapspreventedfurtheruse.Similarly,Wysketal.[27]presentaCAN-Q-basedanalyticalmodeltoestimatethenumberofAGVsinwhichemptyvehicletravelisconsidered.Bakkalbasi[1]formulatestwoanalyticalmodels;oneofhismodelspro-videslowerandupperboundsofemptytravelingtime.Anan-alyticalmodelisprovidedinMahadevanandNarendran[15]fordeterminationofthenumberofAGVsinaexiblemanu-facturingsystem.Srinivasanetal.[20]presentaqueuingmodeltodeterminethethroughputcapacityofanAGVsystem.Kooetal.[11]useaqueuingmodeltodeterminetheAGVeetsizeunderavarietyofvehicleselectionrules.Visetal.[26]developamodelthatcanbeusedatanautomatedcontainerterminal.Chevalieretal.[4]addressaprobleminasystemthatcontainstwostations.JohnsonandBrandeau[10]presentanexcellentoverviewofmodelinganddesignissuespertinenttostochasticmaterial-handlingsystems.ThonemannandBrandeau[24]usequeuingapproximationsfromGendreau[8]andPowell[17]tostudyanAGVsysteminwhichthereisonedepotandmultiplemachinesthatrequirematerialfromthedepot.Markovchainap-proximationsarepopularinuniformizationofcontinuous-timeMarkovchains[21],diffusionapproximations(see[12,Ch.4]),nancialengineering[5]buthavenotbeenusedintheanal-ysisofmaterial-handlingsystemsorqueuing,tothebestofourknowledge.Contributionsofthispaper:ThecapacityoftheAGVisanimportantdesignissuefromamanagerialperspective.Inthispaper,wepresentforthersttime,tothebestofourknowl-edge,aMarkovchainapproximationmodelfordetermining:1)anumberofimportantperformancemeasuresofanAGVsystemwithonevehicleandaclosedlooppathand2)theop-timalcapacityoftheAGV.Efcacyofthemodelisdemon-stratedwithnumericalexperiments.Thelatterindicatethatitsperformanceiscomparabletothatofasimulationmodel;notethatsimulationmodelsareusuallyguaranteedtobeexactinanalmostsuresense.OurMarkovchainmodelrequireslesscom-putationtimeincomparisontothesimulationmodel,therebymakingitusefulforoptimizationpurposes(optimizationofthevehiclescapacity).Also,wewereabletoprovethattheun-derlyingMarkovchainhasaspecialstructurewhichfacilitatesdecompositionoftheMarkovchainassociatedwiththeclosed-loopsystemintoanitenumberofMarkovchainsthathaveasmallerstatespace.ThespecialstructuremakestheanalysiseasierandsimpliesthecomputationsconsiderablybecauseitiseasiertohandlethesmallerMarkovchainsassociatedwiththesubsystems.Furthermore,themodelcanbeusedforsomedis-tributionsinthearrivalofjobsatmachinesthatsatisfyaprop-ertythatweidentify.Dependingonthenatureofthesystem,thepick-uppointscouldeitherbeproductionmachines(jobs)orofces(paperwork),andthedropoffpointcouldbeacon-veyorbeltorthemainofce.TheworkofThonemannandBran-deau[24]isclosesttoourworkinspiritbecausetheyalsoan-alyzeasinglevehicleinaclosedlooppath,buttheyprimarilydealwithadropoffsystem,i.e.,asysteminwhichtheAGVdropsoffmaterialateachpointandpicksupmaterialatadepot.Therandomnessintheirsystemisinthearrivalofjobstothedepot.TheydonotoptimizetheAGVscapacityandconsideronlyPoissonarrivals.Wefocusonapick-upAGV,(foundinlocalindustriesinColorado)i.e.,anAGVthatpicksuploadsatvariousstationsanddropsthemoffatonepoint.TheMarkovchainapproachenablesustocomputesomeQoSmeasures,e.g.,theprobabilityoftheAGVdepartingfromastationleaving numberofjobsstranded,anddeterminetheoptimalcapacityoftheAGV.Tothebestofourknowledge,thisistherstattemptatbothofthesetasksintheliterature.Therestofthispaperisorganizedasfollows.SectionIIdescribestheproblem.SectionIIIdevelopstheMarkovchainmodel.TheperformancemeasuresandtheoptimizationmodelaredescribedinSectionIV.NumericalresultsarepresentedinSectionV.ThelastsectionpresentssomeconclusionsdrawnfromourworkandsomedirectionsforfurtherresearchinthisII.PROBLEMrstdiscusssomeimportantfeaturesoftheproblemunderconsideration.TheAGVtravelsinaxedcircuitfromadropoffpoint(e.g.,conveyorbelt)toeachmachineinasequence(Ma-chine1,thenMachine2,andsoonuntilallthemachineshavebeenvisited)pickingupjobsfromeachofthemachines.TheAGVthenreturnstothedropoffpointtodropthejobsoffandthenrepeatsthecircuit.TheAGVtakesaxed(deterministic)amountoftimetotravelfromonelocationtoanother;thistimeincludestheunloadingorloadingtime.TheAGVemptiesitselfcompletelyatthedropoffpoint.Also,weassumethattherouteoftheAGVisnotinuencedbywhetheritisfull,althoughwhenitisfull,itcannotpickupanymorejobsinthattrip.Jobsarriveateachlocationwithrandominterarrivaltimesthatareinde-pendentandidenticallydistributed;theamountofspace(outputbuffer)nearthepick-uppoint(machine)isxed.Inotherwords,whenthisbufferisfull,themachinestopsproducingandthusthenumberofjobswaitingateachmachinehasanupperlimit.Althoughtheamountofspace(buffer)atthemachineisxed,itisassumedthatthebufferisofasufcientsizethatthemax-imumnumberofjobsthatcanbewaitingatthemachineisrarelyreached.Fig.1presentsaschematicofthesystem. etal.:STOCHASTICMODELINGOFANAUTOMATEDGUIDEDVEHICLESYSTEMWITHONEVEHICLEANDACLOSED-LOOPPATH507forthesystemweconsider.However,numericalresults(pre-sentedlater)willdemonstratethatwiththeapproximationwestillhavereasonablyaccurateresults.Moreover,theapproxima-tionallowsustousethepowerfulframeworkofMarkovchains.2)Denition3.2: .Astateofthe thMarkovchaininthe thepochisdenedbythefollowing4-tuple: (5)where denotesthenumberofjobswaitingatmachine whenthe thepochbegins, denotestheavailablecapacityintheAGVwhenthe thepochbegins,and iftheAGVistravelingfromthedropoffpointtoanymachinewhile iftheAGVistravelingfromthe thmachinetothedropoffpointwhenthe thepochbegins.Sincethe epochcouldbeginwhiletheAGVistravelingeitherbetweenthedropoffpointandthemachine orbetweenthemachine thedropoffpoint,welet denotethenumberofmultiplesof thathaveelapseduntilthebeginningofthe thepochsincetheAGVsdeparture.Clearly,theAGVsdeparturecouldeitherbefromthedropoffpointorthe thmachine.B.TransitionProbabilitiesWiththeabovediscretizationoftime,wehavethefollowingproperty.Foranysmallvalueof andforany ,itisapproximatelytruethat Thisimpliesthatfornonzero,butsmall, ,oursystemcanbeapproximatelymodeledbyaMarkovchain.ItistobenotedthattheMarkovchainsconstructedareessentiallyparametrizedbythenon-negativescalar .However,tokeepthenotationsimple,andbecause xed,wewillsuppressthisparameter.Thus, willbedenotedby .Bythedenition,the Markovchainisconcernedwithjobarrivalsatthe thma-Thedenitionof requiresustoanalyzewhethertheprobabilityoftwoormorearrivalsinanepochcanbeignoredincomparisontothatofzerooronearrival.Wejustifytheuseofasmall withthefollowingproperties.Considerthefollowingtwopropertiesassuming tobethedurationofanepoch.Property1: Property2: Whenthearrivaldistributionsatisesthesetwoproperties,onecanignoretheprobabilityoftwoormorearrivalsincomparisontothoseofzeroarrivalsandonearrivalas tendstozerobecausetheprobabilityoftwoormorearrivalsconvergestozerothaneitheroftheothertwoprobabilities.Inotherwords,forasmallvalueof ,onemaypracticallyignoretheeventassociatedwithmorethantwoarrivals.Withasmall ,wehaveforevery asmallenough ,i.e., inthetwopropertiesabove.Thus,withasufcientlysmalldurationoftheepoch,onecansafelyignoretheprobabilityoftwoormorearrivalsforcertaindistributions.Wewillprovethiswhentheinterarrivaltimeisexponentiallydistributedanduniformlydistributed.Note,how-ever,thatevenforthesedistributions,e.g.,exponential,ourap-proachremainsapproximate.1)Theorem3.1:Iftheinterarrivaltimeisexponentiallydis-tributedwithmean ,Properties1and2aresatisProof: therebysatisfyingProperty1.Similarly byL'Hospital'srule 2)Theorem3.2:Iftheinterarrivaltimeisuniformlydis-tributedwith ,Properties1and2aresatisProof:Fornon-Poissonarrivals,todeterminetheproba-bilitydistributionofthenumberofarrivals,onehastocompute ,the -foldconvolutionofthedistributionoftheinterarrivaltime.If denotesthetimeofthe tharrival(ofjob),thenfortheuniformdistribution Also and Then Then,itfollowsthat: 512IEEETRANSACTIONSONAUTOMATIONSCIENCEANDENGINEERING,VOL.5,NO.3,JULY2008TABLEI.)PALUEOFACHYSTEMANDACHYSTEMHASACHINES TABLEII.)P TABLEIIIVERAGEUMBEROFAITINGATACHINERRIVALS TABLEIVVERAGEUMBEROFAITINGATACHINERRIVALS TABLEVVERAGEUMBEROFAITINGATACHINEISTRIBUTEDNTERARRIVAL 1000tripsperreplicationandweusedtenreplications.Themeanestimatewasassumedtohaveconvergedwhenitremainedwithin0.05%inthenextiteration.Themeanreportedisanav-erageoverallreplications.Thestandarddeviationwasnomorethan0.1%ofthemeanineachcase.1)2-MachineSystems:TablesIIIandIVprovide forPoissonarrivalsfor and ,respectively.Sim-ilarly,TablesVandVIshowthecorrespondingvaluesforagamma-distributedinterarrivaltime.Inthesesystems,i.e.,inthesystemswithtwomachines,forthegammadistribution weusedthesamearrivalrate,buttheparameter wassettoeight.TablesVIIandVIIIshowthevaluesof Poissonarrivalsfor and ,respectively;TablesIXandXshowthecorrespondingvaluesforagamma-distributedinterarrivaltime. 508IEEETRANSACTIONSONAUTOMATIONSCIENCEANDENGINEERING,VOL.5,NO.3,JULY2008Property1isthussatised.Similarly,forProperty2,wehave byL'Hospital'srule Weshowthattheconditions(6)and(7)holdforthetriangulardistributioninAppendixI.Forinterarrivaldistributions(ofjobs)thatsatisfy(6)and(7),wecandevelopexpressionsforthetransitionprobabilitieswithnitebutsmallvalueof .Thetransitionsarecharacterizedby12conditions.ForeachMarkovchainthatweconsider,wewillassume,forthetimebeing,thattheavailablecapacity known.Subsequently,wewillpresentaresult(Theorem4.2)tocomputethedistributionof .Thestateinthe thepochwillbedenotedby .Theone-steptransitionprobabilitywillbedenotedby Fig.2presentsapictorialrepresentationoftheunderlyingMarkovchaininourmodel.TheavailablecapacityoftheAGV, ,equals whenitar-rivesattherstmachineandisarandomvariableforallothermachines.Then, , if , if see3.2 and .ThetransitionprobabilitiesforthisMarkovchainwillnowbede-nedfor12conditions(cases).Cases16(712)areassociatedwiththeAGVtravelingfromthedropoffpointtoamachine(fromamachinetothedropoffpoint).Wenowdescribeallthecasesindetail.Considerthefollowingscenario:theAGVtravelsfromthedropoffpointtomachine duringthe thepoch,thearrivaloftheAGVatmachine doesnotoccurbytheendoftheepoch,andthebufferatmachine isnotfull.Then,ifnojobarrivesduringthe thepoch,wehavethefollowing.Case1:If , , , , , ,and ,then Sincenojobarrivaloccursinunittime,theabovetransi-tionprobabilityisequaltotheprobabilityofnojobarrival.Similarly,ifajobarrivaldoesoccur,wehavethefollowing.Case2:If , , , , , ,and ,then Sincejobarrivaloccursinunittime,theabovetransitionprobabilityisequaltotheprobabilityofjobarrival.Considerthefollowingscenario:theAGVtravelsfromthedropoffpointtothemachine duringthe thepoch,thearrivaloftheAGVatmachine doesnotoccurbytheendoftheepochandthebufferatmachine isfull.Then,wehavethefollowing. Fig.2.Markovchainunderlyingourmodel.BingurestandsforCase3:If , , , , , ,and ,then Sincethebufferatmachine isfull,wedonottakejobarrivalsintoconsideration.Considerthefollowingscenario:theAGVtravelsfromthedropoffpointtothemachine duringthe thepoch,itarrivesatmachine bytheendoftheepoch,andthebufferatmachine isnotfull.Then,ifnojobarrivesduringthecurrentepoch,wehaveCase4,andifajobarriveswehaveCase5. 510IEEETRANSACTIONSONAUTOMATIONSCIENCEANDENGINEERING,VOL.5,NO.3,JULY2008 Fig.3.ComputationalschemebasedonTheorem4.2.assumptionthatcapacityoftheAGVwhenitarrivesatthe machineisknown( ).Wenowneedtocomputethetransitionprobabilitiesoftheentiresystemthetransitionprobabilitiesthatweneedforourperformanceevaluationmodel.Wewillprovidearesult,Theorem4.2,tothisend.Themainideaunder-lyingtheresultisasfollows.Thetransitionprobabilitiesofthe thMarkovchainyieldthedistributionoftheAGVsavailablecapacityofthe thMarkovchain.BecausetheAGVstartsempty,fortherstMarkovchain,theavailablecapacityisknowntobe .Fromthatpointonwards,onecancomputethetransitionprobabilitiesinarecursivestyle.ThecomputationalschemebasedonthenextresultisdepictedinaowchartinFig.3.2)Theorem4.2:Letthelimitingprobabilityofstate inthe thMarkovchainbedenotedby .Let denotetheavailablecapacityoftheAGVwhenitarrivesatthe thma-chine,andfurtherlet denotethelimitingprob-abilityforthestate inthe thMarkovchainwhen Then,wehavethat (8)and (9)for , ,where inwhich and aredenedasfollows: and Proof:Fortheproof,weneedtojustify(8)(10).Equa-tion(8)followsfromthefactthattheAGVemptiesitselfatthedropoffpoint(seeFig.1).Hence,whenitcomestoMachine1,theavailablecapacityisnotarandomvariablebutaconstantequalto ,whichisthemaximumcapacityoftheAGV.Equa-tion(9)followsfromthefactthatwhentheAGVarrivesatsta-tionsnumbered2through ,itscapacityisarandomvariablewhosedistributionhastobecalculated.Thedistributionisgivenin(10).Forthelaststatement,weargueasfollows:WhentheAGVarrivesatthe thstation,for ,itsca-pacityistheavailablecapacityoftheAGVwhenitleavesthe thstation.ThedistributionfortheavailablecapacityontheAGVwhenitleavesthe thstationcanbecomputedfromthelimitingprobabilitiesassociatedwiththe MarkovchainofthosestatesinwhichtheAGVhasdeparted thmachineandisonthewaytothe thmachine.(Notethat denotesthesetofstatesinthe thMarkovchaininwhichtheAGVhasdepartedthe thmachineanditsavailablecapacityis .Similarly, denotesthesetofstatesinthe thMarkovchaininwhichtheAGVhasdepartedfromthe thmachineanditsavailablecapacityassumesallpossiblevalues.) Exploitingthetransitionprobabilities,onecanderiveexpres-sionsforsomeusefulperformancemeasuresofthesystem.Theyareconsiderednext.3)AverageInventoryatEachMachine: theaveragenumberofjobswaitingatthe thmachine,then (11)where and if if 4)QoSPerformanceMeasureAssociatedWithDownsideTheprobabilitythatthenumberofjobswaitingatthe thmachine,denotedby ,whentheAGVdepartsfrom thmachineexceeds canbecalculatedasfollows: (12)where 5)AverageLong-RunCost:Theaveragelong-runcostofrunningthesystemiscomposedoftwoelements,whichare:theholdingcostofthejobsnearthemachineandthecostofoperatingtheAGV.Tocalculatetheholdingcost,wewillneedtheaveragenumberofjobswaitingateachmachine.Then,theaveragecostforoperatingasystemdescribedinthispaper,with 516IEEETRANSACTIONSONAUTOMATIONSCIENCEANDENGINEERING,VOL.5,NO.3,JULY2008 Fig.4.TotalcostversuscapacityofAGVforSystem2. Fig.5.Effectofabigger W(n)]insystem30.Errorsinotherparam-eters,i.e., [W W(5)]areevensmallerforplottedvaluesof Fig.6.Effectofabigger W(n)]inSystem31.Errorsinotherparam-eters,i.e., [W(4)]and W(5)]areevensmallerforplottedvaluesofThesimulationmodelandthetheoreticalmodelmakeidenticalassumptionsaboutthesystem.Thecomputercodescanberequestedfromtherstauthor. Fig.7.EffectofabiggerbiggerD(n)�1]inSystem30.Errorsinotherparameters,i.e.,i.e.,D(1)�1],P[D(2),andandD(5)�1]areevensmallerforplottedvaluesof Fig.8.EffectofabiggerbiggerD(n)�1]inSystem31.Errorsinotherparameters,i.e.,i.e.,D(1)�1],P[D(2)�1],P[D(4)�1],andandD(5)�1]areevensmallerforplottedvaluesofVI.CInthispaper,westudiedaversionofaproblemcommonlyfoundinsmall-scalemanufacturingindustries,e.g.,pharmaceu-rms,whichrequirenomorethanoneAGVthatoperatesinaclosed-looppath.WeusedaMarkovchainapproachforde-velopingtheperformance-analysismodel.Somespecialproper-tiesofthesystemwereexploitedtodeneasimplestructuretothecompleteproblem;thestructureallowsustodecomposethe -machinesystem(Markovchain)into individualsystems(Markovchains)andprovidesuswithamechanismtosimplifythecomputations.Thisleadstoastate-spacecollapsethatiscomputationallyenjoyable.Also,thediscretizationofthestatespaceinourapproachyieldedsimpleexpressionsforthetransi-tionprobabilities;acommoncriticismofmanytransition-prob-abilitymodelsisthattheyhavecomplicatedexpressionswithmultipleintegrals(thatrequirenumericalintegration,whichisslow)astransitionprobabilities. 504IEEETRANSACTIONSONAUTOMATIONSCIENCEANDENGINEERING,VOL.5,NO.3,JULY2008StochasticModelingofanAutomatedGuidedVehicleSystemWithOneVehicleandaClosed-LoopPathAykutF.Kahraman,AbhijitGosavi,Member,IEEE,andKarlaJ.OtyTheuseofautomatedguidedvehicles(AGVs)inmate-rial-handlingprocessesofmanufacturingfacilitiesandwarehousesisbecomingincreasinglycommon.AcriticaldrawbackofanAGVisitsprohibitivelyhighcost.CostconsiderationsdictateaneconomicdesignofAGVsystems.ThispaperpresentsananalyticalmodelthatusesaMarkovchainapproximationapproachtoevaluatetheperformanceofthesystemwithrespecttocostsandtheriskassociatedwithit.Thismodelalsoallowstheanalyticoptimization 518IEEETRANSACTIONSONAUTOMATIONSCIENCEANDENGINEERING,VOL.5,NO.3,JULY2008[25]J.A.Tompkins,J.A.White,Y.A.Bozer,E.H.Frazelle,J.M.A.Tanchoco,andTrevino,FacilitiesPlanning.NewYork:Wiley,1995.[26]I.Vis,R.deCoster,K.Roodbergen,andL.Peeters,DeterminationofthenumberofAGVsrequiredatasemi-automatedcontainerterminal,J.OperationalRes.Soc.,vol.52,no.4,pp.409417,2001.[27]R.A.Wysk,P.J.Egbelu,C.Zhou,andB.K.Ghosh,UseofspreadsheetanalysisforevaluatingAGVsystems,Mater.Flow,vol.4,pp.64,1987. AykutF.KahramanreceivedtheB.S.degreefromtheElectricalandElectronicsEngineeringDepartment,BilkentUniversity,in2002,theM.S.degreefromtheIndustrialandSystemsEngineeringDepartment,ColoradoStateUniversity,Pueblo,in2003,andthePh.D.degreefromtheIndustrialandSystemsEngineeringDepartment,StateUniversityofNewYork,Buffalo,in2006.Hisresearchinterestsincludelogistics,serviceandmanufacturingsystems,andperformanceevaluationinstochasticsystems.Applicationdomainsthatin-teresthimareprimarilyintheaviationindustryandthemanufacturingindustry.Intheeldofstochasticoptimization,hehasspecializedintheareaofdis-crete-timeMarkovchains,Markovdecisionprocesses,semi-Markovdecisionprocesses,andstochasticdynamicprogramming.Currently,heisworkingasanOperationsResearchDeveloperatDeccanInternational,SanDiego,CA. AbhijitGosavi06)receivedtheB.S.degreefromJadavpurUniversity,theM.S.degreefromtheIndianInstituteofTechnology,Madras,andthePh.D.de-greeinindustrialengineeringfromtheUniversityofSouthFlorida,in1999.HeiscurrentlyanAssistantProfessorintheDepartmentofIndustrialandSystemsEngineering,StateUniversityofNewYork,Buffalo.Hisresearchinterestsincludetheapplicationsandmethodsofcontroltheoryandsimulation-basedoptimization.HispublicationshaveappearedinleadingjournalslikeManagementScience,Automatica,MachineLearning KarlaJ.OtyreceivedtheB.S.degreeinmath-ematicsfromTrinityUniversity,SanAntonio,TX,andthePh.D.degreefromtheUniversityofColorado,Boulder,underthedirectionofProf.A.Ramsay.Herdissertationinanalysiswasen-Fourier-StieltjesAlgebrasforR-discreteHavingtaughtfor13yearsatschoolsinOklahomaandColorado,sheiscurrentlyservingasInterimDeanoftheSchoolofScienceandTechnology,CameronUniversity,Lawton,OK.Dr.OtyhasreceivedvariousawardsincludingCameronUniversityfessoroftheYearfortheacademicyear2005 etal.:STOCHASTICMODELINGOFANAUTOMATEDGUIDEDVEHICLESYSTEMWITHONEVEHICLEANDACLOSED-LOOPPATH513TABLEVIVERAGEUMBEROFAITINGATACHINEISTRIBUTEDNTERARRIVAL TABLEVIIROBABILITYHATAGVLEAVESWOORRRIVALS 2)Example:ConsiderSystem5,whoseparametersaregiveninTablesIandII.Inthissystem,jobsarriveattherateof2perunittimeateachmachine.Wechoose .Here, for .TheAGVtravelsfromthedropoffpointtoMachine1,fromMachine1toMachine2,andfromMachine2tothedropoffpointintimeequaling , ,and ,respectively.ThemaximumcapacityoftheAGVis2,andthebuffercapacitiesofeachmachineare4.Weprovidesomesampletransitionproba-bilitiesfortheMarkovchainassociatedwithMachine1Case1: Case3: Case5: Case10: TABLEVIIIROBABILITYHATAGVLEAVESWOORRRIVALS TABLEIXROBABILITYHATAGVLEAVESWOOREHINDFORISTRIBUTEDNTERARRIVAL ThecapacityoftheAGVwhenitarrivesatmachine1is andthecapacitydistributionoftheAGVwhenitarrivesatMa-chine2iscalculatedusingTheorem4.2 and Then,thetransitionprobabilitiesforthesecondMarkovchaincanbecalculatedfromthedistributionof .Thelimitingprob-abilitiescanthenbeusedtocalculatethevaluesoftheperfor-mancemeasures,suchastheaverageinventoryandthedown-siderisk. 514IEEETRANSACTIONSONAUTOMATIONSCIENCEANDENGINEERING,VOL.5,NO.3,JULY2008TABLEXROBABILITYHATAGVLEAVESWOORISTRIBUTEDNTERARRIVAL TABLEXI.)PALUEOFACHNTERARRIVALi.i.d.,EXPONENTIALLYISTRIBUTEDACHACHINES TABLEXII.)PALUEOFACHNTERARRIVALi.i.d.,GISTRIBUTEDANDACHACHINESENOTES 3)Five-MachineSystems:TablesXIIIandXIVlistthevaluesof for ,andTablesXVIIandXVIIIlistthevaluesof for TABLEXIIIVERAGEUMBEROFAITINGATACHACHINERRIVALS TABLEXIVXIII(VERAGEUMBEROFACHACHINEFORRRIVALS TABLEXVVERAGEUMBEROFAITINGATACHACHINEFORISTRIBUTEDNTERARRIVAL Poissonarrivals.Thecorrespondingvaluesforagamma-dis-tributedinterarrivaltimeareshowninTablesXVandXVI(fortheaveragenumberwaiting)andTablesXIXandXX(fortheprobability).TablesXXIandXXIIpresenttheresultsfromoptimizationperformedwiththeMarkovchainmodel.Fig.4