Production,ManufacturingandLogistics
Atwo-echeloninventorymodelwithlostsales
R.M.Hill
aa,*,M.Seifbarghyb,D.K.Smith
aDepartmentofMathematicalSciences,UniversityofExeter,LaverBuilding,NorthParkRoad,ExeterEX44QE,UK
bDepartmentofIndustrialEngineering,SharifUniversityofTechnology,Tehran,Iran
Received3June2004;accepted1August2006
Availableonline18October2006
Abstract
Thispaperconsidersasingle-item,two-echelon,continuous-reviewinventorymodel.Anumberofretailershavetheirstockreplenishedfromacentralwarehouse.Thewarehouseinturnreplenishesstockfromanexternalsupplier.ThedemandprocessesontheretailersareindependentPoisson.Demandnotmetataretailerislost.TheorderquantityfromeachretaileronthewarehouseandfromthewarehouseonthesuppliertakesthesamefixedvalueQ,anexogenousvariabledeterminedbypackagingandhandlingconstraints.Retailerifollowsa(Q,Ri)controlpolicy.Thewarehouseoperatesan(SQ,(SÀ1)Q)policy,withnon-negativeintegerS.IfthewarehouseisinstockthentheleadtimeforretaileriisthefixedtransportationtimeLifromthewarehousetothatretailer.Otherwiseretailerordersaremet,afteradelay,onafirst-comefirst-servedbasis.Theleadtimeonawarehouseorderisfixed.Twofurtherassumptionsaremade:thateachretailermayonlyhaveoneorderoutstandingatanytimeandthatthetransportationtimefromthewarehousetoaretailerisnotlessthanthewarehouseleadtime.Theperformancemeasuresofinterestaretheaveragetotalstockinthesystemandthefrac-tionofdemandmetintheretailers.Proceduresfordeterminingtheseperformancemeasuresandoptimisingthebehaviourofthesystemaredeveloped.
Ó2006ElsevierB.V.Allrightsreserved.
Keywords:Inventory;Multi-echelon;Lostsales;Batch-ordering;Poissondemand;Non-identicalretailers;Iterativeoptimisation
1.Introduction
Weconsiderasingle-item,two-echelon,inventorysystemwithonewarehousereplenishingthestocksofNindependentretailers.Stockreviewiscontinuousinallinstallations.ThedemandprocessinretaileriisPois-sonwithrateki.Thestockineachretaileriscontrolledusinga(Q,Ri)policy,wherebyanorderforQisplacedonthewarehousewheneverthestocklevelfallstoRi.ThebatchreplenishmentquantityQisfixedforallinstallations,havingbeendeterminedbypackagingandhandlingconstraints.ForexampleQmayrepresentthenumberofunitspackedintoacartonorshrink-wrappedontoapallet–thecartonorpalletrepresentingtheproductunitforpurchase,storage,handlingandshipmentpurposes.
*Correspondingauthor.Tel.:+44139226475;fax:+441392264460.E-mailaddress:R.M.Hill@ex.ac.uk(R.M.Hill).
0377-2217/$-seefrontmatterÓ2006ElsevierB.V.Allrightsreserved.doi:10.1016/j.ejor.2006.08.017
754R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766
Thewarehouseisnotifiedimmediatelyaretailerorderisplaced.IfstocktomeetanorderfromaretailerisavailableinthewarehousethenthatretailerreceivesthestockafterafixedtransportationtimeLi.Warehousestockiscontrolledusinga(modified)basestockpolicywithabasestockofSQ,wherebyanorderforQisplacedonanoutsidesupplierwheneveranorderforQisreceivedfromaretailer.Theoutsidesupplierisassumedalwaystohavesufficientstock.NotethatitcannotbeoptimalfortheintegerStobenegative.IfS=0thenthewarehouseholdsnostock.DecreasingSbelow0increasesthedelayinretailersreceivingtheirorderswithnocounter-balancingbenefit.WarehouseordersalwaysarriveafterafixedleadtimeLw.
Unsatisfieddemandataretailerislost.Technicallythismaymeaneitherthatademandislosttothesystem(e.g.losttoacompetitor)orthatitisexpedited(satisfied,butbysomemeansexternaltothenormalreplenishmentsystem).Ordersonthewarehousearemetonafirst-comefirst-servedbasis.Weconsiderthesteadystatebehaviourofthissystem.Theobjectivesaretodetermine,foracontrolpolicyspecifiedby(S,R)=(S,R1,...,RN):themeanstockinthewarehouse;themeanstockintransitbetweentheware-houseandtheretailers;themeanstockattheretailers;theservicelevel(fractionofdemandmet)ateachretailer.
Weassumethroughoutthat06Ri Ifthewarehouseisinstockthentheleadtimeonaretailerorderisthetransportationtimebutiftheware-houseisoutofstockthentherewillbeadelayaddedtothistransportationtime.Theleadtimeexperiencedbyaretailerismadeupofthefixedtransportationtimeplusthisaddeddelay.Theincidenceoflostsalesattheretailerduringthisleadtimewill,inturn,dependonthedistributionoftheleadtime. R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766755 1.1.Briefliteraturereview Sincethelate1950sresearchershaveworkedonmathematicalmodelsforsupplychains.Thepublishedresearchonthemodellingoftwo-echelonchainswithcaptive(backordered)demandatthelowestechelonisfairlyconsiderableandsowewillconcentratehereonwhatismostrelevant.Inafamouspaper,ClarkandScarf(1960)setoutamethodologyforunderstandingthebehaviourofandcontrolofmulti-levelchainsbyconcentratingon‘echelonstocks’.Theechelonstockassociatedwithaparticularlevelina(divergent)sup-plychainisthestockavailableatthatlevelplusanystockheldatanylowerlevelplusanystockintransitbetweenthoseinstallations.Thereisastreamofresearchwhichdevelopstheseideas,butthesewillnotbecon-sideredfurtherhere. Inanotherwell-knownpaper,Sherbrooke(1968)describedamethodologyformanagingatwo-levelsystemforanitem.Thisissetinthecontextofrecoverable(repairable)itemsbuttheprinciplesapplyequallytosolelyconsumableitems.Forconvenience,inwhatfollowsthehigherechelonisassumedtobeawarehouseandthelowerechelonisassumedtoconsistofanumberofindependentretailers,alldependentonthewarehouseforreplenishmentstock.Ifend-customer(customer)demandwereasimplePoissonprocessandtheretailersweretooperatebasestockcontrolpoliciesthenthewarehousewouldobservePoissondemandandcouldbeana-lysedasaqueue.Itwouldthenbepossibletodeterminetheexplicitdistributionoftheextradelaywhicharetailerorderwouldexperience(see,forexample,Kruse,1980).However,manyauthors,fromSherbrookeonwards,usethemeanextradelaytodeterminethedistributionofthetotalleadtimeobservedbyaretailer.SvoronosandZipkin(1988)obtainmuchbetterresultsbyfittingagammadistributiontothefullleadtime.Aswellasofferingabetterfitthisfacilitatesthecomputationofthedistributionofretailerdemandduringleadtimebecausethelatterthentakesanegativebinomialform.WhenretailersfollowabatchorderingpolicythenthearrivalprocessatthewarehouseisErlangratherthanPoisson.However,itisconvenienttocontinuetomodeltheorderarrivalprocessasPoisson,asisdonein,forexampleMoinzadehandLee(1986)(seealsoLeeandMoinzadeh,1987).Albin(1982)carriedoutaninvestigationtodetermineunderwhatcircumstancesaPoissonprocessprovidesanacceptablemodelforwhatisactuallythesuperpositionofdifferentprocessesandconcludedthatitprovidedagoodfitunderawiderangeofconditions.Axsa¨ter(1990,1993,1998),employedadifferentlineofapproach.Hefollowedthepathofaunitofdemandthroughthesystemandusedthisapproachtodeterminethemeancostsassociatedwiththatunitofdemandandhencethetotalsteadystateaveragesystemcost.Thisapproachenabledtheexactcostofbatchorderingpoliciestobedetermined(seealsoForsberg,1996). Acommonassumptionwhichrunsthroughalloftheaboveresearchisthatend-customerdemandiscap-tive.Thisassumptionisapplicablewhenend-customerordersaredocumentedinsomeway.Insuchacase,whenthereisastockoutatthelowestlevelofthechain,itispossibletoretaintheordersinthesystemuntilmorestockisavailable.Onereasonforthewidespreaduseofthisassumptionisthatmuchearlyworkcon-centratedon‘manufacturing’supplychainsinwhichtheend-customerwasitselfanorganisationoranoper-atingunit.Asecondandpossiblymoresignificantreasonisthatmodelswhichassumecaptivedemandarefrequentlytechnicallyeasiertoanalysethanmodelswhichassumenon-captivedemand. Wheredemandisnon-captivetheusualassumptionisthatdemandswhichoccurwhilethelowestlevelofthechainisoutofstockare‘lost’inthesensethattheyarenotmetthroughthenormalchannelsofthesupplychainandarethereforelostasfarasthecontrolsystemisconcerned.Inpracticethismaymeanthatthedemandistransferredtoanother(substitute)productortoanotherstoreoritmaymeanthatthedemandis‘expedited’(metbutbysomemeansoutsidetheusualcontrolsystem).Modelswhichassumethatdemandisnon-captiveareparticularlyappropriatefor‘distribution’supplychains,highstreetretailingandalsoforcontrollingstocksofimportantspareparts,wheretheprospectofmachinedown-timemaymakeitinfeasibletowaitforaunittoarrivethroughthenormalstockreplenishmentchannels. Weareawareofonlytwopaperswhichaddressthesameproblemsbutwithnon-captive(lost)customerdemand.NahmiasandSmith(1994)consideraperiodicreviewmodelwithpartiallostsalesbut,inordertomaketheirmodeltractable,theyassumeinstantaneousdeliveriesfromthewarehousetotheretailers,anassumptionwhichdoesnotgenerallyhold.AnderssonandMelchiors(2001)deriveanapproximatesolutionforthecasewherecustomerdemandisaPoissonprocess,demandduringastockoutislost,retailersoperate(oneforone)basestockcontrolpoliciesandthewarehouseleadtimeisfixed.Theyadjustthearrivalrateof 756R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766 retailerorderstoallowfortheeffectofstockoutsattheretailersandtheyestimateandthenusethemeanextradelayatthewarehouse. Wefollowtheapproach,commonintheliterature,ofapproximatingthearrivalofordersatthewarehousefromtheretailersasaPoissonprocess.Simulationsuggeststhatthisapproximationgivesexcellentresultsindescribingthebehaviourofthesystem.Thekeyfeatureusedisthefinitestatespaceatthewarehouse.Aniter-ativeschemefordeterminingthesteadystatebehaviourisproposed.TheanalysisissetoutinSection2.InSection3arangeofnumericalexamplesareintroducedandusedtoexplorehowthebehaviourofthesystemdependsontheparameters,tocomparetheanalyticalresultswithsimulatedresultsandtocomparethebehav-iouroflostsalesmodelswiththatofacorrespondingbackordermodel.SomeconcludingcommentsaremadeinSection4.Thekeycontributionofthispaperliesinextendingtheanalysisoftwo-levelsupplychainstothelostsalescontext.2.Analysis TheleadtimeexperiencedbyanorderfromretaileriisLi+Ti,whereLiisthefixedtransportationtimefromthewarehousetoretaileriandTiistheaddeddelaywhichwillresultifthewarehouseisoutofstockwhentheorderisplacedandhastowaitformorestocktoarrivebeforeretaileri’sordercanbeshipped.ThemeandelayisE[Ti].LetXibethedemandduringleadtimeatretaileriandletUibethesaleslostduringleadtimeatthatretailer,withmeanwi=E[Ui].Theordercycletimeataretailerrunsfromtheplacementofoneordertotheplacementofthenextorder.ThemeandemandinanordercycleisthemeansalesQplusthemeanlostsalesduringtheorderleadtimewi.Theservicelevel(fractionofdemandmet)atretaileriisQ/(Q+wi)(thesalesperordercycledividedbythemeandemandperordercycle).Themeanordercycletimeforretaileriis(Q+wi)/ki(themeandemandperordercycledividedbythemeandemandperunittime).WheneveraretailerorderistransmittedtothewarehousethewarehouseimmediatelyplacesanequivalentorderonthesupplierandthisarrivesaftertimeLw.Duringthistimetheretailerorderisbeing‘processed’bythewarehouseandthe‘processing’timeisLw.Onemightthinkofthisorderonthesupplierashavingthelabeliattachedtoitanditisbeing‘processed’whileitisintransitbetweenthesupplierandthewarehouse.Theprobabilitythat,atanarbitrarypointintime,retaileri’sorderisbeingprocessedbythewarehouseis pi¼kiLw=ðQþwiÞ; ð1Þ thisbeingtheprocessingtimedividedbythemeanordercycletimeforretaileri.Thekeypointisthatthewarehousecannotreceiveasecondorderfromretaileriwhileretaileri’sfirstorderisbeing‘processed’,be-causetheretailerwillnotyethavereceivedthefirstorder. ItisassumedthatQPkiLiandhenceQPkiLw(foralli),withtheconsequencethatpiisalwaysavalidprobability.ThisassumptionisconsistentwiththeassumptionthatQ>Riandisvalidunderthesamecircumstances. Thenumberofretailerordersbeingprocessedatanarbitrarypointintime(whichisthesameasthenum-bernofwarehouseordersoutstandingonthesupplier),specifiesthestateofthewarehousesystemandhastheprobabilityfunctionP(n).Ifalltheretailerswereidenticalthenthisdistributionwould,droppingthesuffixi,bebinomial(N,p).Whentheretailersarenon-identicalaverysimpleandefficientschemeforcomputingthevaluesofP(n)isasfollows.Labeltheretailers1toNandleta(u,v)betheprobabilitythatthefirstvretailershaveuordersbeingprocessedatthewarehouse,sothatP(n)=a(n,N).Startingwitha(0,0)=1anda(u,v)=0otherwise,valuesarecomputedrecursively,forv=1,...,N,using aðu;vÞ¼pvaðuÀ1;vÀ1Þþð1ÀpvÞaðu;vÀ1Þ; u¼0;...;v: ð2Þ Theprobabilitythatthewarehouseisinstaten(06n6NÀ1)whenanorderarrivesfromretaileriisPi(n)=a(n,NÀ1)whereretaileriisgiventhelabelN.WemayalternativelydefinePi(n)astheprobabilitythatthewarehouseisinstatengiventhatitisnotcurrentlyprocessinganorderfromretaileri. Theprocedure(tobedescribedlater)willinvolvecomputingP(n),computingPi(n)foragivenretaileri,updatingthemeasurepi,re-computingP(n)usingthisnewmeasureandthenrepeatingtheprocessforthenextretailer.TodothisefficientlyweneedtobeabletocomputePi(n)fromP(n).Thefollowingisasimplemeansofdoingthis: R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766757 PiðNÀ1Þ¼PðNÞ=pi; PiðnÞ¼fPðnþ1ÞÀð1ÀpiÞPiðnþ1Þg=pi; n¼NÀ2;...;0: ð3aÞð3bÞ IfS=Nthenallretailerordersaremetfromwarehousestock,theleadtimeforretaileriisalwaysLi,thesteadystatebehaviourateachretailercanbedetermineddirectlyandhencesocanthesteadystatebehaviouratthewarehouse.IfS=0thenallretailerordersaresatisfiedexworks(i.e.directfromthesup-plierbuttransshippedthroughthewarehouse),theleadtimeforretaileriisalwaysLi+Lw,andonceagainthesteadystatebehaviourateachretailerandatthewarehousecanbedetermineddirectly.Wenowassumethat0 fið0Þ¼and fiðtÞ¼ NÀ1Xn¼SSÀ1Xn¼0 PiðnÞð4aÞ nÀSSÀ1 n!tLwÀt PiðnÞ; LwðnÀSÞ!ðSÀ1Þ!LwLw 0 retaileri.Thesecondcomponentgivesthedensityfunctionforthedelaytandcomesfromthebeta-distributedtimettothearrivalofthe(nÀS+1)thorderfromthesuppliergiventhattherearenorderscurrentlyout-standingonthesuppliermultipliedbytheprobabilitythattherearenordersoutstandingatthewarehouse(noneofwhichrelatetoretaileri),summedovern. ThemeanofTi(notingthatthemeanofthebetadistributionisLw(nÀS+1)/(n+1))is E½Ti¼Lw NÀ1Xn¼S PiðnÞ ðnÀSþ1Þ : ðnþ1Þ ð5Þ Notethatwecanrecoverfrom(5)thecorrectexpressionsE[Ti]=LwwhenS=0(becausePi(n)isonlydefinedovern=0,...,NÀ1)andE[Ti]=0whenS=N(followingtheconventionthatasummationis0ifitistakenovertheemptyset). TheprobabilityfunctionfordemandZiinretaileriduringthisperiodofdelay(whenS>0)isgivenby PZiðzÞ¼dðzÞfið0Þþ Z 0Lw nÀSSÀ1zNÀ1 ðkitÞeÀkitXn!tLwÀt PiðnÞdt; LwðnÀSÞ!ðSÀ1Þ!LwLwz!n¼S ð6Þ whered(z)=1ifz=0andd(z)=0otherwise.Theintegrandistheprobabilitythatdemandiszgiventhatthe delayistmultipliedbythedensityfunctionforthedelayt. Inturnmakingthesubstitutions=t/Lw,expandingtheexponentialasapowerseries,re-arrangingtheresultandintegratingoutthebetadensityfunctionweobtain 758R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766 PZiðzÞ¼dðzÞfið0Þþ ¼dðzÞfið0Þþ Z 0 1 zNÀ1 ðkiLwsÞeÀkiLwsXn!SÀ1 snÀSð1ÀsÞds;PiðnÞ ðnÀSÞ!ðSÀ1Þ!z!n¼S z1kNÀ1 ðkiLwsÞXðÀkiLwsÞXn!SÀ1 snÀSð1ÀsÞds;PiðnÞ ðnÀSÞ!ðSÀ1Þ!z!k!n¼Sk¼0 ð7aÞð7bÞ Z 0 1 zNÀ1k1 ðkiLwÞXPiðnÞn!XðÀkiLwÞðzþkþnÀSÞ! ¼dðzÞfið0Þþ ðnÀSÞ!k¼0k!ðzþkþnÞ!z!n¼S Z1 ðzþkþnÞ!szþkþnÀSð1ÀsÞSÀ1 Âds; ðzþkþnÀSÞ!ðSÀ1Þ!0 ð7cÞð7dÞ ¼dðzÞ¼dðzÞ where SÀ1Xn¼0SÀ1Xn¼0 zNÀ1k1XðkiLwÞXn!ðÀkiLwÞðzþkþnÀSÞ! ;PiðnÞþPiðnÞ ðnÀSÞ!ðzþkþnÞ!z!k!n¼Sk¼0 PiðnÞþ NÀ1Xn¼S PiðnÞviðn;zÞ;z¼0;1;...;RiÀ1;ð7eÞ zk1XðkiLwÞn!ðÀkiLwÞðzþkþnÀSÞ! :viðn;zÞ¼ ðzþkþnÞ!z!ðnÀSÞ!k¼0k! ð8Þ Notethat(7d)givesthecorrectPoisson(kiLw)distributionforZiwhenS=0.IfS>0thentheconvergence ofthesummationoverkin(7d)isguaranteed,because(z+k+nÀS)!/(z+k+n)!<1,andwillbemuchfas-terthantheconvergenceofexp(ÀkiLw).ForgivenvaluesofSandRitherequired(NÀS)Rivaluesofvi(n,z)donotchangeandthereforeonlyneedtobecomputedonce. DemandYiduringLihasaPoisson(kiLi)distribution.DemandXiduringtheleadtimeforretaileriisgivenbyXi=Yi+ZiandsotheprobabilitydistributionofXi,PXiðxÞ,isreadilyobtained,forx=0,...,RiÀ1,byconvolvingthedistributionsofYiandZi.Fromthiswecancomputetheexpectedlostsalesduringleadtimeforretaileri,wi,by RiÀ111XXX wi¼ðxÀRiÞPXiðxÞ¼ðxÀRiÞPXiðxÞÀðxÀRiÞPXiðxÞ; x¼Ri x¼0 RiÀ1Xx¼0 x¼0 ð8aÞð8bÞð8cÞð8dÞ ¼E½XiÀRiþðRiÀxÞPXiðxÞ; RiÀ1Xx¼0 ¼E½YiþE½ZiÀRiþðRiÀxÞPXiðxÞ;ðRiÀxÞPXiðxÞ: ¼kifLiþE½TigÀRiþ RiÀ1Xx¼0 GivenwjforalltheotherretailerswecanuseEqs.(1)throughto(8)todeterminewi.Onceallthewiareknownsoisthefullstateofthesystem.Thevaluesofwiarecomputediteratively.Startingwithretailer1andsettingwj=0foralltheotherretailerswedeterminewi.Theprocessisrepeatedforeachoftheotherretailersinturn,alwaysusingtheupdatedvaluesforalltheotherwj.Theiterationcyclesroundalltheretailersuntilnosignificantchangeisnotedinanyofthewi.Theaveragestockinretaileriis QðQþ1Þ ÀkiðLiþE½TiÞþwiRiþð9aÞ Qþwi2 R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766759 or QQþwi ( ) RiÀ1XðQþ1ÞþðRiÀxÞPXiðxÞ2x¼0 ð9bÞ (seeAppendixA). Theaveragestockinthewarehouseis SÀ1XQðSÀnÞPðnÞ:n¼0 ð10Þ TheaveragestockintransitbetweenthewarehouseandretaileriiskiLiQ : Qþwi From(9)–(11)wecancomputethemeantotalstockinthesystemorinanypartofit.2.1.Algorithm Wenowconstructaniterativeschemefordeterminingthesteadystatebehaviourofthesystem. Step1.InputS,Q,N,Lwand(ki,Li,Ri)fori=1,...,N. IfS=0thencomputethePoisson(ki(Lw+Li))probabilitiesPXiðxÞforx=0,...,RiÀ1,i=1,...,NandgotoStep3. ComputethePoisson(kiLi)probabilitiesPYiðyÞfory=0,...,RiÀ1,i=1,...,N.SetPXiðxÞ¼PYiðxÞforx=0,...,RiÀ1,i=1,...,N.Setwi=0fori=1,...,N. Computepi,fori=1,...,N,using(1).ComputeP(n),forn=0,...,N,using(2).Seti=1. Setcount=0. Setepstoaverysmallrealnumber. Step2.SetELS=wi. ComputePi(n)forn=0,...,NÀ1,using(3)andE[Ti]using(5).ComputePZiðzÞforz=0,...,RiÀ1,i=0,...,n,using(7).ComputePXiðxÞfromPYiðxÞandPZiðzÞ,forx=0,...,RiÀ1.Computewifrom(7). IfjwiÀELSj Step3.Computetheaveragesystemstockusing(8)–(10)andtheservicelevelforretaileriasQ/(Q+wi).2.2.Convergence TheiterationusedisaformofPicarditeration(see,forexample,Pizer,1975,Sections3.3.5and3.5.1)andtakestheform þ1nnwn¼g1ðwn11;w2;...;wNÞ;þ1nþ1nwn¼g2ðw1;wn22;...;wNÞ;þ1þ1nþ1nþ1nwn¼gNðwnN1;w2;...;wNÀ1;wNÞ; ð11Þ ð12aÞð12bÞð12cÞ 760R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766 startingfrom 00 w0¼ðw01;w2;...;wNÞ¼ð0;0;...;0Þ; ð12dÞ wheregi(.)encapsulatesthefunctionalwaywherebywiiscomputedfromtheotherwj(notingthatwniisnot nþ1 useddirectlyincomputingwi). EachgiisalinearsumofPi(0),...,Pi(NÀ1).EachPi(n)is,inturn,alinearfunctionofpjforsomej5i(regardingtheotherpkasfixed).Thereforegiislinearinpjandsotakestheformgi=a+bpjwithastrictlypositiveb,sinceincreasingpjincreaseswj.From(1)pjisdecreasinginwjandisaconvexfunctionofwj.There-forewi(andhencegi)isastrictlydecreasingconvexfunctionofeachwj(j5i)takenoneatatime.Convergenceforthetwo-retailerproblemcaneasilybedemonstrateddiagrammatically.2.3.Optimisation Thespecificconditionsimposedonourmodelmeanthat06Ri Foranycontrolparametersetdefinedby(S,R)wecandeterminethebehaviourofthesystemandhencethe‘cost’(averagetotalcostperunittime).Asimpleiterativeschemeforfindingtheoptimalpolicyoperatesasfollows.ThevalueofSissetto0.TheRiaresettotheirinitialvaluesof0.ThevalueofR1whichgivesaminimumcostsubjecttoSandalltheotherRjbeingfixed,isdeterminedandthisvalueforR1isthenretained.ThisprocessisrepeatedforR2,R3,..andbacktoR1.WecyclethroughthecompletesetofRvaluesuntilthereisnofurtherchange.Inourexperimentsthistypicallyinvolved3or4completecyclesthroughtheR’s.WethenincreaseSby1andrepeattheprocess.ThiscarriesonforS=0,...,N.Thelowestcostsolutionobtainedgivestheoptimalpolicy.Thisisclearlygreatlymoreefficientthanthecompleteevaluationofallpossiblepol-icycombinations.ThetimesinvolvedarenegligibleforaC++programrunningonaPC.Theprocesscanbefurtherspeededupifweassumethat,givenSandtheotherRj,thecostisconvexinRiandalsothattheopti-malcostgivenSisconvexinS,butwehavenotbeenabletoshowthateitheroftheseconvexitypropertieshold.Althoughwefoundthatthisiterativeschemegavetheoptimalpolicy(foundbycompleteenumeration)ithasnotbeenpossibletoshowthatsuchaniterativeschemeconvergestoauniqueoptimum–indeedwebelievethatthispropertymaynotalwayshold.Forthisreasonwedonotconsideroptimisationfurther.3.Numericalexamples ThealgorithmdevelopedinSection2wasappliedtoatestsetofproblems.Weconsideredidenticalretail-ers,partlybecausewewishedtocomparetheresultswiththoseofanequivalentidentical-retailerbackordermodel.Themainobjectiveincarryingoutnumericalworkwastoassessthequalityoftheapproximation(basedontheassumptionthatretailerordersarriveatthewarehouseasaPoissonstreamandhencethatthearrivaltimesofordersfromthesupplierfollowabetadistribution).Wefoundthatexamplesinvolvingnon-identicalretailersgaveverysimilarresultsinthesenseofverifyingthelevelofaccuracyofthemodel.Thebaseproblemhasthefollowingparameters:N=10,Q=6,S=4,R=2,k=1,Lw=1,Lr=2,whereRandkarethecommonreorderlevelanddemandraterespectivelyintheretailers.Inordertodeterminetheeffectoftheparametersthebaseproblemismodifiedbyvaryingeachparameterinturn,asfollows:N=5,10,20;Q=4,6,8;S=2,4,8;R=1,2,4;k=0.5,1,2;Lw=0.5,1,2;Lr=1,2,4.ForeachproblemthesteadystatebehaviourisdeterminedusingthemethodologydescribedinSection2.Theresultsaresumma-risedinTable1. R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766 Table1 ApproximateresultsandhowthesedependonthemodelparametersN51020101010101010101010101010101010101010 Q666468666666666666666 S444444248444444444444 R222222222124222222222 k1.01.01.01.01.01.01.01.01.01.01.01.00.51.02.01.01.01.01.01.01.0 Lw1.01.01.01.01.01.01.01.01.01.01.01.01.01.01.00.51.02.01.01.01.0 Lr222222222222222222124 iters345543542443346346344 r-stock3.7073.7013.6442.6603.7014.7203.5983.7013.7073.0543.7015.4964.5253.7012.6443.7073.7013.6124.5163.7012.669 w-stock19.4114.917.487.4614.9122.674.4314.9138.8315.6514.9114.2319.0914.919.8919.4214.917.2014.2814.9116.63 i-stock9.1718.3336.3617.5518.3318.7318.0518.3318.3416.8118.3319.759.8318.3329.3418.3418.3318.089.8218.3329.58 t-stock47.1270.25116.7351.6170.2588.5958.4670.2594.2462.9970.2588.9474.1770.2565.6674.8370.2561.4069.2670.2572.91 761 s-level0.91720.91650.90900.87740.91650.93640.90250.91650.91720.84030.91650.98730.98300.91650.73340.91720.91650.90380.98250.91650.7395 Key:N,Q,S,R,k,LwandLrareparametersdefinedinthetext.Inaddition:itersisthenumberofiterationsrequired,r-stockistheaveragestockperretailer,w-stockistheaveragewarehousestock,i-stockistheaveragestockintransitbetweenwarehouseandretailer,t-stockistheaveragetotalstockands-levelisthefractionofdemandmet.Thefigureshighlightedinbolddrawattentiontotheparametervaluesundergoingchangeinthatparticularsectionofthetable. Thevalueofepsusedtospecifyconvergenceis0.000001.Theprocedurerequiresatleast2iterations.Thenumberofiterationsgenerallyincreasesastheserviceleveldecreases.Thenumberofiterationsrequiredfortheproblemsetissmallandtheruntime(foraprogramwritteninC++andrunningonaPC)isnegligible. If,forthebaseproblem,thewarehousewerealwaysinstockthenwecanshowdirectlythattheservicelevelwouldbe0.9172andtheaveragestockataretailerwouldbe3.707.Theresultssuggestthatthisisclosetobeingthecase.AsNisreducedto5,Sincreasedto8,kreducedto0.5orLwreducedto0.5thenweeffectivelyreachthisstate.Thedependenceofthesystembehaviourontheproblemparametersfollowsthedirectionswhichintuitionwouldsuggest.3.1.Simulation Resultswerealsoobtainedusingacontinuoustimesimulationmodel(writteninC++).Foreachparametersetthesimulationwasrun100times,withdifferentseedsandwitharun-inperiodof10,000followedbyarecordingperiodof100,000.Foreachrunthemeanstockvaluesandtheservicelevelwererecorded.Thesewerethenusedtoproduceoverallmeanvaluesand,assumingnormality,95%confidenceintervalsfortheoverallmeans.ThesimulationresultsaresummarisedalongsidethoseoftheanalyticalmodelinTable2.Theyclearlydemonstratethequalityoftheapproximationusedindeterminingtheanalyticalresults.3.2.Equivalentbackordermodel Inadditiontocomparingtheanalyticalresultswithsimulatedresultstheywerecomparedwithresultsforthe‘nextbestavailablemodel’.Inthiscasethenextbestapproachisassumedtobeusing 762R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766 Table2 ComparisonbetweenanalyticalandsimulatedvaluesN10520101010101010101010101010 Q666486666666666 S444442844444444 R222222214222222 k1.01.01.01.01.01.01.01.01.00.52.01.01.01.01.0 Lw1.01.01.01.01.01.01.01.01.01.01.00.52.01.01.0 Lr222222222222214 r-stock3.7013.7023.7073.7073.6443.6442.6602.6604.7204.7193.5983.5983.7073.7083.0543.0555.4965.4974.5254.5262.6442.6453.7073.7073.6123.6124.5164.5172.6702.670 CI±0.001±0.002±0.001±0.001±0.002±0.001±0.001±0.001±0.002±0.002±0.001±0.001±0.001±0.001±0.001 w-stock14.9114.9119.4119.427.487.497.467.4722.6722.674.434.4438.8338.8315.6515.6514.2314.2319.0919.099.899.8719.4219.427.207.2214.2814.2816.6316.64 CI±0.01±0.00±0.01±0.00±0.01±0.01±0.01±0.01±0.01±0.00±0.01±0.00±0.01±0.01±0.00 i-stock18.3318.329.179.1736.3636.3617.5517.5418.7318.7218.0518.0418.3418.3416.8116.8019.7519.749.839.8329.3429.3418.3418.3418.0818.079.829.8229.5829.57 CI±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01 t-stock70.2570.2647.1247.12116.72116.7351.6151.6188.5988.5958.4658.4694.2494.2462.9963.0088.9488.9474.1774.1865.6665.6674.8374.8361.4061.4169.2669.2772.9172.91 CI±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01±0.01 s-level0.91650.91650.91720.91720.90900.90900.87740.87750.93640.93650.90250.90240.91720.91720.84030.84040.98730.98730.98300.98300.73340.73340.91720.91720.90380.90380.98250.98250.73950.7395 CI±0.0002±0.0003±0.0002±0.0003±0.0002±0.0002±0.0002±0.0003±0.0001±0.0001±0.0002±0.0002±0.0003±0.0001±0.0003 Key:AsTable1except:thesecondrowineachpairinggivestheaveragevaluesobtainedfromthesimulationrunsforthatparameterset.give;theCIcolumnentryspecifiesthe95%confidenceintervalforthesimulatedmeanvalueinthepreviouscolumn. theexactsteadystatebehaviourforthecorrespondingidentical-retailerbackordermodel.TheprocedurefortacklingthebackordermodelisdescribedinAxsa¨ter(1993).BecauseourbackorderapproachandthemeasuresweareinterestedindifferslightlyfromAxsa¨ter’swegivesomedetailsinAppendixB.Theresultsofthebackordermodel,setalongsidethoseofthelostsalesmodel,aresummarisedinTable3.Theaveragestockintransitbetweenwarehouseandretailerisnotofinterestinthebackordermodelbecausealldemandismetandsothisrepresentsafixedoverheadwhichdoesnotdependonthecontrolpolicy.Itis,however,ofinterestinthelostsalesmodelbecauseitisdirectlyproportionaltotheservicelevel.Theservicelevelforthebackordermodelisdefinedasthefractionofretailerdemandmetimmediatelyfromstock.Asexpectedthetwomodelsdifferleastathighservicelevels(R=4,k=0.5orLr=1)andmostatlowservicelevels(k=2orLr=4).Ifstockoutsrarelyoccurthenthedistinctionbetweenlostsalesandbackordersisnotsignificant.Thelostsalesmodelgivesahigherservicelevelthanthebackordermodel,becausethelossofsalesdecreasestheeffectivedemandrateandsolimitsfurtherlossofsales. R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766 Table3 ComparisonbetweenlostsalesandbackordermodelsN10520101010101010101010101010 Q666486666666666 S444442844444444 R222222214222222 k1.01.01.01.01.01.01.01.01.00.52.01.01.01.01.0 Lw1.01.01.01.01.01.01.01.01.01.01.00.52.01.01.0 Lr222222222222214 ModelLSBOLSBOLSBOLSBOLSBOLSBOLSBOLSBOLSBOLSBOLSBOLSBOLSBOLSBOLSBO r-stock3.7013.5443.7073.5543.6443.4522.6602.5454.7204.5373.5983.3893.7073.5543.0542.6355.4965.4944.5254.5042.6441.8633.7073.5533.6123.3904.5164.4942.6691.969 w-stock14.9114.1119.4119.007.486.317.466.4222.6722.044.433.8838.8338.0015.6514.1114.2314.1119.0919.009.895.8919.4219.007.205.8914.2814.1116.6314.11 i-stock18.3320.009.1710.0036.3640.0017.5520.0018.7320.0018.0520.0018.3420.0016.8120.0019.7520.009.8310.0029.3440.0018.3420.0018.0820.009.8210.0029.5840.00 t-stock70.2569.5547.1246.77116.73115.3451.6151.8788.5987.4158.4657.7694.2493.5462.9960.4688.9489.0574.1774.0565.6664.5274.8374.5461.4059.7969.2669.0572.9173.80 763 s-level0.91650.90870.91720.90980.90900.89740.87740.85970.93640.93200.90250.88920.91720.90980.84030.80940.96730.98720.98300.98270.73340.62710.91720.90980.90380.88830.98250.98210.73950.6524 Key:AsTable1except:thefirstrowineachpairinggivesthevaluesforthelostsales(LS)analyticalmodelandthesecondrowgivesthevaluesfortheequivalentbackorder(BO)model;theservicelevelforthebackordermodelisthefractionofdemandmetimmediatelyfromstock. 4.Discussion Multi-echelonmodelswithlostsalesduringastockoutatthelowestechelonremainsanunder-researchedtopic,whichissurprisingbecausethesearethesupplychainmodelswhicharemostlikelytobeofinteresttoincreasinglylargeandpowerfulhighstreetretailcompaniesandthosesparepartsupplychainsinwhichthedemandforimportantsparesduringastockoutisexpeditedratherthanbacklogged.Thispaperhaspresentedameansofstudyingthesteadystatebehaviourofamulti-levelbatchorderingmodelwhich,onthebasisofsimulationresults,providesveryaccurateanswers.Wehavemadeanumberofassumptionsbutmostoftheseareconsistentwithourassumedcontextofthestockingofslow-moving,highvaluecapitalgoodsintheretailsectororthestockingofslow-movingbuttechnicallyvitalspareparts.Theoneassumption,whichisrequiredbytheanalysisaspresented,isthatthewarehouseleadtimecannotexceedaretailertransportationtime.Per-hapsthemostimportantnextstepistoexplorehowtheanalysiscanbemodifiedtoenablethisassumptiontoberelaxed. 764R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766 Acknowledgements ThisresearchwascarriedoutwhileDr.SeifbarghywasvisitingtheUniversityofExeteronaBritishCoun-cilScholarship. Theauthorswouldliketothankthethreeanonymousreviewersforanumberofveryhelpfulsuggestions.AppendixA.Averagestockexpressions Inthisappendixanexpressionfortheaveragestocklevelforalostsalescontinuousreview(Q,R)modelisderived.DemandarrivesasaPoissonstreamwithfixedratek,successiveleadtimesareindependentandiden-ticallydistributed,withmeanE[T]anddensityfunctionf(t),andQ>R,sothatatmostonereplenishmentordermaybeoutstandingatanytime.ThederivationforafixedleadtimeisgiveninHadleyandWhitin(1963,p.200).Thecorrespondingderivationforstochasticleadtimesmaywellexistintheliteraturebutwehavenotcomeacrossitandsoweincludethisderivationforcompleteness.Letp(i)betheprobabilitythatdemandinleadtimeisi,givenby pðiÞ¼ Z 01 ðktÞeÀkt fðtÞdt;i!i fori¼0;1;...ðA:1Þ (Forexample,ifleadtimesaregammadistributedthendemandinleadtimefollowsageneralisednegativebinomialdistribution.) ThemeantimebetweensuccessivedemandsgeneratedbyaPoissonprocesswithratekis1/k.Ifwehavexunitsinstockthenthemeantime-weightedstockholdinguntilstockisdepletedisx/k+(xÀ1)/k+ÁÁÁ+1/k=x(x+1)/(2k).ThestocklevelimmediatelyafterthearrivalofanorderisRÀi+Q,fori=0,...,RÀ1,withprobabilityp(i)andisQwithprobabilityp(R)+p(R+1)+...Toworkoutthemeantime-weightedstockholdingduringanordercyclewetakethetime-weightedstockholdingtodepletionbasedonthestocklevelatthebeginningofanordercycle(justafterthearrivalofanorder)andsubtractfromthisthetime-weightedstockholdingtodepletionbasedonthestocklevelattheendofthatordercycle(justbeforethearrivalofthenextorder).Thisgives RÀ11RÀ1XXXðRÀiþQÞðRÀiþQþ1ÞQðQþ1ÞðRÀiÞðRÀiþ1Þ pðiÞþpðiÞÀpðiÞ 2k2k2ki¼Ri¼0i¼0 RÀ11XXððRÀiÞþQÞððRÀiþ1ÞþQÞÀðRÀiÞðRÀiþ1ÞQðQþ1Þ pðiÞþpðiÞ;¼ 2k2ki¼Ri¼0RÀ11XXQðQþ1Þþ2QðRÀiÞQðQþ1ÞpðiÞþpðiÞ;¼ 2k2ki¼Ri¼0RÀ1 QðQþ1ÞQX þ¼ðRÀiÞpðiÞ;2kki¼0 ! 1XQQ1 þþRÀkE½TþðiÀRÞpðiÞ;¼ k22i¼RQQ1 þþRÀkE½Tþw;¼ k22 ðA:2aÞðA:2bÞðA:2cÞðA:2dÞðA:2eÞðA:2fÞ wherewisthemeanlostsalesduringleadtime.Theaveragestocklevelisgivenbythemeantime-weightedstockholdingperordercycledividedbythemeanordercycletime,followingtheresultgenerallyattributedtoRoss(1970)(seealsoTijms,1994).Themeandemandperordercycleisthemeansalesperordercycle,Q,plusthemeanlostsalesperordercycle,w.Themeanordercycletimeistherefore(Q+w)/k.Thisgivestheaveragestocklevelas R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766765 QQ1 þþRÀkE½Tþw: ðQþwÞ22 ðA:3Þ ThetermQ/(Q+w)isthefractionofdemandmetor‘fillrate’andsoisthefractionoftimeforwhichthesystemisinstock.Theaveragestockexpressionmaythereforebethoughtofasthefractionoftimeforwhichthesystemisinstockmultipliedbytheaveragestocklevelwhilethesystemisinstock. Acontinuousreviewpolicyonlymakessense,mathematically,ifwehaveaninfinitelydivisibledemandarrivalprocess.InpracticethismeanseitherthatdemandsarriveasaPoissonstream(althoughdemandsizesmaybedistributed)orthatdemandarrivesasaconstantcontinuousstream(withratek).Iftheapproachaboveisappliedtothecasewheredemandarrivesasaconstantcontinuousstreamthenasimilarderivationgivestheexpression QQ þRÀkE½Tþw; ðQþwÞ2 ðA:4Þ whichisthesameas(A.3)apartfromthefactor1/2,whichispresentbecauseofthediscretenessofthedemandprocessinthatcase. AppendixB.Thecorrespondingexactbackordermodel Inthisappendixweconsidertheexactbackordermodel,basedverymuchonAxsa¨ter(1993),correspondingtothelostsalesmodeldescribedinthetext.TheretailershereareidenticalwithcommonparameterskandLrandcommoncontrolvariableR.Axsa¨ter’sapproachistofollowaunitofstockthroughthesystem.AretailerorderforQisplaced,when-evertheinventorypositionfallstoR,andthewarehouseimmediatelyplacesanorderforQ.Wefollowthestockunitsinthiswarehousebatchorder.Under‘firstcomefirstserved’,thisbatchwillbedispatchedtomeettheSthnextretailerordertobeplaced.Letpi,SbetheprobabilitythattheSthretailerorderistriggeredbytheithnextdemandplacedonthesystem.Axsa¨terdescribesanefficientprocedureforcomputingtherequiredvaluesofpi,S.ThetimettotheithsystemdemandhasanErlang(i,Nk)distribution.Ift PZðzÞ¼dðzÞPð0Þþ Z 0Lw ðkðLwÀtÞÞzeÀkðLwÀtÞ z! ðNÀ1ÞðQÀ1ÞþSQX i¼Si ðkNÞitiÀ1eÀkNt dt;pi;S ðiÀ1Þ! ðB:1aÞ ðNÀ1ÞðQÀ1ÞþSQzÀjXpi;SðkNÞXkzeÀkLwz!ðÀ1ÞjLzwðjþiÀ1Þ!¼dðzÞPð0Þþz!ðiÀ1Þ!j¼0j!ðzÀjÞ!ðkðNÀ1ÞÞjþii¼S ZLwjþi ðkðNÀ1ÞÞtjþiÀ1eÀkðNÀ1Þt dt; ðjþiÀ1Þ!0 iXðNÀ1ÞðQÀ1ÞþSQzjzXðkLwÞeÀkLwNz!ðÀ1ÞðjþiÀ1Þ! ¼dðzÞPð0Þþpi;S NÀ1j¼0j!ðzÀjÞ!ðiÀ1Þ!ðkðNÀ1ÞLwÞjz!i¼S () jXþiÀ1kÀkðNÀ1ÞLw ðkðNÀ1ÞLwÞe Â1À;z¼0;...;RþQÀ1; k!k¼0 ðB:1bÞ ðB:1cÞ whered(z)=1ifz=0andd(z)=0otherwiseand 766R.M.Hilletal./EuropeanJournalofOperationalResearch181(2007)753–766 ðNÀ1ÞðQÀ1ÞþSQX i¼S Pð0Þ¼ ! iÀ1XðkNLwÞjeÀkNLw pi;S: j!j¼0 ðB:2Þ FromtheprobabilitydistributionforX,PX(x),wecandeterminetheaverageretailerstockas minfQÀ1;RþQÀxÀ1gRþQÀ1XNX PXðxÞðRþQÀxÀjÞ: Qx¼0 j¼0 ðB:3Þ Theaveragewarehousestockis ()ðNÀ1ÞðQÀ1ÞþQSjÀkNLwjÀkNLwiiÀ1XXXðkNLwÞeðkNLwÞe ÀkNLwpi;Si: j!j!i¼Sj¼0j¼0 ðB:4Þ Theretailerstockservicelevel(fractionofdemandmetfromstock),determinedbythefractionoftheunits inthearrivingbatchwhichgointostock(ratherthanbeingdispatchedimmediatelytoclearbackorders),is RþQÀ11X PXðxÞminfQ;RþQÀxg: Qx¼0 ðB:5Þ Thisistheclosestbackorderequivalenttothelostsalesservicelevel.TheaveragestockintransitiskNLr. Thedevelopmentin(B1)makestheimplicitassumptionthatN>1andS>0.IfS=0thentheleadtimeoneachretailerorderisLr+Lwandtheanalysisisstraightforward.IfN=1thenP(0)istheprobabilitythatdemandinLwislessthanSQandPZ(z)isd(z)P(0)plustheprobabilitythatdemandinLwisz+QS. IfQislargeenough(inrelationtoleadtimedemand)fortheprobabilityofaretailerplacingasecondorderbeforethefirstorderhasarrivedtobenegligiblethenthebackordermodelmaybehandledinthesamewayasthelostsalesmodeldescribedinthemaintext,althoughtheperformancemeasureswill,ofcourse,bedifferent.References Albin,S.L.,1982.OnPoissonapproximationsforsuperpositionarrivalprocessesinqueues.ManagementScience28,126–137. Andersson,J.,Melchiors,P.,2001.Atwo-echeloninventorymodelwithlostsales.InternationalJournalofProductionEconomics69,307–315.Axsa¨ter,S.,1990.Simplesolutionproceduresforaclassoftwo-echeloninventoryproblems.OperationsResearch38,64–69.Axsa¨ter,S.,1993.Exactandapproximateevaluationofbatchorderingpoliciesfortwo-levelinventorysystems.OperationsResearch41,777–785.Axsa¨ter,S.,1998.Evaluationofinstallationstockbased(R,Q)-policiesfortwo-levelinventorysystemswithPoissondemand.OperationsResearch46,135–145. Clark,A.J.,Scarf,H.,1960.Optimalpoliciesforamulti-echeloninventoryproblem.ManagementScience6,475–490(SeealsoVeinott,A.F.,Jr(Ed.),MathematicalStudiesinManagementScience.TheMacmillanCo.,NewYork,1965,pp.445–460).Forsberg,R.,1996.Exactevaluationof(R,Q)policiesfortwo-levelinventorysystemswithPoissondemand.Hadley,G.,Whitin,T.M.,1963.AnalysisofInventorySystems.Prentice-Hall,EnglewoodCliffs,NJ. Kruse,W.K.,1980.WaitingtimeinanSÀ1,Sinventorysystemwitharbitrarilydistributedleadtimes.OperationsResearch28,348–352.Lee,H.L.,Moinzadeh,K.,1987.Operatingcharacteristicsofatwo-echeloninventorysystemforrepairableandconsumableitemsunderbatchorderingandshipmentpolicy.NavalResearchLogistics34,365–380. Moinzadeh,K.,Lee,H.L.,1986.Batchsizeandstockinglevelsinmulti-echelonrepairablesystems.ManagementScience32,1567–1581.Nahmias,S.,Smith,S.A.,1994.Optimizinginventorylevelsinatwo-echelonretailersystemwithpartiallostsales.ManagementScience40,582–596. Pizer,S.M.,1975.NumericalComputingandMathematicalAnalysis.ScienceResearchAssociates,Henley-on-Thames,UK.Ross,S.M.,1970.AppliedProbabilityModelswithOptimalityApplications.Holden-Day,SanFrancisco,CA. Sherbrooke,C.C.,1968.METRIC:Amulti-echelontechniqueforrecoverableitemcontrol.OperationsResearch16,122–141.Svoronos,A.,Zipkin,P.,1988.Estimatingtheperformanceofmulti-levelinventorysystems.OperationsResearch36,57–72.Tijms,H.C.,1994.StochasticModels:AnAlgorithmicApproach.JohnWiley&Sons,Chichester,UK. 因篇幅问题不能全部显示，请点此查看更多更全内容