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Orderquantitiesforperishableinventorycontrolwithnon-stationarydemandandafillrateconstraintKarinG.J.Pauls-Worma,,,EligiusM.T.Hendrixb,,AlejandroG.AlcobabAbstract:Westudythepracticaldecisionproblemoffreshfoodproductionwithalongproductionleadtimetodecideeveryperiod(e.g.week)howmanyitemstoproduce.Whenabatchisreadyforuse,itsitemshaveafixedshelflife,afterwhichtheitemsbecomewasteinthesensethattheycannotbesoldanymore.Thedemandfor(fresh)foodproductsisuncertainandhighlyfluctuating,mainlycausedbypricepromotionsofretailorganisations.Wefocusoncaseswhereaso-calledcyclefillrateservicelevelrequirementapplies.Weinvestigatethegenerationofaproductionplanthatfixesthetimingandquantityoftheproductionforafinitetimehorizon.Tominimisewaste,oneissuestheoldestitemsfirst,i.e.aFIFOissuingpolicy.Incaseofout-of-stock,salesarelost.WemodelthisdecisionproblemasaStochasticProgramming(SP)model.TheobjectiveofourstudyistofindorderquantitiesfortheSPmodel,thatapproximatelymeetcyclefillrateservicelevelrequirementswhilekeepingoutdatinglow.TofindapproximatesolutionsfortheSPmodel,anMILPmodelisdeveloped.TheMILPmodelisadeterministicapproximationthatgeneratesfeasiblereplenishmentquantitiesinlessthanasecond.Withascenario-basedMINLPapproach,optimalsolutionsaregeneratedforalargesampleofdemandpathsasabenchmarkfortheMILPsolutions.WeshowthattheMILPmodelissuitableforpracticaluseifthesetupcostissuchthatthereplenishmentcyclesintheproductionplanareclosetoorofthesamelengthasthemaximumshelflife.Inthosecases,theexpectedtotalcostsareclosetothecostsoftheoptimalsolutionandtheaveragefillrateisclosetotherequiredone.Keywords:Production-inventoryplanning;Longleadtime;Perishableproduct;Non-stationarystochasticdemand;Fillrate1.IntroductionAproducerofafreshfoodproductwithalongleadtimehastodecideeveryperiod(e.g.week)howmanyitemstoproduce.Multiplebatchesofthesameproductofdifferentagesareinproduction,sotherearemultipleoutstandingorders.Whenabatchisreadyforuse,theitemsgeta‘bestbefore’ora‘useby’date,resultinginafixedshelflifefortheproduct.Generally,producershaveacontractwithretailorganisationsabouttheminimumremainingshelflifeoftheitemsdelivered.Thetimebetween‘readyforuse’andtheminimumremainingshelflifeisthemaximuminternalshelflifetheproducercanusetoorganiseproductionefficiently.Afterthemaximuminternalshelflifeofseveralperiods,theproductbecomeswasteinthesensethatitcannotbesoldanymorewiththeaimedremainingshelflife.Itstillmaybeusedforotherpurposes,soitmayhaveasalvagevalue.Theseproductioncharacteristicscanbefoundine.g.thematurationofcheese,meatfrombreedingtoslaughteringandcropsfromseedtoharvesting.Intheremainderofthispaperwewillusetheshorterterm‘maximumshelflife’todenotethemaximuminternalshelflife.Thedemandfor(fresh)foodproductsisuncertain,sotheproductionquantityisdeterminedbasedonforecasts.Acomplicatingfactoristhatdemandishighlyfluctuating,mainlyduetopricepromotionsoftheretailorganisations,i.e.demandisnon-stationary.Competitioninretailisverystrong,soretailorganisationsarereluctanttoshareinformationabouttheirpromotionalactivities.Becauseofthelongleadtime,plannedpromotionssometimesbecomeonlyknowntothefoodproducerafterproductionoftheitems.However,retailorganisationscontinuouslyworkonimprovingtheirdemandforecasts.Highlyfluctuatingdemandisnotnecessarilyhighlyuncertain.Itislikelythatthefoodproducerhasacontractwiththeretailorganisationaboutthedeliveryoftheproduct,withrespecttoremainingshelflifeandfillrate.Thefillrateindicatesthatapredefinedpercentageofthedemandperreplenishmentcyclehastobefulfilledfromstock,aso-calledcyclefillrate.Accordingtothefoodproducer,demandthatcannotbefulfilledfromstockislost.Thefoodproducerhascontrolovertheissuingoftheitems.Inordertominimisewasteduetooutdating,theoldestitemsareissuedfirst,sothereisafirstin–firstout(FIFO)issuingpolicy.Thefoodproducerwantstomakeaproductionplanminimisingtheexpectedtotalcosts,indicatingwhentoproduceandhowmuch,forafixedtimehorizonofTweeks.Havingalongleadtime,theinventorylevelsatthetimeofdeliveryareunknownatthetimethereplenishmentquantityhastobedetermined.Sothereplenishmentquantityshouldbedecidedonbeforehand,independentlyoftheinventorylevelI.Therefore,weconsideraso-calledstaticuncertaintypolicydenotedby(Yt,Qt),whereYtdenotesinwhichperiodstodeliver,andQtthecorrespondingreplenishmentquantitiestodeliveratthebeginningofperiodt.TheresearchquestioniswhetheritispossibletogenerateaproductionplanforTperiodsusingexistingsolversandforwhichinstancesthesolutionmightbeclosetotheoptimalsolution.Therestofthepaperisorganisedasfollows.InSection2,weexplainhowthepapercontributestoliteraturebydiscussingthemaincharacteristicsoftheproblem.InSection3,theproblemisformulatedasaStochasticProgramming(SP)model.Section4describesthestepstowardsaproductionplanthatweusetoformulateadeterministicMILPmodelthatgeneratesfeasibleproductionplans.Themodeldeterminesthetimingofdeliveries,thereplenishmentcyclelengthandthereplenishmentordeliveryquantity,inordertomakeaproductionplanforleadtimeLperiodsbeforet.ThismodelispresentedinSection5.Section6investigatestheapplicabilityoftheMILPm