Chapter3InventoryModels★deterministicmodel★stochasticmodelInventoryModelsThestudyofinventorymodelsisconcernedwithtwobasicquestions:HowmuchshouldbeorderedeachtimeWhenshouldthereorderingoccurTheobjectiveistominimizetotalvariablecostoveraspecifiedtimeperiod(assumedtobeannualinthefollowingreview).InventoryCostsOrderingcost--salariesandexpensesofprocessinganorder,regardlessoftheorderquantityHoldingcost--usuallyapercentageofthevalueoftheitemassessedforkeepinganitemininventory(includingfinancecosts,insurance,securitycosts,taxes,warehouseoverhead,andotherrelatedvariableexpenses)Backordercost--costsassociatedwithbeingoutofstockwhenanitemisdemanded(includinglostgoodwill)Purchasecost--theactualpriceofthe

itemsOthercostsDeterministicModelsThesimplestinventorymodelsassumedemandandtheotherparametersoftheproblemtobedeterministicandconstant.Thedeterministicmodelscoveredinthischapterare:Economicorderquantity(EOQ)EconomicproductionlotsizeEOQwithplannedshortagesEOQwithquantitydiscountsEconomicOrderQuantity(EOQ)Themostbasicofthedeterministicinventorymodelsistheeconomicorderquantity(EOQ).Thevariablecostsinthismodelareannualholdingcostandannualorderingcost.FortheEOQ,annualholdingandorderingcostsareequal.DeterministicModel★EconomicOrderQuantity,（EOQ）

Asimplemodelrepresentingthemostcommoninventorysituationfacedbymanufacturers,retailers,andwholesalersistheEOQ(EconomicOrderQuantity)model.(Itsometimesisalsoreferredtoastheeconomiclot-sizemodel.)EconomicOrderQuantityAssumptionsA:Annualnumberofitemsdemanded,c:Unitcostofprocuringanitem,T:Timebetweenorders,h:Annualcostperdollarvalueofholdingitemsininventory.K：fixedcostperorderTheorderquantity(Q)toreplenishinventoryarrivesallatoncejustwhendesired,namely,whentheinventoryleveldropsto0.PlannedshortagesarenotallowedTheobjectiveistodeterminewhenandhowmuchtoreplenishinventorysoastominimizethetotalcost.EconomicOrderQuantityTheMathematicalModelT=Q/AOrderpointsQTime(years)InventoryLevelSlope=-A

Figure3-1InventorysystemforthesimpleeconomicorderquantitymodelDeterministicModel

OrderquantityisQ.Numberofannualorderisn，whichequalsA/Q。Costperorderisk,sothatcostofannualOrder.WhenUnitcostofprocuringanitemisc,theannualprocurementcostisACAverageinventorylevelholdingcost

AverageholdingcostAsummption：DeterministicModel

TotalAnnualCost:transformed：DeterministicModelHoldingcostcurve：Orderingcostcurve：Totalcostcurve：DeterministicModelQ0AnnualCostTotalcost:TCOrderingcost:k(A/Q)QOrderingcost:hc(Q/2)DeterministicModelOptimalquantity：Optimalcycletime

：Optimalnumberoforder：ThevalueofQ,sayQ0,thatminimizesTC,isfoundbysettingthefirstderivativetozeroWhentheunitholdingcostincreases,bothQ0andt0decrease(smallerinventorylevels).Asthedemandrateincreases,Q0increases(largerbatches)butt0decreases(morefrequentsetups).Example:Bart’sBarometerBusinessEconomicOrderQuantityModel

Bart'sBarometerBusinessisaretailoutletthatdealsexclusivelywithweatherequipment.Bartistryingtodecideonaninventoryandreorderpolicyforhomebarometers. BarometerscostBart$50eachanddemandisabout5000peryeardistributedfairlyevenlythroughouttheyear.Reorderingcostsare$80perorderandholdingcostsarefiguredat20%ofthecostoftheitem.DeterministicModel

SupposethataMallsells12,000pairofskiseachyear.Forsimplicity,wewillassumethattheskiissoldataconstantratethroughouttheyear.ThenetcostofeachpairtotheMallis$30.Thewholesalesuppliercharges$10foreachdelivery,regardlessofhowmanypairshavebeenordered,anddeliveryalwaysoccursthedayaftertheorderisplaced.Theowner’sonlyworkingcapitalistiedupininventory,andthesefundshavebeenborrowedfromthelocalbankatasimpleannualinterestrateof10%.Inaddition,theownermustpayastatefranchisetaxof5%oftheannualinventoryvalue,andanother5%fortheftinsurance.Allotheroperatingcostsareeitherfixedinnatureordonotdependontheamountofskisordered.Theownerwantstoevaluatethepresentprocedureofordering1,000pairseachmonthandtoestablishabetterinventorypolicythatwillminimizetheannualcostsofdoingbusinessinskis.Atelevisionmanufacturingcompanyproducesitsownspeakers,whichareusedintheproductionofitstelevisionsets.Thetelevisionsetsareassembledonacontinuousproductionlineatarateof8,000permonth,withonespeakerneededperset.Eachtimeabatchisproduced,asetupcostof$12,000isincurred.Theunitproductioncostofasinglespeakeris$10.Theestimatedholdingcostofkeepingaspeakerinstockis$0.30permonth.Theproductquantitytoreplenishinventoryarrivesallatoncejustwhendesired.Theappropriateparametervaluesfromgivendatasarek=12,000,hc=0.30,A=8,000=25,928DeterministicModel

Costofproductsis$5.Reorderingcostsare$10perorderandholdingcostisfiguredat0.1%ofthecostoftheitem.demandisabout100perdayandbackorderingofitemsisnotallowed。theneededmaterialsarrivejust-in-timewhentheinventoryleveldropsto0.DeterministicModel(1)Plannedshortagesnowareallowed.(2)Whenashortageoccurs,theaffectedcustomerswillwaitfortheproducttobecomeavailableagain.(3)Theirbackordersarefilledimmediatelywhentheorderquantityarrivestoreplenishinventory.Model2：TheEOQModelwithPlannedShortagesDeterministicModelS:actualorderquantityQ：fictitiousorderquantityororderquantitythatshortageisprohibitedp：shortagecostperunitshortperunitoftimeshortQ-S:shortageininventoryjustbeforeabatchofQunitsisadded.Model2：TheEOQModelwithPlannedShortagesLet:DeterministicModelSQ-SADeterministicModelaverageinventorylevel

=Totalinventorylevel

/T=（Totalinventorylevelthatshortagedonotoccur+Totalinventorylevelthatshortageisprohibit

）/TDeterministicModel

timeshortagesdonotoccur

:

Tisasfollowedas:DeterministicModelaverageinventorylevel:DeterministicModelaverageshortagelevel

：

Shortagetime:DeterministicModelaverageshortagelevel

：Totalannualcost：DeterministicModelTherearetwodecisionvariables(SandQ)inthismodel,sotheoptimalvalues(S0andQ0)arefoundbysetting==0Therefore,

Atelevisionmanufacturingcompanyproducesitsownspeakers,whichareusedintheproductionofitstelevisionsets.Thetelevisionsetsareassembledonacontinuousproductionlineatarateof8,000permonth,withonespeakerneededperset.Eachtimeabatchisproduced,asetupcostof$12,000isincurred.Theunitproductioncostofasinglespeakeris$10.Theestimatedholdingcostofkeepingaspeakerinstockis$0.30permonth.Shortagecostofeachspeakeris$1.10permonth.Ifshortagesarepermitted.Howmanyspeakertoproduceineachbatch.EconomicProductionLotSizeModel

Model3：

EconomicProductionLotSizeModelT：timebetweentimesA：demandrateQ：orderquantityhc:holdingcostK：orderingcostB：productionrateEconomicProductionLotSizeModelfeatures：BackordersarenotallowedB:annualproductionratet1timeInventorylevelmaximuminventorylevelaverageinventorylevelOneyeart2Demand-onlyphaseTQSBB-A-AProductionphaseEconomicProductionLotSizeModel

tt1t2S＝(B－A)t1Q-SQ=Bttime圖3－1Ot3features：BackordersareallowedB:annualproductionrateP:shortagecostMaximuminventoryMaximumshortageEOQwithquantitydiscountsExample:WeightyTrashBagCompany’spricingscheduleforitslargetrashcanliners:?Forordersoflessthan500bags,charges30centsperbag;?forordersof500ormorebutfewerthan1,000bags,charges29centsperbag;?andforordersof1,000ormore,charges28centsperbag.EOQwithquantitydiscounts

Thebreakpointsare500and1,000.TheordercostfunctionC(Q)isdefinedasEOQwithquantitydiscounts

IfWeightyusestrashbagsatafairlyconstantrateof600peryr,howtoplaceorder?Supposethatfixedcostofplacinganorderis$8,andholdingcostsarebasedon20%annualinterestrate.FirstcomputeEOQvaluescorrespondingtoeachoftheunitcost.EOQwithquantitydiscountsEachcurveisvalidonlyforcertainvaluesofQ,thustheaverageannualcostfunctionisgivenbydiscontinuouscurves.AnEOQvalueisrealizable,ifitfallswithintheintervalofEOQthatcorrespondstotheunitcostthathasbeenusedtocomputeit.(Q0fortheexample);EOQwithquantitydiscountsConclusion:theoptimalsolutionistoplaceastandingorderfor500unitswithWeightyatanannualcostof$198.10.EOQwithquantitydiscountsEOQwithquantitydiscounts-IncrementalQuantityDiscountsTheaverageannualcostfunctionfollowsTC(Q)hasbeendividedintothreesegmentsTC1(Q),TC2(Q),andTC3(Q),eachofwhichisobtainedbyusingoneofthethreesegmentsofC(Q).EOQwithquantitydiscounts-IncrementalQuantityDiscountsEOQwithquantitydiscounts-IncrementalQuantityDiscountsEOQwithquantitydiscounts-IncrementalQuantityDiscounts?Theoptimalsolutionoccursattheminimumofoneofthethreeaverageannualcostcurves.?Procedures:(1)DetermineanalgebraicexpressionforC(Q)correspondingtoeachpriceinterval.UsethattodetermineanalgebraicexpressionforC(Q)/Q;(2)SubstitutetheexpressionsderivedforC(Q)/QintodefiningequationforTC(Q).ComputetheminimumvalueofQcorrespondingtoeachpriceintervalseparately.(3)Determinewhichminimacomputedin(2)arerealizable(thatisfallintothecorrectinterval).ComparethevaluesoftheaverageannualcostsattherealizableEOQvaluesandpickthelowest.EOQwithquantitydiscountsThestoreplanstopurchaseaproductfromthefactory,annualdemandis500items,orderingcostis$50,unitpurchasecostcis：holdingcostsis$20.Q:what’stheoptimalpurchasequantityQ。ModelsforUncertainDemandAimsofthesectionIntroduceuncertaintyanddevelopsomemodelswherearenotknownexactlybutfollowknownprobabilitydistributions.Inparticular,wefocusonvariabledemand.Wecanclassifyproblemsaccordingtovariablesthatare:known(andeitherconstantorvariable)-inwhichcaseweknowthevaluestakenbyparametersandcanusedeterministicmodels;uncertain-inwhichcasewehaveprobabilitydistributionsforthevariablesandcanuseprobabilisticorstochastic(隨機的)models.Newsboyproblem

Thenewsboyhastodecidehowmanypaperstobuyfromhissupplierwhencustomerdemandisuncertain.Ifhebuystoomanypapers,heisleftwithunsoldstockwhichhasnovalueattheendoftheday:ifhebuystoofewpapershehasunsatisfieddemandwhichcouldhavegivenahigherprofit.Newsboyproblem

Becauseofthisillustration,singleperiodproblemsareoftencallednewsboyproblems.Although,itisawidelyoccurringproblem,wewillsticktotheoriginaldescriptionofanewsboysellingpapers.Newsboyproblem:letprepresentsthelostsalespenalty.Thedemand,beinguncertain,willbedescribedbyarandomvariableDwhoseprobabilitydistributionisspecifiedbyPD(x).WemaythinkofPD(x)astheprobabilitythattotaldemandwillequalx.Anewsboypayscdollarsforeachpaperandsellsforsdollarseach.Attheendofday,theunsoldnewspapers