运筹学与供应链管理库存管理

第讲库存管理（II）,Multi-EchelonInventoryinSupplyChain,TwoStageEchelonInventory,SequentialstockingpointswithleveldemandTwo-stageprocess,TwoStageEchelonInventory,Two-stageprocess:Alittlereflectionshowsthatatleastforthecaseofdeterministicdemanditneverwouldmakesensetohavebeanythingbutanintegermultipleof.Therefore,wecanthinkoftwoalternativedecisionvariablesandwhere(4.1),TwoStageEchelonInventory,Two-stageprocess:ThefirststagecostThesecondstagecostThetotalcost,TwoStageEchelonInventory,Two-stageprocess:Thewarehouseecheloninventoryisvaluedatwhiletheretailerecheloninventoryisvaluedatonly,TwoStageEchelonInventory,Two-stageprocess:Thetotalrelevant(setuppluscarrying)costsperunittimearegivenby=averagevalueofthewarehouseecheloninventory,inunits=averagevalueoftheretailerecheloninventory,inunits,TwoStageEchelonInventory,Two-stageprocess:Substitutingfromequation(4.1)andnotingthattheechelonstocksfollowsawtoothpatterns,TwoStageEchelonInventory,Select(aninteger)andinordertominimizePartialderivationofTRC,TwoStageEchelonInventory,SubstitutetheresultintothecostequationWerecognizethatthenthatminimizesthesimplerexpression,TwoStageEchelonInventory,AconvenientwayistofirstsetwhichgivesThissolvesfor,TwoStageEchelonInventory,AscertainandwhereandarethetwointegerssurroundingtheWhichevergivesthelowervalueofFistheappropriatentouse(becausetheFfunctionisconvexinn).,TwoStageEchelonInventory,Two-stageprocess:Step1ComputeStep2Ascertainthetwointegervalues,and,thatsurround.,TwoStageEchelonInventory,Two-stageprocess:Step3,TwoStageEchelonInventory,Two-stageprocess:Step4Step5,TwoStageEchelonInventory,Example1:Letusconsideraparticularliquidproductthatafirmbuysinbulk,thenbreaksdownandrepackages.Sointhiscase,thewarehousecorrespondstotheinventorypriortotherepackagingoperation,andtheretailercorrespondstotheinventoryaftertherepackagingoperation.Thedemandforthisitemcanbeassumedtobeessentiallydeterministicandlevelatarateof1000litersperyear.,TwoStageEchelonInventory,Example1:Theunitvalueofthebulkmaterialoris$1/liter,whilethevalueaddedbythetransforming(breakandpackage)operationis$4/liter.Thefixedcomponentofthepurchasecharge()is$10,whilethesetupcostforthebreakandrepackageoperation()is$15.Finally,theestimatedcarryingchargeis0.24$/$/yr.,TwoStageEchelonInventory,Example1:Step1:Step2:,TwoStageEchelonInventory,Example1:Step3:thatis,Thus,usen=2.,TwoStageEchelonInventory,Example:Step4:Step5:,TwoStageEchelonInventory,Example1:Inotherwords,wepurchase334litersatatime;one-halfoftheseor167litersareimmediatelybrokenandrepackaged.Whenthese167(finished)litersaredepleted,asecondbreakandrepackagerunof167litersismade.Whenthesearedepleted,westartanewcyclebyagainpurchasing334litersofrawmaterial.,InventoryControlwithUncertainDemand,Thedemandcanbedecomposedintotwoparts,where=Deterministiccomponentofdemandand=Randomcomponentofdemand.,InventoryControlwithUncertainDemand,Thereareanumberofcircumstancesunderwhichitwouldbeappropriatetotreatasbeingdeterministiceventhoughisnotzero.Someoftheseare:Whenthevarianceoftherandomcomponent,issmallrelativetothemagnitudeof.Whenthepredictablevariationismoreimportantthantherandomvariation.Whentheproblemstructureistoocomplextoincludeanexplicitrepresentationofrandomnessinthemodel.,InventoryControlwithUncertainDemand,However,formanyitems,therandomcomponentofthedemandistoosignificanttoignore.Aslongastheexpecteddemandperunittimeisrelativelyconstantandtheproblemstructurenottoocomplex,explicittreatmentofdemanduncertaintyisdesirable.,InventoryControlwithUncertainDemand,Example2:AnewsstandpurchasesanumberofcopiesofTheComputerJournal.Theobserveddemandsduringeachofthelast52weekswere:,InventoryControlwithUncertainDemand,Example2:,InventoryControlwithUncertainDemand,Example2:EstimatetheprobabilitythatthenumberofcopiesoftheJournalsoldinanyweek.Theprobabilitythatdemandis10isestimatedtobe2/52=0.0385,andtheprobabilitythatthedemandis15is5/52=0.0962.Cumulativeprobabilitiescanalsobeestimatedinasimilarway.TheprobabilitythattherearenineorfewercopiesoftheJournalsoldinanyweekis(1+0+0+0+3+1+2+2+4+6)/52=19/52=0.3654.,InventoryControlwithUncertainDemand,Wegenerallyapproximatethedemandhistoryusingacontinuousdistribution.Byfar,themostpopulardistributionforinventoryapplicationsisthenormal.Anormaldistributionisdeterminedbytwoparameters:themeanandthevariance,InventoryControlwithUncertainDemand,Thesecanbeestimatedfromahistoryofdemandbythesamplemeanandthesamplevariance.,InventoryControlwithUncertainDemand,ThenormaldensityfunctionisgivenbytheformulaWesubstituteastheestimatorforandastheestimatorfor.,InventoryControlwithUncertainDemand,OptimizationCriterion,Ingeneral,optimizationinproductionproblemsmeansfindingacontrolrulethatachievesminimumcost.However,whendemandisrandom,thecostincurredisitselfrandom,anditisnolongerobviouswhattheoptimizationcriterionshouldbe.Virtuallyallofthestochasticoptimizationtechniquesappliedtoinventorycontrolassumethatthegoalistominimizeexpectedcosts.,TheNewsboyModel(ContinuousDemands),Thedemandisapproximatelynormallydistributedwithmean11.731andstandarddeviation4.74.Eachcopyispurchasedfor25centsandsoldfor75cents,andheispaid10centsforeachunsoldcopybyhissupplier.Oneobvioussolutionisapproximately12copies.SupposeMacpurchasesacopythathedoesntsell.Hisout-of-pocketexpenseis25cents10cents=15cents.Supposeontheotherhand,heisunabletomeetthedemandofacustomer.Inthatcase,heloses75cents25cents=50centsprofit.,TheNewsboyModel(ContinuousDemands),Notation:=Costperunitofpositiveinventoryremainingattheendoftheperiod(knownastheoveragecost).=Costperunitofunsatisfieddemand.Thiscanbethoughtofasacostperunitofnegativeendinginventory(knownastheunderagecost).Thedemandisacontinuousnonnegativerandomvariablewithdensityfunctionandcumulativedistributionfunction.Thedecisionvariableisthenumberofunitstobepurchasedatthebeginningoftheperiod.,TheNewsboyModel(ContinuousDemands),Determiningtheoptimalpolicy:ThecostfunctionTheoptimalsolutionequation,TheNewsboyModel(ContinuousDemands),Determiningtheoptimalpolicy:,TheNewsboyModel(ContinuousDemands),Example2(continued):Normallydistributedwithmean=11.73andstandarddeviation=4.74.SinceMacpurchasesthemagazinesfor25centsandcansalvageunsoldcopiesfor10cents,hisoveragecostis=2510=15cents.Hisunderagecostistheprofitoneachsale,sothat=7525=50cents.,TheNewsboyModel(ContinuousDemands),Example2(continued):Thecriticalratiois=0.50/0.65=0.77.Purchaseenoughcopiestosatisfyalloftheweeklydemandwithprobability0.77.Theoptimalisthe77thpercentileofthedemanddistribution.,TheNewsboyModel(ContinuousDemands),Example2(continued):,TheNewsboyModel(ContinuousDemands),Example2(continued):Usingthedataofthenormaldistributionweobtainastandardizedvalueof=0.74.TheoptimalisHence,heshouldpurchase15copieseveryweek.,TheNewsboyModel(DiscreteDemands),Optimalpolicyfordiscretedemand:Theprocedureforfindingtheoptimalsolutiontothenewsboyproblemwhenthedemandisassumedtobediscreteisanaturalgeneralizationofthecontinuouscase.Theoptimalsolutionprocedureistolocatethecriticalratiobetweentwovaluesofandchoosethecorrespondingtothehighervalue.Thatis,TheNewsboyModel(DiscreteDemands),Example2:,TheNewsboyModel(DiscreteDemands),Example2:Thecriticalratioforthisproblemwas0.77,whichcorrespondstoavalueofbetween=14and=15.Sinceweroundup,theoptimalsolutionis=15.Noticethatthisisexactlythesameorderquantityobtainedusingthenormalapproximation.,TheNewsboyModel(DiscreteDemands),ExtensiontoIncludeStartingInventory:Theoptimalpolicywhenthereisastartinginventoryofis:Orderif.Dontorderif.Notethatshouldbeinterpretedastheorder-up-topointratherthantheorderquantitywhen.Itisalsoknownasatargetorbasestocklevel.,MultiproductSystems,ABCanalysis:Thetrade-offsbetweenthecostofcontrollingthesystemandthepotentialbenefitsthataccruefromthatcontrol.Inmultiproductinventorysystemsnotallproductsareequallyprofitable.Alargeportionofthetotaldollarvolumeofsalesisoftenaccountedforbyasmallnumberofinventoryitems.,MultiproductSystems,ABCanalysis:,MultiproductSystems,ABCanalysis:SinceAitemsaccountforthelionsshareoftheyearlyrevenue,theseitemsshouldbewatchedmostclosely.InventorylevelsforAitemsshouldbemonitoredcontinuously.Moresophisticatedforecastingproceduresmightbeusedandmorecarewouldbetakenintheestimationofthevariouscostparametersrequiredincalculatingoperatingpolicies.,MultiproductSystems,ABCanalysis:ForBitemsinventoriescouldbereviewedperiodically,itemscouldbeorderedingroupsratherthanindividually,andsomewhatlesssophisticatedforecastingmethodscouldbeused.,MultiproductSystems,ABCanalysis:TheminimumdegreeofcontrolwouldbeappliedtoCitems.ForveryinexpensiveCitemswithmoderatelevelsofdemand,largelotsizesarerecommendedtominimizethefrequencythattheseitemsareordered.ForexpensiveCitemswithverylowdemand,thebestpolicyisgenerallynottoholdanyinventory.Onewouldsimplyordertheseitemsastheyaredemanded.,LotSize-ReorderPointSystems,Inwhatfollows,weassumethattheoperatingpolicyisoftheform.However,whengeneralizingtheEOQanalysistoallowforrandomdemand,wetreatandasindependentdecisionvariables.,LotSize-ReorderPointSystems,AssumptionsThesystemiscontinuous-reviewDemandisrandomands