運籌學(xué)與供應(yīng)鏈管理-第4講

第４講

庫存管理（II）Multi-EchelonInventoryinSupplyChainTwoStageEchelonInventorySequentialstockingpointswithleveldemandTwo--stageprocessTwoStageEchelonInventoryTwo--stageprocess:

Alittlereflectionshowsthatatleastforthecaseofdeterministicdemanditneverwouldmakesensetohavebeanythingbutanintegermultipleof..Therefore,wecanthinkoftwoalternativedecisionvariablesandwhere(4.1)

TwoStageEchelonInventoryTwo--stageprocess:ThefirststagecostThesecondstagecost

ThetotalcostTwoStageEchelonInventoryTwo--stageprocess:Thewarehouseecheloninventoryisvaluedatwhiletheretailerecheloninventoryisvaluedatonly

TwoStageEchelonInventoryTwo--stageprocess:Thetotalrelevant((setuppluscarrying))costsperunittimearegivenby=averagevalueofthewarehouseecheloninventory,inunits=averagevalueoftheretailerecheloninventory,inunits

TwoStageEchelonInventoryTwo--stageprocess:Substitutingfromequation((4.1)andnotingthattheechelonstocksfollowsawtoothpatterns,

TwoStageEchelonInventorySelect((aninteger))andinordertominimizePartialderivationofTRCTwoStageEchelonInventorySubstitutetheresultintothecostequationWerecognizethatthenthatminimizesthesimplerexpressionTwoStageEchelonInventoryAconvenientwayistofirstsetwhichgivesThissolvesforTwoStageEchelonInventoryAscertainandwhereandarethetwointegerssurroundingtheWhichevergivesthelowervalueofFistheappropriatentouse((becausetheFfunctionisconvexinn).TwoStageEchelonInventoryTwo--stageprocess:Step1ComputeStep2Ascertainthetwointegervalues,,and,,thatsurround..TwoStageEchelonInventoryTwo--stageprocess:

Step3TwoStageEchelonInventoryTwo--stageprocess:

Step4Step5TwoStageEchelonInventoryExample1::Letusconsideraparticularliquidproductthatafirmbuysinbulk,thenbreaksdownandrepackages.Sointhiscase,,thewarehousecorrespondstotheinventorypriortotherepackagingoperation,andtheretailercorrespondstotheinventoryaftertherepackagingoperation.Thedemandforthisitemcanbeassumedtobeessentiallydeterministicandlevelatarateof1000litersperyear.TwoStageEchelonInventoryExample1::Theunitvalueofthebulkmaterialoris$$1/liter,whilethevalueaddedbythetransforming((breakandpackage)operationis$4/liter.Thefixedcomponentofthepurchasecharge(())is$$10,whilethesetupcostforthebreakandrepackageoperation())is$$15..Finally,theestimatedcarryingchargeis0.24$/$$/yr.TwoStageEchelonInventoryExample1::Step1::Step2::TwoStageEchelonInventoryExample1::Step3::

thatis,Thus,usen=2..TwoStageEchelonInventoryExample:Step4::Step5::TwoStageEchelonInventoryExample1::Inotherwords,wepurchase334litersatatime;;one-halfoftheseor167litersareimmediatelybrokenandrepackaged.Whenthese167((finished))litersaredepleted,,asecondbreakandrepackagerunof167litersismade.Whenthesearedepleted,westartanewcyclebyagainpurchasing334litersofrawmaterial.InventoryControlwithUncertainDemandThedemandcanbedecomposedintotwoparts,where=Deterministiccomponentofdemandand=Randomcomponentofdemand.InventoryControlwithUncertainDemandThereareanumberofcircumstancesunderwhichitwouldbeappropriatetotreatasbeingdeterministiceventhoughisnotzero.Someoftheseare:Whenthevarianceoftherandomcomponent,issmallrelativetothemagnitudeof..Whenthepredictablevariationismoreimportantthantherandomvariation.Whentheproblemstructureistoocomplextoincludeanexplicitrepresentationofrandomnessinthemodel.InventoryControlwithUncertainDemandHowever,formanyitems,therandomcomponentofthedemandistoosignificanttoignore.Aslongastheexpecteddemandperunittimeisrelativelyconstantandtheproblemstructurenottoocomplex,explicittreatmentofdemanduncertaintyisdesirable.InventoryControlwithUncertainDemandExample2::AnewsstandpurchasesanumberofcopiesofTheComputerJournal.Theobserveddemandsduringeachofthelast52weekswere:InventoryControlwithUncertainDemandExample2::InventoryControlwithUncertainDemandExample2::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:themeanandthevarianceInventoryControlwithUncertainDemand

Thesecanbeestimatedfromahistoryofdemandbythesamplemeanandthesamplevariance..InventoryControlwithUncertainDemand

ThenormaldensityfunctionisgivenbytheformulaWesubstituteastheestimatorforandastheestimatorfor..InventoryControlwithUncertainDemand

OptimizationCriterion

Ingeneral,optimizationinproductionproblemsmeansfindingacontrolrulethatachievesminimumcost.However,whendemandisrandom,,thecostincurredisitselfrandom,,anditisnolongerobviouswhattheoptimizationcriterionshouldbe.Virtuallyallofthestochasticoptimizationtechniquesappliedtoinventorycontrolassumethatthegoalistominimizeexpectedcosts..TheNewsboyModel((ContinuousDemands)Thedemandisapproximatelynormallydistributedwithmean11.731andstandarddeviation4..74..Eachcopyispurchasedfor25centsandsoldfor75cents,,andheispaid10centsforeachunsoldcopybyhissupplier.Oneobvioussolutionisapproximately12copies.SupposeMacpurchasesacopythathedoesn''tsell.Hisout-of-pocketexpenseis25cents10cents==15cents.Supposeontheotherhand,heisunabletomeetthedemandofacustomer..Inthatcase,heloses75cents25cents==50centsprofit.TheNewsboyModel((ContinuousDemands)Notation:=Costperunitofpositiveinventoryremainingattheendoftheperiod((knownastheoveragecost)).=Costperunitofunsatisfieddemand.Thiscanbethoughtofasacostperunitofnegativeendinginventory(knownastheunderagecost)..Thedemandisacontinuousnonnegativerandomvariablewithdensityfunctionandcumulativedistributionfunction..Thedecisionvariableisthenumberofunitstobepurchasedatthebeginningoftheperiod.TheNewsboyModel((ContinuousDemands)Determiningtheoptimalpolicy::ThecostfunctionTheoptimalsolutionequationTheNewsboyModel((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..ThatisTheNewsboyModel((DiscreteDemands)Example2::TheNewsboyModel((DiscreteDemands)Example2::Thecriticalratioforthisproblemwas0.77,whichcorrespondstoavalueofbetween==14and=15.Sinceweroundup,theoptimalsolutionis==15.Noticethatthisisexactlythesameorderquantityobtainedusingthenormalapproximation.

TheNewsboyModel((DiscreteDemands)ExtensiontoIncludeStartingInventory::Theoptimalpolicywhenthereisastartinginventoryofis:Orderif..Don''torderif..

Notethatshouldbeinterpretedastheorder--up--topointratherthantheorderquantitywhen..Itisalsoknownasatargetorbasestocklevel..MultiproductSystemsABCanalysis::Thetrade--offsbetweenthecostofcontrollingthesystemandthepotentialbenefitsthataccruefromthatcontrol..Inmultiproductinventorysystemsnotallproductsareequallyprofitable.Alargeportionofthetotaldollarvolumeofsalesisoftenaccountedforbyasmallnumberofinventoryitems..

MultiproductSystemsABCanalysis::

MultiproductSystemsABCanalysis::SinceAitemsaccountforthelion'sshareoftheyearlyrevenue,theseitemsshouldbewatchedmostclosely..InventorylevelsforAitemsshouldbemonitoredcontinuously.Moresophisticatedforecastingproceduresmightbeusedandmorecarewouldbetakenintheestimationofthevariouscostparametersrequiredincalculatingoperatingpolicies.

MultiproductSystemsABCanalysis::ForBitemsinventoriescouldbereviewedperiodically,,itemscouldbeorderedingroupsratherthanindividually,andsomewhatlesssophisticatedforecastingmethodscouldbeused..MultiproductSystemsABCanalysis::TheminimumdegreeofcontrolwouldbeappliedtoCitems..ForveryinexpensiveCitemswithmoderatelevelsofdemand,largelotsizesarerecommendedtominimizethefrequencythattheseitemsareordered.ForexpensiveCitemswithverylowdemand,,thebestpolicyisgenerallynottoholdanyinventory.Onewouldsimplyordertheseitemsastheyaredemanded.LotSize-ReorderPointSystemsInwhatfollows,,weassumethattheoperatingpolicyisoftheform..However,,whengeneralizingtheEOQanalysistoallowforrandomdemand,,wetreatandasindependentdecisionvariables..LotSize-ReorderPointSystemsAssumptionsThesystemiscontinuous--reviewDemandisrandomandstationaryThereisafixed