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第４講

庫存管理（II）第４講

庫存管理（II）1Multi-EchelonInventoryinSupplyChainMulti-EchelonInventoryinSup2TwoStageEchelonInventorySequentialstockingpointswithleveldemandTwo-stageprocessTwoStageEchelonInventorySeq3TwoStageEchelonInventoryTwo-stageprocess:

Alittlereflectionshowsthatatleastforthecaseofdeterministicdemanditneverwouldmakesensetohave beanythingbutanintegermultipleof.Therefore,wecanthinkoftwoalternativedecisionvariablesandwhere (4.1)

TwoStageEchelonInventoryTwo4TwoStageEchelonInventoryTwo-stageprocess: Thefirststagecost Thesecondstagecost

ThetotalcostTwoStageEchelonInventoryTwo5TwoStageEchelonInventoryTwo-stageprocess: Thewarehouseecheloninventoryisvaluedat whiletheretailerecheloninventoryisvaluedatonly

TwoStageEchelonInventoryTwo6TwoStageEchelonInventoryTwo-stageprocess:

Thetotalrelevant(setuppluscarrying)costsperunittimearegivenby =averagevalueofthewarehouseecheloninventory,inunits =averagevalueoftheretailerecheloninventory,inunits

TwoStageEchelonInventoryTwo7TwoStageEchelonInventoryTwo-stageprocess:

Substitutingfromequation(4.1)andnotingthattheechelonstocksfollowsawtoothpatterns,

TwoStageEchelonInventoryTwo8TwoStageEchelonInventorySelect(aninteger)andinordertominimizePartialderivationofTRCTwoStageEchelonInventorySel9TwoStageEchelonInventorySubstitutetheresultintothecostequationWerecognizethatthenthatminimizesthesimplerexpressionTwoStageEchelonInventorySub10TwoStageEchelonInventoryAconvenientwayistofirstsetwhichgivesThissolvesforTwoStageEchelonInventoryAc11TwoStageEchelonInventoryAscertainandwhereandarethetwointegerssurroundingtheWhichevergivesthelowervalueofFistheappropriatentouse(becausetheFfunctionisconvexinn).TwoStageEchelonInventoryAsc12TwoStageEchelonInventoryTwo-stageprocess:

Step1 Compute Step2 Ascertainthetwointegervalues,and,thatsurround.TwoStageEchelonInventoryTwo13TwoStageEchelonInventoryTwo-stageprocess:

Step3TwoStageEchelonInventoryTwo14TwoStageEchelonInventoryTwo-stageprocess:

Step4 Step5TwoStageEchelonInventoryTwo15TwoStageEchelonInventoryExample1:Letusconsideraparticularliquidproductthatafirmbuysinbulk,thenbreaksdownandrepackages. Sointhiscase,thewarehousecorrespondstotheinventorypriortotherepackagingoperation,andtheretailercorrespondstotheinventoryaftertherepackagingoperation. Thedemandforthisitemcanbeassumedtobeessentiallydeterministicandlevelatarateof1000litersperyear.TwoStageEchelonInventoryExa16TwoStageEchelonInventoryExample1:Theunitvalueofthebulkmaterialoris$1/liter,whilethevalueaddedbythetransforming(breakandpackage)operationis$4/liter.Thefixedcomponentofthepurchasecharge()is$10,whilethesetupcostforthebreakandrepackageoperation()is$15.Finally,theestimatedcarryingchargeis0.24$/$/yr.TwoStageEchelonInventoryExa17TwoStageEchelonInventoryExample1: Step1: Step2:TwoStageEchelonInventoryExa18TwoStageEchelonInventoryExample1:Step3:

thatis, Thus,usen=2.TwoStageEchelonInventoryExa19TwoStageEchelonInventoryExample:Step4:Step5:TwoStageEchelonInventoryExa20TwoStageEchelonInventoryExample1:Inotherwords,wepurchase334litersatatime;one-halfoftheseor167litersareimmediatelybrokenandrepackaged.Whenthese167(finished)litersaredepleted,asecondbreakandrepackagerunof167litersismade.Whenthesearedepleted,westartanewcyclebyagainpurchasing334litersofrawmaterial.TwoStageEchelonInventoryExa21InventoryControlwithUncertainDemandThedemandcanbedecomposedintotwoparts,where=Deterministiccomponentofdemandand=Randomcomponentofdemand.InventoryControlwithUncerta22InventoryControlwithUncertainDemand Thereareanumberofcircumstancesunderwhichitwouldbeappropriatetotreatasbeingdeterministiceventhoughisnotzero.Someoftheseare:Whenthevarianceoftherandomcomponent, issmallrelativetothemagnitudeof.Whenthepredictablevariationismoreimportantthantherandomvariation.Whentheproblemstructureistoocomplextoincludeanexplicitrepresentationofrandomnessinthemodel.InventoryControlwithUncerta23InventoryControlwithUncertainDemand However,formanyitems,therandomcomponentofthedemandistoosignificanttoignore. Aslongastheexpecteddemandperunittimeisrelativelyconstantandtheproblemstructurenottoocomplex,explicittreatmentofdemanduncertaintyisdesirable.InventoryControlwithUncerta24InventoryControlwithUncertainDemand

Example2:

AnewsstandpurchasesanumberofcopiesofTheComputerJournal.Theobserveddemandsduringeachofthelast52weekswere:InventoryControlwithUncerta25InventoryControlwithUncertainDemand

Example2:InventoryControlwithUncerta26InventoryControlwithUncertainDemand

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.

InventoryControlwithUncerta27InventoryControlwithUncertainDemand

Wegenerallyapproximatethedemandhistoryusingacontinuousdistribution.

Byfar,themostpopulardistributionforinventoryapplicationsisthenormal.

Anormaldistributionisdeterminedbytwoparameters:themeanandthevariance

InventoryControlwithUncerta28InventoryControlwithUncertainDemand

Thesecanbeestimatedfromahistoryofdemandbythesamplemeanandthesamplevariance.InventoryControlwithUncerta29InventoryControlwithUncertainDemand

Thenormaldensityfunctionisgivenbytheformula

Wesubstituteastheestimatorforandastheestimatorfor.InventoryControlwithUncerta30InventoryControlwithUncertainDemand

InventoryControlwithUncerta31OptimizationCriterion

Ingeneral,optimizationinproductionproblemsmeansfindingacontrolrulethatachievesminimumcost. However,whendemandisrandom,thecostincurredisitselfrandom,anditisnolongerobviouswhattheoptimizationcriterionshouldbe. Virtuallyallofthestochasticoptimizationtechniquesappliedtoinventorycontrolassumethatthegoalistominimizeexpectedcosts.OptimizationCriterion 32TheNewsboyModel(ContinuousDemands) Thedemandisapproximatelynormallydistributedwithmean11.731andstandarddeviation4.74. Eachcopyispurchasedfor25centsandsoldfor75cents,andheispaid10centsforeachunsoldcopybyhissupplier. Oneobvioussolutionisapproximately12copies. SupposeMacpurchasesacopythathedoesn'tsell.Hisout-of-pocketexpenseis25cents10cents=15cents. Supposeontheotherhand,heisunabletomeetthedemandofacustomer.Inthatcase,heloses75cents25cents=50centsprofit.TheNewsboyModel(Continuous33TheNewsboyModel(ContinuousDemands)

Notation: =Costperunitofpositiveinventoryremainingattheendoftheperiod(knownastheoveragecost). =Costperunitofunsatisfieddemand.Thiscanbethoughtofasacostperunitofnegativeendinginventory(knownastheunderagecost). Thedemandisacontinuousnonnegativerandomvariablewithdensityfunctionandcumulativedistributionfunction.

Thedecisionvariableisthenumberofunitstobepurchasedatthebeginningoftheperiod.TheNewsboyModel(Continuous34TheNewsboyModel(ContinuousDemands)

Determiningtheoptimalpolicy:

Thecostfunction TheoptimalsolutionequationTheNewsboyModel(Continuous35TheNewsboyModel(ContinuousDemands)

Determiningtheoptimalpolicy:TheNewsboyModel(Continuous36TheNewsboyModel(ContinuousDemands)

Example2(continued):

Normallydistributedwithmean=11.73andstandarddeviation=4.74. SinceMacpurchasesthemagazinesfor25centsandcansalvageunsoldcopiesfor10cents,hisoveragecostis=2510=15cents. Hisunderagecostistheprofitoneachsale,sothat=7525=50cents.

TheNewsboyModel(Continuous37TheNewsboyModel(ContinuousDemands)

Example2(continued):

Thecriticalratiois=0.50/0.65=0.77. Purchaseenoughcopiestosatisfyalloftheweeklydemandwithprobability0.77.Theoptimalisthe77thpercentileofthedemanddistribution.

TheNewsboyModel(Continuous38TheNewsboyModel(ContinuousDemands)

Example2(continued):

TheNewsboyModel(Continuous39TheNewsboyModel(ContinuousDemands)

Example2(continued):

Usingthedataofthenormaldistributionweobtainastandardizedvalueof=0.74.Theoptimalis Hence,heshouldpurchase15copieseveryweek.TheNewsboyModel(Continuous40TheNewsboyModel(DiscreteDemands)

Optimalpolicyfordiscretedemand:

Theprocedureforfindingtheoptimalsolutiontothenewsboyproblemwhenthedemandisassumedtobediscreteisanaturalgeneralizationofthecontinuouscase.

Theoptimalsolutionprocedureistolocatethecriticalratiobetweentwovaluesofandchoosethecorrespondingtothehighervalue.ThatisTheNewsboyModel(DiscreteDe41TheNewsboyModel(DiscreteDemands)

Example2:

TheNewsboyModel(DiscreteDe42TheNewsboyModel(DiscreteDemands)

Example2:

Thecriticalratioforthisproblemwas0.77,whichcorrespondstoavalueofbetween=14and=15. Sinceweroundup,theoptimalsolutionis=15.Noticethatthisisexactlythesameorderquantityobtainedusingthenormalapproximation.

TheNewsboyModel(DiscreteDe43TheNewsboyModel(DiscreteDemands)

ExtensiontoIncludeStartingInventory: Theoptimalpolicywhenthereisastartinginventoryof is: Order if.Don'torderif.

Notethatshouldbeinterpretedastheorder-up-topointratherthantheorderquantitywhen.Itisalsoknownasatargetorbasestocklevel.TheNewsboyModel(DiscreteDe44MultiproductSystems ABCanalysis:Thetrade-offsbetweenthecostofcontrollingthesystemandthepotentialbenefitsthataccruefromthatcontrol.Inmultiproductinventorysystemsnotallproductsareequallyprofitable.Alargeportionofthetotaldollarvolumeofsalesisoftenaccountedforbyasmallnumberofinventoryitems.

MultiproductSystems ABCanaly45MultiproductSystems ABCanalysis:

MultiproductSystems ABCanaly46MultiproductSystems ABCanalysis:

SinceAitemsaccountforthelion'sshareoftheyearlyrevenue,theseitemsshouldbewatchedmostclosely. InventorylevelsforAitemsshouldbemonitoredcontinuously. Moresophisticatedforecastingproceduresmightbeusedandmorecarewouldbetakenintheestimationofthevariouscostparametersrequiredincalculatingoperatingpolicies.

MultiproductSystems ABCanaly47MultiproductSystems ABCanalysis:

ForBitemsinventoriescouldbereviewedperiodically,itemscouldbeorderedingroupsratherthanindividually,andsomewhatlesssophisticatedforecastingmethodscouldbeused.MultiproductSystems ABCanaly48MultiproductSystems ABCanalysis:TheminimumdegreeofcontrolwouldbeappliedtoCitems.ForveryinexpensiveCitemswithmoderatelevelsofdemand,largelotsizesarerecommendedtominimizethefrequencythattheseitemsareordered.ForexpensiveCitemswithverylowdemand,thebestpolicyisgenerallynottoholdanyinventory.Onewouldsimplyordertheseitemsastheyaredemanded.MultiproductSystems ABCanaly49LotSize-ReorderPointSystemsInwhatfollows,weassumethattheoperatingpolicyisoftheform.However,whengeneralizingtheEOQanalysistoallowforrandomdemand,wetreatandasindependentdecisionvariables.LotSize-ReorderPointSystems50LotSize-ReorderPointSystemsAssumptionsThesystemiscontinuous-reviewDemandisrandomandstationaryThereisafixedpositiveleadtimeforplacinganorderThefollowingcostsareassumedSetupcostat$perorder.Holdingcostat$perunitheldperyear.Proportionalordercostof$peritem.Stock-outcostof$perunitofunsatisfieddemandLotSize-ReorderPointSystems51LotSize-ReorderPointSystems

Describingdemand:

Thedemandduringtheleadtimeisacontinuousrandomvariablewithprobabilitydensityfunction(orpdf),andaccumulativedistributionfunction(orcdf) .Letandbethemeanandstandarddeviationofdemandduringleadtime.LotSize-ReorderPointSystems52LotSize-ReorderPointSystems

Decisionvariables:

Therearetwodecisionvariablesforthisproblem, and, where=thelotsizeororderquantityand =thereorderlevelinunitsofinventory.

LotSize-ReorderPointSystems53LotSize-ReorderPointSystems

Decisionvariables:LotSize-ReorderPointSystems54AdditionalDiscussionofPeriodic-ReviewSystems

Definetwonumbers,and,tobeusedasfollows: Whenthelevelofonhandinventoryislessthanorequalto,anorderforthedifferencebetweentheinventoryandisplaced. Ifisthestartinginventoryinanyperiod,thenthe policyis:If,order.If,don'torder.AdditionalDiscussionofPerio55AdditionalDiscussionofPeriodic-ReviewSystems

DeterminingoptimalvaluesofAdditionalDiscussionofPerio56第４講

庫存管理（II）第４講

庫存管理（II）57Multi-EchelonInventoryinSupplyChainMulti-EchelonInventoryinSup58TwoStageEchelonInventorySequentialstockingpointswithleveldemandTwo-stageprocessTwoStageEchelonInventorySeq59TwoStageEchelonInventoryTwo-stageprocess:

Alittlereflectionshowsthatatleastforthecaseofdeterministicdemanditneverwouldmakesensetohave beanythingbutanintegermultipleof.Therefore,wecanthinkoftwoalternativedecisionvariablesandwhere (4.1)

TwoStageEchelonInventoryTwo60TwoStageEchelonInventoryTwo-stageprocess: Thefirststagecost Thesecondstagecost

ThetotalcostTwoStageEchelonInventoryTwo61TwoStageEchelonInventoryTwo-stageprocess: Thewarehouseecheloninventoryisvaluedat whiletheretailerecheloninventoryisvaluedatonly

TwoStageEchelonInventoryTwo62TwoStageEchelonInventoryTwo-stageprocess:

Thetotalrelevant(setuppluscarrying)costsperunittimearegivenby =averagevalueofthewarehouseecheloninventory,inunits =averagevalueoftheretailerecheloninventory,inunits

TwoStageEchelonInventoryTwo63TwoStageEchelonInventoryTwo-stageprocess:

Substitutingfromequation(4.1)andnotingthattheechelonstocksfollowsawtoothpatterns,

TwoStageEchelonInventoryTwo64TwoStageEchelonInventorySelect(aninteger)andinordertominimizePartialderivationofTRCTwoStageEchelonInventorySel65TwoStageEchelonInventorySubstitutetheresultintothecostequationWerecognizethatthenthatminimizesthesimplerexpressionTwoStageEchelonInventorySub66TwoStageEchelonInventoryAconvenientwayistofirstsetwhichgivesThissolvesforTwoStageEchelonInventoryAc67TwoStageEchelonInventoryAscertainandwhereandarethetwointegerssurroundingtheWhichevergivesthelowervalueofFistheappropriatentouse(becausetheFfunctionisconvexinn).TwoStageEchelonInventoryAsc68TwoStageEchelonInventoryTwo-stageprocess:

Step1 Compute Step2 Ascertainthetwointegervalues,and,thatsurround.TwoStageEchelonInventoryTwo69TwoStageEchelonInventoryTwo-stageprocess:

Step3TwoStageEchelonInventoryTwo70TwoStageEchelonInventoryTwo-stageprocess:

Step4 Step5TwoStageEchelonInventoryTwo71TwoStageEchelonInventoryExample1:Letusconsideraparticularliquidproductthatafirmbuysinbulk,thenbreaksdownandrepackages. Sointhiscase,thewarehousecorrespondstotheinventorypriortotherepackagingoperation,andtheretailercorrespondstotheinventoryaftertherepackagingoperation. Thedemandforthisitemcanbeassumedtobeessentiallydeterministicandlevelatarateof1000litersperyear.TwoStageEchelonInventoryExa72TwoStageEchelonInventoryExample1:Theunitvalueofthebulkmaterialoris$1/liter,whilethevalueaddedbythetransforming(breakandpackage)operationis$4/liter.Thefixedcomponentofthepurchasecharge()is$10,whilethesetupcostforthebreakandrepackageoperation()is$15.Finally,theestimatedcarryingchargeis0.24$/$/yr.TwoStageEchelonInventoryExa73TwoStageEchelonInventoryExample1: Step1: Step2:TwoStageEchelonInventoryExa74TwoStageEchelonInventoryExample1:Step3:

thatis, Thus,usen=2.TwoStageEchelonInventoryExa75TwoStageEchelonInventoryExample:Step4:Step5:TwoStageEchelonInventoryExa76TwoStageEchelonInventoryExample1:Inotherwords,wepurchase334litersatatime;one-halfoftheseor167litersareimmediatelybrokenandrepackaged.Whenthese167(finished)litersaredepleted,asecondbreakandrepackagerunof167litersismade.Whenthesearedepleted,westartanewcyclebyagainpurchasing334litersofrawmaterial.TwoStageEchelonInventoryExa77InventoryControlwithUncertainDemandThedemandcanbedecomposedintotwoparts,where=Deterministiccomponentofdemandand=Randomcomponentofdemand.InventoryControlwithUncerta78InventoryControlwithUncertainDemand Thereareanumberofcircumstancesunderwhichitwouldbeappropriatetotreatasbeingdeterministiceventhoughisnotzero.Someoftheseare:Whenthevarianceoftherandomcomponent, issmallrelativetothemagnitudeof.Whenthepredictablevariationismoreimportantthantherandomvariation.Whentheproblemstructureistoocomplextoincludeanexplicitrepresentationofrandomnessinthemodel.InventoryControlwithUncerta79InventoryControlwithUncertainDemand However,formanyitems,therandomcomponentofthedemandistoosignificanttoignore. Aslongastheexpecteddemandperunittimeisrelativelyconstantandtheproblemstructurenottoocomplex,explicittreatmentofdemanduncertaintyisdesirable.InventoryControlwithUncerta80InventoryControlwithUncertainDemand

Example2:

AnewsstandpurchasesanumberofcopiesofTheComputerJournal.Theobserveddemandsduringeachofthelast52weekswere:InventoryControlwithUncerta81InventoryControlwithUncertainDemand

Example2:InventoryControlwithUncerta82InventoryControlwithUncertainDemand

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.

InventoryControlwithUncerta83InventoryControlwithUncertainDemand

Wegenerallyapproximatethedemandhistoryusingacontinuousdistribution.

Byfar,themostpopulardistributionforinventoryapplicationsisthenormal.

Anormaldistributionisdeterminedbytwoparameters:themeanandthevariance

InventoryControlwithUncerta84InventoryControlwithUncertainDemand

Thesecanbeestimatedfromahistoryofdemandbythesamplemeanandthesamplevariance.InventoryControlwithUncerta85InventoryControlwithUncertainDemand

Thenormaldensityfunctionisgivenbytheformula

Wesubstituteastheestimatorforandastheestimatorfor.InventoryControlwithUncerta86InventoryControlwithUncertainDemand

InventoryControlwithUncerta87OptimizationCriterion

Ingeneral,optimizationinproductionproblemsmeansfindingacontrolrulethatachievesminimumcost. However,whendemandisrandom,thecostincurredisitselfrandom,anditisnolongerobviouswhattheoptimizationcriterionshouldbe. Virtuallyallofthestochasticoptimizationtechniquesappliedtoinventorycontrolassumethatthegoalistominimizeexpectedcosts.OptimizationCriterion 88TheNewsboyModel(ContinuousDemands) Thedemandisapproximatelynormallydistributedwithmean11.731andstandarddeviation4.74. Eachcopyispurchasedfor25centsandsoldfor75cents,andheispaid10centsforeachunsoldcopybyhissupplier. Oneobvioussolutionisapproximately12copies. SupposeMacpurchasesacopythathedoesn'tsell.Hisout-of-pocketexpenseis25cents10cents=15cents. Supposeontheotherhand,heisunabletomeetthedemandofacustomer.Inthatcase,heloses75cents25cents=50centsprofit.TheNewsboyModel(Continuous89TheNewsboyModel(ContinuousDemands)

Notation: =Costperunitofpositiveinventoryremainingattheendoftheperiod(knownastheoveragecost). =Costperunitofunsatisfieddemand.Thiscanbethoughtofasacostperunitofnegativeendinginventory(knownastheunderagecost). Thedemandisacontinuousnonnegativerandomvariablewithdensityfunctionandcumulativedistributionfunction.

Thedecisionvariableisthenumberofunitstobepurchasedatthebeginningoftheperiod.TheNewsboyModel(Continuous90TheNewsboyModel(ContinuousDemands)

Determiningtheoptimalpolicy:

Thecostfunction TheoptimalsolutionequationTheNewsboyModel(Continuous91TheNewsboyModel(ContinuousDemands)

Determiningtheoptimalpolicy:TheNewsboyModel(Continuous92TheNewsboyModel(ContinuousDemands)

Example2(continued):

Normallydistributedwithmean=11.73andstandarddeviation=4.74. SinceMacpurchasesthemagazinesfor25centsandcansalvageunsoldcopiesfor10cents,hisoveragecostis=2510=15cents. Hisunderagecostistheprofitoneachsale,sothat=7525=50cents.

TheNewsboyModel(Continuous93TheNewsboyModel(ContinuousDemands)

Example2(continued):

Thecriticalratiois=0.50/0.65=0.77. Purchaseenoughcopiestosatisfyalloftheweeklydemandwithprobability0.77.Theoptimalisthe77thpercentileofthedemanddistribution.

TheNewsboyModel(Continuous94TheNewsboyModel(ContinuousDemands)

Example2(continued):

TheNewsboyModel(Continuous95TheNewsboyModel(ContinuousDemands)

Example2(continued):

Usingthedataofthenormaldistributionweobtainastandardizedvalueof=0.74.Theoptimalis Hence,heshouldpurchase15copieseveryweek.TheNewsboyModel(Continuous96TheNewsboyModel(DiscreteDemands)

Optimalpolicyfordiscretedemand:

Theprocedureforfindingtheoptimalsolutiontothenewsboyproblemwhenthedemandisassumedtobediscreteisanaturalgeneralizationofthecontinuouscase.

Theoptimalsolutionprocedureistolocatethecriticalratiobetweentwovaluesofandchoosethecorrespondingtothehighervalue.ThatisTheNewsboyModel(DiscreteDe97TheNewsboyModel(DiscreteDemands)

Example2:

TheNewsboyModel(DiscreteDe98TheNewsboyModel(DiscreteDemands)

Example2:

Thecriticalratioforthisproblemwas0.77,whichcorrespondstoavalueofbetween=14and=15. Sinceweroundup,theoptimalsolutionis=15.Noticethatthisisexactlythesameorderquantityobtainedusingthenormalapproximation.

TheNewsboyModel(DiscreteDe99TheNewsboyModel(DiscreteDemands)

ExtensiontoIncludeStartingInventory: Theoptimalpolicywhenthereisastartinginventoryof is: Order if.Don'torderif.

Notethatshouldbeinterpretedastheorder-up-topointratherthantheorderquantitywhen.Itisalsoknownasatargetorbasestocklevel.TheNewsboyModel(DiscreteDe100MultiproductSystems ABCanalysis:Thetrade-offsbetweenthecostofcontrollingthesystemandthepotentialbenefitsthataccruefromthatcontrol.Inmultiproductinventorysystemsnotallproductsareequallyprofitable.Alargeportionofthetotaldollarvolumeofsalesisoftenaccountedforbyasmallnumberofinventoryitems.

MultiproductSystems ABCanaly101MultiproductSystems ABCanalysis:

MultiproductSystems ABCanaly102MultiproductSystems ABCanalysis:

SinceAitemsaccountforthelion'sshareoftheyearlyrevenue,theseitemsshouldbewatchedmostclosely. InventorylevelsforAitemsshouldbemonitoredcontinu

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