Chapter 6 Inventory Control

8/2/2019 Chapter 6 Inventory Control

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SupplyChainManagementInventoryControlSupplyChainManagementInventoryControlInventorySystemInventoryisthestockofanyitemorresourceusedinanorganizationandcaninclude:rawmaterials,finishedproducts,componentparts,supplies,andwork-in-processAninventorysystemisthesetofpoliciesandcontrolsthatmonitorlevelsofinventoryanddetermineswhatlevelsshouldbemaintained,whenstockshouldbereplenished,andhowlargeordersshouldbeTypesofInventoryRawmaterialRawmaterialPurchasedbutnotprocessedPurchasedbutnotprocessedWorkWork--inin--processprocessUndergonesomechangebutnotcompletedUndergonesomechangebutnotcompletedAfunctionofcycletimeforaproductAfunctionofcycletimeforaproduct

Maintenance/repair/operating(MRO)Maintenance/repair/operating(MRO)NecessarytokeepmachineryandprocessesNecessarytokeepmachineryandprocessesproductiveproductiveFinishedgoodsFinishedgoodsCompletedproductawaitingshipmentCompletedproductawaitingshipmentOBJECTIVES

.InventorySystemDefined.TypesofInventory.Independentvs.DependentDemand

.InventorySystemModels

.Multi-PeriodInventoryModels:BasicFixed-OrderQuantityModels.InventoryCosts.Multi-PeriodInventoryModels:BasicFixed-TimePeriodModel.Single-PeriodInventoryModel.

MiscellaneousSystemsandIssuesInventory

Oneofthemostexpensiveassetsofmanycompaniesrepresentingasmuchas50%oftotalinvestedcapital


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Inventorymanagersmustbalanceinventoryinvestmentandcustomerservice

PurposesofInventory

1.Tomaintainindependenceofoperations2.Tomeetvariationinproductdemand3.Toallowflexibilityinproductionscheduling4.Toprovideasafeguardforvariationinrawmaterialdeliverytime5.Totakeadvantageofeconomicpurchase-ordersize


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TypesofInventory-2

.CycleInventory.SafetyStockInventory.AnticipatoryInventory.PipelineInventoryCycleInventory

.Inventorythatvariesdirectlywithlotsize..Lotsizevarieswithelapsedtimebetweenorders..Thequantityorderedmustmeetthedemandduringtheorderingperiod..Longgapsintheorderingperiodwillrequirelargercycleinventory.

.TheinventorymayvarybetweenordersizeQtozerojustbeforethenewlotisdelivered..AverageinventorysizeisthereforeQ/2SafetyStockInventory

.Safetystockinventoryprotectsagainstuncertaintiesindemand,leadtime,andsupply..Itensuresthatoperationsarenot

disruptedwhenproblemsoccur..Tobuildsafetystockanorderisplacedearlierthantheitemisneededortheorderedquantityislargerthanthequantityrequiredtillthenextdeliveryschedule.AnticipationInventory

.Inventoryusedtoabsorbunevenrateofdemandorsupply.

Predictableseasonaldemandpatternmayjustifyanticipationinventory..Unevendemandoftenmakesthefirmtostockpileduringlowproductiondemandtomakebetteruseofproductionfacilitiesandavoidvaryingoutputratesandlaborforce..Uncertaintiessuchasthreatenedstrikes,problematsuppliersfacilitiesetcalsojustify


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anticipationinventory.Inventorymovingfrompointtopointinthematerialflowsystemiscalledpipelineinventory-fromsupplierstoplant,fromoneoperationtothenextinprocessing,fromplanttodistributioncenterandfromdistributioncentertoretailerPipelineInventorybetweentwopoints,canbeexpressedintermsofleadtimeandaveragedemand(d)duringtheleadtime(L).PipelineInventory=dLPipelineInventoryIndependentvs.DependentDemandIndependentDemand(Demandforthefinalend-productordemandnotrelatedtootheritems)DependentDemand(Deriveddemanditemsforcomponentparts,subassemblies,rawmaterials,

etc)FinishedproductComponentparts


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InventorySystemsModelsInventoryModelsfor

-----Multi-PeriodInventoryModels-Fixed-OrderQuantityModelsEventtriggered(Example:runningoutofstock)-Fixed-TimePeriodModelsTimetriggered(Example:Monthlysalescallbysalesrepresentative)Single-PeriodInventoryModels-

-Onetimepurchasingdecision(Example:

-

vendorsellingt-shirtsatafootballgame)

-

-Seekstobalancethecostsofinventoryoverstockandunderstock

-

IndependentDemand

Needtodeterminewhenandhowmuchtoorder

.Basiceconomicorderquantity

.Productionorderquantity.QuantitydiscountmodelHoldingCosts

HHHooouuus

ssiiinnngggcccooossstttsss(((i


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iinnncccllluuudddiiinnngggrrreeennntttooorrrdddeeeppprrreeeccciiiaaattti

iiooonnn,,,ooopppeeerrraaatttiiinnng

ggcccooossstttsss,,,tttaaaxxxeeesss,,,i

iinnnsssuuurrraaannnccceee)))


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MMMaaattteeerrriiiaaalllhhhaaannndddllliiinnngggcccooossstttsss(((e

eeqqquuuiiipppmmmeeennntttllleeeaaassse

eeooorrrdddeeeppprrreeeccciiiaaattti

iiooonnn,,,pppooowwweeerrr,,,o


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oopppeeerrraaatttiiinnngggcccooosssttt)))LLLaaabbbooorrrcccooossst

ttIIInnnvvveeessstttmmmeeennntttc

ccooossstttsss(((bbbooorrrrrrooowwwiiin

nngggcccooossstttsss,,,tttaaax


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xxeeesss,,,aaannndddiiinnnsssuuurrraaannnccceeeooonnniiinnnvvveeennnt

ttooorrryyy)))PPPiiilllfffeeerrra

aagggeee,,,ssspppaaaccceee,,,aaannndddo

oobbbsssooollleeesssccceeennnc


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cceeeMulti-PeriodModels:Fixed-OrderQuantityModel

ModelAssumptions(Contd.)

.Inventoryholdingcostisbasedonaverageinventory.Orderingorsetupcostsareconstant.Alldemandsfortheproductwillbesatisfied(Nobackordersareallowed)Holding,Ordering,andSetupCosts

.Holdingcosts-thecostsofholdingorcarryinginventoryovertime.

Orderingcosts-thecostsofplacinganorderandreceivinggoods.Setupcosts-costtoprepareamachineorprocessformanufacturinganorderMulti-PeriodModels:Fixed-OrderQuantityModel

Assumptions

.Demandfortheproductisconstantanduniformthroughouttheperiod.Leadtime(timefromorderingtoreceipt)isconstant.Priceperunitofproductisconstant


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BasicFixed-OrderQuantityModelandReorderPointBehaviorR=ReorderpointQ=[Economic]orderquantityL=LeadtimeLLQQQRTimeNumberofunitsonhand1.YoureceiveanorderquantityQ.2.Youstartusingthemupovertime.3.WhenyoureachdowntoalevelofinventoryofR,youplaceyournextQsizedorder.4.Thecyclethenrepeats.CostMinimizationGoalOrderingCostsHoldingCosts

OrderQuantity(Q)COSTAnnualCostofItems(DC)TotalCostQOPTByaddingtheitem,holding,andorderingcoststogether,wedeterminethetotalcostcurve,whichinturnisusedtofindtheQoptinventoryorderpointthatminimizestotalcosts

TheEOQModelQQ=Numberofpiecesperorder=NumberofpiecesperorderQ*Q*=Optimalnumberofpiecesperorder(EOQ)=Optimalnumberofpiecesperorder(EOQ)DD=AnnualdemandinunitsfortheInventoryitem=AnnualdemandinunitsfortheInventoryitemSS=Setupororderingcostforeachorder=SetupororderingcostforeachorderHH=Holdingorcarryingcostperunitperyear=HoldingorcarryingcostperunitperyearAnnualsetupcostAnnualsetupcost==((NumberofordersplacedperyearNumberofordersplacedperyear))x(x(SetuporordercostperorderSetuporordercostperorder))AnnualdemandAnnualdemand

NumberofunitsineachorderNumberofunitsineachorderSetupororderSetuporordercostperordercostperorder===(=(SS))DDQQAnnualsetupcost=SDQTheEOQModel


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QQ=Numberofpiecesperorder=NumberofpiecesperorderQ*Q*=Optimalnumberofpiecesperorder(EOQ)=Optimalnumberofpiecesperorder(EOQ)DD=AnnualdemandinunitsfortheInventoryitem=AnnualdemandinunitsfortheInventoryitemSS=Setupororderingcostforeachorder=SetupororderingcostforeachorderHH=Holdingorcarryingcostperunitperyear=HoldingorcarryingcostperunitperyearAnnualholdingcostAnnualholdingcost==((AverageinventorylevelAverageinventorylevel))x(x(HoldingcostperunitperyearHoldingcostperunitperyear))OrderquantityOrderquantity22=(=(HoldingcostperunitperyearHoldingcostperunitperyear))=(=(HH))QQ22Annualsetupcost=SDQAnnualholdingcost=HQ2TheEOQModelQ=NumberofpiecesperorderQ*=Optimalnumberofpiecesperorder(EOQ)

D=AnnualdemandinunitsfortheInventoryitemS=SetupororderingcostforeachorderH=HoldingorcarryingcostperunitperyearOptimalorderquantityisfoundwhenannualsetupcostequalsannualholdingcostAnnualsetupcost=SDQAnnualholdingcost=HQ2DDQQSS==HHQQ

22SolvingforQ*SolvingforQ*2DS=Q2HQ2=2DS/HQ*=2DS/HBasicFixed-OrderQuantity(EOQ)ModelFormulaH2Q+SQD

+DC=TCTotalAnnual=CostAnnualPurchaseCostAnnualOrderingCost


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AnnualHoldingCost++========TC=TotalannualcostD=DemandC=CostperunitQ=OrderquantityS=CostofplacinganorderorsetupcostR=ReorderpointL=LeadtimeH=Annualholdingandstoragecost

perunitofinventory


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DerivingtheEOQDerivingtheEOQTheEconomicOrderingQuantity(EOQ)==Q=2DSH=2(AnnualDemand)(OrderorSetupCost)AnnualHoldingCostOPT=_Reorderpoint,R=dL_d=averagedailydemand(constant)L=Leadtime(constant)_WealsoneedareorderpointtotelluswhentoplaceanorderEOQExample-1DetermineoptimalnumberofunitstoorderD=1,000unitsS=$10perorder

H=$.50perunitperyear========EOQExample-1a

Determineexpectednumberofordersif:D=1,000unitsQ*=200unitsS=$10perorderH=$.50perunitperyear

=======

==

======

EOQExample-1bDeterminetimebetweenordersif:Determinetimebetweenordersif:DD=1,000=1,000unitsQ*unitsQ*=200=200unitsunitsSS=$10=$10perorderNperorderN=5=5ordersperyearordersperyearHH=$.50=$.50perunit/yrworkingdays=250days/yrperunit/yrworkingdays=250days/yr====

====EOQExample-1c

DDDeeettteeerrrmmmi


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iinnneeecccaaarrrrrryyyiiinnngggcccooossstttiiifff:::DDD===111,,,0000

00000uuunnniiitttsssQQQ***===222000000u

uunnniiitttsssSSS===$$$111000pppeeer

rrooorrrdddeeerrrNNN===555ooor


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rrdddeeerrrssspppeeerrryyyeeeaaarrrHHH===$$$...555000pppeeerrruuunnni

iitttpppeeerrryyyeeeaaarrrTTT===555000d

ddaaayyysss

TTToootttaaalllcccaaar

rrrrryyyiiinnngggcccooosssttt=


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==SSSeeetttuuupppcccooosssttt+++HHHooollldddiiinnngggcccooosssttt

=

==

============


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Price-BreakModelFormulaorQuantityDiscountModelBasedonthesameassumptionsastheEOQmodel,theprice-breakmodelhasasimilarQoptformula:

QuantityDiscountModelorPrice-BreakModel

AnnualHoldingCost2(AnnualDemand)(OrderorSetupCost)=iC2DS=QOPTi=percentageofunitcostattributedtocarryinginventoryC=costperunitSinceCchangesforeachprice-break,theformulaabovewillhavetobeusedwitheachprice-breakcostvalue

Price-BreakExample-2ProblemData(Part1)-

-Acompanyhasachancetoreducetheirinventoryorderingcostsbyplacinglargerquantityordersusingtheprice-breakorderquantityschedulebelow.Whatshouldtheiroptimalorderquantitybeifthiscompanypurchasesthissingleinventoryitemwithane-mailorderingcostof$4,acarryingcostrateof2%oftheinventorycostoftheitem,andanannualdemandof10,000units?

Price-BreakExample-2Solution(Part2)units1,826=0.02(1.20)

4)2(10,000)(=iC2DS=QOPTAnnualDemand(D)=10,000unitsCosttoplaceanorder(S)=$4First,plugdataintoformulaforeachprice-breakvalueofCunits2,000=0.02(1.00)4)2(10,000)(=iC

2DS=QOPTunits2,020=0.02(0.98)4)2(10,000)(=iC2DS=QOPTCarryingcost%oftotalcost(i)=2%


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Costperunit(C)=$1.20,$1.00,$0.98Intervalfrom0to2499,theQoptvalueisfeasibleIntervalfrom2500-3999,theQoptvalueisnotfeasibleIntervalfrom4000&more,theQoptvalueisnotfeasibleNext,determineifthecomputedQoptvaluesarefeasibleornotOrderQuantityunits)Price/unit($)0to2,499$1.202,500to3,999$1.004,000ormore$0.98

Price-BreakExample-3Solution(Part3)--Sincethefeasiblesolutionoccurredinthefirstprice-break,itmeansthatalltheothertrueQoptvaluesoccuratthebeginningsofeachprice-breakinterval.Why?Totalannualcosts-Sothecandidates

fortheprice-breaksare1826,2500,and4000unitsBecausethetotalannualcostfunctionisaushapedfunctionPrice-BreakExample-3Solution(Part4)

-Next,weplugthetrueQoptvaluesintothetotalcostannualcostfunctiontodeterminethetotalcostundereachprice-break

iC2Q+SQD+DC=TC-==-==TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)=$12,043.82

TC(2500-3999)=$10,041TC(4000&more)=$9,849.20Finally,weselecttheleastcostlyQopt,whichisthisproblemoccursinthe4000&moreinterval.Insummary,ouroptimalorderquantityis4000units0182625004000OrderQuantity


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Price-BreakExample-3Solution(Part5)Price-BreakExample-3Solution(Part5)HowImportantistheItem?SegmentationofInventory-Notallinventoryiscreatedequally-Differentclassesofinventory-Resultindifferentlevelsofprofitability/revenue-Havedifferentdemandpatternsandmagnitudes-RequiredifferentcontrolpoliciesABCAnalysisCommonlyusedinpracticeClassifyitemsbyrevenueorvalueCombinationofusage,salesprice,etc.ABCAnalysisIdentifytheitemsthatmanagementshouldspendtimeonPrioritizeitemsbytheirvaluetofirmCreatelogicalgroupingsAdjustasneededABCAnalysisMiscellaneousSystemsandIssues

ABCAnalysis

.Whatisdifferentbetweentheclasses?AItems

VeryfewhighimpactitemsareincludedRequirethemostmanagerialattentionandreviewExpectmanyexceptionstobemade

BItems

Manymoderateimpactitems(sometimesmost)Automatedcontrolw/managementbyexceptionRulescanbeusedforA(butusuallytoomanyexceptions)

CItems

ManyifnotmostoftheitemsthatmakeupminorimpactControlsystemsshouldbeassimpleaspossibleReducewastedmanagementtimeandattentionGroupintocommonregions,suppliers,endusers

.

Butthesearearbitraryclassifications


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MiscellaneousSystems:BinSystemsTwo-BinSystemFullEmptyOrderOneBinofInventoryOne-BinSystemPeriodicCheckOrderEnoughtoRefillBinMiscellaneousSystems:OptionalReplenishmentSystemMActualInventoryLevel,Iq=M-IIQ=minimumacceptableorderquantityIfq>Q,orderq,otherwisedonotorderany.MiscellaneousSystems:BinSystemsTwo-BinSystemFullEmptyOrderOneBinof

InventoryOne-BinSystemPeriodicCheckOrderEnoughtoRefillBinMiscellaneousSystems:OptionalReplenishmentSystemMActualInventoryLevel,Iq=M-IIQ=minimumacceptableorderquantityIfq>Q,orderq,otherwisedonotorderany.

InventoryAccuracyandCycleCountingInventoryaccuracyreferstohowwelltheinventoryrecordsagreewithphysicalcountCycleCountingisaphysicalinventory-takingtechniqueinwhichinventoryiscountedonafrequentbasisratherthanonceortwiceayearSupplyChainManagementInventoryControlSafetyStock,FixedPeriodModeland

SinglePeriodModelQuestion

Onaverage,Isell150,000unitsayear,whichIobtainfromawholesaler.Iestimatethatthecosttomeofplacinganorderis$50withtheaverageinventorystoragecostbeing20%peryearofthecostofaunit($5).

1.


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Whatwouldbetheoptimalorderquantity?2.Icurrentlyorder5timesayear.HowmuchwouldIsavebyswitchingtotheoptimalorderquantityascomparedwithmycurrentpolicyofordering5timesayear?PlannedShortageswithBack-Orders

.Shortage:whencustomerdemandcannotbemet.PlannedshortagescouldbebeneficialCostofkeepingitemismoreexpensivethantheprofitfromsellingite.g.car


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UncertainDemandUncertainDemandUncertainDemand-SafetyStock

.Bufferaddedtoonhandinventoryduringleadtime.Extrareservedstock.Topreventstock-outunderuncertaindemand.Safetystockwillnotnormallybeused,butitisavailableunderuncertaindemandHowmuchsafetystockshouldwehold?Judgmentonservicelevel

ReorderLevelReorderLevel(ROL)=LTxD

ReorderLevel(ROL)=(LTxD)+SafetyStockSafetyStockServiceLevel

.Atargetfortheproportionofdemandthatismetdirectlyfromstock.Themaximumacceptableprobabilitythatademandcanbemetfromthestock.Forexample90%servicelevel90%chanceofmeetingdemandduringleadtimeor

10%chanceofnotmeetingdemand(havingback-orderorlostsales)

ProbabilisticModels

.Sofarweassumedthatdemandisconstantanduniform..However,InProbabilisticmodels,demandisspecifiedasaprobabilitydistribution..Uncertaindemandraisesthepossibilityof

astockout(orshortage).ProbabilisticModels

.Onemethodofreducingstockoutsistoholdextrainventory(calledSafetyStock)..Inthiscase,wechangetheROPformulatoincludethatsafetystock(ss).


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SafetyStockExample

.ROP=50unitsStock-outcost=$40perunit.Ordersperyear=6Carryingcost=$5perunitperyearNumberofUnitsProbability

300.2400.2ROP500.3600.2700.11.0ExampleofProbabilityCurve

Demand(No.ofBuns):4005006007008009001000ProbabilityofDemand0.050.10.20.30.20.10.05

SafetyStockExample

.ROP=50unitsStock-outcost=$40perunit

.Ordersperyear=6Carryingcost=$5perunitperyearAsafetystockof20unitsgivesthelowesttotalcostROP=50+20=70units

ExampleProbabilisticDemand


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ReorderPointforaServiceLevelUsingtheStandardNormalProbabilityTableReorderPointforaServiceLevelUsingtheStandardNormalProbabilityTableUsingtheStandardNormalProbabilityTable=ProbabilisticDemand

Demandisvariableandleadtimeisconstant

.Safetystock,SS:=Zstandarddeviationofleadtime=ZsLT

=Zdlt

.Reorderlevel:ROL=leadtimedemand+safetystock=LTD+ZsLT

.wheres

=standarddeviationofdemandperdayand.dlt=sLTStandarddeviationofdemandduringleadtimeProbabilisticDemand

1.IfmeandemandLandvariationovertheleadtime,dltareknownyoucanusethisequation.

whereX=ROPorROL2)Ifdailydemand,daily,d,dailyvariation,sandleadtimeLTisknown

.ROP(ROL)=leadtimedemand+safetystock=(LTd)+(ZsLT)


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.Safetystock,SS:=Zstandarddeviationofleadtime=ZsLT.wheres=standarddeviationofdemandperdayandZX-

=L

dlt

.dlt=s

LTStandarddeviationofdemandduringleadtime.=L*dLt


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Example-3:SafetyStockMeandailydemand,D=500gm/dayLeadTime,LT=7daysStandarddeviation,=50gm/dayServicelevelrequired=98%or0.98FromnormaldistributionlevelZisdeterminedasz=2.05ROL=(LTD)+zLT=(500x7)+2.05*50*7=3771gmSafetyStock=zLT=2.05*50*7=271gmExample-3:SafetyStockMeandailydemand,D=500gm/dayLeadTime,LT=7daysStandarddeviation,=50gm/dayServicelevelrequired=98%or0.98FromnormaldistributionlevelZisdeterminedasz=2.05ROL=(LTD)+zLT=(500x7)+2.05*50*7=3771gmSafetyStock=zLT=2.05*50*7=271gmSupplyChain

ManagementMaximumInventoryLevel,MPeriodicReviewSystemMActualInventoryLevel,Iq=M-IIExample-3:SafetyStock

.Dailyusageatadrugstorefollowsanormaldistributionwithameanof500gmandastandarddeviationof50gm.Ifthe

leadtimeforprocurementis7daysandthedrugstorewantsariskofonly2%determinea)reorderpointandb)safetystocknecessary

Example:SafetyStockusingZ-Score

.MeanDemandinleadperiod,L=3500gm.

Standarddeviation,s=50gm/day.L=vLt=507gm.Z=2.05fromTableX-


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Z=L

sL

whereXisanormalrandomvariable

.X=3771gm.Safetystock=3771gm-3500gm=271gmInventoryControl

PeriodicReviewSystem

P-System:PeriodicReviewSystem

.Inthissystem,costsarenotexplicitlyconsideredandorderquantityisnotfixed..Timeistakenintoaccountandgivenmore

emphasis.Inventoryisperiodicallyreviewedatfixedintervalsandanydifferencebetweenthepresentandthelastreviewismadeupbyreplenishmentorder..Theorderquantityisthusequaltoreplenishmentlevelminusactualinventoryonhand.


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P-System:PeriodicReviewSystem-2Inthissystem,weareinterestedinactualandaverageconsumptionoveraperiodoftimei.e.timebetweentworeviewsandleadtime.Orderquantitycanbecomputedasfollows:IfLRthenQ=MI-QordWhereL=LeadTimeR=ReviewPeriodM=ReplenishmentLevelinUnitsI=InventoryonhandinUnitsQ=QuantitytobeOrderedQord=Quantityonorder(inpipeline)Example:FixedPeriodInventoryControlSystem(P-System)Theaveragemonthlyconsumptionofanitemis40units,SafetyStockis20units,reviewtimeis1monthandleadtimeis15days,calculatereplenishment

levelMSafetyStock=BRReplen.Lvl.=MLT204060123Example-Solution:P-SystemLreviewtimeBuffer/safetystock=50unitsD=100units/monthReviewTime=1monthL=2monthsM=replinsh.Lvl.=B+D(1+2)=50+100*3M=350UnitsI=B+D/2=50+50=100unitsOrderQtyQ=MI=250units

IfQtyalreadyonorderis100units(reviewafter1mth)Q=M-I-Qord=150unitsSingle-PeriodInventoryModelDecisionunderuncertainity&riskIninventorycontrol,sometimesmanagementhastotakeriskunderuncertainity,thoughwantingtokeeptheriskfactortoaminimum.HowmanyWorldCupshirtstoproduce,whentheshirtswillbeoflittleornovalueaftertheCup.


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HowmanysuitstostockforEidorXmasseason,profitmarginishighbuttheleftoverstockwillprobablybeofnovalueSingle-periodinventorymodelAppliesinthesecasesP-System:PeriodicReviewSystem-2Inthissystem,weareinterestedinactualandaverageconsumptionoveraperiodoftimei.e.timebetweentworeviewsandleadtime.Orderquantitycanbecomputedasfollows:IfLRthenQ=MI-QordWhereL=LeadTimeR=ReviewPeriodM=ReplenishmentLevelinUnitsI=InventoryonhandinUnitsQ=QuantitytobeOrderedQord=Quantityonorder(inpipeline)Example:FixedPeriodInventoryControlSystem(P-System)Theaveragemonthlyconsumptionofanitemis40units,SafetyStockis20units,reviewtimeis1monthand

leadtimeis15days,calculatereplenishmentlevelMSafetyStock=BRReplen.Lvl.=MLT204060123Example-Solution:P-System

LreviewtimeBuffer/safetystock=50unitsD=100units/monthReviewTime=1monthL=2monthsM=replinsh.Lvl.=B+D(1+2)=50+100*3

M=350UnitsI=B+D/2=50+50=100unitsOrderQtyQ=MI=250unitsIfQtyalreadyonorderis100units(reviewafter1mth)Q=M-I-Qord=150unitsSingle-PeriodInventoryModelDecisionunderuncertainity&riskIninventorycontrol,sometimesmanagementhastotakeriskunderuncertainity,thoughwantingtokeeptherisk


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factortoaminimum.HowmanyWorldCupshirtstoproduce,whentheshirtswillbeoflittleornovalueaftertheCup.HowmanysuitstostockforEidorXmasseason,profitmarginishighbuttheleftoverstockwillprobablybeofnovalueSingle-periodinventorymodelAppliesinthesecases


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Single-PeriodInventoryModelThismodelstatesthatweshouldstockuptothepointwhereincrementalgain(IG)isequaltoincrementalloss(IL)=IGistheprofitperitemtimestheprobabilityofsellingxitemsIG=m.P(x)=IListhecostperitemtimestheprobabilitythatxitemswillnotbesoldIL=C.[1-P(x)].EquatingIG&ILandsolvingtheequationweget:===m=marginofprofititemP(x)=probabilityofsellingtheitem

C=Costoftheitem=P(x)=Cm+CSingle-PeriodInventoryModelThismodelstatesthatweshouldstockuptothepointwhereincrementalgain(IG)isequaltoincrementalloss(IL)=IGistheprofitperitemtimestheprobabilityofsellingxitemsIG=m.P(x)

=IListhecostperitemtimestheprobabilitythatxitemswillnotbesoldIL=C.[1-P(x)].EquatingIG&ILandsolvingtheequationweget:===m=marginofprofititemP(x)=probabilityofsellingtheitem

C=Costoftheitem=P(x)=Cm+CSinglePeriodModelExample-4

.Ourcollegebasketballteamisplayinginatournamentgamethisweekend.Basedonourpastexperiencewesellonaverage2,400shirtswithastandarddeviationof350.Wemake$10onevery


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shirtwesellatthegame,butlose$5oneveryshirtnotsold.Howmanyshirtsshouldwemakeforthegame?Cu=$10andCo=$5;P=$10/($10+$5)=.667

=.432thereforeweneed2,400+.432(350)=2,551shirts

Z.667

Single-PeriodInventoryModeluouCCCP+Thismodelstatesthatweshouldcontinuetoincreasethesizeoftheinventorysolongastheprobabilityofsellingthelastunitaddedisequaltoorgreaterthanthe

ratioof:Cu/Co+CuWhere:Co=CostperunitofdemandoverestimatedCu=CostperunitofdemandunderestimatedP=Probabilitythattheunitwillbesold

Example-5(Solution)uouCC

CP+soldbeunitwillythattheProbabilitestimatedunderdemandofunitperCostCestimatedoverdemandofunitperCostC:Whereuo===PCo=Rs1.5[(Cost)Lossifdemandisoverestimated]Cu=Rs2.5[(Cost)ProfitLossifdemandisunderestimated]P[2.5/(2.5+1.5)]

P0.625Probabilityofmeetingdemandis0.65at700buns.Thebakershouldmake700buns.Example-6

.AhmedJuicesmakesavarietyofjuicesforon-thecountersales.Ahmedusesice,whichhegratesinmakingthesedrinks.IceissuppliedtoAhmedinlarge


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blocks,eachcostingRs10.Iceblocksnotusedduringadaygetswastedastheicemeltsandcannotbeusedthenextday.IfAhmedisshortoficeblocksonanyday,hebuysthemfromelsewhere,butatapremiumofRs5perblock.Eachblockoficecanbeusedfor20glassesofjuice.Theprobabilitydistributionforthedemandoficeblocksisasfollows.WhatistheleastcoststockingpolicyforAhmedJuices?.xiceblocks:202122232425262728.pProbability00.050.100.200.250.200.150.050Theend


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Chapter 6 Inventory Control

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