ctl.sc1x - supply chain fundamentals key concept... · these are meant to complement, not replace,...

Summer 2017�CTL.SC1x – Supply Chain Fundamentals Key Concepts�MITx MicroMasters in Supply Chain Management MIT Center for Transportation & Logistics�Cambridge, MA 02142 USA �scm_mm@mit.edu ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-ShareAlike4.0InternationalLicense.

CTL.SC1x-SupplyChainFundamentals

KeyConceptsDocumentThisdocumentcontainstheKeyConceptsfortheSC1xcourse,version2.Thesearemeanttocomplement,notreplace,thelessonvideosandslides.Theyareintendedtobereferencesforyoutousegoingforwardandarebasedontheassumptionthatyouhavelearnedtheconceptsandcompletedthepracticeproblems.ThedraftwasupdatedandrevisedbyDr.AlexisBatemanintheSummerof2017.Thisisadraftofthematerial,sopleasepostanysuggestions,corrections,orrecommendationstotheDiscussionForumunderthetopicthread“KeyConceptDocumentsImprovements.Thanks,ChrisCaplice,EvaPonceandtheSC1xTeachingCommunity

Summer 2017�CTL.SC1x – Supply Chain Fundamentals Key Concepts�MITx MicroMasters in Supply Chain Management MIT Center for Transportation & Logistics�Cambridge, MA 02142 USA �scm_mm@mit.edu This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

TableofContentsCoreSupplyChainConcepts.................................................................................................................3

DemandForecasting.............................................................................................................................9TimeSeriesAnalysis...............................................................................................................................12ExponentialSmoothing..........................................................................................................................14ExponentialSmoothingwithHolt-Winter..............................................................................................16SpecialCases..........................................................................................................................................17

InventoryManagement......................................................................................................................23EconomicOrderQuantity(EOQ)............................................................................................................25EconomicOrderQuantity(EOQ)Extensions..........................................................................................28SinglePeriodInventoryModels..............................................................................................................32SinglePeriodInventoryModels-ExpectedProfitability..........................................................................37ProbabilisticInventoryModels...............................................................................................................37InventoryModelsforMultipleItems&Locations..................................................................................45InventoryModelsforClassA&CItems.................................................................................................50

Warehousing......................................................................................................................................57WarehousingBasics...............................................................................................................................57CoreOperationalFunctions...................................................................................................................58Profiling&AssessingPerformance........................................................................................................61

FundamentalsofFreightTransportation............................................................................................64LeadTimeVariability&ModeSelection................................................................................................65

AppendixA&BUnitNormalDistribution,PoissonDistributionTables...............................................69

CoreSupplyChainConcepts

SummaryVirtuallyallsupplychainsareacombinationofpushandpullsystems.Apushsystemiswhereexecutionisperformedaheadofanactualordersothattheforecasteddemand,ratherthanactualdemand,hastobeusedinplanning.Apullsystemiswhereexecutionisperformedinresponsetoanordersothattheactualdemandisknownwithcertainty.Thepointintheprocesswhereasupplychainshiftsfrombeingpushtopullissometimescalledthepush/pullboundaryorpush/pullpoint.Inmanufacturing,thepush/pullpointisalsoknownasthedecouplingpoint(DP)orcustomerorderdecouplingpoint(CODP).TheCODPcoincideswithanimportantstockpoint,wherethecustomerorderarrives(switchinginventorybasedonaforecasttoactualdemand),andalsoallowstodifferentiatebasicproductionsystems:make-to-stock,assemble-to-order,make-to-order,orengineer-to-order.Postponementisacommonstrategytocombinethebenefitsofpush(productreadyfordemand)andpull(fastcustomizedservice)systems.Postponementiswheretheundifferentiatedraworcomponentsare“pushed”throughaforecast,andthefinalfinishedandcustomizedproductsarethen“pulled”.Segmentationisamethodofdividingasupplychainintotwoormoregroupingswherethesupplychainsoperatedifferentlyandmoreefficientlyand/oreffectively.Whiletherearenoabsoluterulesforsegmentation,therearesomerulesofthumb,suchas:itemsshouldbehomogenouswithinthesegmentandheterogeneousacrosssegments;thereshouldbecriticalmasswithineachsegment;andthesegmentsneedtobeusefulandcommunicable.Asegmentonlymakessenseifitdoessomethingdifferent(planning,inventory,transportationetc.)fromtheothersegments.ThemostcommonsegmentationisforproductsusinganABCclassification.InanABCsegmentation,theproductsdrivingthemostrevenue(orprofitorsales)areClassAitems(theimportantfew).Productsdrivingverylittlerevenue(orprofitorsales)areClassCitems(thetrivialmany),andtheproductsinthemiddleareClassB.Acommonbreakdownisthetop20%ofitems(ClassA)generate80%oftherevenue,ClassBis30%oftheproductsgenerating15%oftherevenue,andtheClassCitemsgeneratelessthan5%oftherevenuewhileconstituting50%oftheitems.Supplychainsoperateinuncertainty.Demandisneverknownexactly,forexample.Inordertohandleandbeabletoanalyzesystemswithuncertainty,weneedtocapturethedistributionofthevariableinquestion.Whenwearedescribingarandomsituation,say,theexpecteddemandforpizzasonaThursdaynight,itishelpfultodescribethepotentialoutcomesintermsofthecentraltendency(meanormedian)aswellasthedispersion(standarddeviation,range).Wewilloftencharacterizethedistributionofpotentialoutcomesasfollowingawell-knownfunctionsuchasNormalandPoisson.

KeyConcepts

Pullvs.PushProcess• Push—workperformedinanticipationofanorder(forecasteddemand)

• Pull—executionperformedinresponsetoanorder(demandknownwithcertainty)

• HybridorMixed—pushrawproducts,pullfinishedproduct(postponementordelayed

differentiation)

• Push/pullboundarypoint—pointintheprocesswhereasupplychainshiftsfrombeing

pushtopull

• Inmanufacturing,alsoknownas“decouplingpoint”(DP)or“customerorderdecoupling

point”(CODP)—thepointinthematerialflowwheretheproductislinkedtoaspecific

customer

• Masscustomization/Postponement—todelaythefinalassembly,customization,or

differentiationofaproductuntilaslateaspossible

Segmentation• Differentiateproductsinordertomatchtherightsupplychaintotherightproduct

• Productstypicallysegmentedon

o Physicalcharacteristics(value,size,density,etc.)

o Demandcharacteristics(salesvolume,volatility,salesduration,etc.)

o Supplycharacteristics(availability,location,reliability,etc.)

• Rulesofthumbfornumberofsegments

o Homogeneous—productswithinasegmentshouldbesimilar

o Heterogeneous—productsacrosssegmentsshouldbeverydifferent

o CriticalMass—segmentshouldbebigenoughtobeworthwhile

o Pragmatic—segmentationshouldbeusefulandcommunicable

• Demandfollowsapowerlawdistribution,meaningalargevolumeofsalesis

concentratedinfewproducts

PowerLawThedistributionofpercentsalesvolumetopercentofSKUs(StockKeepingUnits)tendstofollowaPowerLawdistribution(y=axk)whereyispercentofdemand(unitsorsalesorprofit),xispercentofSKUs,andaandkareparameters.Thevalueforkshouldobviouslybelessthan1sinceifk=1therelationshipislinear.Inadditiontosegmentingaccordingtoproducts,manyfirmssegmentbycustomer,geographicregion,orsupplier.Segmentationistypicallydoneusingrevenueasthekeydriver,butmanyfirmsalsoincludevariabilityofdemand,profitability,andotherfactors,toinclude:

• Revenue=averagesales*unitsalesprice;

• Profit=averagesales*margin;

• Margin=unitsalesprice–unitcost.

HandlingUncertaintyUncertaintyofanoutcome(demand,transittime,manufacturingyield,etc.)ismodeledthroughaprobabilitydistribution.Wediscussedtwointhelesson:PoissonandNormal.NormalDistribution~N(µ,σ)ThisistheBellShapeddistributionthatiswidelyusedbybothpractitionersandacademics.Whilenotperfect,itisagoodplacetostartformostrandomvariablesthatyouwillencounterinpracticesuchastransittimeanddemand.Thedistributionisbothcontinuous(itcantakeanynumber,notjustintegersorpositivenumbers)andissymmetricarounditsmeanoraverage.Beingsymmetricadditionallymeansthemeanisalsothemedianandthemode.ThecommonnotationthatwewillusetoindicatethatsomevaluefollowsaNormalDistributionis~N(µ,σ)wheremu,µ,isthemeanandsigma,σ,isthestandarddeviation.Somebooksusethenotation~N(µ,σ2)showingthevariance,σ2,insteadofthestandarddeviation.Justbesurewhichnotationisbeingfollowedwhenyouconsultothertexts.

TheNormalDistributionisformallydefinedas:

WewillalsomakeuseoftheUnitNormalorStandardNormalDistribution.Thisis~N(0,1)wherethemeaniszeroandthestandarddeviationis1(asisthevariance,obviously).Thechartbelowshowsthestandardorunitnormaldistribution.WewillbemakinguseofthetransformationfromanyNormalDistributiontotheUnitNormal(SeeFigure1).

Figure1.StandardNormalDistribution

Wewillmakeextensiveuseofspreadsheets(whetherExcelorLibreOffice)tocalculateprobabilitiesundertheNormalDistribution.Thefollowingfunctionsarehelpful:

f x x0( ) = e−(x0−µ )

2σ x2

σ x 2π

-6 -4 -2 0 2 4 6

StandardNormalDistribution(μ=0,σ2=1)

• NORMDIST(x,µ,σ,true)=theprobabilitythatarandomvariableislessthanorequalto

xundertheNormalDistribution~N(µ,σ).So,thatNORMDIST(25,20,3,1)=0.952

whichmeansthatthereisa95.2%probabilitythatanumberfromthisdistributionwill

belessthan25.

• NORMINV(probability,µ,σ)=thevalueofxwheretheprobabilitythatarandom

variableislessthanorequaltoitisthespecifiedprobability.So,NORMINV(0.952,20,3)

TousetheUnitNormalDistribution~N(0,1)weneedtotransformthegivendistributionbycalculatingakvaluewherek=(x-µ)/σ.Thisissometimescalledazvalueinstatisticscourses,butinalmostallsupplychainandinventorycontextsitisreferredtoasakvalue.So,inourexample,k=(25–20)/3=1.67.WhydoweusetheUnitNormal?Well,thekvalueisahelpfulandconvenientpieceofinformation.Thekisthenumberofstandarddeviationsthevaluexisabove(orbelowifitisnegative)themean.Wewillbelookingatanumberofspecificvaluesforkthatarewidelyusedasthresholdsinpractice,specifically:

• Probability(x≤0.90)wherek=1.28

BecausetheNormalDistributionissymmetric,therearealsosomecommonconfidenceintervals:

• μ±σ 68.3%—meaningthat68.3%ofthevaluesfallwithin1standarddeviationofthe

• μ±2σ 95.5%—95.5%ofthevaluesfallwithin2standarddeviationsofthemean,and

• μ±3σ 99.7%—99.7%ofthevaluesfallwithin3standarddeviationsofthemean.

Inaspreadsheetsyoucanusethefunctions:• NORMSDIST(k)=theprobabilitythatarandomvariableislessthankunitsabove(or

below)mean.Forexample,NORMSDIST(2.0)=0.977meaningthe97.7%ofthe

distributionislessthan2standarddeviationsabovethemean.

• NORMSINV(probability)=thevaluecorrespondingtothegivenprobability.SothatNORMSINV(0.977)=2.0.IfIthenwantedtofindthevaluethatwouldcover97.7%ofa

specificdistribution,saywhere~N(279,46)Iwouldjusttransformit.Sincek=(x-µ)/σ

forthetransformation,Icansimplysolveforxandget:x=µ+kσ=279+(2.0)(46)=

371.Thismeansthattherandomvariable~N(279,46)willbeequalorlessthan371for

97.7%ofthetime.

Poissondistribution~Poisson(λ)WewillalsousethePoisson(pronouncedpwa-SOHN)distributionformodelingthingslikedemand,stockouts,andotherlessfrequentevents.ThePoisson,unliketheNormal,isdiscrete(itcanonlybeintegers≥0),alwayspositive,andnon-symmetric.Itisskewedright–thatis,it

hasalongrighttail.Itisverycommonlyusedforlowvaluedistributionsorslowmovingitems.WhiletheNormalDistributionhastwoparameters(muandsigma),thePoissononlyhasone,lambda,λ.Formally,thePoissonDistributionisdefinedasshownbelow:

Thechartbelow(Figure2)showsthePoissonDistributionforλ=3.ThePoissonparameterλisboththemeanandthevarianceforthedistribution!Notethatλdoesnothavetobeaninteger.

Figure2.PoissonDistribution

Inspreadsheets,thefollowingfunctionsarehelpful:• POISSON(x0,λ,false)=>P(x=x0)=theprobabilitythatarandomvariableisequaltox0

underthePoissonDistribution~P(λ).So,thatPOISSON(2,1.56,0)=0.256whichmeans

thatthereisa25.6%probabilitythatanumberfromthisdistributionwillbeequalto2.

• POISSON(x0,λ,true)=>P(x≤x0)=theprobabilitythatarandomvariableislessthanor

equaltox0underthePoissonDistribution~P(λ).So,thatPOISSON(2,1.56,1)=0.793

whichmeansthatthereisa79.3%probabilitythatanumberfromthisdistributionwill

belessthanorequalto2.Thisissimplyjustthecumulativedistributionfunction.

Uniformdistribution~U(a,b)WewillsometimesusetheUniformdistribution,whichhastwoparameters:aminimumvalueaandamaximumvalueb.Eachpointwithinthisrangeisequallylikelytooccur.TofindthecumulativeprobabilityforsomevalueC,theprobabilitythatx≤c=(c-a)/(b-a),thatis,theareafromatocminusthetotalareafromatob.Theexpectedvalueorthemeanissimply(a+b)/2

whilethestandarddeviationis=(" − $)/√12.

p[x0 ]= Prob x = x0!" #$=e−λλ x0

x0 !for x0 = 0,1,2,...

F[x0 ]= Prob x ≤ x0!" #$=e−λλ x

0 1 2 3 4 5 6 7 8 9

PoissonDistribution(λ=3)

LearningObjectives

• Identifyandunderstanddifferencesbetweenpushandpullsystems.

• Understandwhyandhowtosegmentsupplychainsbyproducts,customers,etc.

• Abilitytomodeluncertaintyinsupplychains,primarily,butnotexclusively,indemand

uncertainty.

ReferencesPush/PullProcesses:Chopra&MeindlChpt1;NahmiasChpt7;Segmentation:NahmiasChpt5;Silver,Pyke,&PetersonChpt3;BallouChpt3ProbabilityDistributions:Chopra&MeindlChpt12;NahmiasChpt5;Silver,Pyke,&PetersonAppB

Fisher,M.(1997)“WhatIstheRightSupplyChainforYourProduct?,”HarvardBusinessReview.

Olavson,T.,Lee,H.&DeNyse,G.(2010)“APortfolioApproachtoSupplyChainDesign,”SupplyChainManagementReview.

DemandForecasting

SummaryForecastingisoneofthreecomponentsofanorganization’sDemandPlanning,Forecasting,andManagementprocess.DemandPlanninganswersthequestion“Whatshouldwedotoshapeandcreatedemandforourproduct?”andconcernsthingslikepromotions,pricing,packaging,etc.DemandForecastingthenanswers“Whatshouldweexpectdemandforourproducttobegiventhedemandplaninplace?”Thefinalcomponent,DemandManagement,answersthequestion,“Howdoweprepareforandactondemandwhenitmaterializes?”ThisconcernsthingslikeSales&OperationsPlanning(S&OP)andbalancingsupplyanddemand.WithintheDemandForecastingcomponent,youcanthinkofthreelevels,eachwithitsowntimehorizonandpurpose.Strategicforecasts(years)areusedforcapacityplanning,investmentstrategies,etc.Tacticalforecasts(weekstomonthstoquarters)areusedforsalesplans,short-termbudgets,inventoryplanning,laborplanning,etc.Finally,operationsforecasts(hourstodays)areusedforproduction,transportation,andinventoryreplenishmentdecisions.Thetimeframeoftheactiondictatesthetimehorizonoftheforecast.Forecastingmethodscanbedividedintobeingsubjective(mostoftenusedbymarketingandsales)orobjective(mostoftenusedbyproductionandinventoryplanners).SubjectivemethodscanbefurtherdividedintobeingeitherJudgmental(someonesomewhereknowsthetruth),suchassalesforcesurveys,Delphisessions,orexpertopinions,orExperimental(samplinglocalandthenextrapolating),suchascustomersurveys,focusgroups,ortestmarketing.ObjectivemethodsareeitherCausal(thereisanunderlyingrelationshiporreason)suchasleadingindicators,etc.orTimeSeries(therearepatternsinthedemand)suchasexponentialsmoothing,movingaverage,etc.Allmethodshavetheirplaceandtheirrole.Wewillspendalotoftimeontheobjectivemethodsbutwillalsodiscussthesubjectiveonesaswell.Regardlessoftheforecastingmethodused,youwillwanttomeasurethequalityoftheforecast.Thetwomajordimensionsofqualityarebias(apersistenttendencytoover-orunder-predict)andaccuracy(closenesstotheactualobservations).Nosinglemetricdoesagoodjobcapturingbothdimensions,soitisworthhavingmultiple.

KeyConceptsForecastingisbothanartandascience.Therearemany“truisms”concerningforecastingincluding:

ForecastingTruisms

1. Forecastsarealwayswrong–Yes,pointforecastswillneverbecompletelyperfect.The

solutionistonotrelytotallyonpointforecasts.Incorporaterangesintoyourforecasts.

Alsoyoushouldtrytocaptureandtracktheforecasterrorssothatyoucansenseand

measureanydriftorchanges.

2. Aggregatedforecastsaremoreaccuratethandis-aggregatedforecasts–Theideaisthatcombiningdifferentitemsleadstoapoolingeffectthatwillinturnlessenthevariability.

Thepeaksbalanceoutthevalleys.Thecoefficientofvariation(CV)iscommonlyusedto

measurevariabilityandisdefinedasthestandarddeviationoverthemean(*+ = -/.).ForecastsaregenerallyaggregatedbySKU(afamilyofproductsversusanindividual

one),time(demandoveramonthversusoverasingleday),orlocation(demandfora

regionversusasinglestore).

3. Shorterhorizonforecastsaremoreaccuratethanlongerhorizonforecasts–Essentiallythismeansthatforecastingtomorrow’stemperature(ordemand)iseasierandprobably

moreaccuratethanforecastingforayearfromtomorrow.Thisisnotthesameas

aggregating.Itisallaboutthetimebetweenmakingtheforecastandtheevent

happening.Shorterisalwaysbetter.Thisiswherepostponementandmodularization

helps.Ifwecansomehowshortentheforecastingtimeforanenditem,wewill

generallybemoreaccurate.

ForecastingMetricsThereisacosttrade-offbetweencostoferrorsinforecastingandcostofqualityforecaststhatmustbebalanced.Forecastmetricsystemsshouldcapturebiasandaccuracy.

Notation

At:Actualvalueforobservationt

Ft:Forecastedvalueforobservationt

et:Errorforobservationt,/0 = 10 − 20n:numberofobservationsµ:mean

σ:standarddeviation

CV:CoefficientofVariation–ameasureofvolatility–*+ = 34

Formulas:MeanDeviation: 56 = ∑ 89:

MeanAbsoluteDeviation: 516 = ∑ |89|:9;<=

MeanSquaredError: 5@A = ∑ 89B:9;<=

RootMeanSquaredError: C5@A = D∑ 89B:9;<=

MeanPercentError: 5EA =∑ F9

G9:9;<

MeanAbsolutePercentError: 51EA =∑ HF9H

G9:9;<

StatisticalAggregation: -IJJK = -LK + -KK + -NK +⋯+ -=K -IJJ = P-LK + -KK + -NK + ⋯+ -=K .IJJ = .L + .K + .N + ⋯+ .=StatisticalAggregationofnDistributionsofEqualMeanandVariance:

-IJJ = D-LK + -KK + -NK +⋯+ -=K = -Q=R√S

.IJJ = .L + .K + .N +⋯+ .= = S.Q=R

*+IJJ =-√S.S =

-.√S

=*+Q=R√S

TimeSeriesAnalysisTimeSeriesisanextremelywidelyusedforecastingtechniqueformid-rangeforecastsforitemsthathavealonghistoryorrecordofdemand.Timeseriesisessentiallypatternmatchingofdatathataredistributedovertime.Forthisreason,youtendtoneedalotofdatatobeabletocapturethecomponentsorpatterns.Businesscyclesaremoresuitedtolongerrange,strategicforecastingtimehorizons.Threeimportanttimeseriesmodels:

• Cumulative–whereeverythingmattersandalldataareincluded.Thisresultsinavery

calmforecastthatchangesveryslowlyovertime–thusitismorestablethan

responsive.

• Naïve–whereonlythelatestdatapointmatters.Thisresultsinverynervousorvolatile

forecastthatcanchangequicklyanddramatically–thusitismoreresponsivethan

stable.

• MovingAverage–wherewecanselecthowmuchdatatouse(thelastMperiods).This

isessentiallythegeneralizedformforboththeCumulative(M=∞)andNaïve(M=1)

models.

Allthreeofthesemodelsaresimilarinthattheyassumestationarydemand.Anytrendintheunderlyingdatawillleadtoseverelagging.Thesemodelsalsoapplyequalweightingtoeachpieceofinformationthatisincluded.Interestingly,whiletheM-PeriodMovingAveragemodelrequiresMdataelementsforeachSKUbeingforecast,theNaïveandCumulativemodelsonlyrequire1dataelementeach.

Componentsoftimeseries

• Level(a)o Valuewheredemandhovers(mean) o Capturesscaleofthetimeserieso Withnootherpatternpresent,itisa

constantvalue

• Trend(b)o Rateofgrowthordeclineo Persistentmovementinonedirectiono Typicallylinearbutcanbeexponential,

quadratic,etc.

• SeasonVariations(F)

o Repeatedcyclearoundaknownandfixedperiod

o Hourly,daily,weekly,monthly,quarterly,etc.o Canbecausedbynaturalorman-madeforces

• RandomFluctuation(eorε)

o Remainderofvariabilityafterothercomponentso Irregularandunpredictablevariations,noise

Notationxt: ActualdemandinperiodtT0,0VL:Forecastfortimet+1madeduringtimeta: Levelcomponentb: LineartrendcomponentFt: Seasonindexappropriateforperiodtet: Errorforobservationt,/0 = 10 − 20t: Timeperiod(0,1,2,…n)LevelModel: T0 = $ + /0TrendModel: T0 = $ + "X + /0MixLevel-SeasonalityModel: T0 = $20 + /0MixLevel-Trend-SeasonalityModel:T0 = ($ + "X)20 + /0

Formulas

ExponentialSmoothingExponentialsmoothing,asopposedtoCumulative,Naïve,andMovingAverage,treatsdatadifferentlydependingonitsage.Theideaisthatthevalueofdatadegradesovertimesothatnewerobservationsofdemandareweightedmoreheavilythanolderobservations.Theweightsdecreaseexponentiallyastheyage.Exponentialmodelssimplyblendthevalueofnewandoldinformation.Thealphafactor(rangingbetween0and1)determinestheweightingforthenewestinformationversustheolderinformation.The“α”valueindicatesthevalueof“new”informationversus“old”information:

• Asα→1,theforecastbecomesmorenervous,volatile,andnaïve• Asα→0,theforecastbecomesmorecalm,staid,andcumulative• αcanrangefrom0≤α≤1,butinpractice,wetypicallysee0≤α≤0.3

Themostbasicexponentialmodel,orSimpleExponentialmodel,assumesstationarydemand.Holt’sModelisamodifiedversionofexponentialsmoothingthatalsoaccountsfortrendin

TimeSeriesModels(StationaryDemandonly):CumulativeModel: TY0,0VL =

∑ Z[9[0

NaïveModel: TY0,0VL = T0

M-PeriodMovingAverageForecastModel: TY0,0VL =∑ Z[9[;9\<]^

• IfM=t,wehavethecumulativemodelwherealldataisincluded• IfM=1,wehavethenaïvemodel,wherethelastdatapointisusedtopredictthe

nextdatapoint

additiontolevel.Anewsmoothingparameter,β,isintroduced.Itoperatesinthesamewayastheα.

Wecanalsouseexponentialsmoothingtodampentrendmodelstoaccountforthefactthattrendsusuallydonotremainunchangedindefinitelyaswellasforcreatingamorestableestimateoftheforecasterrors.

Notationxt: Actualdemandinperiodtxa,aVL: Forecastfortimet+1madeduringtimetα: Exponentialsmoothingfactorforlevel(0≤α≤1)β: Exponentialsmoothingfactorfortrend(0≤β≤1)φ: Exponentialsmoothingfactorfordampening(0≤φ≤1)ω: MeanSquareErrortrendingfactor(0.01≤ω≤0.1)

ForecastingModels

SimpleExponentialSmoothingModel(LevelOnly)–Thismodelisusedforstationarydemand.The“new”informationissimplythelatestobservation.The“old”informationisthemostrecentforecastsinceitencapsulatestheolderinformation.

TY0,0VL = bT0 + (1 − b)TY0cL,0

DampedTrendModelwithLevelandTrend–Wecanuseexponentialsmoothingtodampenalineartrendtobetterreflectthetaperingeffectoftrendsinpractice.

TY0,0Vd = $Y0 +efQ"g0

$Y0 = bT0 + (1 − b)i$Y0cL + f"g0cLj"g0 = k($Y0 − $Y0cL) + (1 − k)f"g0cL

ExponentialSmoothingforLevel&Trend–alsoknownasHolt’sMethod,assumesalineartrend.Theforecastfortimet+τmadeattimetisshownbelow.Itisacombinationofthelatestestimatesofthelevelandtrend.Forthelevel,thenewinformationisthelatestobservationandtheoldinformationisthemostrecentforecastforthatperiod–thatis,thelastperiod’sestimateoflevelplusthelastperiod’sestimateoftrend.Forthetrend,thenewinformationisthedifferencebetweenthemostrecentestimateofthelevelminusthesecondmostrecentestimateofthelevel.Theoldinformationissimplythelastperiod’sestimateofthetrend.

TY0,0Vd = $Y0 + l"g0 $Y0 = bT0 + (1 − b)($Y0cL + "g0cL)"g0 = k($Y0 − $Y0cL) + (1 − k)"g0cL

ExponentialSmoothingwithHolt-Winter

Seasonality• Formultiplicativeseasonality,thinkoftheFias“percentofaveragedemand”fora

periodi• ThesumoftheFiforallperiodswithinaseasonmustequalP• Seasonalityfactorsmustbekeptcurrentortheywilldriftdramatically.Thisrequiresa

lotmorebookkeeping,whichistrickytomaintaininaspreadsheet,butitisimportanttounderstand

ForecastingModelParameterInitializationMethods• Whilethereisnosinglebestmethod,therearemanygoodones• SimpleExponentialSmoothing

o Estimatelevelparameter$mbyaveragingdemandforfirstseveralperiods• HoltModel(trendandlevel)—mustestimateboth$mand"m

o Findabestfitlinearequationfrominitialdatao Useleastsquaresregressionofdemandforseveralperiods

§ Dependentvariable=demandineachtimeperiod=xt§ Independentvariable=slope=β1§ Regressionequation:xt=β0+β1t

• SeasonalityModelso Muchmorecomplicated,youneedatleasttwoseasonofdatabutpreferably

fourormoreo Firstdeterminethelevelforeachcommonseasonperiodandthenthedemand

forallperiodso Setinitialseasonalityindicestoratioofeachseasontoallperiods

Notation

xt: Actualdemandinperiodtxa,aVL: Forecastfortimet+1madeduringtimetα: Exponentialsmoothingfactor(0≤α≤1)β: Exponentialsmoothingtrendfactor(0≤β≤1)γ: Seasonalitysmoothingfactor(0≤γ≤1)Ft: MultiplicativeseasonalindexappropriateforperiodtP: Numberoftimeperiodswithintheseasonality(note: Fo = Pq

MeanSquareErrorEstimate–Wecanalsouseexponentialsmoothingtoprovideamorerobustorstablevalueforthemeansquareerroroftheforecast.

5@A0 = r(T0 − TY0cL,0)K + (1 − r)5@A0cL

ForecastingModels

SpecialCasesTherearedifferenttypesofnewproductsandtheforecastingtechniquesdifferaccordingtotheirtype.Thefundamentalideaisthatifyoudonothaveanyhistorytorelyon,youcanlookforhistoryofsimilarproductsandbuildone.

DoubleExponentialSmoothing(SeasonalityandLevel)–Thisisamultiplicativemodelinthattheseasonalityforeachperiodistheproductofthelevelandthatperiod’sseasonalityfactor.Thenewinformationfortheestimateofthelevelisthe“de-seasoned”valueofthelatestobservation;thatis,youaretryingtoremovetheseasonalityfactor.Theoldinformationissimplythepreviousmostrecentestimateforlevel.Fortheseasonalityestimate,thenewinformationisthe“de-leveled”valueofthelatestobservation;thatis,youtrytoremovethelevelfactortounderstandanynewseasonality.Theoldinformationissimplythepreviousmostrecentestimateforthatperiod’sseasonality.

TY0,0Vd = $Y02g0Vdcs

$Y0 = b tT02g0cs

u + (1 − b)($Y0cL)

2g0 = v wT0$Y0x + (1 − v)2g0cs

NormalizingSeasonalityIndices–Thisshouldbedoneaftereachforecasttoensuretheseasonalitydoesnotgetoutofsynch.Iftheindicesarenotupdated,theywilldriftdramatically.Mostsoftwarepackageswilltakecareofthis–butitisworthchecking.

2g0yz{ = 2g0|}~ tE

∑ 2g0|}~0Qh0cs

Holt-WinterExponentialSmoothingModel(Level,Trend,andSeasonality)–Thismodelassumesalineartrendwithamultiplicativeseasonalityeffectoverbothlevelandtrend.Forthelevelestimate,thenewinformationisagainthe“de-seasoned”valueofthelatestobservation,whiletheoldinformationistheoldestimateofthelevelandtrend.TheestimateforthetrendisthesameasfortheHoltmodel.TheSeasonalityestimateisthesameastheDoubleExponentialsmoothingmodel.

TY0,0Vd = ($Y0 + l"g0)2g0Vdcs

$Y0 = b tT02g0cs

u + (1 − b)($Y0cL + "g0cL)

"g0 = k($Y0 − $Y0cL) + (1 − k)"g0cL2g0 = v w

T0$Y0x + (1 − v)2g0cs

Whenthedemandisverysparse,suchasforspareparts,wecannotusetraditionalmethodssincetheestimatestendtofluctuatedramatically.Croston’smethodcansmoothouttheestimateforthedemand.

NewProductTypes• Notallnewproductsarethesame.Wecanroughlyclassifythemintothefollowing

sixcategories(listedfromeasiesttoforecasttohardest):

o CostReductions:Reducedpriceversionoftheproductfortheexisting

market

o ProductRepositioning:Takingexistingproducts/servicestonewmarketsor

applyingthemtoanewpurpose(aspirinfrompainkillertoreducingeffects

ofaheartattack)

o LineExtensions:Incrementalinnovationsaddedtocomplementexisting

productlines(VanillaCoke,CokeZero)orProductImprovements:New,

improvedversionsofexistingofferingtargetedtothecurrentmarket—

replacesexistingproducts(nextgenerationofproduct)

o New-to-Company:Newmarket/categoryforthecompanybutnottothe

market(AppleiPhoneoriPod)

o New-to-World:Firstoftheirkind,createsnewmarket,radicallydifferent

(SonyWalkman,Post-itnotes,etc.)

• Whiletheyareapaintoforecastandtolaunch,firmsintroducenewproductsallthe

time–thisisbecausetheyaretheprimarywaytoincreaserevenueandprofits(See

Table1)

*Majorrevisions/incrementalimprovementsaboutevenlysplit

Table1.Newproductintroductions.Source:AdaptedfromCooper,Robert(2001)WinningatNewProducts,Kahn,Kenneth(2006)NewProductforecasting,andPDMA(2004)NewProductDevelopmentReport.)

NewProductDevelopmentProcessAllfirmsusesomeversionoftheprocessshownbelowtointroducenewproducts.Thisissometimescalledthestage-gateorfunnelprocess.Theconceptisthatlotsofideascomeinontheleftandveryfewfinalproductscomeoutontheright.Eachstageorhurdleintheprocesswinnowsoutthewinnersfromthelosersandisusedtofocusattentionontherightproducts.ThescopeandscaleofforecastingchangesalongtheprocessasnotedinFigure3.

Figure3.Newproductdevelopmentprocess

ForecastingModelsDiscussedNewProduct–“Looks-Like�orAnalogousForecasting

• Performbylookingatcomparableproductlaunchesandcreateaweek-by-weekormonth-by-monthsalesrecord.

• Thenusethepercentoftotalsalesineachtimeincrementasatrajectoryguide.• Eachlaunchshouldbecharacterizedbyproducttype,seasonofintroduction,price,

targetmarketdemographics,andphysicalcharacteristics.

IntermittentorSparseDemand–Croston’sMethod• Usedforproductsthatareinfrequentlyorderedinlargequantities,irregularlyordered,

ororderedindifferentsizes.• Croston’sMethodseparatesoutthedemandandmodel—unbiasedandhaslower

variancethansimplesmoothing.• Cautions:infrequentordering(andupdatingofmodel)inducesalagtorespondingto

magnitudechanges.

Notationxt: Demandinperiodtyt: 1iftransactionoccursinperiodt,=0otherwise

zt: Size(magnitude)oftransactionintimetnt: Numberofperiodssincelasttransactionα: Smoothingparameterformagnitudeβ: Smoothingparameterfortransactionfrequency

Formulas

Croston’sMethodWecanuseCroston’smethodwhendemandisintermittent.Itallowsustousethetraditionalexponentialsmoothingmethods.WeassumetheDemandProcessisxt=ytztandthatdemandisindependentbetweentimeperiods,sothattheprobabilitythatatransactionoccursinthecurrenttimeperiodis1/n:

E�Ä" Å0 = 1 =1S$SÇE�Ä" Å0 = 0 = 1 −

UpdatingProcedure:Ifxt=0(notransactionoccurs),then

Ñ0 = Ñ0cL$SÇS0 = S0cLIfxt>0(transactionoccurs),then

Ñ0 = bT0 + (1 − b)Ñ0cLS0 = kS0 + (1 − k)S0cL

Forecast:

ÖÜ,ÜVá =àÜâÜ

LearningObjectives

• ForecastingispartoftheentireDemandPlanningandManagementprocess.

• Rangeforecastsarebetterthanpointforecasts,aggregatedforecastsarebetterthan

dis-aggregated,andshortertimehorizonsarebetterthanlonger.

• Forecastingmetricsneedtocapturebiasandaccuracy.

• Understandhowtoinitializeaforecast.

• UnderstandthatTimeSeriesisausefultechniquewhenwebelievedemandfollows

certainrepeatingpatterns.

• Recognizethatalltimeseriesmodelsmakeatrade-offbetweenbeingnaïve(usingonly

thelastmostrecentdata)orcumulative(usingalloftheavailabledata).

• Understandhowexponentialsmoothingtreatsoldandnewinformationdifferently.

• Understandhowchangingthealphaorbetasmoothingfactorsinfluencestheforecasts.

• Understandhowseasonalitycanbehandledwithinexponentialsmoothing.

• Understandwhydemandfornewproductsneedtobeforecastedwithdifferent

techniques.

• LearnhowtousebasicDiffusionModelsfornewproductdemandandhowtoforecast

intermittentdemandusingCroston’sMethod.

• Understandhowthetypicalnewproductpipelineprocess(stage-.-gate)worksandhow

forecastingfitsin.

ReferencesGeneralDemandForecasting

• Makridakis,Spyros,StevenC.Wheelwright,andRobJ.Hyndman.Forecasting:MethodsandApplications.NewYork,NY:Wiley,1998.ISBN9780471532330.

• Hyndman,RobJ.andGeorgeAthanasopoulos.Forecasting:PrinciplesandPractice.OTexts,2014.ISBN0987507109.

• Gilliland,Michael.TheBusinessForecastingDeal:ExposingBadPracticesandProvidingPracticalSolutions.Hoboken,NJ:Wiley,2010.ISBN0470574437.

Withinthetextsmentionedearlier:Silver,Pyke,andPetersonChapter4.1;Chopra&MeindlChapter7.1-7.4;NahmiasChapter2.1-2.6.Also,IrecommendcheckingouttheInstituteofBusinessForecasting&Planning(https://ibf.org/)andtheirJournalofBusinessForecasting.ForTimeSeriesAnalysisWithinthetextsmentionedearlier:Silver,Pyke,andPetersonChapter4.2-5.5.1&4.6;Chopra&MeindlChapter7.5-7.6;NahmiasChapter2.7.Also,IrecommendcheckingouttheInstituteofBusinessForecasting&Planning(https://ibf.org/)andtheirJournalofBusinessForecasting.

• Makridakis,Spyros,StevenC.Wheelwright,andRobJ.Hyndman.Forecasting:MethodsandApplications.NewYork,NY:Wiley,1998.ISBN9780471532330.

• Hyndman,RobJ.andGeorgeAthanasopoulos.Forecasting:PrinciplesandPractice.OTexts,2014.ISBN0987507109.

ForExponentialSmoothing• Silver,E.A.,Pyke,D.F.,Peterson,R.InventoryManagementandProductionPlanningand

Scheduling.ISBN:978-0471119470.Chapter4.• Chopra,Sunil,andPeterMeindl.SupplyChainManagement:Strategy,Planning,and

Operation.5thedition,PearsonPrenticeHall,2013.Chapter7.• Nahmias,S.ProductionandOperationsAnalysis.McGraw-HillInternationalEdition.

ISBN:0-07-2231265-3.Chapter2.

ForSpecialCases

• Cooper,RobertG.WinningatNewProducts:AcceleratingtheProcessfromIdeatoLaunch.Cambridge,MA:PerseusPub.,2001.Print.

• Kahn,KennethB.NewProductForecasting:AnAppliedApproach.Armonk,NY:M.E.Sharpe,2006.

• Adams,Marjorie.PDMAFoundationNPDBestPracticesStudy:ThePDMAFoundation’s2004ComparativePerformanceAssessmentStudy(CPAS).OakRidge,NC:ProductDevelopment&ManagementAssociation,2004.

InventoryManagement

SummaryInventorymanagementisatthecoreofallsupplychainandlogisticsmanagement.Therearemanyreasonsforholdinginventoryincludingminimizingthecostofcontrollingasystem,bufferingagainstuncertaintiesindemand,supply,deliveryandmanufacturing,aswellascoveringthetimerequiredforanyprocess.Havinginventoryallowsforasmootheroperationinmostcasessinceitalleviatestheneedtocreateproductfromscratchforeachindividualdemand.Inventoryistheresultofapushsystemwheretheforecastdetermineshowmuchinventoryofeachitemisrequired.Thereis,however,aproblemwithhavingtoomuchinventory.Excessinventorycanleadtospoilage,obsolescence,anddamage.Also,spendingtoomuchoninventorylimitstheresourcesavailableforotheractivitiesandinvestments.Inventoryanalysisisessentiallythedeterminationoftherightamountofinventoryoftherightproductintherightlocationintherightform.Strategicdecisionscovertheinventoryimplicationsofproductandnetworkdesign.Tacticaldecisionscoverdeploymentanddeterminewhatitemstocarry,inwhatform(rawmaterials,work-in-process,finishedgoods,etc.),andwhere.Finally,operationaldecisionsdeterminethereplenishmentpolicies(whenandhowmuch)oftheseinventories.WeseektheOrderReplenishmentPolicythatminimizesthesetotalcostsandspecificallytheTotalRelevantCosts(TRC).Acostcomponentisconsideredrelevantifitimpactsthedecisionathandandwecancontrolitbysomeaction.AReplenishmentPolicyessentiallystatestwothings:thequantitytobeordered,andwhenitshouldbeordered.Aswewillsee,theexactformoftheTotalCostEquationuseddependsontheassumptionswemakeintermsofthesituation.Therearemanydifferentassumptionsinherentinanyofthemodelswewilluse,buttheprimaryassumptionsaremadeconcerningtheformofthedemandfortheproduct(whetheritisconstantorvariable,randomordeterministic,continuousordiscrete,etc.).

KeyConcepts

ReasonstoHoldInventory• Coverprocesstime

• Allowforuncouplingofprocesses

• Anticipation/Speculation

• Minimizecontrolcosts

• Bufferagainstuncertaintiessuchasdemand,supply,delivery,and

manufacturing.

InventoryDecisions• Strategicsupplychaindecisionsarelongtermandincludedecisionsrelatedto

thesupplychainsuchaspotentialalternativestoholdinginventoryandproduct

design.

• Tacticalaremadewithinamonth,aquarterorayearandareknownas

deploymentdecisionssuchaswhatitemstocarryasinventory,inwhatformto

carryitemsandhowmuchofeachitemtoholdandwhere.

• Operationalleveldecisionsaremadeondaily,weeklyormonthlybasisand

replenishmentdecisionssuchashowoftentoreviewinventorystatus,how

oftentomakereplenishmentdecisionsandhowlargereplenishmentshouldbe.

Thereplenishmentdecisionsarecriticaltodeterminehowthesupplychainisset

InventoryClassification• Financial/AccountingCategories:RawMaterials,WorkinProgress(WIP),

Components/Semi-FinishedGoodsandFinishedGoods.Thiscategorydoesnot

helpintrackingopportunitycostsandhowonemaywishtomanageinventory.

• Functional(SeeFigure4):

o CycleStock–Amountofinventorybetweendeliveriesorreplenishments

o SafetyStock–Inventorytocoverorbufferagainstuncertainties

o PipelineInventory–Inventorywhenorderisplacedbuthasnotyet

arrived

Figure4.Inventorychart:Depictionoffunctionalinventoryclassifications

RelevantCostsTheTotalCost(TC)equationistypicallyusedtomakethedecisionsofhowmuchinventorytoholdandhowtoreplenish.ItisthesumofthePurchasing,Ordering,Holding,andShortagecosts.ThePurchasingcostsareusuallyvariableorper-itemcostsandcoverthetotallandedcostforacquiringthatproduct–whetherfrominternalmanufacturingorpurchasingitfromoutside.Totalcost=Purchase(UnitValue)Cost+Order(SetUp)Cost+Holding(Carrying)Cost+Shortage(stock-out)Cost

• Purchase:Costperitemortotallandedcostforacquiringproduct.

• Ordering:Itisafixedcostandcontainscosttoplace,receiveandprocessa

batchofgoodincludingprocessinginvoicing,auditing,labor,etc.In

manufacturingthisisthesetupcostforarun.

• Holding:Costsrequiredtoholdinventorysuchasstoragecost(warehouse

space),servicecosts(insurance,taxes),riskcosts(lost,stolen,damaged,

obsolete),andcapitalcosts(opportunitycostofalternativeinvestment).

• Shortage:Costsofnothavinganiteminstock(on-handinventory)tosatisfya

demandwhenitoccurs,includingbackorder,lostsales,lostcustomers,and

disruptioncosts.Alsoknownasthepenaltycost.

Acostisrelevantifitiscontrollableanditappliestothespecificdecisionbeingmade.

Notationc: Purchasecost($/unit)ct: OrderingCosts($/order)h: Holdingrate–usuallyexpressedasapercentage($/$value/time)ce: ExcessholdingCosts($/unit-time);alsoequaltochcs: Shortagecosts($/unit)TRC: TotalRelevantCosts–thesumoftherelevantcostcomponentsTC: TotalCosts–thesumofallfourcostelements

EconomicOrderQuantity(EOQ)TheEconomicOrderQuantityorEOQisthemostinfluentialandwidelyused(andsometimesmisused!)inventorymodelinexistence.Whileverysimple,itprovidesdeepandusefulinsights.Essentially,theEOQisatrade-offbetweenfixed(ordering)andvariable(holding)costs.ItisoftencalledLot-Sizingaswell.TheminimumoftheTotalCostequation(whenassumingdemandisuniformanddeterministic)istheEOQorQ*.TheInventoryReplenishmentPolicybecomes“OrderQ*everyT*timeperiods”whichunderourassumptionsisthesameas“OrderQ*whenInventoryPosition(IP)=0”.

LikeWikipedia,theEOQisaGREATplacetostart,butnotnecessarilyagreatplacetofinish.Itisagoodfirstestimatebecauseitisexceptionallyrobust.Forexample,a50%increaseinQovertheoptimalquantity(Q*)onlyincreasestheTRCby~8%!Whileveryinsightful,theEOQmodelshouldbeusedwithcautionasithasrestrictiveassumptions(uniformanddeterministicdemand).Itcanbesafelyusedforitemswithrelativelystabledemandandisagoodfirst-cut“backoftheenvelope”calculationinmostsituations.Itishelpfultodevelopinsightsinunderstandingthetrade-offsinvolvedwithtakingcertainmanagerialactions,suchasloweringtheorderingcosts,loweringthepurchaseprice,changingtheholdingcosts,etc.

EOQModel• Assumptions

o Demandisuniformanddeterministic.

o Leadtimeisinstantaneous(0)–althoughthisisnotrestrictiveatallsince

theleadtime,L,doesnotinfluencetheOrderSize,Q.

o Totalamountorderedisreceived.

• InventoryReplenishmentPolicy

o OrderQ*unitseveryT*timeperiods.

o OrderQ*unitswheninventoryonhand(IOH)iszero.

• Essentially,theQ*istheCycleStockforeachreplenishmentcycle.Itisthe

expecteddemandforthatamountoftimebetweenorderdeliveries.

Notationc: Purchasecost($/unit)ct: OrderingCosts($/order)ce: ExcessholdingCosts($/unit/time);equaltochcs: Shortagecosts($/unit)D: Demand(units/time)DA: ActualDemand(units/time)DF: ForecastedDemand(units/time)h: Carryingorholdingcost($/inventory$/time)Q: ReplenishmentOrderQuantity(units/order)Q*: OptimalOrderQuantityunderEOQ(units/order)Q*A: OptimalOrderQuantitywithActualDemand(units/order)Q*F: OptimalOrderQuantitywithForecastedDemand(units/order)T: OrderCycleTime(time/order)T*: OptimalTimebetweenReplenishments(time/order)N: OrdersperTimeor1/T(order/time)TRC(Q): TotalRelevantCost($/time)TC(Q):TotalCost($/time)

FormulasTotalCosts: TC=Purchase+Order+Holding+ShortageThisisthegenerictotalcostequation.Thespecificformofthedifferentelementsdependsontheassumptionsmadeconcerningthedemand,theshortagetypes,etc.

ä* ã = å6 + å06ã

+ å8ã2

+ åçA[èSêXë@ℎÄ�X]

TotalRelevantCosts:TRC=Order+HoldingThepurchasingcostandtheshortagecostsarenotrelevantfortheEOQbecausethepurchasepricedoesnotchangetheoptimalorderquantity(Q*)andsincewehavedeterministicdemand,wewillnotstockout.

äC* ã = å06ã

+ å8ã2

OptimalOrderQuantity(Q*)RecallthatthisistheFirstOrderconditionoftheTRCequation–whereitisaglobalminimum.

ã∗ =2å06å8

OptimalTimebetweenReplenishmentsRecallthatT*=Q*/D.Thatis,thetimebetweenordersistheoptimalordersizedividedbytheannualdemand.Similarly,thenumberofreplenishmentsperyearissimplyN*=1/T*=D/Q*.PluggingintheactualQ*givesyoutheformulabelow.

ä∗ =2å06å8

Note:BesuretoputT*intounitsthatmakesense(days,weeks,months,etc.).Don’tleaveitinyears!OptimalTotalCostsAddingthepurchasecosttotheTRC(Q*)costsgivesyoutheTC(Q*).Westillassumenostockoutcosts.

ä* ã∗ = å6 + 2å0å86

OptimalTotalRelevantCostsPluggingtheQ*backintotheTRCequationandsimplifyinggivesyoutheformulabelow.

äC* ã∗ = 2å0å86

SensitivityAnalysisTheEOQisveryrobust.Thefollowingformulasprovidesimplewaysofcalculatingtheimpactofusinganon-optimalQ,anincorrectannualDemandD,oranon-optimaltimeinterval,T.EOQSensitivitywithRespecttoOrderQuantityTheequationbelowcalculatesthepercentdifferenceintotalrelevantcoststooptimalwhenusinganon-optimalorderquantity(Q):

äC*(ã)äC*(ã∗)

ã+ãã∗

Note:Ifoptimalquantitydoesnotmakesense,itisalwaysbettertoorderlittlemoreratherorderinglittleless.EOQSensitivitywithRespecttoDemandTheequationbelowcalculatesthepercentdifferenceintotalrelevantcoststooptimalwhenassuminganincorrectannualdemand(DF)wheninfacttheactualannualdemandisDA:

äC*(ãï∗)äC*(ãñ∗)

6ñ6ï

+6ï6ñ

EOQSensitivitywithRespecttoTimeIntervalbetweenOrdersTheequationbelowcalculatesthepercentdifferenceintotalrelevantcoststooptimalwhenusinganon-optimalreplenishmenttimeinterval(T).Thiswillbecomeveryimportantwhenfindingrealisticreplenishmentintervals.ThePowerofTwoPolicyshowsthatorderinginincrementsof2ktimeperiods,wewillstaywithin6%oftheoptimalsolution.Forexample,ifthebasetimeperiodisoneweek,thenthePowerofTwoPolicywouldsuggestorderingeveryweek(20)oreverytwoweeks(21)oreveryfourweeks(22)oreveryeightweeks(23)etc.Selecttheintervalclosesttooneoftheseincrements.

äC* ääC* ä∗

ää∗

+ä∗

EconomicOrderQuantity(EOQ)ExtensionsTheEconomicOrderQuantitycanbeextendedtocovermanydifferentsituations,threeextensionsinclude:lead-time,volumediscounts,andfinitereplenishmentorEPQ.WedevelopedtheEOQpreviouslyassumingtheratherrestrictive(andridiculous)assumptionthatlead-timewaszero.Thatis,instantaneousreplenishmentlikeonStarTrek.However,includinganon-zeroleadtimewhileincreasingthetotalcostduetohavingpipelineinventorywillNOTchangethecalculationoftheoptimalorderquantity,Q*.Inotherwords,lead-timeisnotrelevanttothedeterminationoftheneededcyclestock.

Volumediscountsaremorecomplicated.Includingthemmakesthepurchasingcostsrelevantsincetheynowimpacttheordersize.Wediscussedthreetypesofdiscounts:All-Units(wherethediscountappliestoallitemspurchasedifthetotalamountexceedsthebreakpointquantity),Incremental(wherethediscountonlyappliestothequantitypurchasedthatexceedsthebreakpointquantity),andOne-Time(whereaone-time-onlydiscountisofferedandyouneedtodeterminetheoptimalquantitytoprocureasanadvancebuy).Discountsareexceptionallycommoninpracticeastheyareusedtoincentivizebuyerstopurchasemoreortoorderinconvenientquantities(fullpallet,fulltruckload,etc.). Apricebreakpointistheminimumquantityrequiredtogetapricediscount.FiniteReplenishmentisverysimilartotheEOQmodel,exceptthattheproductisavailableatacertainproductionrateratherthanallatonce.Inthelessonweshowthatthistendstoreducetheaverageinventoryonhand(sincesomeofeachorderismanufacturedoncetheorderisreceived)andthereforeincreasestheoptimalorderquantity.• Leadtimeisgreaterthan0(ordernotreceivedinstantaneously)

o InventoryPolicy:

§ OrderQ*unitswhenIP=DL

§ OrderQ*unitseveryT*timeperiods

• Discounts

o AllUnitsDiscount—Discountappliestoallunitspurchasediftotalamountexceeds

thebreakpointquantity

o IncrementalDiscount—Discountappliesonlytothequantitypurchasedthatexceeds

thebreakpointquantity

o One-Time-OnlyDiscount—Aone-time-onlydiscountappliestoallunitsyouorder

rightnow(noquantityminimumorlimit)

• FiniteReplenishment

o InventorybecomesavailableatarateofPunits/timeratherthanallatonetime

o IfProductionrateapproachinfinity,modelconvergestoEOQ

Notationc: Purchasecost($/unit)ci: Discountedpurchasepricefordiscountrangei($/unit)cei: Effectivepurchasecostfordiscountrangei($/unit)[forincrementaldiscounts]ct: OrderingCosts($/order)ce: ExcessholdingCosts($/unit/time);Equaltochcs: Shortagecosts($/unit)cg: OneTimeGoodDealPurchasePrice($/unit)Fi: FixedCostsAssociatedwithUnitsOrderedbelowIncrementalDiscountBreakpointi

D: Demand(units/time)DA: ActualDemand(units/time)DF: ForecastedDemand(untis/time)h: Carryingorholdingcost($/inventory$/time)L: OrderLeadtimeQ: ReplenishmentOrderQuantity(units/order)Q*: OptimalOrderQuantityunderEOQ(units/order)Qi: Breakpointforquantitydiscountfordiscounti(unitsperorder)Qg: OneTimeGoodDealOrderQuantityP: Production(units/time)T: OrderCycleTime(time/order)T*: OptimalTimebetweenReplenishments(time/order)N: OrdersperTimeor1/T(order/time)TRC(Q): TotalRelevantCost($/time)TC(Q):TotalCost($/time)

FormulasInventoryPositionInventoryPosition(IP)=InventoryonHand(IOH)+InventoryonOrder(IOO)–BackOrders(BO)–CommittedOrders(CO)InventoryonOrder(IOO)istheinventorythathasbeenordered,butnotyetreceived.ThisisinventoryintransitandalsoknowsasPipelineInventory(PI).AveragePipelineInventoryAveragePipelineInventory(API),onaverage,istheannualdemandtimestheleadtime.Essentially,everyitemspendsLtimeperiodsintransit.

1Eó = 6ò

TotalCostincludingPipelineInventoryTheTCequationchangesslightlyifweassumeanon-zeroleadtimeandincludethepipelineinventory.

ä* ã = å6 + å06ã

+ å8ã2+ 6ò + åçA[èSêXë@ℎÄ�X]

Notethatasbefore,though,thepurchasecost,shortagecosts,andnowpipelineinventoryisnotrelevanttodeterminingtheoptimalorderquantity,Q*:

ã∗ =2å06å8

DiscountsIfweincludevolumediscounts,thenthepurchasingcostbecomesrelevanttoourdecisionoforderquantity.AllUnitsDiscountsDiscountappliestoallunitspurchasediftotalamountexceedsthebreakpointquantity.TheprocedureforasinglerangeAllUnitsquantitydiscount(wherenewpriceisc1iforderingatleastQ1units)isasfollows:

1. CalculateQ*C0,theEOQusingthebase(non-discounted)price,andQ*C1,theEOQusing

thefirstdiscountedprice

2. IfQ*C1≥Q1,thebreakpointforthefirstallunitsdiscount,thenorderQ*C1sinceit

satisfiestheconditionofthediscount.Otherwise,gotostep3.

3. ComparetheTRC(Q*C0),thetotalrelevantcostwiththebase(non-discounted)price,

withTRC(Q1),thetotalrelevantcostusingthediscountedprice(c1)atthebreakpointfor

thediscount.IfTRC(Q*C0)<TRC(Q1),selectQ*C0,otherwiseorderQ1.

Notethatiftherearemorediscountlevels,youneedtocheckthisforeachone.å = åmôÄ�0 ≤ ã ≤ ãL$SÇå = åLôÄ�ãL ≤ ã

äC* = 6åm + å06ã

+ åmℎã2

ôÄ�0 ≤ ã ≤ ãL

äC* = 6åL + å06ã

+ åLℎã2

ôÄ�ãL ≤ ã

Note:Allunitsdiscounttendtoraisecyclestockinthesupplychainbyencouragingretailerstoincreasethesizeofeachorder.Thismakeseconomicsenseforthemanufacturer,especiallywhenheincursaveryhighfixedcostperorder.IncrementalDiscountsDiscountappliesonlytothequantitypurchasedthatexceedsthebreakpointquantity.Theprocedureforamulti-rangeIncrementalquantitydiscount(whereiforderingatleastQ1units,thenewpricefortheQ-Q1unitsisc1)isasfollows:

1. CalculatetheFixedcostperbreakpoint,Fi,

2. CalculatetheQ*iforeachdiscountrangei(toincludetheFi)

3. CalculatetheTRCforalldiscountrangeswheretheQi-1<Q*i<Qi+1,thatis,ifitisin

range.

4. SelectthediscountthatprovidesthelowestTRC.

Theeffectivecost,cei,canbeusedfortheTRCcalculations.2m = 0;2Q = 2QcL + (åQcL − åQ)ãQ

ã∗ =26(å0 + 2Q)

ℎåQ

åQ8 = åQ +2QãQ∗

OneTimeDiscountThisisalesscommondiscount–butitdoeshappen.Aonetimeonlydiscountappliestoallunitsyouorderrightnow(nominimumquantityorlimit).SimplycalculatetheQ*gandthatisyourorderquantity.IfQ*g=Q*thenthediscountdoesnotmakesense.IfyoufindthatQ*g<Q*,youmadeamathematicalmistake–checkyourwork!

ä* = *Ååú/äêù/ ä*∗ + Eû�åℎ$ë/*ÄëX =ãJ6

2å0ℎå6 +ãJ6

å6@$üêS†ë = ä* − ä*°s

@$üêS†ë =ãJ6

2å0ℎå6 +ãJ6

å6 − åJãJ + ℎåJãJ2

ãJ∗ =ã∗åℎ + 6(å − åJ)

ℎåJ

FiniteReplenishmentorEconomicProductionQuantityOnecanthinkoftheEPQequationsasgeneralizedformswheretheEOQisaspecialcasewhereP=infinity.Astheproductionratedecreases,theoptimalquantitytobeorderedincreases.However,notethatifP<D,thismeanstherateofproductionisslowerthantherateofdemandandthatyouwillneverhaveenoughinventorytosatisfydemand.

äC* ã =å06ã

+ã 1 − 6

E ℎå2

AEã =2å06

ℎå 1 − 6E

=A¢ã

1 − 6E

SinglePeriodInventoryModelsThesingleperiodinventorymodelissecondonlytotheeconomicorderquantityinitswidespreaduseandinfluence.AlsoreferredtoastheNewsvendor(Newsboy)model,thesingleperiodmodeldiffersfromtheEOQinthreemainways.First,whiletheEOQassumesuniformanddeterministicdemand,thesingle-periodmodelallowsdemandtobevariableandstochastic(random).Second,whiletheEOQassumesasteadystatecondition(stabledemandwithessentiallyaninfinitetimehorizon),thesingle-periodmodelassumesasingleperiodoftime.Allinventoriesmustbeorderedpriortothestartofthetimeperiodandtheycannotbereplenishedduringthetimeperiod.Anyinventoryleftoverattheendofthetimeperiodisscrappedandcannotbeusedatalatertime.Ifthereisextrademandthatisnotsatisfied

duringtheperiod,ittooislost.Third,forEOQweareminimizingtheexpectedcosts,whileforthesingleperiodmodelweareactuallymaximizingtheexpectedprofitability.Aplannedbackorderiswherewestockoutonpurposeknowingthatcustomerswillwait,althoughwedoincurapenaltycost,cs,forstockingout.Fromthis,wedeveloptheideaofthecriticalratio(CR),whichistheratioofthecs(thecostofshortageorhavingtoolittleproduct)totheratioofthesumofcsandce(thecostofhavingtoomuchoranexcessofproduct).Thecriticalratio,bydefinition,rangesbetween0and1andisagoodmetricoflevelofservice.AhighCRindicatesadesiretostockoutlessfrequently.TheEOQwithplannedbackordersisessentiallythegeneralizedformwherecsisessentiallyinfinity,meaningyouwillnevereverstockout.Ascsgetssmaller,theQ*PBOgetslargerandalargerpercentageisallowedtobebackordered–sincethepenaltyforstockingoutgetsreduced.Thecriticalratioappliesdirectlytothesingleperiodmodelaswell.Weshowthattheoptimalorderquantity,Q*,occurswhentheprobabilitythatthedemandislessthanQ*=theCriticalRatio.Inotherwords,theCriticalRatiotellsmehowmuchofthedemandprobabilitythatshouldbecoveredinordertomaximizetheexpectedprofits.

MarginalAnalysis:SinglePeriodModelTwocostsareassociatedwithsingleperiodproblems

• Excesscost(ce)whenD<Q($/unit)i.e.toomuchproduct

• Shortagecost(cs)whenD>Q($/unit)i.e.toolittleproduct

Ifweassumecontinuousdistributionofdemand• ceP[X≤Q]=expectedexcesscostoftheQthunitordered

• cs(1-P[X≤Q])=expectedshortagecostoftheQthunitordered

ThisimpliesthatifE[ExcessCost]<E[ShortageCost]thenincreaseQandthatweareatQ*

whenE[ShortageCost]=E[ExcessCost].Solvingthisgivesus:E T ≤ ã = £§(£FV£§)

Inwords,thismeansthatthepercentageofthedemanddistributioncoveredbyQshouldbeequaltotheCriticalRatioinordertomaximizeexpectedprofits.

NotationB: Penaltyfornotsatisfyingdemandbeyondlostprofit($/unit)b: BackorderDemand(units)b*: Optimalunitsonbackorderwhenplacinganorder(unit)c: Purchasecost($/unit)ct: OrderingCosts($/order)ce: ExcessholdingCosts($/unit/time);Equaltochcs: ShortageCosts($/unit)

D: AverageDemand(units/time)g: Salvagevalueforexcessinventory($/unit)h: Carryingorholdingcost($/inventory$/time)L: ReplenishmentLeadTime(time)Q: ReplenishmentOrderQuantity(units/order)Qq¶ß∗ :OptimalOrderQuantitywithPlannedbackorders

T: OrderCycleTime(time/order)TRC(Q): TotalRelevantCost($/time)TC(Q): TotalCost($/time)

FormulasEOQwithPlannedBackordersThisisanextensionofthestandardEOQwiththeabilitytoallowforbackordersatapenaltyofcs.

äC* ã, " = å06ã

+ å8ã − " K

2ã+ åç

ãs®|∗ =2å06å8

åçå8åç

= ã∗ (åç + å8)åç

= ã∗ 1*C

"∗ =å8ãs®|∗

åç + å8= 1 −

åçåç + å8

ãs®|∗

äs®|∗ =6

ãs®|∗

OrderQq¶ß∗ whenIOH=-b*;OrderQq¶ß

∗ everyTq¶ß∗ timeperiods

SinglePeriod(Newsvendor)ModelTomaximizeexpectedprofitability,weneedtoordersufficientinventory,Q,suchthattheprobabilitythatthedemandislessthanorequaltothisamountisequaltotheCriticalRatio.Thus,theprobabilityofstockingoutisequalto1–CR.

E T ≤ ã =åç

(å8 + åç)

Forthesimplestcasewherethereisneithersalvagevaluenorextrapenaltyofstockingout,thesebecome:

cs=p–c,thatisthelostmarginofmissingapotentialsaleand,ce=c,thatis,thecostofpurchasingoneunit.

TheCriticalRatiobecomes:*C = £§£§V£F

= (™c£)(™c£V£)

= ™c£™whichissimplythemargindividedby

theprice!Whenweconsideralsosalvagevalue(g)andshortagepenalty(B),thesebecome:

cs=p–c+B,thatisthelostmarginofmissingapotentialsaleplusapenaltyperitemshortandce=c–g,thatis,thecostofpurchasingoneunitminusthesalvagevalueIcangainback.

Nowthecriticalratiobecomes

*C =åç

åç + å8=

(´ − å + ¨)(´ − å + ¨ + å − †)

=(´ − å + ¨)(´ + ¨ − †)

SinglePeriodInventoryModels-ExpectedProfitabilityWeexpandouranalysisofthesingleperiodmodeltobeabletocalculatetheexpectedprofitabilityofagivensolution.Inthepreviouslesson,welearnedhowtodeterminetheoptimalorderquantity,Q*,suchthattheprobabilityofthedemanddistributioncoveredbyQ*isequaltotheCriticalRatio,whichistheratiooftheshortagecostsdividedbythesumoftheshortageandexcesscosts.Inordertodeterminetheprofitabilityforasolution,weneedtocalculatetheexpectedunitssold,theexpectedcostofbuyingQunits,andtheexpectedunitsshort,E[US].CalculatingtheE[US]istricky,butweshowhowtousetheNormalTablesaswellasspreadsheetstodeterminethisvalue.

NotationB: Penaltyfornotsatisfyingdemandbeyondlostprofit($/unit)c: Purchasecost($/unit)ct: OrderingCosts($/order)ce: ExcessholdingCosts($/unit);Forsingleperiodproblemsthisisnotnecessarily

equaltoch,sincethatassumesthatyoucankeeptheinventoryforlateruse.cs: ShortageCosts($/unit)D: AverageDemand(units/time)g: Salvagevalueforexcessinventory($/unit)k: SafetyFactorx: UnitsDemandedE[x]: ExpectedunitsdemandedE[US]:ExpectedUnitsShort(units)Q: ReplenishmentOrderQuantity(units/order)TRC(Q): TotalRelevantCost($/period)TC(Q): TotalCost($/period)

Formulas

ProfitMaximizationInwords,theexpectedprofitfororderingQunitsisequaltothesalesprice,p,timestheexpectednumberofunitsdemanded,E[x]),minusthecostofpurchasingQunits,cQ,minustheexpectednumberofunitsIwouldbeshorttimesthesalesprice.Thedifficultpartofthisequationistheexpectedunitsshort,ortheE[US].

A[E�ÄôêX(ã)] = ´A[T] − åã − ´A[èSêXë@ℎÄ�X]

ExpectedProfitswithSalvageandPenaltyIfweincludeasalvagevalue,g,andashortagepenalty,B,thenthisbecomes:

E(ã) = ≠−åã + ´T + †(ã − T)êôT ≤ ã−åã + ´ã − ¨(T − ã)êôT ≥ ã

A[E(ã)] = (´ − †)A[T] − (å − †)ã − (´ − † + ¨)A[è@]Rearrangingthisbecomes:

A[E(ã)] = ´(A[T] − A[è@]) − åã + †iã − (A[T] − A[è@])j − ¨(A[è@])Inwords,theexpectedprofitfororderingQunitsisequaltofourterms.Thefirsttermisthesalesprice,p,timestheexpectednumberofunitsdemanded,E[x]),minustheexpectedunitsshort.ThesecondtermissimplythecostofpurchasingQunits,cQ.ThethirdtermistheexpectednumberofitemsthatIwouldhaveleftoverforsalvage,timesthesalvagevalue,g.Thefourthandfinaltermistheexpectednumberofunitsshorttimestheshortagepenalty,B.

ExpectedValuesE[UnitsDemanded]Continuous: ∫ TôZ(T)ÇT

∞Zhm = TY Discrete: ∑ TE[T] =∞

Zhm TYE[UnitsSold]Continuous: ∫ TôZ(T)ÇT

±Zhm + ã ∫ ôZ(T)ÇT

∞Zh± Discrete: ∑ TE[T]±

Zhm +ã ∑ E[T]∞

Zh±VL

E[UnitsShort]

Continuous: ∫ (T − ã)ôZ(T)ÇT∞Zh± Discrete: ∑ (T − ã)E[T]∞

Zh±VL

ProbabilisticInventoryModelsWedevelopinventoryreplenishmentmodelswhenwehaveuncertainorstochasticdemand.WebuiltoffofboththeEOQandthesingleperiodmodelstointroducethreegeneralinventory

ExpectedUnitsShortE[US]Thisisatrickyconcepttogetyourheadaroundatfirst.ThinkoftheE[US]astheaverage(meanorexpectedvalue)ofthedemandABOVEsomeamountthatwespecifyorhaveonhand.AsmyQgetslarger,thenweexpecttheE[US]togetsmaller,sinceIwillprobablynotstockoutasmuch.Luckilyforus,wehaveanicewayofcalculatingtheE[US]fortheNormalDistribution.TheExpectedUnitNormalLossFunctionisnotedasG(k).Tofindtheactualunitsshort,wesimplymultiplythisG(k)timesthestandarddeviationoftheprobabilitydistribution.

A[è@] = ≤ (T − ã)ôZ(T)ÇT∞

Zh±= -≥ w

ã − .- x = -≥(¥)

YoucanusetheNormaltablestofindtheG(k)foragivenkvalueoryoucanusespreadsheetswiththeequationbelow:

≥(¥) = µ¢C56ó@ä(¥, 0,1,0) − ¥ ∗ (1 − µ¢C5@6ó@ä(¥))

E[Excess]E[Excess]andE[US]/E[UnitsShort]aredifferentthingsbuttheyarerelatedtoeachother.

E[Excess]isadistancefromQ.So,onaverage,howmuchofQwehaveleftafterwehavesoldalltheunitstobesold.E[UnitsShort]isadistancefromtheDemand.So,onaverage,howmanyunitsshortoftheDemandwereweoncewe'vesoldalltheunitsthatweresoldthatperiod.Youdon'thavetouseE[Excess].Everyproblemcanbesolvedusingeitherone,andyoucanuseeithertocalculateprofits.Ifyouwanttoseehowwegetfromonetotheother,herearesomesteps:

policies:theBaseStockPolicy,the(s,Q)continuousreviewpolicyandthe(R,S)periodicreviewpolicy(theR,Smodelwillbeexplainedinthenextlesson).Thesearethemostcommonlyusedinventorypoliciesinpractice.Theyareimbeddedwithinacompany’sERPandinventorymanagementsystems.Toputthemincontext,hereisthesummaryofthefiveinventorymodelscoveredsofar:

• EconomicOrderQuantity—DeterministicDemandwithinfinitehorizon

o OrderQ*everyT*periods

o OrderQ*whenIP=μDL

• SinglePeriod/Newsvendor—ProbabilisticDemandwithfinite(singleperiod)horizon

o OrderQ*atstartofperiodwhereP[x≤Q]=CR

• BaseStockPolicy—ProbabilisticDemandwithinfinitehorizon

o Essentiallyaone-for-onereplenishment

o Orderwhatwasdemandedwhenitwasdemandedinthequantityitwas

demanded

• ContinuousReviewPolicy(s,Q)—ProbabilisticDemandwithinfinitehorizon

o Thisisevent-based–weorderwhen,andif,inventorypassesacertainthreshold

o OrderQ*whenIP≤s

• PeriodicReviewPolicy(R,S)—ProbabilisticDemandwithinfinitehorizon

o Thisisatime-basedpolicyinthatweorderonasetcycle

o OrderuptoSunitseveryRtimeperiods

Allofthemodelsmaketrade-offs:EOQbetweenfixedandvariablecosts,Newsvendorbetweenexcessandshortageinventory,andthelatterthreebetweencostandlevelofservice.Theconceptoflevelofservice,LOS,isoftenmurkyandspecificdefinitionsandpreferencesvarybetweenfirms.However,forourpurposes,wecanbreakthemintotwocategories:targetsandcosts.Wecanestablishatargetvalueforsomeperformancemetricandthendesigntheminimumcostinventorypolicytoachievethelevelofservice.ThetwometricscoveredareCycleServiceLevel(CSL)andItemFillRate(IFR).Thesecondapproachistoplaceadollaramountonaspecifictypeofstockoutoccurringandthenminimizethetotalcostfunction.ThetwocostmetricswecoveredwereCostofStockOutEvent(CSOE)andCostofItemShort(CIS).Theyarerelatedtoeachother.Regardlessofthemetricsused,theendresultisasafetyfactor,k,andasafetystock.ThesafetystockissimplykσDL.ThetermσDLisdefinedasthestandarddeviationofdemandoverleadtime,butitismoretechnicallytherootmeansquareerror(RMSE)oftheforecastovertheleadtime.Mostcompaniesdonottracktheirforecasterrortothegranularlevelthatyourequireforsettinginventorylevels,sodefaultingtothestandarddeviationofdemandisnottoobadofanestimate.Itisessentiallyassumingthattheforecastisthemean.

NotationB1: Costassociatedwithastockoutevent($/event)c: Purchasecost($/unit)ct: OrderingCosts($/order)ce: ExcessholdingCosts($/unit/time);Equaltochcs: Shortagecosts($/unit)D: AverageDemand(units/time)DS: Demandovershorttimeperiod(e.g.week)DL: Demandoverlongtimeperiod(e.g.month)h: Carryingorholdingcost($/inventory$/time)L: ReplenishmentLeadTime(time)Q: ReplenishmentOrderQuantity(units/order)T: OrderCycleTime(time/order)μDL: ExpectedDemandoverLeadTime(units/time)σDL: StandardDeviationofDemandoverLeadTime(units/time)k: SafetyFactors: ReorderPoint(units)S: OrderuptoPoint(units)R: ReviewPeriod(time)N: OrdersperTimeor1/T(order/time)IP: InventoryPosition=InventoryonHand+InventoryonOrder–BackordersIOH: InventoryonHand(units)IOO: InventoryonOrder(units)IFR: ItemFillRate(%)CSL: CycleServiceLevel(%)CSOE: CostofStockOutEvent($/event)CIS: CostperItemShortE[US]: ExpectedUnitsShort(units)G(k): UnitNormalLossFunction

BaseStockPolicyTheBaseStockpolicyisaone-for-onepolicy.IfIsellfouritems,Iorderfouritemstoreplenishtheinventory.Thepolicydetermineswhatthestockinglevel,orthebasestock,isforeachitem.Thebasestock,S*,isthesumoftheexpecteddemandovertheleadtimeplustheRMSEoftheforecasterroroverleadtimemultipliedbysomesafetyfactork.TheLOSforthispolicyissimplytheCriticalRatio.Notethattheexcessinventorycost,ce,inthiscase(andallmodelshere)assumesyoucanuseitlaterandistheproductofthecostandtheholdingrate,ch.

• OptimalBaseStock,S*: S∗ = µ∏π + kπßªσ∏π• LevelofService(LOS): LOS=P[μDL≤S*]=CR=

ΩæΩæVΩø

Formulas

LevelofServiceMetricsHerearefourmethodsfordeterminingtheappropriatesafetyfactor,k,foruseinanyoftheinventorymodels.TheyareCycleServiceLevel,CostperStockOutEvent,ItemFillRate,andCostperItemShort.

ContinuousReviewPolicies(s,Q)ThisisalsoknownastheOrder-Point,Order-Quantitypolicyandisessentiallyatwo-binsystem.Thepolicyis“OrderQ*unitswhenInventoryPositionislessthanthere-orderpoints”.There-orderpointisthesumoftheexpecteddemandoverthelead-timeplustheRMSEoftheforecasterroroverlead-timemultipliedbysomesafetyfactork.

• ReorderPoint: ë = .~} + ¥-~} • OrderQuantity(Q): QistypicallyfoundthroughtheEOQformula

CycleServiceLevel(CSL)TheCSListheprobabilitythattherewillnotbeastockoutwithinareplenishmentcycle.ThisisfrequentlyusedasaperformancemetricwheretheinventorypolicyisdesignedtominimizecosttoachieveanexpectedCSLof,say,95%.Thus,itisoneminustheprobabilityofastockoutoccurring.IfIknowthetargetCSLandthedistribution(wewilluseNormalmostofthetime)thenwecanfindthesthatsatisfiesitusingtablesoraspreadsheetwheres=NORMINVDIST(CSL,Mean,StandardDeviation)andk=NORMSINV(CSL).

CSL = 1 − P[Stockout] = 1 − P[X > s] = P[X ≤ s]

Notethataskincreases,itgetsdifficulttoimproveCSLanditwillrequireenormousamountofinventorytocovertheextremelimits.

CostPerStockoutEvent(CSOE)orB1CostTheCSOEisrelatedtotheCSL,butinsteadofdesigningtoatargetCSLvalue,apenaltyischargedwhenastockoutoccurswithinareplenishmentcycle.Theinventorypolicyisdesignedtominimizethetotalcosts–sothisbalancescostofholdinginventoryexplicitlywiththecostofstockingout.Minimizingthetotalcostsfork,wefindthataslongas

®<~£F3… ±√KÀ

>1,thenweshouldset:

¥ = Ã2 ln t¨L6

å8-~}ã√2œu

If®<~

£F3… ±√KÀ<1,weshouldsetkaslowasmanagementallows.

SummaryoftheMetricsPresented Metric Howtofindk

%ServiceBased CycleServiceLevel(CSL) K=NORMSINV(1 − P X > s )

%ServiceBased ItemFillRate(IFL) Findkfrom≥ ¥ = ±3…

(1 − ó2C)

$CostBased CostperStockOutEvent(CSOE)¥ = 2 ln

¨L6å8-~}ã 2œ

$CostBased CostperItemShort(CIS) K=NORMSINV(1-±£F~£§

Table2.Summaryofmetricspresented

ItemFillRate(IFR)TheIFRisthefractionofdemandthatismetwiththeinventoryonhandoutofcyclestock.ThisisfrequentlyusedasaperformancemetricwheretheinventorypolicyisdesignedtominimizecosttoachieveanexpectedIFRof,say,90%.IfIknowthetargetIFRandthedistribution(wewilluseNormalmostofthetime)thenwecanfindtheappropriatekvaluebyusingtheUnitNormalLossFunction,G(k).

ó2C = 1 −A[è@]ã = 1 −

-~}≥[¥]ã

≥(¥) =ã-~}

(1 − ó2C)

G(k)istheUnitNormalLossFunction,whichcanbecalculatedinSpreadsheetsas≥(¥) = µ¢C56ó@ä(¥, 0,1,0) − ¥ ∗ (1 − µ¢C5@6ó@ä(¥))

Oncewefindthekusingunitnormaltables,wecanplugthevaluesinë = .~} +¥-~}toframethepolicy.

CostperItemShort(CIS)TheCISisrelatedtotheIFR,butinsteadofdesigningtoatargetIFRvalue,apenaltyischargedforeachitemshortwithinareplenishmentcycle.Theinventorypolicyisdesignedtominimizethetotalcosts–sothisbalancescostofholdinginventoryexplicitlywiththe

costofstockingout.Minimizingthetotalcostsfork,wefindthataslongas±£F~£§

≤ 1,thenweshouldfindksuchthat:

E[@XÄå¥¢ûX] = E[T ≥ ¥] =ãå86åç

Otherwise,weshouldsetkaslowasmanagementallows.Inaspreadsheet,this

becomesk=NORMSINV(1-±£F~£§

ATiponConvertingTimesYouwilltypicallyneedtoconvertannualforecaststoweeklydemandorviceversaorsomethinginbetween.Thisisgenerallyveryeasy–butsomestudentsgetconfusedattimes:Convertinglongtoshort(nisnumberofshortperiodswithinlong):

A[6°] = A[6}]/S+1C 6° = +1C 6} /S

-ç = -}/ SConvertingfromshorttolong:

A 6} = SA[6°]+1C 6} = S+1C 6°

-} = S-ç

PeriodicReviewPoliciesTherearetrade-offsbetweenthedifferentperformancemetrics(bothcost-andservice-based).Wedemonstratethatonceoneofthemetricsisdetermined(orexplicitlyset)thentheotherthreeareimplicitlyset.Becausetheyallleadtotheestablishmentofasafetyfactor,k,theyaredependentoneachother.Thismeansthatonceyouhavesetthesafetystock,regardlessofthemethod,youcancalculatetheexpectedperformanceimpliedbytheremainingthreemetrics.PeriodicReviewpoliciesareverypopularbecausetheyfittheregularpatternofworkwhereorderingmightoccuronlyonceaweekoronceeverytwoweeks.Thelead-timeandthereviewperiodarerelatedandcanbetraded-offtoachievecertaingoals.

NotationB1: Costassociatedwithastockouteventc: Purchasecost($/unit)ct: OrderingCosts($/order)ce: ExcessholdingCosts($/unit/time);Equaltochcs: Shortagecosts($/unit)cg: OneTimeGoodDealPurchasePrice($/unit)D: AverageDemand(units/time)h: Carryingorholdingcost($/inventory$/time)L: ReplenishmentLeadTime(time)Q: ReplenishmentOrderQuantity(units/order)T: OrderCycleTime(time/order)μDL: ExpectedDemandoverLeadTime(units/time)σDL: StandardDeviationofDemandoverLeadTime(units/time)μDL+R: ExpectedDemandoverLeadTimeplusReviewPeriod(units/time)σDL+R: StandardDeviationofDemandoverLeadTimeplusReviewPeriod(units/time)k: SafetyFactors: ReorderPoint(units)S: OrderuptoPoint(units)

R: ReviewPeriod(time)N: OrdersperTimeor1/T(order/time)IP: InventoryPosition=InventoryonHand+InventoryonOrder–BackordersIOH: InventoryonHand(units)IOO: InventoryonOrder(units)IFR: ItemFillRate(%)CSL: CycleServiceLevel(%)CSOE: CostofStockOutEvent($/event)CIS: CostperItemShortE[US]: ExpectedUnitsShort(units)G(k): UnitNormalLossFunction

FormulasInventoryPerformanceMetricsSafetystockisdeterminedbythesafetyfactor,k.Sothat:ë = .~} + ¥-~}andtheexpectedcostofsafetystock=å8¥-~}.Twowaystocalculatek:ServicebasedorCostbasedmetrics:

SafetyStockLogic–relationshipbetweenperformancemetricsTherelationshipbetweenthefourmetrics(2costand2servicebased)isshownintheflowchartbelow(Figure5).Onceonemetric(CSL,IFR,CSOE,orCIS)isexplicitlyset,thentheotherthreemetricsareimplicitlydetermined.

• ServiceBasedMetrics—setktomeetexpectedlevelofserviceo CycleServiceLevel(*@ò = E[T ≤ ¥])o ItemFillRate(ó2C = 1 − 3… –[—]

Note:IFRisalwayshigherthanCSLforthesamesafetystocklevel.

• CostBasedMetrics—findkthatminimizestotalcosts

o CostperStockoutEvent(E[CSOE] = (¨L)E[T ≥ ¥] ‘~±’)

o CostperItemsShort(A[*ó@] = åç-~}≥(¥) ‘~±’)

Figure5.Relationshipamongthefourmetrics

PeriodicReviewPolicy(R,S)ThisisalsoknownastheOrderUpTopolicyandisessentiallyatwo-binsystem.Thepolicyis“OrderUpToS*unitseveryRtimeperiods”.ThismeanstheorderquantitywillbeS*-IP.Theorderuptopoint,S*,isthesumoftheexpecteddemandoverthelead-timeandthereplenishmenttimeplustheRMSEoftheforecasterroroverleadplusreplenishmenttimemultipliedbysomesafetyfactork.

• OrderUpToPoint: @ = .~}V÷ + ¥-~}V÷

Periodic(R,S)versusContinuous(s,Q)Review• Thereisaconvenienttransformationof(s,Q)to(R,S)

o (s,Q)=Continuous,orderQwhenIP≤so (R,S)=Periodic,orderuptoSeveryRtimeperiods

• Allowsfortheuseofallprevious(s,Q)decisionruleso Reorderpoint,s,forcontinuousbecomesOrderUpTopoint,S,forperiodic

systemo QforcontinuousbecomesD*Rforperiodico LforacontinuousbecomesR+Lforperiodic

• Approacho Maketransformationso Solvefor(s,Q)usingtransformationso Determinefinalpolicysuchthat@ = T~}V÷ + ¥-~}V÷

(s,Q) (R,S)s ↔ S

Q ↔ D*RL ↔ R+L

RelationshipBetweenL&RTheleadtime,L,andthereviewperiod,R,bothinfluencethetotalcosts.Notethattheaverage

inventorycostsfora(R,S)systemis= å8[~÷K+ ¥-~}V÷ + ò6].ThisimpliesthatincreasingLead

Time,L,willincreaseSafetyStocknon-linearlyandPipelineInventorylinearlywhileincreasingtheReviewPeriod,RwillincreasetheSafetyStocknon-linearlyandtheCycleStocklinearly.

InventoryModelsforMultipleItems&LocationsThereareseveralproblemswithmanagingitemsindependently,including:

• Lackofcoordination—constantlyorderingitems

• Ignoringofcommonconstraintssuchasfinancialbudgetsorspace

• Missedopportunitiesforconsolidationandsynergies

• Wasteofmanagementtime

ManagingMultipleItemsTherearetwoissuestosolveinordertomanagemultipleitems:

1. CanweaggregateSKUstousesimilaroperatingpolicies?a. Groupusingcommoncostcharacteristicsorbreakpointsb. GroupusingPowerofTwoPolicies

2. Howdowemanageinventoryundercommonconstraints?a. Exchangecurvesforcyclestockb. Exchangecurvesforsafetystock

AggregationMethodsWhenwehavemultipleSKUstomanage,wewanttoaggregatethoseSKUswherewecanusethesamepolicies.GroupingLikeItems—BreakPoints

• BasicIdea:Replenishhighervalueitemsfaster• Usedforsituationswithmultipleitemsthathave

o Relativelystabledemando Commonorderingcosts,ct,andholdingcharges,ho Differentannualdemands,Di,andpurchasecostci

• Approacho Pickabasetimeperiod,w0,(typicallyaweek)o Createasetofcandidateorderingperiods(w1,w2,etc.)o FindDicivalueswhereTRC(wj)=TRC(wj+1)o GroupSKUsthatfallincommonvalue(Dici)buckets

PowerofTwoFormula• Orderintimeintervalsofpowersoftwo• Selectarealisticbaseperiod,Tbase(day,week,month)

GroupingLikeItemsExampleSelectedw0=1weekNumberofweeksofsupply(WOS)toorderforitemiorderingattimeperiodj=Qij=Di(wj/52)Selectingbetweenoptionsw1&w2(wherew1<w2<w3etc.)becomes:

ctDi/Qi1+(cihQi1)/2=ctDi/Qi2+(cihQi2)/252ctDi/Diw1+cihDiw1/104=52ctDi/Diw2+cihDiw2/104

(cihDi/104)(w1-w2)=(52ct)(1/w2–1/w1)Dici=[(104)(52ct)/(h(w1-w2))](1/w2–1/w1)

Dici=5408ct/(hw1w2) RuleifDici≥5408ct/(hw1w2)thenselectw1 Else:ifDici≥5408ct/(hw2w3)thenselectw2 Else:ifDici≥5408ct/(hw3w4)thenselectw3 Else:......

GroupingLikeItemsExampleSupposeyouneedtosetupreplenishmentschedulesforseveralhundredpartsthathaverelativelystable(yetnotnecessarilythesame)demand.Theyallhavesimilarordercosts(ct=$5)andholdingcharge(h=0.20).Youhavethefollowingpotentialorderingperiods(inweeks):w1=1,w2=2,w3=4,w4=13,w5=26,andw6=52.Whatbreak-evenorderingpointsshouldyouestablish?Break-pointforselectingbetween1weekor2weeksis:Dici=5408t/(hw1w2)=5408(5)/(.2)(1)(2)=$67,600IfDici≥$67,600thenorder1week’swortheachweekBreak-pointforselectingbetween2weeksor4weeksis:Dici=5408ct/(hw2w3)=5408(5)/(.2)(2)(4)=$16,900If$67,600>Dici≥$16,900thenorder2week’sworthevery2weeksFinalOrderingBreakPoints:Orderevery1weekifDici≥$67,600Orderevery2weeksifDici≥$16,900Orderevery4weeksifDici≥$2,600Orderevery13weeksifDici≥$400Orderevery26weeksifDici≥$100Orderevery52weeksotherwise

• GuaranteesthatTRCwillbewithin6%ofoptimal!

ManagingUnderCommonConstraintsThereistypicallyabudgetorspaceconstraintthatlimitstheamountofinventorythatyoucanactuallykeeponhand.Managingeachinventoryitemseparatelycouldleadtoviolatingthisconstraint.Exchangecurvesareagoodwaytousethemanagerialleversofholdingcharge,orderingcost,andsafetyfactortosetinventorypoliciestomeetacommonconstraint.ExchangeCurves:CycleStock

• HelpsdeterminethebestallocationofinventorybudgetacrossmultipleSKUs• RelevantCostparameters

o HoldingCharge(h)§ Thereisnosinglecorrectvalue§ Costallocationsfortimeandsystemsdifferbetweenfirms§ Reflectionofmanagement’sinvestmentandriskprofile

o OrderCost(ct)§ Notknowwithprecision§ Costallocationsfortimeandsystemsdifferbetweenfirms

• ExchangeCurveo Depictstrade-offbetweentotalannualcyclestock(TACS)andnumberof

replenishments(N)o Determinesthect/hvaluethatmeetsbudgetconstraints

ExchangeCurves:SafetyStock

• Needtotrade-offcostofsafetystockandlevelofservice• Keyparameterissafetyfactor(k)–usuallysetbymanagement• Estimatetheaggregateservicelevelfordifferentbudgets• Theprocessisasfollows:

1. Selectaninventorymetrictotarget2. Startingwithahighmetricvaluecalculate:

a. TherequiredkitomeetthattargetforeachSKUb. TheresultingsafetystockcostforeachSKUandthetotalsafetystock

(TSS)c. TheotherresultinginventorymetricsofinterestforeachSKUandtotal

3. Lowerthemetricvalue,gotostep24. ChartresultingTSSversusInventoryMetrics

ManagingMultipleLocationsManagingthesameiteminmultiplelocationswillleadtoahigherinventorylevelthanmanagingtheminasinglelocation.Consolidatinginventorylocationstoasinglecommonlocationisknownasinventorypooling.PoolingreducesthecyclestockneededbyreducingthenumberofdeliveriesrequiredandreducesthesafetystockbyriskpoolingthatreducestheCVofthedemand.Thisisalsocalledthesquareroot“law”–whichisinsightfulandpowerful,but

alsomakessomerestrictiveassumptions,suchasuniformlydistributeddemand,useofEOQorderingprinciples,andindependenceofdemandindifferentlocations.

Notationci: Purchasecostforitemi($/unit)ct: OrderingCosts($/order)ce: ExcessholdingCosts($/unit/time);Equaltochcs: Shortagecosts($/unit)Di: AverageDemandforitemi(units/time)h: Carryingorholdingcost($/inventory$/time)Q: ReplenishmentOrderQuantity(units/order)T: OrderCycleTime(time/order)TPractical:PracticalOrderCycleTime(time/order)k: SafetyFactorw0: BaseTimePeriod(time)s: ReorderPoint(units)R: ReviewPeriod(time)N: NumberofInventoryReplenishmentCyclesTACS: TotalAnnualCycleStockTSS: TotalValueofSafetyStockTVIS: TotalValueofItemsShortG(k): UnitNormalLossFunction

Formulas

PowerofTwoPolicyTheprocessisasfollows:

1. CreatetableofSKUs2. CalculateT*foreachSKU3. CalculateTpracticalforeachSKU

ä∗ =ã∗

6 =D2å06å86 = Ã

2å06å8

ä™◊I£0Q£Iÿ = 2Ÿ⁄w¤

√Kx/ Ÿ⁄(K)

Inaspreadsheetthisis:Tpractical=2^(ROUNDUP(LN(Toptimal/SQRT(2))/LN(2)))

PooledInventories

Chart1.Comparisonbetweenindependentandpooledinventories

ExchangeCurves:CycleStock

ä1*@ = ∑ ±[£[K= D£9

‹L√K∑ P6QåQ=QhL =

QhL µ = ∑ ~[±[==

QhL D‹£9

L√K∑ P6QåQ=QhL

Process

• CreateatableofSKUswith“AnnualValue”(Dici)andPDoco• FindthesumofPDocotermforSKUsbeinganalyzed

• CalculateTACSandNforrangeof(ct/h)values• ChartNvsTACS

ExchangeCurves:SafetyStockä@@ = ∑ ¥Q-~}QåQ=

QhL ä+ó@ = ∑ (~[±[

=QhL åQ-~}Q≥(¥Q))

Process:1. Selectaninventorymetrictotarget2. Startingwithahighmetricvaluecalculate:

a. TherequiredkitomeetthattargetforeachSKUb. TheresultingsafetystockcostforeachSKUandthetotalsafetystock

(TSS)c. TheotherresultinginventorymetricsofinterestforeachSKUandtotal

3. Lowerthemetricvalue,gotostep24. ChartresultingTSSversusInventoryMetrics

InventoryModelsforClassA&CItemsInventoryManagementbySegment

AItems BItems CItemsTypeofRecords Extensive,Transactional Moderate None-usearule

LevelofManagementReporting

Frequent(MonthlyorMore)

Infrequently—Aggregated

OnlyasAggregate

InteractionwithDemand DirectInput,HighDataIntegrity,Manipulate

(pricing,etc.)

ModifiedForecast(promotions,etc.)

SimpleForecastatBest

InteractionwithSupply ActivelyManage ManagebyException NonInitialDeployment MinimizeExposure(high

v)SteadyState SteadyState

FrequencyofPolicyReview

VeryFrequent(MonthlyorMore)

Moderate(Annually/EventBased)

VeryInfrequent

ImportanceofParameterPrecision

VeryHigh—AccuracyWorthwhile

Moderate—RoundingandApproximationok

VeryLow

ShortageStrategy ActivelyManage(Confront)

SetServiceLevel&ManagebyException

Set&ForgetServiceLevels

DemandDistribution ConsiderAlternativestoNormalasSituationFits

Normal N/A

ManagementStrategy Active Automatic PassiveTable3.Inventorymanagementbysegment

InventoryPolicies(RulesofThumb)TypeofItem ContinuousReview PeriodicReviewAItems (s,S) (R,s,S)BItems (s,Q) (R,S)CItems Manual~(R,S)

Table4.Inventorypolicies(rulesofThumb)

ManagingClassAItemsTherearetwogeneralwaysthatitemscanbeconsideredClassA:

• FastMovingbutCheap(LargeD,Smallc→Q>1)

• SlowMovingbutExpensive(Largec,SmallD→Q=1

ThisdictateswhichProbabilityDistributiontouseformodelingthedemand• FastMovers

o NormalorLognormalDistribution

o GoodenoughforBitems

o OKforAitemsifµDLorμDL+R≥10

• SlowMovers

o PoissonDistribution

o Morecomplicatedtohandle

o OKforAitemsifµDLorμDL+R<10

ManagingClassCItemsClassCitemshavelowcDvaluesbutcomprisethelion-shareoftheSKUs.Whenmanagingthemweneedtoconsidertheimplicit&explicitcosts.Theobjectiveistominimizemanagementattention.Regardlessofpolicy,savingswillmostlikelynotbesignificant,sotrytodesignsimplerulestofollowandexploreopportunitiesfordisposingofinventory.Alternatively,trytosetcommonreorderquantities.ThiscanbedonebyassumingcommonctandhvaluesandthenfindingDicivaluesfororderingfrequencies.DisposingofExcessInventory

• Whydoesexcessinventoryoccur?o SKUportfoliostendtogrowo Poorforecasts-Shorterlifecycles

• Whichitemstodispose?o LookatDOS(daysofsupply)foreachitem=IOH/Do ConsidergettingridofitemsthathaveDOS>xyears

• Whatactionstotake?o Converttootheruseso Shiptomoredesiredlocationo Markdownpriceo Auction

RealWorldInventoryChallengesWhilemodelsareimportant,itisalsoimportanttounderstandwheretherearechallengesimplementingmodelsinreallife.

• Modelsarenotusedexactlyasintextbooks• Dataisnotalwaysavailableorcorrect• Technologymatters• Businessprocessesmatterevenmore• Inventorypoliciestrytoanswerthreequestions:

o HowoftenshouldIcheckmyinventory?o HowdoIknowifIshouldordermore?o Howmuchtoorder?

• Allinventorymodelsusetwokeynumberso InventoryPositiono OrderPoint

NotationB1: CostAssociatedwithaStockoutEventc: PurchaseCost($/unit)ct: OrderingCosts($/order)

ce: ExcessHoldingCosts($/unit/time);Equaltochcs: ShortageCosts($/unit)cg: OneTimeGoodDealPurchasePrice($/unit)D: AverageDemand(units/time)h: CarryingorHoldingCost($/inventory$/time)L[Xi]: DiscreteUnitLossFunctionQ: ReplenishmentOrderQuantity(units/order)T: OrderCycleTime(time/order)μDL: ExpectedDemandoverLeadTime(units/time)σDL: StandardDeviationofDemandoverLeadTime(units/time)μDL+R: ExpectedDemandoverLeadTimeplusReviewPeriod(units/time)σDL+R: StandardDeviationofDemandoverLeadTimeplusReviewPeriod(units/time)k: SafetyFactors: ReorderPoint(units)S: OrderUptoPoint(units)R: ReviewPeriod(time)N: OrdersperTimeor1/T(order/time)IP: InventoryPosition=InventoryonHand+InventoryonOrder(IOO)–BackordersIOH: InventoryonHand(units)IOO: InventoryonOrder(units)IFR: ItemFillRate(%)CSL: CycleServiceLevel(%)E[US]: ExpectedUnitsShort(units)

G(k): UnitNormalLossFunction

Formulas

LearningObjectives

• Understandthereasonsforholdinginventoryandthedifferenttypesofinventory.

• Understandtheconceptsoftotalcostandtotalrelevantcosts.

• Identifyandquantifythefourmajorcostcomponentsoftotalcosts:Purchasing,

Ordering,Holding,andShortage.

• AbletoestimatetheEconomicOrderQuantity(EOQ)andtodeterminewhenitis

appropriatetouse.

• AbletoestimatesensitivityofEOQtounderlyingchangesintheinputdataand

understandingofitsunderlyingrobustness.

• UnderstandhowtodeterminetheEOQwithdifferentvolumediscountingschemes.

FastMovingAItems

äC* = å0 w6ãx + å8 w

ã2 + ¥-~}x + ¨L w

6ãxE[T > ¥]

ã∗ = A¢ãÃ1+¨LE[T > ¥]

¥∗ = Ã2 ln t6¨L

√2œãå8-~}u

• Iterativelysolvethetwoequations• StopwhenQ*andk*convergewithinacceptablerange

SlowMovingAItemsUseaPoissondistributiontomodelsales

• Probabilityofxeventsoccurringwithinatimeperiod• Mean=Variance=λ

´[Tm] = E�Ä"[T = Tm] =/cfiflZ‡Tm!

ôÄ�Tm

2[Tm] = E�Ä"[T ≤ Tm] = e/cfiflZ

Foradiscretefunction,thelossfunctionL[Xi]canbecalculatedasfollows(Cachon&Terwiesch)

ò[‚Q] = ò[‚QcL] − (‚Q − ‚QcL)(1 − 2[‚QcL])

• UnderstandhowtodeterminetheEconomicProductionQuantity(EPQ)whenthe

inventorybecomesavailableatacertainrateoftimeinsteadofallatonce.

• AbilitytousetheCriticalRatiotodeterminetheoptimalorderquantitytomaximize

expectedprofits.

• AbilitytoestablishedinventorypoliciesforEOQwithplannedbackordersaswellas

singleperiodmodels.

• Abilitytodetermineprofitability,expectedunitsshort,expectedunitssoldofasingle

periodmodel.

• Understandingofsafetystockanditsroleinprotectingforexcessdemandoverlead

• Abilitytodevelopbasestockandorder-point,order-quantitycontinuousreviewpolicies.

• Abilitytodeterminepropersafetyfactor,k,giventhedesiredCSLorIFRorthe

appropriatecostpenaltyforCSOEorCIS.

• Abletoestablishaperiodicreview,OrderUpTo(S,R)ReplenishmentPolicyusinganyof

thefourperformancemetrics.

• Understandrelationshipsbetweentheperformancemetrics(CSL,IFR,CSOE,andCIS)

andbeabletocalculatetheimplicitvalues.

• Abletousetheinventorymodelstomaketrade-offsandestimateimpactsofpolicy

changes.

• UnderstandhowtousedifferentmethodstoaggregateSKUsforcommoninventory

policies.

• UnderstandhowtouseExchangeCurves.

• Understandhowinventorypoolingimpactsbothcyclestockandsafetystock.

• UnderstandhowtousedifferentinventorymodelsforClassAandCitems.

ReferencesForGeneralInventoryManagementTherearemorebooksthatcoverthebasicsofinventorymanagementthantherearegrainsofsandonthebeach!InventorymanagementisalsousuallycoveredinOperationsManagementandIndustrialEngineeringtextsaswell.Awordofwarning,though.Everytextbookusesdifferentnotationforthesameconcepts.Getusedtoit.Alwaysbesuretounderstandwhatthenomenclaturemeanssothatyoudonotgetconfused.

• Nahmias,S.ProductionandOperationsAnalysis.McGraw-HillInternationalEdition.ISBN:0-07-2231265-3.Chapter4.

• Silver,E.A.,Pyke,D.F.,Peterson,R.InventoryManagementandProductionPlanningandScheduling.ISBN:978-0471119470.Chapter1

• Ballou,R.H.BusinessLogisticsManagement.ISBN:978-0130661845.Chapter9.

ForEOQ• Schwarz,LeroyB.,“TheEconomicOrder-Quantity(EOQ)Model”inBuildingintuition:

insightsfrombasicoperationsmanagementmodelsandprinciples,editedbyDilipChhajed,TimothyLowe,2007,Springer,NewYork,(pp135-154).

• Silver,E.A.,Pyke,D.F.,Peterson,R.InventoryManagementandProductionPlanningandScheduling.ISBN:978-0471119470.Chapter5

ForEOQExtensions• Nahmias,S.ProductionandOperationsAnalysis.McGraw-HillInternationalEdition.

ISBN:0-07-2231265-3.Chapter4.• Silver,E.A.,Pyke,D.F.,Peterson,R.InventoryManagementandProductionPlanningand

Scheduling.ISBN:978-0471119470.Chapter5.• Ballou,R.H.BusinessLogisticsManagement.ISBN:978-0130661845.Chapter9.• Schwarz,LeroyB.,“TheEconomicOrder-Quantity(EOQ)Model”inBuildingintuition:

insightsfrombasicoperationsmanagementmodelsandprinciples,editedbyDilipChhajed,TimothyLowe,2007,Springer,NewYork,(pp135-154).

• Muckstadt,JohnandAmarSapra"ModelsandSolutionsinInventoryManagement".,2006,SpringerNewYork,NewYork,NY.Chapter2&3.

ForSinglePeriodInventoryModels• Nahmias,S.ProductionandOperationsAnalysis.McGraw-HillInternationalEdition.

Scheduling.ISBN:978-0471119470.Chapter10.• Porteus,EvanL.,“TheNewsvendorProblem”inBuildingintuition:insightsfrombasic

operationsmanagementmodelsandprinciples,editedbyDilipChhajed,TimothyLowe,2007,Springer,NewYork,(pp115-134).

• Muckstadt,JohnandAmarSapra"ModelsandSolutionsinInventoryManagement".,2006,SpringerNewYork,NewYork,NY.Chapter5.

ForProbabilisticInventoryModels• Nahmias,S.ProductionandOperationsAnalysis.McGraw-HillInternationalEdition.

Scheduling.ISBN:978-0471119470.Chapter7.• Ballou,R.H.BusinessLogisticsManagement.ISBN:978-0130661845.Chapter9.• Muckstadt,JohnandAmarSapra"ModelsandSolutionsinInventoryManagement".,

2006,SpringerNewYork,NewYork,NY.Chapter9,10

ForInventoryModelswithMultipleItemsandLocations• Silver,E.A.,Pyke,D.F.,Peterson,R.InventoryManagementandProductionPlanningand

Scheduling.ISBN:978-0471119470.Chapter7&8.

ForInventoryModelsforClassA&ClassCItems

• Cachon,Gérard,andChristianTerwiesch.MatchingSupplywithDemand:AnIntroductiontoOperationsManagement.Boston,MA:McGraw-Hill/Irwin,2005.

• Silver,E.A.,Pyke,D.F.,Peterson,R.InventoryManagementandProductionPlanningandScheduling.ISBN:978-0471119470.Chapter8&9.

Warehousing

SummaryWenowmoveintoanimportant,yetoften-underexploredcomponentofthesupplychain,warehouses.Warehousesasareoftenoverlookedbecausetheygenerallydonotaddvaluetoaproduct,butareintermediarystationsinthesupplychain.Warehousesstore,handleand/orflowproduct.Theirprimaryoperationfunctionsaretoreceive;putaway;store;pick,packandshipproduct.InsomecasestheyplayaroleValue-AddServicessuchaslabeling,tagging,specialpackaging,minorassembling,kitting,re-pricing,etc.Inaddition,sometimetheyplayaroleinreturns.Themainapproachtoassessingwarehouseperformanceistoprofileitsactivityandbenchmark.Withcontinuousassessmentandfeedback,efficienciesatawarehousecanbeimproved.Businessestypicallyhavewarehousestobettermatchsupplyanddemand.Supplyisnotalwaysinsyncwithwhatisdemandedatthestore.Havingwarehousesservesasabufferforunexpectedshortagesanddemands.Inthislessonwewillcoverwarehousebasicsofthedifferenttypesofwarehouses,theircoreoperationalfunctions,andcommonflowpatterns.Wethenrevieweachofthemajorfunctions,whatisentailed,andhowbesttooptimizepractices.Weconcludewithdifferentwaysofassessingperformanceforbestperformanceofwarehouses.

WarehousingBasicsBasedonneeds,companiesselectdifferentwarehouses.Thewarehousescansimplybeaplacetostoreadditionalproductorcangoallthewaytoservingapartialassemblyandfinishingstage.Withinthewarehousetherearetwocompetingpriorities:spaceandtime.Thismeansthattheywanttomaximizetheirutilizationofspaceandoptimizethroughput.Thesearethetypesofwarehousesandtheirfunction:

§ RawMaterialStorage–closetoasourceormanufacturingpoints§ WIPWarehouses–partiallycompletedassembliesandcomponents§ FinishedGoodswarehouses–bufferslocatednearpointofmanufacture§ LocalWarehouses–inthefieldnearcustomerlocationstoproviderapidresponseto

customers§ FulfillmentCenters–holdsproductandshipssmallorderstoindividualconsumers

(casesoreaches)–predominatelyfore-commerce§ DistributionCenters–accumulateandconsolidateproductsfrommultiplesourcesfor

commonshipmenttocommondestination/customer§ MixingCenters–receivesmaterialfrommultiplesourcesforcross-dockingand

shipmentofmixedmaterials(palletstopallets)

PackageSizeBecausewarehousesareconstantlyconcernedaboutsavingspace,thismeansthatthepackagesizeisofgreatconcern.Thereareseveralprinciplesinpackagesizing.ThegeneralHandling

Ruleis:Thesmallerthehandlingunit,thegreaterthehandlingcost.Aswellasthatingeneral,theunitofstorageforaproductgetssmallerasitmovesdownstreamfromcontainertopallettocasetoeaches.Sizeimpactsdesignandoperationswithaninboundandoutboundflowfrompalletstopallets,palletstocasesandpalletstoeaches.

CoreOperationalFunctionsThereareseveralcoreoperationalfunctionsinwarehousesbeyondstorage.Theseincludereceiving;put-away;pick;check,pack;ship;aswellasadditionalbutnotalwaysincludedstepssuchasvalue-addedservicesandreturns.Seefigurebelow:

MIT Center for Transportation & Logistics

CoreWarehouseFuncSons

Receive Put-Away Check,Pack,ShipPick

Receive• Schedulingarrivals• Dockmanagement• Receiptofmaterials• Unloading&staging•  InspecSonfordamage,short,incomplete,etc.

Put-Away• Materialhandling• VerifystoragelocaSon• MovematerialinstoragelocaSon• Recordlevel&locaSon• SetsloznglocaSon

Store• Physicallyholdthematerial• ConsumesspacemorethanSme• MulSpleformsofstorage(pallet,case,each)

Pick• Movingitemsfromstoragefororders• Verifyinventoryonhand• CreateshippingdocumentaSon• Consistsoftravel,search,&extract

Check-Pack-Ship• Checkorderforcompleteness• Confirmdocuments• Placeinpackage(s)• Collectcommonorders• Schedulepickups•  Loadvehicle

Value-AddServices Returns

• CustomizaSonofproducts:•  Labeling&tagging,Specialpackaging,Minorassembly,Kizng,Re-pricing,etc.

• Postponementofcomponents

• HandlingproductreverseflowsformulSplereasons(damage,expired,returned,etc.)• Canrun5%(retail)andupto30%(e-commerce)ofvolume• StepscanincludeinspecSon,repair,reuse,refurbish,recycle,and/ordispose

~10% ~15% ~55% ~20%PercentofLaborCosts

ReceivingThereceivingfunctionofthewarehouseisoneofthemostimportantbecauseitsetsuptheinteractionatthewarehouseaswellasnextsteps.Therearesomegenerallyagreeduponbestpracticeswhichinclude:

• UseASNs(advancedshippingnotice)• Integrateyardanddockscheduling• Prepareforshipmentatreceiving• Thebestoptionistominimizereceivingactivity• Pursuedropshippingwheneverpossible• IFdropshipisnotpossible,explorecross-docking

PutawayTheputawayfunctionisessentiallyorderpickinginreverse.TheWarehouseManagementSystem(WMS)(willbediscussedingreaterdetailinTechnology&SystemsSC4X)playsasignificantroleinthisstepbydeterminingstoragelocationforreceiveditems(slotting).Italsodirectsstaffwheretoplaceproductandrecordsinventorylevel.RequireddataforWMSinclude:size,weight,cube,height,segmentationstatus,currentorders,currentstatusofpickfaceaswellasidentificationofproductsandlocations.

MIT Center for Transportation & Logistics

CommonFlowPaNerns

Receiving

PalletReserve

CasePick

EachesPick

SorSng

UniSzing

Shipping

Crossdock

Pallets

Eaches

adaptedfromBartholdi,J.andS.Hackman(2016)Warehouse&DistribuSonScience(Release0.97)

DropShiporDirect

Therearedifferentapproachestotheputawayfunction,whichcanbedirected(specificlocationaheadoftime).Itcanbebatched&sequencedwhichmeansthereisapre-sortatstagingforcommonlylocateditems.Oritcanbechaotic,wheretheuserpicksanylocationandrecordsitem-location.

OrderPickingOrderPickingisthemostlabor-intensivetask~50-60%.Pickingstrategieschangebasedonthesizeoftheobjectbeingpicked.Forinstance,fullpalletretrievalistheeasiestandfastest.Casepickingbeingthenextineaseandsmallitempickingbeingthemostexpensiveandtimeconsuming.Thebreakdownoforderpickingeffortis:

§ Traveling 55%§ Searching 15%§ Extracting 10%§ Othertasks 20%

LayoutWhenselectingaplacementforitems,thewarehouseistypicallysetupwithaflowbetween

receivingandshipping.TheFlow-ThroughDesignplacesthemostconvenientitemsdirectlyin

linewithreceivingandshipping.Aconvenientlocationisonethatminimizestotallabortime

(distance)toputawayandretrieve.

Minc*Σi(dini)where:c=laborcostperdistancedi=distanceforpalletlocationifromreceivingtolocationtoshippingni=averagenumberoftimeslocationiisvisitedperyear�#palletssold/#palletsinorderSimpleHeuristic:

1. Rankallpositionsfromlowtohighdi2. RankallSKUsfromhightolownj3. AssignnexthighestSKU(nj)tonextlowestlocation(di)

AisleLayoutIntermsofaislelayout,itistypicallybesttohaveaislesparalleltotheflowtoavoidinconveniencesinflow.Crossaislesinawarehousecanshortendistances,buttheytakeupsignificantlymorespaceandalsoincreasetheamountofaislecrossing.Angledorfishboneaislecanincreaseefficiencyespeciallywhenthereisacentraldispatchpoint.Inaddition,fastmovingitemsshouldbeputinconveniencelocations.

PickingStrategiesSinglePicker–SingleOrder:goodforlownumberoflines/order;suitableforshortpickpaths;

noneedtomarrytheordersafterwards;traveltimecanbehigh

SinglePicker–MultipleOrders:expectedtraveltime(distance)peritemisreduced;requiressortingand“marrying”items;cansort“on-cart”oraftertour;worksforbothpicker-to-stockandstock-to-picker.MultiplePicker-MultipleOrders:wellsuitedfororderswithhighlinecount;expectedtraveltimeperitemisreduced;minimizescongestion&socializinginpickaisles;pickerscanbecome

“experts”inazonebutloseordercompletionaccountability;requiressortingandconsolidation

ofitems;allowsforsimultaneousfillingoforders;difficulttobalanceworkloadacrosszones.

Check,Pack&ShipThefinalfunctionofstandardwarehousefunctionsischeck,pack&ship.Checkingincludescreatingandverifyingshippinglabelsandconfirmingweightandcube.Packingconsistsofensuredamageprotectionandunitizepallets.Shippingisthefinalstep;itisessentiallythereverseofreceiving.Shippingactivitiesincludedockdoorandyardmanagement;minimizingstagingrequirements;andcontainer/trailerloadingoptimization.

Profiling&AssessingPerformance

WarehouseActivityProfileWhenorganizationsareeitherdesigninganewwarehouse/DCorrevampinganexistingone,thereneedstobesomeadvancecriticalthinking.Forinstance,afewdatapointsthatareworthlookingintoinclude:

§ NumberofSKUsinthewarehouse§ Numberofpick-linesperday&numberofunitsperpick-line§ Number&sizeofcustomerordersshippedandshipmentsreceivedperday§ RateofnewSKUintroductionsandrespectivelifecycle

Whenevaluatingthesedatapoints,weneedtomakesuretolookatthedistributionnotjusttheaveragestounderstandpeaksanddipsovertime.ThedatasourcesaretypicallyintheMasterSKUdata,orderhistory,andwarehouselocation.

SegmentationAnalysisWewerefirstintroducedtosegmentationanalysisfordemandplanning,however,itcanbeappliedtowarehousingaswell.Segmentationcanprovidesomeimportantinsightsforwarehousedesign.Differentsegmentationviewsgivedifferentinsights:

FrequencyofSKUssold:TopsellingSKUsinfluenceretailoperations–notnecessarilywarehouseoperations.

Frequencyofpallets/cases/cartonsbySKU:WillnotnecessarilyfollowSKUfrequency;

providesinsightsintoreceiving,putaway,andrestocking;SKUswithfewpiecespercasewillrise

tothetop.

FrequencyofpicksbySKU:Orderpickingdrivesmostlaborcosts;determinesslottingand

forwardpicklocations.

Itisalsoimportanttonotethatthereiscommonvariabilityofdemandthatisaffectedby

seasonality-suchasbyyear,quarter,dayorweek,timeofday.Thereisalsocorrelationto

otherproducts(affinitybetweenitemsandfamilies).Thesecanallhaveaninfluenceonhowa

warehouse/DCshouldbedesigned.

MeasuringandBenchmarkingTobestunderstandtheeffectiveoperationofawarehouse,thereareafewwaystomeasureandbenchmarkactivity.First,itisimportanttounderstandwherethemajorcostdriversare:labor,space,andequipment.Regularassessmentofthesecanprovidefeedbackonsurgesanddipsofspend.Nexttherearekeyperformancemeasuresthatprovideinformationabouttheactivityinthewarehouseandcanbeusedtomakeoperationaldecisions.MajorWarehousingCostDriversLabor= (person-hours/year) x (laborrate)Space= (areaoccupied) x (costofspace)Equipment= (moneyinvested) x (amortizationrate)PerformanceMeasuresProductivity/Efficiency:Ratioofoutputtotheinputsrequired;e.g.,labor=(units,cases,orpallets)/(laborhoursexpended).Utilization:Percentageofanassetbeingactivelyused;e.g.,storagedensity=(storagecapacityinWH)/(totalareaofWH)Quality/Effectiveness:Accuracyinputaway,inventory,picking,shipping,etc.CycleTime:Dock-to-Stocktime–timefromreceipttobeingreadytobepicked;OrderCycleTime–timefromwhenorderisdroppeduntilitisreadytoship.

KeyPoints• Reviewtypesofwarehouseandrecognizetheirprimaryuse.• Understandthecorefunctionsofthewarehouse.• Becomefamiliarwithcommonflowpatternsofawarehouse.• Reviewtheactivitieswithineachofthefunctionsandhowtooptimizethem.• Recognizehowtoassessandbenchmarkwarehouseactivity.

References• Bartholdi,J.andS.Hackman(2016).Warhouse&DistributionScience(Release0.97)• Frazelle,E.(2011).WorldClassWarehousingandMaterialHandling.

FundamentalsofFreightTransportation

SummaryThefundamentalsoffreighttransportationprovidesandoverviewofdifferentmodesoftransportationandsomedifferentwaystomakedecisionsofthemodechoice,analyzingthetrade-offsbetweencostandlevelofservice.Therearedifferentlevelsoftransportationnetworks(fromstrategictophysical).Physicalnetworkrepresentshowtheproductphysicallymoves,theactualpathfromorigintodestination.Costsanddistancescalculationsaremadebasedonthislevel.Decisionsfromnodes(decisionpoints)andarcs(aspecificmode)aremadeintheOperationalnetwork.Thethirdnetwork,thestrategicorservicenetwork,representsindividualpathsfromend-to-end,andthosedecisionsthattieintotheinventorypoliciesaremadeintheStrategicorServicenetworklevel.Freighttransportationalsoincludestheimportantcomponentofpackaging.ThePrimarypackaging,hasdirectcontactwiththeproductandisusuallythesmallestunitofdistribution(e.g.abottleofwine,acan,etc.).TheSecondarypackagingcontainsproductandalsoamiddlelayerofpackagingthatisoutsidetheprimarypackaging,mainlytogroupprimarypackagestogether(e.g.aboxwith12bottleofwines,cases,cartons,etc.).TheTertiarypackagingisdesignedthinkingmoreontransportshipping,warehousestorageandbulkhandling(e.g.pallets,containers,etc.).

KeyConceptsTrade-offsbetweenCostandLevelofService(LOS):

• ProvidespathviewoftheNetwork• Summarizesthemovementincommonfinancialandperformanceterms• Usedforselectingoneoptionfrommanybymakingtrade-offs

Packaging

• Levelofpackagingmirrorshandlingneeds• Pallets—standardsizeof48x40inintheUSA(120x80cminEurope)• ShippingContainers

o TEU(20ft)33m3volumewith24.8kkgtotalpayloado FEU(40ft)67m3volumewith28.8kkgtotalpayloado 53ftlong(DomesticUS)111m3volumewith20.5kkgtotalpayload

TransportationNetworks• PhysicalNetwork:Theactualpaththattheproducttakesfromorigintodestination

includingguideways,terminalsandcontrols.Basisforallcostsanddistancecalculations–typicallyonlyfoundonce.

• OperationalNetwork:Theroutetheshipmenttakesintermsofdecisionpoints.Eacharcisaspecificmodewithcosts,distance,etc.Eachnodeisadecisionpoint.Thefourprimarycomponentsareloading/unloading,local-routing,line-haul,andsorting.

• StrategicNetwork:Aseriesofpathsthroughthenetworkfromorigintodestination.Eachrepresentsacompleteoptionandhasend-to-endcost,distance,andservicecharacteristics.

Notation

TL:TruckloadTEU:TwentyFootEquivalent(cargocontainer)FEU:FortyFootEquivalent(cargocontainer)

LeadTimeVariability&ModeSelectionVariabilityintransittimeimpactsthetotalcostequationforinventory.Thereareimportantlinkagesbetweentransportationreliability,forecastaccuracy,andinventorylevels.Modeselectionisheavilyinfluencednotonlybythevalueoftheproductbeingtransported,butalsotheexpectedandvariabilityofthelead-time.ImpactonInventoryTransportationaffectstotalcostvia

• Costoftransportation(fixed,variable,orsomecombination)

• Leadtime(expectedvalueaswellasvariability)

• Capacityrestrictions(astheylimitoptimalordersize)

• Miscellaneousfactors(suchasmaterialrestrictionsorperishability)

TransportationCostFunctionsTransportationcostscantakemanydifferentforms,toinclude:

• Purevariablecost/unit

• Purefixedcost/shipment

• Mixedvariable&fixedcost

• Variablecost/unitwithaminimumquantity

• Incrementaldiscounts

Lead/TransitTimeReliabilityTherearetwodifferentdimensionsofreliabilitythatdonotalwaysmatch:

• Credibility(reserveslotsareagreed,stopatallports,loadallcontainers,etc.)

• Scheduleconsistency(actualvs.quotedperformance)

Contractreliabilityinprocurementandoperationsdonotalwaysmatchastheyaretypicallyperformedbydifferentpartsofanorganization.Contractreliabilitydiffersdramaticallyacrossdifferentroutesegments(originportdwellvs.port-to-porttransittimevs.destinationportdwellforinstance).Formostshippers,themosttransitvariabilityoccursintheorigininlandtransportationlegsandattheports.

ModeSelectionTransportationmodeshavespecificnichesandperformbetterthanothermodesincertainsituations.Also,inmanycases,thereareonlyoneortwofeasibleoptionsbetweenmodes.CriteriaforFeasibility

• Geographyo Global:AirversusOcean(truckscannotcrossoceans!)o Surface:Trucking(TL,LTL,parcel)vs.Railvs.Intermodalvs.Barge

• Requiredspeedo >500milesin1day—Airo <500milesin1day—TL

• Shipmentsize(weight/density/cube,etc.)o Highweight,cubeitemscannotbemovedbyairo Largeoversizedshipmentsmightberestrictedtorailorbarge

• Otherrestrictionso Nuclearorhazardousmaterials(HazMat)o Productcharacteristics

Trade-offswithinthesetoffeasiblechoicesOnceallfeasiblemodes(orseparatecarrierfirms)havebeenidentified,theselectionwithinthisfeasiblesetismadeasatrade-offbetweencosts.Itisimportanttotranslatethe“non-cost”elementsintocostsviathetotalcostequation.Thetypicalnon-costelementsare:

• Time(meantransittime,variabilityoftransittime,frequency)

• Capacity

• LossandDamage

Notationci: Purchasecostforitemi($/unit)ct: OrderingCosts($/order)ce: ExcessholdingCosts($/unit/time);Equaltochcs: Shortagecosts($/unit)D: AverageDemand(units/time)h: Carryingorholdingcost($/inventory$/time)Q: ReplenishmentOrderQuantity(units/order)T: OrderCycleTime(time/order)μD: ExpectedDemand(Items)duringOneTimePeriodσD: StandardDeviationofDemand(Items)duringOneTimePeriodμL: ExpectedNumberofTimePeriodsforLeadTime(UnitlessMultiplier)σL: StandardDeviationofTimePeriodsforLeadTime(UnitlessMultiplier)μDL: ExpectedDemand(Items)overLeadTimeσDL: StandardDeviationofDemand(Items)overLeadTimeN: RandomVariableAssumingPositiveIntegerValues(1,2,3…)xi: IndependentRandomVariablessuchthatE[xi]=E[X]S: Sumofxifromi=1toN

Formulas

LearningObjectives

• Understandcommonterminologyandconceptsofglobalfreighttransportation.

• Understandingofphysical,operational,andstrategicnetworks.

• AbilitytoselectmodebytradingoffLevelofService(LOS)andcost.

• Understandtheimpactoftransportationoncycle,safety,andpipelinestock.

• Understandhowthevariabilityoftransportationtransittimeimpactsinventory

• Abletousecontinuousapproximationtomakequickestimatesofcostsusingaminimal

amountofdata.

References• Ballou,RonaldH.,BusinessLogistics:SupplyChainManagement,3rdedition,Pearson

PrenticeHall,2003.Chapter6.• Chopra,SunilandPeterMeindl,SupplyChainManagement,Strategy,Planning,and

Operation,5thedition,PearsonPrenticeHall,2013.Chapter14.

RandomSumsofRandomVariables

A[@] = A „e‚Q

‰ = A[µ]A[‚]

+$�[@] = +$� „e‚Q

‰ = A[µ]+$�[‚] + (A[‚])K+$�[µ]

LeadTimeVariability

.~} = .}.~

-~} = D.}-~K + (.~)K-}K

AppendixA&BUnitNormalDistribution,PoissonDistributionTables

Unit%Normal%distribution Example,%for%k=1.67,%the%Probability%that%u<k%=%0.9525%and%the%Expected%Unit%Normal%Loss%is%0.0197k P[u<k] G(k) k P[u<k] G(k) k P[u<k] G(k) k P[u<k] G(k) k P[u<k] G(k) k P[x≤k] G(k) k P[x≤k] G(k) k P[x≤k] G(k)0.00 0.5000 0.3989 0.50 0.6915 0.1978 1.00 0.8413 0.0833 1.50 0.9332 0.0293 2.00 0.9772 0.0085 2.50 0.9938 0.00200%%% 3.00 0.9987 0.000382%%% 3.50 0.9998 0.0000580.01 0.5040 0.3940 0.51 0.6950 0.1947 1.01 0.8438 0.0817 1.51 0.9345 0.0286 2.01 0.9778 0.0083 2.51 0.9940 0.00194%%% 3.01 0.9987 0.000369%%% 3.51 0.9998 0.0000560.02 0.5080 0.3890 0.52 0.6985 0.1917 1.02 0.8461 0.0802 1.52 0.9357 0.0280 2.02 0.9783 0.0080 2.52 0.9941 0.00188%%% 3.02 0.9987 0.000356%%% 3.52 0.9998 0.0000540.03 0.5120 0.3841 0.53 0.7019 0.1887 1.03 0.8485 0.0787 1.53 0.9370 0.0274 2.03 0.9788 0.0078 2.53 0.9943 0.00183%%% 3.03 0.9988 0.000344%%% 3.53 0.9998 0.0000520.04 0.5160 0.3793 0.54 0.7054 0.1857 1.04 0.8508 0.0772 1.54 0.9382 0.0267 2.04 0.9793 0.0076 2.54 0.9945 0.00177%%% 3.04 0.9988 0.000332%%% 3.54 0.9998 0.0000500.05 0.5199 0.3744 0.55 0.7088 0.1828 1.05 0.8531 0.0757 1.55 0.9394 0.0261 2.05 0.9798 0.0074 2.55 0.9946 0.00171%%% 3.05 0.9989 0.000320%%% 3.55 0.9998 0.0000480.06 0.5239 0.3697 0.56 0.7123 0.1799 1.06 0.8554 0.0742 1.56 0.9406 0.0255 2.06 0.9803 0.0072 2.56 0.9948 0.00166%%% 3.06 0.9989 0.000309%%% 3.56 0.9998 0.0000460.07 0.5279 0.3649 0.57 0.7157 0.1771 1.07 0.8577 0.0728 1.57 0.9418 0.0249 2.07 0.9808 0.0070 2.57 0.9949 0.00161%%% 3.07 0.9989 0.000298%%% 3.57 0.9998 0.0000440.08 0.5319 0.3602 0.58 0.7190 0.1742 1.08 0.8599 0.0714 1.58 0.9429 0.0244 2.08 0.9812 0.0068 2.58 0.9951 0.00156%%% 3.08 0.9990 0.000287%%% 3.58 0.9998 0.0000420.09 0.5359 0.3556 0.59 0.7224 0.1714 1.09 0.8621 0.0700 1.59 0.9441 0.0238 2.09 0.9817 0.0066 2.59 0.9952 0.00151%%% 3.09 0.9990 0.000277%%% 3.59 0.9998 0.0000410.10 0.5398 0.3509 0.60 0.7257 0.1687 1.10 0.8643 0.0686 1.60 0.9452 0.0232 2.10 0.9821 0.0065 2.60 0.9953 0.00146%%% 3.10 0.9990 0.000267%%% 3.60 0.9998 0.0000390.11 0.5438 0.3464 0.61 0.7291 0.1659 1.11 0.8665 0.0673 1.61 0.9463 0.0227 2.11 0.9826 0.0063 2.61 0.9955 0.00142%%% 3.11 0.9991 0.000258%%% 3.61 0.9998 0.0000380.12 0.5478 0.3418 0.62 0.7324 0.1633 1.12 0.8686 0.0659 1.62 0.9474 0.0222 2.12 0.9830 0.0061 2.62 0.9956 0.00137%%% 3.12 0.9991 0.000249%%% 3.62 0.9999 0.0000360.13 0.5517 0.3373 0.63 0.7357 0.1606 1.13 0.8708 0.0646 1.63 0.9484 0.0216 2.13 0.9834 0.0060 2.63 0.9957 0.00133%%% 3.13 0.9991 0.000240%%% 3.63 0.9999 0.0000350.14 0.5557 0.3328 0.64 0.7389 0.1580 1.14 0.8729 0.0634 1.64 0.9495 0.0211 2.14 0.9838 0.0058 2.64 0.9959 0.00129%%% 3.14 0.9992 0.000231%%% 3.64 0.9999 0.0000330.15 0.5596 0.3284 0.65 0.7422 0.1554 1.15 0.8749 0.0621 1.65 0.9505 0.0206 2.15 0.9842 0.0056 2.65 0.9960 0.00125%%% 3.15 0.9992 0.000223%%% 3.65 0.9999 0.0000320.16 0.5636 0.3240 0.66 0.7454 0.1528 1.16 0.8770 0.0609 1.66 0.9515 0.0201 2.16 0.9846 0.0055 2.66 0.9961 0.00121%%% 3.16 0.9992 0.000215%%% 3.66 0.9999 0.0000310.17 0.5675 0.3197 0.67 0.7486 0.1503 1.17 0.8790 0.0596 1.67 0.9525 0.0197 2.17 0.9850 0.0053 2.67 0.9962 0.00117%%% 3.17 0.9992 0.000207%%% 3.67 0.9999 0.0000290.18 0.5714 0.3154 0.68 0.7517 0.1478 1.18 0.8810 0.0584 1.68 0.9535 0.0192 2.18 0.9854 0.0052 2.68 0.9963 0.00113%%% 3.18 0.9993 0.000199%%% 3.68 0.9999 0.0000280.19 0.5753 0.3111 0.69 0.7549 0.1453 1.19 0.8830 0.0573 1.69 0.9545 0.0187 2.19 0.9857 0.0050 2.69 0.9964 0.00110%%% 3.19 0.9993 0.000192%%% 3.69 0.9999 0.0000270.20 0.5793 0.3069 0.70 0.7580 0.1429 1.20 0.8849 0.0561 1.70 0.9554 0.0183 2.20 0.9861 0.0049 2.70 0.9965 0.00106%%% 3.20 0.9993 0.000185%%% 3.70 0.9999 0.0000260.21 0.5832 0.3027 0.71 0.7611 0.1405 1.21 0.8869 0.0550 1.71 0.9564 0.0178 2.21 0.9864 0.0047 2.71 0.9966 0.00103%%% 3.21 0.9993 0.000178%%% 3.71 0.9999 0.0000250.22 0.5871 0.2986 0.72 0.7642 0.1381 1.22 0.8888 0.0538 1.72 0.9573 0.0174 2.22 0.9868 0.0046 2.72 0.9967 0.00099%%% 3.22 0.9994 0.000172%%% 3.72 0.9999 0.0000240.23 0.5910 0.2944 0.73 0.7673 0.1358 1.23 0.8907 0.0527 1.73 0.9582 0.0170 2.23 0.9871 0.0045 2.73 0.9968 0.00096%%% 3.23 0.9994 0.000166%%% 3.73 0.9999 0.0000230.24 0.5948 0.2904 0.74 0.7704 0.1334 1.24 0.8925 0.0517 1.74 0.9591 0.0166 2.24 0.9875 0.0044 2.74 0.9969 0.00093%%% 3.24 0.9994 0.000160%%% 3.74 0.9999 0.0000220.25 0.5987 0.2863 0.75 0.7734 0.1312 1.25 0.8944 0.0506 1.75 0.9599 0.0162 2.25 0.9878 0.0042 2.75 0.9970 0.00090%%% 3.25 0.9994 0.000154%%% 3.75 0.9999 0.0000210.26 0.6026 0.2824 0.76 0.7764 0.1289 1.26 0.8962 0.0495 1.76 0.9608 0.0158 2.26 0.9881 0.0041 2.76 0.9971 0.00087%%% 3.26 0.9994 0.000148%%% 3.76 0.9999 0.0000200.27 0.6064 0.2784 0.77 0.7794 0.1267 1.27 0.8980 0.0485 1.77 0.9616 0.0154 2.27 0.9884 0.0040 2.77 0.9972 0.00084%%% 3.27 0.9995 0.000143%%% 3.77 0.9999 0.0000190.28 0.6103 0.2745 0.78 0.7823 0.1245 1.28 0.8997 0.0475 1.78 0.9625 0.0150 2.28 0.9887 0.0039 2.78 0.9973 0.00081%%% 3.28 0.9995 0.000137%%% 3.78 0.9999 0.0000190.29 0.6141 0.2706 0.79 0.7852 0.1223 1.29 0.9015 0.0465 1.79 0.9633 0.0146 2.29 0.9890 0.0038 2.79 0.9974 0.00079%%% 3.29 0.9995 0.000132%%% 3.79 0.9999 0.0000180.30 0.6179 0.2668 0.80 0.7881 0.1202 1.30 0.9032 0.0455 1.80 0.9641 0.0143 2.30 0.9893 0.0037 2.80 0.9974 0.00076%%% 3.30 0.9995 0.000127%%% 3.80 0.9999 0.0000170.31 0.6217 0.2630 0.81 0.7910 0.1181 1.31 0.9049 0.0446 1.81 0.9649 0.0139 2.31 0.9896 0.0036 2.81 0.9975 0.00074%%% 3.31 0.9995 0.000123%%% 3.81 0.9999 0.0000160.32 0.6255 0.2592 0.82 0.7939 0.1160 1.32 0.9066 0.0436 1.82 0.9656 0.0136 2.32 0.9898 0.0035 2.82 0.9976 0.00071%%% 3.32 0.9995 0.000118%%% 3.82 0.9999 0.0000160.33 0.6293 0.2555 0.83 0.7967 0.1140 1.33 0.9082 0.0427 1.83 0.9664 0.0132 2.33 0.9901 0.0034 2.83 0.9977 0.00069%%% 3.33 0.9996 0.000114%%% 3.83 0.9999 0.0000150.34 0.6331 0.2518 0.84 0.7995 0.1120 1.34 0.9099 0.0418 1.84 0.9671 0.0129 2.34 0.9904 0.0033 2.84 0.9977 0.00066%%% 3.34 0.9996 0.000109%%% 3.84 0.9999 0.0000140.35 0.6368 0.2481 0.85 0.8023 0.1100 1.35 0.9115 0.0409 1.85 0.9678 0.0126 2.35 0.9906 0.0032 2.85 0.9978 0.00064%%% 3.35 0.9996 0.000105%%% 3.85 0.9999 0.0000140.36 0.6406 0.2445 0.86 0.8051 0.1080 1.36 0.9131 0.0400 1.86 0.9686 0.0123 2.36 0.9909 0.0031 2.86 0.9979 0.00062%%% 3.36 0.9996 0.000101%%% 3.86 0.9999 0.0000130.37 0.6443 0.2409 0.87 0.8078 0.1061 1.37 0.9147 0.0392 1.87 0.9693 0.0119 2.37 0.9911 0.0030 2.87 0.9979 0.00060%%% 3.37 0.9996 0.000097%%% 3.87 0.9999 0.0000130.38 0.6480 0.2374 0.88 0.8106 0.1042 1.38 0.9162 0.0383 1.88 0.9699 0.0116 2.38 0.9913 0.0029 2.88 0.9980 0.00058%%% 3.38 0.9996 0.000094%%% 3.88 0.9999 0.0000120.39 0.6517 0.2339 0.89 0.8133 0.1023 1.39 0.9177 0.0375 1.89 0.9706 0.0113 2.39 0.9916 0.0028 2.89 0.9981 0.00056%%% 3.39 0.9997 0.000090%%% 3.89 0.9999 0.0000120.40 0.6554 0.2304 0.90 0.8159 0.1004 1.40 0.9192 0.0367 1.90 0.9713 0.0111 2.40 0.9918 0.0027 2.90 0.9981 0.00054%%% 3.40 0.9997 0.000087%%% 3.90 1.0000 0.0000110.41 0.6591 0.2270 0.91 0.8186 0.0986 1.41 0.9207 0.0359 1.91 0.9719 0.0108 2.41 0.9920 0.0026 2.91 0.9982 0.00052%%% 3.41 0.9997 0.000083%%% 3.91 1.0000 0.0000110.42 0.6628 0.2236 0.92 0.8212 0.0968 1.42 0.9222 0.0351 1.92 0.9726 0.0105 2.42 0.9922 0.0026 2.92 0.9982 0.00051%%% 3.42 0.9997 0.000080%%% 3.92 1.0000 0.0000100.43 0.6664 0.2203 0.93 0.8238 0.0950 1.43 0.9236 0.0343 1.93 0.9732 0.0102 2.43 0.9925 0.0025 2.93 0.9983 0.00049%%% 3.43 0.9997 0.000077%%% 3.93 1.0000 0.0000100.44 0.6700 0.2169 0.94 0.8264 0.0933 1.44 0.9251 0.0336 1.94 0.9738 0.0100 2.44 0.9927 0.0024 2.94 0.9984 0.00047%%% 3.44 0.9997 0.000074%%% 3.94 1.0000 0.0000090.45 0.6736 0.2137 0.95 0.8289 0.0916 1.45 0.9265 0.0328 1.95 0.9744 0.0097 2.45 0.9929 0.0023 2.95 0.9984 0.00046%%% 3.45 0.9997 0.000071%%% 3.95 1.0000 0.0000090.46 0.6772 0.2104 0.96 0.8315 0.0899 1.46 0.9279 0.0321 1.96 0.9750 0.0094 2.46 0.9931 0.0023 2.96 0.9985 0.00044%%% 3.46 0.9997 0.000069%%% 3.96 1.0000 0.0000090.47 0.6808 0.2072 0.97 0.8340 0.0882 1.47 0.9292 0.0314 1.97 0.9756 0.0092 2.47 0.9932 0.0022 2.97 0.9985 0.00042%%% 3.47 0.9997 0.000066%%% 3.97 1.0000 0.0000080.48 0.6844 0.2040 0.98 0.8365 0.0865 1.48 0.9306 0.0307 1.98 0.9761 0.0090 2.48 0.9934 0.0021 2.98 0.9986 0.00041%%% 3.48 0.9997 0.000063%%% 3.98 1.0000 0.0000080.49 0.6879 0.2009 0.99 0.8389 0.0849 1.49 0.9319 0.0300 1.99 0.9767 0.0087 2.49 0.9936 0.0021 2.99 0.9986 0.00040%%% 3.49 0.9998 0.000061%%% 3.99 1.0000 0.000007

k P[u<k] G(k) k P[u<k] G(k) k P[u<k] G(k) k P[u<k] G(k) k P[u<k] G(k) k P[x≤k] G(k) k P[x≤k] G(k) k P[x≤k] G(k)0.00 0.5000 0.3989 I0.50 0.3085 0.6978 I1.00 0.1587 1.0833 I1.50 0.0668 1.5293 I2.00 0.0228 2.0085 I2.50 0.0062 2.50200%%% I3.00 0.0013 3.000382%%% I3.50 0.0002 3.500058I0.01 0.4960 0.4040 I0.51 0.3050 0.7047 I1.01 0.1562 1.0917 I1.51 0.0655 1.5386 I2.01 0.0222 2.0183 I2.51 0.0060 2.51194%%% I3.01 0.0013 3.010369%%% I3.51 0.0002 3.510056I0.02 0.4920 0.4090 I0.52 0.3015 0.7117 I1.02 0.1539 1.1002 I1.52 0.0643 1.5480 I2.02 0.0217 2.0280 I2.52 0.0059 2.52188%%% I3.02 0.0013 3.020356%%% I3.52 0.0002 3.520054I0.03 0.4880 0.4141 I0.53 0.2981 0.7187 I1.03 0.1515 1.1087 I1.53 0.0630 1.5574 I2.03 0.0212 2.0378 I2.53 0.0057 2.53183%%% I3.03 0.0012 3.030344%%% I3.53 0.0002 3.530052I0.04 0.4840 0.4193 I0.54 0.2946 0.7257 I1.04 0.1492 1.1172 I1.54 0.0618 1.5667 I2.04 0.0207 2.0476 I2.54 0.0055 2.54177%%% I3.04 0.0012 3.040332%%% I3.54 0.0002 3.540050I0.05 0.4801 0.4244 I0.55 0.2912 0.7328 I1.05 0.1469 1.1257 I1.55 0.0606 1.5761 I2.05 0.0202 2.0574 I2.55 0.0054 2.55171%%% I3.05 0.0011 3.050320%%% I3.55 0.0002 3.550048I0.06 0.4761 0.4297 I0.56 0.2877 0.7399 I1.06 0.1446 1.1342 I1.56 0.0594 1.5855 I2.06 0.0197 2.0672 I2.56 0.0052 2.56166%%% I3.06 0.0011 3.060309%%% I3.56 0.0002 3.560046I0.07 0.4721 0.4349 I0.57 0.2843 0.7471 I1.07 0.1423 1.1428 I1.57 0.0582 1.5949 I2.07 0.0192 2.0770 I2.57 0.0051 2.57161%%% I3.07 0.0011 3.070298%%% I3.57 0.0002 3.570044I0.08 0.4681 0.4402 I0.58 0.2810 0.7542 I1.08 0.1401 1.1514 I1.58 0.0571 1.6044 I2.08 0.0188 2.0868 I2.58 0.0049 2.58156%%% I3.08 0.0010 3.080287%%% I3.58 0.0002 3.580042I0.09 0.4641 0.4456 I0.59 0.2776 0.7614 I1.09 0.1379 1.1600 I1.59 0.0559 1.6138 I2.09 0.0183 2.0966 I2.59 0.0048 2.59151%%% I3.09 0.0010 3.090277%%% I3.59 0.0002 3.590041I0.10 0.4602 0.4509 I0.60 0.2743 0.7687 I1.10 0.1357 1.1686 I1.60 0.0548 1.6232 I2.10 0.0179 2.1065 I2.60 0.0047 2.60146%%% I3.10 0.0010 3.100267%%% I3.60 0.0002 3.600039I0.11 0.4562 0.4564 I0.61 0.2709 0.7759 I1.11 0.1335 1.1773 I1.61 0.0537 1.6327 I2.11 0.0174 2.1163 I2.61 0.0045 2.61142%%% I3.11 0.0009 3.110258%%% I3.61 0.0002 3.610038I0.12 0.4522 0.4618 I0.62 0.2676 0.7833 I1.12 0.1314 1.1859 I1.62 0.0526 1.6422 I2.12 0.0170 2.1261 I2.62 0.0044 2.62137%%% I3.12 0.0009 3.120249%%% I3.62 0.0001 3.620036I0.13 0.4483 0.4673 I0.63 0.2643 0.7906 I1.13 0.1292 1.1946 I1.63 0.0516 1.6516 I2.13 0.0166 2.1360 I2.63 0.0043 2.63133%%% I3.13 0.0009 3.130240%%% I3.63 0.0001 3.630035I0.14 0.4443 0.4728 I0.64 0.2611 0.7980 I1.14 0.1271 1.2034 I1.64 0.0505 1.6611 I2.14 0.0162 2.1458 I2.64 0.0041 2.64129%%% I3.14 0.0008 3.140231%%% I3.64 0.0001 3.640033I0.15 0.4404 0.4784 I0.65 0.2578 0.8054 I1.15 0.1251 1.2121 I1.65 0.0495 1.6706 I2.15 0.0158 2.1556 I2.65 0.0040 2.65125%%% I3.15 0.0008 3.150223%%% I3.65 0.0001 3.650032I0.16 0.4364 0.4840 I0.66 0.2546 0.8128 I1.16 0.1230 1.2209 I1.66 0.0485 1.6801 I2.16 0.0154 2.1655 I2.66 0.0039 2.66121%%% I3.16 0.0008 3.160215%%% I3.66 0.0001 3.660031I0.17 0.4325 0.4897 I0.67 0.2514 0.8203 I1.17 0.1210 1.2296 I1.67 0.0475 1.6897 I2.17 0.0150 2.1753 I2.67 0.0038 2.67117%%% I3.17 0.0008 3.170207%%% I3.67 0.0001 3.670029I0.18 0.4286 0.4954 I0.68 0.2483 0.8278 I1.18 0.1190 1.2384 I1.68 0.0465 1.6992 I2.18 0.0146 2.1852 I2.68 0.0037 2.68113%%% I3.18 0.0007 3.180199%%% I3.68 0.0001 3.680028I0.19 0.4247 0.5011 I0.69 0.2451 0.8353 I1.19 0.1170 1.2473 I1.69 0.0455 1.7087 I2.19 0.0143 2.1950 I2.69 0.0036 2.69110%%% I3.19 0.0007 3.190192%%% I3.69 0.0001 3.690027I0.20 0.4207 0.5069 I0.70 0.2420 0.8429 I1.20 0.1151 1.2561 I1.70 0.0446 1.7183 I2.20 0.0139 2.2049 I2.70 0.0035 2.70106%%% I3.20 0.0007 3.200185%%% I3.70 0.0001 3.700026I0.21 0.4168 0.5127 I0.71 0.2389 0.8505 I1.21 0.1131 1.2650 I1.71 0.0436 1.7278 I2.21 0.0136 2.2147 I2.71 0.0034 2.71103%%% I3.21 0.0007 3.210178%%% I3.71 0.0001 3.710025I0.22 0.4129 0.5186 I0.72 0.2358 0.8581 I1.22 0.1112 1.2738 I1.72 0.0427 1.7374 I2.22 0.0132 2.2246 I2.72 0.0033 2.72099%%% I3.22 0.0006 3.220172%%% I3.72 0.0001 3.720024I0.23 0.4090 0.5244 I0.73 0.2327 0.8658 I1.23 0.1093 1.2827 I1.73 0.0418 1.7470 I2.23 0.0129 2.2345 I2.73 0.0032 2.73096%%% I3.23 0.0006 3.230166%%% I3.73 0.0001 3.730023I0.24 0.4052 0.5304 I0.74 0.2296 0.8734 I1.24 0.1075 1.2917 I1.74 0.0409 1.7566 I2.24 0.0125 2.2444 I2.74 0.0031 2.74093%%% I3.24 0.0006 3.240160%%% I3.74 0.0001 3.740022I0.25 0.4013 0.5363 I0.75 0.2266 0.8812 I1.25 0.1056 1.3006 I1.75 0.0401 1.7662 I2.25 0.0122 2.2542 I2.75 0.0030 2.75090%%% I3.25 0.0006 3.250154%%% I3.75 0.0001 3.750021I0.26 0.3974 0.5424 I0.76 0.2236 0.8889 I1.26 0.1038 1.3095 I1.76 0.0392 1.7758 I2.26 0.0119 2.2641 I2.76 0.0029 2.76087%%% I3.26 0.0006 3.260148%%% I3.76 0.0001 3.760020I0.27 0.3936 0.5484 I0.77 0.2206 0.8967 I1.27 0.1020 1.3185 I1.77 0.0384 1.7854 I2.27 0.0116 2.2740 I2.77 0.0028 2.77084%%% I3.27 0.0005 3.270143%%% I3.77 0.0001 3.770019I0.28 0.3897 0.5545 I0.78 0.2177 0.9045 I1.28 0.1003 1.3275 I1.78 0.0375 1.7950 I2.28 0.0113 2.2839 I2.78 0.0027 2.78081%%% I3.28 0.0005 3.280137%%% I3.78 0.0001 3.780019I0.29 0.3859 0.5606 I0.79 0.2148 0.9123 I1.29 0.0985 1.3365 I1.79 0.0367 1.8046 I2.29 0.0110 2.2938 I2.79 0.0026 2.79079%%% I3.29 0.0005 3.290132%%% I3.79 0.0001 3.790018I0.30 0.3821 0.5668 I0.80 0.2119 0.9202 I1.30 0.0968 1.3455 I1.80 0.0359 1.8143 I2.30 0.0107 2.3037 I2.80 0.0026 2.80076%%% I3.30 0.0005 3.300127%%% I3.80 0.0001 3.800017I0.31 0.3783 0.5730 I0.81 0.2090 0.9281 I1.31 0.0951 1.3546 I1.81 0.0351 1.8239 I2.31 0.0104 2.3136 I2.81 0.0025 2.81074%%% I3.31 0.0005 3.310123%%% I3.81 0.0001 3.810016I0.32 0.3745 0.5792 I0.82 0.2061 0.9360 I1.32 0.0934 1.3636 I1.82 0.0344 1.8336 I2.32 0.0102 2.3235 I2.82 0.0024 2.82071%%% I3.32 0.0005 3.320118%%% I3.82 0.0001 3.820016I0.33 0.3707 0.5855 I0.83 0.2033 0.9440 I1.33 0.0918 1.3727 I1.83 0.0336 1.8432 I2.33 0.0099 2.3334 I2.83 0.0023 2.83069%%% I3.33 0.0004 3.330114%%% I3.83 0.0001 3.830015I0.34 0.3669 0.5918 I0.84 0.2005 0.9520 I1.34 0.0901 1.3818 I1.84 0.0329 1.8529 I2.34 0.0096 2.3433 I2.84 0.0023 2.84066%%% I3.34 0.0004 3.340109%%% I3.84 0.0001 3.840014I0.35 0.3632 0.5981 I0.85 0.1977 0.9600 I1.35 0.0885 1.3909 I1.85 0.0322 1.8626 I2.35 0.0094 2.3532 I2.85 0.0022 2.85064%%% I3.35 0.0004 3.350105%%% I3.85 0.0001 3.850014I0.36 0.3594 0.6045 I0.86 0.1949 0.9680 I1.36 0.0869 1.4000 I1.86 0.0314 1.8723 I2.36 0.0091 2.3631 I2.86 0.0021 2.86062%%% I3.36 0.0004 3.360101%%% I3.86 0.0001 3.860013I0.37 0.3557 0.6109 I0.87 0.1922 0.9761 I1.37 0.0853 1.4092 I1.87 0.0307 1.8819 I2.37 0.0089 2.3730 I2.87 0.0021 2.87060%%% I3.37 0.0004 3.370097%%% I3.87 0.0001 3.870013I0.38 0.3520 0.6174 I0.88 0.1894 0.9842 I1.38 0.0838 1.4183 I1.88 0.0301 1.8916 I2.38 0.0087 2.3829 I2.88 0.0020 2.88058%%% I3.38 0.0004 3.380094%%% I3.88 0.0001 3.880012I0.39 0.3483 0.6239 I0.89 0.1867 0.9923 I1.39 0.0823 1.4275 I1.89 0.0294 1.9013 I2.39 0.0084 2.3928 I2.89 0.0019 2.89056%%% I3.39 0.0003 3.390090%%% I3.89 0.0001 3.890012I0.40 0.3446 0.6304 I0.90 0.1841 1.0004 I1.40 0.0808 1.4367 I1.90 0.0287 1.9111 I2.40 0.0082 2.4027 I2.90 0.0019 2.90054%%% I3.40 0.0003 3.400087%%% I3.90 0.0000 3.900011I0.41 0.3409 0.6370 I0.91 0.1814 1.0086 I1.41 0.0793 1.4459 I1.91 0.0281 1.9208 I2.41 0.0080 2.4126 I2.91 0.0018 2.91052%%% I3.41 0.0003 3.410083%%% I3.91 0.0000 3.910011I0.42 0.3372 0.6436 I0.92 0.1788 1.0168 I1.42 0.0778 1.4551 I1.92 0.0274 1.9305 I2.42 0.0078 2.4226 I2.92 0.0018 2.92051%%% I3.42 0.0003 3.420080%%% I3.92 0.0000 3.920010I0.43 0.3336 0.6503 I0.93 0.1762 1.0250 I1.43 0.0764 1.4643 I1.93 0.0268 1.9402 I2.43 0.0075 2.4325 I2.93 0.0017 2.93049%%% I3.43 0.0003 3.430077%%% I3.93 0.0000 3.930010I0.44 0.3300 0.6569 I0.94 0.1736 1.0333 I1.44 0.0749 1.4736 I1.94 0.0262 1.9500 I2.44 0.0073 2.4424 I2.94 0.0016 2.94047%%% I3.44 0.0003 3.440074%%% I3.94 0.0000 3.940009I0.45 0.3264 0.6637 I0.95 0.1711 1.0416 I1.45 0.0735 1.4828 I1.95 0.0256 1.9597 I2.45 0.0071 2.4523 I2.95 0.0016 2.95046%%% I3.45 0.0003 3.450071%%% I3.95 0.0000 3.950009I0.46 0.3228 0.6704 I0.96 0.1685 1.0499 I1.46 0.0721 1.4921 I1.96 0.0250 1.9694 I2.46 0.0069 2.4623 I2.96 0.0015 2.96044%%% I3.46 0.0003 3.460069%%% I3.96 0.0000 3.960009I0.47 0.3192 0.6772 I0.97 0.1660 1.0582 I1.47 0.0708 1.5014 I1.97 0.0244 1.9792 I2.47 0.0068 2.4722 I2.97 0.0015 2.97042%%% I3.47 0.0003 3.470066%%% I3.97 0.0000 3.970008I0.48 0.3156 0.6840 I0.98 0.1635 1.0665 I1.48 0.0694 1.5107 I1.98 0.0239 1.9890 I2.48 0.0066 2.4821 I2.98 0.0014 2.98041%%% I3.48 0.0003 3.480063%%% I3.98 0.0000 3.980008I0.49 0.3121 0.6909 I0.99 0.1611 1.0749 I1.49 0.0681 1.5200 I1.99 0.0233 1.9987 I2.49 0.0064 2.4921 I2.99 0.0014 2.99040%%% I3.49 0.0002 3.490061%%% I3.99 0.0000 3.990007

Poisson&distribution Columns&are&means&(λ)&while&rows&are&cumulative&probabilibites&(F(x).&&For&example,&the&P[x≤2]&for&~P(λ=0.5)&=&0.98561

F(x) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 F(x)0 0.99005&& 0.98020&& 0.97045&& 0.96079&& 0.95123&& 0.94176&& 0.93239&& 0.92312&& 0.91393&& 0.90484&& 0.86071&& 0.81873&& 0.77880&& 0.74082&& 0.70469&& 0.67032&& 0.63763&& 0.60653&& 01 0.99995&& 0.99980&& 0.99956&& 0.99922&& 0.99879&& 0.99827&& 0.99766&& 0.99697&& 0.99618&& 0.99532&& 0.98981&& 0.98248&& 0.97350&& 0.96306&& 0.95133&& 0.93845&& 0.92456&& 0.90980&& 12 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99998&& 0.99997&& 0.99995&& 0.99992&& 0.99989&& 0.99985&& 0.99950&& 0.99885&& 0.99784&& 0.99640&& 0.99449&& 0.99207&& 0.98912&& 0.98561&& 23 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99998&& 0.99994&& 0.99987&& 0.99973&& 0.99953&& 0.99922&& 0.99880&& 0.99825&& 34 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99998&& 0.99997&& 0.99994&& 0.99989&& 0.99983&& 45 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99999&& 56 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 6

F(x) 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 F(x)0 0.47237&& 0.36788&& 0.28650&& 0.22313&& 0.17377&& 0.13534&& 0.10540&& 0.08208&& 0.06393&& 0.04979&& 0.03877&& 0.03020&& 0.02352&& 0.01832&& 0.01426&& 0.01111&& 0.00865&& 0.00674&& 01 0.82664&& 0.73576&& 0.64464&& 0.55783&& 0.47788&& 0.40601&& 0.34255&& 0.28730&& 0.23973&& 0.19915&& 0.16479&& 0.13589&& 0.11171&& 0.09158&& 0.07489&& 0.06110&& 0.04975&& 0.04043&& 12 0.95949&& 0.91970&& 0.86847&& 0.80885&& 0.74397&& 0.67668&& 0.60934&& 0.54381&& 0.48146&& 0.42319&& 0.36957&& 0.32085&& 0.27707&& 0.23810&& 0.20371&& 0.17358&& 0.14735&& 0.12465&& 23 0.99271&& 0.98101&& 0.96173&& 0.93436&& 0.89919&& 0.85712&& 0.80943&& 0.75758&& 0.70304&& 0.64723&& 0.59141&& 0.53663&& 0.48377&& 0.43347&& 0.38621&& 0.34230&& 0.30189&& 0.26503&& 34 0.99894&& 0.99634&& 0.99088&& 0.98142&& 0.96710&& 0.94735&& 0.92199&& 0.89118&& 0.85538&& 0.81526&& 0.77165&& 0.72544&& 0.67755&& 0.62884&& 0.58012&& 0.53210&& 0.48540&& 0.44049&& 45 0.99987&& 0.99941&& 0.99816&& 0.99554&& 0.99087&& 0.98344&& 0.97263&& 0.95798&& 0.93916&& 0.91608&& 0.88881&& 0.85761&& 0.82288&& 0.78513&& 0.74494&& 0.70293&& 0.65973&& 0.61596&& 56 0.99999&& 0.99992&& 0.99968&& 0.99907&& 0.99780&& 0.99547&& 0.99163&& 0.98581&& 0.97757&& 0.96649&& 0.95227&& 0.93471&& 0.91372&& 0.88933&& 0.86169&& 0.83105&& 0.79775&& 0.76218&& 67 1.00000&& 0.99999&& 0.99995&& 0.99983&& 0.99953&& 0.99890&& 0.99773&& 0.99575&& 0.99265&& 0.98810&& 0.98174&& 0.97326&& 0.96238&& 0.94887&& 0.93257&& 0.91341&& 0.89140&& 0.86663&& 78 1.00000&& 1.00000&& 0.99999&& 0.99997&& 0.99991&& 0.99976&& 0.99945&& 0.99886&& 0.99784&& 0.99620&& 0.99371&& 0.99013&& 0.98519&& 0.97864&& 0.97023&& 0.95974&& 0.94701&& 0.93191&& 89 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99998&& 0.99995&& 0.99988&& 0.99972&& 0.99942&& 0.99890&& 0.99803&& 0.99669&& 0.99469&& 0.99187&& 0.98801&& 0.98291&& 0.97636&& 0.96817&& 910 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99998&& 0.99994&& 0.99986&& 0.99971&& 0.99944&& 0.99898&& 0.99826&& 0.99716&& 0.99557&& 0.99333&& 0.99030&& 0.98630&& 1011 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99997&& 0.99993&& 0.99985&& 0.99971&& 0.99947&& 0.99908&& 0.99849&& 0.99760&& 0.99632&& 0.99455&& 1112 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99998&& 0.99996&& 0.99992&& 0.99985&& 0.99973&& 0.99952&& 0.99919&& 0.99870&& 0.99798&& 1213 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99998&& 0.99996&& 0.99992&& 0.99986&& 0.99975&& 0.99957&& 0.99930&& 1314 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99998&& 0.99996&& 0.99993&& 0.99987&& 0.99977&& 1415 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99998&& 0.99996&& 0.99993&& 1516 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99999&& 0.99998&& 1617 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 1718 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 18

F(x) 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 F(x)0 0.00525&& 0.00409&& 0.00318&& 0.00248&& 0.00193&& 0.00150&& 0.00117&& 0.00091&& 0.00071&& 0.00055&& 0.00043&& 0.00034&& 0.00026&& 0.00020&& 0.00016&& 0.00012&& 0.00010&& 0.00007&& 01 0.03280&& 0.02656&& 0.02148&& 0.01735&& 0.01400&& 0.01128&& 0.00907&& 0.00730&& 0.00586&& 0.00470&& 0.00377&& 0.00302&& 0.00242&& 0.00193&& 0.00154&& 0.00123&& 0.00099&& 0.00079&& 12 0.10511&& 0.08838&& 0.07410&& 0.06197&& 0.05170&& 0.04304&& 0.03575&& 0.02964&& 0.02452&& 0.02026&& 0.01670&& 0.01375&& 0.01131&& 0.00928&& 0.00761&& 0.00623&& 0.00510&& 0.00416&& 23 0.23167&& 0.20170&& 0.17495&& 0.15120&& 0.13025&& 0.11185&& 0.09577&& 0.08177&& 0.06963&& 0.05915&& 0.05012&& 0.04238&& 0.03576&& 0.03011&& 0.02530&& 0.02123&& 0.01777&& 0.01486&& 34 0.39777&& 0.35752&& 0.31991&& 0.28506&& 0.25299&& 0.22367&& 0.19704&& 0.17299&& 0.15138&& 0.13206&& 0.11487&& 0.09963&& 0.08619&& 0.07436&& 0.06401&& 0.05496&& 0.04709&& 0.04026&& 45 0.57218&& 0.52892&& 0.48662&& 0.44568&& 0.40640&& 0.36904&& 0.33377&& 0.30071&& 0.26992&& 0.24144&& 0.21522&& 0.19124&& 0.16939&& 0.14960&& 0.13174&& 0.11569&& 0.10133&& 0.08853&& 56 0.72479&& 0.68604&& 0.64639&& 0.60630&& 0.56622&& 0.52652&& 0.48759&& 0.44971&& 0.41316&& 0.37815&& 0.34485&& 0.31337&& 0.28380&& 0.25618&& 0.23051&& 0.20678&& 0.18495&& 0.16495&& 67 0.83925&& 0.80949&& 0.77762&& 0.74398&& 0.70890&& 0.67276&& 0.63591&& 0.59871&& 0.56152&& 0.52464&& 0.48837&& 0.45296&& 0.41864&& 0.38560&& 0.35398&& 0.32390&& 0.29544&& 0.26866&& 78 0.91436&& 0.89436&& 0.87195&& 0.84724&& 0.82038&& 0.79157&& 0.76106&& 0.72909&& 0.69596&& 0.66197&& 0.62740&& 0.59255&& 0.55770&& 0.52311&& 0.48902&& 0.45565&& 0.42320&& 0.39182&& 89 0.95817&& 0.94622&& 0.93221&& 0.91608&& 0.89779&& 0.87738&& 0.85492&& 0.83050&& 0.80427&& 0.77641&& 0.74712&& 0.71662&& 0.68516&& 0.65297&& 0.62031&& 0.58741&& 0.55451&& 0.52183&& 910 0.98118&& 0.97475&& 0.96686&& 0.95738&& 0.94618&& 0.93316&& 0.91827&& 0.90148&& 0.88279&& 0.86224&& 0.83990&& 0.81589&& 0.79032&& 0.76336&& 0.73519&& 0.70599&& 0.67597&& 0.64533&& 1011 0.99216&& 0.98901&& 0.98498&& 0.97991&& 0.97367&& 0.96612&& 0.95715&& 0.94665&& 0.93454&& 0.92076&& 0.90527&& 0.88808&& 0.86919&& 0.84866&& 0.82657&& 0.80301&& 0.77810&& 0.75199&& 1112 0.99696&& 0.99555&& 0.99366&& 0.99117&& 0.98798&& 0.98397&& 0.97902&& 0.97300&& 0.96581&& 0.95733&& 0.94749&& 0.93620&& 0.92341&& 0.90908&& 0.89320&& 0.87577&& 0.85683&& 0.83643&& 1213 0.99890&& 0.99831&& 0.99749&& 0.99637&& 0.99487&& 0.99290&& 0.99037&& 0.98719&& 0.98324&& 0.97844&& 0.97266&& 0.96582&& 0.95782&& 0.94859&& 0.93805&& 0.92615&& 0.91285&& 0.89814&& 1314 0.99963&& 0.99940&& 0.99907&& 0.99860&& 0.99794&& 0.99704&& 0.99585&& 0.99428&& 0.99227&& 0.98974&& 0.98659&& 0.98274&& 0.97810&& 0.97257&& 0.96608&& 0.95853&& 0.94986&& 0.94001&& 1415 0.99988&& 0.99980&& 0.99968&& 0.99949&& 0.99922&& 0.99884&& 0.99831&& 0.99759&& 0.99664&& 0.99539&& 0.99379&& 0.99177&& 0.98925&& 0.98617&& 0.98243&& 0.97796&& 0.97269&& 0.96653&& 1516 0.99996&& 0.99994&& 0.99989&& 0.99983&& 0.99972&& 0.99957&& 0.99935&& 0.99904&& 0.99862&& 0.99804&& 0.99728&& 0.99628&& 0.99500&& 0.99339&& 0.99137&& 0.98889&& 0.98588&& 0.98227&& 1617 0.99999&& 0.99998&& 0.99997&& 0.99994&& 0.99991&& 0.99985&& 0.99976&& 0.99964&& 0.99946&& 0.99921&& 0.99887&& 0.99841&& 0.99779&& 0.99700&& 0.99597&& 0.99468&& 0.99306&& 0.99107&& 1718 1.00000&& 0.99999&& 0.99999&& 0.99998&& 0.99997&& 0.99995&& 0.99992&& 0.99987&& 0.99980&& 0.99970&& 0.99955&& 0.99935&& 0.99907&& 0.99870&& 0.99821&& 0.99757&& 0.99675&& 0.99572&& 1819 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99999&& 0.99998&& 0.99997&& 0.99996&& 0.99993&& 0.99989&& 0.99983&& 0.99975&& 0.99963&& 0.99947&& 0.99924&& 0.99894&& 0.99855&& 0.99804&& 1920 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99999&& 0.99998&& 0.99996&& 0.99994&& 0.99991&& 0.99986&& 0.99979&& 0.99969&& 0.99956&& 0.99938&& 0.99914&& 2021 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99999&& 0.99998&& 0.99997&& 0.99995&& 0.99992&& 0.99988&& 0.99983&& 0.99975&& 0.99964&& 2122 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99999&& 0.99998&& 0.99997&& 0.99996&& 0.99993&& 0.99990&& 0.99985&& 2223 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99999&& 0.99998&& 0.99998&& 0.99996&& 0.99994&& 2324 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 0.99999&& 0.99999&& 0.99998&& 2425 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 0.99999&& 2526 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 1.00000&& 26

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