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On the Categorization of Demand Patterns

Author(s): A. A. Syntetos, J. E. Boylan and J. D. CrostonSource: The Journal of the Operational Research Society, Vol. 56, No. 5 (May, 2005), pp. 495-503Published by: Palgrave Macmillan Journals on behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/4102103 .

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Journal of the Operational Research Society (2005) 56, 495-503 ( 2005 OperationalResearchSocietyLtd.All rightsreserved. 0160-5682/05 $30.00www.palgrave-journals.com/jors

On the categorization of demand patternsAA Syntetos'*, JE Boylan2 and JD Croston3'University f Salford,GreaterManchester,UK;2BuckinghamshirehilternsUniversityCollege,Bucks,UK;and3lndependentystemsand StatisticalConsultant,Herts,UKThecategorizationf alternativeemand atternsacilitatesheselection f a forecasting ethod nd t is anessentialelement fmany nventoryontrol oftware ackages. hecommon racticen theinventoryontrol oftwarendustryis to arbitrarilyategorizehosedemand atterns ndthenproceedo selectan estimation rocedurendoptimizeheforecastparameters. lternatively,orecastingmethodscan be directlycompared,based on some theoreticallyquantifiedrrormeasure,or thepurpose f establishingegions f superior erformancend hendefine hedemandpatterns asedon theresults. t is thisapproachhat s discussedn thispaperand ts applications demonstratedyconsidering WMA,Croston'smethod ndanalternativeo Croston's stimator eveloped ythe first woauthors fthispaper.Comparisonesults rebasedon a theoreticalnalysis f the meansquare rrordueto itsmathematicallytractable ature.Thecategorizationulesproposed reexpressednterms f theaveragenter-demandnterval nd hesquared oefficientf variation f demand izes.Thevalidity f the results s testedon 3000real-intermittentemanddataseries oming rom heautomotivendustry.Journalof the OperationalResearchSociety (2005) 56, 495-503. doi:10.1057/palgrave.jors.2601841Publishednline25August2004Keywords:ategorization;orecasting;nventoryontrol;ntermittentemand

IntroductionThe categorizationof alternative demand patterns is anessential element of many inventory control softwarepackages.The commonpractice n the software ndustry sto arbitrarily ategorizedemandpatternsand then selectanestimationprocedureand stock controlmethod in ordertoforecast future requirementsand manage stock efficiently.For example,certainarbitrary utoff valuesmay be giventothe numberof demandoccurringperiods n a year,averagedemandper unit timeperiodand standarddeviation of thedemand sizes in order to definedemandpatternsas slow,intermittent,umpy,fast, etc.In this paper,we are concernedwith the selection of themost appropriate estimation procedure as a first steptowards a more systematicand meaningful approach tothe categorizationproblem.Typically,for fast moving items, analysisof the demanddata series will lead to the selection of an appropriateforecastingmodel(or a class of models)and optimizationofthe forecast parameterswill be based on minimizingthemeansquareerror MSE), Bayesianrecursionor someformof residualauto-correlation nalysis.For the remainderofthe categories Moving Averages, simple ExponentiallyWeightedMovingAverages EWMA)or Croston'smethod'are themostcommonlychosenestimationprocedures. n the

case of the sporadicand/or irregulardemandpatterns,thesparsenessof data and/or the aggregatepolicies required,may result in an empiricaldeterminationof the controlparameters, so that, for example, a 13-period MovingAverageor a smoothingconstantequalto 0.15 arespecifiedacrossa whole category.Re-categorization ccursat fixed time intervals annuallyfor example)and the sensitivityof thecategorization ulestooutliers is usually examined so that the categorizationschemedoes not allowproducts o move from one categoryto the othersimplybecausea few extremeobservationshavebeen recorded.Logicalinconsistencies re also exploredsothat demand for a stock keeping unit (SKU) is alwaysclassified in the intended category. However, the moreconsistent the categorization scheme is, the greater thenumberof thedemandcategories hat needto be considered.Theoreticaland practicalrequirementshat should be takeninto account when developing rules for the purpose ofdistinguishingbetween alternative demand patterns havebeen discussed n Williams.2Ultimately,the objectiveof the categorizationexercise sthe selectionof the most appropriate stimationprocedure.However, if this is the real objective it seems moremeaningful to compare possible forecastingmethods forthe purposeof establishing egionsof superiorperformanceand then categorize the demand patterns based on theresults.The theoreticalcomparisonsneedto be based on anaccuracy measure, and the MSE is the most obviouscandidate because of its mathematically ractable nature.

*Correspondence:AA Syntetos, The School of Management, Universityof Salford,Maxwell Building,TheCrescent,Salford,GreaterManchesterM5 4WT, UK.E-mail: [email protected]

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496 Journalf heOperationalesearchocietyol. 6,No.Directcomparisonof the theoreticalMSEs,over a fixed leadtime, associatedwith any two estimationprocedureswillresultin establishingrules that indicateregionsof superiorperformance f one methodover the other. When there aremore than two estimatorstaken into account there will besomeoverlapping reasregarding uperiorperformance ndthe analysiswill need to be extendedin order to proposeruleswhich are valid across all methods.We demonstrate our approach by considering threemethods: (a) Croston'smethod, that has been specificallydesigned to deal with intermittence, b) an alternative toCroston'sestimator,developedby the first two authors ofthis paper and (c) the more generallyapplicableEWMA.The categorization ules areexpressed n terms of the series'average nter-demandntervaland the squaredcoefficientofvariationof the demand sizes (when demandoccurs).Thevalidity of our results is tested on 3000 real-intermittentdemanddata series.

Categorizationschemes proposedin the literatureWilliams2proposeda method of categorizationof demandpatternsbased on an idea that is called variancepartition(we splitthevarianceof thedemandduring ead timeinto itsconstituentparts). The purpose of categorizationwas theidentification of the most appropriate forecasting andinventorycontrolmethods for the resultingcategories.Categorization f the items takesplacein accordancewiththe matrix shown in Figure 1 (with the cutoff valuesbeingthe resultof managerialdecision).In the matrix shown below, 2 is the mean (Poisson)demand arrivalrate, L the mean lead time duration andCV2(x) he squaredcoefficientof variationof demand sizes.

LumpinessCV2 (X)

0.5A C

0.7 IntermittenceB D1

2.8LD2D2

Figure1 Williams' ategorizationcheme.

1/,L indicates he numberof lead timesbetween successivedemands (how often demand occurs or how intermittentdemand is). The higher the ratio, the more intermittentdemand is. CV2(x)/)L indicates how lumpy demand is.Lumpiness depends on both the intermittence and thevariabilityof the demandsize, when demand occurs. Thehigherthe ratio the more lumpydemand is: categoryDl-sporadic(lumpy); ategoryD2-highly sporadic(lumpy).Inthat case we have few very irregulartransactions accom-panied by highly variabledemand sizes;categoryB-slowmoving;others-smooth.Eaves3analyseddemanddata from the Royal Air Force(RAF) and concludedthat Williams'conceptualclassifica-tion scheme did not adequately describe the observeddemand structure. In particular, it was not consideredsufficient o distinguisha smooth demandpatternfrom theremainder implyon the basis of the transactionvariability.Consequently,a revisedclassification chemewas proposed(see Figure 2) that categorizes demand based on thevariabilityof the transactions'rate, demandsize variabilityand lead timevariability.The line items with low transactionvariability are sub-divided into smooth and irregularaccordingto the demandsize variability.The smooth andslow-movingdemandcategoriesare distinguished rom therest based on the varianceof the demandsizes,and the leadtime variance s used only for distinguishingbetweenerraticand highly erraticdemand.The cutoff valueswere decidedbased on: (a) the characteristics f the particulardemanddata set and (b) sufficientsub-sample withineach demandclass) size considerations.In particular,the cutoff pointswere as follows: transactionvariability-0.74; demandsizevariability--.10; lead timevariability-0.53. CategoryAsmooth; category C-irregular; category B-slow-moving;categoryDl-erratic; categoryD2-highly erratic.The above-discussed ategorizationschemes are the onlyones to appear in the academic literaturethat take intoconsiderationalternativedemandpatternsand consequentlydistinguishbetweenthem for the purpose of selectingthemost appropriate orecasting and stock control) procedure.Nevertheless, he cutoff valueshave been arbitrarily hosenso that they make sense only for the particularempirical

Demandizevariability0.10

A C0.74

Transaction B Dlvariability 0.53 Lead-timeariabilityD2

Figure2 Eaves'categorizationcheme.



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AA yntetosta/-Categorizationfdemandatterns 97situationsthat wereanalysed n the correspondingpiecesofresearch.There s no doubt that the criteriaused for categorizationare meaningful.In fact, similarcriteriawill be used for thecategorizationrules that will be proposed later on in thispaper(withthe difference hat lead times are assumedto beconstant).It is rather he arbitrary utoff valuesassigned othem that create certain doubts about the potentialapplicabilityof the proposedcategorization chemes to anydifferentpossiblecontext.

An alternativeapproachto the demandcategorizationproblemJohnston and Boylan4 compared Croston's intermittentdemand estimationprocedurewith EWMA on theoreticallygenerated demand data over a wide range of possibleconditions. (Croston'sapproachto forecasting ntermittentdemand and the latest research findings regarding hismethod are presented in Appendix A). Many differentaverage inter-demand ntervals (negative exponential dis-tribution), smoothing constant values, lead times anddistributionsof the size of demand (negativeexponential,Erlangand rectangular),were considered.The comparisonexercise was extendedto cover not only Poisson but alsoErlangdemandprocesses.The resultswere reported n theformof the ratio of the MSE of one methodover that of theother. For the differentfactor combinations used in thissimulationexperiment,Croston's method was superiortoEWMA for inter-demandntervalsgreater han 1.25reviewperiods. This result was the first responseto the questionraised in Anders Segerstedt'spaper:5 'When is best toseparate the forecasts like Croston suggests and when isit best with traditional treatment (ie simple exponentialsmoothing)?'.Definition of intermittentdemand now results from adirectcomparisonof possibleestimationprocedures o thatregionsof relativeperformancecan be identified.It seemsmorelogical indeedworking n the following way:1. Comparealternative stimationprocedures.2. Identify he regionsof superiorperformanceor eachone

of them.3. Define the demand patterns based on the methods'comparativeperformance ather han arbitrarily efiningdemand patterns and then testing which estimationprocedure performs best on each particular demandcategory.The approachdiscussed above is taken forward in theresearchdescribed n thispaper, n that thedecisionruleswillbe the outcome of a comparison between MSE perfor-mances.Thereare somesignificantdifferenceshoughwhichare the following:(a) the fact that theoreticalrather thansimulatedMSEs areconsidered, b)we proposea two, rather

than one, parameter classification scheme, taking intoaccount not only the demand intervalbut also the demandsize variabilityand (c) we validateempiricallyour results.MSE is similar o the statisticalmeasureof the varianceofforecast errors (which consists of the variance of theestimatesproducedby theforecastingmethod underconcernand the varianceof the actualdemand)but is not quite thesame since bias can also be explicitlyconsidered.MSE is aquadratic error measure and as such it may be undulyinfluencedby outliersbut it is the only accuracymeasurethat allows theoreticalresults to be tested in practice.Inaddition, MSE can be defined for all demand data seriesrather han ratio-scaleddata only, an issue that is obviouslyof great importance n an intermittentdemand context.

The lead time MSE assumingerror auto-correlationIn an inventory control application, the forecast errorsproducedby any estimationprocedureare usuallyassumedto be normal and independent or Poisson for slow movingitems). Usually the effect of non-normality is small.'However, for lead times greater than 1 the errors willtypicallybe auto-correlated nd this issue (the issueof auto-correlationof the forecasterrors)has receivedvery limitedattention'.6

By ignoring the auto-correlationterm, we are mostprobably overstating the performanceof the estimationproceduresunderconcernsinceauto-correlationsnducedbybias or lumpinessare generallypositive.The MSE over a fixedlead time of durationL, assumingerrorauto-correlation, or unbiasedforecastingmethods iscalculatedas follows:7MSEL.T. L{L Var(Estimates) (1)+ Var(ActualDemand)}

where the varianceterms on the right-handside referto asingleperiod.For any biaseddemandestimationprocedure,t is easytoshow that the lead-timeMSE is given by (2):MSEL.T. L {LVar(Estimates)+ LBias2 (2)+ Var(ActualDemand)

It immediately ollows, from Equation (2), that the MSEover lead time L for method A is greater han the MSE overlead time L for methodB if and only ifVar(Estimates)A BiasA> Var(Estimates)B BiasB (3)

where the subscripts refer to the forecasting methodsemployed.



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AA yntetostal-Categorizationfdemandatterns 99

or exact MSE of one other method, and we try to specifyunderwhatconditionsthe inequalityunderconcern s valid.Considering nequality 3) it is obvious that the comparisonbetweenany two estimationprocedures s only in termsofthe bias and the varianceof the one step ahead estimatesassociatedwith their application.That is, the length of thefixed lead time and the varianceof demand itself do notaffectthe final results.At this point, it is important o note that all the pairwisecomparisonresults are generatedassumingthat the samesmoothing constant value is employed by all of theestimation proceduresunder concern. We recognize thattheuse of the samesmoothingconstantmayput one ormoremethodsat a relativeadvantage/disadvantageut the issueof sensitivityof the comparativeperformance esultsto theapplication of the same a value has not been furtherexplored.Comparisonresults (MSE Croston'smethod-MSESyntetos and Boylan method)We firstcompare he MSEof Croston'smethodwiththatofthe Syntetosand Boylan method over a fixed lead time oflengthL (L> 1):

MSECROSTONMSESYNTETOSBOYLAN=

"2(p-)4(p - 1)2)- p(~ -4) (8)

2-2

forp>1,Oo, 1.(All detailed derivationsof our results are availableuponrequestfrom the firstauthor.)The theoretical uledevelopedaboveis expressedn termsof the squaredcoefficientof variation(CV2)and the averageinter-demandnterval.The rule can then be furtheranalysed(consideringdifferentpossible values of the control para-meters: L, y, p and a2) so that cut-off values can bedetermined.For any p> 1.32, inequality(8) holds (superior perfor-mance is theoreticallyexpectedby the new method). Forp< 1.32* If CV2> 0.48 then MSECROSTONMSESYNTETOS&

BOYLAN* If CV2 1.32unittimeperiodsand/orCV2>0.49.In the area that corresponds o p< 1.32 and CV2 MSESYNTETOS&BOYLANf and only if

2 p( 1)[4p - (2 - )2] (2 - a)12 (2 - )(2p2- p) - 4p3 (9)forp>l, 0ocl1.Table1 MSECroston'smethod-MSESyntetosandBoylanmethoda Smoothingconstant value p-Cutoff value CV2cutoff value0.05 1.32 0.490.10 1.32 0.490.15 1.32 0.480.20 1.31 0.47



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AA yntetosta/l--Categorizationfdemandatterns 01

covers the usual values of practicalusage. In this researchfour values are simulated:0.05, 0.10, 0.15 and 0.20.The lead time is also introducedas a control parameter.The lead times consideredare 1, 3 and 5 periods.

MSE resultsA Z2-testwas decided to be the most appropriateway ofassessing he validityof the theoretical ulesproposed n theprevioussection.The average nter-demandntervaland thesquaredcoefficientof variationof the demand sizes weregenerated or all 3000 files and the proposedcategorizationrules were used in order to indicate which method istheoretically xpectedto performbetteror best in each oneof those files. The null hypothesis developed is that theperformanceof the methods is independent of what isexpectedfrom the theory.Croston'smethod is always(on everydemanddataseries)expectedto performbetterthan EWMA, independentlyofthe demand data series characteristics.As such, the Z-teststatisticfor the population proportionwill be used to testwhether or not the number of files on which Croston'smethod out-performsEWMA is significantlygreaterfromthe numberof files on which EWMAperformsbetter.For testingthe categorization uleregardingall methods'performancea 3 x 2 (2 degrees of freedom) or 4 x 2 (3degreesof freedom) ablewill be useddependingon whetheror not ties occur.The Z2-teststatistic values are indicated, for differentsmoothing constant values, in Table 2. In brackets wepresent the number of files that the methods performasexpected.The shaded area refers to the Croston-EWMApairwisecomparisonand the valuespresentedare the Z-teststatistic values for the population proportion.Statisticallysignificant results at 1% level are emboldened whilesignificanceat 5% level is presented n italics.

The results generated on MSEs indicate the practicalvalidityof the rulesproposed n thispaper.Theperformanceof the methods is clearly not independent of what thecategorization rules suggest. This is true at a pairwisecomparisonlevel and when the overall rule (regardingallmethods' performance) s considered.The results, thoughregarding he EWMA-Croston comparison,are not whatone mayhaveexpectedempiricallyor the 'smoother'SKUs.There is some empirical evidence to attribute thatdifference o the high variabilityof the errorsproducedbythe EWMA estimator.'2Thedecompositionof theempiricalMSE into its constituentcomponents(bias squared+ varia-variabilityof demand+ variabilityof the estimates)wouldobviouslyenabletherigorousassessmentof the contributionof thesecomponents o theempiricalMSE.Nevertheless,nosuch resultshave beengenerated n our empirical tudyandthis issue requires urthersimulationson real data.

ConclusionsThe categorizationof alternativedemandpatterns acilitatesthe selection of a forecastingmethod and it is an essentialelement of many inventory control software packages.Despite the importanceof this issue though, the problemof categorizingdemand patternshas receivedvery limitedattentionso far in the academic iterature.Some work hasbeen performed n this area,which neverthelessacks eitheruniversalapplicabilityor empiricalvalidationof the results.The common practice in inventory control softwareindustry is to arbitrarilycategorizedemand patterns andthen select an estimation procedurein order to forecastfuture requirementsand manage stock efficiently. Ulti-mately,the objective s the selectionof a forecastingmethod.Based on Johnston and Boylan4we proposean alternativeapproach o the categorizationproblemaccording o whichdirect comparison of the forecasting methods results inspecifying he demandcategories.Thepairwisecomparisons

Table2 X2est resultsa L.T. Croston-EWMA EWMA-S.B.a Croston-S.B.a Overallcategorizationrules0.05 1 0.33(1509) 0.84(1633) 16.45 1634) 9.19(979)3 5.29 (1645) 1.10 (1691) 15.77 (1631) 12.80 (1111)5 4.35 (1619) 4.41 (1726) 13.18 (1624) 20.47 (1160)0.1 1 3.83 (1605) 6.96 (1709) 20.75 (1653) 1.60 (1081)3 10.11 (1777) 17.98 (1827) 17.22 (1640) 11.94 (1238)5 9.02 (1747) 19.02 (1826) 13.02 (1625) 17.43 (1271)0.15 1 6.83 (1687) 17.34 (1753) 30.56 (1688) 6.20 (1160)3 13.04 (1857) 53.58 (1925) 28.27 (1678) 13.05 (1339)5 13.04 (1857) 44.66 (1913) 16.79 (1644) 22.20 (1364)0.2 1 8.47(1732) 28.99(1802) 53.58(1744) 16.22 1230)3 14.83 (1906) 84.10 (1991) 31.96 (1694) 20.86 (1405)5 15.12 (1914) 60.77 (1949) 18.70 (1654) 21.37 (1418)aS.B.stands orSyntetos ndBoylanmethod.



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502 Journalf heOperationalesearchocietyol. 6,No.are based on the theoretical MSEs and they can indicateuniversallyvalid regions of expected superiorityof onemethod over the other. We illustrate our approach byconsideringCroston'smethod,Syntetosand Boylanmethodand EWMA. Thevalidityof the rulesproposed s confirmedby means of simulationon 3000 real-intermittent emanddata series. The empiricalresults also demonstrate he lackof sensitivityof the cutoffpoints proposedto the smoothingconstantvaluebeingused.The resulting categories of any demand classificationscheme for inventorymanagementare ultimatelymeant toservethe inventoryobjectivesof improvedcustomerservicelevelsand/orreductionof stockholdings.In thatrespect heforecastaccuracyof an estimator and subsequentlyts bias,or the lack of it, and its samplingerrorof the mean)is notthe only theoretical and practical concern regardingcategorization.Thatis to say, specificdemanddistributionalassumptionsas well as certain stock control models shouldalso, eventually,be considered.As far as the former ssueisconcerned, t is important o note that an interesting venuefor furtherresearchcould also be the comparisonof non-parametric (bootstrapping)versusparametricapproaches,the lattercomprisinga forecastingmethod and a particularassumeddemanddistribution.The coefficientof variationofdemand sizes has been shown in this paper to be veryimportantfrom a forecastingperspective.Nevertheless,weshouldmention that the use of this criterionalso impliesthenon-complianceof the true demand size distributionwithstandard heoreticaldistributions such as the geometricorthe logarithmic).As the coefficientof variationrises, so itbecomesmore difficult to justify the use of any standarddistribution and that, as pointed out before, opens up afurtheravenueof research: nderwhat circumstanceshoulda distributionalapproach be used at all. What has beenpresented n this papercan be perceivedas only a firststeptowardsa moresystematicand meaningfulapproach o thecategorizationproblem.

References1 Croston JD (1972). Forecastingand stock control for inter-mittentdemands.OplRes Q 23: 289-304.2 WilliamsTM (1984). Stock control with sporadicand slow-movingdemand.J OplRes Soc 35: 939-948.3 Eaves AHC (2002). Forecasting or the orderingand stockholding of consumablespare parts. PhD thesis, LancasterUniversity,UK.4 JohnstonFR and BoylanJE (1996).Forecasting or items withintermittentdemand.J OplRes Soc 47: 113-121.5 SegerstedtA (1994). Inventorycontrol with variationin leadtimes, especially when demand is intermittent.Int J ProdEconom35: 365-372.6 FildesR and BeardC (1991).Productionandinventory ontrol.IJOPM 12:4-27.7 StrijboschLWG,Heuts RMJ and van derSchootEHM (2000).A combined forecast-inventory ontrol procedurefor spareparts.J OplRes Soc 51: 1184-1192.

8 WillemainTR, Smart CN, Shockor JH and DeSautels PA(1994). Forecasting ntermittentdemand in manufacturing:comparative valuationof Croston'smethod.IntJ Forecast10:529-538.9 SaniB andKingsman G (1997).Selectinghe bestperiodicinventorycontrol and demand forecastingmethods for lowdemand tems.J OplRes Soc 48: 700-713.10 SyntetosAA andBoylan E(2001).Onthe biasof intermittentdemand estimates. nt J Prod Econom71: 457-466.11 Syntetos AA and Boylan JE (1999). Correctingthe bias inforecastsof intermittent emand. Presentedat the 19thInternationalymposiumn Forecasting,une1999,Washing-ton DC, USA.12 SyntetosAA (2001). Forecastingof intermittent emand.PhDthesis, BrunelUniversity-BuckinghamshireChilterns Univer-sityCollege,UK.13 Eaves AHC and Kingsman BG (2004). Forecasting for theorderingand stock-holdingof spareparts. J Opl Res Soc 55:431-437.14 Schultz CR (1987). Forecasting and inventory control forsporadic demand under periodic review. J Opl Res Soc 38:453-458.15 Rao AV (1973).A commenton: forecastingand stock controlfor intermittentemands.OplResQ24: 639-640.

Appendix A: Croston'smethod, Syntetos and BoylanmethodCroston,' as corrected by Rao,'5 proved the inappropriate-ness of exponential smoothing and he also expressed, in aquantitative orm, the bias associatedwith the use of thismethod,when dealingwith intermittentdemands.

Moreover, assuming a stochastic model of arrival and sizeof demand, Croston introduced a new method for char-acterising he demandper periodby modellingthe demandin one periodfrom constituentevents.According o his method,separateexponential moothingestimatesof the averagesize of the demandand the averageintervalbetweendemand ncidencesaremade after demandoccurs.If no demandoccursthe estimatesremainexactlythesame.Let: 1/p,be the Bernoulliprobabilityof demandoccurringin period t; pt the geometricallydistributed(includingthefirst success, ie demand occurring period) inter-demandinterval; p,' the exponentially smoothed inter-demandinterval, updated only if demand occurs in period t,E(p,)= E(pt')=p; z, the normallydistributeddemand size,whendemandoccurs;z,' the exponentially moothedsize ofdemand, updated only if demand occurs in period t,E(z,)= E(zt')= i; o~he common smoothing constant valueused and Y, the demand in unit time period t.Under theseconditions heexpecteddemandperunit timeperiod is E(YI,)= /p. Following Croston's estimationprocedure, he forecast, Y, for the next time periodis givenby: Y,' z,'/p,' and, according to Croston, the expectedestimate of demand per period in that case would be:E( Y,') =E(zt'/p,')= E(z,')/E(p,')= t/p (ie the method isunbiased).



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AA yntetostal--Categorizationfdemandatterns 03The varianceof the forecastsproduced by this methodiscalculatedby Crostonas follows:

Vart42-a p +-

(A.1)

Syntetos and Boylan" showed that Croston's method isbiased and that the expectedestimate of demandper unittime periodis not as calculatedby Croston,but rather) a (p-) (A.2)(A.2)

Moreover, hesamplingerrorof the mean was also foundtobe different:t)I a- P(P - 1) ( 2 ) 02

2-Va zl2-ap4

2 22p2-,-The researchersproposeda new estimator that takes intoaccount the smoothingconstantvalue used and which is the

following:

Y-(=1-2 zt (A.4)

The expectedestimate of demandper unit time period aswell as the variance of the estimates produced by thismethod are givenby (A.5) and (A.6), respectivelyE(Yt )E ((I -2 P -2p2 (A.5)

Var(Y) --Var- 2 (1 Var (A.6)

Received November2003;acceptedJune 2004 after one revision

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