STOCHASTIC MODELS FOR INTERMITTENT DEMANDS FORECASTING AND STOCK CONTROL W. Sadma, O. Bober Uiversity of Bamberg, Germay Correspodig author: W. Sadma Uiversity of Bamberg, Dep. Iformatio Systems ad Applied Computer Sciece 96045 Bamberg, Feldkirchestraße 21, Germay, werer.sadma@ui-bamberg.de Abstract. Demad forecastig with regard to stock cotrol is a cetral issue of ivetory maagemet. Serious difficulties arise for itermittet demads, that is, if there are slow-movig items demaded oly sporadically. Prevalet methods the usually perform poorly as they do ot properly take the stochastic ature of itermittet demad patters ito accout. They ofte rely o theoretically ufouded heuristic assumptios ad apply iappropriate determiistic smoothig techiques. We overcome these weakesses by meas of systematically built ad validated stochastic models that properly fit to real (idustrial) data. Iitially, o assumptios are made but statistical methods are ivoked for model fittig. Reasoable model classes are foud by summary statistics ad correlatio aalysis. Specific models are obtaied by parameter estimatio ad validated by goodess-of-fit tests. Fially, based o the stochastic models, stock cotrol strategies are proposed to facilitate service levels guaratees i terms of probability bouds for beig out of stock. 1 Itroductio Ay orgaizatio or compay that offers, sells ad delivers items to others has to take care about proper ivetory maagemet. Success ad efficiecy substatially deped o the ability to provide ad deliver demaded items withi reasoable time. Stock cotrol is crucial ad the ivetory policy maages how may uits of a item must be i stock subject to certai costraits. Ivetory capacities are limited ad ivetory costs should be as low as possible but at the same time a desired level of item availability should be assured. Typically, i order to be well prepared, stock cotrol relies o forecastig future demads by meas of time series aalysis based o past demad patters. Comprehesive treatmets of time series aalysis ad forecastig ca be foud i, e.g., [1, 2, 4, 5, 8]. For the broader scope of ivetory maagemet we refer the reader to [14, 18, 19]. I practice, the most commo forecastig techique is simple expoetial smoothig (SES), that is forecasts are made by meas of a weighted sum of past observatios i that based o give time series data x 1,...,x t a forecast ˆx t+1 for the ext data poit x t+1 is computed recursively by ˆx 1 = x 1 ad ˆx t+1 = αx t + (1 α) ˆx t for t > 0 where α (0, 1) is a smoothig costat that eeds to be chose appropriately. Ufortuately, SES does ot provide satisfactory forecasts for itermittet demads, i.e. i the case of so-called slow-movig items or low-demad items that are oly demaded sporadically. Istead, Crosto s method [6] is most widely applied to itermittet demads. Crosto separates the time itervals betwee successive demads (iterdemad times) ad the umber of uits that are demaded (demad sizes). He argues that the time periods (measured i days, weeks, or moths) betwee successive demads as well as the demad sizes are idepedet ad idetically distributed (iid) radom variables, which meas that itermittet demads essetially appear at radom without idetifiable treds, seasoality, or the like. He heuristically assumes the geometric distributio for the iterdemad times ad the ormal distributio for the demad sizes. If a demad occurs, separate forecasts for both the iterdemad time ad the demad size are updated accordig to SES usig the same smoothig costat for both forecasts ad the curret demad per period forecast is obtaied by the ratio of these two forecasts. Obviously, the critical issue of choosig a appropriate smoothig costat remais ope. A couple of drawbacks such as biased forecasts or potetial violatios of the idepedece assumptios have bee reported i the literature ad may correctios ad modificatios, repectively, have bee proposed, e.g. [3, 11, 12, 15, 16, 17]. However, though specifically targeted to itermittet demads ad ofte more accurate tha SES, i some cases Crosto s method ad its various modificatios do ot provide proper forecasts ad ot eve outperform pure SES. After all, there is still o satisfactory approach to deal with stock cotrol for slow-movig items based o forecastig itermittet demads. We argue that the problems essetially stem from the applicatio of determiistic smoothig techiques to radom patters. Stochastic models appear to be more appropriate ad promisig for tacklig the peculiarities of itermittet demads forecastig ad stock cotrol.
2 Stochastic Modelig Approach I order to improve forecastig ad stock cotrol for slow-movig items we first have to figure out the weakesses of existig methods ad the requiremets for overcomig these weakesses. The mixture of assumig stochastic behavior ad applyig SES as a forecastig techique is iappropriate. The modificatios of Crosto s origial versio, adoptig the idepedece assumptios as well as the assumptio of geometrically distributed iterdemad times, are maily cocered with oormal (but still cotiuous) distributed demad sizes ad modified forecasts. I particular, they still produce determiistic poit forecasts though the demad patter is essetially radom. Mathematically, they work with determiistic realizatios rather tha with stochastic processes which are supposed to be the data geeratig mechaism. It has bee recetly show by Shestoe & Hydma [13] that the applicatio of these determiistic forecastig techiques caot be cosistet with stochastic models, because ay uderlyig stochastic model must be o-statioary ad defied o a cotiuous sample space with egative values, which both does ot match to the real properties of itermittet demad patters. We believe that startig with a forecastig techique ad buildig a accordig uderlyig model is exactly the reversed order of what is required. I particular, the major problem lies i the iappropriate forecastig techique rather tha i approachig itermittet demads via stochastic models. Additioally, we poit out that i the previously cited literature specific probability distributios are heuristically assumed ad - if at all - checked agaist artificial simulated data. It is ofte just a matter of luck whether or ot the assumptios well fit to real itermittet demad data. Cosequetly, we argue that oe should first build a adequate stochastic model based o real data, the validate its goodess of fit ad fially derive forecasts ad stock cotrol strategies to meet certai requiremets such as service level guaratees. 2.1 Stochastic Time Series Models A time series is a ordered sequece x 1,...,x, iterpreted as a realizatio of a stochastic process (X t ) t T. As we are cocered with discrete time poits (moths), we shall assume that the idex set T is a subset of the oegative itegers. Note that, though ofte eglected i the literature, there is a importat differece betwee a time series ad its "geeratig" stochastic process. Other tha a stochastic process, which is a esemble of time series, a sigle time series is just oe sequece of determiistic data. Time series properties are characterized by correspodig properties of stochastic processes. We briefly preset those that are most importat with regard to stochastic time series models. Defiitio 1 (Momet Fuctios of Stochastic Processes) For a stochastic process (X t ) t T its mea fuctio is defied by µ(t) := E[X t ],t T, variace fuctio is defied by σ 2 (t) := Var[X t ],t T, autocovariace fuctio is defied by γ(s,t) := Cov(X s,x t ) = E[(X s µ(s))(x t µ(t))], s,t T. Note that γ(t,t) = σ 2 (t) for all t T. Defiitio 2 (Strict Statioarity) A stochastic process (X t ) t T is called strictly statioary, if its fiite dimesioal distributios are time ivariat. That is, for all,t 1,...,t,s the radom vectors (X t1,x t2,...,x t ) ad (X t1 +s,x t2 +s,...,x t +s) have the same distributio. Defiitio 3 (Weak Statioarity) A stochastic process (X t ) t T is called weakly statioary, if it has costat mea ad variace fuctio ad its autocovariace fuctio does ot depedet o specific time poits but oly o the time differece, the so-called lag τ. That is, for all s, t T, τ := t s : µ(t) = µ <, σ 2 (t) = σ 2 <, γ(s,t) = γ(s + τ,t + τ). The the autocovariace fuctio for lag τ is defied as γ(τ) = γ(t s) := γ(s,t). It follows for the autocovariace fuctio of a weakly statioary stochastic process, that for all τ : γ(0) = σ 2, γ(τ) = γ( τ), γ(τ) γ(0). Defiitio 4 (Autocorrelatio Fuctio) The autocorrelatio fuctio of a weakly statioary stochastic process (X t ) t T is defied by ρ(τ) := γ(τ) γ(0) = γ(τ) σ 2. It follows for the autocorrelatio fuctio of a weakly statioary stochastic process, that for all τ : ρ(0) = 1, ρ(τ) = ρ( τ), ρ(τ) 1. Now, beig equipped with the most importat properties of stochastic processes, we itroduce the most importat stochastic processes with regard to time series aalysis ad i particular with regard to our modelig approach.
Defiitio 5 (White Noise) A white oise is a sequece (Z t ) t T of idepedet ad idetically distributed (iid) radom variables. If all these radom variables are ormally distributed with expectatio µ = 0 ad variace σ 2 <, the (Z t ) t T is called a Gaussia white oise. Obviously, a white oise is a strictly statioary stochastic process. I time series aalysis, white oise is used for costructig more complex stochastic processes. I the followig, let (Z t ) t T deote a white oise with expectatio µ = 0 ad variace σ 2 <. Defiitio 6 (Autoregressive Process) A autoregressive process of order p, deoted by AR[p], is a stochastic process (X t ) t T defied by X t = α 1 X t 1 +... + α p X t p + Z t, t T, where α 1,...,α p are costat coefficiets. Defiitio 7 (Movig Average Process) A movig average process of order q, deoted by MA[q], is a stochastic process (X t ) t T defied by X t = β 0 Z t + β 1 Z t 1 + β 2 Z t 2 +... + β q Z t q, t T, where β 0,...,β q are costat coefficiets. The radom variables Z t,t T costitutig the uderlyig white oise are usually ormalized such that β 0 = 1. Defiitio 8 (ARMA-Process) A autoregressive movig average process of order (p, q), deoted by ARMA[p,q], is a stochastic process (X t ) t T defied by X t = α 1 X t 1 +... + α p X t p + Z t + β 1 Z t 1 +... + β q Z t q, t T, where α 1,...,α p ad β 1,...,β q are costat coefficiets. Hece, ARMA processes are composed of AR processes ad MA processes ad both AR processes ad MA processes are specific ARMA processes. A AR[p] process is a ARMA[p, 0] process ad a MA[q] process is a ARMA[0,q] process. Note that a white oise also fits this framework i that it is a ARMA[0,0] process. 3 Buildig ad Validatig Stochastic Models We have developed a systematic procedure for model fittig to real data by courtesy of Siemes AG Healthcare Sector Customer Services Material Logistics (aka Siemes Medical Solutios), Erlage, Germay. Iitially, o idepedece assumptio are made but after computig comprehesive statistics, the idepedece of data is checked. Depedet o the outcome of suitable tests, either the time series correspodig to iterdemad times ad demad sizes are fitted to autoregressive movig average (ARMA) processes or fitted to adequate probability distributios. The essetial steps of our modelig procedure ad their order i a automated algorithmic applicatio startig with the raw data are outlied below. We emphasize that this procedure is flexible ad well accessible to practitioers. May steps are supported by statistical software packages, which is importat for beig viable as a part of real ivetory maagemet withi idustrial compaies. 1. Study summary statistics ad the correlatio structure of iterdemad times ad demad sizes 2. Test the idepedece hypothesis: Ljug-Box test 3. If idepedet, fit data to appropriate probability distributio (a) Select potetially appropriate distributio families based o summary statistics (b) For each cadidate distributio, obtai parameters by maximum likelihood estimatio (c) Validate goodess-of-fit by visualizatio: graphical plots (d) Validate goodess-of-fit by statistical tests: χ 2, Aderso-Darlig, Kolmogorov-Smirov 4. If ot idepedet, fit data to ARMA model (a) Select potetially appropriate class of ARMA based o (partial) autocorrelatio fuctios (b) For each cadidate class, obtai parameters by (b1) least squares estimatio for the AR part (b2) umerical iteratio for the MA part (c) Validate goodess-of-fit by residual aalysis
This procedure has bee applied to the demad patters of 54 differet slow-movig items, each recorded from September 1994 to May 2008. I additio to model fittig for each of these items, oe goal was to idetify similarities i order to build a aggregated model that itegrates as much slow-movig items as possible (ivetory of Siemes Medical Solutios takes care bout altogether about 8500 slow-movig items). I the followig we describe some more details of the steps ad outlie the mai fidigs that we obtaied i this maer. Note that we could have formulated our model fittig procedure without explicitly distiguishig betwee idepedet ad depedet data by just fittig to a ARMA process ad keepig i mid that a ARMA[0,0] process is a white oise which meas that the data is idepedet. However, this would be overly geeralized. We make the distictio with regard to the specific fittig methodologies, which are much easier for idepedet data. 3.1 Summary Statistics ad Correlatio Structure Summary statistics are usually ot commo i time series aalysis whe treds, seasoality, depedecies or correlatios are preset. Nevertheless, they should be computed as a first step i ay statistical data aalysis because they almost always give useful isights, i particular i the case of itermittet demads where completely radom patters i the sese of iid data or white oise are very likely. I additio, the correlatio structure is of major importat i time series aalysis. Therefore, we cosider a variety of statistical measures that give us a first quatitative impressio of the data ad its correlatio structure. More precisely, we compute the followig empirical measures from the time series data x 1,...,x. Empirical Mea Empirical Variace Empirical Stadard Deviatio Empirical Coefficiet of Variatio Empirical Stadard Error Empirical Skewess Empirical Kurtosis Empirical Covariace Empirical Coefficiet of Correlatio x = 1 s 2 = 1 1 s = s 2, c = s x, σ = 1 x i, (x i x) 2, s, (x i x) 3 ( ), 3 1 (x i x) 2 1 ( 1 s xy = 1 1 (x i x) 4 (x i x) 2 ) 2 3, (x i x)(y i y), r xy = s xy s x s y. Note that the latter two measures ca be computed with regard to differet time series as well as withi a sigle time series. Oe ca shift the time series by a lag τ ad iterpret the resultig pairs (x i,y i ) = (x i,x i+τ ) i the same way as data poits from differet time series. Cocered with a sigle time series oe also speaks of autocorrelatio as for stochastic processes. I particular, whe cosiderig such autocorrelatios for differet lags, oe ca get useful isights o the stregth of potetially preset depedecies, which the yields guidelies for the choice of appropriate ARMA processes.
3.2 Idepedece Test After computig comprehesive summary statistics ad studyig the correlatio structure of the give data, our ext step is to test the hypothesis of idepedece. More specifically, we wat to examie whether it is appropriate to assume that successive iterdemad times ad demad sizes are idepedet. I the termiology of stochastic time series models, this idepedece hypothesis correspods to the hypothesis that the data geeratig stochastic processes are white oise. Hece, we are cocered with the idepedece of successive data from oe time series (ot with the idepedece of two or more data sets). I order to test it, the Ljug-Box test [9] ca be applied. With the Ljug-Box test all autocorrelatio coefficiets are cosidered simultaeously. The idepedece hypothesis H 0 ad its alterative H 1 are defied as ad the test statistic is computed by H 0 : ρ(s) = 0, for all s N, s > 0, H 1 : {s ρ(s) 0,s N,s > 0} = /0 T ( ˆρ) = ( + 1) h j=1 ˆρ 2 j j. where is the sample size, ˆρ j is the empirical coefficiet of autocorrelatio o lag j ad h is the umber of lags. The for a give sigificace level α the critical sectio is defied by T ( ˆρ) > χ 2 1 α,h where χ1 α,h 2 is the 1 α-quatile of the chi square distributio with h degrees of freedom. The Ljug-Box test has bee specified as a S-Plus program ad applied to the itermittet demad patters, that is to the iterdemad times ad the demad sizes of all slow-movig items. Oe of our mai fidigs is that for almost all itermittet demad patters the idepedece assumptio is valid. More specifically, accordig to the Ljug-Box test o a statistical sigificace level of 0.05, the idepedece hypothesis was oly rejected for two out of the 54 items cosidered, ad the ru tests that we additioally performed did ot idicate ay depedece. 3.3 Fittig Idepedet Data If the data is assumed to be idepedet, our modelig procedure proceeds by fittig the data to a probability distributio. I order to fid appropriate probability distributios that well represet the measured data, we performed a parametric fit. Some cadidate distributios were selected ad for each of these distributios a maximum likelihood estimatio (MLE) was carried out to fid those parameters that fit best to the data. The goodess of fit was tested by the χ 2 test, the Kolmogorov-Smirov (KS) test, ad the Aderso-Darlig (AD) test. I a utshell, give a parametric distributio ad measuremet data, MLE determies the parameter values of the distributio that are most likely with respect to the data. Goodess-of-fit tests check the assumptio that the data is accordig to a specific distributio ad compute test statistics that idicate how well justified this assumptio actually is. Though a statistical test is maily desiged for rejectig a hypothesis that is ot well justified, it ca also provide some sort of rakig, which is importat i cases where several distributios are ot rejected ad may be appropriate. Roughly speakig, the goodess of fit is typically evaluated by the rule that the smaller the test statistic, the better the fit. Also p-values ca be take ito accout ad graphical methods such as a variety of plots visualize how close the fitted distributio is to the data. For the details of the statistical methods we refer the reader to, e.g., [7, 10]. As for the specific distributios that have bee fitted to the data, our focus was o discrete probability distributios sice both the iterdemad times (measured i moths for our data) ad the demad sizes are iteger umbers. We foud that for all 54 items the geometric distributio fits best for the iterdemad times ad that the discrete logarithmic distributio fits best to the demad sizes i the majority of cases ad at least fits well i all cases. Hece, Crosto s iterdemad time distributio has tured out to be appropriate ad is ow validated o the basis of real data. Oly his assumptio of ormally distributed demad sizes has bee substituted by validated logarithmic distributio. This supports our belief that the weakesses are maily due to iappropriate forecastig techiques. 3.4 Fittig Depedet Data If the data is assumed to be depedet, our modelig procedure proceeds by fittig the data to a ARMA process. I order to fid appropriate ARMA processes that well represet the measured data, we first selected a potetially appropriate class of ARMA models based o partial autocorrelatio fuctios, estimated the correspodig parameters ad validated the fit by residual aalysis. Roughly speakig, the fit is made iteratively such that processes of differet orders are successively fitted ad the sum of squares of differeces betwee the fitted process ad the time series data is computed. Fially, the order providig the least square sum is chose.
Iformally, the partial correlatio betwee two variables is the correlatio that remais if the possible impact of all other radom variables has bee elimiated ad it is much easier to determie the order of a ARMA process via the partial autocorrelatio fuctio tha via the origial autocorrelatio fuctio. For our itermittet demad data, ispectio of the partial autocorrelatio fuctios suggested that pure AR processes are most appropriate. We have specified a accordig procedure i S-Plus ad specifically used the Akaike iformatio criterio (AIC) where AIC(k) = log( ˆσ 2 ε,k ) + 2k is cosidered ad the k for which AIC(k) is miimal yields the estimated order of the AR process. For the details we refer the reader to, e.g., [1, 2, 4, 5, 8]. Eve for the two potetially critical items that did ot pass the Ljug-Box test for idepedece, the best fits to ARMA processes resulted i eglectig the MA part ad low orders of the AR part, that is, these data were best fitted to purely autoregressive processes of order 2 ad 4, respectively, which idicates a very weak depedece. Takig the data as idepedet ad fittig to probability distributios resulted i very accurate fits. Hece, it seems reasoable to assume idepedece eve for these two items. Note that statistical tests give oly statemets with certai statistical sigificace, either proofs or disproofs of hypotheses. 4 Forecastig ad Stock Cotrol Oce we have built stochastic models ad validated their appropriateess, it is clear that determiistic poit forecasts are ot very meaigful but stock cotrol strategies are possible which do ot rely o simple poit forecasts. We obtai service level guaratees i terms of probability bouds o stock out or item availability, respectively. More specifically, we ca compute the probability of a demad size beig greater tha some give threshold, which is closely related to quatiles ad tail probabilities of the fitted demad size distributio. Furthermore, to be useful for ivetory maagemet i practice where typically ot every moth each item stock is checked we also eed to cosider a larger time horizo as the plaig period. Therefore, we compute the probability of more tha a give umber of demads withi a certai time period, essetially via tail probabilities of sums of radom variables. After all, we ed up with stock cotrol strategies guarateeig that, give a desired service level i terms of probabilities ad the costrait of miimized ivetory costs, for every ivetory period sufficietly may uits of all items are i stock. Fially, thaks to stochastic similarities, items ca be aggregated yieldig a itegrated ivetory cotrol system. Hece, altogether we have a mathematically well-fouded model fittig procedure for practicable stock cotrol of slow-movig items that seems to be promisig ad ad overcomes some of the weakesses of curretly practiced methods. 4.1 Stock Cotrol Exemplificatio We demostrate a possible applicatio i practice by a simple example where we assume that the iterdemad times ad demad sizes are idepedet ad have bee properly fitted to probability distributios. We further assume that for a example item a prescribed service levels should be achieved withi a time horizo of moths. The service level is cosidered to be the item availability, that is the probability η that demads for this item ca be immediately served by the uits i stock. The questio arises how may uits of this item must be i stock for a give desired availability. Though the demad size has o theoretical upper boud, the demad size distributio allows statemets o the probability of certai demad sizes. We argue that for practicable stock cotrol oly demad sizes with some miimum probability ca be reasoably take ito accout. I other words, we cosider a lower probability boud for the demad sizes such that all demad sizes with a smaller probability are eglected. The, wheever a demad occurs, the uits to be hold i stock should equal the η-quatile Q η of the demad size distributio times the umber of moths i which the item is actually demaded. Hece, by cosiderig a time horizo of more tha oe moth for the plaig period, we are eve able to serve extreme demads with ulikely large demad sizes, provided that they do ot occur multiple times withi the time horizo of the plaig period. The remaiig questio is how ofte (i how may moths) withi the ext moths we should be prepared for demads. We do ot simply take the expectatio of the umber of demads withi moths, because this does ot accout for the specific probability distributio. Similarly as for the demad sizes, we cosider a upper boud such that oly with a small probability demads occur i more moths tha suggested by the boud. This probability is determied by ad ca be obtaied from the iterdemad size distributio. For umerical illustratio, we cosider a example item that has accordig to our fittig procedure geometrically distributed iterdemad times with parameter p = 0.21168. Hece, the probability that the item is demaded i the ext moth equals p, which is relatively large. This meas we should be prepared for a demad i the ext moth. Moreover, the umber of demads is biomially distributed, that is the probability of exactly j demads withi the ext moths is give by ( ) P j, := p j (1 p) j. j
Now, assume that our time horizo of the plaig period is six moths. The the probability that the item is demaded i each of these ext six moths is ( ) 6 P 6,6 = p 6 (1 p) 0 = p 6 = 0.00009, 6 which is extremely small. Such a demad patter will happe o average every thousad years such that it is reasoable to eglect it. Clearly, other patters are more likely. For istace, the probability of exactly oe demad withi the ext six moths is P 1,6 = 0,38667. Hece, we have to choose a boud ζ such that ay probability below this boud will be cosidered too small to realistically assume the correspodig demad patter. Let ζ = 0,01. From the remaiig demad patters with larger probability, choose those with the largest umber of moths with demad. Multiplyig this umber with the η-quatile Q η of the demad size distributio the gives the umber m of uits required i stock. For our example item the demad sizes are logarithmically distributed with parameter ϑ = 0,54295. With the choice of η = 0.95, the 95% quatile of the demad sizes computes as { 1 Q 0.95 = mi x : l(1 ϑ) x ϑ i i 0,95 Now, for demostratio purposes, successively compare the probabilities P j, with ζ = 0,01: P 1,6 = 0.38667 > ζ, P 2,6 = 0.25957 > ζ, P 3,6 = 0.09293 > ζ, P 4,6 = 0.01872 > ζ, P 5,6 = 0.00201 < ζ. Hece, we take four demads withi the ext six moths as sufficietly likely to be prepared for it ad the umber of uits required i stock is therefore m = Q 0.95 max{ j : P j, > ζ } = 4 4 = 16. The outlied approach is astoishigly simple ad provides a useful way of assurig certai service levels. Nevertheless, a couple of issues require further ivestigatio such as a reasoable plaig period (time horizo) over which stock cotrol should be cosidered. Besides, a additioal optio to react withi the plaig period whe the umber of uits i stock falls below a critical level would be surely useful. However, with regard to these poits, there are usually practical costraits ad ot everythig what is theoretically desirable is practically possible. 5 Coclusio We have preseted a stochastic modelig approach for the demad patters of slow movig items with regard to itermittet demads forecastig ad stock cotrol. The key problem i itermittet demads forecastig ad stock cotrol is the demad patter of slow movig items which reders traditioal determiistic expoetial smoothig techiques iappropriate. Stochastic models are required to capture the itermittet demad patter. It turs out that stochastic time series models are well suited. I may cases the itermittet demad patters eve appear to be purely radom i the sese that iterdemad times ad demad sizes are iid radom variables correspodig to a white oise process. A procedure has bee suggested for fittig real data to suitable stochastic models based o which forecastig ad stock cotrol become well-fouded ad automated. While this modelig procedure already seems to be mature eough to address items separately without ay substatial improvemets ecessary, further research should deal with aggregated models for multiple items. With regard to stock cotrol for sigle items, oe potetial applicatio of the stochastic model has bee demostrated ad similar approaches with a aggregated model are highly desirable. Hece, stochastic models for itermittet demads forecastig ad stock cotrol appear to be promisig ad have already prove useful but a lot of future work is still required ad already ogoig. 6 Refereces [1] G. E. P. Box ad G. M. Jekis. Time Series Aalysis: Forecastig ad Cotrol. Holde Day, 1970. [2] G. E. P. Box, G. M. Jekis, ad G. C. Reisel. Time Series Aalysis: Forecastig ad Cotrol. Joh Wiley & Sos, 4th editio, 2008. [3] J. E. Boyla ad A. A. Sytetos. The accuracy of a modified Crosto procedure. Iteratioal Joural of Productio Ecoomics, 507:511 517, 2007. } = 4.
[4] P. J. Brockwell ad R. A. Davis. Itroductio to Time Series ad Forecastig. Spriger, 2d editio, 2002. [5] C. Chatfield. The Aalysis of Time Series: A Itroductio. Chapma & Hall, 6th editio, 2004. [6] J. D. Crosto. Forecastig ad stock cotrol for itermittet demads. Operatioal Research Quarterly, 23(3):289 303, 1972. [7] R. V. Hogg, A. T. Craig, ad J. W. McKea. Itroductio to Mathematical Statistics. Pretice Hall, 6th editio, 2004. [8] G. Kirchgässer ad J. Wolters. Itroductio to Moder Time Series Aalysis. Spriger, 2007. [9] G. M. Ljug ad G. E. P. Box. O a measure of lack of fit i time series models. Biometrika, 2(65):297 303, 1978. [10] W. C. Navidi. Statistics for Egieers ad Scietists. McGraw Hill, 2d editio, 2008. [11] A. V. Rao. A commet o: Forecastig ad stock cotrol for itermittet demads. Operatioal Research Quarterly, 24(4):639 640, 1973. [12] B Sai ad B. G. Kigsma. Selectig the best periodic ivetory cotrol ad demad forecastig methods for low demad items. Joural of the Operatioal Research Society, 48:700 713, 1997. [13] L. Shestoe ad R. J. Hydma. Stochastic models uderlyig Crosto s method for itermittet demad forecastig. Joural of Forecastig, 24:389 402, 2005. [14] E. A. Silver, D. F. Pyke, ad R. Peterso. Ivetory Maagemet ad Productio Plaig ad Schedulig. Joh Wiley & Sos, 1998. [15] A. A. Sytetos ad J. E. Boyla. O the bias of itermittet demad estimates. Iteratioal Joural of Productio Ecoomics, 71:457 466, 2001. [16] A. A. Sytetos ad J. E. Boyla. The accuracy of itermittet demad estimates. Iteratioal Joural of Forecastig, 21:303 314, 2005. [17] R. Teuter ad B. Sai. O the bias of Crosto s forecastig method. Europea Joural of Operatioal Research, Article i Press, 2008. [18] T. Wild. Best Practice i Ivetory Maagemet. Butterworth-Heiema, 2d editio, 2002. [19] P. H. Zipki. Foudatios of Ivetory Maagemet. McGraw-Hill, 2000.