Beta-Biomial Model ad Its Geeralizatios i the Demad Forecastig for Multiple Slow-Movig Alexadre Dolgui Idustrial Egieerig ad Computer Sciece, Ecole des Mies de Sait-tiee, Frace Maksim Pashkevich Exploratory Program Medical Statistics Eli Lilly ad Compay, USA 1 Itroductio Slow-movig stock-keepig uits (SKU) are commo i spare parts ivetory systems ad are usually held to avoid the udesirable cosequeces of the uavailability of items whe requested. These cosequeces could mea ay damage or risk associated with ot beig able to operate the equipmet requested. Due to the extremely low cosumptio rate, slow-movig spares are exceedigly iflexible with respect to overstockig. As such, overestimatig the demad for these items ca result i extra storage costs ad complete losses whe these items become obsolete. I additio, with the umber of SKUs usually beig very high, simple solutios, like keepig extra safety stocks for all items of a certai type, are ot satisfactory. Thus, a accurate demad forecastig must be performed to esure effective ivetory ivestmets. However, the available records of demad are ofte very limited, some of which oly have ull values over observatio itervals. For this reaso, the ivetory maagemet of such items is performed o a group basis, whe multiple stock-keepig uits (SKUs) are used to develop the populatio-averaged leadtime demad probability distributio. This distributio is the used for each item whe decidig o the repleishmet order size ad the re-order level. Air force ivetory systems are examples that rely o this type of approach, particularly because of the large percetage of their slow-movig uits kept stock as a result of extremely low demad. While accepted i practice, the described straightforward method has a clear disadvatage: due to the averagig of demad patters, it uderestimates the demad for items with larger cosumptio ad overestimates the demad for those with lower cosumptio. This chapter deals with a empirical Bayesia method to estimate the demad distributio of multiple slow-movig items i case of extremely low demad ad short history of requests. A extesio of the betabiomial probability distributio is give for flexible empirical Bayesia forecastig of SKUs with zero demad records while relyig o the demad for similar items. To accurately determie the practical sigificace of the problem, cosider the followig real-life example of the UK Royal Air Force (RAF) ivetory system [Eaves & Kigsma, 2004], oe of the largest ad most diverse ivetories i the world. At the begiig of 2000, the RAF kept about 684,000 cosumable lie items resultig i approximately 145 millio SKUs ad a total stock value of 1.2 billio pouds. From the overall umber of lie items, about 8.5% accouted for 90% of the aual demad value (fast-movig items). Moreover, 37.3% of the SKUs had fewer tha 10 demad trasactios, of which 40.5% had zero
2 Chapter 1 Beta-Biomial Model ad Its Geeralizatios i the Demad Forecastig for Multiple Slow-Movig demad over the observatio period of six years. I total, about 60% of the SKUs could be classified as slow-movig with extremely low cosumptio, for which the demad per period (oe moth) is biary, ad the maximum lead-time demad is fiite ad small. It is clear that a accurate demad forecastig for items of the cosidered type is very importat for a ivetory system like that of the RAF. This chapter is based o our previous publicatios [Dolgui et al., 2004; Dolgui & Pashkevich, 2008a; Dolgui & Pashkevich, 2008b; Dolgui & Pashkevich, 2008c]. The rest of this chapter is orgaized as follows: Sectio 2 reviews related literature; Sectio 3 deals with the problem statemet; Sectio 4 describes the usage of the stadard beta-biomial model to solve the problem; a extesio of this model that icorporates prior iformatio for the maximum probability of demad per period is preseted i Sectio 5; parameter estimatio ad the correspodig Bayesia forecastig issues for the proposed model are cosidered i Sectios 6 ad 7, respectively; fially, we preset our coclusio i Sectio 8. 2 Related Publicatios Accurate demad forecastig is a cetral issue i successful ivetory maagemet ad is critical i achievig realistic estimates of the overall service level, or a total fill rate i the case of multiple SKUs [Silver et al., 1998]. Itermittet demad aalysis is a challege for ivetory cotrol due to the specific ature of the uderlyig demad process, which makes the forecastig problem especially difficult [Willemai et al., 2004; Fildes & Beard, 1992]. Slow-movig demad is a sigificat type of itermittet demad i which a request is made for a sigle uit i most of the cases [Sytetos et al., 2005]. As a example, this is a commo situatio i service parts ivetory systems [Cochra & Lewis, 2002]. For slow-movig demad, the major problems are (i) the lack of order records to estimate the past cosumptio reliably ad (ii) zero cosumptio over a log period of time [Mitchell et al., 1983]. Forecastig techiques developed for smooth ad cotiuous demad are ot applicable here because their assumptios of cotiuity ad ormal demad distributio are ot appropriate. A umber of techiques that relax these assumptios have bee developed, ad of these, the most widely used is Crosto s method [Crosto, 1972]. Willemai et al. [1994] evaluated Crosto s method uder the relaxatio of the assumptios o which it is based usig simulated data; the authors the compared it with sigle expoetial smoothig usig idustrial data. Modificatios of this approach with various improvemets have likewise bee proposed [Johsto & Boyla, 1996; Sytelos et al., 2005]. Aother way of hadlig the demad for slow-movig SKUs is based o the Poisso probability distributio [Dusmuir & Syder, 1989; Schultz, 1987; Ward, 1978; Watso, 1987]. Recetly, the bootstrap techique has become popular whe modelig itermittet demad [Willemai et al., 2004]. Nevertheless, i the cases whe some of the ivetory items uder cosideratio have o records of past requests, the problem of demad forecastig becomes eve more complex. This is because commo techiques like Crosto s method ad its extesios caot be applied to estimate the probability distributio of demad. Aother challege is the limited demad history available to model lead-time cosumptio, a difficulty that has bee overcome through the Bayesia paradigm. First proposed by Scarf [1959] ad Silver [1965], it was later successfully applied to various ivetory maagemet problems [Hill, 1997, 1999; Arois et al., 2004]. Still, the empirical Bayesia techique, oe which uses the heterogeeity amog multiple ivetory items to overcome the problem of short historical records, has ot received eough attetio i scietific literature [Bradford & Sugrue, 1990], eve though it is potetially importat for practical applicatios. Most of the work i the Bayesia ivetory modelig has bee doe by assumig a Poisso demad distributio with a gamma prior for the rate parameter. However, the beta-biomial distributio is a better
A. Dolbui & M. Pashkevich 3 choice i the case of extremely low cosumptio, as the upper boud of the lead-time demad is fiite ad small. This mixture probability distributio was previously applied to a umber of ivetory maagemet problems [Petrovic et al., 1989]. We explai the beta-biomial model (BBM) i this chapter. A extesio of the beta-biomial probability distributio is also preseted with the objective of cosiderig additioal iformatio regardig extremely low demad probability for multiple slow-movig SKUs. I [Dolgui & Pashkevich, 2008a], we compared the performace of the biomial (BM) ad BBM models of demad forecastig for multiple slow-movig ivetory items usig the most commo demad patters correspodig to the uiform case, a bell-like distributio i which all values are close to zero or oe, two sub-groups of items with high probability of demad i oe group ad low i the secod oe, ad so o. I that paper, we cocluded that the BBM model sigificatly decreased the holdig costs required to achieve a desired service level whe compared with the BM model for all tested patters (8 66% of gai depedig o the patter). I additio, we foud that the greatest gai was obtaied for the U-shaped probability distributio. As metioed earlier, we preset the BBM model i this chapter as well as its extesio called Exteded Beta-Biomial Model (EBBM). We use the same empirical Bayesia approach ad the BBM, as i our previous work, but this time, we performed a more precise estimatio of the demad usig a additioal parameter [Dolgui & Pashkevich, 2008b]. 3 Problem Descriptio Let us itroduce the followig otatios: k Number of ivetory items i the maaged group Number of periods with available historical demad data m Lead-time legth B = (b ij ) Biary k -matrix with past demad records b ij Equal to oe if demad has occurred for item i i period j; zero otherwise s i Total umber of requests for ivetory item i i the past (b i1,, b i2,..., b i ) p i Ukow probability of a request per period for ivetory item i p BM Populatio-averaged probability of a request per period π Exteded beta-biomial model parameter; expert s estimate for maximum of {p i } L i Ukow re-order level for ivetory item i c i Uit holdig cost for ivetory item i D i Radom lead-time demad for ivetory item i D A Radom populatio-averaged lead-time demad F Target weighted probability of o shortage durig the lead-time B(α, β) Beta-fuctio with parameters α ad β Assume that the ivetory cotrol is performed over a group of similar SKUs, with k beig the group size. Let deote the umber of periods for which the demad B data are available, where B = (b ij ) is a biary k -matrix ad b ij = 1 if the demad has occurred for item i i the period j, ad b ij = 0 if otherwise. The demad is assumed to be biary due to the specific properties of the SKUs uder cosideratio, that is, it is supposed that there is always a possibility to select the period legth so that o more tha oe request occurs i each period. The availability of a item is crucial for the system to work effectively as a stock-out leads to udesirable cosequeces. For example, the correspodig equipmet might ot operate or the usatisfied demads could result i a backlog with a backloggig cost [Chauha et al., 2009]. The correspodig expeses
4 Chapter 1 Beta-Biomial Model ad Its Geeralizatios i the Demad Forecastig for Multiple Slow-Movig (backloggig cost) are difficult to estimate ad are ot comparable with the holdig cost. Thus, here a service level criterio is employed istead of the backloggig cost, cosiderig all items are slow-movig ad with some of them havig o record of demad over the etire observatio period. Presumably, the repleishmet orderig costs are isigificat with respect to the ivetory storage costs. Therefore, the size of the repleishmet order is always oe. For a group of slow-movig items, a straightforward approach estimates the average request probability as follows: p BM = k i=1 j=1 b i j k, (1) ad assigs the biomial distributio Bi(m, p BM ) to all SKUs i the group, where m is the lead time. Afterwards, the followig problem is solved for oe aggregated item preseted as: Miimize: L, subject to: P{D L} F. (2) The expressio (2) presets a usual service-level problem for a sigle ivetory item, ad L is the reorder level used for all SKUs i the group. The described method will be called the BM of the lead-time demad hereafter. This approach ca solve the data availability problem of the idustrial data sets that are short ad wide [Willemai et al., 2004]. The problem of havig few observatios () for each SKU is overcome by utilizig a large umber of items i a group (k). Nevertheless, the approach sigified by expressios (1) ad (2) does ot cosider the heterogeeity of item demads. This uderestimates the re-order level for the items with larger cosumptio ad overestimates the re-order level for the oes with lower cosumptio. Therefore, i this chapter, a more sophisticated ivetory optimizatio approach is utilized [Grage, 1998], the aim of which is to satisfy the target weighted probability of o shortage durig the lead-time over the whole group of k items. The ultimate goal is to fid the ukow re-order levels {L i } from the followig costraied optimizatio problem: Miimize: L1,...,L k k i=1 c k L i, subject to: k i=1 P{D i L i } E{D i } k i=1 E{D i } F, (4) where D i deotes the radom demad for the item i over the lead-time, c i is the uit holdig cost for this item, ad F is the target weighted probability of o shortage durig the lead-time. The problem (4) is NP-hard, with k decisio variables i the geeral case. To solve the optimizatio problem (4), oe must kow the probability distributios of the radom product demads D i, i = 1, 2,..., k. These distributios must be assessed usig the biary k -matrix B which cotais the historical demad records, where k is large (hudreds or thousads), ad is small (5 10, for example). I the approach (1) (2), there is a commo lead time m for all items; i the method preseted further i this chapter, we also rely o a sigle lead time m for all SKUs [Dolgui & Pashkevich, 2008b]. A accurate estimatio of the probability distributios for {D i } is performed through the Bayesia forecastig of these distributios while cosiderig the history of idividual requests for each item. The SKUs are ow forecasted based ot oly o the demad for all items i the group but also o idividual requests history (empirical Bayesia approach). Problem (4) is solved with respect to each item type based o their past requests. Each class cotais items with the same demad level (umber of requested items) i the past ad is assiged its ow lead-time demad distributio, as opposed to the biomial model with oe aggregated item. The umber of decisio
A. Dolbui & M. Pashkevich 5 variables is reduced to a maximum of + 1 which makes the exhaustive search for (4) applicable (sice is small). A separate ote should be made i assessig the quality of forecastig. As the records of demad are limited ad may cotai may zeroes, the performace of the forecastig approach must be evaluated ot for each item, but over a group of items uder review. The shortage of ay item has a key cosequece, ad it is impossible to compare the outcomes of shortages for differet items a priori. It is also difficult to state which items are more importat tha others. I additio, the simulatios show that global system performaces are cosistet with practical cosideratios. Ideed, for this type of ivetory system, the average performace is more importat tha accuracy for a particular item [Hopp & Spearma, 1995]. Let us ow preset a overview of the forecastig cocept employed i a BBM. 4 BBM for Slow-Movig Items with Low Demad ad Short History This sectio provides a framework for the solutio of the problem beig cosidered by relyig o ideas culled from logitudial statistical data aalysis [Diggle et al., 2002], that have bee successfully used to solve a umber of problems with similar data structure i other applicatio areas [Pashkevich & Khari, 2004]. We would like to poit out that the beta-biomial distributio, which serves as a foudatio i our approach, is ofte justified i ivetory maagemet i a slightly differet maer. It is usually supposed that a demad distributio for a sigle ivetory item is of this kid because the request probabilities of the cosumers have a beta distributio, ad the umber of cosumers is limited ad small (otherwise, a egative biomial distributio is used). Here, we use the beta-biomial distributio because there is heterogeeity of demad withi the group of SKUs, similar to the approach proposed by Bradford ad Sugrue [1990] for the mixed Poisso distributio. The BBM [Collet, 2002], as applied to the lead-time demad forecastig problem uder cosideratio, is formulated usig three assumptios: A 1, A 2, ad A3. The first assumptio (A 1 ) is that the probability p i of a item i beig ordered durig the overall review iterval is ivariat with respect to period umber j: (A 1 ) P{b ij = 1} = p i, j = 1, 2,...,. This assumptio of the statioary demad over the review period is realistic from a practical poit of view, as i our case i which the cosumptio rate is low, ad the umber of periods with available data is small. The latter ca be iterpreted as the overall review period beig egligibly small whe compared with the demad dyamics. The secod assumptio (A 2 ) itroduces both the relatio ad the heterogeeity amog the SKUs by supposig that the demad probabilities {p i } for all i are draw from the same beta distributio [Johso et al., 1995] with the parameters α ad β: (A 2 ) L{p i } = B(α, β), i = 1, 2,..., k. This statistical distributio will be called the prior distributio hereafter. This assumptio allows creatig a populatio-averaged demad model of SKUs, which will later be used for the Bayesia forecastig of the lead-time demad probability distributio for each particular item. The beta distributio is very flexible ad ca represet differet heterogeeities i the demad patter ad take a variety of shapes depedig o the values of the parameters α ad β [Collet, 2002]. The probability desity fuctio is expressed as follows: f pi = (y α,β) = y α 1 (1 y) β 1 /B(α,β). (5)
6 Chapter 1 Beta-Biomial Model ad Its Geeralizatios i the Demad Forecastig for Multiple Slow-Movig For example, if α ad β are both less tha or equal to 1, the distributio will be U- or J-shaped. These shapes represet polarized distributios, i which some items have small request probabilities ad others have large, although few SKUs ca be foud i betwee. O the other had, if α ad β are both large, the distributio will resemble a spike so that all items have more or less the same request probabilities. If the values of α ad β are just a little larger tha 1, the the beta distributio looks like a iverted U or like the cetral part of the ormal curve. If α ad β are both equal to oe, the distributio becomes uiform. If oe of the parameters is larger tha 1 ad the secod is smaller, the probability desity fuctio is L-shaped. Beig a cojugate prior for the biomial distributio, the beta distributio is capable of flexibly describig the heterogeeity of demad probabilities for the items withi the group [Collet, 2002]. It is also capable of providig computatioally simple forecastig expressios. The third assumptio (A 3 ) is that the radom demad probabilities {p i } are idepedetly draw from the beta distributio, ad are thus idepedet i total: (A 3 ) Probabilities p 1, p 2,..., p k are i.i.d. radom variables. Uder the assumptios A 1 A 3 ad with the lead time equal to m periods, the populatio-averaged leadtime demad probability distributio for every item of the group is the beta-biomial with the parameters m, α, ad β [Collet, 2002]: ( ) m P{D A = x} = x B(α + x,β + m x), x = 0,1,...,m, (6) B(α, β) where D A deotes the populatio-averaged lead-time demad. We ca the divide all SKUs ito (S + 1) classes (or sub-groups) based o the previous requests {s i }, S = max(s 1, s 2,..., s k ). Afterwards, we ca deal with S + 1 aggregated items. Each class cotais items with the same umber of demaded items i the past. Uder the proposed assumptios, the maximum possible value of S is equal to. For each aggregated item, we will apply the above-formulated BBM with the empirical Bayesia approach. The major advatage of the Bayesia approach i forecastig is that it leads to the adjustmet of the populatio-averaged probability distributio (6) usig observatios specific to a particular item. As the beta distributio is cojugate prior to the biomial model, the posterior distributio of the demad probability p i for the item i is also beta but with shifted parameters: L{p i s i } = B(α + s i,β + s i ), s i = j=1 b i j, (7) where s i is the total demad over the observed period for the i-th SKU. Hece, the forecast of the leadtime demad D i for the item i also follows the beta-biomial distributio but with the parameters modified accordig to (7): ( ) m P{D i = x s i } = x B( ˆα + s i + x, ˆβ + m + s i x) B( ˆα + s i, ˆβ, x = 0,1,...,m, (8) + s i ) where the hats deote the estimates of the correspodig parameters (see Sectio 6). Expressio (8) ca be used to calculate the re-order levels for the SKUs i the correspodig sub-group. Estimatio of the model parameters is performed usig explicit expressios based o the method of momets (MM) or the iterative umerical estimators based o the maximum likelihood (ML) (Tripathi et al., 1994). The procedure of modelig the demad distributio for the group of slow-movig items with low demad ca be outlied as follows:
A. Dolbui & M. Pashkevich 7 Obtai the historical demad data for the group of items as a (k ) biary matrix B. Estimate the parameters α ad β usig the matrix B. Divide SKUs i (S+1) sub-groups, with each sub-group cotaiig items with the same umber of requests i the past. Compute the Bayesia posterior lead-time demad probability distributio (8) for each sub-group i based o the demad s i observed for the correspodig items. Fially, problem (4), which balaces the weighted probability of o shortage durig the lead-time ad the holdig costs, takes the followig form: Miimize: L 1,..., L k k i=1 r c x L x, subject to: s x=0 r c xp{ D x L x } E{ D x } s x=0 r c xe{ D x } F, (9) where r c x = k i=1 c i I(s i = x), ad I(.) is the uit fuctio that takes the value of oe if the argumet coditio holds, ad zero if otherwise. The variables L x ad D x deote the re-order level ad radom leadtime demad for the sub-group x, respectively. As aforemetioed, the sub-group x is defied as a subset of SKUs with exactly x requests for the last periods. As i our case the past demad record is very short, the problem (9) ca be efficietly solved usig a simple Brach-ad-Boud algorithm with the followig coditio for cuttig upromisig braches for a icomplete solutio { L 0, L 1,..., L q }, q < S : q x=0 rc x P{ D x L x } E{ D x } + S x=q+1 rx c E{ D x } S < F. (10) x=0 rx c E{ D x } 5 Geeralizatios i the Case of Extremely Low Demad Although the BBM explaied i the previous sectio provides a reasoable framework for hadlig demad for slow-movig SKUs with short ad zero demad histories, it does ot cosider the fact that the demad is very low for all items. I this sectio, we preset a extesio of the beta-biomial model that ca cosider this iformatio. First, a correspodig probability distributio is preseted. Afterwards, parameter estimatio issues are cosidered. Fially, we preset a empirical Bayesia forecastig procedure for the ew model, which esures mea square optimal predictio. The followig extesio to the assumptio A 2 is proposed. Suppose that there exists a expert estimate π (0, 1) of the maximum demad probability per period for the cosidered group of SKUs, the prior distributio of the demad probabilities the becomes the followig special case of the geeralized beta distributio: fp g i (y α,β,π) = yα 1 (π y) β 1. (11) B(α,β) πα+β 1 The parameter π is assumed to be small due to the specific properties of the cosidered demads. I the followig, we preset a populatio-averaged lead-time demad probability distributio uder prior (11), ad the show the expressios for its mathematical expectatio ad variace. The populatio-averaged lead-time demad probability distributio for the EBBM with prior distributio (11) ca be expressed as a weighted sum of the shifted beta-biomial probabilities preseted as: P{D A = x} = m l=x w xl (π,m) P 0 (l m,α,β), x = 0,1,...,m, (12)
8 Chapter 1 Beta-Biomial Model ad Its Geeralizatios i the Demad Forecastig for Multiple Slow-Movig where the weights w xl are computed as: {( l ) w xl (π,m) = x π x (1 π) l x, 0 x l, 0, l < x m, (13) ad P 0 (l m, α, β) deotes the beta-biomial probability with the correspodig parameters: ( ) m B(α + l,β + m l) P 0 (l m,α,β) =. (14) l B(α, β) The mathematical expectatio ad variace of the developed probability distributio (12) are calculated as follows: mα E{D A } = π α + β, V {D A} = π 2 mαβ(α + β + m) (α + β) 2 (α + β + 1) + π(1 π) mα (15) α + β Proofs of the results preseted above ad i subsequet sectios of this chapter ca be foud i our previous work [Dolgui & Pashkevich, 2008b]. 6 Parameter Estimatio To determie the parameters of the proposed EBBM, two traditioal approaches are employed. First, the explicit estimators based o the MM are preseted. The correspodig estimates ca the be itroduced as the iitial approximatio for maximum likelihood estimators, which are based o the umerical optimizatio routies. The parameter π is assumed to be kow. The raw data for the parameter estimatio are the biary k -matrix B preseted i the sufficiet form {s 1, s 2,..., s k } preseted i expressio (7). The MM estimators for the parameters α ad β of the probability distributio (12) with the kow parameter π ca be expressed as: α = λ s π, β = λ (π s ) s (π s ), λ = ( 1) π v s (1 s ) 1, s = s, v = v, (16) where s ad v are the mea ad variace of the sample {s 1, s 2,..., s k }. To obtai the maximum likelihood estimates, the followig optimizatio problem must be solved: ( k ( ) ) max : L(α,β) = B(α + l,β + l) w si l(π). (17) α,β l i=1 l=s i B(α, β) This ca be accomplished usig a modificatio of the steepest descet method. Followig a stadard path, we ca obtai the iitial approximatio via the MM, ad the apply a umerical optimizatio algorithm to compute the maximum likelihood estimates of the parameters α ad β. If there is o available expert estimate for the parameter π, it ca be estimated from the available data o past requests i the followig way. Oe approach is to implemet a grid search over a reasoable rage of values for π by estimatig the parameters α ad β for each value ad the selectig the estimates that lead to the best fit based o χ 2 -statistics. Aother approach, which is more precise but proe to overfittig, is to joitly estimate the parameters α, β, ad π by maximizig the likelihood fuctio (17) with respect to all three parameters. To surmout overfittig i the latter method, the fial estimate of π ca be selected as the value which is greater tha the obtaied maximum likelihood estimate ad is reasoable from a practical poit of view.
A. Dolbui & M. Pashkevich 9 7 Bayesia Forecastig for EBBM Oce the model parameters are estimated, a empirical Bayesia forecastig approach [Wikler, 2003] ca be applied to obtai the idividual probability distributio of the lead-time demad for each ivetory item. I the followig, we preset the posterior distributio of the demad probability uder the assumptio of prior (11). The proofs are provided i our previous work [Dolgui & Pashkevich, 2008b]. For prior (11), the posterior distributio of the demad probability ca be represeted as a weighted sum of the shifted geeralized beta distributios: h g p i (y s i ) = l=0 ω si l(π,) yα+l 1 (1 y) β+ l 1 B(α + l,β + l). (18) The correspodig mea-square-error optimal poit forecast is computed as follows: ˆp g i (s i) = The weights have the followig form: l=0 ω si l(π,) π (α + l) α + β +. (19) ω rl (π,) = ( 1 w r j (π,) P 0 ( j,α,β)) w rl (π,) P 0 (l,α,β), (20) j=0 where w rl is defied by (13). Oce the distributio of the demad probability p i is kow, the ext step is to develop the posterior lead-time demad probability distributio, which adjusts the populatio-averaged cosumptio for a particular item with respect to its ow demad history. We ow show expressios for the correspodig probability row ad the mea ad variace of the radom lead-time demad. The proofs are preseted i our previous work [Dolgui & Pashkevich, 2008b]. For the EBBM based o prior (11), the probability distributio of the lead-time demad D i for the ivetory item i is expressed as a weighted sum of the lead-time demad probability distributios for the EBBM (12) with shifted parameters, where the weights are give by expressio (20): P{D i = x s i } = ω si r(π,) r=s i The mea demad ad its variace are computed as follows: V {D i s i } = ω si r r=s m l=x w xl (π,m) P 0 (l m,α + r,β + r), x = 0,1,...,m. (21) E{D i s i } = m π ω si r(π,) r=s i α + r α + β +, (22) (π 2 m[2 ] α [2+] ) (α + β) +π mα E 2 {D [2+] i s i }, m [2 ] = m (m 1). (23) α + β These results ca be used to estimate the re-order poits for the group of slow-movig SKUs with extremely low demad ad with the past records characterized by a large percetage of zeros ad short observatio period. Its major advatage over the classic BBM is the additioal parameter π, which allows for a more precise specificatio of prior demad probability distributios.
10 Chapter 1 Beta-Biomial Model ad Its Geeralizatios i the Demad Forecastig for Multiple Slow-Movig 8 Coclusios This chapter cosidered the problem of modelig lead-time demad for multiple slow-movig ivetory items whe the available demad history is very short, ad a large percetage of items have oly zero records. A empirical Bayesia forecastig approach was chose to predict the lead-time demad probability distributio. As the demad ca be cosidered biary, ad the past history records are short, the BBM ca be successfully employed. This model is very flexible ad ca represet heterogeeities of item demads. Furthermore, a extesio of the BBM which cosidered the prior iformatio o the maximum expected probability of demad per period was preseted. Parameter estimatio ad Bayesia forecastig routies were proposed for this model as well. To be applicable for ivetory systems, our forecastig techiques should be itegrated i a appropriate ivetory cotrol model, with a example provided for referece. A optimizatio algorithm based o Brach ad Boud approach ca optimally solve this model. Our experiece i the use of these models has led to the followig coclusios: For the ivetory cotrol problem cosidered i this paper, the stadard biomial model teds to overestimate the ivetory eeded to achieve the target weighted probability of o shortage durig the lead-time ad is ot reliable with respect to the actual probability of o shortage. The beta-biomial model demostrates stable ad robust performace, esures lower holdig costs whe compared with the BM, ad guaratees the actual probability of o shortage durig the leadtime beig cosistet with the selected target. The EBBM esures cosiderably lower holdig costs tha the BBM ad also satisfies the target weighted probability of o shortage durig the lead-time, while the additioal computatioal complexity over the BBM is ot sigificat. Usually, items i the same group are supposed to have a related demad, which meas that they possess similar properties. For example, these items ca be spare parts from the same assembly uit or items that are historically kow to have correlated demads. Thus, this aspect reiforces the borrow-stregth properties of the empirical Bayesia method used i our approach, cosequetly improvig its performace. The preseted approaches are valid ot oly for the problems metioed i the itroductio, such as military ivetory systems, but also for a wide class of spare part ivetory systems as those i automobile ad aircraft idustries, amog others. Although the assumptio of the same lead-time for all items is ofte realistic for the type of ivetory system cosidered, the results preseted are ot restricted by this assumptio ad ca be directly exteded to the case of differet lead-times. The assumptio that o more tha oe request ca occur i each period poses some restrictios o the applicability of our methods. However, i most of the cosidered cases, this assumptio was acceptable, thereby illustratig the importace ad complexity of the problems ivolved. Refereces [Arois et al., 2004] Arois, P.-K., Magou, I., Dekker, R., & Tagaras, G. (2004). Ivetory cotrol of spare parts usig a Bayesia approach: A case study. Europea Joural of Operatioal Research, 154, 730 739 [Bradford & Sugrue, 1990] Bradford, J.W., & Sugrue, P.K. (1990). A Bayesia approach to the two-period style-goods ivetory problem with sigle repleishmets ad heterogeeous Poisso demads. Joural of the Operatioal Research Society, 41, 211 218
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