On the demand distributions of spare parts

Transcription

1 DePaul University From the SelectedWorks of Nezih Altay Aril, On the demand distributions of sare arts A A Syntetos, University of Salford M Z Babai, King Saud University Nezih Altay, DePaul University Available at: htts://works.beress.com/nezih_altay//

2 International Journal of Production Research Vol. 5, No. 8, 5 Aril, 7 On the demand distributions of sare arts A.A. Syntetos a *, M.Z. Babai bc and N. Altay d a University of Salford, Salford, UK; b BEM-Bordeaux Management School, Bordeaux, France; c King Saud University, Riyadh, Saudi Arabia; d DePaul University, Chicago, USA (Received May ; final version received January ) Sare arts have become ubiquitous in modern societies, and managing their requirements is an imortant and challenging task with tremendous cost imlications for the organisations that are holding relevant inventories. Demand for sare arts arises whenever a comonent fails or requires relacement, and as such the relevant atterns are different from those associated with tyical stock keeing units. Such demand atterns are most often intermittent in nature, meaning that demand arrives infrequently and is intersersed by time eriods with no demand at all. A number of distributions have been discussed in the literature for reresenting these atterns, but emirical evidence is lacking. In this aer, we address the issue of demand distributional assumtions for sare-arts management, conducting a detailed emirical investigation on the goodness-of-fit of various distributions and their stock-control imlications in terms of inventories held and service levels achieved. This is an imortant contribution from a methodological ersective, since the validity of demand distributional assumtions (i.e. their goodness-of-fit) is distinguished from their utility (i.e. their real-world imlications). Three emirical datasets are used for the uroses of our research that collectively consist of the individual demand histories of aroximately, SKUs from the military sector (UK and USA) and the Electronics Industry (Euroe). Our investigation rovides evidence in suort of certain demand distributions in a real-world context. The natural next stes of research are also discussed, and these should facilitate further develoments in this area from an academic ersective. Keywords: demand distributions; inventory; sare arts management; emirical study. Introduction Parametric aroaches to stock control rely uon a lead-time demand distributional assumtion and the emloyment of an aroriate forecasting rocedure for estimating the moments of such a distribution (tyically mean and variance). For the case of fast demand items, the Normality assumtion is tyically sufficient. However, Stock Keeing Units (SKUs) often exhibit intermittent or irregular demand atterns that may not be reresented by the Normal distribution. This is almost invariably the case for service/sare arts. Such roducts have become ubiquitous in modern societies, and managing their requirements is an imortant and challenging task with tremendous cost imlications for the organisations that are holding relevant inventories. Intermittent demand aears at random, with some time eriods having no demand at all. Moreover, demand, when it occurs, is not necessarily for a single unit or a constant demand size. In the academic literature, intermittent demand is often referred to as lumy, soradic, or erratic demand. A concetual framework that serves the urose of distinguishing between such non-normal demand atterns has been discussed by Boylan et al. (7). As mentioned above, intermittent demand items are tyically sare arts; those may be engineering sares (e.g. Mitchell 96, Hollier 98, Strijbosch et al. ) or arts ket at the wholesaling/retailing level (e.g. Sani 995, Syntetos et al. 9c). Demand for such items arises whenever a comonent fails or requires relacement as oosed to being generated according to buying behaviours of end-consumers (and the way that demand moves ustream in a suly chain). However, intermittent demand may indeed be characterising any SKU within the range of roducts offered by all organisations at any level of the suly chain (e.g. Croston 97, Willemain et al. 99). Such items are tyically the so-called slow-movers, and although they contribute little to the sales of an organisation, they may collectively account for u to 6% of the total stock value (Johnston et al. ). Consequently, small imrovements regarding their management may be translated into substantial organisational savings. Such items are articularly revalent in the aerosace, automotive, and IT sectors, and they are often *Corresonding author. a.syntetos@salford.ac.uk ISSN 75 rint/issn X online ß Taylor & Francis htt://dx.doi.org/.8/ htt://

3 A.A. Syntetos et al. the SKUs at greatest risk of obsolescence. As will be discussed later in this section, the emirical data available for the uroses of our investigation relate to sare arts; although such SKUs constitute the focus of our research, natural extensions with regards to our findings can be made to all roducts characterised by intermittent demand structures (lease refer also to the last section of our aer). Research in the area of forecasting and stock control for intermittent demand items has develoed raidly in recent years (e.g. Syntetos and Boylan 5, Chatfield and Hayya 7, Quintana and Leung 7, Altay et al. 8, Gutierrez et al. 8) with new results imlemented into software roducts because of their ractical imortance. Key issues remaining in this area relate to () the further develoment of robust oerational definitions of intermittent demand for forecasting and stock-control uroses and () a better modelling of the underlying demand characteristics for the urose of roosing more owerful estimators useful in stock control. Both issues link directly to the hyothesised distribution used for reresenting the relevant demand atterns. Surrisingly though, not much has been contributed in this area in the academic literature, and this constitutes the main urose of our work.. Objectives and structure of the aer Intermittent demand atterns are built from constituent elements, namely a demand arrival rocess and a distribution of the demand sizes, when demand occurs. As such, the very nature of such atterns necessitates, concetually at least, the emloyment of comound distributions, such as the negative binomial distribution (NBD). Although a number of distributions have been ut forward in the literature (lease refer also to the second section of this aer) there is still more emirical evidence needed on the goodness-of-fit of these distributions to real data. That is to say, there is a need for more insights into the emirical validity of various distributional assumtions. In addition, a distinction needs to be made between the goodness-of-fit on demand-er-eriod data and the validity of a distributional assumtion for reresenting lead-time demand. The latter is what matters in a stock-control context, and this distinction has been overlooked in the academic literature. Finally, it is also imortant to distinguish between the emirical validity and utility of a demand distributional assumtion, the latter being related to the stock-control imlications of using a articular distribution. Goodness-of-fit tests focus on the entire demand distribution, whereas stock control erformance is exlicitly deendent uon the fit on the right-hand tail of a distribution. All the above-discussed issues are addressed as art of our work. In articular, the objective of this aer is threefold: first, we conduct an emirical investigation that enables the analysis of the goodness-of-fit of various two- (or single-) arameter statistical distributions used in the literature in the context of intermittent demand; second, we extend the analysis discussed above to assess the linkage between the validity of various demand distributional assumtions and their utility (i.e. actual imlications for stock control in terms of inventory volumes and service levels achieved); third, we discuss the imlications of our work for Oerational Research (OR) theory and ractice, and we rovide an agenda for further research in this area. We use three emirical datasets for the uroses of our analysis that collectively constitute the individual demand histories of aroximately, SKUs. Two datasets come from the military sector (Royal Air Force, RAF UK and US Defense Logistics Agency, DLA) and one from the Electronics industry. In all cases, the SKUs are sare/service arts. At this oint, it is imortant to note that some non-arametric rocedures have also been suggested in the literature to forecast intermittent demand requirements (e.g. Willemain et al., Porras and Dekker 8). Such aroaches tyically rely uon bootstraing rocedures that ermit a re-construction of the emirical distribution of the data, thus making distributional assumtions and estimation of the moments of such distributions redundant. Although it has been claimed that such aroaches have an advantage over arametric methodologies, more emirical research is needed to evaluate the conditions under which one aroach outerforms the other. In this aer, we will be focusing solely on arametric aroaches to inventory management. The remainder of this aer is organised as follows. In Section, an introduction to forecasting and stock control issues in the context of intermittent demand is resented, along with a review of the demand distributions discussed in the literature and/or used by ractitioners. In Section, we resent the datasets used for the urose of this emirical investigation followed by a discussion on the statistical goodness-of-fit tests that we have considered for analysing the erformance of various distributions. The emirical validity and utility of distributional assumtions are then exlored in Sections and 5 for demand-er-eriod data and lead-time demands resectively.

4 International Journal of Production Research Finally, the conclusions of our research along with its imlications and some natural extensions for further work in this area are resented in Section 6.. Research background In this section, a brief review of the literature on issues related to arametric inventory forecasting is resented. First, we establish the need for estimating the mean and variance of demand in an inventory control context, followed by a discussion of various suggestions that have been made in the literature with regards to the hyothesised distribution of demand atterns, focusing on the case of intermittence. The second sub-section culminates with the selection of a number of distributions to be considered in the emirical art of our work.. Parametric forecasting Practical arametric aroaches to inventory management rely uon estimates of some essential demand distribution arameters. The decision arameters of the inventory systems (such as the Re-Order Point or the Order-U-To-Level) are then based on these estimates. Different inventory systems require different variables to be forecast. Some of the most cited, for examle (R, s, S) olicies (Naddor 975, Ehrhardt and Mosier 98), require only estimates of the mean and variance of demand. (In such systems, the inventory osition is reviewed every R eriods, and if the stock level dros to the re-order oint s, enough is ordered to bring the inventory osition u to the order-u-to-level S.) In other cases, and deending on the objectives or constraints imosed on the system, such estimates are also necessary, although they do not constitute the key quantities to be determined. We may consider, for examle, an (R, S) oran(s, Q) olicy oerating under a fill-rate constraint known as P. (In the former case, the inventory osition is reviewed eriodically, every R eriods, and enough is ordered to bring it u to S. In the latter case, there is a continuous review of the inventory osition and as soon as that dros to, or below, s an order is laced for a fixed quantity Q.) In those cases we wish to ensure that x% of demand is satisfied directly off the shelf, and estimates are required for the robabilities of any demands exceeding S or s (for the (R, S) an(s, Q) olicy resectively). Such robabilities are tyically estimated indirectly, based on the mean demand and variance forecast in conjunction with a hyothesised demand distribution. Similar comments aly when these systems oerate under a different service-driven constraint: there is no more than an x% chance of a stock-out during the relenishment cycle (this service measure is known as P ). Consequently, we need to estimate the ( x)th ercentile of the demand distribution. In summary, arametric aroaches to forecasting involve estimates of the mean and variance of demand. In addition, a demand distribution needs also to be hyothesised, in the majority of stock-control alications, for the urose of estimating the quantities of interest. Issues related to the hyothesised demand distribution are addressed in the following sub-section. For a detailed account on issues related to the estimation of mean demand and its variance in an intermittent demand context, interested readers are referred to Syntetos and Boylan (8) and Syntetos et al. (9b).. Demand distribution Intermittent demand atterns are characterised by infrequent demands, often of variable size, occurring at irregular intervals. Consequently, it is referable to model demand from constituent elements, i.e. the demand size and interdemand interval. Therefore, comound theoretical distributions (that exlicitly take into account the size-interval combination) are tyically used in such contexts of alication. We first discuss some issues related to modelling demand arrivals and hence inter-demand intervals. We then extend our discussion to comound demand distributions. If time is treated as a discrete (whole number) variable, demand may be generated based on a Bernoulli rocess, resulting in a geometric distribution of the inter-demand intervals. When time is treated as a continuous variable, the Poisson demand generation rocess results in negative exonentially distributed inter-arrival intervals. There is sound theory in suort of both geometric and exonential distribution for reresenting the time interval between successive demands. There is also emirical evidence in suort of both distributions (e.g. Dunsmuir and Snyder 989, Kwan 99, Willemain et al. 99, Janssen 998, Eaves ). With Poisson

5 A.A. Syntetos et al. arrivals of demands and an arbitrary distribution of demand sizes, the resulting distribution of total demand over a fixed lead time is comound Poisson. Inter-demand intervals following the geometric distribution in conjunction with an arbitrary distribution for the sizes results in a comound binomial distribution. Regarding the comound Poisson distributions, the stuttering Poisson, which is a combination of a Poisson distribution for demand occurrence and a geometric distribution for demand size, has received the attention of many researchers (for examle: Gallagher 969, Ward 978, Watson 987). Another ossibility is the combination of a Poisson distribution for demand occurrence and a normal distribution for demand sizes (Vereecke and Verstraeten 99), although the latter assumtion has little emirical suort. Particularly for lumy demands, the demand size distribution is heavily skewed to the right, rendering the normality assumtion far from aroriate. Quenouille (99) showed that a Poisson-Logarithmic rocess yields a negative binomial distribution (NBD). When event arrivals are assumed to be Poisson-distributed, and the order size is not fixed but follows a logarithmic distribution, total demand is then negative binomially distributed over time. Another ossible distribution for reresenting demand is the gamma distribution. The gamma distribution is the continuous analogue of the NBD and although not having a riori suort [in terms of an exlicit underlying mechanism such as that characterising comound distributions], the gamma is related to a distribution which has its own theoretical justification (Boylan 997,. 68). The gamma covers a wide range of distribution shaes; it is defined for non-negative values only, and it is generally mathematically tractable in its inventory control alications (Burgin and Wild 967, Burgin 975, Johnston 98). Nevertheless if it is assumed that demand is discrete, then the gamma can be only an aroximation to the distribution of demand. At this oint, it is imortant to note that the use of both NBD and gamma distribution requires estimation of the mean and variance of demand only. In addition, there is emirical evidence in suort of both distributions (esecially the former), and therefore they are recommended for ractical alications. If demand occurs as a Bernoulli rocess, and orders follow the Logarithmic-Poisson distribution (which is not the same as the Poisson-Logarithmic rocess that yields NBD demand), then the resulting distribution of total demand er eriod is the log-zero-poisson (Kwan 99). The log-zero-poisson is a three-arameter distribution and requires a rather comlicated estimation method. Moreover, it was found by Kwan (99) to be emirically outerformed by the NBD. Hence, the log-zero-poisson cannot be recommended for ractical alications. One other comound binomial distribution that aeared in the literature is that involving normally distributed demand sizes (Croston 97, 97). However, and as discussed above, a normality assumtion is unrealistic, and therefore the distribution is not recommended for ractical alications. Desite the inaroriateness of the normal distribution for reresenting demand sizes, it may in fact constitute a reasonable assumtion for lead-time demand itself, when lead times are long (see also Syntetos and Boylan 8). This is because long lead times ermit central limit theorem effects for the sum of demands over the corresonding eriod, thus making the normality assumtion more lausible. In addition, the assumtion of normality may rove to be good when the coefficient of variation (CV) of the distribution of demand er eriod is small. Finally, algorithms based on normality are simle to imlement, making the normal distribution a very commonly assumed one among ractitioners. For very slow-moving items, such as those commonly encountered in a military context, for examle, the Poisson distribution is known to offer a very good fit, and much of the stock control theory in this area has been develoed uon the exlicit assumtion that demand er eriod is Poisson distributed (see, for examle, Silver et al. 998). In this case, demand is assumed to arrive as a Poisson rocess coule with unit-sized transactions. In an early work, Friend (96) also discussed the use of a Poisson distribution for demand occurrence, combined with demands of constant size. Vereecke and Verstraeten (99) resented an algorithm develoed for the imlementation of a comuterised stock-control system for sare arts in a chemical lant. The demand was assumed to occur as a Poisson rocess with a ackage of several ieces being requested at each demand occurrence. The resulting distribution of demand er eriod was called a Package Poisson distribution. The same distribution has aeared in the literature under the name hyothetical SKU (h-sku) Poisson distribution (Williams 98), where demand is treated as if it occurs as a multile of some constant, or clumed Poisson distribution, for multile item orders for the same SKU of a fixed clum size (Ritchie and Kingsman 985). The Package Poisson distribution requires, as the Poisson distribution itself, an estimate of the mean demand only. The review of the literature resented above indicates that it is worth while assessing the emirical validity and utility of the following demand distributional assumtions: () Poisson; () NBD; () Stuttering Poisson; () Gamma; and (5) Normal. In the next sections we conduct relevant tests and comment on the lausibility of the relevant assumtions for alications in an intermittent demand context.

6 International Journal of Production Research 5. Emirical investigation In this section, we first describe the datasets used for the uroses of this emirical investigation, followed by a discussion on statistical goodness-of-fit testing.. Emirical data The emirical databases available for the uroses of our research come from the US Defense Logistics Agency (DLA), Royal Air Force (RAF), and Electronics Industry and they consist of the individual monthly demand histories of 588, 5, and 55 SKUs resectively. Some information regarding these datasets is resented in Table, followed by detailed descritive statistics (Mean and Standard Deviation SD) on the demand data series characteristics for each of the datasets resented in Tables. (At this oint it should be noted that the time series considered have not been tested for stationarity.) Tables indicate that the three datasets considered for the urose of our emirical investigation reflect a wide range of ossible underlying demand atterns. Datasets no. and no. are associated with a somewhat low degree of intermittence; the average demand intervals have a median value of aroximately and.5 resectively. However, there is a relatively high degree of erraticness; demand sizes are highly variable with a median standard deviation of aroximately 5 (6) but a maximum one equal to 7 (99) for dataset no. (no. ). On the other hand, dataset no. consists of average demand intervals that are very large, since their median value is almost nine months, but the demand sizes are associated with a comaratively lower degree of erraticness. Table. Emirical datasets. No. Country Industry Table. Dataset No. : US Defense Logistics Agency (DLA). 588 SKUs No. of SKUs Time bucket History length Lead-time info USA Military/DLA 588 Month 6 No UK Military/RAF 5 Month 8 Yes Euroe IT 55 Month 8 Constant ¼,6 Demand intervals Demand sizes Demand er eriod Mean SD Mean SD Mean SD Min th ercentile Median th ercentile Max Table. Dataset No. : Royal Air Force (RAF). Demand intervals Demand sizes Demand er eriod 5 SKUs Mean SD Mean SD Mean SD Min th ercentile Median th ercentile Max Notes: Dataset No. has been used in the studies conducted by Syntetos et al. (9a) and Teunter et al. ()

7 6 A.A. Syntetos et al. Table. Dataset No. : electronics. Demand intervals Demand sizes Demand er eriod 55 SKUs Mean SD Mean SD Mean SD Min th ercentile Median th ercentile Max Note: Dataset No. has been used in the study conducted by Babai et al. ().. Statistical goodness-of-fit tests Two tests have been mainly used and discussed in the literature for checking statistically significant fit, namely: the chi-square test and the Kolmogorov Smirnov (K-S) test (see, for examle, Harnett and Soni 99). These tests measure the degree of fit between observed and exected frequencies. Problems often arise with the standard chisquare test through the requirement that data needs to be groued together in categories to ensure that each category has an exected frequency of at least a minimum of a certain number of observations. Some modifications of this test have also been considered in the literature. A modified chi-square test has been develoed for the urose of testing the goodness-of-fit for intermittent demands (Eaves ). This test differs in that boundaries are secified by forming a certain number of categories with similar exected frequencies throughout, rather than combining grous just at the margins. However, the imlementation of this test requires the secification of the number of categories to be used. We encountered difficulty in using the standard or modified chi-square test in our research, namely that of deciding how to secify the categories intervals or the number of categories. On the other hand, the K-S test does not require grouing of the data in any way, so no information is lost; this eliminates the troublesome roblem of categories interval secification. In an inventory context one could argue that measures based on the entire distribution can be misleading (Boylan and Syntetos 6). A good overall goodness-of-fit statistic may relate to the chances of low demand values, which can mask oor forecasts of the chances of high-demand values. However, for inventory calculations, attention should be restricted to the uer end of the distribution (say, the 9th or 95th ercentiles). The develoment of modified goodness-of-fit tests for alication in inventory control, and even more secifically in an intermittent demand context, is a very imortant area but not one considered as art of this research. (We return to this issue in the last section of the aer where the natural next stes of research are discussed.) Consequently, we have selected the K-S test for the urose of assessing goodness of fit. At this oint, it should be noted that the K-S test assumes that the data are continuous, and the standard critical values are exact only if this assumtion holds. Several researchers (e.g. Noether 96, 967, Walsh 96, Slakter 965) have found that the standard K-S test is conservative when alied to data that are discrete. The standard exact critical values rovided for the continuous data are larger than the true exact critical values for discrete data. Consequently, the test is less owerful if the data are discrete as in the case of this research; it could result in acceting the null hyothesis at a given significance level while the correct decision would have been not to accet the null hyothesis. As discussed in the revious section, we are considering five distributions, the fit of which is tested on the demand data related to,6 SKUs. First, the distribution of the demand er eriod has been considered, followed by tests on the lead-time demand. This distinction is a very imortant one, since the latter distribution may in fact be considerably different from the former and yet is the one required for stock control uroses. Critical values have been comuted based on K-S statistical tables for % and 5% significance levels. We consider that:. there is a Strong Fit if the P-value is less than the critical value for 5%;. there is Good Fit if the P-value is less than the critical value for % but larger than that for 5%;. there is No Fit if the P-value is larger than the critical value for %.

8 International Journal of Production Research 7 Table 5. Goodness-of-fit results: demand er eriod. Percentage of SKUs Dataset No. No. of SKUs Distribution Strong fit Good fit No fit 588 Poisson NBD Stuttering Poisson Normal Gamma Poisson NBD Stuttering Poisson Normal Gamma Poisson NBD Stuttering Poisson Normal Gamma Emirical results: demand er eriod In Table 5 we resent the ercentage of SKUs that satisfy the various degrees of goodness-of-fit taken into account in our research, for each of the datasets and statistical distributions considered. As shown in Table 5, the discrete distributions, i.e. Poisson, NBD and Stuttering Poisson, rovide overall a better fit than the continuous ones, i.e. Normal and Gamma. More recisely, and with regards to Strong Fit, the Stuttering Poisson distribution erforms best in all three datasets considered in our research. This is followed by the NBD and then by the Poisson distribution. On the other hand, the Normal distribution is judged to be far from aroriate for intermittent demand items; as will be discussed in the next section, this is artly, at least, due to the exerimental structure emloyed for this art of our analysis that relied uon the distribution of demand er time eriod rather than the distribution of the lead time demand. Contrary to our exectations, the Gamma distribution has also been found to erform oorly. This may be exlained in terms of the inconsistency between the distribution under concern, which is continuous in nature, and the discreteness of the (demand) data emloyed in our goodness-of-fit tests. We return to this issue in the last section of the aer where the next stes of our research are discussed in detail.. Linking the goodness-of-fit to demand characteristics Johnston and Boylan (996) offered for the first time an oerationalised definition of intermittent demand for forecasting uroses (demand atterns associated with an average inter-demand interval ( ) greater than.5 forecast revision eriods). They comared, on a wide range of theoretically generated demand atterns, the Mean Squared Error erformance of Croston s method that has been secifically designed for intermittent demand estimation, and Single Exonential Smoothing (SES) that is tyically used for fast-moving items. They found that the former method erforms better than the latter on demand atterns associated with an average inter-demand interval () greater than.5 forecast revision eriods. Their contribution lies on the identification of the average inter-demand interval as a demand classification arameter to determine intermittence rather than the secification of an exact cutoff value. Syntetos et al. (5) took this work forward by develoing a demand classification scheme that relies uon both and the squared coefficient of variation of demand sizes (CV ). In their work, they comared (assuming fixed lead-times) the theoretical MSE erformance of three estimators: Croston, SES, and SBA (Syntetos-Boylan Aroximation; Syntetos and Boylan 5) and they identified regions where one method outerforms all others. Both schemes serve the urose of identifying the best erforming estimation rocedure in each category; however, inventory control issues and demand distributional assumtions were not addressed. Boylan et al. (7)

9 8 A.A. Syntetos et al. assessed the stock control imlications of the work conducted by Syntetos et al. (5) by means of exerimentation on an inventory system develoed by a UK-based software manufacturer. The researchers demonstrated, emirically, the insensitivity of the cutoff value, for demand classification uroses, in the aroximate range.8.86 eriods. In this section, we attemt to exlore the otential linkages between demand distributional assumtions and the classification scheme develoed by Syntetos et al. (5). By doing so, we aim at roviding some further insights into the otential usefulness of such distributions in an intermittent demand context. In Figure we resent for dataset no. and each of the distributions considered the SKUs associated with a Strong Fit as a function of the inter-demand intervals (, exressed as a number of zero demand eriods: average interval between successive demand occurrences) and the squared demand coefficient of variation (CV, dimensionless measure) of the demand sizes. (The results for the other two datasets lead to similar insights, and they are resented searately in the Aendix.) As shown in Figure (and in the Aendix) and theoretically exected, both the Stuttering Poisson and the Negative Binomial distribution erform comaratively better for all the datasets considered. This is true for the SKUs with both high inter-demand intervals (e.g. SKUs with being u to in dataset no. or SKUs with a value u to in datasets no. and no. ) and low demand intervals (e.g. SKUs with values starting from in datasets no. and no. ). Moreover, it should be noted that there is a strong fit of NBD and Stuttering Poisson to all the SKUS that are also associated with a strong fit of the Poisson distribution, which is exected since both distributions under concern are comound Poisson ones. The SKUs where there is commonly a strong fit of those three distributions are those characterised by relatively low CV values. The Normal distribution seems to have the narrowest CV range among all tested distributions, showing a good fit roughly between zero and one. This imlies that the Normal distribution is not a good fit for erratic data. Furthermore, the Normal distribution erforms well Goodness of fit - Poisson 5 5 Goodness of fit - Stuttering Poisson Goodness of fit - Gamma Goodness of fit - NBD Goodness of fit - Normal 5 5 Figure. Dataset no. : goodness-of-fit results.

10 International Journal of Production Research 9 for the SKUs with relatively low inter-demand intervals (e.g. SKUs with values close to in datasets no. and no. and ¼.8 in the dataset no. ). However, there are also a few SKUs with high inter-demand intervals ( going u to in dataset no. and in datasets no. and no. ) for which the Normal distribution rovides a strong fit. Those latter SKUs have a minimum CV (i.e. CV ¼ ) which can be exlained by the fact that their demand is very low (in most of the cases, the demand is equal to one) and can fit to the Normal distribution with low mean (i.e. equivalently high values of ) and variance. Finally, in addition to the SKUs where there is a fit to the Normal distribution (those with low values of ), the Gamma distribution rovides also a strong fit to the SKUs with very high values of (i.e. SKUs with an inter-demand interval going u to eriods in dataset no. and eriods in datasets no. and no. ) and high CV values (i.e. SKUs with CV u to 6 in dataset no., CV ¼ in the dataset no. and CV ¼ 8 in the dataset no. ). This is also exected since the Gamma distribution is known to be very flexible in terms of its mean and variance, so it can take high values for its and CV and can be reduced to the Normal distribution for certain arameters of the mean and the variance. 5. Emirical results: lead-time demand In this section, we are concerned with the goodness-of-fit of the various distributions considered in our work to leadtime demand data. The emirical utility of the relevant demand distributional assumtions is also assessed by means of exerimentation with a eriodic order-u-to level system. At this oint it is imortant to note that no lead-time information has been made available to us for dataset no., and consequently this has not been considered for this art of our research. All SKUs included in dataset no. are associated with a fixed lead time of three eriods (months). The goodness-of-fit results are resented in Table 6. Although the searation among the tested distributions is not as clear as in the case of demand er eriod, the results indicate that NBD erforms, overall, best with about 6% (weighted average) of 855 SKUs showing a strong fit. Stuttering Poisson comes in a close second. Desite the general exectation that the Normal distribution would be a good reresentative of lead-time demand owing to the effect of Central Limit Theorem, our results do not demonstrate dominance by the Normal distribution. On the contrary, the Normal distribution seems to have a slit between strong fit and no fit; only about 7% of the 855 SKUs showing a strong fit for Normal, and about % showing no fit. Although the Gamma distribution still comares unfavourably to the comound distributions, it does erform better than the demand-er-eriod case. On the contrary, the Poisson distribution is associated with an inferior erformance when comared with the demand-er-eriod case; this was theoretically exected since lead-times are overall rather lengthy resulting in high lead-time demand cumulative sizes which cannot be reresented by the Poisson distribution. In the next sub-section, we design a simulation study to test the utility of these results. 5. Linking the goodness-of-fit to stock control erformance In this sub-section we are concerned with the emirical utility of the demand distributional assumtions discussed thus far in our aer. (The Poisson distribution will not be considered in this art of the analysis, as reliminary Table 6. Goodness-of-fit results: lead-time demand. Percentage of SKUs Dataset No. No. of SKUs Distribution Strong fit Good fit No fit 5 Poisson NBD Stuttering Poisson Normal Gamma Poisson NBD Stuttering Poisson Normal Gamma

11 A.A. Syntetos et al. results indicated a very oor erformance as comared with that of all other distributions.) The imlications of using these distributions are evaluated through an order-u-to-level (R, S) olicy that oerates under a Cycle Service Level (CSL) objective (roortion of demand satisfied directly from stock). Mean demand is estimated by using the SBA estimator, and the variance of demand is estimated through the smoothed MSE rocedure. By keeing all asects of the stock-control system fixed and varying only the hyothesised demand distribution, the effects of using such distributions may be evaluated and contrasted with each other. Three target CSL values have been considered: 9%, 95%, and 99%. Performance is evaluated by means of the average holding volumes and the average backlog volumes (er time eriod for each SKU) as well as the achieved CSLs. In Tables 7 and 8, we show, for datasets no. and no., resectively, for every demand distribution and for every target CSL, the relevant quantities, averaged across all SKUs. Overall, the results in Tables 7 and 8 indicate that the Normal distribution is the worst-erforming one for higher target CSLs (i.e. CSL ¼ 95% and 99%). However, for a target CSL equal to 9%, the Normal comares favourably to the other distributions. The Normal distribution may not aroximate well the tail of the emirical distribution of data, but erforms very satisfactorily otherwise. For high target CSLs, the NBD rovides the best results, followed by the Gamma distribution and Stuttering Poisson. The results resented in the above tables are somewhat difficult to interret in terms of deriving tangible conclusions on the comarative erformance of the various distributions. Higher holding volumes are associated with higher achieved CSLs, making it difficult to areciate the comarative merits of using one distribution instead of another. To that end, we have also considered the resentation of our results through the use of efficiency curves. In Figures and, we lot (for datasets no. and no. resectively) the holding volumes required to suort a certain achieved CSL for each of the three target values and each of the distributions considered in our investigation. Table 7. Stock control-erformance: Dataset No.. Distribution Percentage Holding (volumes) Backlog (volumes) CSL (%) Normal Gamma Stuttering Poisson NBD Table 8. Stock-control erformance: Dataset No.. Distribution Percentage Holding (volumes) Backlog (volumes) CSL (%) Normal Gamma Stuttering Poisson NBD

12 International Journal of Production Research In these figures, for a fixed holding volume, a curve that is further from the x-axis shows that the CSL is higher and indicates that the corresonding demand distribution is more efficient. The efficiency curves show that the NBD and Gamma are the more efficient distributions, followed by the Stuttering Poisson. In fact, for a fixed holding volume, the former two distributions rovide the highest CSL. Although the Stuttering Poisson rovides a good overall fit (as demonstrated in the beginning of this section) the stock control erformance of this distribution is somewhat inferior to that associated with Gamma (that erformed less satisfactorily in terms of its goodness-of-fit results). This reinforces otentially earlier comments that we have made with resect to the imortance of fitting the tail of the distribution as oosed to the entire distribution of lead-time demand. In addition, the results demonstrate that the Normal distribution is the least efficient one for targets above 9%, and this is in accordance with theoretical exectations. 9% 9% 9% CSL 9% 89% 88% 87% NBD 86% Gamma 85% Normal StPoisson 8% 8% Holding volumes Figure. Efficiency curves: CSL against holding volume (dataset no. ). 95% 9% 9% CSL 9% 9% 9% 89% NBD Gamma Normal StPoisson 88% Holding volumes Figure. Efficiency curves: CSL against holding volume (dataset no. ).

13 A.A. Syntetos et al. In summary, our results for the lead-time demand distribution tests are overall different from the findings emerged from the demand-er-eriod analysis. The Poisson distribution was found to erform considerably better in the latter case, whereas the oosite is true for the Gamma distribution. The Normal distribution roved to be a weak candidate for the demand-er-eriod distribution. It was found to erform better for leadtime demand (robably due to Central Limit Theorem effects), although it may still be judged as far from aroriate for imlementation in a sare-arts context. The results have also indicated that the goodness-of-fit erformance on lead-time demand may not translate to similar insights with regards to the stock-control imlications of using the various distributions. Most noticeably, the Gamma distribution was found to comare favourably to the Stuttering Poisson with regards to stock control, whereas the goodness-of-fit tests oint to the oosite direction. 6. Conclusions, imlications, and further research Practical arametric aroaches to inventory management rely uon an exlicit demand distributional assumtion. Although the Normal distribution is tyically adequate for fast demand items, this is not true for sare arts that are tyically characterised by intermittent demand structures. Although a number of distributions have been discussed in the academic literature for reresenting such atterns, evidence on the emirical validity of the relevant assumtions (i.e. emirical goodness of fit) is lacking. Moreover, the linkage between the validity and the utility of these distributional assumtions, the latter referring to the actual stock control imlications of using a articular distribution, has not been exlored in the literature. An excetion to that is erhas a PhD thesis conducted at Lancaster University in the early 99s (Kwan 99). Unfortunately, no arts of that thesis have ever been ublished in the academic literature. Finally, an imortant distinction to be made is that between the erformance of a hyothesised distribution when fitting demand-er-eriod data and that when alied to lead-time demands, the latter being what matters in a real stock control system. In this aer a number of distributions are first selected as otential candidates for reresenting intermittent demands, on the basis of: () theoretical arguments, () intuitive aeal, and () emirical suort. These distributions are: the Poisson, Negative Binomial Distribution (NBD), Stuttering Poisson, Normal, and Gamma. The emirical goodness-of-fit of these distributions is then assessed for both demand-er-eriod data and lead-time demands by emloying the K-S test. Conclusions are being drawn on the emirical validity of the various distributional assumtions followed by an assessment of their stock control imlications (i.e. their emirical utility), under a eriodic order-u-to-level system. The emirical databases available for the uroses of our investigation come from the US DLA, RAF, and Electronics Industry, and they consist of the individual monthly demand histories of 588, 5, and 55 SKUs resectively. When we consider demand-er-eriod data, the results indicate that both the NBD and Stuttering Poisson rovide the most frequent fit. Both these distributions are comound in nature, meaning that they account exlicitly for a demand arrival rocess (Poisson) and a different distribution for the transaction sizes (Log series and Geometric for the NBD and Stuttering Poisson resectively). Desite revious claims, the Gamma distribution does not erform very well, and the same is true for the Normal distribution. This may be attributed to the continuous nature of these distributions (since their fit is tested on discrete observations) but also to the fact that demand er unit time eriod is considered as oosed to lead time demand. The Poisson distribution rovides a reasonable fit and this is theoretically exected for slow moving items. We have also attemted to link the goodness-of-fit of the various distributions to two key characteristics that have been shown to be collectively sufficient for defining intermittent demands (Syntetos et al. 5). These characteristics are the inter-demand interval and the squared coefficient of variation of the demand sizes. Goodness-of-fit erformance was then analysed across all series as a function of those characteristics, and some interesting insights were generated into the comarative erformance of the distributions. Recognising that the lead-time demand distribution is more imortant from an inventory control ersective, we also tested the goodness-of-fit of the above-mentioned distributions on lead-time demand. NBD and Stuttering Poisson were found to erform best also in this case; however, the remaining distributions were associated with differences in erformance. The Poisson distribution was found to erform considerably worse in that case, and this may be exlained in terms of the relatively long lead times associated with our emirical dataset. The Gamma distribution though was found to erform better for lead-time demand; similarly, the Normal distribution

14 International Journal of Production Research roved also to erform better (Central Limit Theorem effects), although it may still be judged as far from aroriate for imlementation in a sare-arts context. Subsequently, simulation exeriments using already-roven forecasting (Syntetos Boylan Aroximation) and inventory control methods ((R, S ) tye eriodic review olicy) were used to assess the stock-control imlications of using the various distributions. Although NBD was indeed confirmed as the best-erforming distribution, the results have also indicated that the goodness-of-fit erformance on lead-time demand may not translate to similar insights with regards to the stock-control imlications of using a articular distribution. Most noticeably, the Gamma distribution was found to comare favourably to the Stuttering Poisson with regards to stock control, whereas the goodness-of-fit tests ointed to the oosite direction. In summary, the results of our emirical investigation suggest that the NBD erforms best in an inventory context. This is followed by the Gamma and Stuttering Poisson distribution, the former found to outerform the latter in terms of their stock-control imlications. We exect further academic studies to take note of our findings and build sare-arts management research on these distributions. We also exect these findings to be of interest to ractitioners dealing with sare arts and being interested in imroving automated rocedures towards the most effective management of such items. At this oint we should also note that although sare arts constituted the focus of our research, natural extensions with regards to our findings can be made to all roducts characterised by intermittent demand structures. Our research has demonstrated that the choice of a demand distribution has considerable effects on stockcontrol erformance in a sare arts context. In addition, the work resented in this aer has revealed a number of interesting themes for further research. Distributional assumtions lay a critical role in ractical inventory management alications, and further work on the following issues should rove to be valuable from both a theoretical and ractitioner ersective:. The develoment of modified goodness-of-fit tests for alication in inventory control, and even more secifically in an intermittent demand context, is a very imortant area. In articular, utting more emhasis on the right-hand tail of the distribution seems aroriate for stock-control alications.. Quantifying the effect that the inconsistency between the discrete nature of demand data and the continuous nature of certain distributions (e.g. Gamma) may have on goodness-of-fit statistics constitutes an interesting research question.. The inconsistency between the discrete nature of demand observations and the imlicit assumtion of continuous data emloyed by various goodness-of-fit tests should be further exlored.. Relication of the analysis conducted in this aer in larger demand datasets couled with the assessment of the emirical utility of various distributional assumtions under other stock control formulations should hel advance knowledge in this area. Also, it would be worth while exloring the sensitivity of our findings under the realistic scenario of variable lead times; see, for examle, Babai et al. (9) and, for continuity uroses, also Babai and Dallery (9).. Emirical studies are needed to determine the comarative merits of non-arametric aroaches (such as bootstraing) in a sare-arts context. Such aroaches do not rely uon any underlying distributional assumtion. They rather consider the emirical distribution function and emhasise the direct estimation of the relevant ercentiles (say the 95th or the 99th ercentile). Although this is a very aealing roerty, the current emirical knowledge base is not extended enough to suggest how such aroaches comare to arametric inventory management in real terms.. In addition, and given that the inverse of the distribution function is the quantile function, another relevant aroach to the issues discussed in this aer relates to quantile forecasting. Directly forecasting quantiles avoids the need for assumtions regarding the shae and sread of the underlying distribution (Taylor 7) and should be given more attention in the inventory literature.. Finally, assessing Value at Risk (VaR) in inventory management is arguably also an area that has not been given sufficient attention. VaR is increasingly being used in financial management as a natural measure of the risk taken with a given osition. In the framework of inventory management it can work as well (Taiero 5) and is articularly relevant, since VaR roblems involve only one tail of the distribution (of the financial returns). Extensions into the distribution of inventory costs are feasible and intuitively aealing (Luciano et al. ) and further work into the issue of the efficient comutation of VaR would constitute an imortant addition to the current state of knowledge in the area of inventory control.