Inventories of optional components in discrete manufacturing are often subject to so-called low-count demand patterns. Quantities demanded from such inventories in any given period are sufficiently small that it may be unrealistic to forecast them with conventional models based on the normal distribution, and specialized models may be required. Fortunately, the statistical treatment of low-count time series has been the focus of much recent research. This paper recounts an attempt to apply some of this research to forecasting demands for optional parts at Sun Microsystems, a manufacturer and vendor of network computer products. Specifically, we compare the forecast performance of three simple state-space models using demand data obtained from Sun's inventory management records. The models are estimated using Bayesian methods, producing forecasts in the form of full predictive distributions. The accuracy of these probabilistic forecasts is compared using techniques borrowed from the field of meteorology, allowing us to assess the suitability of the candidate models for this type of application.
Fig. 1 displays a number of time series comprising the units of a selection of manufacturing parts used over a 78-week period in the operations of Sun Microsystems Inc., a manufacturer and vendor of network computing products. The parts in the figure are a subset of a larger sample of some 100 optional parts (that is, parts whose inclusion in a product depends upon configuration choices), drawn at random from Sun's inventory records.
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Fig. 1.
Sample inventory demands.
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Managing inventories for optional parts can be troublesome. In principle, with firm orders in hand for finished goods, materials requirements planning can be used to calculate parts demand over the short term through a straightforward bill-of-materials (BOM) “explosion” (Clement et al., 1995). In practice there are normally very many different types of such thinly demanded components, so that the administrative overhead of entering them into the BOM and maintaining the correct BOM entries in the face of product changes, changes in parts specification or supplier, etc., is often prohibitive. Planning at longer horizons could also be achieved by BOM explosion of finished good demand forecasts, but even if all the parts are actually in the BOM, their presence or absence in the final product depends on particular configuration choices, which must themselves be forecast. In many instances, therefore, it is often expedient to forecast demands for optional parts directly.
As Fig. 1 illustrates, the time series in the sample are of a fairly idiosyncratic nature—an impression corroborated by Fig. 2—which displays the marginal distribution of weekly demand across the entire sample. From the latter, it is clear that the bulk of the values in the series are positive integers between 0 and 4, with zero occurring quite frequently (in fact, weeks with 0 units of demand constitute approximately 40% of all the weeks in the sample). None of the parts experienced weekly demands in excess of nine units during the period of observation. McCabe and Martin (2005) refer to time series of this type as low-count series, distinguished in that low counts are poorly approximated by that staple of mainstream forecasting models, the normal distribution. (In contrast, series comprising larger count values are approximated much more felicitously.)
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Fig. 2.
Marginal distribution of demand.
From the preliminary investigations described in this paper, it would appear that in this application at least, the gamma-Poisson local level model is the superior forecaster, significantly improving upon the performance of the baseline model, and of the dynamic log-linear model, too. By contrast, ad hoc adjustment of a linear-normal state-space model produces forecasts that are inferior not only to the specialized state-space models, but to the baseline model as well. The latter results should sound a note of caution regarding the routine use of conventional exponential smoothing methods to forecast inventory demands of the sort examined here. The performance of the dynamic log-linear model (which barely improves upon the baseline) is somewhat disappointing; the prospect of employing a direct analog of generalized linear models in this application is an appealing one, and significant effort has been expended on their efficient Bayesian estimation (see de Jong and Shephard, 1995 and Durbin and Koopman, 2001, for example). We are investigating whether the performance of the PL model might be improved, by using a non-canonical link (rather than the exponential used here) or by imposing an informative prior on the evolution variance, to reduce the right tails of the model's predictive distributions.
As we noted in the previous section, all of the candidate models tend to produce forecasts in excess of the actual demands. Two explanations for this tendency—and consequent remedies—suggest themselves: (1) though Fig. 2 shows that all demands are bounded to 9 units or less, all of the state-space models considered here have essentially unbounded level components. We are currently working to establish whether models with finite discrete state spaces (McDonald and Zucchini, 1997) produce less inflated forecasts in situations where a definite bound can be placed on all future demands. (2) Case-by-case examination of the forecasts made during the comparison exercise indicates that all of the models fail to anticipate phases of a part's lifecycle—particularly the diminution in demand that accompanies phase-out. Unfortunately, given the highly decentralized nature of Sun's operations, basic facts such as part end-of-life dates are rarely collected together in a systematic fashion, making it far from trivial to predict the commencement of a part's phase-out. We are, however, exploring more sophisticated model structures built on the work described in Yelland (2004), to see if it is possible statistically either to predict or rapidly detect transitions in part lifecycles.
Finally, though we observed that long periods without demand occur relatively rarely in the sample series, the distinction between “low-count” and “intermittent” demands is not a sharp one, and it is quite reasonable to consider inclusion of forecast model components specifically to address demand intermittance. Since the latter normally center on estimating the length of breaks in demand, and since such breaks are themselves likely to be low counts in many instances, it is also possible that the results of our investigations here might be useful in the development of forecasting techniques for intermittent demand, too.
