Organizations with large-scale inventory systems typically have a large proportion of items for which demand is intermittent and low volume. We examine various different approaches to demand forecasting for such products, paying particular attention to the need for inventory planning over a multi-period lead-time when the underlying process may be non-stationary. This emphasis leads to the consideration of prediction distributions for processes with time-dependent parameters. A wide range of possible distributions could be considered, but we focus upon the Poisson (as a widely used benchmark), the negative binomial (as a popular extension of the Poisson), and a hurdle shifted Poisson (which retains Croston’s notion of a Bernoulli process for the occurrence of active demand periods). We also develop performance measures which are related to the entire prediction distribution, rather than focusing exclusively upon point predictions. The three models are compared using data on the monthly demand for 1046 automobile parts, provided by a US automobile manufacturer. We conclude that inventory planning should be based upon dynamic models using distributions that are more flexible than the traditional Poisson scheme.
Modern inventory control systems may involve thousands of items, many of which show very low levels of demand. Furthermore, such items may be requested only on an occasional basis. When events corresponding to positive demands occur only sporadically, we refer to the demand as intermittent. When the average size of a customer order is large, a continuous distribution is a suitable description, but when it is small, a discrete distribution is more appropriate.
In this paper, our interest focuses upon intermittent demand with low volume. On occasion, such stock keeping units (SKUs) may be of very high value, such as, for example, spare aircraft engines. However, even when the individual units are of low value, it is not unusual for them to represent a large percentage of the number of SKUs, so that they collectively represent an important element in the planning process. Johnston and Boylan (1996a, p. 121) cite an example where the average number of purchases of an item by a customer was 1.32 occasions per year, and “For the slower movers, the average number of purchases was only 1.06 per item [per] customer”. Similarly, in the study of car parts discussed in Section 6, out of 2509 series with complete records for 51 months, only 1046 had (a) ten or more months with positive demands, and (b) at least some positive demands in the first 15 and the last 15 months.
Demand forecasting for high volume products can be handled successfully using exponential smoothing methods, for which a voluminous body of literature exists; see for example Hyndman, Koehler, Ord, and Snyder (2008) and Ord, Koehler, and Snyder (1997). When volumes are low, the exponential smoothing framework must be based upon a distribution that describes count data, rather than the normal distribution. Further, as was recently emphasized by Syntetos, Nikolopoulos, and Boylan (2010), it is not sufficient to look at point forecasts when making inventory decisions. Instead, they recommend the use of stock control metrics. We accept their viewpoint completely, but since such metrics depend upon the underlying prediction distribution, we have opted to work with such distributions directly. This choice is reinforced by the observation that prediction distributions are applicable to count problems beyond inventory control. Moreover, the information on costs and lead times required when using inventory criteria was not available for the data considered in Section 6.
The remainder of the paper is structured as follows. It begins in Section 2 with a review of the literature on forecasting intermittent demand. The focus here is on models that allow for both non-stationary and stationary features. For example, the demand for spare parts may increase over time as the machines age and then decline as they fail completely or are withdrawn from service. In Section 3, we summarize the different models which will be considered in the empirical analysis and examine how they might be estimated and how they might be used to simulate various prediction distributions. Since our particular focus is on the ability of a model to furnish the entire prediction distribution, not just point forecasts, we examine suitable performance criteria in Section 4. Issues relating to model selection are examined briefly in Section 5. In Section 6 we present an empirical study using data on the monthly demand for 1046 automobile parts. Then, in Section 7, we examine the links between forecasting and management decision making, with an illustration of the use of prediction distributions in inventory management. Finally, various conclusions from our research are summarized briefly in Section 8.
In this paper we have introduced some new models for forecasting intermittent demand time series based on a variety of count probability distributions, coupled with a variety of dynamic specifications to account for potential serial correlation. These models were then compared with established forecasting procedures using a database of car parts demands. Particular emphasis was placed on prediction distributions from these models rather than point forecasts, because the latter ignore features such as variability and skewness, which can be important for safety stock determination.
The empirical results suggest that although many series may be able to be modeled adequately using traditional static schemes, substantial gains may be achievable by using dynamic versions for many of the others. A similar argument favors the use of richer models than the Poisson. Thus, an effective forecasting framework for SKUs that have low volume intermittent demands must look beyond the traditional static Poisson format.
Our study has indicated that simple exponential smoothing can work well in conjunction with an unrestricted negative binomial distribution. It has also indicated that there is little advantage in using a multi-model approach with information criteria for model selection. The usual caveat that such results are potentially data dependent must be made. Nevertheless, we expect that similar results would emerge for other datasets. This being the case, it appears that the Croston method, even in an adapted form, should possibly be replaced by exponential smoothing, coupled with a negative binomial distribution.
There are a number of series, most of which were excluded from the sample of 1046 SKUs used here, for which the demand is very low, perhaps in the order of one or two units per year. In such cases, a static model might be preferable, although from the stock control perspective the decision will often lie between holding one unit of stock and holding zero stock and submitting replenishment orders as needed.
There still remains the important issue of what to do with new parts which have no or only limited demand data. In such cases, maximum likelihood methods applied to single series are going to be ineffective. This question is the subject of an on-going investigation into the forecasting of demand for slow moving inventories, which explores the possibility of extending the maximum likelihood principle to multiple time series in a quest to overcome the paucity of data.
