1. ¹University of Salford
2. ²Buckinghamshire Chilterns University College
3. ³Independent Systems and Statistical Consultant

We thank AV Kostenko and RJ Hyndman for their comments and constructive criticism on our paper published last year in JORS (Syntetos et al, 2005), where a demand categorization scheme was developed to facilitate the selection of forecasting methods.

As we pointed out in our paper, demand categorization has received very limited attention in the academic literature despite its importance for many inventory management computerized applications. It is pleasing, therefore, that Kostenko and Hyndman (KH) have developed an intuitively appealing extension of our work. In fact, the theoretically coherent delineation of the 'smooth' demand region was identified as an area of practical importance and worthy of further research by Syntetos (2001). He considered some possible forms of the 'optimal lines' for delineating the regions (see Figure 1), but without analyzing their functional equations further.

Figure 1.

Delineation of the 'smooth' demand quadrant (Syntetos, 2001, p 316).

Full figure and legend (36K)

We expect that the new rule proposed by KH will yield greater forecast accuracy than the original rule. Simulations are required on theoretically generated and empirical data in order to confirm this. In a more recent research project (Boylan et al, 2006), we have assessed the empirical forecasting and stock control performance of the four quadrants approach to categorization by considering a real inventory management software package. Our study has verified the choice of the categorization variables previously proposed based on theoretical derivations. Both the average inter-demand interval (order frequency) and the squared coefficient of variation of the demand sizes have been shown to capture performance differences effectively, albeit with cutoff values not matching those previously suggested. In a similar way, the parameters of the linear function derived by KH should be validated on real data, before being employed by practitioners.

In fact, caution should be exercised in using the limiting value for CV², since the inequality (1) cited by KH is the result of an approximate rather than exact analysis, with the approximation becoming less accurate as p approaches unity. Turning to the average inter-demand interval, p, KH stated that 'the limiting value of p is obtained when =0 and =0 giving p=4/3 (not 1.32 as given by SBC)'. This implies that our objective was to identify the limiting value of the categorization parameter under concern, which was not the case. Our analysis was restricted to a minimum realistic value equal to 0.05, thus explaining the small discrepancy between the p cutoff points. The same is true for the other categorization parameter.

KH discussed the demand independence assumption underlying our research (Syntetos et al, 2005) suggesting that such an assumption is never true. Nevertheless, there is empirical evidence in favour of demand independence for some intermittent series (eg Willemain et al, 1994). Moreover, KH stated that if demand is independent then the historical mean is the best predictor. For a stationary mean, and considering all points in time, this is true, but in a re-order level context (ie, utilization of forecasts at issue points only), the overall mean produces a biased forecast. This bias may be substantial for short demand series.

KH emphasized that neither Croston's method (Croston, 1972) nor any related estimators (eg, the Syntetos–Boylan Approximation; Syntetos and Boylan, 2005) are optimal for any demand model. Consequently, they suggested that a useful line of research may be to find forecasting methods that are optimal for a realistic model. We certainly agree that such an approach may be fruitful. Nevertheless, one needs to consider the following: a method that is optimal for one particular model may be severely sub-optimal for another model. We believe that, for intermittent demand, robustness of a method across a wide range of possible underlying demand models is more important than optimality under one particular model. The very scarcity of demand observations presents a significant inherent difficulty in identifying the demand model. In our view, the best approach to this problem would bring together consideration of optimality, as advocated by Kostenko and Hyndman, with examination of robustness.