Abstract
The categorization of alternative demand patterns facilitates the selection of a forecasting method and it is an essential element of many inventory control software packages. The common practice in the inventory control software industry is to arbitrarily categorize those demand patterns and then proceed to select an estimation procedure and optimize the forecast parameters. Alternatively, forecasting methods can be directly compared, based on some theoretically quantified error measure, for the purpose of establishing regions of superior performance and then define the demand patterns based on the results. It is this approach that is discussed in this paper and its application is demonstrated by considering EWMA, Croston's method and an alternative to Croston's estimator developed by the first two authors of this paper. Comparison results are based on a theoretical analysis of the mean square error due to its mathematically tractable nature. The categorization rules proposed are expressed in terms of the average inter-demand interval and the squared coefficient of variation of demand sizes. The validity of the results is tested on 3000 real-intermittent demand data series coming from the automotive industry.
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Appendix A: Croston's method, Syntetos and Boylan method
Appendix A: Croston's method, Syntetos and Boylan method
Croston,1 as corrected by Rao,15 proved the inappropriateness of exponential smoothing and he also expressed, in a quantitative form, the bias associated with the use of this method, when dealing with intermittent demands.
Moreover, assuming a stochastic model of arrival and size of demand, Croston introduced a new method for characterising the demand per period by modelling the demand in one period from constituent events.
According to his method, separate exponential smoothing estimates of the average size of the demand and the average interval between demand incidences are made after demand occurs. If no demand occurs the estimates remain exactly the same.
Let: 1/p t be the Bernoulli probability of demand occurring in period t; p t the geometrically distributed (including the first success, ie demand occurring period) inter-demand interval; p t ′ the exponentially smoothed inter-demand interval, updated only if demand occurs in period t, E(p t )=E(p t ′)=p; z t the normally distributed demand size, when demand occurs; z t ′ the exponentially smoothed size of demand, updated only if demand occurs in period t, E(z t )=E(z t ′)=μ; α the common smoothing constant value used and Y t the demand in unit time period t.
Under these conditions the expected demand per unit time period is E(Y t )=μ/p. Following Croston's estimation procedure, the forecast, Y t ′ for the next time period is given by: Y t ′=z t ′/p t ′ and, according to Croston, the expected estimate of demand per period in that case would be: E(Y t ′)=E(z t ′/p t ′)=E(z t ′)/E(p t ′)=μ/p (ie the method is unbiased).
The variance of the forecasts produced by this method is calculated by Croston as follows:
Syntetos and Boylan11 showed that Croston's method is biased and that the expected estimate of demand per unit time period is not as calculated by Croston, but rather
Moreover, the sampling error of the mean was also found to be different:
The researchers proposed a new estimator that takes into account the smoothing constant value used and which is the following:
The expected estimate of demand per unit time period as well as the variance of the estimates produced by this method are given by (A.5) and (A.6), respectively
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Syntetos, A., Boylan, J. & Croston, J. On the categorization of demand patterns. J Oper Res Soc 56, 495–503 (2005). https://doi.org/10.1057/palgrave.jors.2601841
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DOI: https://doi.org/10.1057/palgrave.jors.2601841