Intermittent Demand Forecasting. John E. Boylan

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upper R plus upper L"/> demand over the protection interval (Weeks 1 and 2). Probability is denoted by the symbol .

      To find the probabilities in Table 3.3, we first consider all the possible ways of achieving the demand values over two weeks. For example, a total demand of two can be achieved in three ways (two in Week 1, zero in Week 2; or one in Week 1, one in Week 2; or zero in Week 1 and two in Week 2). The full listings are given in the second and third columns of Table 3.3.

The probabilities in the fourth and fifth columns are taken directly from Table 3.2. The product of these probabilities in the sixth column represents the chance of a particular sequence of demands in Weeks 1 and 2 (assuming independence of demands). In the final column, the probabilities are summed appropriately, for each potential value of total demand over two weeks. For example, the probability of having a total demand of two is the sum of 0.10, 0.09, and 0.10, giving a value of 0.29. Now that the probabilities of demand over two weeks have been calculated, we can find the cumulative probability distribution, which represents the probabilities of observing particular demand values, or less than those values, as shown in Table 3.4.

The cumulative distribution of total demand over the protection interval is often used as an approximation to the cycle service level in inventory systems (Cardós and Babiloni 2011). (The reasons why this is approximate, rather than exact, will be explained in Section 3.5.3.) Therefore, the approximate service levels may be read directly from the cumulative probabilities in Table 3.4, which shows a CSL of 84% if the OUT level is set at two units and a CSL of 96% if the OUT level is set at three units. So, knowledge of all the probabilities of demand (the demand distribution) is sufficient to allow the calculation of the approximate CSL for different OUT levels if demand is independent and identically distributed.

      3.5.3 Cycle Service Levels Based on Cycles with Demand

The CSL measure is appropriate for non‐intermittent demand but suffers from some drawbacks for intermittent demand. Cardós et al. (2006) gave the example of a product, without any stock at all, facing a demand once every 10 weeks. Even with no stock, there would be no stockouts in 90% of the weeks because there is no demand in 90% of the weeks. If the replenishment cycle is one week, then this would mean that the CSL is 90%. However, as the authors commented, with no stock, there is no service!

      Examples such as this motivate the development of a revised method of calculating the cycle service level for intermittent demand items. Instead of looking at all replenishment cycles, we focus on those replenishment cycles that contain some demand. In this book, the revised cycle service level will be denoted as , where the plus sign indicates restriction to review intervals with a positive (non‐zero) demand. Cardós et al. (2006) argued that the usual calculation of demand probabilities would require amendment to recognise this restriction. Teunter and Duncan (2009) also took this into account in their non‐parametric analysis, to be discussed in Chapter 13. Finding will require separate evaluation of:

      1 The probability of demand over the review interval, conditional on this demand being strictly positive.

      2 The (unconditional) probability of demand over the lead time.

These calculations are illustrated in Table 3.5.

      The notation in Table 3.5 is the same as in Table 3.3. The probability in the fifth column is unconditional, and may be written as , as there is no restriction on demand during lead time. The probabilities in the fourth and final columns are conditional, as they are subject to the condition that is strictly positive (), and are written as and , where the vertical lines indicate ‘subject to the condition that’.

      In Table 3.5, the review interval (Week 1) and the lead time (Week 2) are considered separately. In the second column, we exclude the possibility of zero demand, as the measure is restricted to review intervals with non‐zero demand. The values in the fourth column have also changed. In the long run, only half of the weeks have non‐zero demands, according to Table 3.2. Therefore, although only 30% of weeks have a demand of one unit, 60% of weeks with some demand have a demand of one unit. Similarly, 40% of weeks with some demand have a demand of two units. The fifth column uses the same probabilities as in Table 3.2 because the possibility of zero demand is not excluded in Week 2. The final two columns in Table 3.5 are calculated in the same way as in Table 3.3.

Table 3.5 Distribution of total demand over two weeks conditional on non‐zero demand in first week.

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Total	Week 1	Week 2	Week 1 Probability	Week 2 Probability	Product	Total Probability
1	1	0	0.6	0.5	0.30