John E. Boylan - Intermittent Demand Forecasting

In this chapter, we make two important assumptions, which correspond to the representation of chance events by a physical or virtual die:

1 Independent: The probability of demand in one period does not depend on the demand in previous periods (as each roll of the die is independent of previous rolls). This may not always be true in practice if ‘streaks’ of non‐zero demands are observed more frequently than would be expected if demands were truly independent.

2 Identically distributed: The probabilities are not changing over time (as the faces on the die do not change). In practice, it is possible that the chance of a zero demand may decrease or increase over time. For example, as original equipment is withdrawn from production, the demand for spares will eventually decline, leading to a higher chance of zero demand.

For the remainder of this chapter, these two assumptions are maintained. In Chapters 6and 7, we look at situations where demand is not identically distributed over time. In Chapters 13and 14, we examine non‐independent demand processes leading to streaks of demand.

In addition to the two assumptions of independence and identical distribution over time, we also make a third assumption:

1 Non‐negative: Demand cannot be negative, although it can be zero.

This is a natural assumption for demand itself, although it is not appropriate for ‘net demand’, which is found by subtracting returns from demand (Kelle and Silver 1989). This is relevant in closed‐loop supply chains where items are returned for refurbishment or remanufacturing.

In Chapter 2, we indicated why the distribution of demand over the whole protection interval ( ) is needed to determine OUT levels in periodic review systems. To recap, suppose that the stock on hand is at the OUT level just after a review and no order is triggered. In that case, the stock must last not just until the time of the next review (an interval of R time units), but until any stock is received after that review. This necessitates a further delay of L time units, to allow for the supplier's lead time. Care is needed in counting the length of the lead time. The use of an protection interval assumes that an order placed at the end of period arrives in time to satisfy demands of period . If it arrives in time to satisfy the demands of period , then the effective lead time is and, for review intervals of length one period, the protection interval is of length rather than (Teunter and Duncan 2009).

Suppose that the demand distribution in Table 3.2accurately represents the probabilities of future demand values over a single week, and that demand is independent and identically distributed. We are using a periodic review system, with a review interval of one week and a lead time of one week. Therefore, the protection interval is two weeks, and the distribution of demand over two weeks is shown in Table 3.3.

Table 3.3 Probability distribution of total demand over two weeks.

Total	Week 1	Week 2	Week 1	Week 2	Product	Total
0	0	0	0.5	0.5	0.25	0.25
1	1	0	0.3	0.5	0.15
	0	1	0.5	0.3	0.15	0.30
2	2	0	0.2	0.5	0.10
	1	1	0.3	0.3	0.09
	0	2	0.5	0.2	0.10	0.29
3	2	1	0.2	0.3	0.06
	1	2	0.3	0.2	0.06	0.12
4	2	2	0.2	0.2	0.04	0.04

Table 3.4 Cumulative distribution of total demand over two weeks.

In Table 3.3, denotes demand over the review interval (Week 1), demand over the lead time (Week 2), and 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.