John E. Boylan - Intermittent Demand Forecasting

Johnson et al. (1995) pointed out that the traditional fill rate calculation can suffer from double counting. This arises from the same shortage being counted in two separate periods. To appreciate how this happens, we continue with our example in Table 3.8.

Let us look again at the results corresponding to a demand of 12 (over three weeks), shown in the bottom row of the middle section of Table 3.8. For , the traditional formula gives a shortage of five units at the end of the third week. However, this is not accurate. A total demand of 12 can have arisen only from a demand of four in each of the three weeks because the distribution in Table 3.7shows that four is the maximum weekly demand. Therefore, the demand in the first two weeks must have been for eight units, giving a shortage (and backorder) of one unit at the end of the second week if . The shortage of five units at the end of the third week is actually the sum of one unit backordered in the second week and a further four units backordered in the third week. To count this as five would be to double count the unit that was short in the second week and is still short in the third week.

More generally, the traditional fill rate formula is appropriate if there are no backorders at the end of periods (still assuming that ). However, if there are some backorders, then these should not be added on to any further backorders that may arise in the next period. This motivated the development of a revised formula, proposed by Sobel (2004), for calculating the fill rate when the review interval is of one period:

(3.4)

where is the demand over the lead time, is the demand in the single period just after the completion of the lead time, and the other notation is unchanged. This formula overcomes the problem of double counting if demand is always non‐negative, and is independent and identically distributed. To show how the formula works in practice, we recalculate the fill rate for using Sobel's formula, keeping the lead time as two weeks, as shown in Table 3.9.

Table 3.9 Sobel's fill rate calculation ( , , ).

Demand,	SOH	Demand,	Not satisfied	Probability )	Probability )	Expected not satisfied
0	7	1	0	0.25	0.3	0.000
0	7	4	0	0.25	0.2	0.000
1	6	1	0	0.30	0.3	0.000
1	6	4	0	0.30	0.2	0.000
2	5	1	0	0.09	0.3	0.000
2	5	4	0	0.09	0.2	0.000
4	3	1	0	0.20	0.3	0.000
4	3	4	1	0.20	0.2	0.040
5	2	1	0	0.12	0.3	0.000
5	2	4	2	0.12	0.2	0.048
8	0	1	1	0.04	0.3	0.012
8	0	4	4	0.04	0.2	0.032
					Total	0.132
					Fill rate ( )	88.0%

In Table 3.9, the first and third columns give all the potential combinations of demands during the lead time (two periods) and the review interval (one period), excluding the possibility of zero demand in the review interval, as this cannot yield unsatisfied demand in that period. The second column shows the stock on hand (SOH) for , after depletion of the lead time demand (zero if the lead time demand is for eight units).

The fourth column shows the unsatisfied demand, given the stock on hand in the second column and the review interval demand in the third column. Let us return to the case of a demand of eight units in the first two weeks and four units in the third week, shown in the bottom row of the middle section of the table. Now, the unsatisfied demand in the third period is shown as four units, avoiding double counting of the one unit of unsatisfied demand in the second week.

The probabilities in the fifth and sixth columns are multiplied together to give the chance of the combination of demand values in the lead time and the review interval (assuming independence of demands over time). This is then multiplied by the unsatisfied demand, in the fourth column, to give the expected unsatisfied demand. (The probabilities of demand over two weeks, shown in the fifth column, are calculated in the same way as for Table 3.3. The probabilities in the sixth column are taken directly from Table 3.7.) Finally, the overall expected unsatisfied demand per period of 0.132 and the mean demand of 1.1 per single time period are substituted into Eq. (3.4)to give a fill rate of 88.0%. As anticipated, Sobel's formula gives a higher fill rate than the traditional formula (84.4%) because it avoids the problem of double counting of backorders.