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 3.7 shows 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:
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.
3.8.
