3.8 show how much demand would not be satisfied for the specified OUT levels. For example, for an OUT level of seven units, a demand of six units can be fully satisfied, but demands of eight, nine, or twelve units will be only partly satisfied, with unsatisfied demand of one, two, and five units, respectively.
The fourth and sixth columns contain the expected shortages, corresponding to different demand values. These expected shortages are calculated by multiplying the number of items not satisfied (third and fifth columns) by the probability of demand over the protection interval (second column). The values in the fourth and sixth columns are summed to give the total expected shortages for OUT levels of seven and eight units.
The final calculation of fill rates uses Eq. (3.3). The mean value, , is found as a weighted average of the probabilities of demand in a single period (see Table 3.7). The calculation is: . This value, and the overall values for expected unsatisfied demand per period are substituted into Eq. (3.3) to give the fill rates of 84.4% and 93.8% for OUT levels of seven and eight units, respectively.
3.6.3 Fill Rates: Sobel's Formula
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 3.7 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.
