Table 3.8 Traditional fill rate calculation (
| Demand | Probability | Not satisfied | Expected | Not satisfied | Expected |
|---|---|---|---|---|---|
|
|
|
|
|
|
|
| 1 | 0.225 | 0 | 0.000 | 0 | 0.000 |
| 2 | 0.135 | 0 | 0.000 | 0 | 0.000 |
| 3 | 0.027 | 0 | 0.000 | 0 | 0.000 |
| 4 | 0.150 | 0 | 0.000 | 0 | 0.000 |
| 5 | 0.180 | 0 | 0.000 | 0 | 0.000 |
| 6 | 0.054 | 0 | 0.000 | 0 | 0.000 |
| 8 | 0.060 | 1 | 0.060 | 0 | 0.000 |
| 9 | 0.036 | 2 | 0.072 | 1 | 0.036 |
| 12 | 0.008 | 5 | 0.040 | 4 | 0.032 |
| Total | 0.172 | Total | 0.068 | ||
|
Fill rate ( |
84.4% |
Fill rate ( |
93.8% |
Continuing this example, we summarise in Table 3.8 the probabilities of demand over a protection interval of three weeks (first two columns) and traditional fill rate calculations for OUT levels of seven units (middle two columns) and eight units (final two columns).
The first column of Table 3.8 lists all of the possible total demands over three weeks, given possible demands of zero, one, and four units in one week. The possibility of zero demand over the whole three weeks has not been included. It is not relevant from a fill rate perspective because there is no demand to be fulfilled. Some demand values are omitted, such as seven, as there is no combination of three weeks of demand, in this example, that can give this number. The detailed calculations for the second column are not given but they follow exactly the same approach as in Table 3.3, where all combinations of demands are identified, and probabilities are calculated accordingly.
The third and fifth columns of Table 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,
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
