Читать книгу Intermittent Demand Forecasting - John E. Boylan - Страница 84

To calculate the unit fill rate (FR), we need to evaluate how much demand is satisfied, over a given period of time, as a proportion of the demand over the same period. Equivalently, we can evaluate the average unsatisfied demand per period, as a proportion of the average demand per period, and subtract this ratio from one, as shown in Eq. (3.2).

(3.2)

where represents the stock on hand at the start of time period , after receipt of any orders and dispatch of any outstanding backorders, is demand during time period , and is the number of time periods over which the fill rate is being measured. The superscript indicates a result of zero if the expression in the brackets is negative, and unchanged otherwise. For example, and .

In Eq. (3.2), the expression represents the backorders generated at the end of period as a consequence of demand in that period not being satisfied. If there is sufficient stock ( less than or equal to ), then there are no backorders. If there is insufficient stock ( strictly greater than ), then units are backordered. In the numerator of the ratio in Eq. (3.2), the backorders are summed over all periods and divided by the number of periods () to give the average unsatisfied demand per period. In the denominator, we have the average demand per period. The ratio represents the average unsatisfied demand per period as a proportion of the average demand per period.

For ease of exposition, from this point on, we assume that the review interval is one period (). At the end of this section, we return to the more general case when it can be longer.

Equation (3.2) is an exact calculation of the fill rate. It can be used to find the historical fill rates, providing that we maintain records of the demands in each period, and the backorders in each period. Suppose that, for a particular SKU, the average demand per period was for 10 units and the average backorder quantity per period was one unit. Then, it follows immediately that the historical fill rate for that SKU was .

Now suppose that we wish to experiment with different OUT levels. Equation (3.2) is not helpful because, even if the average demand per period remains unchanged, the equation does not reveal the effect on backorders of changing the OUT level (). To assess the effect of different OUT levels () on the fill rate (), the formula given in Eq. (3.3) is often used.

(3.3)

where is the demand over the protection interval () and is the long‐run average demand per (single) period. We shall refer to this as the traditional fill rate calculation.

As we shall see later, this traditional fill rate calculation suffers from some drawbacks, whether demand is intermittent or not. However, it is often used in practice, and so it is important to understand its calculation, including its flaws and how they can be rectified.

To illustrate the traditional calculation of fill rates, we now look at another example. The review interval is set as one week and the lead time as two weeks. The distribution of demand is similar to the example in Table 3.2, except that it is lumpier, with a spike of demand at four units, as shown in Table 3.7.

Table 3.7 Distribution of lumpy demand over one week.

Demand	Probability
0	0.5
1	0.3
4	0.2
5 or more	0.0

Table 3.8 Traditional fill rate calculation ( and ; ).

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, , 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.