Intermittent Demand Forecasting. John E. Boylan

periods with each period showing some demand, as shown in Table 3.6.

      How should we define the overall fill rate? There are two possible approaches. The first is to total the satisfied demand (8 units) and divide by the total demand (16 units) to give an overall fill rate of 50%, as shown in Table 3.6. The second approach is to average the fill rates over all four periods, giving an overall fill rate of 56.25%, which is somewhat higher than the first calculation. If all four periods had a fill rate of 50%, then the two calculation methods would agree. The disagreement arises because the average of the fill rates in the second and third periods is 62.5%, whereas only 50% of the total demand over these two periods is fulfilled. The second method can be applied to intermittent demand only if periods with zero demands are excluded from the calculation. For further discussion of this method, please refer to Guijarro et al. (2012). We will base our analysis on the first method, as it is simpler for intermittent demand, and is the standard method in the literature and in practice.

Table 3.6 Fill rates per time period.

Period	Demand	Satisfied	Fill rate
1	2	1	0.50
2	4	1	0.25
3	2	2	1.00
4	8	4	0.50
Total	16	8	0.50

      3.6.2 Fill Rates: Standard Formula

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)