# 11.3.4 Determining the Service Level and the Relation of Service Level to Fill Rate

### Intended learning outcomes: Describe the order point technique where the length of order cycle provided by the batch size is a multiple of the lead time. Explain the probability of stockout in dependency on stockout costs per unit. Present the service function (of the stockout quantity coefficient) P(s) in dependency upon the safety factor s. Produce an overview on and examples of the relation between fill rate and service level.

Figure 11.3.4.1 shows a typical order cycle using the order point technique shown in Figure 11.3.1.1, in which the length of order cycle, that is, the length of time the batch size will provide stockout coverage, is a multiple of the lead time. The batch size itself is a multiple of the expected demand during the lead time.

Fig. 11.3.4.1       Order point technique with an order cycle where the length of order cycle provided by the batch size is a multiple of the lead time.

If the length of order cycle divided by lead time equals 10, for example, and demand is not too discontinuous, then 90% of the batch size can be covered without stockout. Stockout will only occur for demand during the lead time, or for 10% of the batch size. If no safety stock were carried (safety factor is 0, that is, a service level of only 50%), the fill rate would be approximately 90% and higher. This shows that service level can usually be a percentage that is significantly smaller than the desired fill rate (which in most cases must be set at close to 100%; see the discussion in Section 5.3.1).

As mentioned above, determining the desired fill rate and service level has to be the quantitative application of the qualitative answer to the question of what stockouts will cost. Thus, fill rate and service level express an estimation of opportunity cost.

Stockout costs are the economic consequences of stockouts.

Stockout costs can include extra costs for express/emergency production or procurement or customer delivery, but also penalty costs, loss of sales, loss of contribution margin, loss of customer goodwill, and all kinds of associated costs. See the discussion in Section 1.3.1.

The following shows the derivation of two methods of determining the desired service level:

1. The first method is based on the assumption that opportunity costs can be assigned directly to each unit not filled.
2. The second method is based on the assumption that the total opportunity costs can be assigned to the fill rate during a particular time period (a year, for instance).

1. Determine service level on the basis of stockout costs for each unit of an item not filled.

Where stockout costs can be expressed as costs per unit (of measure) not delivered, [Cole00], [SiPy98], and [Ters93] offer the following direct calculation of the optimum probability of stockout (see Figure 11.3.4.2). Because a stockout can only happen at the end of an order cycle, the number of stockouts cannot be greater than the number of order cycles. Often the period chosen for the calculation is one year.

Fig. 11.3.4.2       Probability of stockout in dependency on stockout costs per unit.

As a consequence, the optimum service level results directly from the relation in Figure 11.3.3.3. Section 11.4 discusses determination of batch size, which often precedes safety stock calculation.

For example, if there are five order cycles per year (the average annual consumption is five times the batch size) and stockout costs per unit are four times greater than carrying cost, the resulting optimum probability of stockout is 0.05 and the optimum service level is 95%.[note 1104]

2. Determine service level on the basis of fill rate.

If a certain stockout percentage or backorder percentage has been set on the basis of estimated annual stockout costs, then the service level can be derived from the fill rate by estimating the stockout quantity per order cycle. See also [Brow67] and [Stev14].

For a particular safety factor, from now on called s, the stockout quantity is the product of all possible not-filled quantities and their probability of occurrence. A specific not-filled quantity is the quantity m, which exceeds the expected quantity of demand plus s times the standard deviation of demand during the lead time. Proportional to the standard deviation, this quantity can be expressed as (t–s) times the standard deviation s, for each t  ≥  s. p(t) is then, for example, the normal probability density function. Instead of the quantity, the factor of proportionality with its probability of occurrence yields the stockout quantity coefficient.[note 1105]

The stockout quantity coefficient P(s) is the factor that, multiplied by the standard deviation of demand per lead time, yields the expected stockout quantity in dependency on the safety factor s.

The formula for the stockout quantity coefficient in Figure 11.3.4.3 is similar to the formula in Figure 11.3.3.5. P(s) is the integral, for all possible t  ≥  s, of the factor of proportionality (t – s) of the standard deviation of demand during lead time multiplied with p(t).

Fig. 11.3.4.3       Service function (of the stockout quantity coefficient) P(s) in dependency upon the safety factor s.

Figure 11.3.4.4 shows examples of corresponding values of safety factor s and stockout quantity coefficient P(s). The values can be determined by table look-up; see, for example, tables in [Brow67], p. 110, or [Stev14].

Fig. 11.3.4.4       Safety factor s and stockout quantity coefficient P(s) with normally distributed demand. (Following [Brow67] or [Stev14].)

Thus, the expected stockout quantity per order cycle can be calculated from safety factor s via the stockout quantity coefficient P(s).

According to the definition in Section 5.3.1, the stockout quantity per order cycle is also the product of batch size and stockout percentage (that is, the complement of fill rate). This yields formulas, shown in Figure 11.3.4.5, that relate service level to fill rate.

Fig. 11.3.4.5       Relation between fill rate and service level.

Let us look at an example that illustrates the relation between fill rate and service level.

Say the batch size is 100 units, and the standard deviation of demand during the lead time is 10 units. What safety stock should be carried to provide a desired fill rate of 99.9%? The stockout quantity coefficient P(s) is 0.01 (Figure 11.3.4.5), and the safety factor is thus 1.92 (Figure 11.3.4.4). Therefore, the resulting safety stock is 1.92 times 10 = 19.2 units (Figure 11.3.3.7).[note 1106]

Figure 11.3.4.6 shows that the quotient resulting from the standard deviation of demand during lead time divided by batch size (following Figure 11.3.4.5) has a leverage between service level and fill rate. The smaller this quotient is, the higher — at a constant service level — the expected fill rate. That means that with a service level of 50% (that is, no safety stock) and a quotient of 1/5, a fill rate of more than 92% is achieved, while with a quotient of 1/10 (as in the example above), the fill rate achieved is about 96%. With a service level of 80%, a quotient of 1/10 results in a fill rate of over 98.8%.

Fig. 11.3.4.6       Examples of the relation between service level and fill rate.

And finally, consider an example that links stockout costs per unit, via the optimal service level derived using method 1 above, with the fill rate calculated with method 2 above.

In this example, annual carrying cost per unit is 1, the batch size is 100, average annual demand is 500, and the standard deviation of demand during the lead time is 10. What is the expected fill rate based on the given carrying cost per unit of 4? The optimum probability of stockout in each order cycle is 0.05 (Figure 11.3.4.2), which results in an optimum service level of 95% following Figure 11.3.3. Following Figure 11.3.4.4, this corresponds to the stockout quantity coefficient P(s) = 0.021. Following Figure 11.3.4.5, this yields a fill rate of 99.79%.

According to the formulas in both method 1 and method 2 above for calculating the desired service level, the service level and safety stock both decrease with increasing batch size. For this reason, it would be desirable to set the batch size as large as possible. For production orders in particular, however, as Chapter 13 will show, the cumulative lead time often grows disproportionately as batch size increases, making it necessary to apply stochastic models of demand and to include the standard deviation. From this perspective, a small batch size is desirable. In practice, then, batch sizes and safety stock must be determined simultaneously (de facto in iteration).

## Course section 11.3: Subsections and their intended learning outcomes

• ##### 11.3.4 Determining the Service Level and the Relation of Service Level to Fill Rate

Intended learning outcomes: Describe the order point technique where the length of order cycle provided by the batch size is a multiple of the lead time. Explain the probability of stockout in dependency on stockout costs per unit. Present the service function (of the stockout quantity coefficient) P(s) in dependency upon the safety factor s. Produce an overview on and examples of the relation between fill rate and service level.