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

### Intended learning outcomes: Produce an overview on and examples of the relation between fill rate and service level.

Continuation from previous subsection (11.3.4b)

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 ROP (Re)-Order Point Technique, and Safety Stock Calculation

Intended learning outcomes: Explain the (re-)order point technique and variants thereof. Describe the safety stock calculation with continuous demand. Disclose the determination of the service level and the relation of service level to fill rate.

• ##### 11.3.1 The ROP (Re)-Order Point Technique

Intended learning outcomes: Present in detail characteristic data for the (re-)order point technique.

• ##### 11.3.1b Order Point Calculation

Intended learning outcomes: Explain the (re-)order point calculation. Identify the criterion for the release of a production or procurement order.

• ##### 11.3.2 Variants of the Order Point Technique

Intended learning outcomes: Identify the criterion for the release of a production or procurement order, if the customer allows a minimum delivery lead time. Explain the criterion for an early issuance of a production or procurement order.

• ##### 11.3.2b The Min-Max Reorder System and the Double Order Point System

Intended learning outcomes: Produce an overview on the min-max (reorder) system. Describe the double order point system.

• ##### 11.3.3 Safety Stock Calculation with Continuous Demand

Intended learning outcomes: Describe different techniques for determining safety stock. Identify different patterns of the deviation of demand from forecast.

• ##### 11.3.3b Service Level, Safety Factor, and Service Function

Intended learning outcomes: Explain safety stock in relation to service level. Identify the safety factor and the service function.

• ##### 11.3.3c Safety Stock Calculation with Continuous Demand Following a Normal Distribution

Intended learning outcomes: Disclose the normal integral distribution function (service function) to determine the safety factor that corresponds to a desired service level. Present the formula for safety stock.

• ##### 11.3.3d Safety Stock Calculation with Continuous Demand Following a Poisson Distribution

Intended learning outcomes: Disclose the Poisson distribution integral function to determine the safety factor that corresponds to a desired service level.

• ##### 11.3.4 Determining the Service Level on the Basis of Stockout Costs

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.

• ##### 11.3.4b Determining the Relation of Service Level to Stockout Quantity per Order Cycle

Intended learning outcomes: 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.

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

Intended learning outcomes: Produce an overview on and examples of the relation between fill rate and service level.