# 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.

Continuation from previous subsection (11.3.4)

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 [Stev21].

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 [Stev21].

Fig. 11.3.4.4       Safety factor s and stockout quantity coefficient P(s) with normally distributed demand. (Following [Brow67] or [Stev21].)

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

Continuation in next subsection (11.3.4c).

## 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.