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

*.*

*the safety factor*that corresponds to a desired*service level**Continuation from previous subsection (11.3.3c)*

In particular for small demand quantities, we cannot always assume that demand is normally distributed. Sometimes, we could assume a Poisson distribution instead. However, with a mean value (average demand quantity) of merely 9 units, the upper part of the Poisson distribution curve is very close to the curve of the normal distribution. This is particularly true for larger safety factors and high service levels. See also Figure 10.5.5.1.

Figure 11.3.3.8 shows an example of the *Poisson distribution* and its integral function. Depending upon the mean value λ, a different curve and a different inverse function will result.

**Fig. 11.3.3.8** Poisson distribution integral function.

Figures 11.3.3.9 and 11.3.3.10 show pairs of values of the service level and safety factor for means of λ = 4 and λ = 9, respectively.

For small consumption quantities, the cost of a stockout often does not depend so much on the quantity not delivered as upon the fact that there is a failure to meet the full quantity. Thus, with small usage quantities the tendency is to choose a high service level, which in turn results in a high safety factor. The calculated safety factor that is based on a Poisson distribution is then generally fairly equivalent to the one based on a normal distribution.

**Fig. 11.3.3.9** Table of values for the Poisson cumulative distribution with a mean demand value of λ = 4 and standard deviation SQRT(λ) = 2 units per period. (From [Eilo62], p. 84 ff.)

**Fig. 11.3.3.10** Table of values for the Poisson cumulative distribution with a mean demand value of λ = 9 and standard deviation SQRT(λ) = 3 units per period. (From [Eilo62], p. 84 ff.)

However, based on *probability* of stockout alone, we cannot say anything about the stockout *quantity*, the stockout *percentage*, or backorder *percent-age*. Thus, service level is not the same as *fill rate*, which only measures what actually happens when demand occurs. See also [Bern99], [Chap06].

Like fill rate (see the definition in Section 5.3.1), service level is the quantitative application of the answer to the following question: What are the costs of not meeting customer demands from stock? Both measures, fill rate and service level, are thus estimates of opportunity cost. To achieve a specific fill rate, however, it is generally sufficient to set a smaller number as the service level, or desired probability that demand can be met from stock. The relationship between the two measures, fill rate and service level, as well as ways of determining the appropriate service level, are examined in Section 11.3.4.

## 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.