*Intended learning outcomes: Identify variables for a consumption distribution. Present expected value and variance of the consumption distribution. Explain distribution function, expected value, and variance of the consumption distribution under the assumption of a Poisson distribution for the frequency of events.*

The *distribution of forecast errors* is a tabulation of the forecast errors according to the frequency of occurrence of each error value (cf. [ASCM22]).

The errors in forecasting are, in many cases, normally distributed, even when the observed data do not come from a normal distribution. Therefore, we now take a closer look into the origin of the observed values.

A *consumption distribution*, such as a statistic for order receipts allocated by time periods, can be understood as an aggregation of multiple individual events during each period. These individual events can be described by:

- The *distribution of the frequency of the events* themselves

- A *distribution of characteristic values for an event*, that is, order quantities

A combination of these two distributions results in consumption distribution.

Given the definitions in Figure 10.5.2.1 and a constant process (e.g., for constant demand), the formulas contained in Figure 10.5.2.2 are valid according to [Fers64]. Here, E stands for the *expected value*; VAR stands for the *variance*.[note 1009]

**Fig. 10.5.2.1** Definitions for a consumption distribution.

**Fig. 10.5.2.2** Expected value and variance of the consumption distribution.

In a purely random process, the number of events per period has a Poisson distribution with distribution function P(n) and expected value = variance = λ. Knowing this, we can derive the formulas in Figure 10.5.2.3, where CV corresponds to the *coefficient of variation* for the distribution, that is, the quotient of standard deviation and expected value.

**Fig. 10.5.2.3** Distribution function, expected value, and variance of the consumption distribution under the assumption of a Poisson distribution for the frequency of events.

A few large issues can greatly influence the coefficient of variation for the *order quantity*. The square can very well take on a value of 3. If all issues are equally large, then the value is clearly at its minimum of 0 (e.g., the order quantity for service parts may always equal 1). Even if the measured values of the consumption distribution allowed, based on the rules of statistics, the assumption of a normal distribution as such, a coefficient of variation of CV ≤ 0.4 is a prerequisite for effective procedures in the stochastic materials management. From the formula in Figure 10.5.2.3, it is possible to say how many issues are necessary, so that such a small coefficient of variation results. Specifically, if 1 is assumed as the mean for the coefficient of variation of the distribution of the *order quantity*, then at least 12.5 orders or issues per period are needed, which can be high for a machine manufacturer (λ ≥ (1+1) / 0.16 = 12.5).

*Continuation in next subsection (10.5.2b).*

## Course section 10.5: Subsections and their intended learning outcomes

##### 10.5 Using Forecasts in Planning

Intended learning outcomes: Produce an overview on the choice of suitable forecasting technique. Describe consumption distributions and their limits, continuous and discontinuous demand. Explain demand forecasting of variants of a product family. Present safety demand calculation for various planning periods. Disclose the translation of forecast into quasi-deterministic demand.

##### 10.5.1 Comparison of Techniques and Choice of Suitable Forecasting Technique

Intended learning outcomes: Differentiate between various areas of applicability of forecasting techniques.

##### 10.5.2 Consumption Distributions and Their Limits

Intended learning outcomes: Identify variables for a consumption distribution. Present expected value and variance of the consumption distribution. Explain distribution function, expected value, and variance of the consumption distribution under the assumption of a Poisson distribution for the frequency of events.

##### 10.5.2b Managing Discontinuous Demand, or Lumpy Demand

Intended learning outcomes: Describe the demand filter to handle a discontinuous demand due to infrequent large issues. Disclose effects of length of statistical period on demand fluctuations.

##### 10.5.3 Demand Forecasting of Variants of a Product Family

Intended learning outcomes: Describe the option percentage. Explain the formulas for forecasting demand for variants.

##### 10.5.4 Safety Demand Calculation for Various Planning Periods

Intended learning outcomes: Identify variables for safety demand calculations. Explain expected value and standard deviation with continuous demand. Describe expected value and standard deviation over n statistical periods.

##### 10.5.5 Translation of Forecast into Quasi-Deterministic Demand and Administration of the Production or Purchase Schedule

Intended learning outcomes: Identify safety demand. Explain independent demand as total demand including safety demand, taken as a function of the planning period to be covered.