# 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.

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.