# 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.

Often variants of a product are derived gradually from one basic type, a standard product with options or a product family. Often, a forecast can predict the total demand for a product family. Deriving the demand for components that are the same for all variants is no longer very difficult. Demand forecasting for variants is more difficult. When the number of delivered variants of a product family is large enough, the use of variant items — related to 100 units of the product family, for example — can be recorded in a statistic and used for management.

`The option percentage OPC is the frequency with which a variant item is used within a product family.`

This percentage varies from time period to time period and is therefore a stochastic variable that can be described with expected value and variance. Often, in practice, the dispersion of the option percentage is not taken into consideration; that is, E(PF) is treated as a quasi-deterministic value. This increases the risk of stock failures. To calculate option percentages, sales are subdivided by statistical period. For each period, we determine the actual frequency of use and calculate mean and standard variation from the results of multiple periods. Linking the forecast for the product family with the option percentage for demand for variants is achieved by using the formulas in Figure 10.5.3.1. These formulas are used for every periodic demand.

Fig. 10.5.3.1      Forecasting demand for variants.

The proportional-factor-weighted demand (expected value and variance) for the product family is the independent demand for a variant. Because of safety demand calculation, the sum of variant demands is greater than the demand for components not dependent upon variants.

For a more in-depth consideration of the formulas in Figure 10.5.3.1, see the footnote.

     The following derivation of Prof. Büchel’s formula for multiplicative coupling x of two independent distribu­tions y and z, x= y*z, provides a more in-depth consideration of the matter. See also [Fers64]. Multiplication of a particular value Y of y by z results in a linear trans­formation of z with the following parameters:

E(Y*z) = Y * E(z) , VAR(Y*z) = Y2 * VAR(z) .

The distribution obtained in this way is weighted by f(y) and summed (or, with continuous distributions, integrated) to create a mixed distribution. The zero mo­ments are to be applied for this. The result of the individual linear transforma­tions for the second zero moment — defined as E(u2) = E2(u) + VAR(u) — are as follows:

E((Y*z)2)     = E2(Y*z) + VAR(Y*z) = Y2 * E2(z) + Y2 * VAR(z)  =  Y2 * (E2(z) + VAR(z)) = Y2 * E(z2) .

The summation produces the following result:

E(x) = E(y) * E(z) , E(x2) = E(y2) * E(z2), and so the following hold:

VAR(x) = E(x2) – E2(x) = E(y2) * E(z2) – E2(y) * E2(z)  =  [E2(y) + VAR(y)] * [E2(z) + VAR(z)] – E2(y) * E2(z)  =  E2(y) * VAR(z) + VAR(y) * E2(z) + VAR(y) * VAR(z).

CV2(x)   = VAR(x) / E2(x)  =  [E2(y) * VAR(z) + VAR(y) * E2(z) + VAR(y) * VAR(z)] / [E2(y) * E2(z)]  =  CV2(z) + CV2(y) + CV2(y) * CV2(z).

Note: The formulas in Figure 10.5.2.2 can be derived analogously. Linear transformations are replaced by the distributions for the sum of multiple issues per period (so-called convolutions), whose parameters are determined as follows:

E(n*z) = n * E(z);       VAR(n*z) = n * VAR(z)

A general statement as to the form of the distribution cannot be made; a log-nor­mal distribution (which becomes a normal distribution with small coefficients of variation) represents a useful approximation for practical application. When there are many periods with zero issues (low issue frequency), special consideration of the choice of “risk” may be required.

This is also true for Section. 10.5.2. However, the planning periods, not the statistical periods, are decisive.