# 10.4.1 Trend Extrapolation Forecast

### Intended learning outcomes: Identify base demand B0(k) for period k>0 known at time t=0. Explain the calculation of the quotient “actual demand in period t+k” divided by “base demand known for period t+k at the end of period t”, k>1. Describe smoothing of quotient means for extrapolation leading to extrapolated forecast values for forecast distance k.

A trend extrapolation forecast attempts to estimate a variable in the future based on the same variable as known at a specific point in time.

In materials management, it may happen that the demand known at a particular point in time t encompasses only a portion of the demand needed for the coming period. Figure 10.4.1.1 provides an example.

Fig. 10.4.1.1      (Base) demand B0(k) for period k>0 known at time t=0.

Extrapolation calculates the total anticipated demand from the demand already known for a product or product family. It compares the base demand Bt(t+k), 1 ≤ k ≤ ∞, known at time t, to the demand Nt+k observed after the closing of a delivery period t+k. This is shown in Figure 10.4.1.2. The variables for the calculation are chosen either as defined in Figure 10.1.3.4 or in a similar fashion. k stands for the forecast distance.

Fig. 10.4.1.2 Actual demand Nt+k, divided by base demand Bt(t + k).

This quotient, λt(k) = Nt+k / Bt(t+k), 1 ≤ k ≤ t is called the extrapolation constant. The dilemma of this definition? Not until the end of period t+k can we determine the actual value of the extrapolation constant, namely λt+k(0) = Nt / Bt-k(t), that we had to estimate k periods ago, that is, as λt(k). Hence, the idea is to smooth the quotients over several periods using exponential smoothing. From now on, let λt(k), 1 ≤ k ≤ t, be the mean after period t for forecast distance k. The previous mean is used to calculate the new mean using exponential smoothing with smoothing constant α according to the formula in Figure 10.4.1.3.

Fig. 10.4.1.3 Smoothing of quotient means for extrapolation.

The extrapolation constant is defined for every forecast distance and can be used to extrapolate total demand, at the moment not completely known, from the base demand. Figure 10.4.1.4 gives the forecast value Pt(t+k) for the forecast distance k at the end of period t.

Fig. 10.4.1.4 Extrapolated forecast values for forecast distance k.

The technique described here assumes that the customers’ basic order behavior does not change on the time axis or that it does so very slowly. This means that from a change in customer orders on hand, we can infer a proportional change in total demand. Since this assumption is often invalid in the average case, the technique will yield useful results only when used in combination with other forecasting techniques, such as intuitive ones.

The planner can use this same technique to forecast seasonal components. In the grocery industry, for example, the retailer must give orders to the producers early enough to ensure that shipments arrive on time. Assuming that the retailers’ order behavior does not change significantly from year to year, the producer can derive standardized quotients from sales over multiple years; the probable total demand for the season in a future year can be extrapolated from the demand already known at a specific point in time.