# 10.2.2 First-Order Exponential Smoothing Forecast

### Intended learning outcomes: Identify the weighted mean as well as exponential demand weighting. Explain first-order exponential smoothing: mean, MAD, and standard deviation. Disclose the average age of the observed values. Describe how the smoothing constant α determines the weighting of the past. Present an example of first-order exponential smoothing.

If we wish to adapt the forecasting technique to actual demand, the demand values for the last periods must be weighted more heavily, according to the principle of the weighted mo­ving average. The formula in Figure 10.2.2.1 takes this weighting into account; the variables were chosen according to the definitions in Figure 10.1.3.4 and include an indefinite number of periods. Gt-i always expresses the weighting of demand in the period (t–i). [note 1005]

Fig. 10.2.2.1       Weighted mean.

In the first-order exponential smoothing forecast technique, or single (exponential) smoothing, the weights are in an exponentially declining relationship and adhere to the definitions in Figure 10.2.2.2.

Fig. 10.2.2.2       Exponential demand weighting.

Figure 10.2.2.3 shows the calculation of Mean smoothed consumption as measure of mean, and Mean absolute deviation (MAD) as measure of dispersion. See also the definitions of indexes and variables in Figure 10.1.3.4.

Fig. 10.2.2.3       First-order exponential smoothing: mean, MAD, and standard Deviation.

Since the weighting Gy follows a geometric series, the recursive calcula­tion indicated in the formulas is self-evident. These formulas allow us to perform the same calculation as in moving average using only the past values for mean and MAD and the demand value for the current period instead of many demand values. With a normal distribution, standard deviation and mean absolute deviation (MAD) stand in the same relationship as that given in Figure 10.2.2.3.

The recursion to Mt-1 results by factoring out (1-a) of the part of the formula that is emphasized by the horizontally cambered bracket. Factual equality between σ and MAD*1.25 requires n>30 or α < 6.5%. Figure 10.2.2.4 shows the average age of the observed values. The age of Nt-i is i for 0 ≤ i ≤ n-1.

Fig. 10.2.2.4       Average age of the observed values.

The choice of smoothing constant α or alpha factor determines the weighting of current and past demand according to the formula in Figure 10.2.2.3.

Figure 10.2.2.5 shows the effect of α = 0.1, a value often chosen for well-established products, and α = 0.5 for products at the beginning or the end of their life cycles.

Fig. 10.2.2.5       The smoothing constant α determines the weighting of the past.

Figure 10.2.2.6 shows the behavior of the forecast curve with various values of the smoothing constant α. A high smoothing constant results in a rapid but also nervous reaction to changes in demand behavior. See also Sections 10.2.3 and 10.5.1.

Fig. 10.2.2.6       Forecasts with various values of the smoothing constant α.

Using exponential smoothing techniques, we can determine the uncertainty of a forecast by extrapolating the forecast error. To do this, we calculate the mean absolute deviation (MAD). Figure 10.2.2.7 is an example of expo­nential smoothing with smoothing constant α = 0.2. It was chosen in a way similar to the example of moving average calculation in Figure 10.2.1.4.

Fig. 10.2.2.7       First-order exponential smoothing with smoothing constant α = 0.2.