Choice of time window

In choosing the time window k, we have to consider a tradeoff between

• The ability to smooth (filter) noise associated with occasionally large or small demand values
• The ability to adapt to changes in market conditions that shift average demand

If k is large, the method has a lot of inertia and is not significantly influenced by occasional variability; however, it will be slow to adapt to systematic changes. On the contrary, if k is low, the algorithm will be very prompt, but also very nervous and prone to chase false signals. The difference in the behavior of a moving average as a function of time window length is illustrated in Fig. 11.4. The demand history is depicted by empty circles, whereas the squares show the corresponding forecasts (calculated one time bucket before; we assume h = 1). The demand history shows an abrupt jump, possibly due to opening another distribution channel. Two simulations are illustrated, with time windows k = 2 and k = 6, respectively. Note that, in plot (a), we start forecasting at the end of time bucket t = 2 for t = 3, since we assume h = 1, whereas in plot (b) we must wait for t = 6. We may also notice that, when k = 2, each forecast is just the vertical midpoint between the two last observations. We see that, with a shorter time window, the forecast tends to chase occasional swings; on the other hand, with a longer time window, the adaptation to the new regime is slower.

Fig. 11.4 Moving average with k = 2 and k = 6.

Moving average is a very simple approach, with plenty of applications beyond demand forecasting.⁹ However, this kind of average can be criticized because of its “all or nothing” character. As we see in Fig. 11.3, the most recent observations have the same weight, 1/k, which suddenly drops to 0. We could try a more gradual scheme in which

• Different weights are attributed to more recent observations
• A small but non-zero weight is associated with older observations

In other words, weights should decrease gradually to zero for older observations. This is exactly what the family of exponential smoothing algorithms accomplishes.