Smoothing with trend and multiplicative seasonality

The last exponential smoothing approach we consider puts everything together and copes with additive trend and multiplicative seasonality. The Holt–Winter method is based on Eq. (11.13), which we repeat for convenience:

The overall scheme uses three smoothing coefficients and it proceeds as follows

All of the remarks we have made about simpler versions of exponential smoothing apply here as well, including initialization. To initialize the method, we need s + 2 parameters, which must be estimated on the basis of at least s + 1 demand observations, since seasonal factors are not independent. Heuristic initialization approaches, based on time series decomposition, are described in the references. Alternatively, we could use a formal approach based on least-squares-like optimization models, as illustrated in Example 11.4.