IIE Exponential Smoothing with Trend and Seasonality – Winter’s Method

This method is appropriate when trend and seasonality are present in the time series. It decomposes the times series down into three components: base, trend and seasonal components.

Let s_i = seasonal factor for period i

If s_i = 1, then season is “typical”

If s_i < 1, then season is smaller than “typical”

If s_i > 1, then season is larger than “typical”

When an actual observation is divided by its corresponding seasonal factor, it is said to be “deseasonalized.” (i.e. the seasonal component has been removed.) This allows us to make meaningful comparisons across time periods.

Let c = the number of periods in a cycle (12 if months of year, 7 if days of week, …)

The relevant formulas for this method follow.

E_i = a( y _i / S _i-c )+(1-a) (E _i-1 + T _i-1)

T_i = b(E_i – L _i-1) + (1-b)T _i-1

S_i = g (y_i / L_i) + (1-g) s _i-c

yhat _i+1 = (E_i + T_i) s _i+1-c

where g is another smoothing constant between 0 and 1

To start this method, we need L₁ , T₁ , and a seasonal factor for each period in the cycle. These will either be given or you can use the method described next in your notes.

An easy way for developing initial estimates of the seasonal factors is to collect c observations and let:

s_i = y_i / [ (1/c)(y₁ + y₂ + … y_c)]

Then assume that E_c = y_c/s_c and that T_c = 0.

So the steps of the process can be summarized as:

1) forecast for period i

2) collect observation for period i

3) calculate smoothed average and the error for period i

4) calculate the trend for period i

5) calculate the seasonal factor for period i

6) i = i+1, go back to step 1

If you want to forecast k periods ahead, use the following logic

yhat _i+k = (E_i + kT_i) s _i+k-c

Work the lawn mower problem (lawn.xls) using an a of .5, a b of .5 and a g of .5. Use Solver to see if there are better weights.

Forecast sales for next August.