II C Simple Exponential Smoothing

Again, this method works best when the time series fluctuates about a constant base level. Simple exponential smoothing is an extension of weighted moving averages where the greatest weight is placed on the most recent value and then progressively smaller weights are placed on the older values.

Recall that y_i = observed value i and yhat_i = forecasted value i

The following equation is used for developing forecasts from the simple exponential smoothing model.

yhat_i+1 = yhat _i + a (y _i – yhat _i)

Which says

forecast for the next period = forecast for this period + smoothing constant * error for this period

where 0<=a<=1

The more common way of expressing this model is

yhat_i+1 = ay _i +(1- a) yhat _i

The forecast for the current period is a weighted average of all past observations. The weight given to past observations declines exponentially.

The larger the a, the more weight is given to recent observations.

To start the process, assume that yhat₁ = y₁ unless told otherwise. Do not use this observation in your error calculations though.

Let’s now rework the MILK problem (MILK_SE.XLS) using simple exponential smoothing with an alpha of 0.3.

Check to see if Solver can find a better value for a.

At this point (day 20), your best forecast for all future days is yhat₂₁, your most recent forecast.

IID Exponential Smoothing with Trend

This method works best when the time series has a positive or negative trend (i.e. upward or downward).

After observing the value of the time series at period i (y_i), this method computes an estimate of the base, or expected level of the time series (E_i) and the expected rate of increase or decrease per period (T_i).

yhat _i+1 = E_i +T_i

where

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

T_i = b(E_i – E _i-1) + (1-b)T _i-1

And b is another smoothing constant where 0<=b<=1.

It is customary to assume that E₁ = y₁ and unless told otherwise, assume T₁ = 0.

To use the method, first calculate the base level E_i for time i. Then compute the expected trend value T_i for time period i. Finally, compute the forecast yhat _i+1. Once an observation y_i is made, calculate the error and continue the process for the next time period.

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

yhat _i+k = E_i +kT_i

Work the scanner problem (SCANNER.XLS) using an alpha of 0.2 and a beta of 0.3. Then check to see if Solver can find a better set of smoothing constants.

What is the expected demand for month 32?