# Question

Consider a monthly series of air conditioner (AC) sales. In the discussion of Winters’ method, a monthly seasonality of 0.80 for January, for example, means that during January, AC sales are expected to be 80% of the sales during an average month. An alternative approach to modeling seasonality, called an additive model, is to let the seasonality factor for each month represent how far above average AC sales are during the current month. For instance, if SJan = 50, then AC sales during January are expected to be 50 fewer than AC sales during an average month. Similarly, if SJuly = 90, then AC sales during July are expected to be 90 more than AC sales during an average month. Let

St = Seasonality for month t after observing month t demand

Lt = Estimate of level after observing month t demand

Tt = Estimate of trend after observing month t demand

Then the Winters’ method equations given in the text should be modified as follows:

Lt = α(I) + (1 - α)(Lt -1 + Tt - 1)

Tt = β(Lt – Lt - 1) + (1 - β)Tt - 1

St = γ(II) + (1 - γ)St -12

a. What should (I) and (II) be?

b. Suppose that month 13 is January, L12 = 30, T12 = -3, S1 = -50, and S2 = -20. Let α = γ = β = 0.5. Suppose 12 ACs are sold during month 13. At the end of month 13, what is the forecast for AC sales during month 14 using this additive model?

St = Seasonality for month t after observing month t demand

Lt = Estimate of level after observing month t demand

Tt = Estimate of trend after observing month t demand

Then the Winters’ method equations given in the text should be modified as follows:

Lt = α(I) + (1 - α)(Lt -1 + Tt - 1)

Tt = β(Lt – Lt - 1) + (1 - β)Tt - 1

St = γ(II) + (1 - γ)St -12

a. What should (I) and (II) be?

b. Suppose that month 13 is January, L12 = 30, T12 = -3, S1 = -50, and S2 = -20. Let α = γ = β = 0.5. Suppose 12 ACs are sold during month 13. At the end of month 13, what is the forecast for AC sales during month 14 using this additive model?

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