Example: Method 12: Exponential Smoothing with Trend and Seasonality

This method is similar to Method 11, Exponential Smoothing, in that a smoothed average is calculated. However, Method 12 also includes a term in the forecasting equation to calculate a smoothed trend. The forecast is composed of a smoothed average that is adjusted for a linear trend. When specified in the processing option, the forecast is also adjusted for seasonality.

Forecast specifications:

• Alpha equals the smoothing constant that is used in calculating the smoothed average for the general level or magnitude of sales.

  Values for alpha range from 0 to 1.

• Beta equals the smoothing constant that is used in calculating the smoothed average for the trend component of the forecast.

  Values for beta range from 0 to 1.

• Whether a seasonal index is applied to the forecast.

  Note: Alpha and beta are independent of one another. They do not have to sum to 1.0.

Minimum required sales history: One year plus the number of time periods that are required to evaluate the forecast performance (periods of best fit). When two or more years of historical data is available, the system uses two years of data in the calculations.

Method 12 uses two Exponential Smoothing equations and one simple average to calculate a smoothed average, a smoothed trend, and a simple average seasonal index.

An exponentially smoothed average:

A_t = α (D_t/S_t-L) + (1 - α)(A_t-1 + T_t-1)

An exponentially smoothed trend:

T_t = β (A_t - A_t-1) + (1 - β)T_t-1

A simple average seasonal index:

The forecast is then calculated by using the results of the three equations:

F_t+m = (A_t + T_tm)S_t-L+m

where:

• L is the length of seasonality (L equals 12 months or 52 weeks).

• t is the current time period.

• m is the number of time periods into the future of the forecast.

• S is the multiplicative seasonal adjustment factor that is indexed to the appropriate time period.

  This table lists history used in the forecast calculation:

  Past Year	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec	Total
  1	128	117	115	125	122	137	140	129	131	114	119	137	1514
  2	125	123	115	137	122	130	141	128	118	123	139	133	1534

  Calculation of Linear and Seasonal Exponential Smoothing, given alpha = 0.3, beta = 0.4

Initializing the Process:

January of past year 1 Seasonal Index, S₁ =

S₁ = (125 + 128 / 1534 + 1514) × 12 = 0.083005 × 12 = 0.9961

January of past year 1 Smoothed Average*, A₁ =

A₁ = (January of past year 1 Actual) / (January Seasonal Index)

A₁ = 128 / 0.9960

A₁ = 128.51

January of past year 1 Smoothed Trend*, T₁ =

T₁ = 0 insufficient information to calculate first smoothed trend

February of past year 1 Seasonal Index, S₂ =

S₂ = (123 + 117 / 1534 + 1514) × 12 = 0.07874 × 12 = 0.9449

February of past year 1 Smoothed Average, A₂ =

A₂ = α(D₂ / S₂) + (1 – α) (A₁ + T₁)

A₂ = 0.3(117 / 0.9449) + (1 – 0.3) (128.51 + 0) = 127.10

February of past year 1 Smoothed Trend, T₂ =

T₂ = β(A₂ - A₁) + (1 - β)T₁

T₂=0.4 (127.10 – 128.51) + (1 – 0.4) × 0 = –0.56

March of past year 1 Seasonal Index, S₃ =

S₃ = (115 + 115 / 1534 + 1514) × 12 = 0.07546 × 12 = 0.9055

March of past year 1 Smoothed Average, A₃ =

A₃ = α(D₃/S₃) + (1 – α)(A₂ + T₂)

A₃ = 0.3 (115 / 0.9055) + (1 – 0.3)(127.10 – 0.56) = 126.68

March of past year 1 Smoothed Trend, T₃ =

T₃ = β(A₃ –A₂) + (1 – β)T₂

T3 = 0.4(126.68 – 127.10) + (1 – 0.4) x – 0.56 = – 0.50

(Continue through December of past year 1)

December of past year 1 Seasonal Index, S₁₂ =

S₁₂ = (133 + 137 / 1534 + 1514) × 12 = 0.08858 × 12 = 1.0630

December of past year 1 Smoothed Average, A₁₂ =

A₁₂ = α (D₁₂/S₁₂)+ (1 – α)( A₁₁ + T₁₁)

A₁₂ = 0.3 (137/1.0630 ) + ( 1 – 0.3)( 124.64 – 1.121 ) = 125.13

December of past year 1 Smoothed Trend, T₁₂ =

T₁₂ = β (A₁₂ – A₁₁) + (1 – β)T₁₁

T₁₂ = 0.4 (125.13 – 124.64)+ ( 1 – 0.4) x – 1.121 = – 0.477

Calculation of linear and seasonal exponentially smoothed forecast is calculated as follows:

F _{t + m} = (A_t +T_t m )S_{t – L + m}

* Calculations for Exponential Smoothing with Trend and Seasonality are initialized by setting the first smoothed average equal to the deseasonalized first actual sales data. The trend is initialized at zero for the first iteration. For subsequent calculations, alpha and beta are set to the values that are specified in the processing options.

This table indicates the Exponential Smoothing with Trend and Seasonality forecast for next year, where alpha = 0.3, beta = 0.4:

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
124.16	117.33	112.01	127.10	117.91	128.52	134.73	122.74	118.45	121.77	121.77	126.92