Question
In: Statistics and Probability
Consider the following given information, determine the 3-year moving averages that would have been used to...
Consider the following given information, determine the 3-year moving averages that would have been used to forecast the sales (in millions) for the years subsequent to each group of three years. Determine the forecast error associated with each forecast.
Year(t) 1993／ 1994 ／1995 ／1996 ／1997 ／1998 ／1999 ／2000
Profit Y(t) 0.9 ／ 1.1 ／1.2 ／ 1.3 ／ 1.4 ／ 1.5 ／ 1.6 ／ 1.7
You may create a table with below information.
Year ／ Sales in millions ／ 3-year Moving Forecast ／ Forecast error ／ Absolute error
b) In the below table, by using the actual level of sales for 1994 of 1.1 million dollars as the “seed” forecast for 1995, determine the forecast for each annual sales amount by the method of simple exponential smoothing. First use a smoothing constant of alpha = 0.8, then use a smoothing constant of alpha = 0.2, and compare the two sets of forecasts.
Year(t) 1993 /1994 /1995 /1996 /1997 /1998 /1999 /2000
Sales in millions Y(t) 0.9 /1.1 /1.5 /1.3 /1.1 /1.7 /1.9 /2.3
Solutions
Expert Solution
Note: The following data table
Year(t) 1993／ 1994 ／1995 ／1996 ／1997 ／1998 ／1999 ／2000
Profit Y(t) 0.9 ／ 1.1 ／1.2 ／ 1.3 ／ 1.4 ／ 1.5 ／ 1.6 ／ 1.7
is not used as it lists the profits. However, both the parts refer to forecasting sales in millions
Let 
be the actual sales in millions and 
be the forecasted sales in millions for period t
a) A 3-year Moving Forecast for period t is
Forecast error is
Absolute error is
Get the following
| Year(t) | Sales in millions Y(t) | 3-year Moving Forecast | Forecast error | Absolute error |
| 1993 | 0.9 | |||
| 1994 | 1.1 | |||
| 1995 | 1.5 | |||
| 1996 | 1.3 | (0.9+1.1+1.5)/3=1.167 | (1.3-1.167)=0.133 | |0.133|=0.133 |
| 1997 | 1.1 | (1.1+1.5+1.3)/3=1.3 | (1.1-1.3)=-0.2 | |-0.2|=0.2 |
| 1998 | 1.7 | (1.5+1.3+1.1)/3=1.3 | (1.7-1.3)=0.4 | |0.4|=0.4 |
| 1999 | 1.9 | (1.3+1.1+1.7)/3=1.367 | (1.9-1.367)=0.533 | |0.533|=0.533 |
| 2000 | 2.3 | (1.1+1.7+1.9)/3=1.567 | (2.3-1.567)=0.733 | |0.733|=0.733 |
ans: A 3-year Moving Forecast is
| Year(t) | Sales in millions Y(t) | 3-year Moving Forecast | Forecast error | Absolute error |
| 1993 | 0.9 | |||
| 1994 | 1.1 | |||
| 1995 | 1.5 | |||
| 1996 | 1.3 | 1.167 | 0.133 | 0.133 |
| 1997 | 1.1 | 1.300 | -0.200 | 0.200 |
| 1998 | 1.7 | 1.300 | 0.400 | 0.400 |
| 1999 | 1.9 | 1.367 | 0.533 | 0.533 |
| 2000 | 2.3 | 1.567 | 0.733 | 0.733 |
| MAD | 0.400 |
The mean absolute deviation is
b) Exponential forecast for period t, with a smoothing constant of alpha is
Get the following table for alpha=0.8
| Year(t) | Sales in millions Y(t) | Exponential forecast , alpha=0.8 | Forecast error | Absolute error |
| 1993 | 0.9 | |||
| 1994 | 1.1 | |||
| 1995 | 1.5 | |||
| 1996 | 1.3 | 0.8*1.5+(1-0.8)*1.1=1.42 | (1.3-1.42)=-0.12 | |-0.12|=0.12 |
| 1997 | 1.1 | 0.8*1.3+(1-0.8)*1.42=1.324 | (1.1-1.324)=-0.224 | |-0.224|=0.224 |
| 1998 | 1.7 | 0.8*1.1+(1-0.8)*1.324=1.145 | (1.7-1.145)=0.555 | |0.555|=0.555 |
| 1999 | 1.9 | 0.8*1.7+(1-0.8)*1.145=1.589 | (1.9-1.589)=0.311 | |0.311|=0.311 |
| 2000 | 2.3 | 0.8*1.9+(1-0.8)*1.589=1.838 | (2.3-1.838)=0.462 | |0.462|=0.462 |
ans: Exponential forecast using a smoothing constant of alpha=0.8 is
| Year(t) | Sales in millions Y(t) | Exponential forecast , alpha=0.8 | Forecast error | Absolute error |
| 1993 | 0.9 | |||
| 1994 | 1.1 | |||
| 1995 | 1.5 | 1.1 | ||
| 1996 | 1.3 | 1.420 | -0.120 | 0.120 |
| 1997 | 1.1 | 1.324 | -0.224 | 0.224 |
| 1998 | 1.7 | 1.145 | 0.555 | 0.555 |
| 1999 | 1.9 | 1.589 | 0.311 | 0.311 |
| 2000 | 2.3 | 1.838 | 0.462 | 0.462 |
| MAD | 0.334 |
The mean absolute deviation is
Get the following table for alpha=0.2
| Year(t) | Sales in millions Y(t) | Exponential forecast , alpha=0.2 | Forecast error | Absolute error |
| 1993 | 0.9 | |||
| 1994 | 1.1 | |||
| 1995 | 1.5 | |||
| 1996 | 1.3 | 0.2*1.5+(1-0.2)*1.1=1.18 | (1.3-1.18)=0.12 | |0.12|=0.12 |
| 1997 | 1.1 | 0.2*1.3+(1-0.2)*1.18=1.204 | (1.1-1.204)=-0.104 | |-0.104|=0.104 |
| 1998 | 1.7 | 0.2*1.1+(1-0.2)*1.204=1.183 | (1.7-1.183)=0.517 | |0.517|=0.517 |
| 1999 | 1.9 | 0.2*1.7+(1-0.2)*1.183=1.287 | (1.9-1.287)=0.613 | |0.613|=0.613 |
| 2000 | 2.3 | 0.2*1.9+(1-0.2)*1.287=1.409 | (2.3-1.409)=0.891 | |0.891|=0.891 |
ans: Exponential forecast using a smoothing constant of alpha=0.2 is
| Year(t) | Sales in millions Y(t) | Exponential forecast , alpha=0.2 | Forecast error | Absolute error |
| 1993 | 0.9 | |||
| 1994 | 1.1 | |||
| 1995 | 1.5 | 1.1 | ||
| 1996 | 1.3 | 1.180 | 0.120 | 0.120 |
| 1997 | 1.1 | 1.204 | -0.104 | 0.104 |
| 1998 | 1.7 | 1.183 | 0.517 | 0.517 |
| 1999 | 1.9 | 1.287 | 0.613 | 0.613 |
| 2000 | 2.3 | 1.409 | 0.891 | 0.891 |
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