Essentials of Business Analytics (MindTap Course List)
Essentials of Business Analytics (MindTap Course List)
2nd Edition
ISBN: 9781305627734
Author: Jeffrey D. Camm, James J. Cochran, Michael J. Fry, Jeffrey W. Ohlmann, David R. Anderson
Publisher: Cengage Learning
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Chapter 8, Problem 8P

(a)

To determine

Obtain exponential smoothing for α=0.1 and α=0.2.

Identify the preferred smoothing constant using MSE measure of forecast accuracy.

(a)

Expert Solution
Check Mark

Explanation of Solution

Exponential smoothing for α=0.1:

Exponential smoothing is obtained using the formula given below:

y^t+1=αyt+(1α)y^t

WeekTime Series ValueForecastForecast ErrorSquared Forecast Error
117
2210.1(17)+(10.1)(17)=174.0016.00
3190.1(21)+(10.1)(17)=17.401.602.56
4230.1(19)+(10.1)(17.4)=17.565.4429.59
5180.1(23)+(10.1)(17.56)=18.10−0.100.01
6160.1(18)+(10.1)(18.10)=18.09−2.094.38
7200.1(16)+(10.1)(18.09)=17.882.124.48
8180.1(20)+(10.1)(17.88)=18.10−0.100.01
9220.1(18)+(10.1)(18.1)=18.093.9115.32
10200.1(22)+(10.1)(18.09)=18.481.522.32
11150.1(20)+(10.1)(18.48)=18.63−3.6313.18
12220.1(15)+(10.1)(18.63)=18.273.7313.94
Total101.78

MSE=|er|2nk=101.7811=9.253

Thus, the mean squared error is 9.253.

Exponential smoothing for α=0.2:

WeekTime Series ValueForecastForecast ErrorSquared Forecast Error
117  
2210.2(17)+(10.2)(17)=174.0016.00
3190.2(21)+(10.2)(17)=17.81.201.44
4230.2(19)+(10.2)(17.8)=18.044.9624.60
5180.2(23)+(10.2)(18.04)=19.03−1.031.07
6160.2(18)+(10.2)(19.03)=18.83−2.837.98
7200.2(16)+(10.2)(18.83)=18.261.743.03
8180.2(20)+(10.2)(18.26)=18.61−0.610.37
9220.2(18)+(10.2)(18.61)=18.493.5112.34
10200.2(22)+(10.2)(18.49)=19.190.810.66
11150.2(20)+(10.2)(19.19)=19.35−4.3518.94
12220.2(15)+(10.2)(19.35)=18.483.5212.38
Total98.80

MSE=|er|2nk=98.8011=8.982

Thus, the mean squared error is 8.982.

MSE when α=0.2 is less than MSE when α=0.1. Thus, α=0.2 is preferred.

(b)

To determine

Check whether the results are same when MAE is used as measure of accuracy.

(b)

Expert Solution
Check Mark

Answer to Problem 8P

No, the results are not same.

Explanation of Solution

The MSE for four-week moving average is obtained as given below:

Exponential smoothing for α=0.1:

Exponential smoothing is obtained using the formula given below:

y^t+1=αyt+(1α)y^t

WeekTime Series ValueForecastForecast ErrorAbsolute Forecast Error
117
2210.1(17)+(10.1)(17)=174.004.00
3190.1(21)+(10.1)(17)=17.401.601.60
4230.1(19)+(10.1)(17.4)=17.565.445.44
5180.1(23)+(10.1)(17.56)=18.10−0.100.10
6160.1(18)+(10.1)(18.10)=18.09−2.092.09
7200.1(16)+(10.1)(18.09)=17.882.122.12
8180.1(20)+(10.1)(17.88)=18.10−0.100.10
9220.1(18)+(10.1)(18.1)=18.093.913.91
10200.1(22)+(10.1)(18.09)=18.481.521.52
11150.1(20)+(10.1)(18.48)=18.63−3.633.63
12220.1(15)+(10.1)(18.63)=18.273.733.73
Total28.25

MAE=|er|nk=28.258=2.568

Thus, the mean absolute error is 2.568.

Exponential smoothing for α=0.2:

WeekTime Series ValueForecastForecast ErrorSquared Forecast Error
117  
2210.2(17)+(10.2)(17)=174.004.00
3190.2(21)+(10.2)(17)=17.81.201.20
4230.2(19)+(10.2)(17.8)=18.044.964.96
5180.2(23)+(10.2)(18.04)=19.03−1.031.03
6160.2(18)+(10.2)(19.03)=18.83−2.832.83
7200.2(16)+(10.2)(18.83)=18.261.741.74
8180.2(20)+(10.2)(18.26)=18.61−0.610.61
9220.2(18)+(10.2)(18.61)=18.493.513.51
10200.2(22)+(10.2)(18.49)=19.190.810.81
11150.2(20)+(10.2)(19.19)=19.35−4.354.35
12220.2(15)+(10.2)(19.35)=18.483.523.52
Total28.56

MAE=|er|nk=28.5611=2.596

Thus, the mean absolute error is 2.596.

MAE when α=0.1 is less than MAE when α=0.2. Thus, α=0.1 is preferred.

Hence, the results are not same when MAE is used as measure of accuracy.

(c)

To determine

Obtain the results when MAPE is used as measure of accuracy.

(c)

Expert Solution
Check Mark

Answer to Problem 8P

MAPE when α=0.1 is 12.95.

MAPE when α=0.2 is 13.40.

Explanation of Solution

Exponential smoothing for α=0.1:

WeekTime Series ValueForecastForecast Error|100×Forecast errorTime series value|
117 
2210.1(17)+(10.1)(17)=174.0019.05
3190.1(21)+(10.1)(17)=17.401.608.42
4230.1(19)+(10.1)(17.4)=17.565.4423.65
5180.1(23)+(10.1)(17.56)=18.10−0.100.58
6160.1(18)+(10.1)(18.10)=18.09−2.0913.09
7200.1(16)+(10.1)(18.09)=17.882.1210.58
8180.1(20)+(10.1)(17.88)=18.10−0.100.53
9220.1(18)+(10.1)(18.1)=18.093.9117.79
10200.1(22)+(10.1)(18.09)=18.481.527.61
11150.1(20)+(10.1)(18.48)=18.63−3.6324.20
12220.1(15)+(10.1)(18.63)=18.273.7316.97
Total142.46

MAPE=|100×Forecast errorTime series value|nk=142.4611=12.95

Thus, the value of MAPE is 12.95.

Exponential smoothing for α=0.2:

WeekTime Series ValueForecastForecast Error|100×Forecast errorTime series value|
117  
2210.2(17)+(10.2)(17)=174.0019.05
3190.2(21)+(10.2)(17)=17.81.206.32
4230.2(19)+(10.2)(17.8)=18.044.9621.57
5180.2(23)+(10.2)(18.04)=19.03−1.035.73
6160.2(18)+(10.2)(19.03)=18.83−2.8317.66
7200.2(16)+(10.2)(18.83)=18.261.748.70
8180.2(20)+(10.2)(18.26)=18.61−0.613.38
9220.2(18)+(10.2)(18.61)=18.493.5115.97
10200.2(22)+(10.2)(18.49)=19.190.814.05
11150.2(20)+(10.2)(19.19)=19.35−4.3529.01
12220.2(15)+(10.2)(19.35)=18.483.5215.99
Total147.43

MAPE=|100×Forecast errorTime series value|nk=147.4311=13.40

Thus, the value of MAPE is 13.40.

MAPE when α=0.1 is less than MAPE when α=0.2. Thus, α=0.1 is preferred.

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