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CHAPTER 4
MOVING AVERAGES AND SMOOTHING METHODS
ANSWERS TO PROBLEMS AND CASES
1. Exponential smoothing
6. a.
t Yt
t
Y
e t et e t
2
t
t
Y
e
t
t
Y
e
1 19.39 19.00 .39 .39 .1521 .020 .020
5 18.43 17.89 .54 .54 .2916 .029 .029
7 19.51 19.98 – .47 .47 .2209 .024 -.024
9 19.78 20.63 – .85 .85 .7225 .043 -.043
11 21.18 21.25 – .07 .07 .0049 .003 -.003
7.
Price AVER1 FITS1 RESI1
17.89 18.3500 18.8500 -0.96000
21.25 20.5533 19.9733 1.27667
Accuracy Measures
8. a. See plot below.
Yt Avg Fits Res
200 * * *
225 219 216 9
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b. & c. See plot below.
Yt Smoothed Forecast
210 204.000 200.000
216 211.440 208.400
220 216.678 214.646
Accuracy Measures
Caution: If Minitab is used, the final result depends on how many
9. a. & c, d, e, f 3-month moving-average (See plot below.)
Month Yield MA Forecast Error
4 10.25 10.133 9.813 0.437
10 10.50 10.797 11.137 -0.637
Accuracy Measures
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b. & c, d, e, f 5-month moving-average (See plot below.)
Month Yield MA Forecast Error
6 11.07 10.416 10.060 1.010
Accuracy Measures
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10. Accuracy Measures (See plot below.)
11. See plot below.
Month Demand Smooth Forecast Error
1 205 205.000 205.000 0.0000
2 251 228.000 205.000 46.0000
3 304 266.000 228.000 76.0000
Forecast for month 13 (Jan. 2007) is 297.309
12. Naïve method – Forecast for 1996 Q2: 25.68 (Actual: 26.47)
13. a. = .4
c. Looking at the error measures, there is not much difference between the two
14. None of the techniques do much better than the naïve method. Simple exponential
15. A time series plot of quarterly Revenues and the autocorrelation function show
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Quarter Forecast Lower Upper
71 2444.63 2275.34 2613.92
16. a. An Excel time series plot for Sales follows
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b. & c. The Excel spreadsheet for calculating MAPE for the naïve forecasts and
the simple exponential smoothing forecasts is shown below.
Note: MAPE2 for simple exponential smoothing in the Excel spreadsheet
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d. Neither the naïve model nor simple exponential smoothing is likely to
e. The results of Winters multiplicative smoothing with = = = .5 is
shown in the Minitab plot below.
f. From part e, MAPE = 2.45%. Prefer Winters multiplicative smoothing
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17. a. The four-week moving average seems to represent the data a little better.
Compare the error measures for the four-week moving average in the figure
b. Simple exponential smoothing with a smoothing constant of
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18. a. As the order of the moving average increases, the smoothed data become more
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quarterly income are shown below. A plot of the residual