6 – 1
Chapter 6
Forecasting
Learning Objectives
1. Understand that the long-run success of an organization is often closely related to how well
management is able to predict future aspects of the operation.
5. Understand how the classical time series model can be used to explain the pattern or behavior of the
data in a time series and to develop a forecast for the time series.
6. Be able to determine and use seasonal indexes for a time series.
7. Know how regression models can be used in forecasting.
Markov Processes
Solutions:
1. The following table shows the calculations for parts (a), (b), and (c).
Week
Time Series
Value
Forecast
Forecast
Error
Absolute
Value of
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute
Value of
Percentage
Error
1
18
2
13
18
-5
5
25
-38.46
38.46
3
16
13
3
3
9
18.75
18.75
4
11
16
-5
5
25
-45.45
45.45
5
17
11
6
6
36
35.29
35.29
6
14
17
-3
3
9
-21.43
21.43
22
104
-51.30
a. MAE = 22/5 = 4.4
b. MSE = 104/5 = 20.8
c. MAPE = 159.38/5 = 31.88
2. The following table shows the calculations for parts (a), (b), and (c).
Week
Time Series
Value
Forecast
Forecast
Error
Absolute
Value of
Forecast
Error
Squared
Forecast
Error
Percentage
Error
Absolute
Value of
Percentage
Error
1
18
2
13
18.00
-5.00
5.00
25.00
-38.46
38.46
3
16
15.50
0.50
0.50
0.25
3.13
3.13
4
11
15.67
-4.67
4.67
21.81
-42.45
42.45
5
17
14.50
2.50
2.50
6.25
14.71
14.71
6
14
15.00
-1.00
1.00
1.00
7.14
13.67
54.31
-70.21
a. MAE = 13.67/5 = 2.73
b. MSE = 54.31/5 = 10.86
c. MAPE = 105.89/5 = 21.18
Markov Processes
3. The following table shows the measures of forecast error for both methods.
Exercise 1
Exercise 2
MAE
4.40
2.73
MSE
20.80
10.86
MAPE
31.88
21.18
4. a.
Month
Time Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
24
2
13
24
–11
121
3
20
13
7
49
4
12
20
64
5
19
12
7
49
6
23
19
4
16
7
15
23
64
Total
363
MSE = 363/6 = 60.5
The forecast for month 8 is
= Y8 = 15.
b.
Week
Time Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
24
2
13
24.00
-11.00
121.00
3
20
18.50
1.50
2.25
4
12
19.00
-7.00
49.00
5
19
17.25
1.75
3.06
6
23
17.60
5.40
29.16
7
15
18.50
-3.50
12.25
Total
216.72
MSE = 216.72/6 = 36.12
Forecast for month 8 is
8
ˆ
y
= (Y1 + Y2 + Y3 + Y4 + Y5 + Y6 +Y7) / 7 = (24 + 13 + 20 + 12 + 19 + 23 + 15) /
7 = 18.
Chapter 17
c. The average of all the previous values is better because MSE is smaller.
5. a.
b. Three-week moving average.
Week
Time Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
18
2
13
3
16
4
11
5
17
6
14
MSE = 35.67/3 = 11.89.
The forecast for week 7 is
= (Y4 + Y5 + Y6) / 3 = (11 + 17 + 14) / 3 = 14.
12
14
16
18
20
Markov Processes
c. Smoothing constant = .2
Week
Time Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
18
2
13
18.00
-5.00
25.00
3
16
17.00
-1.00
4
11
16.80
-5.80
33.64
5
17
15.64
6
14
15.91
-1.91
Total
65.15
MSE = 65.15/5 = 13.03
The forecast for week 7 is
= Y6 + (1-)
= .2(14) + (1 – .2)15.91 = 15.53
d. The three-week moving average provides a better forecast since it has a smaller MSE.
Alpha
0.367694922
Squared
Time Series
Forecast
Forecast
Week
Value
Forecast
Error
Error
1
18
2
13
18
-5.00
25.00
3
16
16.16
-0.16
0.03
4
16.10
-5.10
26.03
5
14.23
2.77
7.69
6
15.25
-1.25
1.55
60.30
Chapter 17
6. a.
The data appear to follow a horizontal pattern.
b. Three-week moving average.
Week
Time Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
24
2
13
3
20
4
12
5
19
6
23
7
15
MSE = 110/4 = 27.5.
The forecast for week 8 is
= (Y5 + Y6 + Y7) / 3 = (19 + 23 + 15) / 3 = 19.
15
20
25
30
Markov Processes
c. Smoothing constant = .2
Week
Time Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
24
2
13
24.00
-11.00
121.00
3
20
21.80
-1.80
3.24
4
12
21.44
-9.44
89.11
5
19
19.55
-0.55
0.30
6
23
19.44
3.56
12.66
7
15
20.15
-5.15
26.56
Total
252.87
MSE = 252.87/6 = 42.15
The forecast for week 8 is
= Y7 + (1-)
=.2(15) + (1 – .2)20.15 = 19.12.
Alpha
0.351404848
Month
Time Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
24
2
13
24
-11.00
121.00
3
20
20.13
-0.13
0.02
4
12
20.09
-8.09
65.40
5
19
17.25
3.08
6
23
17.86
26.40
7
15
19.67
-4.67
21.79
Total
237.69
MSE = 237.69/6 = 39.61428577
Chapter 17
7. a. Four and Five -week moving averages.
Week
Sales
4 Period
Moving
Average
5 period
Moving
Average
1
17
2
21
3
19
4
23
5
18
20.00
6
16
20.25
7
20
19.00
8
18
19.25
9
22
18.00
20
19.00
15
20.00
22
18.75
b. The MSE for the four-week and five-week moving averages.
For the four-week moving average:
Week
Time
Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
17
2
21
3
19
4
23
5
18
20.00
-2.00
4.0000
6
16
20.25
-4.25
18.0625
7
20
19.00
1.00
1.0000
8
18
19.25
-1.25
1.5625
9
22
18.00
4.00
20
19.00
1.00
1.0000
15
20.00
-5.00
22
18.75
3.25
10.5625
Total
77.1875
Markov Processes
For the five-week moving average:
Week
Time
Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
17
2
21
3
19
4
23
5
18
6
16
19.60
-3.60
12.96
7
20
19.40
0.60
0.36
8
18
19.20
-1.20
1.44
9
22
19.00
3.00
9.00
20
18.80
1.20
1.44
15
19.20
-4.20
17.64
22
19.00
3.00
9.00
Total
51.84
c. The MSE for the moving average forecasts are:
three week
27.500
four week
9.648
five week
7.406
8. a.
Week
Time-Series
Value
Weighted Moving
Average Forecast
Forecast
Error
(Error)2
1
17
2
21
3
19
4
23
19.33
3.67
13.47
5
18
21.33
-3.33
11.09
6
16
19.83
-3.83
14.67
7
20
17.83
2.17
4.71
8
18
18.33
-0.33
0.11
9
22
18.33
3.67
13.47
10
20
20.33
-0.33
0.11
11
15
20.33
-5.33
28.41
12
22
17.83
4.17
17.39
Total
b. MSE = 103.43 / 9 = 11.49
Prefer the unweighted moving average here; it has a smaller MSE.
Chapter 17
c. You could always find a weighted moving average at least as good as the unweighted moving average.
Actually the unweighted moving average is a special case of the weighted average for which the
weights are equal.
9. a. Exponential smoothing forecasts using α = .1:
Week
Time
Series
Value
Forecast
1
17
17.00
2
21
17.00
3
19
17.40
4
23
17.56
5
18
18.10
6
16
18.09
7
20
17.88
8
18
18.10
9
22
18.09
20
18.48
15
18.63
22
18.27
For a smoothing constant of α = .2:
Week
Time
Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
17
17.00
2
21
17.00
4.00
16.00
3
19
17.40
1.60
2.56
4
23
17.56
5.44
29.59
5
18
18.10
-0.10
0.01
6
16
18.09
-2.09
4.38
7
20
17.88
2.12
4.48
8
18
18.10
-0.10
0.01
9
22
18.09
3.91
15.32
20
18.48
1.52
2.32
15
18.63
-3.63
13.18
22
18.27
13.94
Total
MSE = 101.78/11 = 9.253
Markov Processes
For a smoothing constant of α = .2:
Week
Time
Series
Value
Forecast
Forecast
Error
Squared
Forecast
Error
1
17
17.00
2
21
17.00
4.00
16.00
3
19
17.80
1.20
1.44
4
23
18.04
4.96
24.60
5
18
19.03
-1.03
1.07
6
16
18.83
-2.83
7.98
7
20
18.26
1.74
3.03
8
18
18.61
-0.61
0.37
9
22
18.49
3.51
12.34
20
19.19
0.81
0.66
15
19.35
-4.35
18.94
22
18.48
3.52
12.38
Total
98.80
MSE = 98.80 / 11 = 8.982
Applying the MSE measure of forecast accuracy, a smoothing constant of α = .2 produces a smaller
MSE and so is preferred.
b. For a smoothing constant of α = .1:
Week
Time
Series
Value
Forecast
Forecast
Error
Absolute
Forecast
Error
1
17
17.00
2
21
17.00
4.00
4.00
3
19
17.40
1.60
1.60
4
23
17.56
5.44
5.44
5
18
18.10
-0.10
0.10
6
16
18.09
-2.09
2.09
7
20
17.88
2.12
2.12
8
18
18.10
-0.10
0.10
9
22
18.09
3.91
3.91
20
18.48
1.52
1.52
15
18.63
-3.63
3.63
22
18.27
3.73
3.73
Total
MAE = 28.25 / 11 = 2.568
Chapter 17
For a smoothing constant of α = .2:
Week
Time
Series
Value
Forecast
Forecast
Error
Absolute
Forecast
Error
1
17
17.00
2
21
17.00
4.00
4.00
3
19
17.80
1.20
1.20
4
23
18.04
4.96
4.96
5
18
19.03
-1.03
1.03
6
16
18.83
-2.83
2.83
7
20
18.26
1.74
1.74
8
18
18.61
-0.61
0.61
9
22
18.49
3.51
3.51
20
19.19
0.81
0.81
15
19.35
-4.35
4.35
22
18.48
3.52
3.52
Total
28.56
MAE = 28.56 / 11 = 2.596
Applying the MAE measure of forecast accuracy, a smoothing constant of α = .1 produces a slightly
smaller MAE and so is preferred.
c. For a smoothing constant of α = .1:
Week
Time
Series
Value
Forecast
Forecast
Error
100*(Forecast Error/
Time Series Value)
Absolute Value of
100*(Forecast Error/
Time Series Value)
1
17
17.00
2
21
17.00
4.00
19.05
19.05
3
19
17.40
1.60
8.42
8.42
4
23
17.56
5.44
23.65
23.65
5
18
18.10
-0.10
-0.58
0.58
6
16
18.09
-2.09
13.09
7
20
17.88
2.12
10.58
8
18
18.10
-0.10
-0.53
0.53
9
22
18.09
3.91
17.79
20
18.48
1.52
7.61
15
18.63
-3.63
24.20
22
18.27
3.73
16.97
Markov Processes
For a smoothing constant of α = .2:
Week
Time
Series
Value
Forecast
Forecast
Error
100*(Forecast Error/
Time Series Value)
Absolute Value of
100*(Forecast Error/
Time Series Value)
1
17
17.00
2
21
17.00
4.00
19.05
19.05
3
19
17.80
1.20
6.32
6.32
4
23
18.04
4.96
21.57
21.57
5
18
19.03
-1.03
-5.73
5.73
6
16
18.83
-2.83
-17.66
17.66
7
20
18.26
1.74
8.70
8.70
8
18
18.61
-0.61
-3.38
3.38
9
22
18.49
3.51
15.97
15.97
20
19.19
0.81
4.05
4.05
15
19.35
-4.35
-29.01
29.01
22
18.48
3.52
15.99
15.99
MAPE = 147.43 / 11 = 13.40
Applying the MAPE measure of forecast accuracy, a smoothing constant of α = .1 produces a smaller
MAPE and so is preferred.
10. a.
13
ˆ
y
= .2Y12 + .16Y11 + .64(.2Y10 + .8
10
ˆ
y
) = .2Y12 + .16Y11 + .128Y10 + .512
10
ˆ
y
13
ˆ
y
= .2Y12 + .16Y11 + .128Y10 + .512(.2Y9 + .8
9
ˆ
y
) = .2Y12 + .16Y11 + .128Y10 + .1024Y9 + .4096
9
ˆ
y
Chapter 17
11. a.
The data appear to follow a horizontal pattern.
b. For the three month moving average:
Month
Time
Series
Value
Forecast
Forecast
Error
Square
Forecast
Error
1
80
2
82
3
84
4
83
82.00
1.00
1.00
5
83
83.00
0.00
0.00
6
84
83.33
0.67
0.44
7
85
83.33
1.67
2.78
8
84
84.00
0.00
0.00
9
82
84.33
-2.33
5.44
83
83.67
-0.67
0.44
84
83.00
1.00
1.00
83
83.00
0.00
0.00
Total
MSE = 11.11 / 9 = 1.235
85
90
Markov Processes
For the exponential smoothing forecast for α = .2:
Month
Time
Series
Value
Forecast
Forecast
Error
Square
Forecast
Error
1
80
80.00
2
82
80.00
2.00
4.00
3
84
80.40
3.60
12.96
4
83
81.12
1.88
3.53
5
83
81.50
1.50
2.26
6
84
81.80
2.20
4.85
7
85
82.24
2.76
7.63
8
84
82.79
1.21
1.46
9
82
83.03
-1.03
1.06
83
82.83
0.17
0.03
84
82.86
1.14
1.30
83
83.09
-0.09
0.01
39.11
MSE = 39.80 / 11 = 3.555
Applying the MSE measure of forecast accuracy, a three-month moving average produces a smaller
MSE and so is preferred.
12. a.
The data appear to follow a horizontal pattern.
b.
Month
Time-Series
Value
3-Month Moving
Average Forecast
(Error)2
4-Month Moving
Average Forecast
(Error)2
1
9.5
2
9.3
3
9.4
4
9.6
9.40
0.04
5
9.8
9.43
0.14
9.45
0.12
6
9.7
9.60
0.01
9.53
0.03
7
9.8
9.70
0.01
9.63
0.03
8
9.77
0.53
9.73
0.59
9
9.9
0.01
9.95
0.00
9.7
0.14
9.98
0.08
9.6
0.18
9.97
0.14
9.6
9.73
0.02
9.92
0.10
1.08
1.09
MSE(3-Month) = 1.08 / 9 = .12
MSE(4-Month) = 1.09 / 8 = .14
The MSE for the 3-Month moving average is smaller, so use the 3-Month moving average.
9.5
10.0
10.5
11.0
Month (t)
Markov Processes
13. a.
Month
Time-Series
Value
3-Month Moving
Average Forecast
(Error)2
= .2
Forecast
(Error)2
1
240
2
350
240.00
12100.00
3
230
262.00
1024.00
4
260
273.33
177.69
255.60
19.36
5
280
280.00
0.00
256.48
553.19
6
320
256.67
4010.69
261.18
3459.79
7
220
286.67
4444.89
272.95
2803.70
8
310
273.33
1344.69
262.36
2269.57
9
240
283.33
1877.49
271.89
1016.97
310
256.67
2844.09
265.51
1979.36
240
286.67
2178.09
274.41
1184.05
230
263.33
1110.89
267.53
17,988.52
MSE(3-Month) = 17,988.52 / 9 = 1998.72
MSE(α = .2) = 27,818.49 / 11 = 2528.95
Based on the above MSE values, the 3-month moving average appears better. However, exponential
smoothing was penalized by including month 2 which was difficult for any method to forecast. Using
only the errors for months 4 to 12, the MSE for exponential smoothing is:
250
300
350
400
Chapter 17
14. a.
b. Smoothing constant = .3.
Month t
Time-Series Value
Yt
Forecast
ˆt
y
Forecast Error
Yt –
ˆt
y
Squared Error
(Yt –
ˆt
y
)2
1
105
2
135
105.00
30.00
900.00
3
120
114.00
6.00
36.00
4
105
115.80
-10.80
116.64
5
90
112.56
-22.56
508.95
6
120
105.79
14.21
201.92
7
145
110.05
34.95
1221.50
8
140
120.54
19.46
378.69
9
100
126.38
-26.38
695.90
80
118.46
-38.46
1479.17
100
106.92
-6.92
47.89
110
104.85
5.15
5613.18
MSE = 5613.18 / 11 = 510.29
The forecast for month 13 is
13
ˆ
y
= Y12 + (1-)
12
ˆ
y
= .3(110) + .7(104.85) = 106.4.
100
120
140
160
Markov Processes
c. The value of that yields the smallest possible MSE is = 0.032564518, which yields an MSE of
459.6929489.
Alpha
0.032564518
Squared
Time Series
Forecast
Forecast
Month
Value
Forecast
Error
Error
1
105
2
135
105.00
30.00
900.00
3
120
105.98
14.02
196.65
4
105
106.43
-1.43
2.06
5
90
106.39
-16.39
268.53
6
120
105.85
14.15
200.13
7
145
106.31
38.69
1496.61
8
140
107.57
32.43
1051.46
9
100
108.63
-8.63
74.47
80
108.35
-28.35
803.65
100
107.43
-7.43
55.14
110
107.18
2.82
7.93
5056.62
15. a.
7.60
7.70
7.80
7.90
8.00