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CHAPTER 4 FO R E C A ST I N G 47
Copyright ©2014 Pearson Education, Inc.
48 CHAPTER 4 FO R E C A S T I N G
4.51
Period
Demand
Exponentially Smoothed Forecast
1
7
5
2
9
5 + 0.2 × (7 – 5) = 5.4
5
13
5.90 + 0.2 × (9 – 5.90) = 6.52
6
8
6.52 + 0.2 × (13 – 6.52) = 7.82
7
Forecast
7.82 + 0.2 × (8 – 7.82) = 7.86
4.52
Actual
Forecast
|Error|
Error2
95
100
5
25
108
110
2
4
123
120
3
9
130
130
0
0
10
38
MAD = 10/4 = 2.5, MSE = 38/4 = 9.5
4.53 (a) 3-month moving average:
3-Month
Absolute
Month
Sales
Moving Average
Deviation
January
11
April
10
(11 + 14 + 16)/3 = 13.67
3.67
May
15
(14 + 16 + 10)/3 = 13.33
1.67
June
17
(16 + 10 + 15)/3 = 13.67
3.33
July
11
(10 + 15 + 17)/3 = 14.00
3.00
August
14
(15 + 17 + 11)/3 = 14.33
0.33
September
17
(17 + 11 + 14)/3 = 14.00
3.00
October
12
(11 + 14 + 17)/3 = 14.00
2.00
November
14
(14 + 17 + 12)/3 = 14.33
0.33
December
16
(17 + 12 + 14)/3 = 14.33
1.67
January
11
(12 + 14 + 16)/3 = 14.00
3.00
February
(14 + 16 + 11)/3 = 13.67
= 22.00
MAD = 2.20
(b) 3-month weighted moving average
(c) Based on a mean absolute deviation criterion, the
3-month moving average with MAD = 2.2 is to be pre-
ferred over the 3-month weighted moving average with
MAD = 2.72.
4.54 (a)
Actual
Cumulative
Cum.
Tracking
Week
Miles
Forecast
Error
Error
|Error|
MAD
Signal
1
17
17.00
0.00
–
0.00
0
2
21
17.00
+4.00
4.00
4.00
2
2
3
19
17.80
+1.20
5.20
5.20
1.73
3
4
23
18.04
+4.96
10.16
10.16
2.54
4
5
18
19.03
–1.03
9.13
11.19
2.24
4
6
16
18.83
–2.83
6.30
14.02
2.34
2.7
7
20
18.26
+1.74
8.04
15.76
2.25
3.6
8
18
18.61
–0.61
7.43
16.37
2.05
3.6
9
22
18.49
+3.51
10.94
19.88
2.21
5
10
20
19.19
+0.81
11.75
20.69
2.07
5.7
11
15
19.35
–4.35
7.40
25.04
2.28
3.2
12
22
18.48
+3.52
10.92
28.56
2.38
4.6
(b) The MAD = 28.56/12 = 2.38
(c) The cumulative error and tracking signals appear to
4.55
y
x
x2
xy
7
1
1
7
9
2
4
18
5
3
9
15
11
4
16
44
10
5
25
50
13
6
36
78
55
21
91
212
9.17
3.5
5.27 1.11
y
x
yx
=
=
=+
Period 7 forecast = 13.07
Period 12 forecast = 18.64, but this is far outside the range
of valid data.
Month
Sales
3-Month Moving Average Moving
Absolute Deviation
January
11
February
14
March
16
April
10
(1 × 11 + 2 × 14 + 3 × 16)/6 = 14.50
4.50
May
15
(1 × 14 + 2 × 16 + 3 × 10)/6 = 12.67
2.33
June
17
(1 × 16 + 2 × 10 + 3 × 15)/6 = 13.50
3.50
July
11
(1 × 10 + 2 × 15 + 3 × 17)/6 = 15.17
4.17
August
14
(1 × 15 + 2 × 17 + 3 × 11)/6 = 13.67
0.33
September
17
(1 × 17 + 2 × 11 + 3 × 14)/6 = 13.50
3.50
October
12
(1 × 11 + 2 × 14 + 3 × 17)/6 = 15.00
3.00
November
14
(1 × 14 + 2 × 17 + 3 × 12)/6 = 14.00
0.00
December
16
(1 × 17 + 2 × 12 + 3 × 14)/6 = 13.83
2.17
January
11
(1 × 12 + 2 × 14 + 3 × 16)/6 = 14.67
3.67
February
(1 × 14 + 2 × 16 + 3 × 11)/6 = 13.17
= 27.17
MAD = 2.72
50 CHAPTER 4 FO R E C A S T I N G
1
Standard error of the estimate:
294 1 20 1 70
2 5 2
3
yx
Y a Y b XY
Sn
− − − −
==
−−
= = =
4.62 Using software, the regression equation is: Games lost =
6.41 + 0.533 × days rain.
1. One way to address the case is with separate forecasting models
for each game. Clearly, the homecoming game (week 2) and the
Forecasts
Game
Model
2013
2014
R2
1
y = 30,713 + 2,534x
48,453
50,988
0.92
2
y = 37,640 + 2,146x
52,660
54,806
0.90
3
y = 36,940 + 1,560x
47,860
49,420
0.91
4
y = 22,567 + 2,143x
37,567
39,710
0.88
5
y = 30,440 + 3,146x
52,460
55,606
0.93
2. Revenue in 2013 = (239,000) ($50/ticket) = $11,950,000
Revenue in 2014 = (250,530) ($52.50/ticket) = $13,152,825
3. In games 2 and 5, the forecast for 2014 exceeds stadium ca-
pacity. With this appearing to be a continuing trend, the time has
come for a new or expanded stadium.
VIDEO CASE STUDIES
FORECASTING TICKET REVENUE FOR
3. Using the multiple regression model in the case:
Revenue = $14,996 + 10,801 (4) + 23,379 (3) + 10,784 (3)
= $160,743
4. Time of day for game, other competing sports events within
100 miles on that date, special half-time or pregame entertainment
planned, date set for a special group night (for example, Boy
Scouts or Rotary). These may be potential independent variable
for Perez’s model.
cafes, (2) retail sales, (3) banquet sales, (4) concert sales, (5) eval-
uating managers, and (6) menu planning. They could also employ
2. The POS system captures all the basic sales data needed to
drive individual cafe’s scheduling/ordering. It also is aggregated
at corporate HQ. Each entrée sold is counted as one guest at a
3. The weighting system is subjective, but is reasonable. More
weight is given to each of the past 2 years than to 3 years ago.
This system actually protects managers from large sales variations
(weather); hotel occupancy; spring break from colleges; beef pric-
es; promotional budget; etc.
5. Y = a + bx
Month
Advertising X
Guest Count Y
X2
Y2
XY
1
14
21
196
441
294
2
17
24
289
576
408
3
25
27
625
729
675
4
25
32
625
1,024
800
5
35
29
1,225
841
1,015
2
15,910 10 36.6
37.6 0.7996 36.6 8.3363 8.3
8.3363 0.7996
a
YX
−
= − =
=+
At $65,000; y = 8.3 + .8 (65) = 8.3 + 52 = 60.3, or 60,300 guests.
For the instructor who asks other questions than this one:
r2 = 0.8869
Std. error = 5.062
ADDITIONAL CASE STUDIES*
THE NORTH-SOUTH AIRLINE
Northern Airline Data
Airframe Cost
Engine Cost
Average
Year
per Aircraft
per Aircraft
Age (hrs)
2003
51.80
43.49
6512
2004
54.92
38.58
8404
2005
69.70
51.48
11077
2006
68.90
58.72
11717
2007
63.72
45.47
13275
2008
84.73
50.26
15215
2009
78.74
79.60
18390
Southeast Airline Data
Airframe Cost
Engine Cost
Average
Year
per Aircraft
per Aircraft
Age (hrs)
2003
13.29
18.86
5107
2004
25.15
31.55
8145
2005
32.18
40.43
7360
2006
31.78
22.10
5773
2007
25.34
19.69
7150
2008
32.78
32.58
9364
2009
35.56
38.07
8259
Utilizing the software package provided with this text, we
can develop the following regression equations for the variables
of interest:
Northern Airlines—Airframe Maintenance Cost:
www.pearsonhighered.com/heizer and www.myomlab.com.
The following graphs portray both the actual data and the re-
gression lines for airframe and engine maintenance costs for both
airlines.
Note that the two graphs have been drawn to the same scale
to facilitate comparisons between the two airlines.
Comparison:
◼ Northern Airlines: There seem to be modest correlations
52 CHAPTER 4 FO R E C A S T I N G
those for Northern Airlines. From the graphs, at least, they
appear to be rising more sharply with age.
From an overall perspective, it appears that Southeast Airlines may
1. A plot of the data indicates a linear trend (least squares) mod-
el might be appropriate for forecasting. Using linear trend you
obtain the following:
x
y
x2
xy
y2
1
480
1
480
230,400
2
436
4
872
190,096
3
482
9
1,446
232,324
4
448
16
1,792
200,704
5
458
25
2,290
209,464
6
489
36
2,934
239,121
7
498
49
3,486
248,004
8
430
64
3,440
184,900
9
444
81
3,996
197,136
10
496
100
4,960
246,016
11
487
121
5,357
237,169
12
525
144
6,300
275,625
13
575
169
7,475
330,625
14
527
196
7,378
277,729
25
608
625
15,200
369,664
26
597
676
15,522
356,409
27
612
729
16,524
374,544
28
603
784
16,884
363,609
29
628
841
18,212
394,384
30
605
900
18,150
366,025
440.85 5.25 (time)
y
=+
r = 0.873, indicating a reasonably good fit
The student should report the linear trend results, but deflate
the forecast obtained based upon qualitative information about
industry and technology trends.
Because there is limited seasonality in the data, the linear
trend analysis above provides a good r2 of .76.
However, a more precise forecast can be developed addressing
the seasonality issue, which is done below. Methods a and c yield
r2 of .85 and .86, respectively, and methods b and d, which also
center the seasonal adjustment, yield r2 of .93 and .94, respectively.
2. Four approaches to decomposition of The Digital Cell Phone
data can address seasonality, as follows:
a) Multiplicative seasonal model,
Cases = 443.87 + 5.08 (time), r2 = .85, MAD = 20.89
b) Multiplicative Seasonal Model, with centered moving averages
Copyright ©2014 Pearson Education, Inc.
CHAPTER 4 FO R E C A ST I N G 47
Copyright ©2014 Pearson Education, Inc.
48 CHAPTER 4 FO R E C A S T I N G
4.51
Period
Demand
Exponentially Smoothed Forecast
1
7
5
2
9
5 + 0.2 × (7 – 5) = 5.4
5
13
5.90 + 0.2 × (9 – 5.90) = 6.52
6
8
6.52 + 0.2 × (13 – 6.52) = 7.82
7
Forecast
7.82 + 0.2 × (8 – 7.82) = 7.86
4.52
Actual
Forecast
|Error|
Error2
95
100
5
25
108
110
2
4
123
120
3
9
130
130
0
0
10
38
MAD = 10/4 = 2.5, MSE = 38/4 = 9.5
4.53 (a) 3-month moving average:
3-Month
Absolute
Month
Sales
Moving Average
Deviation
January
11
April
10
(11 + 14 + 16)/3 = 13.67
3.67
May
15
(14 + 16 + 10)/3 = 13.33
1.67
June
17
(16 + 10 + 15)/3 = 13.67
3.33
July
11
(10 + 15 + 17)/3 = 14.00
3.00
August
14
(15 + 17 + 11)/3 = 14.33
0.33
September
17
(17 + 11 + 14)/3 = 14.00
3.00
October
12
(11 + 14 + 17)/3 = 14.00
2.00
November
14
(14 + 17 + 12)/3 = 14.33
0.33
December
16
(17 + 12 + 14)/3 = 14.33
1.67
January
11
(12 + 14 + 16)/3 = 14.00
3.00
February
(14 + 16 + 11)/3 = 13.67
= 22.00
MAD = 2.20
(b) 3-month weighted moving average
(c) Based on a mean absolute deviation criterion, the
3-month moving average with MAD = 2.2 is to be pre-
ferred over the 3-month weighted moving average with
MAD = 2.72.
4.54 (a)
Actual
Cumulative
Cum.
Tracking
Week
Miles
Forecast
Error
Error
|Error|
MAD
Signal
1
17
17.00
0.00
–
0.00
0
2
21
17.00
+4.00
4.00
4.00
2
2
3
19
17.80
+1.20
5.20
5.20
1.73
3
4
23
18.04
+4.96
10.16
10.16
2.54
4
5
18
19.03
–1.03
9.13
11.19
2.24
4
6
16
18.83
–2.83
6.30
14.02
2.34
2.7
7
20
18.26
+1.74
8.04
15.76
2.25
3.6
8
18
18.61
–0.61
7.43
16.37
2.05
3.6
9
22
18.49
+3.51
10.94
19.88
2.21
5
10
20
19.19
+0.81
11.75
20.69
2.07
5.7
11
15
19.35
–4.35
7.40
25.04
2.28
3.2
12
22
18.48
+3.52
10.92
28.56
2.38
4.6
(b) The MAD = 28.56/12 = 2.38
(c) The cumulative error and tracking signals appear to
4.55
y
x
x2
xy
7
1
1
7
9
2
4
18
5
3
9
15
11
4
16
44
10
5
25
50
13
6
36
78
55
21
91
212
9.17
3.5
5.27 1.11
y
x
yx
=
=
=+
Period 7 forecast = 13.07
Period 12 forecast = 18.64, but this is far outside the range
of valid data.
Month
Sales
3-Month Moving Average Moving
Absolute Deviation
January
11
February
14
March
16
April
10
(1 × 11 + 2 × 14 + 3 × 16)/6 = 14.50
4.50
May
15
(1 × 14 + 2 × 16 + 3 × 10)/6 = 12.67
2.33
June
17
(1 × 16 + 2 × 10 + 3 × 15)/6 = 13.50
3.50
July
11
(1 × 10 + 2 × 15 + 3 × 17)/6 = 15.17
4.17
August
14
(1 × 15 + 2 × 17 + 3 × 11)/6 = 13.67
0.33
September
17
(1 × 17 + 2 × 11 + 3 × 14)/6 = 13.50
3.50
October
12
(1 × 11 + 2 × 14 + 3 × 17)/6 = 15.00
3.00
November
14
(1 × 14 + 2 × 17 + 3 × 12)/6 = 14.00
0.00
December
16
(1 × 17 + 2 × 12 + 3 × 14)/6 = 13.83
2.17
January
11
(1 × 12 + 2 × 14 + 3 × 16)/6 = 14.67
3.67
February
(1 × 14 + 2 × 16 + 3 × 11)/6 = 13.17
= 27.17
MAD = 2.72
50 CHAPTER 4 FO R E C A S T I N G
1
Standard error of the estimate:
294 1 20 1 70
2 5 2
3
yx
Y a Y b XY
Sn
− − − −
==
−−
= = =
4.62 Using software, the regression equation is: Games lost =
6.41 + 0.533 × days rain.
1. One way to address the case is with separate forecasting models
for each game. Clearly, the homecoming game (week 2) and the
Forecasts
Game
Model
2013
2014
R2
1
y = 30,713 + 2,534x
48,453
50,988
0.92
2
y = 37,640 + 2,146x
52,660
54,806
0.90
3
y = 36,940 + 1,560x
47,860
49,420
0.91
4
y = 22,567 + 2,143x
37,567
39,710
0.88
5
y = 30,440 + 3,146x
52,460
55,606
0.93
2. Revenue in 2013 = (239,000) ($50/ticket) = $11,950,000
Revenue in 2014 = (250,530) ($52.50/ticket) = $13,152,825
3. In games 2 and 5, the forecast for 2014 exceeds stadium ca-
pacity. With this appearing to be a continuing trend, the time has
come for a new or expanded stadium.
VIDEO CASE STUDIES
FORECASTING TICKET REVENUE FOR
3. Using the multiple regression model in the case:
Revenue = $14,996 + 10,801 (4) + 23,379 (3) + 10,784 (3)
= $160,743
4. Time of day for game, other competing sports events within
100 miles on that date, special half-time or pregame entertainment
planned, date set for a special group night (for example, Boy
Scouts or Rotary). These may be potential independent variable
for Perez’s model.
cafes, (2) retail sales, (3) banquet sales, (4) concert sales, (5) eval-
uating managers, and (6) menu planning. They could also employ
2. The POS system captures all the basic sales data needed to
drive individual cafe’s scheduling/ordering. It also is aggregated
at corporate HQ. Each entrée sold is counted as one guest at a
3. The weighting system is subjective, but is reasonable. More
weight is given to each of the past 2 years than to 3 years ago.
This system actually protects managers from large sales variations
(weather); hotel occupancy; spring break from colleges; beef pric-
es; promotional budget; etc.
5. Y = a + bx
Month
Advertising X
Guest Count Y
X2
Y2
XY
1
14
21
196
441
294
2
17
24
289
576
408
3
25
27
625
729
675
4
25
32
625
1,024
800
5
35
29
1,225
841
1,015
2
15,910 10 36.6
37.6 0.7996 36.6 8.3363 8.3
8.3363 0.7996
a
YX
−
= − =
=+
At $65,000; y = 8.3 + .8 (65) = 8.3 + 52 = 60.3, or 60,300 guests.
For the instructor who asks other questions than this one:
r2 = 0.8869
Std. error = 5.062
ADDITIONAL CASE STUDIES*
THE NORTH-SOUTH AIRLINE
Northern Airline Data
Airframe Cost
Engine Cost
Average
Year
per Aircraft
per Aircraft
Age (hrs)
2003
51.80
43.49
6512
2004
54.92
38.58
8404
2005
69.70
51.48
11077
2006
68.90
58.72
11717
2007
63.72
45.47
13275
2008
84.73
50.26
15215
2009
78.74
79.60
18390
Southeast Airline Data
Airframe Cost
Engine Cost
Average
Year
per Aircraft
per Aircraft
Age (hrs)
2003
13.29
18.86
5107
2004
25.15
31.55
8145
2005
32.18
40.43
7360
2006
31.78
22.10
5773
2007
25.34
19.69
7150
2008
32.78
32.58
9364
2009
35.56
38.07
8259
Utilizing the software package provided with this text, we
can develop the following regression equations for the variables
of interest:
Northern Airlines—Airframe Maintenance Cost:
www.pearsonhighered.com/heizer and www.myomlab.com.
The following graphs portray both the actual data and the re-
gression lines for airframe and engine maintenance costs for both
airlines.
Note that the two graphs have been drawn to the same scale
to facilitate comparisons between the two airlines.
Comparison:
◼ Northern Airlines: There seem to be modest correlations
52 CHAPTER 4 FO R E C A S T I N G
those for Northern Airlines. From the graphs, at least, they
appear to be rising more sharply with age.
From an overall perspective, it appears that Southeast Airlines may
1. A plot of the data indicates a linear trend (least squares) mod-
el might be appropriate for forecasting. Using linear trend you
obtain the following:
x
y
x2
xy
y2
1
480
1
480
230,400
2
436
4
872
190,096
3
482
9
1,446
232,324
4
448
16
1,792
200,704
5
458
25
2,290
209,464
6
489
36
2,934
239,121
7
498
49
3,486
248,004
8
430
64
3,440
184,900
9
444
81
3,996
197,136
10
496
100
4,960
246,016
11
487
121
5,357
237,169
12
525
144
6,300
275,625
13
575
169
7,475
330,625
14
527
196
7,378
277,729
25
608
625
15,200
369,664
26
597
676
15,522
356,409
27
612
729
16,524
374,544
28
603
784
16,884
363,609
29
628
841
18,212
394,384
30
605
900
18,150
366,025
440.85 5.25 (time)
y
=+
r = 0.873, indicating a reasonably good fit
The student should report the linear trend results, but deflate
the forecast obtained based upon qualitative information about
industry and technology trends.
Because there is limited seasonality in the data, the linear
trend analysis above provides a good r2 of .76.
However, a more precise forecast can be developed addressing
the seasonality issue, which is done below. Methods a and c yield
r2 of .85 and .86, respectively, and methods b and d, which also
center the seasonal adjustment, yield r2 of .93 and .94, respectively.
2. Four approaches to decomposition of The Digital Cell Phone
data can address seasonality, as follows:
a) Multiplicative seasonal model,
Cases = 443.87 + 5.08 (time), r2 = .85, MAD = 20.89
b) Multiplicative Seasonal Model, with centered moving averages
Copyright ©2014 Pearson Education, Inc.
