4
C H A P T E R
Forecasting
DISCUSSION QUESTIONS
1. Qualitative models incorporate subjective factors into the
AACSB: Application of knowledge
4. The steps that should be used to develop a forecasting
system are:
(f) Validate the forecasting model
(g) Make the forecast
8. MAD, MSE, and MAPE are common measures of forecast
accuracy. To find the more accurate forecasting model, forecast
9. The Delphi technique involves:
(a) Assembling a group of experts in such a manner as to pre-
(f) Repeating steps (b) through (e) several times as necessary
10. A time-series model predicts on the basis of the assumption
that the future is a function of the past, whereas an associative
CHAPTER 4 FO R E C A S T I N G 33
ACTIVE MODEL 4.2: Exponential Smoothing
4. At what level of alpha is the mean absolute deviation (MAD)
ACTIVE MODEL 4.4: Trend Projections
374 + 368 + 381
(a) 374.33 pints
34 CHAPTER 4 FO R E C A ST I N G
(d) The 3-year moving average appears to give better
results.
1
0
3,000
2
500
3,500
3
3,450
4
350
175
3,625
5
75
3,663
Total
Year
2
8
9
Demand
Naïve
7.0
Naïve tracks the ups and downs best but lags the data by one
period. Exponential smoothing is probably better because it
smooths the data and does not have as much variation.
=+
= + =
=+
= + − =
July June
August July
(a) 0.2(Forecasting error)
42 0.2(40 – 42) 41.6
(b) 0.2(Forecasting error)
41.6 0.2(45 41.6) 42.3
FF
FF
4.4
September
20
9
81
180
October
20
10
100
200
=
3,700 + 3,800
(a) 3,750 miles
2
4.5
in 6th year.
Year
Mileage
2-Year
Moving Average
Error
|Error|
1
3,000
2
4,000
3
3,400
3,500
–100
100
4
3,800
3,700
100
100
5
3,700
3,600
100
100
Totals
100
300
300
MAD 100.
3
==
4.5 (c) Weighted 2-year M.A. with .6 weight for most recent year.
Year
Mileage
Forecast
Error
|Error|
1
3,000
2
4,000
3
3,400
3,600
–200
200
4
3,800
3,640
160
160
5
3,700
3,640
60
60
420
Forecast for year 6 is 3,740 miles.
==
420
MAD 140
3
4.6
Y Sales
X Period
X2
XY
January
20
1
1
20
February
21
2
4
42
March
15
3
9
45
April
14
4
16
56
May
13
5
25
65
June
16
6
36
96
July
17
7
49
119
August
18
8
64
144
November
21
11
121
231
December
23
12
144
276
Sum
218
78
650
1,474
Average
18.2
6.5
(a)
CHAPTER 4 FO R E C A S T I N G 35
(b) [i] Naïve The coming January = December = 23
[ii] 3-month moving (20 + 21 + 23)/3 = 21.33
[iii] 6-month weighted [(0.1 × 17) + (.1 × 18)
+ (0.1 × 20) + (0.2 × 20)
++ =
(96 88 90)
(a) 91.3
3
4.8
+=
(88 90)
(b) 89
2
36 CHAPTER 4 FO R E C A ST I N G
Copyright ©2017 Pearson Education, Inc.
(c) The forecasts are about the same.
4.9
(d) Table for Problem 4.9(d):
= .1
= .3
= .5
Month
Price per Chip
Forecast
|Error|
Forecast
|Error|
Forecast
|Error|
January
$1.80
$1.80
$.00
$1.80
$.00
$1.80
$.00
February
1.67
1.80
.13
1.80
.13
1.80
.13
March
1.70
1.79
.09
1.76
.06
1.74
.04
April
1.85
1.78
.07
1.74
.11
1.72
.13
May
1.90
1.79
.11
1.77
.13
1.78
.12
June
1.87
1.80
.07
1.81
.06
1.84
.03
July
1.80
1.80
.00
1.83
.03
1.86
.06
August
1.83
1.80
.03
1.82
.01
1.83
.00
September
1.70
1.81
.11
1.82
.12
1.83
.13
October
1.65
1.80
.15
1.79
.14
1.76
.11
November
1.70
1.78
.08
1.75
.05
1.71
.01
December
1.75
1.77
.02
1.73
.02
1.70
.05
Totals
$.86
$.86
$.81
MAD (total/12)
$.072
$.072
$.0675
= .5 is preferable, using MAD, to = .1 or = .3. One could
also justify excluding the January error and then dividing by
n = 11 to compute the MAD. These numbers would be $.078
(for = .1), $.078 (for = .3), and $.074 (for = .5).
Note: If table numbers are not rounded, answers would be $0.077 (for alpha = 0.3) and $0.072 (for alpha = 0.5).
4.10
Year
1
2
3
4
5
6
7
8
9
10
11
Forecast
Demand
4
6
4
5.0
10.0
8.0
7.0
9.0
12.0
14.0
15.0
(a)
3-year moving
4.7
5.0
6.3
7.7
8.3
8.0
9.3
11.7
13.7
(b)
3-year weighted
4.5
5.0
7.3
7.8
8.0
8.3
10.0
12.3
14.0
4.11
(a)
Year
1
2
3
4
5
6
7
8
9
10
11
Forecast
Demand
4
6.0
4.0
5.0
10.0
8.0
7.0
9.0
12.0
14.0
15.0
Exp. smoothing
5
4.7
5.1
4.8
4.8
6.4
6.9
6.9
7.5
8.9
10.4
11.8
Copyright ©2017 Pearson Education, Inc.
11years 11
4.12
t
Day
Actual
Demand
Forecast
Demand
1
Monday
88
88
2
Tuesday
72
88
4
Thursday
48
80
5
Friday
72
Answer
Ft = Ft–1 + (At–1 – Ft–1)
Let = .25. Let Monday forecast demand = 88
F2 = 88 + .25(88 – 88) = 88 + 0 = 88
F3 = 88 + .25(72 – 88) = 88 – 4 = 84
4.13 (a) Exponential smoothing, = 0.6:
Exponential
Absolute
Year
Demand
Smoothing = 0.6
Deviation
1
45
41
4.0
2
50
41.0 + 0.6(45–41) = 43.4
6.6
3
52
43.4 + 0.6(50–43.4) = 47.4
4.6
4
56
47.4 + 0.6(52–47.4) = 50.2
5.8
5
58
50.2 + 0.6(56–50.2) = 53.7
4.3
6
?
53.7 + 0.6(58–53.7) = 56.3
Exponential
Absolute
Year
Demand
Smoothing = 0.9
Deviation
1
45
41
4.0
2
50
41.0 + 0.9(45–41) = 44.6
5.4
3
52
44.6 + 0.9(50–44.6 ) = 49.5
2.5
4
56
49.5 + 0.9(52–49.5) = 51.8
4.2
5
58
51.8 + 0.9(56–51.8) = 55.6
2.4
6
?
55.6 + 0.9(58–55.6) = 57.8
= 18.5
MAD = 3.7
3-Year
Absolute
Year
Demand
Moving Average
Deviation
1
45
2
50
3
52
4
56
(45 + 50 + 52)/3 = 49
7
5
58
(50 + 52 + 56)/3 = 52.7
5.3
6
?
(52 + 56 + 58)/3 = 55.3
(c) Trend projection:
Absolute
Year
Demand
Trend Projection
Deviation
1
45
42.6 + 3.2 × 1 = 45.8
0.8
2
50
42.6 + 3.2 × 2 = 49.0
1.0
3
52
42.6 + 3.2 × 3 = 52.2
0.2
4
56
42.6 + 3.2 × 4 = 55.4
0.6
5
58
42.6 + 3.2 × 5 = 58.6
0.6
6
?
42.6 + 3.2 × 6 = 61.8
38 CHAPTER 4 FO R E C A ST I N G
Copyright ©2017 Pearson Education, Inc.
=+
=
=
22
–
–
–
Y a bX
XY nXY
b
X nX
a Y bX
X
Y
XY
X2
1
45
45
1
2
50
100
4
3
52
156
9
4
56
224
16
5
58
290
25
Then: X = 15, Y = 261, XY = 815, X2 = 55,
X
= 3,
Y
= 52.2
Therefore:
6
815 – 5 3 52.2 3.2
55 – 5 3 3
52.20 – 3.20 3 42.6
42.6 3.2 6 61.8
b
a
Y
==
= =
= + =
Forecast Methodology
MAD
Exponential smoothing, = 0.6
5.06
Exponential smoothing, = 0.9
3.7
3-year moving average
6.2
Trend projection
0.64
Based on a mean absolute deviation criterion, the trend projection
is to be preferred over the exponential smoothing with = 0.6,
exponential smoothing with = 0.9, or the 3-year moving average
forecast methodologies.
4.14
Method 1: MAD: (0.20 + 0.05 + 0.05 + 0.20)/4 = .125 better
2
495
488.4
6.6
3
518
522.0
4.0
4
563
555.6
7.4
5
589.2
5.2
6
4.15 (a)
Forecast 3-Year
Absolute
Year
Sales
Moving Average
Deviation
1
450
2
495
3
518
4
563
(450 + 495 + 518)/3 = 487.7
75.3
5
584
(495 + 518 + 563)/3 = 525.3
58.7
6
(518 + 563 + 584)/3 = 555.0
= 134
(b)
134
MAD 67
2
==
(c) MSE
Year
Sales
Error2
1
450
2
495
3
518
4
563
(75.3)2 = 5,675
5
584
(58.7)2 = 3,442
9,117S=
9,117
MSE 4,558.5
2
==
4.16 (a)
Year
Sales Y
X2
XY
1
450
1
450
2
495
4
990
= 2610
= 55
= 8166
3, 522XY==
Y a bX
XY nXY
b
X nX
a Y bX
yx
y
=+
− −
= = = =
−
−
= − = − =
=+
= + =
22
8166 (5)(3)(522) 336 33.6
55 (5)(9) 10
522 (33.6)(3) 421.2
421.2 33.6
421.2 33.6 6 622.8
(b) MAD
Year
Sales
Forecast Trend
Absolute Deviation
1
450
454.8
4.8
MAD = 5.6
4
563
5
CHAPTER 4 FO R E C A S T I N G 39
(c) MSE
5
4.17
Forecast Exponential
Absolute
Year
Sales
Smoothing = 0.6
Deviation
1
450
410.0
40.0
2
495
410 + 0.6(450 – 410) = 434.0
61.0
3
518
434 + 0.6(495 – 434) = 470.6
47.4
4
563
470.6 + 0.6(518 – 470.6) = 499.0
64.0
5
584
499 + 0.6(563 – 499) = 537.4
46.6
Forecast Exponential
Absolute
Year
Sales
Smoothing = 0.9
Deviation
1
450
410.0
40.0
2
495
410 + 0.9(450 – 410) = 446.0
49.0
3
518
446 + 0.9(495 – 446) = 490.1
27.9
4
563
490.1 + 0.9(518 – 490.1) = 515.2
47.8
5
584
515.2 + 0.9(563 – 515.2) = 558.2
25.8
6
558.2 + 0.9(584 – 558.2) = 581.4
= 190.5
MAD = 38.1
(Refer to Solved Problem 4.1)
For = 0.3, absolute deviations for years 1–5 are 40.0, 73.0, 74.1,
96.9, and 88.8, respectively. So MAD = 372.8/5 = 74.6.
=
=
=
=
=
=
0.3
0.6
0.9
MAD 74.6
MAD 51.8
MAD 38.1
Because it gives the lowest MAD, the smoothing constant of
= 0.9 gives the most accurate forecast.
4.18 We need to find the smoothing constant . We know in
or 48 = 50 + 42 – 50
or –2 = –8
So, .25 =
4.19 Trend adjusted exponential smoothing: = 0.1, = 0.2
Unadjusted
Adjusted
Month
Income
Forecast
Trend
Forecast
|Error|
Error2
February
70.0
65.0
0.0
65
5.0
25.0
March
68.5
65.5
0.1
65.6
2.9
8.4
April
64.8
65.89
0.16
66.05
1.2
1.6
May
71.7
65.92
0.13
66.06
5.6
31.9
June
71.3
66.62
0.25
66.87
4.4
19.7
July
72.8
67.31
0.33
67.64
5.2
26.6
August
68.16
68.60
24.3
113.2
MAD = 24.3/6 = 4.05, MSE = 113.2/6 = 18.87. Note that all
numbers are rounded.
Note: To use POM for Windows to solve this problem, a period 0,
which contains the initial forecast and initial trend, must be added.
40 CHAPTER 4 FO R E C A ST I N G
Unadjusted
Adjusted
Month
Demand (y)
Forecast
Trend
Forecast
Error
|Error|
Error2
February
70.0
65.0
0
65.0
5.00
5.0
25.00
March
68.5
65.5
0.4
65.9
2.60
2.6
6.76
April
64.8
66.16
0.61
66.77
–1.97
1.97
3.87
May
71.7
66.57
0.45
67.02
4.68
4.68
21.89
June
71.3
67.49
0.82
68.31
2.99
2.99
8.91
July
72.8
68.61
1.06
69.68
3.12
3.12
9.76
Totals
419.1
16.42
20.36
76.19
Average
69.85
2.74
3.39
12.70
August forecast
71.30
(Bias)
(MAD)
(MSE)
Based on the MSE criterion, the exponential smoothing with = 0.1, = 0.8 is to be preferred
over the exponential smoothing with = 0.1, = 0.2. Its MSE of 12.70 is lower. Its MAD of 3.39 is
also lower than that in Problem 4.19.
4.20 Trend adjusted exponential smoothing: = 0.1, = 0.8
( )
( )
( )( ) ( )( )
5 4 4 4
1 – 0.2 19 0.8 20.14
3.8 16.11 19.91
F A F T4.21 = + + = +
= + =
()
( ) ( )( )
( )( ) ( )
5 5 4 4
– 1 – 0.4 19.91 – 17.82
0.6 2.32 0.4 2.09
1.39 0.84 1.39 2.23
T F F T= + =
+=
+ = + =
5 5 5 19.91 2.23 22.14FIT F T= + = + =
( )
()
( )( ) ( )( )
6 5 5 5
1 – 0.2 24 0.8 22.14
4.8 17.71 22.51
F A F T= + + = +
= + =
()
( ) ( ) ( )
( )
6 6 5 5
– 1 – 0.4 22.51 – 19.91 0.6 2.23
0.4 2.6 1.34
1.04 1.34 2.38
T F F T= + = +
=+
= + =
6 6 6 22.51 2.38 24.89FIT F T= + = + =
7 6 6 6
7 7 6 6
7 7 7
(1 – )( ) (0.2)(21) (0.8)(24.89)
4.2 19.91 24.11
( – ) (1 – ) (0.4)(24.11 – 22.51)
(0.6)(2.38) 2.07
24.11 2.07 26.18
F A F T
T F F T
FIT F T
4.22 = + + = +
= + =
= + =
+=
= + = + =
8 7 7 7
(1 – )( ) (0.2)(31)
(0.8)(26.18) 27.14
F A F T= + + =
+=
()
( ) ( )
( )
8 8 7 7
– 1 – 0.4 27.14 – 24.11
0.6 2.07 2.45
T F F T= + =
+=
( )
()
( )( )
( )( )
8 8 8
9 8 8 8
27.14 2.45 29.59
1 – 0.2 28
0.8 29.59 29.28
FIT F T
F A F T
= + = + =
= + + =
+=
()
( ) ( )( )
( )( )
9 9 8 8
– 1 – 0.4 29.28 – 27.14
0.6 2.45 2.32
T F F T= + =
+=
9 9 9 29.28 2.32 31.60FIT F T= + = + =
4.23 Students must determine the naïve forecast for the four
months. The naïve forecast for March is the February actual of 83,
etc.
(a)
Actual
Forecast
|Error|
|% Error|
March
101
120
19
100 (19/101) = 18.81%
April
96
114
18
100 (18/96) = 18.75%
May
89
110
21
100 (21/89) = 23.60%
June
108
108
0
100 (0/108) = 0%
58
61.16%
58
MAD (for management) 14.5
4
61.16%
MAPE (for management) 15.29%
4
==
==
(b)
Actual
Naïve
|Error|
|% Error|
March
101
83
18
100 (18/101) = 17.82%
April
96
101
5
100 (5/96) = 5.21%
May
89
96
7
100 (7/89) = 7.87%
June
108
89
19
100 (19/108) = 17.59%
49
48.49%
==
==
49
MAD (for naïve) 12.25
4
48.49%
MAPE (for naïve) 12.12%
4
Naïve outperforms management.
(c) MAD for the manager’s technique is 14.5, while MAD for the
naïve forecast is only 12.25. MAPEs are 15.29% and 12.12%,
respectively. So the naïve method is better.
4.24
Number of
Accidents
Month
(y)
x
xy
x2
January
30
1
30
1
February
40
2
80
4
March
60
3
180
9
April
90
4
360
16
Totals
220
10
650
30
Averages
y
= 55
x
= 2.5