Chapter 03 – Forecasting
3–33
b.
Month
Error
Error2
38.120
9
345
20
1
20
2==
−
n
e
1
–8
64
2
–2
4
3
4
16
4
7
49
The errors may be cyclical, suggesting that there may be a cyclical component in demand
that is being overlooked in the forecast.
31. a. yt = 35.8 + 4.018t
Year
10
11
12
13
14
Forecast
75.98
80.00
84.02
88.04
92.05
c.
Year
Sales
Forecast
Error
10
77.2
75.98
1.22
11
82.1
80.00
2.10
12
87.8
84.02
3.78
13
90.6
88.04
2.56
14
98.9
92.05
6.55
Month
5
9
81
6
5
25
7
0
0
8
–3
9
16
–1
Chapter 03 – Forecasting
3–34
32. a.
Period
Actual
Forecast 1
Forecast 2
error 1
error 2
2
1
e
2
2
e
1
e
2
e
1
37
36
36
+1
+1
1
1
1
1
2
39
38
37
+1
+2
1
4
1
2
3
37
40
38
–3
–1
9
1
3
1
4
39
42
38
–3
+1
9
1
3
1
5
45
46
41
–1
+4
1
16
1
4
–2
+4
34
35
16
15
89.3
9
35
MSE 78.3
9
34
MSE
21
====
b. The errors for alternative 1 cycle (+1, +1, –3, –3, –1, +3, +1, –1, –1), although all are within
2s control limits. The errors for alternative 2 (+1, +2, –1, +1, +4, –3, 0, +1, –1, +1) don’t
6
49
46
52
+3
–3
9
9
3
3
7
47
46
47
1
0
1
0
8
49
48
48
+1
1
1
1
1
9
51
52
52
–1
–1
1
1
1
1
10
54
55
53
–1
+1
1
1
1
1
Chapter 03 – Forecasting
3–35
33.
A
(sales)
F
(Forecast)
A–F
(Error)
Cumulative
Error
Error
Error
e2
MAD
TS
15
15
0
0
0
0
0
0
0
21
20
1
1
1
1
1
.5
2
23
25
–2
–1
2
3
4
1
–1
30
30
0
–1
0
3
0
.75
–1.333
14.5
36
2=
=
=n
e
MSE
No, the forecast is not performing adequately.
As you can see from the tracking signal values,
34.
A
(sales)
T = 10 + 5t
Trend
F = T * S
Forecast
Error
Cumulative
Error
(Error)2
Error
Error
MAD
TS
14
15
13.5
.5
.5
.25
.5
.5
1
20
20
19
1
1.5
1
1
1.5
.75
2
24
25
26.25
–2.25
–.75
5.063
2.25
3.75
1.25
–.6
31
30
33
–2
–2.75
4
2
5.75
1.438
–1.912
31
35
31.5
–.5
–3.25
.25
.5
6.25
1.25
–2.6
37
40
38
1
1
7.25
43
45
47.25
0
4.25
11.50
1.643
0
48
50
55
–7
–7
7
18.50
2.313
–3.026
52
55
49.5
–4.5
6.25
2.5
21.0
–1.929
333.2
9
21
==
MAD
32
35
–4
3
6
9
1.2
–3.333
38
40
–6
2
8
4
1.333
–4.50
42
45
–3
–9
3
9
1.571
–5.729
47
50
3
9
1.75
–6.85
Chapter 03 – Forecasting
3–36
Case: M & L Manufacturing
1. The potential benefit of using a formalized approach to forecasting is that it will be easier to
utilize the computer and easier to quantify the information. A less formalized approach is more
2. Product 1
Plotting the data for Product 1 reveals a linear pattern with the expectation of demand in week 7.
Demand in week 7 is unusually high and does not fit the linear trend pattern of the remaining
Therefore in this case, the demand of 90 in week 7 will be replaced with 71.5 = [(67 + 76)/2].
x
y
x2
xy
1
50
1
50
2
54
4
108
3
57
9
171
4
60
16
240
5
64
25
320
7
49
9
79
81
711
10
82
100
820
11
85
121
935
13
92
169
Chapter 03 – Forecasting
3–37
Product 1 (continued)
498.3
)105()1015)(14(
)5.1020(105()5.8449)(14(
)x()x(n
)y)(x()xy(n
b222
=
−
−
=
−
−
=
The trend line is T = 46.536 + 3.5t
The next four forecasts (t = 15, 16, 17, 18) are:
Period
Forecast (T = 46.658 + 3.498t)
15
T = 46.658 + 3.498 (15) = 99.13
Product 2
Plotting the data for Product 2 yields a more complex pattern: There is a spike once every four weeks; the
values between the spikes are fairly close to each other. In addition, the data appear to be increasing at the
rate of one unit per week. An intuitive approach would be to use the average of the three nonspike periods
plus 1.0 to predict the next three nonspike periods. Doing so for the data up to period 15 yields a very
small average forecast error (MAD = 0.54). Given the fact that we have only two data points following
the last spike, a reasonable forecast might be to use the last three period average plus 1.0 (i.e., 43.33 to
Hence, the forecasts are:
Period
Forecast
15
43.33
16
50.5 or 51
17
44.5
18
44.5
Chapter 03 – Forecasting
3–38
Case: Highline Financial Services, Ltd.
Aligning data by quarters, we can see (in the tables and in the figures) that demand for service A is
increasing, demand for service B is decreasing, and demand for service C is mixed. Note, though, that
total annual demand for service C has changed only slightly.
A Quarter B Quarter C Quarter
Year
1
2
3
4
1
2
3
4
1
2
3
4
Service A
80
100
120
Series1
80
100
Service B
Year 1
Service C
80
100
120
Chapter 03 – Forecasting
3–39
Enrichment Module: Additional Methods for Evaluating Forecast Accuracy
The major problem in determining which forecast accuracy measure to use is that there is no universally
accepted accuracy measure. In Chapter 3, several different accuracy measures are covered. In order to
develop a better understanding of the forecast accuracy measures, first we must understand the nature of
the forecast errors. There are two types of forecast errors.
1. Mean Forecast Error (MFE)
2. Tracking Signal
3. Control Charts
When we sum the error terms, if there is no bias, positive and negative error terms will cancel each other
out and the MFE will be zero. As was pointed out above, negative MFE is an indication of overestimation
and positive MFE is an indication of underestimation. However, if the positive and negative error values
tend to cancel each other out and the MFE or Tracking Signal value is zero or near zero, then we can
conclude that the forecasting method does not result in bias (underestimation or overestimation). Even if
1. Mean Absolute Deviation (MAD)
In order to be able to assess both the overall accuracy and forecast bias, an analyst should probably utilize
at least one method from each category. In the next section we will discuss two additional methods for
evaluating forecast accuracy.
Relative measures of forecast accuracy:
Chapter 03 – Forecasting
3–40
The utilization of MSE as a criterion in determining the accuracy of forecasts has some drawbacks. One
of the drawbacks is that in many cases it is not appropriate to compare MSE values obtained from
different forecasting models because different methods use different ways of obtaining the forecasted
values. Thus, comparison of methods using a single criterion such as MSE becomes questionable.
The relative forecast accuracy measures that we will discuss in the remaining portion of this section are:
1. Mean Percentage Error (MPE)
2. Mean Absolute Percentage Error (MAPE)
MPE measures the forecast bias while MAPE measures overall forecast accuracy. As with any other
)1()100(
A
FA
PE
i
ii
i
−
=
where:
Ai is the actual value from period i, and;
Fi is the forecasted (estimated) value from period i.
Chapter 03 – Forecasting
3–41
Problems for the enrichment module
Problem 1
An analyst must decide between two different forecasting techniques for weekly sales of bicycles: a linear
trend equation and the naïve approach. The linear trend equation is:
ii XY 212
ˆ+=
, and it was developed
using data from periods 1 through 10. Based on the data from periods 11 through 20, calculate the MPE
and MAPE. Based on the values of MPE and MAPE, comment on which of the two methods has the
greater overall accuracy. Compare the two methods in terms of the forecast bias.
T
Units Sold
T
Units Sold
11
25
16
39
Problem 2
In solving problem 26 in the textbook, we calculated both MAD and MSE values. In this exercise we are
going to use the same data and information and calculate MPE and MAPE values. The revised problem is
stated as follows:
Two different forecasting techniques (F1 and F2) were used to forecast demand for cases of bottled water.
Actual demand and the two sets of forecasts are as follows:
Time Period
Actual Demand
Forecasted demand 1
(F1)
Forecasted demand 2
(F2)
1
68
66
66
2
75
68
68
3
70
72
70
4
74
71
72
5
69
72
74
6
72
70
76
7
80
71
78
8
78
74
80
a. Compute MPE for both sets of forecasts. Which of the two forecasting methods has a higher
forecast bias? Explain.
12
28
17
48
13
34
18
50
14
40
19
47
15
44
20
54
Chapter 03 – Forecasting
3–42
Solutions to enrichment module problems
Solution to problem 1
Percentage Error Calculations using the Naïve Method
Period
Actual
Forecast
Error
i
PE
i
PE
11
25
–
–
–
–
12
28
25
3
.1071
.1071
13
34
28
6
.1765
.1765
14
40
34
6
.15
.15
===
%66.7
9
6895.
6895.
MPE PE
i
Percentage Error Calculations using the Linear Regression Equation
Period
Actual
Forecast
Error
i
PE
i
PE
11
25
34
–9
–.36
.36
12
28
36
–8
–.2857
.2857
13
34
38
–4
–.1176
.1176
14
40
40
0
0
0
15
44
42
.0455
16
39
44
–5
–.1282
.1282
17
48
46
2
.0417
.0417
18
50
48
2
.04
.04
19
47
50
–3
–.0638
.0638
20
54
52
2
.037
.037
−=
−
=−=
%91.7
10
7911.
7911.
MPE PE
i
15
44
40
.0909
16
39
44
–5
–.1282
17
48
39
9
.1875
.1875
18
50
48
2
.04
.04
19
47
50
–3
–.0638
20
54
47
7
.1296
.1296
Chapter 03 – Forecasting
3–43
Since the MAPE values for the two methods are approximately equal, the overall accuracy of the two
methods is about the same. Both methods are predicting approximately 11% away from the actual.
Since the MPE is –7.91% for the linear trend equation, the trend equation is overestimating sales by
7.91%. On the other hand, since the MPE is positive for the naïve forecasting method, it is overestimating
the sales by 7.66%. In this situation, in order to decide between the two methods, we have to compare the
Solution to problem 2
a. Forecast Errors and Percentage Errors Using the First Forecasting Method
Time Period
ei
i
PE
i
PE
1
2
.02941
.02941
2
7
.09333
.09333
3
–2
–.02857
.02857
4
3
.04054
.04054
5
.04348
6
.02778
7
9
.11250
.11250
8
4
.05128
.05128
==−=
%535.3
8
28279.
28279.
MPE PE
i
Chapter 03 – Forecasting
3–44
b. Forecast Errors and Percentage Errors Using the Second Forecasting Method
Time Period
ei
i
PE
i
PE
1
2
.02941
.02941
2
7
.09333
.09333
3
–2
0
0
===
%264.0
8
0211.
0211.
MPE PE
i
We recommend the second forecasting method for two reasons:
5
–3
.07246
6
2
.05556
7
9
.025
.025