Chapter 03 – Forecasting
3–48
Education.
Warning on MSE: 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
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
forecast bias measure, when calculating MPE, negative and positive error terms offset each other.
Therefore, for a given time-series data, MPE MAPE.
Before stating the equations for MPE and MAPE, first we need to define Percentage Error. The
Percentage Error (PE) for a given time-series data measures the percentage points deviation of the
forecasted value from the actual value. The equations for PE in period i, MPE, and MAPE are given
below in equations 1, 2, and 3 respectively.
)3(
)2(
)1()100(
1
1
n
PE
MAPE
n
PE
MPE
A
FA
PE
n
i
i
n
i
i
i
ii
i
where:
Ai is the actual value from period i.
Fi is the forecasted (estimated) value from period i.
Both MPE and MAPE are more intuitive and easier to understand and interpret than most of the other
measures because 4.00% has far more meaning to the user than MSE of 224.00 or Tracking Signal value
of 3.50.
Chapter 03 – Forecasting
3–49
Education.
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
12
28
17
48
13
34
18
50
14
40
19
47
15
44
20
54
Problem 2
In solving problem 20 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:
t
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.
b. Compute the MAPE for the two sets of forecasts. Which of the two forecasting methods provides
higher overall accuracy with this data set?
Chapter 03 – Forecasting
3–50
Education.
Solutions to Enrichment Module Problems
Solution to Problem 1 (round % to two decimals)
Percentage Error Calculations using the Naïve Method
Period
Actual
Forecast
Error
i
PE
i
PE
11
25
–
–
–
–
12
28
25
3
10.71%*
10.71%
13
34
28
6
17.65%
17.65%
14
40
34
6
15.00%
15.00%
15
44
40
4
9.09%
9.09%
16
39
44
–5
-12.82%
12.82%
17
48
39
9
18.75%
18.75%
18
50
48
2
4.00%
4.00%
19
47
50
–3
-6.38%
6.38%
20
54
47
7
12.96%
12.96%
Sum
68.96%
107.36%
Chapter 03 – Forecasting
3–51
Education.
Percentage Error Calculations using the Linear Trend Equation
Period
Actual
Forecast
Error
i
PE
i
PE
11
25
34
–9
-36.00%*
36.00%
12
28
36
–8
-28.57%
28.57%
13
34
38
–4
-11.76%
11.76%
14
40
40
0
0.00%
0.00%
15
44
42
2
4.55%
4.55%
16
39
44
–5
-12.82%
12.82%
17
48
46
2
4.17%
4.17%
18
50
48
2
4.00%
4.00%
19
47
50
–3
-6.38%
6.38%
20
54
52
2
3.70%
3.70%
Sum
-79.11%
111.95%
%20.11
10
%95.111
%91.7
10
%11.79
MAPE
MPE
Note: We had 10 periods for which we had both actual demand and forecasts.
a. MPE & Forecast Bias:
Naïve method: Because the MPE is positive for the naïve forecasting method (7.66%), it is
underestimating sales by 7.66%.
Linear trend method: Because the MPE is negative (–7.91%) for the linear trend method, the
b. MAPE & Overall Accuracy
Naïve method: The MAPE 11.93%
Chapter 03 – Forecasting
3–52
Education.
To decide between the two methods, we have to compare the consequences and the cost of
overestimation with the consequences and the cost of underestimation. The cost of overestimation
involves the cost of carrying excess inventories, while the cost of underestimation includes the
cost of shortages, backordering, and lost sales.
If the cost of underestimation is less than the cost of overestimation, then the naïve method
should be selected.
If the cost of overestimation is less than the cost of underestimation, then the linear trend method
should be used.
Solution to Problem 2 (round % to two decimals)
a. Forecast Errors and Percentage Errors Using the First Forecasting Method
Period
ei
i
PE
i
PE
1
2
2.94%
2.94%
2
7
9.33%
9.33%
3
–2
-2.86%
2.86%
4
3
4.05%
4.05%
5
–3
-4.35%
4.35%
6
2
2.78%
2.78%
7
9
11.25%
11.25%
8
4
5.13%
5.13%
Sum
28.27%
42.69%
%34.5
8
%69.42
%53.3
8
%27.28
MAPE
MPE
Chapter 03 – Forecasting
b. Forecast Errors and Percentage Errors Using the Second Forecasting Method
Period
ei
i
PE
i
PE
1
2
2.94%
2.94%
2
7
9.33%
9.33%
3
–2
0.00%
0.00%
4
3
2.70%
2.70%
5
–3
-7.25%
7.25%
6
2
-5.56%
5.56%
7
9
2.50%
2.50%
8
4
-2.56%
2.56%
Sum
2.10%
32.84%
%11.4
8
%84.32
%26.0
8
%10.2
MAPE
MPE
We recommend the second forecasting method for two reasons:
1. The second forecasting method has less forecast bias because (MPE2 = 0.26%) < (MPE1 =
3.53%).