TOOLS
AND
TECHNIQUES
Forecasting
10
Forecasting methods can be classified as either statistical or judgmental. Statistical forecasting is
based on the assumption that the future will be an extrapolation of the past. Judgmental forecasts rely primarily on qualitative information and experience.
JUDGMENTAL FORECASTING
Techniques commonly used in judgmental forecasts are expert opinion, market surveys, and
the Delphi method. Forecasting by expert opinion simply consists of gathering judgments
and opinions of key personnel based on their experience and knowledge of the situation.
Though such forecasts are sometimes very inaccurate, the advantage of using expert judgment is its low cost in comparison to other methods. Market surveys use questionnaires, telephone contacts, or personal interviews as a means of gathering data.The cost of such surveys
is high due to labor costs, postage, low response rates, and postsurvey processing. Moreover,
these surveys are sometimes biased. In the Delphi method, several experts are questioned
individually about their perceptions of future events.The experts are not consulted as a group
to avoid consensus being reached because of dominant personality factors. Instead, the
responses and supporting arguments of each individual are summarized by an outside party
and returned to the experts along with further questions.The process iterates until the group
reaches a consensus, which usually takes only a few rounds.The Delphi method is useful for
long-range forecasting and for predicting technological changes.
STATISTICAL FORECASTING
If historical data are available, they should first be plotted over time.The plot can be used to
answer such questions as, Is there an upward or downward trend? Is there significant variation about the trend line? Is there any evidence of seasonality?
The choice of a forecasting method depends on several criteria.Among them are the time
span for which the forecast is being made, the needed frequency of forecast updating, data
requirements, the level of accuracy desired, the level of aggregation, and the quantitative skills
needed.The time span is one of the most critical criteria. Different techniques are applicable
for long-range, intermediate-range, and short-range forecasts.The level of aggregation often
dictates the appropriate method. Forecasting the total amount of laundry soap to produce
over the next planning period is certainly different from forecasting the amount of each different size to produce. Aggregate forecasts are generally much easier to develop, whereas
detailed forecasts require more time and resources.
Statistical methods of forecasting are based on the analysis of historical data, called a time
series, in which a set of observations is measured at successive points in time or over successive periods of time. We will discuss two popular techniques: moving averages and
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T O O L S
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TOOLS AND TECHNIQUES 10
Forecasting
exponential smoothing, which uses a weighted average of past time-series values to forecast the value of the time series in the next period.
MOVING AVERAGES
(most recent n data values)
n
The moving average for one period is generally used as the forecast for the next period.
EXAMPLE T10.1. Moving Average. To illustrate the moving-averages method, consider
the 12 weeks of data presented in Table T10-1 and Figure T10-1.These data show the number
of gallons of gasoline sold by a gasoline distributor over the past 12 weeks.
TABLE T10-1
|
Gasoline Sales Time Series
Week
Sales
(1,000s of gallons)
Week
Sales
(1,000s of gallons)
1
2
3
4
5
6
17
21
19
23
18
16
7
8
9
10
11
12
20
18
22
20
15
22
FIGURE T10-1
Sales (1000s of gallons)
1 0
The moving average associated with any time period is simply an average of the most recent
n data values in the time series. Mathematically, the moving-average calculation is
|
Graph of Gasoline Sales Time Series
25
20
15
0
1
2
3
4
5
6
7
8
9
10
11
12
Week
To use moving averages to forecast the gasoline sales time series, we must first select the
number of data values to be included in the moving average.As an example, let us compute fore-
Forecasting
21 + 19 +23
= 21
3
This provides a forecast for week 5 of 21. The error associated with this forecast is 18 – 21 =
–3.Thus, we see that the forecast error can be positive or negative, depending on whether the
forecast is too low or too high.
A complete summary of the three-week moving-average calculations for the gasoline
sales time series is shown in Table T10-2 and Figure T10-2.The average squared error for these
forecasts is 92/9 = 10.22.This is called the Mean Squared Error, or MSE.
TABLE T10-2
|
Summary of Moving-Average Calculations
Week
Time-Series
Value
1
2
3
4
5
6
7
8
9
10
11
12
17
21
19
23
18
16
20
18
22
20
15
22
3-Period
MovingAverage
Forecast
19
21
20
19
18
18
20
20
19
Totals
Forecast
Error
4
–3
–4
1
0
4
0
–5
3
0
(Error)2
16
9
16
1
0
16
0
25
9
92
1 0
17 + 21 + 19
= 19
3
This moving-average value is then used as the forecast for week 4. Since the actual value
observed in week 4 is 23, we see that the forecast error in week 4 is 23 – 19 = 4.
The calculation for the second three-week moving average is
T O O L S
casts based on a three-week moving average.The moving-average calculation for the first three
weeks of the gasoline sales time series is
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TOOLS AND TECHNIQUES 10
FIGURE T10-2
1 0
Sales (1000s of gallons)
T O O L S
4
Forecasting
|
Graph of Moving-Average Forecasts
3-week moving-average
forecasts
25
20
15
0
1
2
3
4
5
6
7
8
9
10
11
12
Week
The number of data values to be included in the moving average is arbitrary, and different numbers of data values differ in their ability to forecast the time series accurately. One
way to find the best number is to use trial-and-error to identify the number that minimizes
the average squared error measure of forecast accuracy.
EXPONENTIAL SMOOTHING
The basic exponential-smoothing model is
Ft + 1 = Yt + (1 – )Ft
where Ft + 1 = forecast of the time series for period t + 1
Yt = actual value of the time series in period t
Ft = forecast of the time series for period t
= smoothing constant (0 1)
To see that the forecast for any period is a weighted average of all the previous actual values
for the time series, suppose we have a time series consisting of three periods of data: Y1, Y2,
and Y3.To get the exponential-smoothing calculations started, we let F1 equal the actual value
of the time series in period 1; that is, F1 = Y1. Hence the forecast for period 2 would be written as
F2 = Y1 + (1 – )F1
= Y1 + (1 – )Y1
= Y1
In general, then, the exponential-smoothing forecast for period 2 is equal to the actual value
of the time series in period 1.
To obtain the forecast for period 3, we substitute F2 = Y1 in the expression for F3; the
result is
F3 = Y2 + (1 – )Y1
Forecasting
F4 = Y3 + (1 – )[Y2 + (1 – )Y1]
= Y3 + (1 – )Y2 + (1 – )2Y1
TABLE T10-3
|
Exponential-Smoothing Forecasting
Time-Series
Value (Yt)
ExponentialSmoothing
Forecast (Ft)
for = 0.2
Forecast
Error (Yt – Ft)
1
2
3
4
5
6
7
8
17
21
19
23
18
16
20
18
17.00
17.80
18.04
19.03
18.83
18.26
18.61
4.00
1.20
4.96
–1.03
–2.83
1.74
–0.61
9
10
11
12
22
20
15
22
18.49
19.19
19.35
18.48
3.51
0.81
–4.35
3.52
Week (t)
To generate a forecast for week 13, we obtain
F13 = 0.2Y12 + 0.8F12
= 0.2(22) + 0.8(18.48) = 19.18
Figure T10-3 is the plot of the actual and the forecast time-series values. Note in particular how the forecasts “smooth out” the irregular fluctuations in the time series.
In the preceding smoothing calculations, we used a smoothing constant of = 0.2,
although any value of between 0 and 1 is acceptable. Some values yield better forecasts than
others, however. Some insight into choosing a good value for can be obtained by rewriting the exponential-smoothing model as follows:
1 0
Hence we see that F4 is a weighted average of the first three time-series values.The sum of
the coefficients, or weights, for Y1, Y2, and Y3 will always equal 1. A similar argument can be
made to show that any forecast Ft + 1 is a weighted average of the previous t time-series values. An advantage of exponential smoothing is that it is a simple procedure and requires very
little historical data. Once the smoothing constant, , has been selected, only two pieces of
information are needed to compute the forecast for the next period.
To illustrate the exponential-smoothing approach to forecasting, consider the gasoline
sales data again. Applying the formulas we have described with = 0.2 leads to the forecasts
in the third column of Table T10-3.
T O O L S
Finally, substituting this expression for F3 in the expression for F4, we obtain
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TOOLS AND TECHNIQUES 10
Forecasting
F t + 1 = Yt + (1 – )Ft
Ft + 1 = Yt + F – F t
Ft + 1 = Ft + (Yt – F t )
1 0
Forecast Forecast error
in period t in period t
|
Graph of Time Series and Exponential-Smoothing Forecasts
Sales (1,000s of gallons)
FIGURE T10-3
Actual time series
25
20
15
Forecast time series with a = 2
0
1
2
3
4
5
6
7
8
9
10
11
12
Week
So we see that the new forecast, Ft + 1, is equal to the previous forecast, Ft, plus an adjustment,
which is times the most recent forecast error, Yt – Ft.That is, the forecast in period t + 1
is obtained by adjusting the forecast in period t by a fraction of the forecast error. If the time
series is very volatile and contains substantial random variability, a small value of the smoothing constant is preferred.The reason for this choice is that since much of the forecast error is
due to random variability, we do not want to overreact and adjust the forecasts too quickly.
For a fairly stable time series with relatively little random variability, larger values of the
smoothing constant have the advantage of quickly adjusting the forecasts when forecasting
errors occur and therefore allowing the forecast to react faster to changing conditions.
FORECAST ERROR AND ACCURACY
In Tables T10-2 and T10-3 we calculated the forecast error, which is simply the difference
between the time-series value and the forecast.A common statistical measure of forecast accuracy is the mean square error (MSE), which is the average of the squared forecast errors.
Using a spreadsheet, it is easy to calculate exponential smoothing forecasts and forecast errors
for different smoothing constants to find one that minimizes the MSE. It is good practice to
analyze new data to see whether the smoothing constant should be revised to provide better
forecasts.
Another common measure of forecast accuracy is the mean absolute deviation
(MAD). This measure is simply the average of the sum of the absolute deviations for all the
forecast errors. Using the errors in Table T10-3, we obtain
Forecasting
T O O L S
|Y – Ft|
MAD = t n
A major difference between MSE and MAD is that the MSE measure is influenced much
more by large forecast errors than by small errors (since the errors are squared for the MSE
measure). The selection of the best measure of forecasting accuracy is not a simple matter.
Indeed, forecasting experts often disagree on which measure should be used. MAD, however,
is useful in tracking a forecast; that is, in monitoring the forecast to determine whether the
forecasting technique being used remains adequate.The tracking method most often used is
to compute a moving sum of forecast errors divided by the MAD; that is,
(Y – Ft)
t MAD
where the sums of both forecast errors and MAD are computed over the last n periods.
Problems and Exercises
1. Refer to the gasoline sales time-series data.
a. Compute four- and five-week moving averages for the time series.
b. Compute the mean square error (MSE) for the four- and five-week movingaverage forecasts.
c. What appears to be the best number of weeks of past data to use in computing the
moving average? Remember that the MSE for the three-week moving average is
10.22.
2. Forecasts and actual sales of portable CD players at Just Say Music! are given in the table
below.
Forecast
150
220
205
256
250
Sales
170
229
192
271
238
a. What is the mean absolute deviation (MAD)?
b. What is the forecast for August, using a three-period moving average?
c. If the actual demand for August is 268, what would the forecast be for September?
1 0
4.00 +1.20 + . . . + 4.35 + 3.52
= 11
28.56
= = 2.596
11
Month
March
April
May
June
July
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TOOLS AND TECHNIQUES 10
Forecasting
3. The weekly number of customers calling a technical support hot line for the past 12
weeks is given below.
1 0
Week
1
2
3
4
5
6
Number of Customers
2,750
3,100
3,250
2,800
2,900
3,050
Week
7
8
9
10
11
12
Number of Customers
3,300
3,100
2,950
3,000
3,200
3,150
Use these 12 weeks of data and exponential smoothing with = 0.4 to develop a forecast of demand for the 13th week.
4. The historical number of pizzas sold on Friday nights at a local pizza restaurant is given
below.
Week
1
2
3
4
5
Demand
22
18
23
26
27
Week
6
7
8
9
10
Demand
29
30
29
32
31
a. Use exponential smoothing with = 0.2 to develop a forecast for week 11.
b. Compare your forecasts from part (a) with smoothing constants of 0.5 and 0.8.
Which value provides the best forecasts? Explain.
5. The number of component parts used in a production process each of the last 10 weeks
is reported below.
Week
1
2
3
4
5
Parts
200
350
250
360
250
Week
6
7
8
9
10
Parts
210
280
350
290
320
a. Use a smoothing constant of 0.25 to develop the exponential-smoothing values for
this time series. Indicate your forecast for next week.
b. Find the best value of the smoothing constant to minimize MSE.