Procedures for Selecting a Double Moving Average for Forecasting. The double moving average
is designed to develop a forecast that smoothes random variation and projects any trend exhibited
in the data. As with the single moving average, no averaging period is best for forecasting in all
circumstances. You must identify the best averaging period for each situation:
Step 1. Develop 1-period forecasts using different available periods so that you can
compare forecasts with actual observations to evaluate accuracy.
• Normally, we use at least a three period double moving average. Since a double moving
average is a moving average of moving averages, you cannot begin to calculate a 3-
period double moving average until Period 5. You can calculate 4-period double moving
averages beginning in Period 7. For any value of n, you can calculate an n-period double
moving average beginning in Period 2n – 1.
• To conduct a meaningful evaluation of forecast accuracy, you must have at least two
forecasts and actual data for the same period for accuracy evaluation. As a result, you
must have 2n + 1 data points in order to calculate a double moving average forecast and
the related MADF.
Step 2. Evaluate 1-period forecast accuracy using mean absolute deviations (MADs)
between forecasts and actual observations.
Step 3. Select the averaging period found to produce the most accurate results.
Calculations Required for Forecast Development . Develop 1-period forecasts using available
averaging periods so that you can compare forecasts with actual observations to evaluate
accuracy.
Step 1A. Calculate Double Moving Averages . Calculate double moving averages for available
averaging periods using the following equation:
Where:
M2 n,t = An n-period double moving average calculated
in Period t
M1 n,t = An n-period single moving average calculated
in period t
n = Number of periods in the moving average
Note : You must use the same value of n for calculating both M1 and M2.
Step 1B. Develop Forecasts Using Moving Averages : Once you calculate a double moving
average, you can use that average to develop a forecast.
FM2 n,t,t+h = A n,t + Bn,t + h
Where:
FM2 n,t,t+h = The, n-period, double moving average forecast made in period t for period t+h
A n,t = The intercept fpr an n-period double moving average forecast, calculated:
A n,t = 2M1 n,t – M2 n,t
Bn,t = The slope for an n-period double moving average forecast, calculated:
Bn,t = (M1 n,t = M2 n,t )
n = The number of periods in the moving average
t = The period in which the forecast is made
h = The horizon, the number of periods you are forecasting into the future
Note : Depending on the value of n and the period in which the forecast is made, there is a
unique intercept (A n,t ) and slope (Bn,t ).
Developing 1-Period Forecasts for Example 1 Data .
Step 1A. Calculate Double Moving Averages . In the previous section, we determined that we
should use a double moving average to develop a forecast from the data below. Here we will use
the data to demonstrate the procedures for double moving average forecast development.
Quarter
(t)
Wage Rate
(Y)
1 $12.50
2 $11.80
3 $12.85
4 $13.95
5 $13.30
6 $13.95
7 $15.00
8 $16.20
9 $16.10
3Y M1 3M1 M2
37.15 12.38
38.60 12.87
40.10 13.37 38.62 12.87
41.20 13.73 39.97 13.32
42.25 14.08 41.18 13.73
45.15 15.05 42.86 14.29
47.30 15.77 44.90 14.97
Step 1B. Develop Forecasts Using Moving Averages . Period 5 is the first period that we can
develop a 3-period double moving average forecast, because that is the first period that we have
the values of M1 and M2 that we need to make the forecast. In Period 5, we can make a forecast
for Period 6 as follows:
A 3,5 = 2M1 3,5 – M2 3,5
= 2(13.37) – 12.87
= 26.74 – 12.87
= 13.87
B3,5 = (2 (3-1)) (M1 3,5 – M2 3,5 )
= (2 (3-1)) (13.37 – 12.87)
= (2 2) (13.37 -12.87)
= 13.37 – 12.87
= .50
FM2 3,5,6 = A 3,5 + B3,5 (h)
= 13.87 + .50(1)
= 13.87 + .50
= 14.37
3-Period Double Moving Average Forecast
t Actual M1 M2 A B FM2
1 12.50
2 11.80
3 12.85 12.38
4 13.95 12.87
5 13.30 13.37 12.87 13.87 0.50
6 13.95 13.73 13.32 14.14 0.41 14.37
7 15.00 14.08 13.73 14.43 0.35 14.55
8 16.20 15.05 14.29 15.81 0.76 14.78
9 16.10 15.77 14.97 16.57 0.80 16.57
Forecasts developed using a 4-period moving average and the same procedures are shown in the
following table. Note that only two forecasts can be made for comparison with actual
observations.
4-Period Double Moving Average Forecast
t Actual M1 M2 A B FM2
1 12.50
2 11.80
3 12.85
4 13.95 12.78
5 13.30 12.98
6 13.95 13.51
7 15.00 14.05 13.33 14.77 0.48
8 16.20 14.61 13.79 15.43 0.55 15.25
9 16.10 15.31 14.37 16.25 0.63 15.98
Calculations Required for Evaluating Forecast Accuracy . Evaluate 1-period forecast accuracy
using mean absolute deviations (MADs) between forecasts and actual observations. Here we use
the same formula for calculating MAD F that we used in evaluating the accuracy of single
moving averages.
Where:
MAD F = The mean absolute deviation of the forecast
S = Summation of all the variables that follow the symbol
|D| = The absolute value of the deviation (i.e., the difference, without regard to sign) between the
actual value which occurred and the value forecasted
|D| = |Y – F|
n = The number of deviations (Ds) computed
3-Period Double Moving Average Forecast Evaluation
t Actual M1 M2 A B FM2 D |D|
1 12.50
2 11.80
3 12.85 12.38
4 13.95 12.37
5 13.30 13.37 12.87 13.87 0.50
6 13.95 13.73 13.32 14.14 0.41 14.37 -0.42 0.42
7 15.00 14.08 13.73 14.43 0.35 14.55 0.45 0.45
8 16.20 15.05 14.29 15.81 0.76 14.78 1.42 1.42
9 16.10 15.77 14.97 16.57 0.80 16.57 -0.47 0.47
Total Absolute Deviation 2.76
Mean Absolute Deviation = |D| n = 2.76 4 = 0.69
4-Period Double Moving Average Forecast Evaluation
Actual M1 M2 t B FM2 D |D|
1 12.50
2 11.80
3 12.85
4 13.95 12.78
5 13.30 12.98
6 13.95 13.51
7 15.00 14.05 13.33 14.77 0.48
8 16.20 14.61 13.79 15.43 0.55 15.25 0.95 0.95
9 16.10 15.31 14.37 16.25 0.63 15.98 0.12 0.12
Total Absolute Deviation 1.07
Mean Absolute Deviation = |D| n = 1.07 2 = 0.69
Select Averaging Period
Select the averaging period found to produce the most accurate results.
Summary Of MAD Computations
n MAD F
3 0.69
4 0.54
The lowest MADf in this example was attained using a 4-period double moving average.
Accordingly, you should select a 4-period double moving average for forecasting.
Use a Double Moving Average in Forecasting . Use the moving average with the lowest MAD F
for forecasting. Based on our evaluation of the data in Example 1, we would use the 4-period
double moving average for forecasting. For example our forecast for Period 13 [four periods (h)
into the future] would be $18.77, calculated as follows:
A 3,5 = 2M1 4,9 – M2 4,9
= 2(15.31) – 14.37
= 30.62 – 14.37
= 16.25
B3,5 = (2 (4-1)) (M1 4,9 – M2 4,9 )
= (2 (4-1)) (15.31 – 14.37)
= (2 3) (15.31 – 14.37)
= (2 3) (.94)
= .63
FM2 3,5,6 = A 4,9 + B4,9 (h)
= 16.25 + .63 (4)
= 16.25 + 2.52
= 18.77
The selection of the most accurate averaging period for forecast development is essential. For
example, using different averaging periods and the data in this example, we could have
calculated two very different forecasts.
Forecast Comparison
n FM2 n,9,13
3 $19.77
4 $18.77
6.5 – Identifying Issues And Concerns
Questions to Consider in Analysis . As you perform price or cost analysis, consider the issues
and concerns identified in this section, whenever you use moving averages.
• Is a moving average the best choice for estimate development?
When using a moving average, you assume that the trend experienced over time is the best guide
available to forecast future variable values. If that assumption is not correct, you should use
another technique. Detailed estimates that consider all the facts involved are normally more
defensible in negotiations than the result of any estimating relationship. If an independent
variable (other than time) can be identified and measured, another comparison technique may
provide better results than moving average analysis. For example, estimating parts demand based
on sales and usage data would probably produce better results than an estimate based on use of a
moving average. A moving average can estimate price changes based on recent periods but it
cannot predict a turning point that will alter the historical pattern.
• Is the type of moving average selected appropriate for the situation?
When there is a trend in the data, you should use a double moving average. When there is no
trend, you should use a single moving average. If you use a single moving average in a situation
where a trend exists, your forecast will not consider the trend.
• Is the averaging period the best choice for the data?
You should select the averaging period that provides the best estimates when tested against
actual observations. Take special care in your analysis when the moving average covers a large
number or periods (e.g., 12 months). Selection of an average that covers a large number of
periods is often appropriate because it dampens the effect of random fluctuation. However, an
average that considers a large number of data points will also make it more difficult to identify a
trend in the data. Occasionally, an estimator will use a large number of periods to mask a trend in
the data. When analyzing estimates made using a moving average, you should look at the raw
data and consider appropriate alternative estimating procedures.
• How far into the future can you forecast?
The farther into the future that you forecast, the greater the risk. Remember, that you cannot
predict a change in an historical trend with a moving average.
In this chapter, you will learn improvement curve concepts and their application to cost and price
analysis.
• 7.0 – Chapter Introduction
• 7.1 – Situations for Use
• 7.2 – Analyzing Improvement Using Unit Data and Unit Theory
• 7.3 – Analyzing Improvement Using Lot Data and Unit Theory
• 7.4 – Fitting a Unit Curve
• 7.5 – Estimating Using Unit Improvement Curve Tables
• 7.6 – Identifying Issues and Concerns
7.0 Chapter Introduction
7.0.1 Basic Improvement Curve Concept
You may have learned about improvement curves using the name learning curve analysis.
Today, many experts feel that the term learning curve implies too much emphasis on learning by
first-line workers. They point out that the theory is based on improvement by the entire
organization not just first-line workers. Alternative names proposed for the theory include:
improvement curve, cost-quantity curve, experience curve, and others. None have been
universally accepted. In this text, we will use the term improvement curve to emphasize the need
for efforts by the entire organization to make improvements to reduce costs.
Just as there are many names for the improvement curve, there are many different formulations.
However, in each case the general concept is that the resources (labor and/or material) required
to produce each additional unit declines as the total number of units produced over the items
entire production history increases. The concept further holds that decline in unit cost can be
predicted mathematically. As a result, improvement curves can be used to estimate contract
price, direct labor-hours, direct material cost, or any other recurring contract cost.
7.0.2 Improvement Curve History
The improvement curve is based on the concept that, as a task is performed repetitively, the time
required to perform the task will decrease. Management planners have followed that element of
the concept for centuries, but T. P. Wright pioneered the idea that improvement could be
estimated mathematically. In February 1936, Wright published his theory in the Journal of
Aeronautical Sciences as part of an article entitled “Factors Affecting the Cost of Airplanes.”
Wright’s findings showed that, as the number of aircraft produced in sequence increased, the
direct labor input per airplane decreased in a regular pattern that could be estimated
mathematically.
During the mobilization for World War II, both aircraft companies and the Government became
interested in the theory. Among other considerations, the theory implied that a fixed amount of
labor and equipment could be expected to produce larger and larger quantities of defense
products as production continued.
After World War II, the Government engaged the Stanford Research Institute (SRI) to study the
validity of the improvement curve concept. The study analyzed essentially all World War II
airframe direct labor input data to determine whether there was sufficient evidence to establish a
standard estimating model. The SRI study validated a mathematical model based on the World
War II findings that could be used as a tool for price analysis. However, that model was slightly
different than the one originally offered by Wright.
Since World War II, the improvement curve concept has been used by Government and industry
to aid in pricing contracts. Over the years, the improvement curve has been used as a contract
estimating and analysis tool in a variety of industries including: airframes, electronics systems,
machine tools, shipbuilding, missile systems, and depot level maintenance of equipment.
Improvement curves have also been applied to service and construction contracts where tasks are
performed repetitively.
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