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Since 1936, many different formulations have been proposed to explain and estimate the
improvement that takes place in repetitive production efforts. Of these, the two most popular are
the unit improvement curve and the cumulative average improvement curve
Unit Improvement Curve. The unit improvement curve is the model validated by the postWorld War II SRI study. The formulation is also known by two other names: Crawford curve,
after one of the leaders of the SRI research; and Boeing curve, after one of the firms that first
embraced its use.
Unit curve theory can be stated as follows:
As the total volume of units produced doubles the cost per unit decreases by some constant
percentage.
The constant percentage by which the costs of doubled quantities decrease is called the rate of
learning. The term “slope” in the improvement curve analysis is the difference between 100
percent and the rate of improvement. If the rate of improvement is 20 percent, the improvement
curve slope is 80 percent (100 percent – 20 percent). The calculation of slope is described in
detail later in the chapter.
Unit curve theory is expressed in the following equation:
Y = AX B
Where:
Y = Unit cost (hours or dollars) of the X th unit
X = Unit number
A = Theoretical cost (hours or dollars) of the first unit
B = Constant that is related to the slope and the rate of change of the improvement curve. It is
calculated from the relationship:
In calculating B, the slope MUST be expressed in decimal form rather than percentage form.
Then B will be a negative #, leading to the decreasing property stated above.
7.0.4 Identifying Basic Improvement Curve Theories: Cumulative Average Improvement
Curve
Cumulative Average Improvement Curve. The cumulative average improvement curve is the
model first introduced by Wright in 1936. Like the unit improvement curve, the cumulative
average curve is also known by two other names: Wright Curve, after T.P. Wright; and Northrop
Curve, after one of the firms that first embraced its use.
Cumulative average theory can be stated as follows:
As the total volume of units produced doubles the average cost per unit decreases by some
constant percentage.
As with the unit improvement curve, the constant percentage by which the costs of doubled
quantities decrease is called the rate of improvement. The slope of the improvement curve
analysis is the difference between 100 percent and the rate of learning. However, the rate of
improvement and the slope are measured using cumulative averages rather than the unit values
used in unit improvement curve analysis.
Cumulative average curve theory is expressed in the following equation:
Y = AX B
Where:
= Cumulative average unit cost (hours or dollars) of units
through the X th unit
All other symbols have the same meaning used in describing the unit improvement curve.
Curve Differences. Note that the only difference between definitions of the unit improvement
curve and the cumulative average improvement curve theories is the word average . In the unit
curve, unit cost is reduced by the same constant percentage. In the cumulative average curve, the
cumulative average cost is reduced by the some constant percentage.
The most significant practical difference between the two different formulations is found in the
first few units of production. Over the first few units, an operation following the cumulative
average curve will experience a much greater reduction in cost (hours or dollars) than an
operation following a unit curve with the same slope. In later production, the reduction in cost
for an operation following a cumulative average curve will be about the same as an operation
following a unit curve with the same slope.
Because of the difference in early production, many feel that the unit curve should be used in
situations where the firm is fully prepared for production; and the cumulative average curve
should be used in situations where the firm is not completely ready for production. For example,
the cumulative average curve should be used in situations where significant tooling or design
problems may NOT be completely resolved. In such situations, the production of the first units
will be particularly inefficient but improvement should be rapid as problems are resolved.
In practice, firms typically use one formulation regardless of differences in the production
situation. Most firms in the airframe industry use the cumulative average curve. Most firms in
other industries use the unit curve.
7.1 Identifying Situations for Use
The improvement curve cannot be used as an estimating tool in every situation. Situations that
provide an opportunity for improvement or reduction in production hours are the types of
situations that lend themselves to improvement curve application. Use of the improvement curve
should be considered in situations where there is:
• A high proportion of manual labor.
It is more difficult to reduce the labor input when there is limited labor effort, the labor effort is
machine paced, or individual line workers only touch the product for a few seconds.
• Uninterrupted production.
As more and more units are produced the firm becomes more adept at production and the labor
hour requirements are reduced. If supervisors, workers, tooling, or other elements of production
are lost during a break in production, some improvement will also likely be lost.
• Production of complex items.
The more complex the item the more opportunity there is to improve.
• No major technological change.
The theory is based on continuing minor changes in production and in the item itself. However,
if there are major changes in technology, the benefit of previous improvement may be lost.
• Continuous pressure to improve.
The improvement curve does not just happen; it requires management effort. The management of
the firm must exert continuous pressure to improve. This requires investment in the people and
equipment needed to obtain improvement.
7.1.1 Situations for Use
As you examine situations that appear to have potential for improvement curve application,
consider management emphasis on the following factors affecting the rate of improvement:
• Job Familiarization By Workers.
As noted earlier, many feel that this element has been overemphasized over the years. Still,
workers do improve from repetition and that improvement is an important part of the
improvement curve.
• Improved Production Procedures.
As production continues, both workers and production engineers must constantly be on the
lookout for better production procedures.
• Improved Tooling and Tool Coordination.
Part of the examination of production procedures must consider the tooling used for production.
Tooling improvements offer substantial possibilities for reduction of labor requirements.
• Improved Work Flow Organization.
Improving the flow of the work can substantially reduce the labor effort that does not add value
to the product. Needless movement of work in progress can add significant amounts of labor
effort.
• Improved Product Producibility.
Management and workers must constantly consider product changes that will make the product
easier to produce without degrading the quality of the final product.
• Improved Engineering Support.
The faster production problems can be identified and solved, the less production labor effort will
be lost waiting for problem resolution.
• Improved Parts Support.
As production continues, better scheduling should be possible to eliminate or significantly
reduce worker time lost waiting for supplies. In addition, production materials more appropriate
for production can be identified and introduced to the production process.
7.1.2 Factors that Support Improvement
As you examine situations that appear to have potential for improvement curve application,
consider management emphasis on the following factors affecting the rate of improvement:
• Job Familiarization By Workers . As noted earlier, many feel that this element has
been overemphasized over the years. Still, workers do improve from repetition and that
improvement is an important part of the improvement curve.
• Improved Production Procedures . As production continues, both workers and
production engineers must constantly be on the lookout for better production procedures.
• Improved Tooling and Tool Coordination . Part of the examination of production
procedures must consider the tooling used for production. Tooling improvements offer
substantial possibilities for reduction of labor requirements.
• Improved Work Flow Organization . Improving the flow of the work can substantially
reduce the labor effort that does not add value to the product. Needless movement of
work in progress can add significant amounts of labor effort.
• Improved Product Producibility . Management and workers must constantly consider
product changes that will make the product easier to produce without degrading the
quality of the final product.
• Improved Engineering Support . The faster production problems can be identified and
solved, the less production labor effort will be lost waiting for problem resolution.
• Improved Parts Support . As production continues, better scheduling should be
possible to eliminate or significantly reduce worker time lost waiting for supplies. In
addition, production materials more appropriate for production can be identified and
introduced to the production process.
7.2 Analyzing Improvement Using Unit Data and the Unit Theory
Section Introduction
In this text, we will only consider application of the unit improvement curve in making initial
contract estimates. There are many texts that address other improvement curve theories (e.g.,
cumulative average improvement curves), as well as many advanced issues such as the effects of
contract changes, breaks in production, and retained learning.
Improvement Illustration
To illustrate the effect of the unit curve, assume that the first unit required 100,000 labor-hours to
produce. If the slope of the curve is 80 percent slope, the following table demonstrates the laborhours required to produce units at successively doubled quantities.
Units
Produced
LABORHOURS Per
Unit at
Doubled
Quantities
Difference in
LABORHOURS Per
Unit at
Doubled
Quantities
Rate of
Improvement
(%)
Slope of
Curve
(%)
1 100,000
2 80,000 20,000 20 80
4 64,000 16,000 20 80
8 51,200 12,800 20 80
16 40,960 10,240 20 80
32 32,768 8,192 20 80
Obviously, the amount of labor-hour reduction between doubled quantities is not constant. The
number of hours of reduction between doubled quantities is constantly declining. However, the
rate of change or decline remains constant (20 percent).
Also note that the number of units required to double the quantity produced is constantly
increasing. Between Unit #1 and Unit #2, it takes only one unit to double the quantity produced.
Between Unit #16 and Unit #32, 16 units are needed.
7.2.1 Analysis Techniques Graphing the Data
Improvement curves are modeled using an exponential equation. Exponential equations can be
solved using logarithms. Plotting improvement curve data on a logarithmic scale causes the
points to lie on a straight line, which allows for projection of future costs. Historically this
process was accomplished manually; however, it can now be accomplished using the regression
function on most data analysis tools or using computer modeling programs.
Since it is often helpful to understand the theory behind the use of data analysis tools and
computer models, this text will include information and illustrations on how to manually
calculate the cost of future units as well as introduce the reader to the differences when
calculating using the regression function.
Rectangular Coordinate Graph. A labor-hour graph of this data curve drawn on ordinary graph
paper (rectangular coordinates) becomes a curve as shown in the graph below. In this graph
equal spaces represent equal amounts of change. When thinking of numbers in terms of their
absolute values, this graph presents an accurate picture, but it is difficult to make an accurate
prediction from this curve.
The graph is a curve because the number of hours of reduction between doubled quantities is
constantly declining and an increasing number of units are required to double the quantity
produced. Note that most of the improvement takes place during the early units of production.
The curve will eventually become almost flat. The number of production hours could become
quite small but it will never reach zero.
Log-Log Graph. To examine the data and make predictions using unit improvement curve
theory, we need to transform the data to logarithms. One way of making the transformation is
through the use of log-log graph paper also known as full-logarithmic graph paper.
7.2.2 Log-Log Paper
There are several special elements that we must consider when using log-log graph paper.
• There is already a scale indicated on both the horizontal and vertical axes. Note that there
are no zeros. Values can approach zero but never reach it.
• The scale only goes from “1” to “1”. Each time the number scale goes from “1” to “1”, the
paper depicts a cycle. Each “1” moving up on the vertical axis or to the right on the
horizontal axis is 10 times the “1” before it. You should mark the actual scale you are
using in the margin of the log-log paper before starting to plot points.
• In improvement curve analysis, always graph the number of the unit produced on the
horizontal axis. Assign the first “1” on the left of the page a value of 1 representing the
first unit produced. The second “1” is 10. The third “1” is 100. The fourth “1” is 1,000.
• Always graph the cost in hours or dollars on the vertical axis. The scale will change
depending on the data being graphed. The first “1” can be .001, .01, 1, 100, 1,000 or any
other integral power of 10. Whatever the value assigned to the first “1,” the next “1” is 10
times more, and the next one 10 times more than that. To determine the scale to be used:
• Estimate the largest number to be plotted or read on the Y axis. This figure will probably
be the theoretical cost of the first unit. For example, suppose this is 60,000 hours.
• Determine the next integral power of ten above this number (e.g., the next integral power
above 60,000 is 100,000).
• Assign this value to the horizontal line at the top of the upper cycle on the Y axis. The
horizontal line at the top of the next lower cycle must then represent 10,000 of the same
units, and the line at the bottom of the lower cycle represents 1,000.
• On log-log graph paper, the distances between numbers on each axis are equal for equal
percentage changes. For example, the distance between “1” and “2” is the same as
between “4” and “8;” both represent a 100 percent increase.
7.2.3 Log-Log Graph
You can obtain surprisingly accurate results from a log-log graph, but your accuracy greatly
• Always use a sharp pencil.
• Make points plotted on the paper as small as possible and the lines as narrow as possible.
• When the smallest possible point has been marked on the paper, it may easily be lost
sight of or confused with a blemish in the paper. To avoid this, draw a small ring around
the point. Circles, triangles, and squares are also used to identify points which belong to
different sets of data.
• Exercise great care in drawing a line. If it is supposed to go through a point, it should
pass exactly through it, not merely close to it.
A graph of the data described in the example above forms a perfectly straight line when plotted
on log-log paper. That is, a straight line passes exactly through each of the points. A straight line
on log-log paper indicates that the rate of change is constant.
Since improvement curve theory assumes continuing improvement at a constant rate, the straight
line becomes an excellent estimating tool. Assuming that improvement will continue at the same
rate, the line can be extended to estimate the cost of future units.
7.2.4 Calculating the Theoretical Value of Unit #1
When we discuss improvement curves, we normally describe them in terms of the theoretical
value for Unit #1 and the slope of the curve. With these two values, you can use graph paper,
tables, or computer programs to estimate the cost of future units.
The value of Unit #1 is referred to as a theoretical value (T1), because in most cases you will
not know the actual cost of Unit #1. Instead, T1 is the value indicated by the line-of-best-fit. On
a graph, it is the point where the line-of-best-fit and the vertical line representing Unit #1
intersect. (Remember, the graph of the improvement curve always begins with Unit #1.)
7.2.5 Estimating the Slope
The term “slope” as used for improvement curves is a mathematical misnomer. It cannot be
related to the definition of slope in a straight line on rectangular coordinates. Instead, the slope of
an improvement curve is equal to 100 minus that constant percentage decrease (100 – rate of
improvement).
You can calculate the slope of a curve, by dividing the unit cost (Y X ) at some unit (X) into the
unit cost (Y 2x ) at twice the quantity (2X) and multiplying the resulting ratio by 100.
Therefore, you can measure the slope of an improvement curve drawn on log-log paper by
reading a cost (Y X ) at any quantity, X; reading a cost (Y 2X ) at any quantity, 2X; dividing the
second value by the first; and multiplying by 100.
For example, if the number of hours to make Unit #5 is 70 and the number of hours to make Unit

# 10 is 50, the slope of the improvement curve is:

Slope = 100 (Y 10 Y 5 )
Slope = 100 (50 70)
Slope = 71.4 percent
Slope Research Data. The post-war SRI study revealed that many different slopes were
experienced by different firms, sometimes by different firms manufacturing the same products.
In fact, manufacturing data collected from the World War II aircraft manufacturing industry had
slopes ranging from 69.7 percent to almost 100 percent. These slopes averaged 80 percent.
Research by DCAA in 1970 found curves ranging from less than 75 percent to more than 95
percent. The average slope was 85 percent.
Slope Selection and Verification. Unfortunately, information on industry average curves is
frequently misapplied by practitioners who use them as a standard or norm. Because each
situation is different, you should select a slope based on your analysis of the situation and not on
general averages. The order of preference in slope selection is:
• A curve developed from data pertaining to the production of the same product (as we did
above).
• The median percentage from a group of curves for items having some similarity to the
end item.
• The median percentage from the product category in which the item would most likely be
included.

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