# THEORY AND APPLICATIONS OF THE LEARNING CURVE

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Publication Date:

July 24, 1956

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REPORT

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85
RESEARCH AID
THEORY AND APPLICATIONS
OF THE LEARNING CURVE
CIA/RR RA-7
24 July 1956
CENTRAL INTELLIGENCE AGENCY
IM
OFFICE OF RESEARCH AND RE~Rj
? TO ARCHIVES RECORDS CENTER
IMMEDIATELY AFTER USE
JD BOl(---
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I
This material contains information affecting
the National Defense of the United States
within the meaning of the espionage laws,
Title 18, USC, Sees. 793 and 794, the trans-
mission or revelation of which in any manner
to an unauthorized person is prohibited by law.
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RESEARCH AID
THEORY AND APPLICATIONS OF THE LEARNING CURVE
CIA/RR RA-7
(ORR Project 33.978)
Office of Research and Reports
FOR
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FOREWORD
Important indications of the capabilities, vulnerabilities, and
intentions of a foreign country may be obtained through analysis of
that country's expenditures on key military end items. This type
of analysis is considerably complicated by the decline in labor
costs and in total production costs of military equipment over time.
Many of these items involve repetitive major assembly tasks using
large inputs of labor. Although the first units are produced over
relatively long periods of time and at relatively high costs, the
unit cost declines significantly as experience is gained.
The application of a single price over an extended output could
result in misleading distortions. The trend of learning, on the
other hand, tends to approximate the trend of output costs. The
trend of learning may also aid in evaluating the significance of
cost-saving announcements and in providing the basis for predicting
the probable cost trends of programs which may be undertaken in the
future.
Producers in the US and other Western countries have observed
this downward trend and have ascertained that it follows what has
come to be known as a learning, progress, experience, or improve-
ment curve. Similar trends are probably experienced in the USSR.
This research aid describes the nature of several types of these
curves and indicates how they may be derived and applied.
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CONTENTS
Page
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
II. History and Development . . . . . . . . . . . . . . . . . . . 1
III. The Learning Curve on Arithmetic Graph Paper . . . . . . . . 2
TV. The Learning Curve on Log-Log Graph Paper . . . . . . . . . . 3
V. Learning Curves of Various Percentages . . . . . . . . . . . 4
VI. Mathematical Development . . . . . . . . . . . . . . . . . . 6
A. Class 1 . . . . . . . . . . .
1. Cumulative Average Time Curve . . . . . . . . . . . .
2. Unit Time Curve . . . . . . . . . . . . . . . . . . .
3. Total Time Curve . . . . . . . .
B. Class 2 . . . . . . . .
1. Unit Time Curve . . . . . . . . . . . . . . . . .
2. Total Time Curve . . . . . . . . . . . . . . . . . .
3. Cumulative Average Time Curve . . . . . . . . . . . .
9
9
10
C . Class 3 . . . . . . . . . . . . . . . . . . . . . . . . . 10
1. Unit Time Curve . . . . . . . . . . . . . . . . . . . 10
2. Total Time Curve . . . . . . . . . . . . . . . 11
3. Cumulative Average Time Curve . . . . . . . . . . . . 11
VII. Determination of the Slope of a Theoretical Learning Curve. . 11
VIII. Determination of the Slope of an Actual Learning Curve . . . 12
:X. Uses of the Learning Curve . . . . . . . . . . . . 15
X. Applicability . . . . . . . . . . . . . . . . . . . . . . . . 15
1. Product Innovation . . . . . . . . . . . . . . . . . . . 15
2. Assembly Time as a Proportion of Total Time . . . . . . . 16
3. Advance Planning . . . . . . . . . . . . . . . . . . . . 16
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Appendixes
Page
Appendix A. Solution ofProblems in Which the Cumulative Average
Time Curve is a Straight Line on Log-Log Graph
Paper of the Type y = K . . . . . . . . ? . . . . . 17
xn
Appendix B. Solution of Problems in Which the Unit Time Curve
is a Straight Line on Log-Log Graph Paper
of the Type y = K . . . . . . . . . . . . . . . . . 23
T
1. -Percentage Rates of Learning of Selected US Aircraft Companies
During World War II . . . . . . . . . . . . . . . . . . . . . 5
2. Numerical Value of the Slopes of 50- to 100- Percent Learning
Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Charts
Following Page
Figure 1. The 80-Percent Learning Curve Plotted on Arith-
metic Graph Paper . . . . . . . . . . . . . .
Figure 2. The 80-Percent Learning Curve Plotted on Log-Log
Graph Paper . . . . . . . . . . . . . . . .. . .
Figure 3. Learning Curves of Various Percentages . . . . . 4
Figure 4. Cumulative Average Time, Total Time, and Unit Time 8
Curves -- -Class 1 . . . . . . . . . . . . . .
Figure 5. Cumulative Average Time, Total Time, and Unit Time
Curves -- Class 2 . . . . . . . . . . . . . . . 10
Figure 6. Cumulative Average Time, Total Time, and Unit Time
Curves -- Class 3 . . . . . . . . . . . . . . ?
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CIA/RR RA-7
(ORR Project 33.978)
THEORY AND APPLICATIONS OF THE LEARNING CURVE
I. Introduction.
The purpose of this research aid is to explain the basic theory of
the learning curve and to familiarize analysts with the use of the learn-
ing curve as a tool in research.
The basic theory of the learning curve is simple. A man learns as
he works. The more often he repeats an operation, the more efficient
he becomes, and the less the time required to perform the operation.
The reduction in the time required, however, becomes less with each
successive operation. These facts have long been recognized, but until
a decade ago the fact that the rate of improvement is sufficiently reg-
ular to be predictable was not known.
II. History and Development.
Dr. T.P. Wright developed the theory of the learning curve while
he was employed by the Curtiss-Wright Corporation, Buffalo, New York.
In February 1936, his findings were published in the Journal of Aero-
nautical Sciences in an article entitled "Factors Affecting the Cost
of Airplanes."
By World War II the fact that the direct labor* input per airplane
declined regularly as the cumulative number of airplanes produced in-
creased had been recognized. More important in wartime was the fact
that the unit cost also progressively declined so that more airplanes
could be produced with the same labor force and facilities.
This reduction in direct labor input per airplane might have been
called rising productivity except for the fact that each time a new
airplane model was put into production the process repeated itself:
that is, the direct labor required to produce the first unit of the
new model reverted back to approximately the labor required to produce
the first unit of the preceding model, and the learning process (assum-
ing similar weight and function) had to begin over again. Because of
this repetitive characteristic the phenomenon was called learning.**
* The term direct labor, as used in this research aid, refers to the
labor expended on the rframe>manufacturing process which consists of
machining, processing, Gating, and assembling. Direct labor does
not include engineering, design, supervision, administration, or main-
tenance.
*' The learning curve is sometimes referred to as the improvement curve,
the experience curve, or the progress curve.
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Not long after the publication of Dr. Wright's article, several US
aircraft companies established certain standards of learning, which
they used as bases for predicting direct labor input and cost. The US
armed services became interested in the theory of the learning curve and
after World War II sponsored a statistical study of direct labor input
by the Stanford Research Institute. The conclusions reached in this
study were published in 1949.
By 1949a series of learning curves, which represented the average
experience in the production of various types of airframes such as
fighters and bombers, had been developed. These curves all were dif-
ferent in terms of the direct labor input required to produce the first
unit of a particular type, but they had one characteristic in common,
the rate of improvement.
The average rate of improvement for all US aircraft companies was
approximately 20 percent between doubled quantities of units produced.
This rate of improvement was referred to as an 80-percent learning
curve: that is, the direct labor required to produce the second unit
was 80 percent of that required to build the first; for the fourth,
80 percent of that required to build the second; for the )400th, 80
percent of that required to build the 200th; and so forth.
It may seem unreasonable to expect this reduction in direct labor
input to continue indefinitely because the labor required eventually
would appear to reach an absurdly low figure. The quantity of units
produced, however, must be doubled for every 20-percent reduction in
the required labor input so that, although the labor required per unit
steadily decreases, the cumulative outputapproaches infinity much
faster than does the corresponding labor input. When large quantities
of the same item have been produced, the rate of improvement in rela-
tion to the required production time may be so small as to -seem to
have reached-a plateau where no further improvement is possible.
Most US aircraft companies operate in the range between unit 1 and
unit 1,500, and the rate of improvement therefore is perhaps most
noticeable in the aircraft industry.
III. The Learning Curve on Arithmetic Graph Paper.
If an aircraft company operates on an 80-percent learning curve
and is building -a new model and if 100,000 man-hours are required to
build the first unit, only 80,000 man-hours will be required to build
the second unit. Production has been doubled, and only 80 percent as
much time is required to build the second unit. If production is
doubled again, only 6+,000 man-hours will be required for the fourth
unit. Thus the learning process continues. The following tabulation
shows the direct labor required when the quantities of the theoretical
airframe produced are doubled:
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Direct Labor Required
Unit Number (Man-Hours)
1 100,000
2 80,000
4 64,000
8 51,200
16 40,960
32 32 .,768
64 26,214
128 20,972
Figure 1* shows the preceding, tabulation plotted on arithmetic graph
paper. On arithmetic graph paper the line is a "true curve" and dramat-
ically shows the reduction in direct labor required as succeeding air-
frame units are produced. The line dips sharply at first and then
begins to slope downward more gently as the percentage rate of improve-
ment is spread over an increasingly larger volume of production.
Although the percentage rate of improvement is constant, a constant
rate is difficult to interpret on arithmetic graph paper. As the number
of units produced increases in geometric progression, the variable (time,
price, or the like) decreases in geometric progression. To interpret the
curve, therefore, knowledge of analytical geometry is necessary. Con-
sequently, arithmetic graph paper usually is not used to show the learn-
ing curve. Another disadvantage in the use of arithmetic paper is the
difficulty in showing unit 1 and unit 1,000 on the same graph without
making the graph impractically large. For these and other reasons, the
learning curve usually is plotted on log-loge graph paper.
IV. The Learning Curve on Log-Log Graph Paper.
Figure 2*** shows the same information as is shown in Figure 1
plotted on log-log graph paper. When the learning curve is plotted on
log-log graph paper, it follows a straight line and therefore is easy
to interpret and easy to project.
The curve in Figure?.2 may appear to be too steep, but it is not.
Because of the expanding scales of log-log graph paper, the curve
actually decreases at a decreasing rate and therefore the number of
man-hours approaches zero at infinity.
Following p. 4.
* When the word low appears as an adjective, it is an abbreviation
for logarithmic; when it appears as a noun, it is an abbreviation for
logarithm.
*** Following p. 4.
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The main difference between arithmetic graph paper and log-loggraph
paper is that log-log graph paper is laid out so that the distance be-
tween doubled quantities is equal. Because-of this fact, the learning
curve on log-log graph_paper follows a straight line. If the percentage
rate of decrease between doubled quantities is equal and if the distance
on the graph between doubled,q antities is equal, the line will be
straight. The learning curve can be plotted easily with a ruler on
log-log graph paper.
V. Learning Curves of Various Percentages.
Each US aircraft company operates differently. Each company there-
fore shows learning at different percentage rates. The steepness-of
the learning curve depends upon the percentage rate of learning. Some
companies -show -a rate of learning of 70 or 75 percent, whereas other
companies show -a rate of learning of 85 or 90 percent. -Figure 3* shows
some of the different percentage rates of learning plotted on log-log
graph paper.
The percentage rates of learning, of -selected US aircraft companies
during World War II are shown in Table 1.** Based on these-rates of
learning, the slopes of the learning curves for direct labor input in
series production of airframes generally range from 73, to 88 percent.
The degree of -slope of the learning curve is influenced by a number of
factors such as the following:- job familiarization (-supervisors and
workmen), tool coordination, shop organization (lack of balance, job
-assignments, congestion, coordination between shops, shop load, and
schedule status of other -shops), engineering liaison, jig alignment,
parts supply (availability and quality of work previously done on
parts by other shops), handling time by cranes and the like, inspec-
tion (official and other), time to change jobs (check-in procedure),
personal factors (fatigue, morale), modifications and changes, turnover
of personnel, modernization of tooling, shifting work into -subassembly,
small tools (issuance, quality), tolerances, shift changes, bench and
hand work, types of construction or processing, suggestion systems,
specialization of operations, schedule increases, and quantities pro-
duced.
Although the slopes of the learning curves for direct labor input
in series prbduction of airframes generally range from 73 to 88 per-
cent, the slope trends vary somewhat for different operations. Learn-
ing curves for the production of detail parts are several degrees
flatter than the average curve for the production of the whole air-
frame; curves for theproduction of subassemblies usually are compatible
* Following p. 4.
Table 1 follows on p. 5.
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FOR OFFICIAL USE ONLY
THE 80-PERCENT LEARNING CURVE
PLOTTED ON ARITHMETIC GRAPH PAPER
70
60
0 50
?
40
?
30
I I
10 20 30 40 50
Cumulative Uniis
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FOR OFFICIAL USE ONLY
THE 80-PERCENT LEARNING CURVE
PLOTTED ON LOG-LOG GRAPH PAPER
8 10
Cumulative Units
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LEARNING CURVES OF VARIOUS PERCENTAGES
50% 55x 60%
I r i I i - I I I I IN 1 1.1 w
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Percentage Rates of Learning of Selected US Aircraft Companies
During World War II J
Company
Beech Aircraft Corporation,
Wichita, Kansas
Boeing Airplane Company,
Renton, Washington
Boeing Airplane Company,
Renton, Washington
Boeing Airplane Company,
Wichita, Kansas
Boeing Airplane Company,
Wichita, Kansas
Convair Consolidated Vultee
Aircraft Corporation, Fort
Worth, Texas
Douglas Aircraft Company, In-
corporated, Long Beach,
California
Douglas Aircraft Company, In-
corporated, Tulsa, Oklahoma
Ford Motor Company, Willow Run,
Aircraft Division, Ypsilanti,
Michigan
Lockheed Aircraft Corporation,
Burbank, California
North American Aviation, In-
corporated, Dallas, Texas
North American Aviation, In-
corporated, Dallas, Texas
Republic Aviation Corpora-
tion, Farmingdale, New York
AT-10
First 400 B-29's
Last 700 B-29's
First 900 B-29's
Last 800 B-29's
First 1,000 B-24D's
First 1,000 B-17's
B-24 E
B-24
B-17
B-24
AT,6
P-47 N
76.7
80.5
79.0
71.8
69.5
76.4
77.4
70.8
65.3
75.0
98.o
89.0
a. Army Air Forces, Materiel Command, Wright Field, Dayton, Ohio, and
Industrial Mobilization Office. Source nook of World War II Basic Data
Airframe Industry, vol 1, Direct Man-Hours Progress Curves, April
1952. U.
Percentage Rates
of Learning
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with the'average curve for the whole airframe; and curves for final as-
sembly operations are several degrees steeper than the average curve for
the whole airframe. The slope of the detail parts curve is flatter,
for example, because of the influence of machine operations. Machines
have a definite limit of output. The only improvement which can be ex-
pected in machine operations is a reduction in set-up time and in han-
dling time.
The range of slopes in the selling price curves which result from
the learning curves noted above is not so wide as in the corresponding
learning curves, because all other cost elements (with the exception
of production overhead applied to labor) temper the effect of labor
costs on the total selling price. The reduction in material costs
per unit as a result of series production, for example, is insignifi-
cant as compared with the reduction in direct labor required. The
slope of the -selling price curves for airframes produced in series
generally rangesfrom 88 to 95 percent.
VI. Mathematical Development.*
The mathematical development of the learning curve is not difficult.
The use of correlation and other statistical methods has shown that a
graph of actual performance data may be described by an equation of the
following type:
This equation** is the theoretical learning curve and may be used to
describe the learning curve unit time curve, cumulative average time
curve, and total time curve. Although the equation will describe
actual data, the mathematical exactness of the theoretical learning
curve will not permit these three curves all to be straight lines
simultaneously on log-log graph paper. Learning curves may be con-
sidered in the following three classes:
1. Where the cumulative average time curve and the total
time curve are of the type
* For problems based on the learning curve, see Appendixes A and
B.
** K = a constant = value of y when x = 1.
n = the tangent of the angle which the straight :Line curve forms
with the x axis.
-6-
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2. Where only the unit time curve is of the type
3. Where all three time curves are modifications of classes
1 and 2, above.
A. Class 1.
1. Cumulative Average Time Curve.
The cumulative average time curve, shown in Figure 4,* has
been developed by the equation
(1)
y = cumulative average time in direct
man-hours required to produce any
number of units,
K = number of direct man-hours required
to produce the first unit,
x = any number of units produced, and
n = slope (tangent) of the learning curve.
2. Unit Time Curve.
The unit time curve parallels the cumulative average time
curve shown in Figure 4, from unit 10 on. The unit cost in direct man-
hours can be read along the vertical axis from the unit time curve.
(The two curves meet at unit 1 because the unit time and cumulative
average time are the same at that point.) The unit time curve gradually
approaches a straight line that is parallel to, and somewhat lower than,
the cumulative average time curve. This straight line is called the
asymptote, and the unit time curve is said to be asymptotic to it. The
values of all points on the asymptote are equal to (1 - n) times the
values of the respective points on the cumulative average time curve.
The individual unit time for any unit x is approximately equal to the
time shown on the asymptote at unit x - 1/2. Thus the unit time for
unit 2 is equal to the value of the asymptote at unit number 1 1/2.
This method may be used to approximate the values of the unit time
curve through unit 10. For practical purposes, the unit time curve for
units 10 and above may be considered as equal to the values of the
asymptote.
* Following p. 8.
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The approximate time for a single unit x greater than 10 is
expressed by the following equation:
y = K (1 - n) (2)
x
where y = direct man-hours required to produce a specific unit
and all other symbols are the same as in equation
(1). A closer approximation than equation (2) is
y = K (1 - n)
x - 1/2)"
(3)
3. Total Time Curve.
The total time curve in Figure 3* is expressed by the equa-
y_ (1-n)
where y = total direct man-hours required to produce a
specific number of units and all other symbols
are the same as in equation (1).
(4)
To find the total time curve take any point on the cumula-
tive average time curve -- point B on Figure 4 -- and measure the dis-
tance horizontally back to unit 1. Erect this distance vertically over
point B to produce a point B". Then connect the 2 points from the
direct man-hours for unit 1 -- point A and point B" -- and extend.
From this curve the tot-al number of direct man-hours required to pro-
duce any number-of units may be read. This procedure actually adds
the logarithmic distance of the cumulative average time to the logari-
thmic distance of the units involved, thus multiplying the values.
Because the curve runs off the graph, take point D,
vertically under point C and at the unit 1 level, and draw a new
curve parallel to the one just drawn, which should be read in units
10 times as great. If parallel rulers are not available, choose
another point E, measure back to E' at unit 10, and repeat the pro-
cess.
Following p. 4, above.
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CUMULATIVE AVERAGE TIME,
TOTAL TIME, AND UNIT TIME CURVES-CLASS I
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B. Class 2.
1. Unit Time Curve.
The unit time curve shown in Figure 5* has been developed by
the equation
(5)
y = direct man-hours required to produce
unit x,
K = number of direct man-hours required
to produce the first unit,
x = any number of units produced, and
n = slope (tangent) of the learning
curve.
2. Total Time Curve.
The summation of a series of unit man-hours may be closely
approximated by using the definite integral from the first unit minus
one-half to the last unit plus one-half. Thus the integration will be
between the value one-half and the last unit number plus one-half, as
follows:
Z Y
y K
1 - n
n- (1/2)1-n
(6)
where y = total direct man-hours required to produce all units
through unit x and all other symbols are the same
as in equation (5).
The total time curve shown in Figure 5 has been developed from
equation (6).
* Following p. 10.
Kx -n dx = K / x -n dx
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3. Cumulative Average Time Curve.
The cumulative average time curve in Figure 5 was developed
from the equation
K 1 nl
_ 1 - n I(x + 1/911 n-- (1~2) JJ
x
(7)
where y = cumulative average time in direct man-hours. required
to produce each unit and all other symbols are the
same as in equation (6).
The cumulative average time curve gradually -approaches a
-straight line which is parallel to and somewhat higher than the unit
time curve. The values of all points on the asymptote are equal to
the values of the respective points on the unit time curve divided by
(1 - n). -
1. Unit Time Curve. The unit time curve shown in Figure 6* has been developed
by research at Stanford University and may be determined by the follow-
ing equation
y = a (8)
x + B
y = direct man-hours required to produce.
unit x,
x -= unit number of any unit,
a = constant expressed in terms of direct
man-hours,
B = constant expressed in terms of units,
and
n = slope ,(tangent) of the asymptote of
the learning curve.
* Following p. 10.
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CUMULATIVE AVERAGE TIME,
TOTAL TIME, AND UNIT TIME CURVES-CLASS 2
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FIGURE 6
CUMULATIVE AVERAGE TIME,
TOTAL TIME, AND UNIT TIME CURVES-CLASS 3
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2. Total Time Curve.
The total time curve shown in Figure 6* was developed by inte-
grating equation (8) from the first unit minus one-half to the last unit
plus one-half, as follows:
x + 1/2 x + 1/2
f'
Y LV a (x+B) -ndx C,a(x+B) ax
1/2 1/2
y = a (x + B + 1/2) 1 n - (B + 1/2)
1 - n
(9)
where y = cumulative total direct man-hours required to pro-
duce all units through unit x and all other symbols
are the same as in equation (8).
3. Cumulative Average Time Curve.
The cumulative average time curve shown in Figure 6 was
developed from the equation
a
n
y =l n Lx+B+l/21_n_B+lI21_
(10)
where y = cumulative average direct man-hours required to pro-
duce each unit and all other symbols are the same
as in equation (8).
VII. Determination of the Slope of Theoretical Learning Curve.
Equation (1) can be solved for the slope of a learning curve by taking
the log of both sides of the equation, which gives
log y = log K - n log x (11)
* Following p. 10, above.
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To find the numerical value of slope n for an 80-percent learning
curve, proceed as follows:
By definition, the percent of the learning curve is the ratio
of the man-hours required to produce unit 2 to the man-hours
required to produce unit 1. Let K = 100 and y = 80, where
In equation (5)
Log 80 = log 100 - n log 2
1.90309 = 2.00000 - 0.30103 n
n = 2.00000 - 1.90309 = M9693-= 0.32193
0.30103 0.30103
Table 2* shows the numerical value of the slope n for various
learning curves from 50 to 100 percent.
VIII. Determination of the Slopeof an Actual Learning Curve.
The simplest method of determining the slope of an actual learning
curve is to plot the data on log-log graph paper and draw a straight
line as nearly through all the points as possible. The ratio of unit
2 to unit 1 indicates the-percentage of the curve. From Table 2* the
slope may be read.
Another method of determining the slope of an -actual learning
curve is to solve simultaneously the following equations, which are
a modification of equation (2):
Z n log x + z log y= log K
Z n log xl + 7 log y1 t I log K
* Table 2 follows on p. 13.
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Numerical Value of the Slopes
of 50- to 100-Percent Learning Curves
Percent
Slope
n Percent
Slope
n
50
1.0000
76
0.3959
51
0.9714
77
0.3771
52
0.9434
78
0.3585
53
0.9159
79
0.3401
54
0.8890
80
0.3219
55
0.8625
81
0.3040
56
0.8365
82
0.2863
57
0.8110
83
0.2688
58
0.7859
84
0.2515
59
0.7612
85
0.2345
6o
0.7370
86
0.2176
61
0.7131
87
0.2009
62
0.6897
88
0.1844
63
0.6666
89
0.1681
64
0.6439
9o
0.1520
65
0.6215
91
0.1361
66
0.5996
92
0.1203
67
0.5778
93
0.1047
68
0.5564
94
0.0893
69
0.5353
95
0.0740
70
0.5146
96
0.0589
71
0.4941
97
0.0439
72
0.4739
98
0.0291
73
0.4540
99
0.0145
74
0.4344
100
0.0000
75
0.4150
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This method is illustrated by an example in the following tabulation:
Direct Labor Required
Unit Number (Man-Hours)
1 204,131
2 176,747
3 152,546
4 144,123
5 139,713
6 131,538
7 128,221
8 113,812
9 114,407
10 109,.624
11 103,323
12 102,618
The logs of x and y may then be determined and the equations solved,
as follows:
Log x
Log y
204,131
0.00000
5.30991
176,747
0.30103
5.24736
152,546
0.47712
5.18341
144,123
0.60206
5.15872
139,713
0.69897
5.14523
131,538
0.77815
5.11906
log x = 2.85733
log y =
31.163 69
7
128,221
0.84509
5.io796
8
113,812
0.90309
5.05618
9
114,407
0.95424
5.05846
10
109,624
1.00000
5.03991
11
103,323
1.04139
5.01420
12
102,618
1.07918
5.01122
Z log xi = 5.8 2299
1 log yl =
30.28793
2.85733 n + 31.16369 = 6 log c
5.82292 n + 30.28793 = 6 log c
2.96566 n - 0.7576 = 0
n = 0.29530
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The resultant slope may be used to determine percent, as follows:
log y = log 100 - 0.29530 log 2
log y = 2.00000 - 0.08889
log y = 1.91111
y = 81.5 percent
IX. Uses of the Learning Curve.
The importance of the application of the learning curve may be seen
in the following uses:
1. US aircraft companies use the learning curve to estimate costs
when bidding for new contracts.
2. The US Government uses the learning curve to check the aircraft
companies' bids for accuracy and reasonableness based on statistical anal-
ysis of the companies' records with respect to general production perform-
ance.
3. US aircraft companies use the learning curve to develop labor
requirements, to estimate space and equipment requirements, to prepare
budgets, to measure shop efficiency, and to check the progress of current
contracts.
4. The US Government uses the learning curve to measure the effi-
ciency and production dependability of aircraft companies.
5. US military planners use the learning curve to estimate air-
craft mobilization expansion potential of the US.
X. Applicability.
Although the learning curve was developed and is used principally by
the US aircraft industry, many types of industries should be able to apply
the learning curve profitably. The usefulness of the learning curve de-
pends-upon the following factors:
1. Product Innovation.
Learning is an important factor in the performance of the
workers in industries where major and minor design changes are frequent,
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where new products often are introduced, or where production is charac-
terized by short production-runs at well-separated intervals. These
industries are near the top of their learning curves, where savings be-
tween units of production are significant much of the time.
2. Assembly Time as a Proportion of Total Time.
The more an operation consists of machine time as opposed
to assembly time, the slower is the reduction in direct labor required.
The US aircraft industry has found that there is a 20-percent or greater
reduction in the time required for direct labor between doubled quanti-
ties in assembly operations but only a 10-percent reduction in the time
required for direct labor machine-tool operations.
3. Advance Planning.
The more an operation can be planned in advance, partic-
ularly in terms of methods analysis and tooling, the more predictable
will be the rate of reduction in the time required for direct labor.
Learning in the literal sense tends to produce a smooth curve. Changes
in methods, toolings, and the like during a production run make the
learning curve uneven and probably -give it a-more pronounced slope.
Product innovation is extremely important in the electronics in-
dustry because of the long, complicated as-sembly lines in this industry.
The electronics industry therefore should be able to make good use of
the learning curve because the same conditions exist in this industry
that have made learning curves helpful in the aircraft industry.
The learning curve may also be applicable to the shipbuilding in-
dustry. Direct labor represents a large percentage of the total costs
of ships. Statistics on the direct labor costs of basically similar
bulkheads and panels could be projected to cover units-contemplated
for production.
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SOLUTION OF PROBLEMS IN WHICH THE CUMULATIVE AVERAGE TIME CLEVE
IS A STRAIGhT LINE ON LOG-LOG GRAPH PAPER OF THE TYPE y = K
I. Problem Number One.
A. Problem.
Experience indicates that an average time of 2,100 direct man-
hours per unit is required to produce 25 hydraulic pumps. Experience
also indicates an 84-percent learning curve for this type of production.
How many direct man-hours would be required to produce 75 additional
units?
B. Solution.
The computations would be as follows:
1. Direct man-hours required to produce unit 1 = (y = K)
Then, K = yxn = 2,100 (25) 0.2515 = 4,715 direct man-hours.
2. Cumulative average direct man-hours required to produce
unit 100 = y = 4 1 = 1,871 direct man-hours.
100
3. Direct man-hours required to produce 75 additional units
_ (100 x 1,871) - (25 x 2,100) = 95,600 direct man-hours.
II. Problem Number Two.
A. Problem.
A subcontractor had produced an initial lot of 130 fin assemblies
at an average expenditure of 750 direct man-hours per fin assembly. At
the completion of this lot, experience indicated that these fin assem-
blies were produced on an 82-percent learning curve. One year later the
subcontractor received an order for 165 additional fin assemblies. How
many direct man-hours would be required to produce 165 additional fin
assemblies?
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B. Solution.
Because production was disrupted for 1 year, much of the learning
gained on the previous contract disappeared. It is therefore impossible
to add 165 additional fin assemblies to the end of the initial 130 fin
assemblies for projection purposes.
Experience has shown that the degree of learning "lost" can be
measured to some extent. If a repeat order immediately follows an ini-
tial order and prevents disassembly of the production stations, additional
units can be added immediately to the last unit of the initial order. If
the production-stations were to be disassembled for 4 to 8 months and
then to be re-established to fill a repeat order, the advantage, or credit,
gained from the learning supplied by the initial order would represent
perhaps one-half of the initial learning. The repeat order of 165 fin
assemblies, for example, would be subsequent to 65 units previously pro-
duced. If production were to lapse for about 1 year, the credit gained
from the initial learning would be only one-quarter of the initial learn-
ing. Based on an initial order of 130 fin assemblies and a repeat order
of 165 units 1 year later, the computations to determine the number of
direct man-hours required to produce the 165 additional units would be
as follows:
. 1. Direct man-hours required to produce unit 1 - 750 x 130 0.2863
= 3,020 direct man-hours.
2. Cumulative average direct man-hours required to produce unit
33 (25 percent of the initial order of 130 units = 33 units) 3,020
= 33~.2bb
= 1,110 direct man-hours.
3. Cumulative average direct man-hours required to produce 165
units after learning had been gained on 33 units = 020 = 665 direct
198 0-20b3
4. Direct man-hours required to produce 165 additional units
= 198 x 665 - 33 x 1,110 = 95,000 direct man-hours.
III. Problem Number Three.
A. Problem.
Twenty-five special-type cylinders involving machining and as-
sembly operations have previously been purchased from a vendor-at a price
of US $250* per, cylinder. The prime contractor plans to purchase 200 more
cylinders. What would be a reasonable price for the repeat order?
* All dollar values are given in US dollars throughout this research aid.
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Because the prime contractor does not know the actual cost of
the materials involved. in the assembly of the cylinders, he would apply
a price curve of 88 to 90 percent to the initial order. If no further
minor changes are involved and if the repeat order is exactly the same
as the initial order, an 88-percent price curve would be appropriate.
If minor modifications are involved, a 90-percent price curve would be
used. Assume that the former condition is applicable and that an 88-per-
cent curve is used. The calculation would be as follows:
1. Cost of unit 1 = 250 x 25 o.1844 = $452.65
2. Average cost per unit for 225 units = 2 52.30 _ $166.71
2
3. Total cost of the 200 additional units = 225 x 166.71 - 25
x 250 = $31,259.75
4. Average cost per unit for the 200 additional units
_ .1,259.75 = $156.30
200
Vendors, in attempting to maintain a position in price on a
repeat order often advance problems which are justifiable in many
instances. These problems usually include (1) anticipated increases
in labor and material costs, (2) overtime required on the repeat
order, and (3) necessary changes. If the vendor's cost breakdown is
not available because of the fixed price status or competitive nature
of his quotation, but if his reasons are valid, consideration is given
to the use of an 89- or 90-percent price curve.
IV. Problem Number Four.
Fifty sets (1 right and 1 left) of elevator assemblies have been
purchased from a subcontractor for $1,800 per set. It is planned to
purchase 300 additional elevator sets to follow the initial 50 sets,
with no appreciable gap in production. The subcontractor, however, has
completed only one-half of the initial order of 50 sets and believes that
the quotation of $1,800 will realize slightly less profit than originally
anticipated. What should be the prime contractor's procedure to determine
a realistic price for 300 additional elevator sets?
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B. Solution.
A study of the subcontractor's original quotation and subsequent
thinking in regard to the initial 50 elevator sets and a review of his
position for the production of 300 additional sets disclosed the follow-
ing general information:
1. $12,000 was estimated as the cost of tooling the first 50
sets. This amount was amortized over 50 sets by the subcontractor.
2. Estimated materials for the initial 50 sets cost $120.00
per set. Indications are that material costs will increase by approx-
imately 4 percent during the period in which it is planned to procure
materials for the 300additional sets.
3. The subcontractor is not sure of his learning curve. The
prime contractor's experience for new-type production, however, indi-
cates approximately an 81-percent learning-curve, which is compatible
with the industry. This learning curve, therefore, is being used to
evaluate his quotation.
4. The subcontractor recently had agreed to an 8-percent wage
increase. His average wage rate estimated in the quotation for the
first 50 sets was $1.80 per direct man-hour.
5. The subcontractor's total overhead rate is approximately
150 percent of his direct labor costs.
6. The subcontractor is anticipating a 10-percent profit on
estimated cost for the repeat order but e-stimates that only a 3 per-
cent profit on cost will be realized for the initial order of 50 sets.
Based on this information, computations would be as follows:
1. Original price perset for 50 sets $1,800.00
minus the following items:
Profit at 3 percent on cost $ 52.43
Tooling: $12,000 + 50 sets 240.00
Material cost per set 120.00
$ 412.43
Cost of direct labor plus overhead per set
$1,387.57
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2. Estimated direct man-hour dollars per set in initial 50 sets
is determined at $1,387.57 + 2.5 or $555.03 (1 unit for direct man-hours
and 1.5 units for overhead).
3. Anticipated direct man-hours required per set for 50 sets
_ $555.03 + $1.80 per direct man-hour, or 308.4 direct man-hours.
4. Man-hours for unit 1 = 308.4 x 50 0.3040 = 1,013 direct man-
5. Average man-hours for 350 sets = 1 01 = 170.7 direct
350
6. Estimated average direct man-hours for 300 additional sets
will be: 350 x 170.7 - 50 x 308.4 = 147.75, rounded to 148 direct man-
300
hours per set.
7. Estimated price breakdown per set (excluding any additional
tooling requirements) for 300 additional sets.
a.
Direct material cost of $120 per set plus
4-percent increase
$124.80
b.
c.
Direct man-hours per set, 148
Direct man-hour dollars ($1.80 per hour
plus 8 percent) = $1.94
$287.12
d.
Overhead of 150 percent of direct man-
hours
$430.68
Total estimated cost per set
$842.60
e.
Profit of 10 percent of cost
$ 84.26
Estimated selling price per set (for
300 additional sets)
26.86
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SOLUTION OF PROBLEMS IN WHICH THE UNIT TIME CURVE
IS A STRAIGHT LINE ON LOG-LOG GRAPH' PAPER OF THE TYPE y = K
xn
1. Problem Number One.
A. Problem.
Find the estimated direct man-hours required to produce the
185th unit using 750 direct man-hours for the first unit and an 80-
percent learning curve.
B. Solution.
Direct man-hours for the 155th unit = 750 = 402 direct
185 0 73M7
man-hours.
II. Problem Number Two.
A. Problem.
Find the total estimated direct man-hours'for 185 units, using
750 direct man-hours for the first unit and an 80-percent learning curve.
B. Solution.
Total direct
1 - 0.3219
(0.5)
man-hours = 750 1c185 + 0.5) 1 - 0.3219
1-0.3219 L
0.6781 0 6781
750 (185.5) - (0.5)
0. 7 1
37,500 direct man-hours.
III. Problem Number Three.
A. Problem.
Find the average direct man-hours per unit for 185 units, using
750 direct man-hours for the first unit and an 80-percent learning curve.
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B. Solution.
The average direct man-hours per unit for a gi.ven number of units
can be found by dividing the total direct man-hours for the corresponding
number of units by the number units. man-hours man-hours for
185 units was found to be 37,5of
Average direct man-hours per unit = 3Z,500 = 203 direct man-hours.
185
IV. Problem Number Four.
A. Problem.
Find the average direct man-hours per unit for a block of units
185'through 235, using 750 direct man-hours for the first unit and an
80-percent learning curve.
B. Soltion.
1. Find the total direct man-hours for a block of units 185
through 235?
1 - 0.3219
Total direct man-hours = 750 _ (235 + 0.5)
- (185 - 0.5) 1 - o.j~ly _ 750 (235.5) 0* 6781 - (101+.5) 0 6781
06781
6,,860 direct man-hours
2. Average direct man-hours per unit = 6, = 6,860 = 137
direct man-hours.
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FOR OFFICIAL USE ONLY
FOR OFFICIAL USE ONLY
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