Learning Curves
Chapter 19
Sections:
1. Learning Curve Theory
2. Why the Learning Curve Occurs
3. Determining the Learning Rate
4. Factors Affecting the Learning Curve
5. Learning Curve Applications
6. Time Standards Versus the Learning
Curve
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Learning Curve Phenomenon
Reduction in cycle time that occurs in a repetitive
work activity as the number of cycles increases
 An important topic in time study
 When a worker accomplishes a task over and
over, the time required for each successive
cycle decreases as he or she learns the task
 At first the learning effect is rapid, and the
cycle time decreases significantly
 As more and more cycles are completed,
the cycle time reduction becomes less and
less
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Learning Curve Theory
 According to theory, there is a constant learning rate that
applies to a given repetitive task
 Learning rate = the proportion by which the dependent
variable (e.g., task time) is multiplied every time the
number of task cycles doubles
 Rate of improvement is the proportion by which the
dependent variable is reduced every time the number of
units doubles IR = 1-LR
 Example: If T1 = 10 hr and LR = 80%, then
T2 = 0.80(10) = 8.0 hr,
T4 = 0.80(8.0) = 6.4 hr
T8 = 0.80(6.4) = 5.12 hr
and so on
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Learning Curve for LR = 80%
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Log-Linear Model
When learning curve time data are plotted on
log-log coordinates, the plot yields a straight
line with slope m
y = kxm
where y = dependent variable(usually the time
or cost of a task cycle or work unit), k =
constant representing the value of the
dependent variable for the first work cycle, x =
number of work units completed, and m = slope
lnLR 
m=
ln 2
and LR = 2m
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Same Learning Curve in Log-Log Plot
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Two Widely Used Log-Linear Models
1. Crawford Model
 Dependent variable is unit time TN
2. Wright Model
 Dependent variable is cumulative
average time T N
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Crawford Model
Most widely used learning curve model in
industry today
TN  T1N m
Example: T1 = 10 hr, LR = 80%, how long will
it take to complete the 20th work unit?
m = ln(0.80)/ln 2 = -0.32193
T20 = 10(20)-0.32193 = 3.81 hr
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Crawford Model

N
TTN  T1  i m
i 1
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Crawford Model
Cumulative average time
TN 
TTN
N
Cumulative average time between two values of N
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Cumulative Total Time for Crawford
Model

Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

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A welder produces 7 welded assemblies during the first day on a new job, and
the seventh assembly takes 45 minutes (unit time). The worker produces 10
welded assemblies on the second day, and the 10th assembly on the second day
takes 30 minutes. Given this information, (a) what is the percent learning rate and
(b) what is the total cumulative time to produce all 17 welded assemblies? Use
the Crawford learning curve model.
Solution: (a) Given T7 = 45 min and T7+10 = T17 = 30 min
T7 = T1(7)m = 45
T17 = T1(17)m = 30
T1 = 45/7m = 30/17m
45/30 = 7m/17m = (7/17)m
1.5 = (0.41177)m
ln 1.5 = m ln 0.41177
0.40547 = -0.8873 m
m = -0.45696
LR = 2-0.45696 = 0.7285 = 72.9%
(b) T1 = 45/7-0.45696 = 109.5 hr
E(TT17) = 109.5{(17.51-0.45696 – 0.51-0.45696)/(1-0.45696)}
E(TT17) = 109.5{(4.7317- 0.6863)/0.54304} = 815.7 min = 13.595 hr
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Wright Model
 First learning curve model to be published
 Based on cumulative average time per unit
T N = T1Nm
 Total cumulative time:

TTN = N T N
or TTN = T1Nm+1
Unit time:
TN = TTN - TTN-1
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Advantage of Log-Linear Model
In some cases, a log-linear model turns out not to be the
best model to use and several alternatives to log-linear
formula are explored. The trouble with other alternative
learning curve equation is that they have more complex
forms and require valuation of more parameters than
the log-linear model
Log-linear model Consists of only two parameters:
1. The time associated with the first unit
2. The slope m, from which can be derived the learning
rate LR
 Conversely, if LR is known, the slope m can be
deduced
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Determine the Learning Rate


Read Example 19.6 page 493
The unit time for the first unit of
production is 72 hr and the total
cumulative time for 50 units is 2,347
hr. Determine the learning rate for
(a) the Crawford learning curve
model and (b) the Wright learning
curve model.
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(b) Wright model: 2,347 = 72(50)m+1
32.5972 = (50)m+1
(m+1) ln 50 = ln 32.597
3.912 m + 3.912 = 3.484
3.912 m = -0.428
m=0.1094
LR = 2-0.1094 = 0.927 = 92.7%
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(a) Crawford model: E(TT50) = 2,347 =
72{(50.5m+1 – 0.5m+1)/(m+1)}
2,347/72 = 32.5972 = (50.5m+1 – 0.5m+1)/(m+1)
Trial and error solution required.
Try m = -0.30
32.5972 = (50.50.7 – 0.50.7)/(0.7) = 21.36
Try m = -0.20
32.5972 = (50.50.8 – 0.50.8)/(0.8) = 28.09
Try m = -0.15
32.5972 = (50.50.85 – 0.50.85)/(0.85) = 32.34
Try m = -0.14
32.5972 = (50.50.86 – 0.50.86)/(0.86) = 33.27
Interpolating, (32.597 – 32.34)/(33.27 – 32.34)
= 0.276
Try m = -0.15 – 0.276(0.01) = -0.147
32.5972 = (50.50.853 – 0.50.853)/(0.853) = 32.61
(Close enough)
LR = 2-0.147 = 0.903 = 90.3%
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Comparison of Crawford and Wright
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Why the Learning Curve Occurs
Contributions of the worker
 Worker becomes familiar with the task - the
worker learns the task
 Worker makes fewer mistakes as the task is
repeated
 Hand and body motions become more efficient,
and there is a rhythm and pattern developed
 Minor adjustments in workplace layout to
reduce distances
 Fewer delays that interrupt the operation
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Why the Learning Curve Occurs
Contributions of the larger organization
 Methods improvements by the IE Department
 Fine-tuning of machinery and tooling
 Development of special tooling
 Technological improvements
 Product design improvements
 Improved quality of starting materials
 Better scheduling
 Improved logistical support
 Better motivation of workers
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Contributions to the Learning Curve
Contributions to
learning by the
worker and the
organization
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Typical Learning Rates (Crawford)
Type of work
LR, %
Assembly
84-85
Prototype assembly 65
Clerical ops
75-85
Inspection
86
Machining
90
Welding
85-90
Industry
LR, %
Aerospace
Complex machines
Construction
Electronics mfg
Machine shop
Shipbuilding
85
75-85
70-90
90-95
90-95
80-85
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Determining the Learning Rate

Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

A metal fabrication shop would like to know what the learning rate is for a certain
section of the plant that makes welded assemblies. In one case study, a total of
four assemblies were completed. Records were kept for units 1 and 4, and the
times were 60 hours and 48.6 hours, respectively. Unfortunately, the times for
units 2 and 3 were lost. Use the Crawford learning curve model to determine the
learning rate indicated by the data?
 Solution: Two solutions:
(1) Note the doubling effect:
 48.6 = 60(LR)(LR) = 60(LR)2
 LR2 = 48.6/60 = 0.81
 LR = = 0.90 = 90%
(2) Use the log-linear equation:
 T4 = T1(N)m
 48.6 = 60(4)m
 0.81 = 4m
 m ln 4 = ln 0.81
 1.3863 m = -0.21072
m = -0.152
 LR = 2-0.152 = 0.90 = 90%
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Worker-Machine Systems
 Learning in a work-machine system will be a
composite of the learning rates of the two
components of the system
LRw-m = pw LRw + pm LRm
where pw = proportion of cycle time controlled
by worker, and pm = proportion of cycle time
controlled by machine
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Interruptions in the Learning Curve
 Learning curve effect tends to be disrupted by
interruptions in production
 Possible reasons for interruptions:
 Batch production, in which there are
extended periods between repeat orders
 Labor strikes
 Vacations
 Raw material shortages
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Learning Loss due to Interruption
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Learning Loss due to Interruption
 Assumptions
 If the duration of the interruption exceeds 12
months, then resuming production after the
break would be the same as starting over
for the first time
 If the duration of the interruption is less than
one month, therefor no effect on remission.
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Learning Loss due to Interruption

-
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Other Factors Affecting Learning
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Product complexity
 More opportunities for learning - higher T1
and lower LR
Preproduction planning
 Fewer opportunities for learning - lower T1
and higher LR
Labor turnover - slower learning (higher LR)
Plateau model - learning curve consists of two
phases:
1. Start-up - log-linear model applies
2. Steady state - no further improvement
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Plateau Learning Curve Model
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Why the Plateau Model Occurs
 In a worker-machine system, worker is finally
limited by the speed of the machine
 No incentives for workers to make further
improvements
 Management is unwilling or unable to invest in
newer production technologies that would
sustain the learning curve
 Management does not believe that learning
can continue indefinitely, and this becomes a
self-fulfilling prophecy
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Learning Curve Applications
Accident prevention
Capacity planning
Cost accounting
Cost estimating
Cost reduction
Contract administration
Failure analysis
Labor requirements
estimation
Make-or-buy decisions
Methods analysis
New product introduction
Pricing negotiations
Pricing strategies
Product design changes
Productivity analysis
Quality control
Reliability analysis
Safety
Vendor selection
Wage incentives
Warranty maintenance
Worker performance
analysis
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Time Standards vs. Learning Curve
 Apparent contradiction:
 A time standard is a constant value that is
defined under the following conditions:
 Standard method
 Standard work units
 Performed by a trained average worker
 Task is performed at 100% performance
 But the learning curve predicts that the
actual time for the work cycle will decrease
over time
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
How to Reconcile the Contradiction
 Although the standard time is a constant, the
actual cycle time for a manual task is a variable
 The time varies from one cycle to the next
 One of the reasons why the cycle time varies is
because the worker learns the task
 The cycle time is gradually reduced as the
number of repetitions increases, as predicted
by learning curve theory
 As long as there are no changes in methods or
work unit, it seems fair to give credit to the
worker for reductions in cycle time due to
improvements in skill, effort, pace, etc.
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Standard Reference Quantity (SRQ)
 In batch production, it can be argued that the
quantity of the production run should influence
the value of the time standard
 Surely a worker must be given an
opportunity to learn a task before he or she
is measured against the time standard
 SRQ = the discrete quantity of production (e.g.,
10, 100, 1000 units) for which the standard
time for the task is applicable
 If the actual quantity of production is
different from the SRQ, then an adjustment
in the time standard is appropriate
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
SRQ Adjustment Ratio
Work Systems and the Methods, Measurement, and Management of Work
by Mikell P. Groover, ISBN 0-13-140650-7.
©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.