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Professional Development Learning Event, 27th March 2007
Learning Curves – Some Alternative Approaches
Alan R Jones, BAE Systems
“O! This Learning, what a thing it is.”
William Shakespeare (c.1594, The Taming of The Shrew)
The material presented here is based on a case study presented in the following publication:
Jones, A.R. ‘Case Study - Applying Learning Curves in Aircraft Production - Procedures and Experiences’ in Zandin, K (editor) Maynards
Industrial Engineering Handbook, 5th Edition, McGraw-Hill, New York, 2001
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Learning Curves – An Alternative Approach
Learning Curves – Definition and Basic Properties
– Unit Learning Curve Formula
Constituent Elements of Production Learning
– Segmentation Theory
Applications
– Effect of Output Rate Constraint
– “End of Line” Effect
– Assessing Loss of Learning
Cumulative and Cumulative Average Data
– Formulae
– Examples
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The Basics
Learning Curves – Definition and Basic Properties
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Learning Curve: Definition
The Learning Curve expresses the empirical relationship between experience
and efficiency - hence their alternative name of “experience curve”.
The Learning Curve Effect states that the more times a task is performed,
the less time will be required on each subsequent iteration.
The phenomenon was first quantified in the USA by T.P.Wright* in 1936 in
relation to Aircraft Production.
Equation of a Unit Learning Curve:
TA = T1 Aε
where ε is the learning exponent: ε = log(p)/log(2)
with p = the learning percentage expressed as a decimal
and TA is the time at Unit A
* Source: T. P. Wright, (1936) Learning Curve, Journal of the Aeronautical Sciences, Feb 1936
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Simple Learning Curve: Basic Property
Equation of a Unit Learning Curve:
TA = T1 Aε
where ε is the learning exponent: ε = log(p)/log(2)
with p = the learning percentage expressed as a decimal
and TA is the time at Unit A
This is often expressed by saying that whenever the number of units
produced doubles, the time to produce a unit reduces to p% of the
earlier time.
But the more general property is that for any constant multiplier, k say, the
time taken reduces by a fixed percentage
TkA = T1 (kA)ε
= T1 kε Aε
= kε T1 Aε
= kε TA
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Learning based on any multiplier - Example
Unit
Manhours
1
100.00
2
80.00
3
70.20
4
64.00
5
59.56
6
56.17
7
53.45
8
51.20
9
49.29
10
47.65
11
46.21
12
44.93
Multiple of 2
Multiple of 3
Multiple of 4
80% of 100.00
70.2% of 100.00
80% of 80.00
80% of 70.20
64% of 100.00
70.2% of 80.00
80% of 64.00
64% of 80.00
70.2% of 70.20
80% of 59.56
80% of 56.17
70.2% of 64.00
64% of 70.20
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Man-hours
Unit Learning and Batch Learning are “Equivalent”
Σ
Σ
If you have batch totals only,
do a Best Fit Regression on the
last points in each batch only –
not all the points – that gives a
different result!
Σ Σ
Σ
1
10
Cumulative Units
Unit Average Values
100
Cumulative Batch Values
1000
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Background to the Approach
Constituent Elements of Production Learning
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Constituent Elements of Production Learning
Tooling Improvements
34%
Manufacturing Cost
Improvements
Quality Control
23%
4%
6%
22%
11%
Manufacturing Control
Operator Learning
Engineering Changes to
Assist Production
Source: P Jefferson, ‘Productivity Comparisons with the USA – where do we differ?’ Aeronautical Journal, Vol 85 No.844 May 1981, p.179
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Segmenting the Learning Curve: Mathematical Model
Consider 4 cost driver components with values α, β, γ, and δ
where α + β + γ + δ = 1 (or 100%)
Equation of a Unit Learning Curve:
TA = T1 Aε
where ε is the learning exponent: ε = log(p)/log(2)
with p = the learning percentage expressed as a decimal
and TA is the time at Unit A
Expand the exponent:
TA = T1 A(α + β + γ + δ) ε
TA = T1 Aαε Aβε Aγε Aδε
In order to model data with breakpoints, re-define the variable A:
TA = T1 A1αε A2βε A3γε A4δε
For the primary learning (where all cost drivers are “active”), the values of A1 A2 A3 and A4 are all
equal
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Segmenting the Learning Curve: Mathematical Model
Example based on a production run of 60 units
Impact of design
freeze truncates
relative learning
for this cost
driver
All cost drivers active.
Relative learning points
are all equal
Build No
Design
Operator
Tooling
Logistics
A
A1
A2
A3
A4
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
10
5
10
10
10
45
5
10
45
45
60
5
10
45
45
Impact of
constant output
rate truncates
relative learning
for this cost
driver
“End of Line”
truncates relative
learning for these
cost drivers
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Learning Curve Segmentation: Points to Consider
Benefits of Approach:
•
•
•
Allows discontinuities to be modelled easily (using an on/off switch
approach)
Allows scenarios to be modelled which assume learning rates greater than
or less than the “norm” for a particular process or product type
Allows multiple linear regression techniques to be applied in cost data
analysis
Words of Caution:
•
•
As with all modelling techniques, the approach requires calibration for the
specific environment in which it is to be applied
There should be a logical model or explanation of why particular cost
drivers have been “switched in or out”
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Application Example
Effect of Output Rate Constraint on Learning
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Effect of Output Rate Constraint on Learning
Average
Contents
Number of Operators
x
Average Hours Worked
in Time Period
=
Constant
“Constant”
Every operator
performs same task
on every unit
Constrained by working
hour practices (basic
working week &
sustainable overtime
(For Optimum
Learning)
(Effective Upper
& Lower Limits)
Average Hours spent
per Unit in Time Period
Reducing
(Learning Curve)
x
Number of Units
produced in Time Period
Increasing
(Rate Ramp-up)
The “Reduced Cost : Increased Output” is in
part a natural response of increased product
familiarity, and in part a response to market
expectations of affordability etc
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Effect of Output Rate Constraint on Learning
Average
Contents
Number of Operators
x
Average Hours Worked
in Time Period
=
Average Hours spent
per Unit in Time Period
Reducing
x
Number of Units
produced in Time Period
Reducing
“Constant”
Constant
(Effective Upper
& Lower Limits)
(Learning Curve)
(Fixed Output Rate)
Reducing the number
of operators violates
the premise for
optimum learning
Constrained by working
hour practices (basic
working week &
sustainable overtime
A response to market
expectations of
affordability etc to
drive down costs
Customer contractual
limitation or
constraint
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Example 1: Cumulative Deliveries of Product A
350
Delivery Rate
Build-up
Constant Rate
Deliveries
300
Cumulative Units
250
200
9.75 per month
150
117
100
50
0
Years
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Example 2: Assembly Learning for Product A
Delivery Rate
Build-up
Constant Rate
Deliveries
Man-hours
80.4% Learning
after the breakpoint
75.7% Learning
up to the breakpoint
Swingometer
22%
Breakpoint
@ 117
1
10
Actual
Regression
Cumulative Units
5% Confidence Level
100
78%
1000
95% Confidence Level
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Example 2: Cumulative Deliveries of Product B
250
Delivery Rate
Build-up
Constant Rate
Deliveries
Cumulative Units
200
150
4 per month
100
60
50
0
Years
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Example 2: Assembly Learning for Product B
Constant Rate
Deliveries
Man-hours
Delivery Rate
Build-up
87.8% Learning
after the breakpoint
Swingometer
72.1% Learning
up to the breakpoint
40%
60%
Breakpoint
@ 60
1
10
Actual
Regression
Cumulative Units
5% Confidence Level
100
1000
95% Confidence Level
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Effect of Output Rate Constraint on Learning
Other factors affecting the analysis:
•
•
•
•
•
•
The examples emanate from different factories with little different
management styles and cultural heritage
One product was essentially for a single customer variant/mark initially
followed by small batch export orders
The other product was a multiple variant/mark international collaboration
The level of continued investment was geared around the known and
perceived market opportunities
The level and timing of engineering change required to introduce export
variants and support customer modifications has to be considered
The underlying manufacturing technology used on the two products was
similar but not identical
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Application Example
“End of Line” Effect on Learning Curves
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“End of Line” Effect on Learning Curves
Premise:
To enable ongoing learning curve reduction once a constant rate of
output is achieved requires investment in new or improved technology,
process or logistics etc
Reduced quantity remaining over which investment can be recovered
100
Cumulative Return on Investment
90
Reduced
saving
per unit
80
70
60
50
40
30
1
2
3
4
5
6
7
8
9
10
11
12
Diminishing
Return on
Investment
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“End of Line” Effect on Learning Curves
Factor (Cumulative Return on Investment)
1000
Learning Rate
75%
80%
85%
90%
100
Diminishing Cumulative Return on Investment =
(Unit Learning Curve Reduction) x (Units Remaining)
It would seem that there is a case that a
75%
learning curve will truncate naturally
80%
85%
somewhere between the 60% to 80% point
90%
of the total envisaged production quantity,
regardless of the learning curve rate?
10
1
¾ Quantity
0.1
0.01
¾ Quantity
0.001
1
10
Cumulative Units
100
1000
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“End of Line” Effect on Learning Curves
Factor (Cumulative Return on Investment)
100000
The empirical relationship of the “End of Line”
Effect on a learning curve can be attributed to
the “Law of Diminishing Returns”.
It is not unreasonable to expect that a
learning curve will truncate naturally
somewhere between the 60% to 80% point of
the total envisaged production quantity.
10000
¾ Quantity
Example:
• Constant rate of output at unit 50
• 400 units planned in total
• 75% Learning Curve
1000
1
10
Breakpoint
@ Constant Rate
Cumulative Units
100
1000
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Application Example
Assessing Loss of Learning
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Assessing Loss of Learning: Anderlohr Method
Consider a Break in Production
of 12 months after 50 units
Man-hours
3.
4.
1.
2.
1
This defines the re-start
position for learning
Repeat the learning
process (offset by the
number of units lost)
Determine how many units have been
produced in the previous 12 months
Back track up the learning curve by
this quantity
10
Cumulative Units
100
Source: Anderlohr, G., ‘What production breaks cost’, Journal of Industrial Engineering,
September 1969, pp.34-36
1000
Basic
Anderlohr
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Assessing Loss of Learning: Segmentation Method
Consider a Break in Production
of 12 months after 50 units
Continued Component of Learning
3.
le
mp
a
Ex
Man-hours
69%
4.
31%
Component subject to Re-Learning
1.
2.
After the break the
continued learning
component still applies
Factor this by the
re-learning component
(offset by the number
of units lost)
Determine the proportion of learning
that will continue by considering the
cost drivers that might be affected
This defines the re-start position for
learning after the break
1
10
Basic
Cumulative Units
With Re-learning
100
Continued Learning
1000
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Assessing Loss of Learning: Comparison of Methods
Man-hours
Consider a Break in Production
of 12 months after 50 units
Anderlohr method always lags
the segmentation method for
the same re-start value
1
10
Basic
Cumulative Units
Anderlohr
100
Segmentation
1000
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Assessing Loss of Learning: Comparison of Methods
Practical Considerations:
•
•
•
•
Small breaks in production will be more difficult to detect further down
the curve due to potential “noise” in the actual data
The Anderlohr Method assumes that the rate of learning loss is equivalent
to the rate of learning gain. This is not necessarily the case, but a
modified approach which “backtracks” only a proportion of the “lost”
learning could be adopted
What happens when the break in production occurs during the latter
stages of the production run (often the case)? The learning curve may
have “bottomed out” by this stage
Either approach could be applied to other cases of learning loss other than
time breaks; for example, a physical relocation or new start-up.
Consider the following example using the cost driver segmentation method
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Example 3: Cumulative Deliveries of Product C
3-Year Break
In Production
1000
980
960
Cumulative Units
940
2 per month
@ peak
2 per month
@ end of line
920
900
880
860
840
820
800
Years
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Example 3: Assembly Learning for Product C
Rate restricted learning
Man-hours
Re-learning
Swingometer
Swingometer
22%
29%
78%
700
71%
Break in Production
750
800
850
Cumulative Units
Actual
Regression
900
950
1000
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Alternative Approaches
Cumulative and Cumulative Average Data
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Cumulative Average Data
Cumulative Average Model:
•
The formula for the Cumulative Average version of a Learning Curve is the
same as that for a Unit Learning Curve:
TA = T1 Aε
where ε is the learning exponent: ε = log(p)/log(2)
with p = the learning percentage expressed as a decimal
and TA is the Cumulative Average Time at Unit A (Clearly T1 = T1)
•
The Cumulative Average version will be inherently “smoother” than its Unit
counterpart, but the rate of learning indicated will be very similar for
higher quantities (greater than 30 – depending on the accuracy required)
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Man-hours
Cumulative Average Data
Cumulative Average Curve
runs parallel to the Unit
Curve for larger quantities
1
10
Unit
Cumulative Units
Unit Cum Ave
100
Unit Regression
1000
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Cumulative Data Approximation Formulae
Cumulative Data Approximations for a Unit Learning Curve:
For a positive error1,
CA
~
T1 [ (A + 0.5)ε+1 - 0.5ε+1 ]
(ε + 1)
For a negative error2,
CA ->
TA ->
T1 Aε+1
(ε + 1)
T1 Aε
(ε + 1)
CA
~
T1 (Aε+1 - 1) + T1 (Aε + 1)
(ε + 1)
2
where ε is the learning exponent: ε = log(p)/log(2)
with p = the learning percentage expressed as a decimal
For large values of A
Source:
1.
Conway, R.W. and Schultz, A.Jr., ‘The Manufacturing Progress Function’, Journal of Industrial Engineering, Jan-Feb 1959, pp.39-54
2.
Jones, A.R. ‘Case Study - Applying Learning Curves in Aircraft Production - Procedures and Experiences’ in Zandin, K (editor)
Maynards Industrial Engineering Handbook, 5th Edition, McGraw-Hill, New York, 2001
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Cumulative Data Approximation Formulae Error
2.50%
2.00%
1.50%
% Error
Conway-Schultz Approximation
80% learning curve
1.00%
0.50%
0.00%
-0.50%
75% learning curve
-1.00%
Jones Approximation
-1.50%
-2.00%
-2.50%
0
10
20
30
40
50
60
70
80
90
100
Cumulative Units
Source: Jones, A.R. ‘Case Study - Applying Learning Curves in Aircraft Production - Procedures and Experiences’ in Zandin, K (editor)
Maynards Industrial Engineering Handbook, 5th Edition, McGraw-Hill, New York, 2001
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Cumulative Data: Equivalent Unit Completion Method
50
40
30
20
10
0
Calendar Time
Cumulative Average
Cumulative Average based on
Equivalent Unit Completions
Man-hours
Cumulative Units
60
Unit Learning Curve
• So, by using time-phased (EVM) data, you can test whether the
target learning curve is being met much earlier in the programme
• The method also provides an ‘early warning’ indicator of whether
later units are going to ‘stick’ to the assumed learning curve
0.1
1
Cumulative Units
10
100
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Learning Curves – Some Alternative Approaches
Thank you for you for listening – any questions