1. Learning Curves

2. Learning Curve Introduction
Learning Curves are used to estimate the time saving that is experienced when performing the same task multiple times
The greater the amount of labour involved in the task, the greater the saving experienced with each iteration

3. Learning Curve Introduction
Clearly manufacturing a ship or airline has a steeper learning curve (greater savings per iteration) than an automated process of making widgets (e.g. previous figure which is steep - ships)
Two predominant learning curve theories (though there are more)
Unit Theory
Cumulative Average Theory
Both will be explained using the Canadian Frigate program from the late 1980s and early 1990s

4. Unit Theory Learning Curve
The theory states that the cost decreases by a fixed percentage as the quantity produced doubles
The following table reflects an 80% learning curve which means the cost decreases by 20% for each doubling of production (multiply by .8)
Unit Number
Cost 1
$100 2
$80 4
$64 8
$51.2 16
$40.96
Note: Examples from Cost Estimation by Mislick and Nussbaum

5. Unit Theory Learning Curve
Unit Theory equation
Yx = A * xb where:
Yx = the cost of unit x
A = cost of the first unit
x = the unit number
s = the slope of the learning curve = 2b
Cost of 2x = (cost of x)*s
Substituting A*xb for cost & rearrange
s = A*(2x)b/A*xb = 2b
Using logarithms: b = ln(s)/ln(2)

6. Unit Theory Learning Curve
From the previous example:
A=100
Slope of learning curve = 0.8
0.8 = 2b
Using logarithms:
ln(0.8) = b * ln(2)
b = ln(0.8)/ln(2) =
Therefore, cost unit x = 100 * x
For x = 4, cost = 64.0
For x =3, cost = 70.2

7. Unit Theory Learning Curve
Canadian Frigate Example
Ship number
Number of hours to build
ln(ship number)
ln(hours) 1
5,228,212 2
4,651,262 3
3,892,713 4
3,366,301 5
2,959,776 6
2,538,396 7
2,326,265 8
2,218,333 9
2,142,963

8. Unit Theory Learning Curve
Yx = A * xb
To determine “b”, use logarithms, i.e.
ln(Yx) = ln(A) + b*ln(x)
“b” slope of this equation (learning slope = 2b)
Graph using Excel Scatter with Markers after converting your costs (hours in this case) and number of units using natural logs
Under Chart Tools, select Trendline – more trendline options
Then select display equation on chart and display R-squared value on chart

9. Unit Theory Learning Curve
From the chart equation, the slope is which = b
From before, if s = 2b, then s = or a 73.3% unit learning curve which is steeper than typical but reasonable given SJSL’s inexperience

10. Cumulative Average Theory
Subtly different from Unit Theory
Cumulative average theory is average cost of groups of units while unit theory is based on individual unit cost
Therefore, for cumulative average theory (using the previous 80% example):
The average cost of 2 units is 80% of 1 unit
The average cost of 4 units is 80% of 2 units
The average cost of 50 units is 80% of 25 units
Doubling your production reduces your average cost by your learning curve (80% for this example)

11. Cumulative Average Theory
Cumulative Average Theory equation
YN = A * Nb where:
YN = the cumulative average cost of N units
A = cost of the first unit
N = the cumulative number of units produced
As before:
s = the slope of the learning curve = 2b
Using logarithms: b = ln(s)/ln(2)
Cumulative Average Theory usually used for production lots where batches of units are made

12. Cumulative Average Theory
Canadian Frigate example again:
Ship number
Number of hours to build
Cumulative Quantity
Cumulative average hours
ln(ship number)
ln(cum avg hours) 1
5,228,212 2
4,651,262
4,939,737 3
3,892,713
4,590,729 4
3,366,301
4,284,622 5
2,959,776
4,019,653 6
2,538,396
3,772,777 7
2,326,265
3,566,132 8
2,218,333
3,397,657 9
2,142,963
3,258,247

13. Cumulative Average Theory
From the chart equation, the slope is which = b
From before, if s = 2b, then s = or a 85.6% unit learning curve which is less steep than unit theory as expected

14. Unit vs Cumulative Average Theory
Cumulative average theory is a less steep curve such that unit learning is always below it
Cumulative average is less responsive to variation in costs from unit to unit since it is based on averages
Use cumulative if initial production is expected to have large cost variation unit to unit

15. Unit vs Cumulative Average Theory
For Frigate example, which learning curve is better?
Hard to tell visually, I used stats, went with unit
R2 for unit = vs cumulative =
P-value (not shown) unit=5.2x10-6 cum=2.0x10-5