Learning Curve

Learning Curve Primer
The concept of a Learning Curve is motivated by the observation (in many diverse production environments) that, each time the cumulative production doubles, the hours required to produce the most recent unit decreases by approximately the same percentage. For example, for an 80% learning curve: If cumulative production doubles from 50 to 100, then the hours required to produce the 100th unit is 80% of that for the 50th unit. The learning curve formula can be expressed mathematically as:

Y = aXb
ln(Y) = ln a + b*ln(X)

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where:
Y = avg. cost/unit
X = cumulative production
a = cost of first unit
b = learning slope or coefficient

Using set of actual data

Plot ln(Y) against ln(X). In other words, plot the ln of the cost of the xth unit produced against the ln of x, the cumulative number of units produced.

Fit a straight line to the data in this plot.

Find b, the slope of the straight line you just fit to the data. (Find the “rise” over the “run” using the logarithmic units, not the original units. You will get a negative number, generally between zero and negative one.)

Find the learning rate: s = eb*ln(2)

Adapted from: Schmidt and Wood. 1999. The Growth of the Intel and the Learning Curve, Stanford University

Learning Curve Primer
Doubling production method

Pick any time period of interest for which you know the starting and ending production costs, and the starting and ending cumulative production volumes.

Calculate n, the number of times product volume has doubled over the time period, where n = [ln(cumulative quantity produced at the end of the time period) – ln (cumulative quantity produced at the beginning of time period)]/(ln(2))

Calculate the cost ratio r, where r = (cost at end of period)/(cost at beginning of period)

Calculate the learning rate s, where s = n√r. In other words, s is the nth root of r. That is sn = r

Brute force method

Pick any time period of interest for which you know the starting and ending production costs, and the starting and ending cumulative production volumes.

Create two simultaneous equations where the cost of the first unit a and the learning coefficient b are unknown.

Solve: ln(Y1) = ln a + b*ln(X1) for ln a

Then use that value to solve for b in: ln(Y2) = ln a + b*ln(X2)

Find the learning rate: s = eb*ln(2)

Adapted from: Schmidt and Wood. 1999. The Growth of the Intel and the Learning Curve, Stanford University

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