9.5 Learning Curves: Unit and Cumulative-Average Models, Learning Rates & Applications

9.5 Learning Curves

Workers and organizations get faster with repetition. T.P. Wright documented the pattern in airframe assembly in 1936: each time cumulative production doubled, labor hours per plane fell by a roughly constant percentage. The NCEES specification lists learning curves as a Work Design topic. They matter whenever a standard, a staffing plan, or a price must be set for a new product or process.

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1. The Doubling Rule and the Learning Rate

If the learning rate is $r$ (for example 0.85, an "85% curve"):

$T_{2n} = r \cdot T_n$

• Unit 2 takes 85% of unit 1's time, unit 4 takes 85% of unit 2's, unit 8 takes 85% of unit 4's, and so on.
• A lower percentage means faster learning. An 80% curve improves faster than a 90% curve.
• Typical rates are about 70% to 80% for heavily manual, complex work, and 90% to 95% for highly automated or machine-paced work, where people control little of the cycle.

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2. The Mathematical Model

$T_n = T_1 \cdot n^{\,b}, \qquad b = \frac{\ln r}{\ln 2} = \frac{\log r}{\log 2}$

Two versions of the model:

• Unit model (Crawford): $T_n$ is the time for the $n$-th unit. This is the more common convention in industrial engineering texts.
• Cumulative-average model (Wright): $T_n$ is the average time of the first $n$ units. Then total time for $n$ units is $n \cdot T_1 n^b = T_1 n^{b+1}$.

The two give different numbers for the same $r$. Read the problem carefully; if it says "the 8th unit," use the unit model.

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3. Worked Examples (Unit Model)

First unit $T_1 = 100$ hours, 85% learning curve, so $b = \ln 0.85 / \ln 2 = -0.2345$.

(a) Time for unit 8. Unit 8 is three doublings from unit 1:

$T_8 = 100 \times 0.85^3 = 61.4 \text{ hr} \quad \text{(or } 100 \times 8^{-0.2345} = 61.4\text{)}$

(b) Time for unit 10 (not a power of 2, so use the formula):

$T_{10} = 100 \times 10^{-0.2345} = 58.3 \text{ hr}$

(c) Total hours for the first 10 units. Summing $T_1$ through $T_{10}$ gives 711.6 hours. The continuous approximation is:

$\text{Total}_N \approx T_1 \frac{(N + 0.5)^{b+1} - 0.5^{b+1}}{b + 1} = 100 \times \frac{10.5^{0.7655} - 0.5^{0.7655}}{0.7655} \approx 713 \text{ hr}$

which is close enough for planning.

(d) When does the time reach 40 hours? Solve $40 = 100 \times n^{-0.2345}$:

$n = (0.40)^{1/b} = 0.40^{-4.265} \approx 49.8$

so about the 50th unit.

Finding the Learning Rate from Data

If unit 1 took 50 hours and unit 4 took 32 hours, two doublings separate them:

$r^2 = \frac{32}{50} = 0.64 \Rightarrow r = 0.80$

For units that are not a power of 2 apart, use $b = \ln(T_n/T_1)/\ln n$ and then $r = 2^b$.

Unit vs. Cumulative-Average Model Compared

For $T_1 = 100$ hours on an 80% curve ($b = -0.3219$):

Under the cumulative-average model, unit 2 by itself takes $160 - 100 = 60$ hours, not 80. The same 80% rate therefore implies faster unit-level improvement under the Wright model than under the Crawford model. Mixing the two conventions in one problem is a common source of wrong answers.

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4. Applications in Work Design and Operations

1. Setting time standards. A stopwatch study during start-up captures workers who are still learning. Either wait until performance levels off, or set a temporary standard that follows the curve. A standard set too early is too loose and will be easy to beat later.
2. Staffing and capacity planning. Early units need more labor hours, so the hours needed per week fall as cumulative output grows. Plan crews and overtime to the curve.
3. Cost estimating and bidding. Defense and aerospace contracts often price production lots on a learning curve. The average cost of lot 2 is lower than lot 1.
4. Make-or-buy and outsourcing. A supplier far down its curve may beat a new in-house line on cost.
5. Incentive and training programs. Compare a trainee's actual times with the expected curve to spot those who need help.

Plateaus and limits. Learning slows when the process is machine paced, when operators are replaced (turnover resets part of the learning), or after long interruptions ("forgetting"). Many organizations treat the curve as leveling off at the engineered standard.