• Why are learning curves important to study?
• How do you create and evaluate learning curves?

500 words

Jacobs, F. R. & Chase, R. B. (2014).
Operations and Supply Chain Management (14
^th ed). New York, NY: McGraw-Hill

Understand what a learning curve is and where learning curves are applicable.

A learning curve is a line displaying the relationship between unit production time and the cumulative number of units produced. Learning (or experience) curve theory has a wide range of application in the business world. In manufacturing, it can be used to estimate the time for product design and production, as well as costs. Learning curves are also an integral part in planning corporate strategy, such as decisions concerning pricing, capital investment, and operating costs based on experience curves.

Learning curve

A line displaying the relationship between unit production time and the cumulative number of units produced.

Learning curves can be applied to individuals or organizations. Individual learning is improvement that results when people repeat a process and gain skill or efficiency from their own experience. That is, “practice makes perfect.” Organizational learning results from practice as well, but it also comes from changes in administration, equipment, and product design. In organizational settings, we expect to see both kinds of learning occurring simultaneously and often describe the combined effect with a single learning curve.

Individual learning

Improvement that results when people repeat a process and gain skill or efficiency from their own experience.

Organizational learning

Improvement that comes both from experience and from changes in administration, equipment, and product design.

Learning curve theory is based on three assumptions:

1. The amount of time required to complete a given task or unit of a product will be less each time the task is undertaken.

2. The unit time will decrease at a decreasing rate.

3. The reduction in time will follow a predictable pattern.

Each of these assumptions was found to hold true in the airplane industry, where learning curves were first applied.¹ In this application, it was observed that, as output doubled, there was a 20 percent reduction in direct production worker-hours per unit between doubled units. Thus, if it took 100,000 hours for Plane 1, it would take 80,000 hours for Plane 2, 64,000 hours for Plane 4, and so forth. Because the 20 percent reduction meant that, say, Unit 4 took only 80 percent of the production time required for Unit 2, the line connecting the coordinates of output and time was referred to as an “80 percent learning curve.” (By convention, the percentage learning rate is used to denote any given exponential learning curve.)

A learning curve may be developed from an arithmetic tabulation, by logarithms, or by some other curve-fitting method, depending on the amount and form of the available data.

There are two ways to think about the improved performance that comes with learning curves: time per unit (as in Exhibit 6.1A) or units of output per time period (as in 6.1B). Time per unit shows the decrease in time required for each successive unit. Cumulative average time shows the cumulative average performance times as the total number of units increases. Time per unit and cumulative average times are also called progress curves or product learning and are useful for complex products or products with a longer cycle time. Units of output per time period is also called industry learning and is generally applied to high-volume production (short cycle time).

exhibit 6.1 Learning Curves Plotted as Times and Numbers of Units

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Note in Exhibit 6.1A that the cumulative average curve does not decrease as fast as the time per unit because the time is being averaged. For example, if the time for Units 1, 2, 3, and 4 were 100, 80, 70, and 64, they would be plotted that way on the time per unit graph, but would be plotted as 100, 90, 83.3, and 78.5 on the cumulative average time graph.

HOW ARE LEARNING CURVES MODELED?

There are many ways to analyze past data to fit a useful trend line. We will use the simple exponential curve first as an arithmetic procedure and then by a logarithmic analysis. In an arithmetical tabulation approach, a column for units is created by doubling, row by row, as 1, 2, 4, 8, 16 . . . The time for the first unit is multiplied by the learning percentage to obtain the time for the second unit. The second unit is multiplied by the learning percentage for the fourth unit, and so on. Thus, if we are developing an 80 percent learning curve, we would arrive at the figures listed in column 2 of Exhibit 6.2. Because it is often desirable for planning purposes to know the cumulative direct labor hours, column 3, which lists this information, is also provided. Column 4, the cumulative average direct labor hours, is found by dividing the entry in column 3 by the column 1 entry. (See the next section for the exploration of how to do these calculations for each individual unit.)

LO6–2

Plot and analyze learning curves.

Exhibit 6.3A shows three curves with different learning rates: 90 percent, 80 percent, and 70 percent. Note that if the cost of the first unit was $100, the 30th unit would cost $59.63 at the 90 percent rate and $17.37 at the 70 percent rate. Differences in learning rates can have dramatic effects.

In practice, learning curves are plotted using a graph with logarithmic scales. The unit curves become linear throughout their entire range, and the cumulative curve becomes linear after the first few units. The property of linearity is desirable because it facilitates extrapolation and permits a more accurate reading of the cumulative curve. This type of scale is an option in Microsoft Excel. Simply generate a regular scatter plot in your spreadsheet and then select each axis and format the axis with the logarithmic option. Exhibit 6.3B shows the 80 percent unit cost curve and average cost curve on a logarithmic scale. Note that the cumulative average cost is essentially linear after the eighth unit.

Analytics

exhibit 6.2 Unit, Cumulative, and Cumulative Average Direct Labor Worker-Hours Required for an 80 Percent Learning Curve

For the Excel template, visit www.mhhe.com/jacobs14e.

(1) UNIT NUMBER	(2) UNIT DIRECT LABOR HOURS	(3) CUMULATIVE DIRECT LABOR HOURS	(4) CUMULATIVE AVERAGE DIRECT LABOR HOURS
1	100,000	100,000	100,000
2	80,000	180,000	90,000
4	64,000	314,210	78,553
8	51,200	534,591	66,824
16	40,960	892,014	55,751
32	32,768	1,467,862	45,871
64	26,214	2,392,447	37,382
128	20,972	3,874,384	30,269
256	16,777	6,247,572	24,405

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exhibit 6.3 Learning Curve Plots

Although the arithmetic tabulation approach is useful, direct logarithmic analysis of learning curve problems is generally more efficient because it does not require a complete enumeration of successive time–output combinations. Moreover, where such data are not available, an analytical model that uses logarithms may be the most convenient way of obtaining output estimates.

Logarithmic Analysis

The normal form of the learning curve equation is²

$\begin{array}{cc}{Y}_x=K{x}^n& \left[6.1\right]\end{array}$

where

$\begin{array}{c}x=\text{Unit number}\\ Y_x=\text{Number of direct labor hours required to produce the xth unit}& K=\text{Number of direct labor hours required to produce the first unit}\\ n=logb/log2,\text{where b=Learning percentage}\end{array}$

We can solve this mathematically or by using a table, as shown in the next section. Mathematically, to find the labor-hour requirement for the eighth unit in our example (Exhibit 6.2), we would substitute as follows:

Y₈ = (100,000)(8)ⁿ

Using logarithms:

$\begin{array}{ccc}{Y}_8& =& 100,000{\left(8\right)}^{log0.8/log2}\\ & =& 100,000{\left(8\right)}^{-0.322}=\frac{100,000}{{\left(8\right)}^{0.322}}\\ & =& \frac{100,000}{1.9534}=51,193\end{array}$

Therefore, it would take 51,193 hours to make the eighth unit. Your answer may vary slighty due to rounding. To get the exact answer use the Excel formula = 100000 × 8 (Log (0.8)/ Log (2)), for example. (See the spreadsheet “Learning Curves.”)

Learning Curve Tables

When the learning percentage is known, Exhibits 6.4 and 6.5 can be used to easily calculate estimated labor hours for a specific unit or for cumulative groups of units. We need only multiply the initial unit labor hour figure by the appropriate tabled value.