Incremental Unit-Time Learning Model

In the incremental unit-time learning model, the incremental time needed to produce the last unit declines by a constant percentage each time the cumulative quantity of units produced doubles. Again, consider Rayburn Corporation and an 80% learning curve. With this model, the 80% means that when the quantity of units produced is doubled from X to 2X, the time needed to produce the unit corresponding to 2X is 80% of the time needed to produce the Xth unit. Exhibit 10-12 shows calculations for the incremental unit-time learning model. Note how when the units produced double from 2 to 4 in column A, the time to produce unit 4 (the last unit when 4 units are produced) is 64 hours in column B, which is 80% of the 80 hours needed to produce unit 2 (the last unit when 2 units are produced). We obtain the cumulative total time in column D by summing the individual unit times in column B. For example, to produce 4 cumulative units would require 314.21 labor-hours (100.00+80.00+70.21+64.00).

Exhibit 10-12 

Incremental Unit-Time Learning Model for Rayburn Corporation

Spreadsheet screenshot shows the incremental unit-tune learning model for Rayburn Corporation at 80% learning curve.

Details of the spreadsheet are as follows:

Cumulative number of units (X): 1

(A) Individual unit time for 1st unit (denote by “y”): Labor-hours: 100.00

(B) Cumulative total time–Labor hours: 100.00

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 100.00

Cumulative number of units (X): 2

(A) Individual unit time for 2nd unit (denote by “y”): Labor-hours: 80.00 (product of 100.00 and 0.8)

(B) Cumulative total time–Labor hours: 180.00 (sum of cumulative time 100 and individual unit time 80) 

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 90.00

Cumulative number of units (X): 3

(A) Individual unit time for 3rd unit (denote by “y”): Labor-hours: 70.21

(B) Cumulative total time–Labor hours: 250.21 (sum of 180 and 70.21)

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 83.40

Cumulative number of units (X): 4

(A) Individual unit time for 4th unit (denote by “y”): Labor-hours: 64.00 (product of 80 and 0.8)

(B) Cumulative total time–Labor hours: 314.21

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 78.55

Cumulative number of units (X): 5

(A) Individual unit time for 5th unit (denote by “y”): Labor-hours: 59.56

(B) Cumulative total time–Labor hours: 373.77

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 74.75

Cumulative number of units (X): 6

(A) Individual unit time for 6th unit (denote by “y”): Labor-hours: 56.17

(B) Cumulative total time–Labor hours: 429.94

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 71.66

Cumulative number of units (X): 7

(A) Individual unit time for 7th unit (denote by “y”): Labor-hours: 53.45

(B) Cumulative total time–Labor hours: 483.39 

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 69.06

Cumulative number of units (X): 8

(A) Individual unit time for 8th unit (denote by “y”): Labor-hours: 51.20 (product of 64.00 and 0.8)

(B) Cumulative total time–Labor hours: 534.59

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 66.82

Cumulative number of units (X): 9

(A) Individual unit time for 9th unit (denote by “y”): Labor-hours: 49.29

(B) Cumulative total time–Labor hours: 583.89

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 64.88

Cumulative number of units (X): 10

(A) Individual unit time for 10th unit (denote by “y”): Labor-hours: 47.65

(B) Cumulative total time–Labor hours: 631.54

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 63.15

Cumulative number of units (X): 11

(A) Individual unit time for 11th unit (denote by “y”): Labor-hours: 46.21

(B) Cumulative total time–Labor hours: 677.75

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 61.61

Cumulative number of units (X): 12

(A) Individual unit time for 12th unit (denote by “y”): Labor-hours: 44.93

(B) Cumulative total time–Labor hours: 722.68 

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 60.22

Cumulative number of units (X): 13

(A) Individual unit time for 13th unit (denote by “y”): Labor-hours: 43.79

(B) Cumulative total time–Labor hours: 766.47

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 58.96

Cumulative number of units (X): 14

(A) Individual unit time for 14th unit (denote by “y”): Labor-hours: 42.76

(B) Cumulative total time–Labor hours: 809.23

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 57.80

Cumulative number of units (X): 15

(A) Individual unit time for 15th unit (denote by “y”): Labor-hours: 41.82

(B) Cumulative total time–Labor hours: 851.05

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 56.74

Cumulative number of units (X): 10

(A) Individual unit time for 10th unit (denote by “y”): Labor-hours: 40.96 (product of 51.2 and 0.8)

(B) Cumulative total time–Labor hours: 892.01

(C) Cumulative average time per unit–Labor-hours (calculated by dividing the cumulative total time with number of units, and denoted by “X”): 55.75

Note: The mathematical relationship underlying the incremental unit-time learning model is as follows:

y equals “a” times X superscript “b”

where y equals Time (labor-hours) taken to produce the last single unit 

X equals Cumulative number of units produced

a equals Time (labor-hours) required to produce the first unit

b equals Factor used to calculate incremental unit time to produce units

which equals the quotient of log natural (learning-curve percent in decimal form)

and ln2

For an 80% learning curve, b equals log natural 0.8 divided by log natural 2 which equals negative 0.2231 divided by 0.6931 equals negative 0.3219,

For example, when X equals 3, a equals 100, and b equals negative 0.3219,

y equals 100 times 3 superscript (0.3219) equals 70.21 labor-hours

The cumulative total time when X equals 3 is 100 plus 80 plus 70.21 equals 250.21 labor-hours.

Numbers in the table may not be exact because of rounding.

Exhibit 10-13 shows the cumulative average-time learning model (using data from Exhibit 10-11) and the incremental unit-time learning model (using data from Exhibit 10-12). Panel A plots the cumulative average time per unit as a function of cumulative units produced for (1) the cumulative average-time learning model using data from Exhibit 10-11, column B, and (2) the incremental unit-time learning model using data from Exhibit 10-12, column E. Panel B plots the cumulative total labor-hours again as a function of cumulative units produced for (1) the cumulative average-time learning model using data from Exhibit 10-11, column D and (2) the incremental unit-time learning model using data from Exhibit 10-12, column D.

Exhibit 10-13

Plots for Cumulative Average-Time Learning Model and Incremental Unit-Time Learning Model for Rayburn Corporation

Screenshot of a spreadsheet shows the curves for cumulative average-time learning model and incremental unit-time learning model for Rayburn Corporation.

The spreadsheet shows two panels A and B. All values are approximate here. Details of the curve plots are as follows:

(1) Panel A: Cumulative average time per unit (80% learning curve; First unit takes 100 labor-hours)

(2) Panel B: Cumulative total labor-hours (80% learning curve; first unit takes 100 labor-hours)

Assuming the learning rate is the same for both models, the cumulative average-time learning model represents a faster pace of learning. For example, in Exhibit 10-13, Panel B, the cumulative total labor-hours graph for the 80% incremental unit-time learning model lies above the graph for the 80% cumulative average-time learning model. To produce 4 cumulative units, the 80% incremental unit-time learning model predicts 314.21 labor-hours, whereas the 80% cumulative average-time learning model predicts 256.00 labor-hours. That’s because under the cumulative average-time learning model the average labor-hours needed to produce all 4 units is 64 hours; the labor-hour amount needed to produce unit 4 is much less than 64 hours—it is 45.37 hours (see Exhibit 10-11). Under the incremental unit-time learning model, the labor-hour amount needed to produce unit 4 is 64 hours, and the labor-hours needed to produce each of the first 3 units is more than 64 hours, so the average time needed to produce all 4 units is more than 64 hours.

How do managers choose which model and what percent learning curve to use? They do so on a case-by-case basis. For example, they use the 80% learning-curve cumulative average-time learning model if the behavior of manufacturing labor-hour usage as production levels increase follows a pattern predicted by this model. Engineers, plant managers, and workers are good sources of information on the amount and type of learning actually occurring as production increases. Plotting this information and estimating the model that best fits the data are helpful when selecting the appropriate model.*