Last updated 1996 Sep 29
Accounting 503
Learning
Measurement
| Examples
| References
Measures
| Formulas
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One aspect of learning is being able to do a task faster as the task is done over and over again. The basic measurements are:
| measurement |
symbol |
| the number of times the task is done |
N |
| the total time to do the task this number of times |
T |
A derived measurement is
| cumulative average time |
T/N |
Cumulative average time is average time from the very start of production. Cumulative average time will be greater than the time for producing the last or most recent item.
Experience tells us that cumulative average time should be a decreasing function of N. Experiments involving both individual and group learning support the formula
T/N = a N-b
where a and b are positive constants determined from historical data. Notice that whenever total production doubles, the cumulative average time declines by a set ratio, namely, 2-b:
a (2N)-b = (a N-b ) (2-b )
Percentage Learning with Doubling of Production
- The ratio 2-b is often in the range of 90% to 85%. Correspondingly we speak of a 90% or 85% learning curve. Such learning rates are common in
- aerospace manufacturing
- tract housing construction
- The limiting cases are
- 100% curve means no learning
- 50% curve means instantaneous production
Derivation of the general formula
Let f(x) be cumulative average time for x iterations. Let r be the doubling ratio.
| formula |
derivation |
| f(2x) = rf(x) |
from the doubling formula |
| f(2k x) = rk f(x) |
by iteration for all integers k, positive and negative |
| f(2y) = ry f(1) |
for all y by taking x=1 and using continutity of f(x) |
| f(x) = x (ln r / ln 2) f(1) = ax - b |
by letting 2y = x and f(1) = a |
Calculation of the constants a and b
Budgeting with the Learning Curve. Suppose that a company has made 15 airframes using 9,936 direct labor hours and gone on to make a total of 25 airframes with a total of 14,691 direct labor hours. If the contract calls for 200 airframes, how many direct labor hours can we expect that to require altogether?
Using the formula, we write the historical data as
9936 / 15 = a 15-b
and
14691 / 25 = a 25-b
Dividing the first equation by the second gives us
1.1272 = ( 15 / 25) -b = 0.6-b
Taking the logarithm of both sides, we get
0.1198 = -b ( log 0.6 ) = .5108 b
or
b = .2344
Now substitute this value for b into the first equation and solve for
a = ( 9936 / 15 ) / ( 15-.2344 ) = 1250
The cumulative average time for 200 airframes in direct labor hours is
T/N = 1250 ( 200-.2344 ) = 361
and the total time is
( 361 ) ( 200 ) = 72,200.
Since 2-.2344 = .85, cumulative average time declines to 85% of its former value whenever total production doubles. We call this an 85% learning curve, and we can construct the table:
| Total Air Frames |
Cumulative Average Time |
Total Time |
| 25 |
14691/25 = 588 |
14,691 |
| 50 |
85% x 588 = 500 |
25,000 |
| 100 |
85% x 500 = 425 |
42,500 |
| 200 |
85% x 425 = 361 |
72,200 |
The next part of this exercise is to budget for airframes 26 through 200 with the cost of direct labor at $35 an hour, variable overhead at $25 an hour, and material at $50,000 per airframe. The table above shows the last 200 - 25 = 175 airframes will take 72,200 - 14,691 = 57,309 direct labor hours. The budget is
| cost item |
unit cost |
volume |
extended cost |
| direct material |
$50,000 |
175 |
$8,750,000 |
| direct labor |
$35 |
57,309 |
$2,005,815 |
| variable overhead |
$25 |
57,309 |
$1,432,725 |
CMA Dec 1977
A company has just completed one bridge, its first over a river, in 100 weeks. It would like to continue building similar bridges but must get the average time down to 52 weeks. With an 80% learning curve the company will see
| total bridges |
cumulative average time |
total time |
| 1 |
100 |
100 |
| 2 |
80 |
160 |
| 4 |
64 |
256 |
| 8 |
51.2 |
409.6 |
References
Studies of Learning
- Individual Performance
- nineteenth century telegraph operators
- Cuban cigar makers in the 1950's
- Group Performance
- aircraft manufacturing in the 1930's
Articles
John H. Cushman, Jr., Pentagon overbilling discovered, New York Times News Service, August 9, 1988.
Diane D. Pattison and Charles J. Teplitz, Are Learning Curves Still Relevant?, Management Accounting, February, 1989, pages 37 to 40.
Ken M Boze, Measuring Learning Costs, Management Accounting, August, 1994, pages 48 to 52.
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