LEARNING CURVE
The learning curve is based on the proposition that labor hours decrease in a definite pattern as labor operations are repeated. More specifically, it is based on the statistical findings that as the cumulative production doubles, the cumulative average time required per unit will be reduced by some constant percentage, ranging typically from 10 percent to 20 percent. By convention, learning curves are referred to in terms of the complements of their improvement rates.
For example, an 80 percent learning curve denotes a 20 percent decrease in unit time with each doubling of repetitions. As an illustration, a project is known to have an 80 percent learning curve. It has just taken a laborer 10 hours to produce the first unit. Then each time the cumulative output doubles, the time per unit for that amount should be equal to the previous time multiplied by the learning percentage. An 80 percent learning curve is shown in Figure 1.
The learning curve model is as follows:
y_{n} = a n^{- b}
where y_{n} = Time for the nth unit
a = Time for the first unit (in this example, 10 hours)
b = The index of the rate of increase in productivity during learning (Log learning rate/log 2)
To be able to utilize linear regression, we need to convert this power (or exponential) function form into a linear form by taking a log of both sides, which yields:
Log y_{n} = log a - b log n
The learning rate, which is indicated by b, is estimated using a least-squares regression, with the sample data on y and n. Note that
which means:
Log (learning rate) = b x log 2
The unit time (i.e., the number of labor hours required) for the nth can be computed using the estimated model:
y_{n} = a n^{- b}
NOTE: This learning phenomenon is observed in the behavior of labor and labor driven overhead. Material costs per unit may also be subject to this effect if less scrap and waste occur as a result of learning. |
Example 6
For an 80 percent curve with a = 10 hours, the time for the third unit would be computed as:
y_{3} = 10 (3 ^{- log .8/ log 2 }) = 10 (3^{.3219 }) = 7.02
Fortunately, it is not necessary to grid through this model each time a learning calculation is made; values (nb) can be found using Table 2 of the Appendix (Learning Curve Coefficients). The time for the nth unit can be quickly determined by multiplying the table value by the time required for the first unit.
FIGURE 1
An 80% learning curve
Example 7
NB Contractors, Inc. is negotiating a contract involving production of 20 jets. The initial jet required 200 labor days of direct labor. Assuming an 80 percent learning curve, we will determine the expected number of labor days for (1) the 20th jet, and (2) all 20 jets as follows:
Using Table 2 with n=20 and an 80 percent learning rate, we find: Unit=.381 and Total=10,485. Therefore,
(1) Expected time for the 20th jet = 200 (.381)= 76.2 labor days.
(2) Expected total time for all 20 jets = 200(10.485) =2,097 labor days.
NOTE: The learning curve theory has found useful applications in many areas, including: 1. Budgeting, purchasing, and inventory planning. 2. Scheduling labor requirements. 3. Setting incentive wage rates. 4. Pricing new products. 5. Negotiated purchasing. 6. Evaluating suppliers` price quotations. |
Example 8 illustrates the use of the learning curve theory for the pricing of a contract.
Example 8
Big Mac Electronics Products, Inc. finds that new product production is affected by an 80 percent learning effect. The company has just produced 50 units of output at 100 hours. Costs were as follows:
Materials 50 units @$20 - $1.000.
Labor and labor related costs:
Direct labor 100 hours @$8 - $800.
Variable overhead 100 hours @$2 - $200. Total - $2,000.
The company has just received a contract calling for another 50 units of production. It wants to add a 50 percent markup to the cost of materials and labor and labor related costs. To determine the price for this job, the first step is to build up the learning curve table.
The company has just received a contract calling for another 50 units of production. It wants to add a 50 percent markup to the cost of materials and labor and labor related costs. To determine the price for this job, the first step is to build up the learning curve table.
Thus, for the new 50 unit job, it takes 60 hours in total. The contract price is:
Materials 50 units @$20 - $1.000.
Labor and labor related costs:
Direct labor 60 hours @$8 - $480.
Variable overhead 60 hours @$2 - 120.
Total - $1.600.
50 percent markup - 800.
Contract price - $2.400.