There are following two techniques of simple exponential curve to analyze the past data:

(a) Arithmetical tabulation

(b) Logarithmic analysis

(a) Arithmetical Tabulation:

In an arithmetical tabulation approach, a column for units is created by doubling, row by row as: 1, 2, 3, 4, 8, 16…….. the time for the first unit is multiplied by the learning percent for the fourth unit, and so on.

Thus, if we are developing an 80% learning curve, we would arrive at the figures listed in column 2 of Table 29.1. Since it is often desirable for planning purposes to know the cumulative direct labour hours, column 4, which lists this information, is also provided.

The calculation of these figures is straightforward; for example, for unit 4, cumulative average direct labour hours would be found by dividing cumulative direct labour hours by 4, yielding the figure given in column 4.

Table 29.1-unit, cumulative & cumulative average direct labour worker hours required for an 80% learning curve.

(1) Unit Number


Unit Direct Labour Hours


Cumulative Direct Labour Hours


Cumulative Average Direct Labour Hours





































Fig. 29.3 shows three curves with different learning rates: 90 percent, 80 percent and 70 percent. Note that if the cost of the 1st unit was Rs. 100, the 30th unit would cost Rs. 59.63 at the 90 percent rate and Rs. 17.37 at the 70 percent rate. Differences in learning rates can have dramatic effects.

(b) Logarithmic Analysis:

Yx = Kxn

The normal form of the learning curve equation is:


x = Unit number

Yx = Number of direct labour hours required to produce the xth unit.

K = Number of direct labour hours required to produce the first unit.

n = Log b/log 2, where b = learning percentage.

Thus, to find the labour-hour requirement for the eighth unit in our example (Table 29.2), we would substitute as follows:

Y8 = (1, 00,000) (8) n

This may be solved by using logarithms:

Y8 = 1, 00,000 (8) log 0.8/log 2

= 1, 00,000 (8) – 0.322

= 1, 00,000/(8) 0.322

= 1, 00,000/1.9535

= 51,900

Therefore, it would take 51,900 hours to make the eighth unit.

In practice, learning curves are plotted on log paper, with the result that 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.

Fig. 29.4 shows the 80 percent unit cost curve and average cost curve on logarithmic paper. Note that the cumulative average cost is essentially linear after the eighth unit.

While the arithmetic tabulation approach is useful, direct logarithmic analysis of learning curve problems is generally more efficient since 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.

Implications of Learning Curve:

Learning curve leads to following Improvements:

1. More skilful movements of worker.

2. Improvements in machine and tooling.

3. Less rejection and rework.

4. Improved management control.

5. Less time required to instruct workers.

6. Better operation – sequences, machine feeds speeds.

7. Less set up time due to larger lots.

Estimating the Learning Percentage:

If production has been underway for some time, the learning percentage is easily obtained from production records. Generally speaking, the longer the production history, the more accurate the estimate. Since a variety of other problems can occur during the early stages of production, most companies do not begin to collect data for learning curve analysis until some units have been completed.

Statistical analysis can also be used. For example, in an exponential learning curve to find out how well the curve fits past data, it can be converted to a straight line logarithmic (data plotted on log-log graph paper). If the data are questionable because they do not fit a line well, a correlation of .70 or less when the number of data points is 20 or more would be considered a poor fit.

If production has not started, estimating the learning percentage becomes enlightened guess work. In these cases the analyst has following three options :

1. Assuming that the learning percentage will be the same as it has been for previous applications within the same industry.

2. Assuming that it will be the same as it has been for the same or similar products.

3. Analyzing the similarities and differences between the proposed start up and previous start ups and develop a revised learning percentage that appears best to fit the situation.

In selecting the option, the decision turns to how closely the start up under consideration approximates previous start ups in the same industry or with the same or similar products. In any case while a number of industries have used learning curves extensively, acceptance of the industry norm (such as the 80 percent figure from the airframe industry) is risky. An analysis of the company’s own data should be undertaken even though it may ultimately lead to the industry improvement percentage.

There are two reasons for disparities between a firm’s learning rate and that of its industry. First, there are the inevitable differences in operating characteristics between any two firms, stemming from the equipment methods, product design, plant organisation, and so forth. Second, procedural differences are manifested in the development of the learning percentage itself, such as whether the industry rate is based on a single product or on a product line, and the manner in which the data are aggregated.