According to the Offi cial Terminology:
Learning Curves
Learning curve: The mathematical expression of the phenomenon that, when complex and labour-intensive procedures are repeated, unit labour times tend to decrease at a constant rate. The learning curve models mathematically this reduc- tion in unit production time.
LEARNING CURVES
The recognition of the so-called learning curve phenomenon stems from the experience of aircraft manufacturers, such as Boeing, during the Second World War. They observed that the time taken to assemble an individual aircraft declined as the number of aircraft assembled increased: as workers gained experience of the process, their profi ciency, and hence speed of working, increased. The learning ’ gained on the assembly of one plane was translated into the faster assembly of the next. The actual time taken by the assembly workers was monitored, and it was discovered that the rate at which the learning took place was not random, but rather waspredictable.
It was found that the cumulative average time per unit decreased by a fi xed percentage each time the cumulative production doubled. In the aircraft industry, the percentage by which the cumulative average time per unit declined was typically 80 per cent. For other industries, other rates may be appropriate. Further, the unit of measurement may more sensibly be taken as abatch of product, rather than as an individual unit. This does not, of course, affect the underlying principle.
Let us take as an example a learning rate of 90 per cent:
● If the fi rst batch of a product is produced in 100 hours:
● The cumulative average time taken to produce two batches (a doubling of the cumulative production) would be 90 hours.
● This gives a total production time of 2 90 180 hours.
● The actual time taken to produce the second batch (the batch being the unit of measure in this case) would thus be 80 hours, the cumulative total time taken to produce two batches – 180 hours – less the time taken to produce the fi rst batch – 100 hours.
As a doubling of cumulative production is required, in order to gain the benefi ts of learning in the form of reduced average labour hours per unit of cumulative production, the effects of the learning rate on labour time will become muchless signifi cant as produc- tion increases. Table 9.1 shows this.
The graphs in Figure 9.1 show the data from Table 9.1 plotted both on an ordinary and on a double logarithmic scale. It will be noted that the double logarithmic form is linear;
the ordinary form is not.
The learning percentage is usually somewhere between 70 per cent and 85 per cent. The more complicated the product the steeper the learning curve will be. For example, research on Japanese motorbikes found that the small 50cc bikes had a learning curve of 88 per cent, the 50–125cc bikes, one of 80 per cent and the 250cc bikes, one of 76 per cent.
Table 9.1 Cumulative average time learning rate: 0.90 Batches
Cumulative average time per batch (hours)
Cumulative total (hours)
1 100.00 100
2 90.00 180
4 81.00 324
8 72.90 583
16 65.61 1,050
32 59.05 1,890
64 53.14 3,401
128 47.83 6,122
256 43.05 11,021
512 38.74 19,835
LEARNING CURVES
Table 9.2 illustrates how an 80 per cent learning curve works, as units are doubled from 1 to 2, to 4, to 8.
In constructing Tables 9.1 and 9.2 , it was assumed that we already knew the learning rate that applied to this particular situation. However, it must be appreciated that, in the real world, this rate can only be established by observation. Records must be kept of the number of units/batches produced and the associated time taken, in order to construct the equivalent of Table 9.1 (although it is likely that fewer observations would actually be taken). It is then the job of the engineer or accountant to deduce the learning rate from these observations,
Table 9.2 An 80 per cent learning curve No. of
units
Average time per unit
Total time in hours
Additional units
Additional time
Time per unit
1 10 10
2 8 (10 80%) 16 1 6 hours 6 hours
4 6.4 (8 80%) 25.6 2 9.6 4.8 (6 80%)
8 5.12 (6.4 80%) 40.96 4 15.36 3.8 (4.8 80%)
100 90 80 70 60 50
1 2 4 8 16 32
X Batches (or units) Yx
Cum.
average time per batch, hours
Log of cum.
average time per batch, hours
Log of no. of batches (or units) Figure 9.1 Cumulative data graphs
LEARNING CURVES
which will require the specifi cation of an equation to fi t the data. For example, the observa- tions in Table 9.1 are plotted in Fig. 9.1 , and can be described by the equation:
Learning curve formula:
Y axb where Y cumulative average time taken per unit a time taken for the fi rst unit
x total number of units b index of learning
where b the log of learning
log of 2 i.e. log 0.8 (for 80%)
log 2 0.0969
0.3010 0.32
Thus, in Table 9.2 , the average time taken per unit, for 8 units, is given by:
Y
10 8
5 14
0 32.
.
which, allowing for rounding, is the same average time that was derived in Table 9.2 . Note that the log is easily found by using the log button on your calculator. You are not expected to have a working knowledge of logs for the exam.
Exercise
Derive the learning curve formula for a 90 per cent learning rate, and apply it to confi rm the cumulative average time per unit for 64 units, as shown in Table 9.1 .
Solution
Y axb
b , the index of learning log 0.9
log 2 0 152. *
*This is calculated using a scientifi c calculator, but you can use log tables if you pre- fer. Inserting the relevant data into the formula, we get:
Average time taken per unit
100 64
53 14
0 152.
. This agrees with Table 9.1 above.
The learning curve equation is not normally used to check earlier calculations in the way that we have just done, but is rather used to assess the time that will be required for an output level that does not represent a doubling of the cumulative production total, and thus cannot be determined by simply creating a table such as the one used earlier. For example, let us assume that the manufacturer above has the opportunity to bid for a contract to produce ten batches
LEARNING CURVES
of his product, and wishes to estimate the time it will take to complete the contract, in order to help set the tender price. If the cumulative total production of his product to date is 32 batches, the learning curve equation can be employed to calculate the cumulative average time per batch to produce 42 batches: the 32 already produced, plus the ten under consideration:
Y ax
Y Y Y
x b
x x x
100 42
100 0 5666 56 66
0 152.
. .
that is, the average time per batch to produce 42 batches is 56.66 hours, giving a total pro- duction time of 56.66 42 2,380 hours.
Inspection of Table 9.1 above reveals that the average time per batch to produce 32 batches is 59.05 hours, with a total production time of 1,890 hours. The total time taken to produce the ten batches under consideration will thus be 2,380 – 1,890 490 hours, that is, an average time of 49 hours for each of the batches 33–42. This may be compared with the average of (3,401 – 1,890)/32 47.20 hours indicated by Table 9.1 for the next doubling of a full 32 batches from 33 to 64. Obviously, if the current level of production does not lie on a table such as Table 9.1 , that particular average time, and the correspond- ing cumulative hours to date, will need to be calculated from the same formula.