## CIMA P2: Learning Curves

It’s taken a while for me to get back on track with my CIMA P2 studies but here I am with an article on Learning Curves which is a feature of the CIMA P2 syllabus. ## The Learning Curve

The learning curve relates to the observed tendency that workers become adept at a task, the more often they perform it. Hence the task will take less and less time the more often it is performed.

This is quite a simplistic way to look at the learning curve but there are many factors to consider when applying the learning curve and ultimately, understanding when and how to use it.

The official CIMA terminology for the learning curve can be found below:
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Learning curve is the mathematical expression of the commonly observed effect that, as complex and labour-intensive procedures are repeated, unit labour times tend to decrease. The equation usually relates the average time taken per unit/batch to the cumulative number of units/batches produced.

– CIMA Terminology

History tells us that the “learning curve” was first observed during the construction of World War 2 aircraft. It was discovered, that, as workers gained more experience building complex aircraft it actually took less time to complete each. And not only that, but the rate at which learning took place was actually predictable.

In reality it was found that the cumulative average time per unit decreased by a fixed percentage each time the cumulative production doubled.

### Learning Curve Example

The illustration below is using a learning rate of 80% and gives you a clear overview of the numbers involved.

 Cumulative average time learning rate: 80% Batches Cumulative average time per batch (hours) Cumulative total (hours) 1 100.00 100.00 2 80.00 160.00 4 64.00 256.00 8 51.20 409.60 16 40.96 655.36 32 32.77 1048.58 64 26.21 1677.72 128 20.97 2684.35 256 16.78 4294.97 512 13.42 6871.95

Nevertheless, there is a more scientific way of generating this information above, in the form of an equation, which is expressed as The Learning Curve Formula.

And knowledge of how to use this formula is required for the P2 exam.

### The Learning Curve Formula

The learning curve formula is simply expressed as y=ax^b

• y = cumulative average time taken per unit
• a = time taken for first unit
• x = total number of units
• b = the index of learning
• where b = the log of learning rate/ the log of 2

It’s a simple equation but perhaps the complex part is using the log tables when calculating the index of learning, as you may not be familiar  with how to apply them on your calculator? I know it was relatively new to me.

While the formula itself gives you the answer for the Cumulative Average Time Taken Per Unit (Y), the CIMA P2 examiner might also ask you to calculate the Index of Learning (b). So it’s important to understand each element of the formula.

### P2 Example Question on Learning Curves So to see this formula in action let’s go through it step-by-step.

Q – Company B has a learning rate of 90% and the total time to make the first unit is 8 hours, use the learning curve formula to find out the average time per unit for 16 units

The above question gives us the following information:

a = 8 hours x = 16 units b = ?

• First of all we need to calculate (b) the index of learning by using log tables.
• So b = log of 90%/log of 2
• To calculate this on any scientific calculator you need to press the following:
• LOG, 0.9/LOG, 2 = -0.152
• Therefore, the index of learning (b) is -0.152
• Now we are in a position to complete the question.
• 8 x 16^-0.152 (8 hours multiply 16 units to the power of -0.152) = 5.25
• Therefore, the cumulative average time per unit (y) = 5.25 hours
• Finally, we need to multiply this by 16 units (16 x 5.25) = 84 hours.

Q – Calculate the index of learning when the learning rate is 80% and it takes 100 hours to produce 1 batch (I’ve used the table in the first illustration as a base for this question).

Here we know the learning curve formula is y=ax^b and b is the index of learning which can be calculated using log tables.

• b = log of 80%/log of 2
• Therefore, the index of learning b = -0.32192
• We can cross check this with the table in my first illustration to prove it’s the correct answer.
• So let’s use this data to calculate y based on 64 units.
•  y = 100 (hours taken on first unit) x 64 (number of units) ^-0.321 (the index of learning). Remember the formula y=ax^b.
• Therefore, based on productions of 64 units, the cumulative time per unit (y) would be = 26.21 hours which is the same figure displayed in my original table.
• This confirms that the index of learning we calculated is correct.