CMA Final · Strategic Cost Management

Learning Curve: formula sheet

Full chapter guide

Key formulas

Learning rate (cumulative average): Cumulative average time at 2n units = Learning rate × Cumulative average time at n units; Applies each time cumulative output doubles. An 80% curve means a rate of 0.80.
Total time: Total time for n units = Cumulative average time per unit × n; Use this to convert average time into total time for a cumulative output.
Incremental time: Time for units n+1 to m = Total time for m units − Total time for n units; Always find the time of a batch by subtracting cumulative totals.
Learning rate from data: Learning rate = Cumulative average time at 2n units ÷ Cumulative average time at n units; Use when the question gives two doubling points and asks for the rate.
Learning curve model: Y = aX^b; Y = cumulative average time (or cost) per unit for X units; a = time for first unit; X = cumulative units.
Learning index: b = log r ÷ log 2; r is the learning rate as a decimal (80% = 0.8). b is negative for r below 1.
Learning rate from index: r = 2^b; Use to get back the rate once b is known.
Learning rate from data: b = log(Y ÷ a) ÷ log X, then r = 2^b; Use when the first-unit time and the cumulative average for X units are given.
Total time: Total time for X units = Y × X; Time for units m+1 to n = total for n units minus total for m units.
Doubling shortcut: Average for 2X units = r × average for X units; Valid only when quantity doubles exactly, such as 1, 2, 4, 8, 16.
Learning curve model: Y = aXᵇ; Y = cumulative average time per unit for X units; a = time for the first unit; X = cumulative units.
Index b: b = log r ÷ log 2; r is the learning rate as a decimal (0.8 for 80%). b is negative. For 80%, b ≈ -0.3219. For 90%, b ≈ -0.1520.
Doubling rule: Average time for 2X units = r × average time for X units; Use this when units are 1, 2, 4, 8, 16 and so on. No logs are needed.
Total time: Total time for X units = X × Y; Y is the cumulative average time for X units.
Incremental time: Time for units (m+1) to n = Total time for n units − Total time for m units; For the nth unit alone, take total time for n units less total time for (n−1) units.
Labour cost: Labour cost = Time × Rate per hour; Apply the rate to total or incremental hours, as the question asks.
Learning curve model: Y = aX^b; Y = cumulative average time (or cost) per unit for X units; a = time for the first unit; X = cumulative units; b = index of learning.
Index of learning: b = log r ÷ log 2; r is the learning rate as a decimal (80% = 0.8). b is negative. For an 80% curve, b ≈ −0.3219.
Doubling rule: Average time for 2X units = r × average time for X units; Fast route when units are 1, 2, 4, 8, 16 and so on. Each doubling multiplies the average by r.
Total time for X units: Total time = X × Y; Y is the cumulative average time at X units.
Time for a later batch: Time for units (m+1) to n = Total time for n units − Total time for m units; Use this for repeat orders, the nth unit, and budgets for later batches.
Learning-adjusted cost: Total cost = Materials + (Learning-adjusted hours × labour rate) + (Hours × labour-hour overhead rate); Only labour and labour-hour-driven overheads are adjusted. Materials do not change with learning.

Quick revision

Learning curve: the time per unit falls as workers repeat a task and gain experience.
Standard model: cumulative average time per unit falls by a fixed percentage each time cumulative output doubles.
An 80% curve means the average falls to 80% of its earlier value at each doubling, a 20% reduction.
Model: Y = aX^b, with Y the cumulative average time per unit and a the time for the first unit.
b = log(learning rate) ÷ log 2, and b is negative for a learning rate below 100%.
Total time for X units = Y × X.
Time for the nth unit or a block of units = total time up to that level minus total time before it.
Doubling shortcut: 1 unit at 100 hours gives an average of 80 hours for 2 units on an 80% curve, so total is 160 hours.
The curve applies mainly to new, labour-intensive, repetitive work and fades once the learning is complete.
Only labour time falls; apply the learning effect to labour hours, then price the hours.
Use the curve for pricing, budgeting labour, make-or-buy and tender decisions, and state a recommendation.

Common mistakes

Applying the rate to the time of each unit instead of cumulative average time. Fix: Apply the rate to the cumulative average time only at each doubling of cumulative output, unless the question says incremental.
Applying the rate when output goes up by a fixed number, not a doubling. Fix: Check that cumulative output has doubled. For other quantities use the mathematical model.
Treating the learning rate as the fall in time, so using 0.2 for an 80% curve. Fix: An 80% curve means average time becomes 80% of before. Use r = 0.8 in b = log 0.8 ÷ log 2.
Reading Y as the time for the Xth unit. Fix: Y is the average for all X units. Multiply by X for total time, and subtract totals to get a particular unit or batch.
Treating the 80% as the time of the second unit, so unit 2 is taken as 80% of unit 1 and unit 4 as 80% of unit 2 only. Fix: Remember that 80% applies to the cumulative average time when output doubles. Find the average first, then total.
Using the average time as the total time. Fix: Always do total time = units × average time before computing cost.
Using first-unit hours for the whole order. Fix: Always compute the cumulative average for the order size first, then multiply by the number of units.
Treating the 80% rate as a fall of 80% in time. Fix: An 80% curve means the average falls to 80% of its earlier value on doubling, which is a 20% reduction.

Exam tips

In theory answers, list the assumptions and the limitations separately. Examiners look for both.
In numerical questions, read whether the rate is for cumulative average time or incremental time before you start.
Show the doubling table. Even if you slip on arithmetic, method marks are protected.
For MCQs, test whether output has actually doubled. Options often include the trap answer from misapplied rates.
End decision questions with a short comment, for example that early cost estimates should allow for learning and standards need the steady state.
Show b = log r ÷ log 2 with its value. Marks are often given for the index even if the later arithmetic slips.
Check the question for the model. This topic is the cumulative average model, so Y is an average, not the Xth unit's time.
Use the doubling table when quantities are powers of 2. It is faster and you avoid log errors.