## **Learning Outcomes**

After studying this article, you will be able to describe learning curves as a forecasting technique, explain the difference between cumulative average and incremental (interval) models, calculate labour hours using learning curve equations, and recognise common limitations, ensuring accurate exam performance in ACCA Performance Management.

## **ACCA Performance Management (PM) Syllabus**

For ACCA Performance Management (PM), you are required to understand the application of learning curves and related forecasting models for budgeting and control. Specifically, this article covers:

- Explain and apply the learning curve concept for forecasting labour costs and production times
- Distinguish between cumulative average and incremental (interval) learning curve models
- Calculate cumulative average time per unit and total time using learning curve equations
- Estimate the learning rate from data and interpret learning curve implications for budgeting
- Discuss the limitations and appropriateness of learning curves in different production settings

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. What does an 80% learning curve mean for labour time per unit as output doubles?
2. Which learning curve formula is usually used for ACCA PM exam calculations?
3. If the first unit of a new product takes 100 hours, and cumulative average time for 4 units is 64 hours, what is the total time taken for those 4 units?
4. True or false? The learning curve effect will always continue no matter how many units are produced.

## **Introduction**

Learning curves are a core quantitative technique for forecasting in performance management, particularly where new products or processes involve complex, repetitive manual work. Understanding their application, especially the difference between cumulative average and interval (incremental) models, is essential for ACCA PM candidates.

A learning curve reflects how the time taken to produce each unit decreases as production experience grows. This effect must be captured accurately in budgets and cost estimation, as assuming constant times for each unit can lead to serious under- or over-estimation of labour costs.

> **Key Term: learning curve**
> A mathematical model describing how labour time per unit falls at a predictable rate each time cumulative output doubles, due to increased worker efficiency.
>
> **Key Term: cumulative average model**
> A learning curve approach where each time cumulative output doubles, the average time per unit drops by a fixed percentage.
>
> **Key Term: interval (incremental) model**
> A learning curve approach where the time taken for a specific batch or unit interval is calculated directly, rather than via the cumulative average.

## **LEARNING CURVES IN FORECASTING**

Recognition of the learning effect is especially important in industries where new, labour-intensive processes occur. For instance, manufacturing aircrafts, custom machinery, or certain one-off design jobs.

### The Principle

When a process exhibits a learning effect, the first units produced take the most time. Every time cumulative production doubles, the average labour hour per unit decreases to a fixed proportion (learning rate) of its previous average.

For example, with an 80% learning curve:

- If first unit = 100 hours, at two units average is 80 hours. At four units, average is 64 hours. At eight units, average is 51.2 hours.

Learning curves usually apply to direct labour time (manual tasks), and not to automatic or capital-intensive processes.

### Applicability

Learning effects are most notable when:

- The process is new and repetitive
- Labour input is significant
- There are no major breaks or design changes during production
- Sufficient volume is produced for learning to take effect

If production methods, workforce, or product design change, the learning effect may disappear or be reset.

## **CUMULATIVE AVERAGE VS INTERVAL LEARNING CURVE MODELS**

Two models are commonly encountered:

> **Key Term: learning rate**
> The percentage by which the cumulative average time per unit falls every time total output doubles. Expressed as a decimal (e.g., 0.8 or 80%).

### Cumulative Average Model

This is the most common model for ACCA PM exam questions. The average time per unit to produce all units up to 'x' is determined by:

> **Key Term: learning curve equation**
> The equation:
>
> $$
> Y = a x^b
> $$
>
> where Y is cumulative average time per unit, a is time for the first unit, x is the cumulative number of units, and b is the learning index.

Where:

- $Y$ = cumulative average time per unit to produce $x$ units
- $a$ = time required for the first unit
- $x$ = total units produced
- $b$ = index of learning, calculated as $b = \frac{\log LR}{\log 2}$

The total time for all units = $Y \times x$.

### Incremental (Interval) Model

The interval or incremental model is used less frequently, but may be required when you are asked how much time is needed for the next batch or units (e.g., for units 5–8 rather than 1–8 in total).

This is calculated as:

- Total time to produce up to unit $n$, minus total time for units up to $m$ ($n > m$).

So, time for units $m+1$ to $n$ = (average time per unit for $n$ units × $n$) − (average time per unit for $m$ units × $m$).

## **CALCULATION STEPS**

1. Find the learning rate ($LR$) from the scenario (e.g., 80%, 90%).
2. Calculate the learning index $b$ as $b = \frac{\log LR}{\log 2}$.
3. Plug $a$, $b$, and desired $x$ into $Y = a x^b$ for cumulative average time.
4. Multiply $Y \times x$ for total time to produce $x$ units.
5. For incremental time (a batch): subtract total time taken for previous units from the total time taken for the new batch.

### **Worked Example 1.1**

A company is producing a new product. The first unit takes 100 hours to produce. An 80% learning curve is expected for the first 16 units.

**Required:** Calculate the average time to produce the first 16 units, total time for 16 units, and the time for units 9–16.

> **Answer:**
>
> 1. Calculate (rounded to 4 decimals):
>    $$
>    b = \log 0.8 / \log 2 = -0.3219
>    $$
> 2. Cumulative average for 16 units (hours, rounded to 2 decimals):
>    $$
>    Y = 100 \times 16^{-0.3219} = 40.96
>    $$
> 3. Total time for 16 units: 40.96 × 16 = 655.36 hours.
> 4. Time for units 9–16: Calculate total for 16, subtract total for 8.
>    - Cumulative average for 8 units: 100 \times 8^{-0.3219} = 51.20 hours.
>    - Total for 8 units: 51.20 \times 8 = 409.60 hours.
>    - So, time for units 9–16 = 655.36 - 409.60 = 245.76 hours.

### **Worked Example 1.2**

The first batch of a new assembly takes 40 hours. The next three batches (batches 2–4) are to be produced. An 85% learning curve applies.

**Required:** Find the total time to produce four batches and the time taken for the fourth batch alone.

> **Answer:**
>
> 1. Calculate:
>    $$
>    b = \log 0.85 / \log 2 = -0.234
>    $$
> 2. Y for four batches (average hours):
>    $$
>    Y = 40 \times 4^{-0.234} = 28.49
>    $$
> 3. Total time for four batches: 28.49 \times 4 = 113.96 hours.
> 4. Y for three batches (average hours):
>    $$
>    Y = 40 \times 3^{-0.234} = 31.39
>    $$
>    - Total time for three: 31.39 \times 3 = 94.17 hours.
>    - Time for fourth batch: 113.96 - 94.17 = 19.79 hours.

## **INTERPRETATION AND LIMITATIONS**

Learning curves do not continue indefinitely. Eventually, a steady state or "plateau" is reached, where times no longer decrease further.

The model only works if:

- The task is repetitive and unchanged
- Production is not interrupted
- The workforce remains consistent

Learning curves are less reliable for:

- Automated or highly mechanised processes
- Small batch numbers
- Situations with frequent staff or design changes

> **Key Term: steady state**
> The point in production at which further decreases in labour time per unit no longer occur and the learning curve effect ends.

### **Exam Warning**

> Learning curve questions in ACCA PM will usually require you to use the cumulative average model. Always check whether the question wants the average time, total time, or the time for a specific batch. Careless selection of the model may lead to lost marks.

### **Revision Tip**

> When asked for total time for a range of units, calculate cumulative total time for all and subtract the total time for previous output—that gives you the time for the new batch.

## **Summary**

Learning curves provide a method for forecasting how labour times fall as production experience increases, which is critical in accurate budgeting and cost estimation for new products. The cumulative average model is most common in ACCA PM; it predicts how average times decrease as cumulative output doubles. The interval model allows calculation of time for specific batches but relies on cumulative average calculations. Always consider process changes, steady state, and whether the learning effect is likely before applying these models for forecasting.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Explain the meaning of learning curves in forecasting labour cost and time
- Apply the cumulative average model and its formula ($Y = a x^b$) correctly
- Calculate cumulative average and incremental times for production batches
- Recognise and calculate the learning index ($b$) from a given learning rate
- Differentiate between average and incremental models, and their use-cases in practice
- Identify the practical limitations of the learning curve for accurate forecasting

## **Key Terms and Concepts**

- learning curve
- cumulative average model
- interval (incremental) model
- learning rate
- learning curve equation
- steady state


Forecasting techniques - Learning curves and cumulative average/interval models