## **Learning Outcomes**

After reading this article, you will understand how to optimise a production plan when faced with resource constraints and multiple products. You will be able to calculate and interpret contribution per limiting factor, determine the optimal product mix, recognise the intuition behind shadow pricing, and apply these concepts in decision-making scenarios. These skills are frequently examined in ACCA Advanced Performance Management.

## **ACCA Advanced Performance Management (APM) Syllabus**

For ACCA Advanced Performance Management (APM), you are required to understand the principles of resource allocation where resources are limited and multiple products compete for capacity. In particular, focus your revision on:

- The analysis and solution of multi-product resource constraints using contribution per limiting factor
- Calculating and interpreting shadow prices in the context of scarce resources
- Advising on optimal production plans and the value of additional resources
- Identifying non-financial and practical considerations affecting the resource allocation decision

## **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 is the first step in determining the optimal product mix when a company is faced with a single resource constraint and multiple products?
2. How is the contribution per unit of the limiting factor calculated, and why is it important?
3. In the context of scarce resources, what does the shadow price of a resource measure?
4. True or false? A shadow price always represents the total profit increase if more of a scarce resource is purchased, regardless of any other constraints.

## **Introduction**

Resource planning becomes critical when an organisation cannot meet all demand due to limited resources—such as labour hours, machine time, or materials. In such cases, managers must decide how best to allocate those scarce resources among competing products to maximise profit or contribution. This article explains how to approach these "multi-product mix" problems and introduces the concept of the shadow price, a key metric for evaluating the value of relaxing a constraint.

> **Key Term: Limiting Factor**
> A resource that restricts output because its supply is insufficient to produce the required quantities of all products demanded.
>
> **Key Term: Contribution per Limiting Factor**
> The contribution generated by a product from the use of a single unit of the constrained resource. Calculated as contribution per unit divided by the quantity of the limiting factor required per unit.

## **Resource Constraints and Product Mix**

When only one resource is limited and several products compete for it, a systematic approach is essential.

### Step 1: Identify the Limiting Factor

Common limiting factors include direct labour hours, machine time, or a material under allocation. Only resources in short supply count as true constraints.

### Step 2: Calculate Contribution per Unit

Work out the contribution per unit of each product:
Contribution per unit = Sales price – Variable cost

### Step 3: Find Contribution per Limiting Factor

Divide each product's contribution per unit by the amount of the limiting resource required for one unit of that product.

This identifies the return (contribution) produced per unit of the scarce resource.

Rank products from highest to lowest contribution per limiting factor.

> **Key Term: Scarce Resource**
> A production input available in limited quantity relative to demand for its use.

### Step 4: Allocate Resource to Maximise Contribution

Prioritise products with the highest contribution per limiting factor. Satisfy demand for the highest-ranked product first, then allocate any remaining constraint to the next, and so on, until the resource is exhausted.

### **Worked Example 1.1**

A company produces three products, A, B, and C. Only 1,000 hours of skilled labour are available this week. Data per unit:

|                        | A   | B   | C     |
| ---------------------- | --- | --- | ----- |
| Contribution ($)       | 12  | 15  | 9     |
| Labour hours needed    | 2   | 1   | 0.5   |
| Maximum demand (units) | 300 | 500 | 2,000 |

**Required:** Determine the mix of products that maximises total contribution.

> **Answer:**
>
> 1. Calculate contribution per limiting factor:
>
> - A: 12 / 2 = \$6 per labour hour
> - B: 15 / 1 = \$15 per labour hour
> - C: 9 / 0.5 = \$18 per labour hour
>
> 1. Rank: C highest, then B, then A.
> 2. Allocate the scarce 1,000 hours:
>
> - C: Max demand = 2,000 units × 0.5 hr = 1,000 hrs (uses all hours).
>
> No hours left for B or A. So, produce 2,000 units of C only. Maximum contribution = 2,000 × \$9 = \$18,000.

### **Exam Warning**

> In APM exams, you may face situations where maximum demand is not fully attainable for all products. Always start with the product yielding the highest contribution per limiting factor to maximise total profit.

## **Shadow Price Intuition**

The shadow price reflects the extra value the business gains from one additional unit of the limiting resource, assuming all other factors remain the same.

> **Key Term: Shadow Price**
> The increase in total contribution that would result from having one more unit of the limiting resource available, while all else remains unchanged.

### Interpreting the Shadow Price

A positive shadow price indicates there is unmet demand for the scarce resource. It shows the business would be willing to pay up to this amount for each extra unit.

The shadow price is valid only as long as:

- There is unmet demand for profitable products awaiting the resource.
- No new limiting factor emerges before all demand is met for the products ranking higher by contribution per limiting factor.

### **Worked Example 1.2**

Suppose in the earlier example, the company can buy up to 100 extra labour hours at an additional cost of \$10 per hour. Should this be done, and what is the shadow price of skilled labour?

> **Answer:**
> With 1,000 hours, only product C is made (all 1,000 hours used). If 100 more hours are bought (total 1,100), product C's demand is already met, so the next-highest-ranked product is B. Each unit of B needs 1 hour and contributes \$15, so 100 additional hours allow 100 units of B, adding \$1,500 contribution. Shadow price per hour = \$15. The firm should buy up to 100 hours if the cost per hour is less than \$15. At \$10 per hour, it is profitable.

### Limits of the Shadow Price

The shadow price holds only until the demand for the profitable products is satisfied, or another resource becomes limiting.

### **Worked Example 1.3**

A factory faces a limiting machine time of 800 hours; shadow price was calculated as \$8 per hour for the first 50 extra hours. What happens if more than 50 extra hours are obtained?

> **Answer:**
> After 50 extra hours, another product or resource may become the next limiting factor, so the shadow price may fall. The original shadow price calculation is only valid up to the point where the next constraint or demand limit is reached.

### **Revision Tip**

> When answering shadow price questions, always state the valid range for the calculated shadow price (“valid up to X extra units of the resource”). This is a frequent detail required in exam answers.

## **Summary**

Resource allocation under constraint involves ranking products by contribution per limiting factor and allocating the scarce resource to maximise total contribution. The shadow price shows the marginal value of relaxing the constraint but has limits of validity determined by product demand and the interaction of multiple constraints.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Calculate contribution per limiting factor and use it to rank products under constraint
- Allocate resources to maximise total contribution and satisfy as much profitable demand as possible
- Define and interpret the shadow price of a limiting resource
- Explain when the shadow price is valid and when it may change
- Identify practical and non-financial considerations in multi-product constraint decisions

## **Key Terms and Concepts**

- Limiting Factor
- Contribution per Limiting Factor
- Scarce Resource
- Shadow Price

Resource planning and constraints - Multi product mix and shadow pricing intuition