CVP and linear programming - Sales mix, multi-product CVP, a...

Learning Outcomes

After reading this article, you will be able to calculate weighted average contribution to sales (C/S) ratios, determine break-even and target profit in a multi-product context, interpret multi-product profit-volume charts, and apply linear programming to optimise production plans when faced with multiple limiting factors. You will also be able to rank products based on their contribution per unit of a scarce resource and use this ranking in practical decision scenarios for the ACCA Performance Management (PM) exam.

ACCA Performance Management (PM) Syllabus

For ACCA Performance Management (PM), you are required to understand advanced applications of cost-volume-profit (CVP) analysis and linear programming in environments involving multiple products and scarce resources. This includes:

• Calculation and interpretation of break-even points and target profits when a business sells several products, using the sales mix and weighted-average C/S ratio
• Application of multi-product CVP, including margin of safety and profit-volume graphs
• Determining the optimal sales mix to maximise profit
• Ranking products using contribution per unit of a limiting (scarce) resource
• Formulation and graphical solution of linear programming problems involving multiple limiting factors
• Understanding and calculating shadow prices and slack for scarce resources

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. In a multi-product business, why is it important to calculate a weighted average C/S ratio for CVP analysis?
2. What is the correct calculation for break-even revenue in a two-product scenario with a fixed sales mix?
3. When production is limited by an external constraint (e.g., machine hours), how should products be ranked for production priority?
4. Briefly explain the significance of a shadow price in linear programming.

Introduction

Many businesses offer more than one product, and each product typically yields a different level of contribution and uses resources differently. Standard CVP analysis for a single product is insufficient in these situations. Exam-standard questions will require you to calculate break-even and target profit for a mix of products, interpret multi-product profit charts, and find the best product mix when resources are limited.

Additionally, business decisions must address real-world limitations such as restricted materials or production hours. Key factor analysis and linear programming allow you to determine the profit-maximising mix given these constraints, a core requirement in the ACCA PM exam.

Key Term: sales mix
The proportion or ratio in which different products are budgeted or planned to be sold by a business, expressed as a ratio or percentage.

Key Term: weighted average C/S ratio
The average contribution to sales ratio across all products, weighted according to the planned sales mix.

Key Term: limiting factor (scarce resource)
Any input or resource whose limited availability restricts the output of an organisation.

Key Term: shadow price
The additional contribution earned from one more unit of a scarce resource, calculated within linear programming.

MULTI-PRODUCT CVP ANALYSIS

The Weighted Average Contribution to Sales Ratio

When analysing break-even or target profit in a multi-product situation, use the weighted average C/S ratio based on the standard sales mix. This allows you to find break-even and target sales revenue for the business as a whole.

Formula:

Weighted average C/S ratio =
Total contribution at standard mix / Total revenue at standard mix

Multi-Product Break-Even and Target Profit

Once the weighted average C/S ratio is found:

Break-even revenue:
Break-even revenue = Fixed costs / Weighted average C/S ratio

Target profit revenue:
Target revenue = (Fixed costs + Target profit) / Weighted average C/S ratio

For margin of safety calculations or other analysis, map results back proportionately to each product using the sales mix ratio.

Profit-Volume Graphs in Multi-Product Situations

A multi-product profit-volume graph plots cumulative profit against cumulative revenue assuming a consistent sales mix, or by ranking products in order of C/S ratios for best/worst-case break-even analysis. The break-even point may occur earlier if high-C/S products are sold first.

Worked Example 1.1

A company budgets to sell two products, A and B, in the ratio 3:2. Product A sells for $10 and gives a $4 contribution per unit; Product B sells for $15 and gives a $6 contribution per unit. Fixed costs are $60,000. What is the break-even revenue?

Answer:
Step 1: Calculate mixed total contribution and sales for standard mix of 3A:2B:

• Contribution: (3 × $4) + (2 × $6) = $12 + $12 = $24 (per 5 units)
• Sales: (3 × $10) + (2 × $15) = $30 + $30 = $60 (per 5 units)
• Weighted average C/S ratio = 24/60 = 0.40 (40%) Step 2: Break-even revenue = $60,000 / 0.40 = $150,000

Multi-Product Margin of Safety

Margin of safety % =
(Budgeted sales revenue – Break-even revenue) / Budgeted sales revenue × 100

LIMITING FACTOR (KEY FACTOR) ANALYSIS

When a single resource (labour hours, materials, machine hours) is in short supply, a business must decide which products to prioritise. The process:

1. Calculate each product's contribution per unit.
2. Calculate the usage of the scarce resource per unit.
3. Calculate and rank by contribution per unit of the scarce resource.
4. Produce in descending order of contribution per scarce resource until the resource is exhausted.

Worked Example 1.2

A company can make Product X (contribution $10/unit, uses 5 machine hours/unit) and Product Y (contribution $7/unit, uses 2 machine hours/unit). If machine hours are limited, which is prioritised?

Answer:

• Product X: $10 ÷ 5 = $2 per machine hour
• Product Y: $7 ÷ 2 = $3.50 per machine hour Product Y is prioritised. Produce as much Y as possible, then use any remaining machine hours for X. Exam Warning Never rank by contribution per unit alone when there is a limiting factor. Always calculate contribution per unit of the scarce resource.

LINEAR PROGRAMMING IN MULTIPLE CONSTRAINT SITUATIONS

When more than one resource restricts activity, linear programming provides a mathematical and graphical solution. Only two variables ("products") can be graphed, but the method can be applied to more.

Steps:

1. Define variables (e.g., x = number of Product A, y = number of Product B).
2. Set the objective function (e.g., maximise C = 5x + 8y).
3. Express constraints as inequalities (e.g., 4x + 7y ≤ 12,000 hours).
4. Plot constraints to identify the feasible region.
5. Find the optimal point (usually a vertex of the feasible region) where the highest total contribution occurs.

Key Term: feasible region
The area on a graph representing all possible combinations of activities that satisfy all resource constraints.

Key Term: iso-contribution line
A straight line representing all combinations of products that yield the same total contribution.

Worked Example 1.3

A business makes two products, A and B:

• A: $8 contribution, uses 3kg material, 4 hours labour per unit
• B: $6 contribution, uses 2kg material, 5 hours labour per unit

Materials available: 900 kg
Labour available: 1,200 hours

Max demand: 200 units of each.

Find the optimal production mix.

Answer:
Let x = units of A, y = units of B. Constraints:

• 3x + 2y ≤ 900 (material)
• 4x + 5y ≤ 1,200 (labour)
• x ≤ 200, y ≤ 200
• x, y ≥ 0 Graphically plot the lines to find the feasible region. Using the vertices (using simultaneous equations for intersection), calculate C = 8x + 6y at each. The combination with the highest C is optimal.

SHADOW PRICE AND SLACK IN LINEAR PROGRAMMING

A shadow price quantifies how much extra contribution would be earned if one more unit of the scarce resource were available, provided no other constraint becomes limiting. Slack is the amount of unused resource at the optimal solution.

Worked Example 1.4

In Example 1.3, suppose the optimal solution used up all labour but had 30kg of material left over.

Answer:

• Slack for material = 30kg (resource not fully used, so zero shadow price).
• Shadow price for labour: Calculate the improvement in contribution from one extra hour of labour by resolving the equations (if this hour changes the optimal mix). The gain is the shadow price per hour.

Summary

Multi-product CVP analysis allows you to assess overall performance, break-even, and margin of safety when more than one product is sold. When resources are limited, always rank products by contribution per unit of the scarce resource for optimal profit. For multiple constraints, linear programming—using objective functions, constraints, feasible region, and iso-contribution lines—gives the optimal solution and enables calculation of shadow prices. Graph interpretation and the ability to work step-by-step through both key factor and linear programming solutions are key ACCA skills.

Key Point Checklist

This article has covered the following key knowledge points:

• Calculate and interpret the weighted average C/S ratio in a multi-product environment
• Apply multi-product CVP analysis: break-even, target profit, margin of safety
• Use profit-volume graphs to interpret performance under different sales mixes
• Optimise product mix where a single limiting factor exists (key factor analysis)
• Rank products by contribution per unit of scarce resource
• Apply linear programming for multiple limiting factors, define constraints and objective
• Interpret feasible region and iso-contribution lines in graphical solutions
• Calculate and explain shadow prices and slack in linear programming

Key Terms and Concepts

• sales mix
• weighted average C/S ratio
• limiting factor (scarce resource)
• shadow price
• feasible region
• iso-contribution line