Contribution Margin Under Constraint: Which Product to Push
Harlow Components Ltd makes three products - Alpha, Beta and Gamma - using a single machine that is the binding constraint. Machine hours available next month are capped at 1,800 hours. The following per-unit data applies:
| Product | Selling price | Direct materials | Direct labour | Machine hours | Max demand |
|---|---|---|---|---|---|
| Alpha | £120 | £40 | £20 | 3.0 | 300 units |
| Beta | £90 | £25 | £15 | 1.5 | 500 units |
| Gamma | £150 | £55 | £30 | 4.0 | 250 units |
Fixed overheads are £60,000 per month and are absorbed at £30 per machine hour. The sales director wants to prioritise Gamma because it has the highest selling price and highest unit contribution. The production manager wants to prioritise Alpha because it uses fewer hours than Gamma.
1. Calculate the contribution per unit and contribution per machine hour for each product, and determine the optimal production plan.
Read the worked answer
Contribution per unit (price − materials − labour):
- Alpha: £120 − £40 − £20 = £60
- Beta: £90 − £25 − £15 = £50
- Gamma: £150 − £55 − £30 = £65
Contribution per machine hour (÷ machine hours):
- Alpha: £60 ÷ 3.0 = £20.00/hr
- Beta: £50 ÷ 1.5 = £33.33/hr
- Gamma: £65 ÷ 4.0 = £16.25/hr
Ranking: Beta (£33.33) > Alpha (£20.00) > Gamma (£16.25).
Optimal plan (allocate scarce hours in rank order):
- Beta: 500 units × 1.5 = 750 hrs (uses 750 of 1,800)
- Alpha: 300 units × 3.0 = 900 hrs (uses 900; total 1,650)
- Remaining: 150 hrs → Gamma: 150 ÷ 4.0 = 37.5 → 37 units (148 hrs)
Total contribution: (500×£50) + (300×£60) + (37×£65) = £25,000 + £18,000 + £2,405 = £45,405.
Note: the £30/hr overhead absorption is irrelevant to the ranking - it is a fixed cost and does not change with the mix.
2. The sales director argues Gamma must be prioritised because it has the highest unit contribution. Evaluate this argument and explain what would change your answer.
Read the worked answer
Evaluation: The sales director is confusing unit contribution with scarcity value. When a resource is constrained, the correct decision metric is contribution per unit of the scarce resource (here, machine hours), not contribution per unit. Gamma earns £65 per unit but consumes 4 hours (£16.25/hr); Beta earns £50 but consumes only 1.5 hours (£33.33/hr). Prioritising Gamma would leave capacity idle or under-used and reduce total contribution.
Quantify the cost of the error: If Gamma were prioritised first (250 units = 1,000 hrs), then Beta (750 hrs) = 1,750 hrs, leaving 50 hrs → 16 Alpha units. Contribution = (250×£65) + (500×£50) + (16×£60) = £16,250 + £25,000 + £960 = £42,210 - i.e. £3,195 worse than the optimal plan.
What would change the answer:
- A change in the binding constraint (e.g. labour hours become scarce, or machine hours increase) - re-rank on the new scarce resource.
- A change in selling prices, material costs or labour rates that alters contribution per hour.
- Minimum contract quantities or customer commitments forcing a minimum output of Gamma.
- Quality or strategic considerations (e.g. Gamma is a loss-leader that opens a market) - these are qualitative and may override the quantitative ranking.
- If demand for Beta and Alpha were fully met and hours remained, Gamma would then be produced.