Contribution Mix Under a Machine-Hour Constraint
Brackenfield Ltd makes three products on a single CNC machine that is available for 4,000 hours per quarter. Per unit data:
- Product A: selling price £120, variable cost £72, 0.5 machine hours
- Product B: selling price £90, variable cost £45, 0.25 machine hours
- Product C: selling price £200, variable cost £130, 0.8 machine hours
Demand is A 6,000 units, B 9,000 units, C 3,000 units. Fixed costs are £210,000 per quarter. The sales director has promised the largest customer a mixed order totalling at least 2,000 units per quarter across the three products, and says the mix should simply favour the highest contribution per unit. The production manager wants to know the contribution-maximising plan and whether the promised order can be met in full.
1. Determine the optimal production plan and the resulting quarterly profit. Show your ranking logic.
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Step 1 - Contribution per unit and per constrained resource (machine hour):
| Product | Cont/unit | Hours/unit | Cont per machine hour | Rank |
|---|---|---|---|---|
| A | £48 | 0.5 | £96 | 1 |
| B | £45 | 0.25 | £180 | 1 (best) |
| C | £70 | 0.8 | £87.50 | 3 |
Correct ranking by contribution per machine hour: B, then A, then C. Ranking by contribution per unit (C £70 > A £48 > B £45) is the classic error - C consumes 3.2× the hours of B.
Step 2 - Allocate 4,000 hours:
- B: 9,000 × 0.25 = 2,250 hrs → contribution 9,000 × £45 = £405,000
- Remaining 1,750 hrs to A: 1,750 / 0.5 = 3,500 units → contribution 3,500 × £48 = £168,000
- No hours left for C.
Total contribution = £573,000. Profit = £573,000 − £210,000 = £363,000.
Step 3 - Check the 2,000-unit promise: plan produces 9,000 + 3,500 = 12,500 units, comfortably above 2,000, so the promise is met provided the customer accepts A and B. If the customer specifically requires C, each unit of C displaces 1.6 units of A (0.8/0.5), costing 1.6 × £48 = £76.80 of contribution against £70 earned - a net loss of £6.80 per unit of C. Any C requirement should be quantified and negotiated, not accepted silently.
2. A supplier offers to supply Product C externally at £150 per unit, freeing machine hours. Advise whether to buy in C, assuming demand for C must be satisfied in full (3,000 units) and freed hours go to the next-best use.
Read the worked answer
Make-or-buy with a scarce resource - compare relevant costs including the opportunity cost of the freed hours.
Freed hours per unit of C = 0.8. Next-best use of those hours is Product A (rank 2), earning £96 contribution per machine hour.
- Opportunity cost per unit of C = 0.8 × £96 = £76.80
- Relevant cost to make = variable cost £130 + opportunity cost £76.80 = £206.80
- Relevant cost to buy = £150.00
Buying saves £56.80 per unit × 3,000 units = £170,400 per quarter better off.
Check total: buying C costs 3,000 × £150 = £450,000 and frees 2,400 hrs, which make 4,800 units of A (2,400/0.5) earning 4,800 × £48 = £230,400. Making C in-house earns 3,000 × £70 = £210,000 on the same hours. Buying is better by £20,400 - but this ignores that A demand was only 3,500 units under the original plan. Once A demand is capped at 6,000 units, the freed hours beyond A's unmet demand (2,500 units = 1,250 hrs) have no alternative use, so only 1,250 hrs carry opportunity cost. Recompute: opportunity cost = 1,250 × £96 = £120,000 total, i.e. £40 per unit of C. Relevant cost to make = £130 + £40 = £170 > £150, so still buy. The lesson: the buy decision is robust here, but the size of the benefit depends on whether freed capacity has a genuine alternative use - always test capacity utilisation before quoting an opportunity cost.