Cost-Volume-Profit Under a Capacity Constraint
Kariuki Foods Ltd produces three products from a single packing line that runs a maximum of 10,000 machine hours per month. Current monthly data:
| Product | Selling price (KES) | Variable cost (KES) | Machine hrs/unit | Demand (units) |
|---|---|---|---|---|
| A | 500 | 300 | 2 | 3,000 |
| B | 800 | 560 | 4 | 2,000 |
| C | 1,200 | 900 | 5 | 1,200 |
Fixed costs are KES 1,800,000 per month. The sales manager wants to push Product C because it has the highest selling price and the highest absolute contribution per unit (KES 300). The production manager says the line is already at capacity and any push for C means dropping something else. There is also a special order enquiry for 400 units of B at KES 700 each, from a customer who will not affect existing sales.
1. Determine the optimal production mix and the resulting monthly profit, showing your ranking logic.
Read the worked answer
Contribution per unit and per constraint
| Product | Contribution/unit | Hrs/unit | Contribution per machine hour | Rank |
|---|---|---|---|---|
| A | 200 | 2 | 100 | 1 |
| B | 240 | 4 | 60 | 2 |
| C | 300 | 5 | 60 | 3 (tie with B) |
The sales manager is wrong to rank by absolute contribution. The scarce resource is machine hours, so rank by contribution per machine hour. C looks best per unit but is the least efficient use of the line.
Optimal mix (10,000 hours available)
- Product A: 3,000 units × 2 hrs = 6,000 hrs; contribution 3,000 × 200 = 600,000
- Remaining hours = 4,000. B and C tie at 60/hr, so either mix gives the same contribution. Take B first as demand is larger:
- B: 2,000 units × 4 hrs = 8,000 hrs - exceeds remaining 4,000 hrs. So produce 1,000 units of B (4,000 hrs).
- Contribution = 1,000 × 240 = 240,000
- C: zero units.
Total contribution = 600,000 + 240,000 = KES 840,000 Profit = 840,000 − 1,800,000 = loss of KES 960,000
The company is loss-making at full capacity. That is the real finding: the constraint is not the problem, the cost structure is.
Check the tie If instead 800 units of C (4,000 hrs) were made: contribution = 800 × 300 = 240,000 - identical. The tie means B vs C is a commercial choice (customer relationships, risk), not a profit choice.
2. Evaluate the special order for 400 units of B at KES 700, and state what qualitative factors could change the decision.
Read the worked answer
Quantitative test
- Special price 700 vs variable cost 560 → contribution KES 140/unit
- 400 units × 140 = KES 56,000 incremental contribution
- Machine hours needed: 400 × 4 = 1,600 hrs
Is capacity available? No - the line is fully utilised at the optimal mix. Accepting the order means displacing the lowest-ranked use of hours. The displaced contribution is 60/hr (B or C).
- Contribution from special order = 56,000 / 1,600 hrs = 35/hr
- Displaced contribution = 60/hr
- Net loss = (60 − 35) × 1,600 = KES 40,000
So the order is not profitable at full capacity. It would only be attractive if there were idle hours (e.g., demand shortfall), in which case the full 56,000 would be incremental.
Qualitative factors
- Will the customer expect KES 700 as the new normal price? Price erosion risk.
- Is this a strategic entry customer with future volume?
- Does it use the same line, or could it run in overtime (check overtime cost - if overtime premium pushes variable cost above 700, decline)?
- Contractual risk: penalties, quality standards, payment terms.
- Capacity expansion: if demand is sustained, the real question is whether to add a second line - evaluate payback.
Recommendation Decline at current capacity, or counter-offer at a price that at least matches the displaced contribution per hour (i.e., ≥ 560 + 60×4 = KES 800 - which is the normal price). Alternatively, accept only if overtime capacity is available and the overtime premium keeps contribution positive.