Optimal product mix and a special order at Kijabe Manufacturers Ltd
Kijabe Manufacturers Ltd produces two products on a single machine with a practical capacity of 24,000 machine hours per year, which cannot be increased before 2026. Product P sells for KES 900, has variable costs of KES 520 and uses 3 machine hours per unit; maximum demand is 4,000 units. Product Q sells for KES 1,400, has variable costs of KES 900 and uses 5 machine hours per unit; maximum demand is 3,000 units. Annual fixed costs are KES 1,200,000. A new customer has offered to buy 600 units of P at KES 780 each, which would absorb 1,800 machine hours. The sales manager wants to accept, saying that the price is above variable cost and that the factory is empty at night anyway.
1. Determine the optimal production plan and the resulting total contribution, showing the basis of your ranking.
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The scarce resource is machine hours, so rank by contribution per machine hour, not by unit contribution.
| P | Q | |
|---|---|---|
| Selling price | 900 | 1,400 |
| Variable cost | (520) | (900) |
| Contribution per unit | 380 | 500 |
| Machine hours per unit | 3 | 5 |
| Contribution per hour | 126.67 | 100.00 |
| Rank | 1st | 2nd |
Hours available = 24,000. Hours demanded = (4,000 x 3) + (3,000 x 5) = 12,000 + 15,000 = 27,000. Shortfall = 3,000 hours.
Plan:
- P: 4,000 units x 3 hrs = 12,000 hrs; contribution 4,000 x 380 = 1,520,000
- Q: remaining 12,000 hrs / 5 = 2,400 units; contribution 2,400 x 500 = 1,200,000
- Total contribution 2,720,000; less fixed costs 1,200,000 = profit 1,520,000
Note the shortfall passed on: 600 units of Q (3,000 - 2,400) remain unmet. Q is rationed, not P. Fixed costs are irrelevant to the ranking because they do not change with the mix.
2. Evaluate whether the special order at KES 780 per unit should be accepted, and state the minimum price you would quote.
Read the worked answer
Answer: reject at KES 780. The minimum acceptable price is KES 820 per unit.
Incremental analysis:
- Contribution from the order: 600 x (780 - 520) = KES 156,000
- Hours absorbed: 600 x 3 = 1,800 hours
- There is no idle capacity, so those hours must come from Q (lowest contribution per hour): 1,800 / 5 = 360 units of Q displaced
- Contribution lost: 360 x 500 = KES 180,000
- Net effect: 156,000 - 180,000 = -KES 24,000, so wealth falls if accepted.
Minimum price = variable cost + opportunity cost per unit = 520 + (180,000 / 600) = 520 + 300 = KES 820
Address the sales manager's reasoning directly: the machine is the constraint, not the labour force, so an empty factory at night creates no extra machine hours. A price above variable cost only makes sense where there is genuinely idle capacity. If a second machine could be hired for less than KES 24,000 for the 1,800 hours (or an overtime premium below KES 40,000 against the extra contribution), the decision would flip - that is the negotiation to pursue.