The Overtime Dilemma

Contribution per labour hour:

• Product X: £20 / 2 hours = £10 per hour
• Product Y: £30 / 3 hours = £10 per hour

Both products yield the same contribution per labour hour. Therefore, the optimal production plan is not determined by contribution per labour hour alone; any combination that uses all available labour hours will yield the same total contribution, provided demand is met. To maximise profit, we should produce as much as possible of either product, but since both have the same contribution per hour, the total contribution is maximised by fully utilising the 10,000 hours. However, we must also consider demand constraints.

Let x = units of X, y = units of Y. Maximise 20x + 30y Subject to: 2x + 3y ≤ 10,000 (labour hours) x ≤ 3,000 y ≤ 2,000 x, y ≥ 0

Because contribution per hour is equal, the objective function is 10*(2x+3y), so maximising 2x+3y subject to the constraints. The maximum labour hours used will be 10,000. We can test corner points:

• If we produce only X: max x = min(3000, 10000/2=5000) = 3000, then labour used = 6000, remaining hours = 4000, can produce y = 4000/3 ≈ 1333.33, but y must be integer? Usually units are whole. But for planning, we can use fractional? Typically, we assume continuous. But let's check: if we produce 3000 X and 1333.33 Y, total contribution = 300020 + 1333.3330 = 60,000 + 40,000 = 100,000. Alternatively, produce 2000 Y and then X: labour for 2000 Y = 6000, remaining 4000 hours for X = 2000 units, contribution = 200020 + 200030 = 40,000 + 60,000 = 100,000. Same. So any combination that uses all hours and meets demand gives same contribution. But to maximise profit, we need to ensure fixed costs are covered. Profit = total contribution - fixed costs. Fixed costs = £30,000. Minimum profit required = £50,000, so total contribution must be at least £80,000. With full labour utilisation, total contribution = £10 per hour * 10,000 hours = £100,000, which exceeds £80,000. So any plan that uses all hours yields profit £70,000, which is above minimum. However, the question asks for optimal production plan that maximises profit. Since all full-utilisation plans give same contribution, there is no unique optimal plan. But we should consider that demand for X is 3000 and Y is 2000. The maximum contribution is £100,000, achieved by any plan that uses all 10,000 hours and does not exceed demand. For example, produce 3000 X and 1333.33 Y, or 2000 Y and 2000 X, etc. But since units are typically whole, we might need integer solution. The problem likely expects prioritising based on contribution per limiting factor. Since both are equal, we can choose either. However, the production manager's claim that X has higher contribution per labour hour is incorrect; they are equal. So the optimal plan is to produce any combination that fully utilises labour hours and meets demand, but to maximise profit, we should aim to produce at the point that gives highest total contribution, which is £100,000. A simple plan: produce 2000 units of Y (uses 6000 hours) and 2000 units of X (uses 4000 hours), total hours = 10,000, contribution = 200030 + 200020 = 60,000 + 40,000 = £100,000. Profit = £100,000 - £30,000 = £70,000. This meets the minimum profit requirement. Alternatively, produce 3000 X and 1333 Y (if fractional allowed) gives same contribution. So the optimal plan is to fully utilise labour hours, and since contribution per hour is equal, any such plan is optimal. The key is to recognise that the limiting factor is labour hours, and the contribution per hour is the correct ranking method, not contribution per unit. The sales director's argument is flawed because it ignores the labour hour constraint. Therefore, the optimal production plan is to produce 2000 units of Y and 2000 units of X (or any combination that uses all hours and meets demand).

The production manager argues that product X should be prioritised because it has a higher contribution per labour hour, but actually both products have the same contribution per labour hour (£10). Therefore, the production manager's premise is incorrect. The sales director argues that product Y should be prioritised because it has a higher contribution per unit, but this ignores the labour hour constraint; contribution per unit is not the correct measure when a resource is scarce. The correct approach is to rank products by contribution per unit of the limiting factor (labour hours). Since both products yield the same contribution per hour, neither should be prioritised over the other based on financial grounds alone. Additional factors to consider include:

• Demand constraints: we cannot sell more than the market demands.
• Operational feasibility: can we produce the mix without disrupting production?
• Employee skills and availability: producing more of one product might require different skills.
• Customer relationships: prioritising one product might affect sales of other products or future orders.
• Long-term strategy: which product aligns with the company's growth plans?
• Quality and brand reputation: rushing production might affect quality.
• Fixed costs and minimum profit: the plan must cover fixed costs and achieve the minimum profit target, which it does.
• Potential for overtime or additional shifts: could increase labour hours and allow more production.
• Inventory levels and storage costs: producing more than needed might increase holding costs.
• Cash flow: timing of cash inflows from sales. Therefore, the decision should not be based solely on contribution per unit or per hour; a holistic view is needed.