Production Plan With Binding Machine Constraints
A small workshop produces two components, A and B. Each unit of A requires 3 hours on a lathe and 2 hours on a milling machine. Each unit of B requires 2 lathe hours and 4 milling hours. Weekly capacity is 240 lathe hours and 320 milling hours. A customer contract requires at least 20 units of A each week, and demand limits B to 60 units. Profit is RM45 per unit of A and RM70 per unit of B. The owner can buy up to 20 extra lathe hours at RM8 per hour. The workshop must decide an integer weekly production plan. Current plans are 30 A and 50 B, or 40 A and 60 B; the owner wants to know which is better, whether overtime is worthwhile, and how sensitive the plan is to the profit estimate for B.
1. Formulate the linear programme and determine the profit-maximising integer plan. Compare the two plans mentioned.
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Let x = units of A and y = units of B. Maximise P = 45x + 70y subject to 3x + 2y ≤ 240, 2x + 4y ≤ 320 (or x + 2y ≤ 160), x ≥ 20, y ≤ 60, x and y integers. The intersection of 3x + 2y = 240 and x + 2y = 160 is x = 40, y = 60. Check vertices: (20,0) gives P = 900; (80,0) gives 3,600; (20,60) gives 5,100; (40,60) gives 6,000. The plan 30A, 50B is feasible - lathe 90 + 100 = 190, milling 60 + 200 = 260 - but P = 1,350 + 3,500 = 4,850. The plan 40A, 60B uses all lathe hours (120 + 120 = 240) and all milling hours (80 + 240 = 320), with P = 1,800 + 4,200 = 6,000. So 40 A and 60 B is optimal among the vertices and integer points.
2. Calculate the shadow price of lathe capacity and decide whether to buy the extra hours at RM8.
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At (40,60), both main constraints bind. The dual prices λ for lathe and μ for milling solve 3λ + 2μ = 45 and 2λ + 4μ = 70. Doubling the first: 6λ + 4μ = 90; subtract the second: 4λ = 20, so λ = 5 and μ = 10. One extra lathe hour is worth RM5 at the margin, below the RM8 price. Check the 20-hour offer directly: lathe becomes 260, so 3x + 2y = 260 and x + 2y = 160 give 2x = 100, x = 50, y = 55. Profit is 50×45 + 55×70 = 2,250 + 3,850 = 6,100. The gain is RM100, but 20 hours cost RM160, a net loss of RM60. Do not buy the overtime. Even a smaller purchase would still cost more per hour than its marginal value.
3. If the profit on B rises to RM85, does the optimal plan change? Explain without assuming the answer from the current plan.
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The feasible region does not change because only the objective coefficient changes. Re-evaluate the vertices with P = 45x + 85y. At (40,60): 1,800 + 5,100 = 6,900. At (20,60): 900 + 5,100 = 6,000. At (80,0): 3,600. At (20,0): 900. So the plan remains 40 A and 60 B. The reason is that (40,60) dominates (20,60) by 20 extra A units at the same B level, and it beats (80,0) as long as 60×85 > 40×45, which is true. A switch would require a much lower B profit or a changed constraint, not merely a higher B profit. The plan is therefore stable to this increase.