Muebles Sierra: wood, labour and an extra-wood offer
Muebles Sierra makes two desk models, basic (x) and premium (y). Each basic desk uses 2 units of wood and 4 labour-hours; each premium desk uses 3 units of wood and 2 labour-hours. Weekly resources are 120 units of wood and 160 labour-hours. A contract with a local school requires at least 10 basic desks each week, and demand limits premium desks to at most 30. Profit is €60 per basic desk and €80 per premium desk. The owner first considers a special order for 20 basic and 30 premium desks. A timber supplier then offers up to 20 additional units of wood at €18 per unit, arguing that extra wood will always raise profit because both wood and labour are scarce at the current optimum.
1. Formulate the linear programme, find the feasible vertices and determine the profit-maximising weekly production plan. State which constraints bind.
Read the worked answer
Maximise P = 60x + 80y subject to: 2x + 3y ≤ 120 (wood) 4x + 2y ≤ 160, equivalently 2x + y ≤ 80 (labour) x ≥ 10 (school contract) y ≤ 30 (demand) x ≥ 0, y ≥ 0. The feasible vertices are (10,0), (10,30), (15,30), (30,20) and (40,0). The point (25,30) is not feasible because 2(25) + 3(30) = 140 > 120. Evaluating profit: (10,0): 600 (10,30): 600 + 2400 = 3000 (15,30): 900 + 2400 = 3300 (30,20): 1800 + 1600 = 3400 (40,0): 2400. The maximum is at x = 30, y = 20, with profit €3,400. Both wood and labour bind: 2(30) + 3(20) = 120 and 4(30) + 2(20) = 160. The contract and demand constraints have slack.
2. Is the special order of 20 basic and 30 premium desks feasible? If the school order fixes x = 20, what is the largest integer number of premium desks that can also be made, and how does its profit compare with the optimum?
Read the worked answer
For x = 20, y = 30, wood use is 2(20) + 3(30) = 130, which exceeds 120; labour use is 4(20) + 2(30) = 140 ≤ 160. So the special order is not feasible because it is short of 10 units of wood. If x = 20 is kept, the wood constraint gives 40 + 3y ≤ 120, so y ≤ 80/3 = 26.67. With integer desks, at most 26 premium desks can be made. Profit is then 60(20) + 80(26) = 1,200 + 2,080 = €3,280, which is below the €3,400 optimum. Accepting the special order in full would require buying at least 10 extra units of wood or reducing the premium quantity.
3. The supplier offers up to 20 extra units of wood at €18 per unit. Using the binding constraints at the current optimum, decide whether the offer should be accepted and calculate the net gain from buying all 20 units.
Read the worked answer
At the optimum both wood and labour bind. Let the wood stock be W. Solving 2x + 3y = W and 4x + 2y = 160 gives 2x + y = 80, hence y = (W − 80)/2 and x = 60 − W/4. Profit is P(W) = 60(60 − W/4) + 80((W − 80)/2) = 3600 − 15W + 40W − 3200 = 400 + 25W. Thus, while this pair of constraints remains binding, one extra unit of wood raises profit by €25. Since €25 > €18, buying wood is profitable. For 20 extra units, W = 140, y = 30 and x = 25. Profit is 60(25) + 80(30) = €3,900, an increase of €500. The extra wood costs 20 × €18 = €360, so the net gain is €140. Buy all 20 units, but note that at W = 140 the premium demand limit y ≤ 30 becomes binding, so further wood would not have the same marginal value.