Lumilite Workshop: Product Mix and the Price of Extra Assembly Hours
Lumilite Workshop makes two desk-lamp models, A and B, and sells all it can produce. Model A earns a contribution of RM30 per unit and model B RM40 per unit. Each A needs 2 hours in assembly and 3 hours in finishing; each B needs 4 hours in assembly and 2 hours in finishing. Weekly capacity is 240 assembly hours and 180 finishing hours. A standing contract requires at least 10 units of A each week, and demand for B is capped at 50 units. The workshop currently makes 40 A and 30 B, which uses all finishing time but leaves 40 assembly hours idle. A neighbouring firm offers up to 30 extra assembly hours next week at RM5 per hour. The supervisor wants to know whether to accept, and whether the contract minimum for A is costing money. Formulate and solve the weekly product-mix problem, then advise.
1. Formulate Lumilite's weekly product-mix problem as a linear programme and solve it. State the optimal mix and contribution, and compare the solution with the current plan.
Read the worked answer
Let x = units of A and y = units of B.
Maximise P = 30x + 40y subject to 2x + 4y <= 240 (assembly) 3x + 2y <= 180 (finishing) x >= 10, y <= 50, x,y >= 0.
The feasible corner points are (10,0), (10,50), (20,50), (30,45) and (60,0). Contributions:
| Point | P = 30x + 40y |
|---|---|
| (10,0) | 300 |
| (10,50) | 2,300 |
| (20,50) | 2,600 |
| (30,45) | 2,700 |
| (60,0) | 1,800 |
Optimum: 30 A and 45 B, contribution RM2,700. Both capacity constraints bind. The current plan of 40 A and 30 B gives P = 1,200 + 1,200 = RM2,400, so it is RM300 worse. The current plan also uses finishing fully but leaves assembly idle, which is a sign that the mix is wrong rather than that capacity is short.
2. Calculate the shadow prices of assembly and finishing hours. Use the assembly shadow price to advise on the neighbouring firm's offer, and state the range over which your advice is valid.
Read the worked answer
At the optimum both constraints bind: 2x + 4y = 240 and 3x + 2y = 180, giving x = 30, y = 45.
Assembly: add one hour, so 2x + 4y = 241 with 3x + 2y = 180. Solving gives x = 29.75, y = 45.375. New P = 30(29.75) + 40(45.375) = 892.50 + 1,815 = RM2,707.50. Shadow price = RM7.50 per assembly hour.
Finishing: add one hour, so 3x + 2y = 181 with 2x + 4y = 240. Solving gives x = 30.5, y = 44.75. New P = 915 + 1,790 = RM2,705. Shadow price = RM5 per finishing hour.
The assembly shadow price is valid while the same two constraints determine the solution and y <= 50 is not breached. Writing the assembly capacity as R, solving with 3x + 2y = 180 gives y = (3R - 360)/8. The cap y <= 50 binds when R = 253.33. Thus the range is 240 <= R <= 253.33, i.e. about 13.33 extra hours.
At RM5 per hour, each extra assembly hour adds RM7.50 of contribution, a net gain of RM2.50. Accept up to about 13 hours, not all 30. Beyond roughly 253 assembly hours the extra hours have zero value because B's demand cap and finishing capacity, not assembly, control the mix.
3. The contract minimum for A is 10 units. Determine whether it is binding at the optimum and quantify what relaxing it by one unit would be worth.
Read the worked answer
At the optimum x = 30, so the contract requires 10 and there are 20 units of slack. The constraint is not binding, so its shadow price is zero. Relaxing the minimum from 10 to 9 would not change the optimal mix or contribution; the value is RM0 within the relevant range. The contract is therefore not costing Lumilite money at present. It would start to matter only if the required minimum rose above 30, at which point the firm would be forced away from the unconstrained best mix and contribution would fall.