Lamp Production Under Three Capacity Limits
Sunrise Lamps makes two desk-lamp models, A and B, in weekly batches. Each A needs 4 hours of assembly, 2 hours of finishing and 1 hour of packing; each B needs 3, 1 and 2 hours respectively. The workshop can schedule at most 240 assembly hours, 100 finishing hours and 120 packing hours per week. Contracts require at least 10 A and at least 8 B each week. Profit is ₹350 per A and ₹250 per B. The owner currently plans 24 A and 48 B because that uses all assembly capacity and nearly all finishing capacity, but a supervisor argues that 30 A and 40 B is better. A packing supervisor also warns that packing capacity may be cut to 105 hours in the next quarter. The owner wants a defensible production plan and an idea of which hours are genuinely limiting.
1. Formulate the linear programming problem and evaluate every vertex of the feasible region to determine the optimal weekly plan. Show why the supervisor's proposal of 30 A and 40 B beats 24 A and 48 B.
Read the worked answer
Let x be batches of A and y batches of B. Maximise P = 350x + 250y subject to 4x + 3y ≤ 240 (assembly), 2x + y ≤ 100 (finishing), x + 2y ≤ 120 (packing), x ≥ 10, y ≥ 8.
Vertices come from intersections of boundary lines that satisfy all constraints:
- x = 10, y = 8 gives (10, 8).
- x = 10 with packing x + 2y = 120 gives (10, 55); finishing would allow y ≤ 80 and assembly y ≤ 66.7, so this is feasible.
- y = 8 with finishing 2x + y = 100 gives (46, 8); packing and assembly allow x = 46.
- Finishing and assembly: 2x + y = 100 and 4x + 3y = 240 give x = 30, y = 40. Packing use is 30 + 80 = 110 ≤ 120.
- Assembly and packing: 4x + 3y = 240 and x + 2y = 120 give x = 24, y = 48. Finishing use is 48 + 48 = 96 ≤ 100.
- Finishing and packing give (80/3, 140/3), but assembly use is about 246.7 > 240, so it is infeasible.
Profits: (10,8) → 5,500; (10,55) → 17,250; (46,8) → 18,100; (24,48) → 20,400; (30,40) → 20,500. The maximum is ₹20,500 at 30 A and 40 B. The supervisor wins by ₹100 per week; the 24 A, 48 B plan leaves 4 finishing hours unused.
2. Which constraints bind at the optimum, and what is the marginal value of one extra hour of assembly and one extra hour of finishing? Use these values to advise the owner on overtime.
Read the worked answer
At (30, 40), assembly is 4(30) + 3(40) = 240 of 240, finishing is 60 + 40 = 100 of 100, and packing is 30 + 80 = 110 of 120. Assembly and finishing bind; packing has 10 hours of slack.
Add 1 finishing hour: solve 2x + y = 101 with 4x + 3y = 240. From y = 101 − 2x, 4x + 303 − 6x = 240, so x = 31.5, y = 38. Packing use is 107.5 ≤ 120. Profit = 350(31.5) + 250(38) = 11,025 + 9,500 = ₹20,525, a gain of ₹25.
Add 1 assembly hour: solve 4x + 3y = 241 with 2x + y = 100. From y = 100 − 2x, 4x + 300 − 6x = 241, so x = 29.5, y = 41. Packing use is 111.5 ≤ 120. Profit = 10,325 + 10,250 = ₹20,575, a gain of ₹75.
So an extra assembly hour is worth ₹75 and an extra finishing hour ₹25 at the margin. Packing overtime is worth ₹0 until packing becomes binding, so the owner should buy assembly overtime first and reject packing overtime.
3. If packing capacity is cut to 105 hours, does the current plan remain feasible, and what is the new optimum? Compute and comment on the profit loss.
Read the worked answer
The plan (30, 40) needs 30 + 2(40) = 110 packing hours, so it becomes infeasible under a 105-hour cap.
New candidate vertices include (10,8), (46,8), (10,47.5) from packaging at x = 10, and the intersection of finishing 2x + y = 100 with packaging x + 2y = 105. Solving gives x = 95/3 ≈ 31.67 and y = 110/3 ≈ 36.67. Assembly use is 4(95/3) + 3(110/3) = 710/3 ≈ 236.7 ≤ 240, so this point is feasible. The intersection of assembly and packaging is (33, 36), but finishing use is 102 > 100, so it is infeasible.
Profit at the new vertex is 350(95/3) + 250(110/3) = (33,250 + 27,500)/3 = 60,750/3 = ₹20,250. The new optimum is about 31.67 A and 36.67 B, with profit ₹20,250, a loss of ₹250 per week. The cut makes packing binding and forces the company to give up ₹250 of profit.