Allocating Labor and Machine Hours in a Two-Product Shop
A shop makes two products, 1 and 2. Each unit of product 1 requires 2 labor-hours and 1 machine-hour; each unit of product 2 requires 1 labor-hour and 3 machine-hours. Regular capacity is 100 labor-hours and 90 machine-hours per week. Unit profit is $30 for product 1 and $40 for product 2. Demand is unlimited. The shop can buy up to 10 hours of overtime labor at $8 per hour. A neighboring firm has offered to rent any unused machine time at $9 per hour. The production manager asks whether to buy overtime and whether to rent out machine capacity.
1. Formulate the product-mix problem and solve it graphically. State the optimal quantities and weekly profit.
Read the worked answer
Let x1 and x2 be weekly units of products 1 and 2.
Maximize Z = 30x1 + 40x2 subject to:
- 2x1 + x2 ≤ 100 (labor)
- x1 + 3x2 ≤ 90 (machine)
- x1, x2 ≥ 0
Evaluate the corner points:
- (0, 0): Z = 0
- (50, 0): Z = 1,500
- (0, 30): Z = 1,200
- Intersection of 2x1 + x2 = 100 and x1 + 3x2 = 90: solving gives x1 = 42, x2 = 16, Z = 30(42) + 40(16) = $1,900
The optimal product mix is 42 units of product 1 and 16 units of product 2, with weekly profit of $1,900. Both capacity constraints are binding at the optimum.
2. Compute the shadow prices of labor and machine time. Should the shop buy the 10 overtime hours at $8/hour?
Read the worked answer
At the optimum both constraints are binding. Let y_L and y_M be the dual prices. The dual constraints are:
- 2y_L + y_M ≥ 30
- y_L + 3y_M ≥ 40
Setting both equal and solving gives y_L = $10/hour and y_M = $10/hour. Each additional labor-hour raises profit by $10, which exceeds the $8 overtime cost, so the overtime is worth buying up to the allowed 10 hours (provided the optimum stays on the same basis).
Check by re-solving with 110 labor-hours:
- 2x1 + x2 = 110
- x1 + 3x2 = 90
This gives x1 = 48, x2 = 14, and Z = 30(48) + 40(14) = $2,000. Profit rises by $100; overtime costs $80; net benefit is $20. Buy the overtime, but only because the shadow price exceeds the premium.
3. Should the shop rent out unused machine time at $9/hour? What if the rental rate were $12/hour?
Read the worked answer
The shadow price of machine time is $10/hour. Renting out one machine-hour reduces production profit by $10 while generating only $9, so the net effect is −$1/hour. At $9/hour, do not rent out machine capacity; the shop is better off using it internally. At $12/hour, the net effect is +$2/hour, so renting would be attractive within the range where the shadow price remains $10. The answer changes only if enough machine time is rented out to shift the optimal basis, at which point the problem must be re-solved.