Bakery Capacity Expansion: Shadow Prices and a Rental Offer
Northfield Bakery makes two celebration cakes, Deluxe and Standard. Each Deluxe cake uses 2 hours of oven time and 1 hour of decorating; each Standard uses 1 hour of oven time and 2 hours of decorating. Weekly capacity is 100 oven hours and 80 decorating hours. Deluxe contributes $12 and Standard $8. A contract requires at least 10 Standard cakes each week. The owner has been offered 10 additional oven hours next week at $5 per hour from a neighbouring bakery, but only if booked now. The production planner says the bakery is already fully utilized and that extra oven time is therefore valuable. The owner is unsure whether the offer is worth taking and asks for a marginal analysis, including the assumptions behind it.
1. Formulate the weekly production problem as a linear programme and determine the optimal number of Deluxe and Standard cakes.
Read the worked answer
Let x = Deluxe cakes and y = Standard cakes per week.
Maximise profit: 12x + 8y subject to: 2x + y ≤ 100 (oven hours) x + 2y ≤ 80 (decorating hours) y ≥ 10 (contract) x, y ≥ 0
The two resource constraints intersect where 2x + y = 100 and x + 2y = 80. Solving: from the first, y = 100 − 2x; substituting gives x + 200 − 4x = 80, so −3x = −120, x = 40 and y = 20. Profit = 12(40) + 8(20) = 480 + 160 = $640.
Checking other vertices: (0,10) gives $80; (45,10) gives $620; (0,40) gives $320. The contract constraint is satisfied at (40,20). The optimal mix is 40 Deluxe and 20 Standard, with profit $640 per week.
2. Calculate the shadow price of an oven hour and advise whether the 10-hour offer at $5 per hour should be accepted. Include the effect of integer cakes.
Read the worked answer
Increase oven capacity by one hour and re-solve the binding constraints: 2x + y = 101 and x + 2y = 80. Solving gives x = 40.667 and y = 19.667. Profit = 12(40.667) + 8(19.667) = 488 + 157.33 = $645.33, an increase of $5.33. The shadow price is therefore about $5.33 per oven hour.
For 10 extra hours, re-solve with 2x + y = 110 and x + 2y = 80: x = 46.667, y = 16.667, profit = $693.33. The gain is $53.33; the cost is 10 × $5 = $50; net gain = $3.33. Accept in the continuous model.
With whole cakes only, the best integer mix near the new intersection is x = 47, y = 16, profit = $692. The gain over $640 is $52, so the net gain is $2. It is still marginally worth accepting, but the planner should check that the extra hours can actually be scheduled. The shadow price relies on linearity, fixed prices, divisibility, and no other bottleneck becoming active.
3. The neighbouring bakery also offers decorating hours at $2 per hour. Using the same model, should Northfield buy decorating hours instead?
Read the worked answer
Increase decorating capacity by one hour: 2x + y = 100 and x + 2y = 81. Solving gives x = 39.667 and y = 20.667. Profit = 12(39.667) + 8(20.667) = 476 + 165.33 = $641.33. The shadow price of decorating time is therefore about $1.33 per hour.
At $2 per hour, the marginal cost exceeds the marginal value: each extra decorating hour adds about $1.33 of profit but costs $2, a net loss of about $0.67 per hour. Do not buy decorating hours at that price. If both resources are offered together, the combined offer must be evaluated as a package, because each resource is binding at the optimum and extra amounts of only one resource may not be usable.