Bookcase Production Plan with a Contract and Whole-Number Output
A small furniture workshop produces two bookcase models. Each Standard bookcase requires 2 hours of cutting and 1 hour of assembly; each Deluxe bookcase requires 3 hours of cutting and 2 hours of assembly. In a particular week, the workshop has at most 60 cutting hours and 40 assembly hours available. A contract requires at least 5 Standard bookcases to be made, and the workshop cannot make more than 30 Standard bookcases because of storage. The profit is $400 per Standard and $700 per Deluxe. The workshop manager proposes making only Deluxe bookcases, arguing that each Deluxe earns $300 more. The production planner suspects this is wrong but wants a justified plan. All bookcases are whole units, and unused hours cannot be carried over. The planner also needs to know whether the best continuous solution can simply be rounded down to whole numbers.
1. Formulate the constraints and determine the number of each model that maximises weekly profit. Show why the manager's proposal is not optimal.
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Let x be the number of Standard bookcases and y the number of Deluxe bookcases.
Constraints: 2x+3y ≤ 60 (cutting) x+2y ≤ 40 (assembly) x ≥ 5 (contract) x ≤ 30 (storage) y ≥ 0 x, y integers.
The manager proposes x=0, but this violates x ≥ 5. Even though each Deluxe earns more per unit, the resources used differ.
The continuous feasible corner points are (5,0), (30,0) and the intersection of x=5 with 2x+3y=60, namely (5, 16⅔). Profit P=400x+700y gives: (5,0): $2,000 (30,0): $12,000 (5,16⅔): $13,666.67
The whole-number optimum must be checked nearby. At (5,16): P=2,000+11,200=$13,200. At (6,16): cutting 12+48=60 and assembly 6+32=38 are both feasible, and P=2,400+11,200=$13,600. Other nearby integer points such as (7,15) give $13,300 and (9,14) give $13,400, both lower. Hence the best whole-number plan is 6 Standard and 16 Deluxe, with profit $13,600. The manager's all-Deluxe plan is infeasible under the contract, so it cannot be optimal.
2. The planner first solves the problem without the whole-number restriction and obtains a maximum at x=5, y=16⅔. Explain, using this example, why simply rounding down both coordinates is unsafe, and find a whole-number plan that beats the rounded-down plan.
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At the continuous optimum (5,16⅔), profit is 400(5)+700(16⅔)=$13,666.67. Rounding down gives (5,16), which uses 10+48=58 cutting hours and 5+32=37 assembly hours, leaving slack. Its profit is $13,200.
Rounding down is unsafe because the integer optimum need not lie at the rounded coordinate. Moving one unit right to x=6 uses the 2 spare cutting hours exactly: 2(6)+3(16)=60. Assembly is 6+2(16)=38 ≤ 40, so the point is feasible. Its profit is 400(6)+700(16)=$13,600, which beats the rounded-down plan by $400.
Thus one must test integer points near the continuous optimum, especially points that use up slack resources, rather than rounding blindly.
3. If the contract were removed, what would be the best plan and profit? Does this change your advice about the manager's reasoning?
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Without x ≥ 5, the best plan is all Deluxe. The constraints become 3y ≤ 60 and 2y ≤ 40, so y ≤ 20, and y=20 is feasible. Profit is 700×20=$14,000. This is higher than $13,600.
However, the contract is part of the actual problem, so this plan is unavailable. The manager's instinct that Deluxe is more profitable per unit is correct only when the contract is ignored; it does not justify an infeasible plan when the contract applies. The planner should report the best feasible plan, 6 Standard and 16 Deluxe, and explain that the contract is binding.