A furniture company is producing two types of furniture. Product A requires 8 board feet of wood and 2 lbs of wicker. Product B requires 6 board feet of wood and 6 lbs of wicker. There are 2000 board feet of wood available for product and 1000 lbs of wicker. Product A earns a profit margin of $30 a unit and Product B earns a profit margin of $40 a unit. What should the objective function be?

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BlackTom AI · Accepted Answer

We begin by restating the problem in our own words to ensure clarity: a furniture company produces two products, A and B. Product A uses 8 board feet of wood and 2 lbs of wicker per unit, while Product B uses 6 board feet of wood and 6 lbs of wicker per unit. Available resources are 2000 board feet of wood and 1000 lbs of wicker. The profit per unit is $30 for A and $40 for B. The task is to determine the objective function for the linear programming model.

First, identify the decision variables: let A represent the number of units of Product A to produce and B represent the number of units of Product B to produce.

Next, derive the objective function: the total profit is the sum of profits from each product. For each unit of A, profit is $30, so the contribution from A is 30A. For each unit of B, profit is $40, so the contribution from B is 40B. Therefore, the objective is to maximize total profit: 30A + 40B.

Now consider the constraints that limit production:
- Wood constraint: Each A requires 8 board feet and each B requires 6 board feet. With 2000 board feet available, the wood constraint is 8A + 6B ≤ 2000.
- Wicker constraint: Each A uses 2 lbs of wicker and each B uses 6 lbs. With 1000 lbs available, the wicker constraint is 2A + 6B ≤ 1000.
- Non-negativity constraints: A ≥ 0, B ≥ 0, since you cannot produce negative quantities.

Regarding the provided options, there is only one option listed: the objective function proposed is Maximize 30A + 40B. This aligns with the standard approach for a profit-maximization linear programming model where the objective function sums each product’s unit profit multiplied by the quantity produced.

If one were to consider alternative incorrect options (though none are displayed here), common mistakes could include:
- Using a sum of costs rather than profits (e.g.,  -30A - 40B or 30A - 40B), which misrepresents the goal as minimizing profit or balancing different criteria.
- Misplacing coefficients (e.g., using 8A + 6B or 2A + 6B as the objective) which would confuse resource usage with objective value.
- Treating the objective as minimizing rather than maximizing the total profit.

In this scenario, since the objective is to maximize profit given the resource constraints, the correct form is to maximize 30A + 40B with the constraints 8A + 6B ≤ 2000, 2A + 6B ≤ 1000, A ≥ 0, B ≥ 0. If you had alternative options, you would compare them against this logic to determine which one truly captures profit maximization.

QANT_620-VA1-2025SP-VR Midterm Exam

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