ITPEC FE Morning April 2022 Question 78
A company sells two products, A and B, and makes a profit of 40 dollars and 30 dollars per unit on them respectively. They are produced through a common production process and are sold in two different markets. The production process has a total capacity of 3,000 person-hours. Pruduct A’s production time is 3 person-hours that is three times longer than that of product B. Having surveyed the market, company officials feel that the maximum number of units of product A that can be sold is 8,000, while for product B the maximum number of units is 1,200. Subject to these limitations, products can be sold in any combination. When the linear programming model is applied, which of the following is the formulation result that obtains the production amount that maximizes the company’s total profit? Here, the number of units of products A and B are x and y, respectively.
Explanation
Maximise 40x + 30y subject to 3x + y ≤ 3,000, x ≤ 8,000, y ≤ 1,200 — translate each fact of the story into one constraint.
- •Objective: profit is 40 per unit of A and 30 per unit of B, so maximise 40x + 30y
- •Capacity: A takes 3 person-hours, three times as long as B, so B takes 3 ÷ 3 = 1 person-hour, giving 3x + y ≤ 3,000
- •Market: at most 8,000 of A and 1,200 of B can be sold, so x ≤ 8,000 and y ≤ 1,200
- •Non-negativity: x ≥ 0, y ≥ 0
Answer (d)
Why not others:
- •(a) — gives B nine person-hours, 3 × 3, multiplying where it should divide, and turns the market maximums into minimums with ≥
- •(b) — has the market limits right but the same 9 person-hours for B in the capacity constraint
- •(c) — has the capacity right but writes the market limits as x ≥ 8,000 and y ≥ 1,200, which with 3x + y ≤ 3,000 cannot even be satisfied
Key rule: A "maximum that can be sold" is an upper bound (≤), and "A takes three times as long as B" means B's time is A's divided by three.
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