ITPEC FE Subject A October 2024 Question 58
Two (2) types of raw materials A and B are required to manufacture products X and Y. The quantity of each raw material required per unit and the procurable quantity are listed in the table below. When the profit per unit is $1 for product X and $1.5 for product Y, which of the following is the maximum profit in dollar?
Explanation
$70 — the optimum is the plan that uses up both raw materials.
Let x and y be the units of X and Y. Maximise P = x + 1.5y subject to:
- •Material A: 2x + y ≤ 100
- •Material B: x + 2y ≤ 80
- •x ≥ 0, y ≥ 0
The maximum of a linear program lies at a corner of the feasible region:
The corner (40, 20) solves 2x + y = 100 and x + 2y = 80: doubling the second gives 2x + 4y = 160, subtracting the first leaves 3y = 60, so y = 20 and x = 40. Then P = 40 + 1.5 × 20 = 70.
Answer (c)
Why not others:
- •(a) 50 — makes only X, 50 units limited by material A, leaving part of material B unused
- •(b) 60 — makes only Y, 40 units limited by material B, leaving part of material A unused
- •(d) 80 — no feasible plan reaches it; for example, 80 units of X would need 160 units of material A against 100 available
Key rule: In a two-variable linear program, evaluate the objective at every corner of the feasible region; the best is usually where the binding constraints intersect, not where one resource is used alone.
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