ITPEC FE Subject A April 2024 Question 56

16 — the maximum lies at a corner of the feasible region, here where the whole of x + y = 4 goes to y.

The feasible region is the triangle with corners (0, 0), (4, 0) and (0, 4):

• •(0, 0): Z = 0
• •(4, 0): Z = 3 × 4 = 12
• •(0, 4): Z = 4 × 4 = 16

Each unit of y adds 4 to Z and each unit of x only 3, so the whole allowance of 4 goes to y.

Answer (c)

Why not others:

• •(a) 12 — the corner (4, 0), which spends everything on x, the variable with the smaller coefficient
• •(b) 14 — the point (2, 2), 6 + 8, which splits the allowance instead of giving it all to the better variable
• •(d) 18 — unreachable: 3x + 4y ≤ 4x + 4y = 4(x + y) ≤ 16

Key rule: In a linear program the optimum lies at a vertex of the feasible region; evaluate the objective at every vertex and take the best.

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