The corner points of the feasible region for an LPP are: (0, 2), (5, 3), (6, 8) and (3, 0). Let the objective function be \(Z=4x+6y\). The minimum value of Z occurs at:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → any point on the line segment joining (3, 0) and (0, 2)
By the corner point method, evaluate Z at each corner:
\((0, 2): \ Z = 12\)
\((5, 3): \ Z = 20+18 = 38\)
\((6, 8): \ Z = 24+48 = 72\)
\((3, 0): \ Z = 12\)
The minimum value 12 occurs at two adjacent corners, (0, 2) and (3, 0). Then Z takes the same minimum value at every point of the edge joining them: on that edge \(2x+3y = 6\), so \(Z = 2(2x+3y) = 12\).
Why the other options are incorrect
Option (1) (0, 2) only — Incorrect. Z = 12 at (3, 0) as well, so the minimum is not at (0, 2) only.
Option (2) (3, 0) only — Incorrect. Z = 12 at (0, 2) as well.
Option (3) the mid point of the line segment joining (0, 2) and (3, 0) only — Incorrect. The midpoint (1.5, 1) gives Z = 12, but so does every other point of the segment.