If the corner points of the bounded feasible region for an LPP are (5,0), (0,4), (0,2), (3,4), and (0,7), then the minimum value of the objective function \( z=2x+5y \) occurs at :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → every point on the line segment joining the point (5,0) and (0,2)
By the corner point method:
\( \displaystyle z(5,0)=10 \)
\( \displaystyle z(0,4)=20 \)
\( \displaystyle z(0,2)=10 \)
\( \displaystyle z(3,4)=26 \)
\( \displaystyle z(0,7)=35 \)
(The point (0, 4) lies on the y-axis between (0, 2) and (0, 7), so it is not a true corner of the region; its value 20 does not affect the minimum.)
The minimum value 10 occurs at two corners, (5, 0) and (0, 2). When a linear objective function has the same optimal value at two adjacent corners, it has that value at every point of the segment joining them: for \( 0\le t\le 1 \),
\( \displaystyle z\left(5-5t,\,2t\right)=2(5-5t)+5(2t) \)
\( \displaystyle =10-10t+10t=10 \)
Why the other options are incorrect
Option (1) (5,0) only — Incorrect. The minimum is not at (5, 0) only; z = 10 at (0, 2) as well.
Option (2) (0,2) only — Incorrect. The minimum is not at (0, 2) only; z = 10 at (5, 0) as well.
Option (3) the mid-point of the line segment joining the points (5,0) and (0,2). — Incorrect. The mid-point (2.5, 1) does give z = 10, but so does every other point of the segment, so “the mid-point” alone is incomplete.