The corner points of the bounded feasible region (convex set) determined by the system of linear constraints are (0,0), (40,0), (25, 30) and (0,45). Let \( z=1000x+500y \) be the objective function. The maximum value of occurs at :
(A) (40,0)
(B) (25, 30)
(C) (0,45)
(D) every point on the line segment joining the points (25, 30) and (40,0)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
The maximum of z over a bounded feasible region occurs at a corner point, so z is worked out at every corner:
\( \displaystyle z(0,0)=0 \)
\( \displaystyle z(40,0)=40000 \)
\( \displaystyle z(25,30)=25000+15000=40000 \)
\( \displaystyle z(0,45)=22500 \)
The maximum value 40000 occurs at two adjacent corners, (40, 0) and (25, 30). When this happens, z has the same maximum value at every point of the edge joining them.
So the maximum of z is 40000, at (40, 0), at (25, 30) and at every point of the segment joining them; at (0, 45), z is only 22500.
Checking the statements
(A) (40,0) — Correct. z = 40000, the maximum.
(B) (25, 30) — Correct. z = 40000, the maximum.
(C) (0,45) — Incorrect. z = 22500 only.
(D) every point on the line segment joining the points (25, 30) and (40,0) — Correct. A point t(40, 0) + (1 − t)(25, 30), 0 ≤ t ≤ 1, gives z = 40000t + 40000(1 − t) = 40000.