Two linear programming problems (LPPs) have same bounded feasible region with three corner points (0, 3), (4, 4), (7, 0). If \( Z_1=x-3y \) and \( Z_2=2x+y \) are objective functions of LPPs, then which of the following is/are correct ?
A. only \( Z_1 \) maximizes at (7, 0).
B. both \( Z_1,\ Z_2 \) minimize at (0, 3).
C. both \( Z_1,\ Z_2 \) maximize at (7, 0).
D. \( Z_2 \) maximizes at (4, 4).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → B and C only
For a bounded feasible region, each objective function takes its maximum and minimum at corner points. The values at the corners are
At (0, 3): \( Z_1=-9 \), \( Z_2=3 \)
At (4, 4): \( Z_1=-8 \), \( Z_2=12 \)
At (7, 0): \( Z_1=7 \), \( Z_2=14 \)
So both \( Z_1 \) and \( Z_2 \) have their maximum at (7, 0) (values 7 and 14) and their minimum at (0, 3) (values −9 and 3).
Checking the statements
(A) only \( Z_1 \) maximizes at (7, 0). — Incorrect. \( Z_2 \) also has its maximum, 14, at (7, 0), so “only” is wrong.
(B) both \( Z_1,\ Z_2 \) minimize at (0, 3). — Correct. −9 and 3 are the smallest values.
(C) both \( Z_1,\ Z_2 \) maximize at (7, 0). — Correct. 7 and 14 are the largest values.
(D) \( Z_2 \) maximizes at (4, 4). — Incorrect. \( Z_2(4,4)=12\lt 14=Z_2(7,0) \).