Question

Which of the following statements are correct for Maximize \( Z = 50x + 40y \) subject to \( 1000x + 1200y \le 7600 \), \( 3x + 2y \le 18 \), \( x, y \ge 0 \)?
(A) The LPP has a unique optimal solution at (4, 3) only.
(B) The feasible region is bounded.
(C) The maximum value is unique, but there are infinite optimal solutions.
(D) The feasible region is bounded with corner points (0,0), (6,0), (4,3) and (0,19/3).

• A. (A), (B), (C) and (D)
• B. (A), (B) and (C) only
• C. (A) and (D) only
• D. (A), (B) and (D) only

Hint:

Always plot the constraints to identify the closed polygon defining the feasible region.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Analyze constraints and objective function values at corner points to find the optimal solution.

Step 2: Detailed Explanation:
Constraints: (1) \( 5x + 6y \le 38 \); (2) \( 3x + 2y \le 18 \).
Corner points:
Intersection of (1) and (2): \( 15x + 18y = 114 \) and \( 15x + 10y = 90 \). Subtracting: \( 8y = 24 \implies y = 3 \). \( 3x + 6 = 18 \implies x = 4 \). Point is (4,3).
At (0,0), Z = 0. At (6,0), Z = 300. At (0, 38/6 = 19/3), Z = 40(6.33) = 253.33. At (4,3), Z = 200 + 120 = 320.
The feasible region is bounded (B is true). Corner points match (D). Maximum value is 320 at (4,3), which is unique (A is true).

Step 3: Final Answer:
The correct combination is (d).