Question

The maximum value of the linear programming problem, max. \( z = 3x + 4y \) subject to the constraints: \( x - y \le -1 \), \( x \ge y \), \( x, y \ge 0 \) is

• A. 7
• B. 4
• C. 3
• D. maximum value does not exist

Hint:

Always verify if a feasible region exists before attempting to calculate objective function values at corner points.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Linear programming problems require a bounded feasible region for a maximum value to exist.

Step 2: Detailed Explanation:
Constraints: \( y \ge x + 1 \) and \( y \le x \).
For any \( x \ge 0 \), these two inequalities contradict each other because if \( x \ge 0 \), then \( x + 1 > x \). Therefore, \( y \) cannot be simultaneously \( \ge x + 1 \) and \( \le x \).
The feasible region is empty.

Step 3: Final Answer:
Since there is no feasible region, a maximum value does not exist.