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 involves finding the optimum value of a linear objective function subject to a set of constraints.

Step 2: Detailed Explanation:
The constraints are:
1) \( x - y \le -1 \implies y \ge x + 1 \)
2) \( x \ge y \implies y \le x \)
3) \( x, y \ge 0 \)

If we graph these:
Constraint 1 is the region above the line \( y = x + 1 \).
Constraint 2 is the region below the line \( y = x \).
These two regions do not intersect for any real \( x, y \). The lines \( y = x + 1 \) and \( y = x \) are parallel and have no common points.

Step 3: Final Answer:
Since there is no region that satisfies all constraints simultaneously (the feasible region is empty), the maximum value does not exist.