Question

The L.P.P. to maximize \( z = x + y \), subject to \[ x + y \le 30,\; x \le 15,\; y \le 20,\; x + y \ge 15,\; x \ge 0,\; y \ge 0 \] has

• A. no solution
• B. a unique solution
• C. infinite solutions
• D. unbounded solutions

Hint:

If the objective function is parallel to a boundary of the feasible region, the L.P.P. has infinitely many optimal solutions.

Answer 1

The Correct Option is C

Step 1: Identify the feasible region.
The constraints \( x+y \le 30 \) and \( x+y \ge 15 \) define a strip between two parallel lines. The constraints \( x \le 15 \), \( y \le 20 \), \( x \ge 0 \), \( y \ge 0 \) bound this strip, forming a closed feasible region.

Step 2: Analyze the objective function.
The objective function \( z = x + y \) is parallel to the constraint lines \( x+y = \text{constant} \).

Step 3: Determine optimal solutions.
Since the line \( x+y = 30 \) is part of the feasible region, every point on this segment yields the same maximum value of \( z \).

Step 4: Conclusion.
The maximum value is attained at infinitely many points. Hence, the L.P.P. has infinite solutions.