Question

Assertion (A) : Consider a Linear Programming Problem with minimise Z = x + 2y subject to constraints 2x + y \( \geq \) 3, x + 2y \( \geq \) 6, x, y \( \geq \) 0 which gives minimum Z at infinitely many points. The corner points of feasible region are (0, 3) and (6, 0).
Reason (R) : If two corner points produce the same minimum value of the objective function, then every point on the line segment joining the points will give the same minimum value.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

An objective function \( Z = ax + by \) results in infinite solutions if its slope \( -a/b \) matches the slope of an active constraint boundary.
Infinite solutions only occur on a bounded or boundary segment of the feasible region.

Answer 1

The Correct Option is A

Concept:
• Feasible region corner points are vertices of the polygon formed by constraints.
• Optimal solutions occur at corner points.
• If the objective function line is parallel to one of the active constraint boundaries, there can be infinite optimal solutions.

Step 1: Verify the corner points of the feasible region
Constraint 1: \( 2x + y = 3 \) (Intercepts: (1.5, 0), (0, 3)).
Constraint 2: \( x + 2y = 6 \) (Intercepts: (6, 0), (0, 3)).
For the "greater than or equal to" region, the boundary vertices are \( (0, \infty), (0, 3) \) and \( (6, 0), (\infty, 0) \). The corner points on the boundary are indeed \( (0, 3) \) and \( (6, 0) \).

Step 2: Evaluate the objective function at corner points
\( Z = x + 2y \)
• At \( (0, 3) \): \( Z = 0 + 2(3) = 6 \)
• At \( (6, 0) \): \( Z = 6 + 2(0) = 6 \) Since both corner points yield the same minimum value of \( 6 \), every point on the segment joining \( (0, 3) \) and \( (6, 0) \) will also yield \( Z = 6 \). This gives infinitely many points. Assertion (A) is true.

Step 3: Check Reason (R) and its logical connection
Reason (R) is a fundamental theorem in Linear Programming regarding multiple optimal solutions. It is true. The reason Assertion (A) concludes there are "infinitely many points" is exactly the theorem stated in Reason (R). Thus, Reason (R) is the correct explanation.