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.
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In LPP, if the objective function is parallel to one of the constraint lines forming the feasible region, multiple optimal solutions occur along that constraint boundary.
Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
Assertion (A) is true, but Reason (R) is false.
Assertion (A) is false, but Reason (R) is true.
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The Correct Option isA
Solution and Explanation
Concept: • Optimal solutions in LPP can be unique or infinitely many.
• If an objective function reaches its optimal value at two different vertices, then all points on the line segment joining these vertices are also optimal solutions.
Step 1: Evaluate the objective function at corner points
Objective function: \( Z = x + 2y \).
Evaluate at the given corner points:
1. At \( (0, 3) \): \( Z = 0 + 2(3) = 6 \)
2. At \( (6, 0) \): \( Z = 6 + 2(0) = 6 \)
Both points yield the same value \( Z = 6 \).
Step 2: Verify Assertion (A)
Since two corner points produce the same minimum value, according to the multiple optimal solutions theorem, every point on the segment connecting \( (0,3) \) and \( (6,0) \) is a solution.
There are infinitely many such points. Thus, Assertion (A) is true.
Step 3: Verify Reason (R) and its connection
Reason (R) is a standard theorem of Linear Programming regarding multiple optimal solutions.
It is true.
Since Assertion (A) follows directly from the logic described in Reason (R), the Reason provides the correct explanation.
