Question

Assertion (A): Every point of the feasible region of a Linear Programming Problem is an optimal solution.
Reason (R): The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.

• A. Both Assertion (A) and Reason (R) are true and the 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:

In Linear Programming, the optimal solution is always found at a corner point of the feasible region, not in the interior of the region.

Answer 1

The Correct Option is D

Resolving the stated problem necessitates an understanding of Linear Programming Problems (LPP) and their associated feasible regions.

Analysis of Assertion (A)

Assertion (A): All points within the feasible region of a Linear Programming Problem represent optimal solutions.

The feasible region of an LPP comprises all points satisfying the problem's constraints. However, for an LPP, the optimal solution is not found at every point within this region; it is typically located at the vertices, or corner points, of the feasible region. Consequently, the assertion that every point is an optimal solution is erroneous.

Analysis of Reason (R)

Reason (R): The optimal solution for a Linear Programming Problem is found exclusively at one or more corner point(s) of the feasible region.

This statement is accurate. In the context of LPPs, the "Fundamental Theorem of Linear Programming" posits that if an optimal solution exists, it will occur at one of the corner points (vertices) of the feasible region. Therefore, this reason correctly identifies the location of optimal solutions within the feasible region.

Conclusion

Upon evaluation:

• Assertion (A) is false—it is inaccurate to claim that every point in the feasible region is an optimal solution.
• Reason (R) is true—the optimal solution is indeed located at corner points.

The correct determination is: Assertion (A) is false, but Reason (R) is true.