Question

In a Linear Programming Problem (LPP), the objective function Z is minimized subject to constraints. Where does the minimum value occur?

• A. Inside feasible region
• B. At corner points of feasible region
• C. Outside feasible region
• D. At origin only

Hint:

This is why the "Corner Point Method" is the most efficient way to solve LPPs manually. Calculate the value of \( Z \) at every vertex, and the smallest result is your minimum!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
According to the Fundamental Theorem of Linear Programming, if an optimal solution (maximum or minimum) exists for an LPP, it must occur at one of the vertices (corner points) of the feasible region.

Step 2: Detailed Explanation
1. The Feasible Region is the set of all points that satisfy all the given constraints. 2. The Objective Function \( Z = ax + by \) represents a family of parallel lines. 3. As we move these lines across the feasible region, the first or last point they touch before leaving the region will always be a corner point (or an entire edge if two corners give the same value). 4. Therefore, to find the minimum or maximum, we only need to test the coordinates of the corner points in the function \( Z \).

Step 3: Final Answer
The minimum value occurs at the corner points of the feasible region.