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:
Linear Programming Problems (LPP) involve optimizing (maximizing or minimizing) a linear objective function subject to a set of linear inequalities (constraints).
Step 2: Key Formula or Approach:
According to the Fundamental Theorem of Linear Programming, if a linear objective function $Z = ax + by$ has an optimal value (maximum or minimum) over a feasible region formed by linear constraints, that optimal value must occur at one or more of the corner points (vertices) of the feasible region.
Step 3: Detailed Explanation:

Inside feasible region: The objective function $Z$ represents a family of parallel lines. It monotonically increases or decreases as it sweeps across the feasible region. Therefore, extremes cannot occur in the interior.
Outside feasible region: Points here violate at least one constraint, so they are not valid solutions.
At origin only: The origin may be a corner point, but it is not {always} the optimal solution unless specifically dictated by the function and constraints.
At corner points: The optimal value is found exactly where the "sweeping" line of the objective function touches the very edge of the polygon representing the feasible region, which is inevitably a corner point.

Step 4: Final Answer:
The minimum value occurs at the corner points of the feasible region.