Question

The feasible region of a linear programming problem is always:

• A. Circular
• B. Open
• C. Convex
• D. Irregular

Hint:

Remember the fundamental property of feasible regions in LPPs: they are always convex polygons (or polyhedra). This convexity is what allows the "corner point method" for finding optimal solutions.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Linear Programming (LP) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model whose requirements are represented by linear relationships.
The set of all possible points (values for variables) that satisfy all of the problem's constraints (inequalities and equations) simultaneously is known as the feasible region.
Step 2: Detailed Explanation:
1. Nature of Constraints: In LPP, constraints are typically linear inequalities like \(ax + by \leq c\) or \(ax + by \geq c\).
2. Geometric Representation: In a two-dimensional plane, a linear inequality represents a "half-plane."
3. Defining Convexity: A set is called convex if, for any two points \(P_1\) and \(P_2\) belonging to the set, the entire straight-line segment connecting \(P_1\) and \(P_2\) is also contained within the set.
4. Half-Planes are Convex: A half-plane is a convex set.
5. Intersection Property: A fundamental property of convex sets is that the intersection of any number of convex sets is itself a convex set.

Since the feasible region is formed by the intersection of several half-planes (the constraints), it must necessarily be a convex set.
The region might be bounded (forming a polygon) or unbounded, but its convexity remains a constant property. It cannot have "dents" (concave areas) or be disconnected.
Step 3: Final Answer:
By the definition of linear constraints and the geometric properties of intersections of half-planes, the feasible region of a linear programming problem is always convex.