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 Question:
The question asks about a fundamental property of the feasible region in a linear programming problem (LPP).

Step 3: Detailed Explanation:
Feasible Region: In a linear programming problem, the feasible region is the set of all points that satisfy all the given constraints (linear inequalities). These constraints typically define a polygon or a polyhedral set in 2D or 3D space, respectively.
Convexity: A region is said to be convex if, for any two points within the region, the entire line segment connecting these two points also lies entirely within the region.
The intersection of a set of half-planes (which is what linear inequalities define) always results in a convex set. Therefore, the feasible region of a linear programming problem is always convex. This property is crucial because it ensures that if an optimal solution exists, it will occur at one of the vertices (corner points) of the feasible region.
Other options:
- (A) Circular: The feasible region is generally a polygon, not necessarily circular.
- (B) Open: It can be bounded (closed) or unbounded (open in one direction), but "open" itself is not a defining characteristic.
- (D) Irregular: While it can have many sides, it's always a well-defined convex polygon/polyhedron, not simply "irregular."
Therefore, the feasible region of an LPP is always convex.

Step 4: Final Answer:
The feasible region of a linear programming problem is always Convex.