Which of the following statements is correct?

Options

A. Every L.P.P. admits an optimal solution.
B. A L.P.P. admits a unique optimal solution.
C. If a L.P.P. admits two optimal solutions, it has an infinite number of optimal solutions
D. The set of all feasible solutions of a L.P.P. is not a convex set.

Answer

To determine which of the statements regarding Linear Programming Problems (LPP) is correct, let’s analyze each option step-by-step.

### Step 1: Analyze the first statement
**Statement:** Every LPP admits an optimal solution.

**Analysis:** This statement is incorrect. An LPP may have no solution at all (infeasible) or may be unbounded, meaning it does not have an optimal solution. Therefore, this option is wrong.

### Step 2: Analyze the second statement
**Statement:** An LPP admits a unique optimal solution.

**Analysis:** This statement is also incorrect. An LPP can have a unique optimal solution, but it can also have multiple optimal solutions or no solution at all. Thus, this option is wrong as well.

### Step 3: Analyze the third statement
**Statement:** If an LPP admits two optimal solutions, then it has an infinite number of optimal solutions.

**Analysis:** This statement is correct. If an LPP has two distinct optimal solutions, it implies that there are infinitely many solutions along the line segment connecting these two points in the feasible region. Therefore, this option is correct.

### Step 4: Analyze the fourth statement
**Statement:** The set of all feasible solutions of an LPP is not a convex set.

**Analysis:** This statement is incorrect. The set of feasible solutions of an LPP is always a convex set. Therefore, this option is also wrong.

### Conclusion
After analyzing all the statements, we conclude that the correct answer is:
**C: If an LPP admits two optimal solutions, then it has an infinite number of optimal solutions.**

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