Question

For a linear programming problem, which one is correct?

• A. Optimal solution always exists.
• B. If the feasible region is unbounded, then the maximum or minimum value of the objective function exists.
• C. If the feasible region \((R)\) is bounded, then the objective function has a maximum or a minimum value on \(R\).
• D. If the objective function has an optimal value, then this optimal value must occur at a corner point of the feasible region.

Hint:

In Linear Programming Problems: \[ \text{Optimal Solution} \Longrightarrow \text{Corner Point Solution} \] Therefore, after finding the feasible region, evaluate the objective function only at the corner points.

Answer 1

The Correct Option is D

Concept: Fundamental theorem of Linear Programming: quote If an optimal solution exists, then it occurs at a corner (vertex) point of the feasible region. quote

Step 1: Examine each statement. \[ \begin{aligned} (A) &:\quad \text{False. Some LPPs have no optimal solution.} \\ (B) &:\quad \text{False. An unbounded feasible region may not have a finite optimum.} \\ (C) &:\quad \text{Not always true; feasibility and objective behaviour must be considered.} \\ (D) &:\quad \text{Fundamental theorem of LPP. Correct.} \end{aligned} \]

Step 2: Apply the theorem of linear programming. Whenever an optimal solution exists, Optimal value occurs at a corner point of the feasible region. \[\begin{aligned} \boxed{\text{Option (D)}} \end{aligned}\] Hence, option \(\mathbf{(D)}\) is correct.