Question

A feasible solution to a linear programming problem

• A. Must optimize the value of the objective function
• B. Need not satisfy all the constraints, only some of them
• C. Must be a corner point of the feasible region
• D. Must satisfy all the problem's constraints simultaneously

Hint:

A feasible solution in linear programming satisfies all constraints, forming the feasible region. The optimal solution, which maximizes or minimizes the objective, is a subset of feasible solutions.

Answer 1

The Correct Option is D

Step 1: Understand the concept of a feasible solution in linear programming.
In linear programming (LP), a problem consists of an objective function to be optimized (maximized or minimized) subject to a set of linear constraints (inequalities or equalities) and non-negativity conditions. A feasible solution is any solution that satisfies all the constraints of the problem. Step 2: Define a feasible solution.
A feasible solution must satisfy:
All the linear constraints (e.g., inequalities like \( 2x + y \leq 10 \)).
Any non-negativity constraints (e.g., \( x \geq 0, y \geq 0 \)).
The set of all feasible solutions forms the feasible region, which is a convex polytope in the solution space.
A feasible solution does not necessarily optimize the objective function; that is the role of the optimal solution, which is a subset of feasible solutions.
Step 3: Evaluate the options.
(1) Must optimize the value of the objective function: Incorrect, as a feasible solution only needs to satisfy the constraints, not optimize the objective function. Optimization is a requirement for the optimal solution, not any feasible solution. Incorrect.
(2) Need not satisfy all the constraints, only some of them: Incorrect, as a feasible solution must satisfy all constraints simultaneously. If any constraint is violated, the solution is not feasible. Incorrect.
(3) Must be a corner point of the feasible region: Incorrect, as a feasible solution can be any point within the feasible region, not just a corner point (vertex). Corner points are significant in the Simplex Method because the optimal solution, if it exists, will be at a vertex, but feasible solutions include all points in the region. Incorrect.
(4) Must satisfy all the problem's constraints simultaneously: Correct, as this is the definition of a feasible solution in linear programming. Correct.
Step 4: Select the correct answer.
A feasible solution to a linear programming problem must satisfy all the problem's constraints simultaneously, matching option (4).