Fundamental Theorem of Linear Programming: Version 3 If the optimal value of the objective function in a linear programming problem exists, then that value must occur at one or more of the basic feasible solutions of the initial system.
The correct answer and explanation is :
Correct Answer:
The Fundamental Theorem of Linear Programming (Version 3) states that:
“If the optimal value of the objective function in a linear programming problem exists, then that value must occur at one or more of the basic feasible solutions of the initial system.”
Explanation:
Linear programming (LP) is a mathematical method used to find the best possible outcome in a given mathematical model, subject to constraints. The fundamental theorem of linear programming ensures that if there is an optimal solution to an LP problem, it must be found at a basic feasible solution (BFS).
Key Concepts:
- Feasible Region: This is the set of all points that satisfy the given constraints of an LP problem.
- Extreme Points (Corner Points): These are the points where constraints intersect, and they are candidates for optimality.
- Basic Feasible Solutions (BFS): These are solutions that correspond to extreme points of the feasible region. They are obtained by setting some variables to zero (non-basic variables) and solving for the remaining basic variables.
- Optimal Solution: The best possible value of the objective function, which maximizes or minimizes the function within the feasible region.
Why Does the Optimal Solution Occur at a BFS?
- The objective function in an LP problem is linear, meaning it increases or decreases along straight lines.
- The feasible region, formed by linear constraints, is a convex polyhedron.
- Since a linear function attains its maximum or minimum on a convex polyhedron at extreme points, the optimal solution (if it exists) must be at one of these points.
Thus, instead of checking every possible point in the feasible region, we only need to evaluate BFS, significantly reducing computational effort. The Simplex Method exploits this property by moving from one BFS to another until an optimal solution is found.