Basic Feasible Solution

Definition

A basic feasible solution is a point in the feasible region of a linear programming problem that satisfies all constraints and is formed by setting some variables to zero while solving the remaining equations. This solution represents a corner point of the feasible region, where it is possible to evaluate the objective function for optimization. Understanding this concept is crucial as it lays the foundation for identifying optimal solutions and differentiating between basic and non-basic variables.

5 Must Know Facts For Your Next Test

1. Basic feasible solutions correspond to the vertices of the feasible region and are essential for determining potential optimal solutions in linear programming.
2. In a system with 'm' constraints, a basic feasible solution can involve at most 'm' non-zero variables, while others are set to zero.
3. The process of moving from one basic feasible solution to another is called pivoting, which helps in finding better solutions iteratively.
4. Not every vertex of the feasible region is a basic feasible solution; only those that meet the necessary conditions of feasibility and being formed by the intersection of constraints qualify.
5. Basic feasible solutions can be found using methods such as the Simplex Algorithm, which systematically explores vertices of the feasible region.

Review Questions

• How does a basic feasible solution relate to the feasible region and optimal solutions in linear programming?
  • A basic feasible solution is a specific point within the feasible region where all constraints are satisfied. This point corresponds to a vertex of the feasible region, which is significant because optimal solutions are often found at these vertices. By analyzing basic feasible solutions, one can evaluate which point yields the best value for the objective function, thus linking them directly to finding optimal solutions.
• Compare and contrast basic and non-basic variables in terms of their roles in a basic feasible solution.
  • Basic variables are those that take non-zero values in a basic feasible solution, while non-basic variables are set to zero. The number of basic variables typically equals the number of constraints, indicating which dimensions of the solution space are actively participating. Non-basic variables do not influence that particular solution's outcome but can be altered in the pursuit of better solutions through methods like pivoting. Understanding this distinction is crucial for effectively navigating linear programming problems.
• Evaluate how identifying basic feasible solutions contributes to solving linear programming problems and achieving optimal outcomes.
  • Identifying basic feasible solutions is fundamental in solving linear programming problems because it allows one to systematically explore potential solutions at vertices of the feasible region. Each basic feasible solution provides insight into how changes in variable values affect the objective function. By evaluating these points through algorithms like the Simplex method, one can transition between different solutions until an optimal outcome is reached. This systematic approach emphasizes the importance of understanding both basic and non-basic variables in this process.

Related terms

Feasible Region: The set of all possible points that satisfy the constraints of a linear programming problem, representing potential solutions.

Optimal Solution: The best possible value of the objective function within the feasible region, found at one or more corner points.

Non-basic Variables: Variables in a linear programming problem that are set to zero in a basic feasible solution, playing no active role in that specific solution.