Basic feasible solution

Definition

A basic feasible solution is a solution to a linear programming problem that satisfies all the constraints of the problem while also being feasible, meaning it lies within the feasible region defined by those constraints. This solution is obtained by selecting a subset of the variables and solving the resulting system of equations, ensuring that the non-negativity constraints are met. It is fundamental in determining optimal solutions in the simplex method and other optimization techniques.

5 Must Know Facts For Your Next Test

1. A basic feasible solution corresponds to a vertex of the feasible region, where multiple constraints intersect.
2. In linear programming, there can be multiple basic feasible solutions, but only one may be optimal.
3. Basic feasible solutions are determined by setting non-basic variables to zero and solving for the basic variables.
4. The number of basic feasible solutions is determined by the number of constraints and variables in the system.
5. Identifying basic feasible solutions is essential for applying the simplex method effectively to optimize a given objective function.

Review Questions

• How does a basic feasible solution differ from other types of solutions in linear programming?
  • A basic feasible solution is specifically a solution that not only satisfies all constraints but also resides within the defined feasible region. Unlike other types of solutions, which may not meet all constraints or lie outside the feasible area, basic feasible solutions ensure that all conditions are fulfilled. Additionally, they are associated with vertices of the feasible region, representing points where constraints intersect.
• Discuss the role of basic feasible solutions in the simplex method and how they are utilized to find an optimal solution.
  • In the simplex method, basic feasible solutions serve as stepping stones towards finding an optimal solution. The algorithm begins at an initial basic feasible solution and iteratively moves along edges of the feasible region to explore adjacent vertices. Each movement aims to improve the objective function until no further improvements can be made, leading to an optimal solution. This process underscores the importance of starting from a valid basic feasible solution to ensure efficient navigation through potential solutions.
• Evaluate how the concept of basic feasible solutions impacts decision-making in real-world applications of linear programming.
  • The concept of basic feasible solutions is crucial in real-world linear programming applications, such as resource allocation, transportation, and production scheduling. By identifying these solutions, decision-makers can evaluate different scenarios within the constraints of their systems and optimize outcomes effectively. The ability to pinpoint valid options among potentially numerous choices allows organizations to make informed decisions that maximize efficiency and profitability while adhering to limitations, illustrating its practical significance beyond theoretical understanding.

Related terms

feasible region: The set of all possible points that satisfy the constraints of a linear programming problem.

simplex method:

An algorithm for solving linear programming problems by moving along the edges of the feasible region to find the optimal vertex.

constraint: A condition or limitation that defines the feasible region in a linear programming problem.