Basic feasible solution

Definition

A basic feasible solution is a specific type of solution in linear programming that satisfies all constraints and is formed by setting some of the decision variables to zero while others take on non-negative values. This solution represents a vertex of the feasible region, which is the set of all possible solutions that meet the problem's constraints. Understanding this concept is essential for applying the simplex method, as it helps identify optimal solutions within the feasible region.

5 Must Know Facts For Your Next Test

1. Basic feasible solutions are found at the vertices of the feasible region formed by the constraints of a linear programming problem.
2. In a linear program with m constraints and n decision variables, a basic feasible solution can be derived by selecting m variables to be non-zero and setting the remaining n-m variables to zero.
3. A basic feasible solution may not always be optimal; however, it is an essential starting point for finding optimal solutions using methods like the simplex method.
4. Each basic feasible solution corresponds to a unique combination of decision variables that satisfies all constraints while remaining non-negative.
5. Identifying basic feasible solutions is crucial for determining the possible paths to reach an optimal solution within the simplex method framework.

Review Questions

• How does a basic feasible solution relate to the feasible region in linear programming?
  • A basic feasible solution directly relates to the feasible region as it represents a point at one of its vertices, where all constraints are satisfied. The feasible region itself consists of all points that meet these constraints, forming a polygon or polyhedron. By identifying basic feasible solutions at these vertices, one can explore potential optimal solutions within this region during the optimization process.
• Discuss how the concept of basic feasible solutions is utilized in the simplex method to find optimal solutions.
  • In the simplex method, basic feasible solutions are crucial as they serve as starting points for iterations aimed at finding optimal solutions. The algorithm moves from one basic feasible solution to another along the edges of the feasible region, evaluating adjacent solutions based on their objective function values. By systematically assessing these basic feasible solutions, the simplex method converges towards the most favorable outcome for the linear programming problem.
• Evaluate the importance of basic feasible solutions in understanding constraints and their impact on optimization in linear programming.
  • Basic feasible solutions are vital for grasping how constraints shape optimization in linear programming. They highlight how different combinations of decision variables can meet various constraints while remaining non-negative. This understanding allows for effective exploration of potential solutions and insights into which constraints may limit or facilitate reaching optimal outcomes. Analyzing these basic feasible solutions ultimately aids in designing more effective linear programs and better decision-making strategies.

Related terms

Feasible Region: The feasible region is the set of all possible points that satisfy the constraints of a linear programming problem, typically represented as a polygon or polyhedron.

Simplex Method: The simplex method is an iterative algorithm used to find the optimal solution to a linear programming problem by moving along the edges of the feasible region.

Decision Variables: Decision variables are the unknowns in a linear programming problem that decision-makers will choose values for in order to achieve the best outcome.