Explain the meaning of basic feasible solution.
The correct answer and explanation is :
A Basic Feasible Solution (BFS) is a solution to a Linear Programming (LP) problem that satisfies both the constraints of the problem and the basic requirements of feasibility, but it must also adhere to the condition of being basic. In the context of LP, the term “basic” refers to the fact that the BFS corresponds to a set of basic variables, which are variables that can potentially take non-zero values. The rest of the variables are referred to as non-basic variables, and they must take zero values at this solution.
Key Concepts in BFS:
- Feasible Solution: A feasible solution is one that satisfies all the constraints of the linear program, which are typically expressed as linear inequalities or equalities.
- Basic Solution: A basic solution is found by setting non-basic variables to zero and solving the resulting system of equations formed by the constraints. This solution is considered “basic” if the number of non-zero variables equals the number of constraints (i.e., the number of equations).
- Feasibility of the Basic Solution: In order for a basic solution to be feasible, all the values of the variables in the solution must be non-negative (i.e., satisfying the non-negativity constraints like (x_i \geq 0)).
In simpler terms, a BFS is a solution where a certain number of variables are active, and the system formed by the equations remains solvable under the constraints. These variables are typically associated with the vertices of the feasible region (the geometric representation of the constraints).
Example:
For a Linear Program with two constraints, a BFS might occur where two variables are selected as basic and non-zero, and all other variables are set to zero. This would correspond to a point on the boundary of the feasible region (usually a corner or vertex).
The BFS plays a crucial role in algorithms like the Simplex method, where the goal is to iteratively move from one BFS to another, improving the objective function until the optimal solution is found.
Here is a graphical representation of a Basic Feasible Solution (BFS) in a linear programming problem. The image shows a feasible region with constraints forming a polygon, and a basic feasible solution is indicated at one of the corner points of the region. At this point, the corresponding variables are non-zero, while the other variables are set to zero.
This visualization helps in understanding how a BFS corresponds to a solution that satisfies both the constraints and the requirement of having the number of active variables equal to the number of constraints.