Define: Basic feasible solution, optimum solution, optimum basic feasible solution.
The correct answer and explanation is :
Definitions:
- Basic Feasible Solution (BFS):
A Basic Feasible Solution is a solution to a linear programming (LP) problem that satisfies all the constraints and is associated with a set of basic variables. These basic variables correspond to a set of linearly independent constraints, and the non-basic variables are set to zero. The BFS is a feasible solution, meaning it meets all the problem’s constraints, and the set of basic variables satisfies the condition for linear dependence. - Optimum Solution:
The optimum solution to a linear programming problem is the solution that optimizes (maximizes or minimizes) the objective function while satisfying all the problem’s constraints. In LP, it is the point where the objective function reaches its maximum or minimum value. This solution can be found either at a vertex of the feasible region or along an edge. - Optimum Basic Feasible Solution (OBFS):
An Optimum Basic Feasible Solution is a BFS that not only satisfies the constraints but also optimizes the objective function. It is the best solution, among all BFSs, that gives the optimal value of the objective function. This solution is found at a vertex of the feasible region, and it corresponds to the highest or lowest value of the objective function, depending on whether the problem is a maximization or minimization problem.
Explanation:
In the context of linear programming, the feasible region is a convex set formed by the constraints of the problem. A feasible solution lies within this region and satisfies all constraints. However, not all feasible solutions are optimal. A basic feasible solution is a particular type of feasible solution where a subset of the constraints are active, and the solution lies at one of the vertices (corner points) of the feasible region.
The optimum solution refers to the best possible outcome for the objective function, whether it is maximizing profit or minimizing cost, subject to the constraints. The optimum solution is typically found at a vertex (or along an edge) of the feasible region.
The optimum basic feasible solution is a BFS that also satisfies the optimality condition. It is the solution at a vertex of the feasible region that maximizes or minimizes the objective function.
Linear programming algorithms like the Simplex method are used to iteratively move from one BFS to another, eventually reaching the optimum basic feasible solution.
Here is a diagram illustrating the concepts of Basic Feasible Solution (BFS), Optimum Solution, and Optimum Basic Feasible Solution (OBFS) in a linear programming context. It highlights how the feasible region and vertices relate to these solutions.