Feasible solution

Definition

A feasible solution refers to a set of values that satisfy all constraints of a given optimization problem. In the context of linear programming, these solutions fall within the feasible region, which is defined by the intersection of constraints represented graphically. A feasible solution does not guarantee optimality; it merely indicates that the solution meets the criteria laid out by the inequalities in the problem.

5 Must Know Facts For Your Next Test

1. Feasible solutions can be classified into basic and non-basic solutions depending on whether they correspond to corner points of the feasible region.
2. If a feasible solution exists, there could be multiple feasible solutions within the same region, but only one optimal solution may exist.
3. In cases where no feasible solution exists, it indicates that the constraints are too strict and contradictory.
4. Graphically, feasible solutions can be visualized as points within a bounded or unbounded area defined by intersecting lines representing constraints.
5. When solving linear programming problems using methods like the Simplex method, identifying feasible solutions is a critical first step before determining the optimal solution.

Review Questions

• How do you identify feasible solutions in a linear programming problem?
  • To identify feasible solutions in a linear programming problem, you need to graph the constraints represented by inequalities on a coordinate plane. The feasible region is where all these constraints overlap. Any point within this area represents a feasible solution since it satisfies all constraints. To ensure accuracy, check that these points also align with any non-negativity restrictions, if present.
• Discuss how changes in constraints can affect the set of feasible solutions in a linear programming scenario.
  • Changes in constraints can significantly alter the set of feasible solutions. For instance, tightening a constraint can shrink the feasible region or potentially eliminate all feasible solutions if it creates contradictions. Conversely, relaxing constraints typically enlarges the feasible region, allowing for more possible solutions. Understanding these dynamics is crucial for adjusting models and interpreting outcomes in optimization problems.
• Evaluate the importance of distinguishing between feasible and optimal solutions in linear programming.
  • Distinguishing between feasible and optimal solutions is crucial because while all optimal solutions are feasible, not all feasible solutions are optimal. The goal of linear programming is to maximize or minimize an objective function while adhering to constraints. If one cannot identify an optimal solution among many feasible ones, resources may be wasted pursuing suboptimal outcomes. This understanding emphasizes the need for both feasibility analysis and optimization strategies to achieve effective decision-making in practical applications.

Related terms

Objective Function:

The mathematical expression that defines the goal of the optimization problem, such as maximizing profit or minimizing cost.

Constraints:

The limitations or requirements that define the feasible region in a linear programming problem, typically expressed as linear inequalities.

Optimal Solution:

The best possible outcome within the feasible region, which maximizes or minimizes the objective function.