Feasible Region

Definition

The feasible region is the set of all possible solutions to a linear programming problem that satisfy all given constraints. This region is typically represented graphically as a polygon in two-dimensional space, where each vertex corresponds to a potential solution. The feasible region helps identify the optimal solutions by showing where the objective function can achieve its maximum or minimum value within the defined constraints.

5 Must Know Facts For Your Next Test

1. The feasible region is always bounded by the constraints of the problem, which can be equalities or inequalities.
2. In two-dimensional problems, the feasible region is visualized as a polygon, while in three dimensions, it appears as a polyhedron.
3. If there are no overlapping areas among the constraints, the feasible region may be empty, indicating no possible solutions exist.
4. The optimal solution to a linear programming problem will always occur at one of the vertices of the feasible region.
5. The shape and size of the feasible region can greatly influence the complexity and solvability of a linear programming problem.

Review Questions

• How does the feasible region relate to the constraints in a linear programming problem?
  • The feasible region is directly determined by the constraints set within a linear programming problem. Each constraint limits the possible values for the decision variables, and when these constraints are graphed, they create boundaries that define the feasible region. This area represents all combinations of values that satisfy all constraints simultaneously, showcasing where potential solutions can exist.
• What role do vertices play in identifying optimal solutions within a feasible region?
  • Vertices are crucial because they are the specific points at which the boundaries of the feasible region meet. According to linear programming principles, any optimal solution—whether it's maximizing or minimizing an objective function—will occur at one of these vertices. By evaluating the objective function at each vertex, one can determine which point yields the best outcome within the defined constraints.
• Evaluate how changes in constraints affect the feasible region and potential solutions in linear programming.
  • When constraints in a linear programming problem are altered, the shape and size of the feasible region may change significantly. For instance, tightening a constraint may reduce the feasible region, potentially eliminating some solutions and possibly affecting where the optimal solution lies. Conversely, loosening a constraint could expand the feasible region, introducing new potential solutions. This dynamic interaction emphasizes how sensitive optimal solutions are to modifications in constraints.

Related terms

Constraints: Conditions or limitations that must be satisfied in a linear programming problem, often represented as linear inequalities.

Objective Function: A mathematical expression that defines the goal of a linear programming problem, typically aiming to maximize or minimize some quantity.

Vertices: The corner points of the feasible region, which are critical for determining optimal solutions in linear programming.