Feasible Region

Definition

The feasible region is the set of all possible solutions that satisfy a given set of constraints in a linear programming problem. It is typically represented graphically as a polygon or polyhedron, where each vertex corresponds to a potential solution that meets the constraints. The feasible region plays a crucial role in determining the optimal solution by providing a boundary within which the solutions can be evaluated.

5 Must Know Facts For Your Next Test

1. The feasible region is bounded by the constraint lines and can be either closed (if all constraints form a finite area) or unbounded (if some constraints extend infinitely).
2. In two-dimensional problems, the feasible region can often be represented as a polygon, while in three dimensions, it can be visualized as a polyhedron.
3. Any point within the feasible region satisfies all of the constraints, while points outside do not meet at least one constraint.
4. The optimal solution to a linear programming problem will always occur at one of the vertices of the feasible region.
5. If there are no points in the feasible region, this indicates that the constraints are contradictory and no solution exists.

Review Questions

• How do constraints impact the shape and boundaries of the feasible region in linear programming?
  • Constraints define the limitations within which solutions must fall, and they directly influence the shape and boundaries of the feasible region. Each constraint corresponds to a line (or plane in higher dimensions), and where these lines intersect forms the vertices of the feasible region. If constraints are added or modified, they can either expand or shrink this region, thereby affecting which solutions are possible.
• Discuss how to determine if a feasible region is bounded or unbounded and its implications for finding optimal solutions.
  • To determine if a feasible region is bounded, you can check whether all constraint lines enclose a finite area. If at least one constraint allows for solutions to extend infinitely in any direction, then the feasible region is unbounded. An unbounded feasible region can affect optimal solutions because it may lead to scenarios where there is no maximum or minimum value for the objective function, meaning that it could grow indefinitely.
• Evaluate why vertices of the feasible region are critical when determining the optimal solution in linear programming.
  • Vertices of the feasible region are crucial for determining optimal solutions because linear programming theory states that if an optimal solution exists, it will be found at one of these vertices. This means when evaluating potential solutions, you only need to assess points at these intersections rather than every possible point within the region. By focusing on these key locations, it simplifies the optimization process and efficiently identifies maximum or minimum values for the objective function.

Related terms

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

Constraints: Inequalities or equations that define the limits or requirements within which the solutions must fall in a linear programming problem.

Vertices: The points of intersection of the constraint lines in the feasible region, which are potential candidates for the optimal solution.