Feasible Region

Definition

The feasible region is the set of all possible solutions to a linear programming problem that satisfy the given constraints. It is usually represented graphically as a polygon or polyhedron, where each point within this region represents a potential solution that meets all conditions. Understanding the feasible region is crucial for determining optimal solutions and analyzing linear systems of equations.

5 Must Know Facts For Your Next Test

1. The feasible region can be bounded or unbounded, depending on the constraints; a bounded feasible region has a finite area while an unbounded one extends infinitely in at least one direction.
2. When graphed, the feasible region will typically form a polygon in two dimensions or a polyhedron in three dimensions, defined by the intersection of the constraints.
3. The vertices of the feasible region are critical points where potential optimal solutions may exist; the optimal solution is found at one of these vertices.
4. If there are no points within the feasible region, it indicates that the constraints contradict each other, making it impossible to find a solution.
5. To find the feasible region for a linear programming problem, all inequalities must be graphed, and the overlapping area represents the set of all feasible solutions.

Review Questions

• How do constraints shape the feasible region in a linear programming problem?
  • Constraints determine the boundaries of the feasible region by setting limits on the values that variables can take. Each constraint is represented as a line or plane in graphing, and when these lines intersect, they create an area where all conditions are satisfied. The feasible region is thus shaped by the combined effect of these constraints, defining which combinations of variable values are permissible solutions.
• In what ways can identifying the vertices of the feasible region help in solving a linear programming problem?
  • Identifying the vertices of the feasible region is essential because optimal solutions in linear programming occur at these corner points. By evaluating the objective function at each vertex, one can determine which point provides the maximum or minimum value needed. Therefore, focusing on these intersections allows for a more efficient approach to finding the best solution rather than testing every point within the feasible area.
• Evaluate how an unbounded feasible region affects the outcome of a linear programming problem.
  • An unbounded feasible region means that there are infinite solutions available within certain directions, which can lead to either maximization or minimization outcomes without a limit. This situation poses unique challenges as it may result in either no optimal solution being found or an objective function value approaching infinity. When assessing such problems, it's crucial to analyze whether there are any constraints that ultimately restrict movement toward infinity and ensure that there still exists some viable solution within acceptable limits.

Related terms

Constraints: Limits or restrictions placed on the variables of a linear programming problem that define the boundaries of the feasible region.

Objective Function:

A mathematical expression that needs to be maximized or minimized within the feasible region in a linear programming problem.

Vertex: A corner point of the feasible region, where two or more constraints intersect, which is often evaluated for optimal solutions in linear programming.