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. This region is typically represented graphically as an area on a coordinate system where any point within it corresponds to a valid solution that meets all the inequalities or equalities defined by the constraints.

5 Must Know Facts For Your Next Test

1. The feasible region can be bounded or unbounded, depending on whether the constraints create a finite area or extend infinitely.
2. In two-variable problems, the feasible region is often illustrated as a polygon or a convex shape on a graph, where each vertex can represent a potential optimal solution.
3. Points outside of the feasible region do not satisfy all constraints and are therefore not considered valid solutions.
4. The corners of the feasible region, known as extreme points, are key candidates for determining the optimal solution in linear programming.
5. When constraints change, the shape and size of the feasible region may also change, impacting where the optimal solutions lie.

Review Questions

• How does altering constraints affect the feasible region in a linear programming problem?
  • Altering constraints can significantly change the shape and size of the feasible region. If new constraints are added, they may either narrow down the existing region or expand it, depending on their nature. For instance, adding a stricter inequality can reduce the area of feasible solutions, while loosening a constraint may increase it. This dynamic illustrates how sensitive the feasible region is to changes in constraints, directly influencing potential optimal solutions.
• Explain how the graphical method helps visualize the feasible region and determine optimal solutions for two-variable problems.
  • The graphical method allows for a visual representation of the feasible region by plotting each constraint on a graph and identifying the overlapping area that satisfies all conditions. This method makes it easy to identify extreme points or vertices of the feasible region where potential optimal solutions may exist. By evaluating these points against the objective function, one can find which vertex provides the best outcome. This visual approach simplifies understanding complex relationships between variables and constraints.
• Discuss the implications of using Lagrange multipliers in identifying optimal solutions within a constrained feasible region.
  • Using Lagrange multipliers provides a systematic way to find optimal solutions when dealing with constrained optimization problems. By introducing multipliers corresponding to each constraint, one can transform the problem into one where gradients indicate potential maxima or minima within the feasible region. This method efficiently handles cases where graphical representation is impractical due to higher dimensions. The use of Lagrange multipliers not only identifies optimal solutions but also reflects how changes in constraints influence those solutions within the defined feasible region.

Related terms

Constraints:

Limits or restrictions placed on the decision variables in a mathematical optimization problem, often expressed as inequalities or equalities.

Optimal Solution:

The best possible outcome from the feasible region, typically maximized or minimized based on the objective function in a linear programming problem.

Objective Function:

A mathematical expression that defines the goal of an optimization problem, usually representing profit, cost, or some other measure to be maximized or minimized.