Feasible Region

Definition

The feasible region is the set of all possible solutions to a linear programming problem that satisfy all constraints. It represents the area where all constraints overlap, indicating the combinations of decision variables that lead to valid solutions for optimization problems.

5 Must Know Facts For Your Next Test

1. The feasible region is typically represented graphically as a polygon or polyhedron in two or more dimensions, showing all combinations of decision variables that meet the constraints.
2. Points within the feasible region are known as feasible solutions, while points outside this region are infeasible and do not satisfy the problem's constraints.
3. The optimal solution for a linear programming problem is usually found at one of the vertices (corners) of the feasible region, due to the properties of linear functions.
4. In situations with no overlapping constraints, the feasible region may be empty, indicating that no solution exists that meets all requirements.
5. When dealing with multiple constraints, the shape and size of the feasible region can change significantly, affecting the complexity and solvability of the 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 in a linear programming problem. It includes all combinations of decision variables that meet these constraints, typically represented graphically. Each constraint creates a boundary in the solution space, and where these boundaries intersect defines the area of feasible solutions. Thus, understanding how each constraint influences the shape and size of the feasible region is key to identifying valid solutions.
• What role does the objective function play in relation to the feasible region when solving optimization problems?
  • The objective function is crucial when analyzing the feasible region because it defines what needs to be optimized—whether it's maximization or minimization. Once the feasible region is identified, finding the optimal solution involves evaluating points within this region based on the objective function. The optimal solution will typically occur at one of the vertices of this feasible area, emphasizing how effectively navigating between constraints can lead to achieving business goals.
• Evaluate how changes in constraints might affect both the feasible region and potential solutions in a linear programming scenario.
  • Changes in constraints can significantly alter both the shape and size of the feasible region, potentially leading to new optimal solutions. For instance, tightening a constraint could shrink the feasible region, possibly eliminating previously viable solutions and creating new corners for consideration. Conversely, relaxing a constraint might expand the feasible region, introducing additional options for optimization. This dynamic nature emphasizes the importance of flexibility in decision-making processes based on changing operational conditions.

Related terms

Constraints: Constraints are the conditions or limitations placed on the decision variables in a linear programming model, often expressed as inequalities.

Objective Function: The objective function is the mathematical expression that defines the goal of a linear programming problem, such as maximizing profit or minimizing costs.

Linear Programming: Linear programming is a mathematical method used for determining a way to achieve the best outcome in a given mathematical model, which includes a linear objective function and constraints.