Feasibility Region

Definition

The feasibility region, also known as the feasible set, is the set of all possible solutions that satisfy the constraints of an optimization problem. This region is crucial for understanding the limits of what can be achieved within given parameters and directly relates to concepts like sensitivity analysis and shadow prices, which help assess how changes in constraints impact the solutions.

5 Must Know Facts For Your Next Test

1. The feasibility region is usually represented graphically as a shaded area in a graph, illustrating all combinations of variables that meet the constraints.
2. An empty feasibility region indicates that no combination of variables satisfies the constraints, leading to infeasibility in the optimization problem.
3. The shape and size of the feasibility region can change if constraints are adjusted, which is essential for performing sensitivity analysis.
4. Within the feasibility region, there may be multiple optimal solutions, and sensitivity analysis can help identify how these solutions shift with changes in constraints.
5. Shadow prices are derived from the boundaries of the feasibility region and provide insight into how tightly constrained resources affect the optimal solution.

Review Questions

• How does the concept of the feasibility region relate to finding optimal solutions in an optimization problem?
  • The feasibility region defines all potential solutions that meet given constraints, creating a boundary within which optimal solutions can be identified. Optimal solutions exist at points within this region, and any adjustments to constraints may shift these solutions. Understanding where these points lie within the feasibility region is crucial for effectively optimizing an objective function while adhering to limitations.
• Discuss how changes in constraints impact the feasibility region and its implications for sensitivity analysis.
  • Changes in constraints directly alter the shape and size of the feasibility region. When a constraint is tightened, it may shrink the region, potentially eliminating previously feasible solutions. Conversely, relaxing a constraint can expand the region, introducing new feasible options. Sensitivity analysis examines these shifts and helps decision-makers understand how responsive optimal solutions are to changes in constraints.
• Evaluate the significance of shadow prices in relation to the feasibility region when addressing resource allocation decisions.
  • Shadow prices play a vital role in resource allocation decisions by quantifying the value of relaxing constraints within the feasibility region. By understanding shadow prices, decision-makers can assess which resources are most critical and where investments or adjustments can yield substantial benefits. This analysis provides deeper insights into how optimizing within the feasibility region can lead to improved outcomes when managing limited resources.

Related terms

Constraints: Limitations or requirements placed on the variables of an optimization problem that must be satisfied for a solution to be considered feasible.

Optimal Solution: The best possible outcome in an optimization problem, found within the feasibility region, that maximizes or minimizes the objective function.

Shadow Price: The value of an additional unit of a resource within the context of an optimization problem, indicating how much the objective function would improve if the constraint were relaxed.