Feasible Region

Definition

A feasible region is the set of all possible solutions that satisfy a given set of constraints, typically represented as inequalities in a graphical context. This region is bounded by the intersection of these constraints and is essential for finding optimal solutions in various mathematical applications. Within this region, any point represents a solution that meets all the specified conditions.

5 Must Know Facts For Your Next Test

1. The feasible region can be visualized as a polygon in two-dimensional space formed by the intersection of lines representing the constraints.
2. Points outside the feasible region do not satisfy one or more constraints and are therefore not valid solutions.
3. If the feasible region is empty, it indicates that there are no possible solutions that meet all the constraints.
4. The optimal solution to a linear programming problem is often found at one of the vertices of the feasible region.
5. In higher dimensions, the feasible region can take on complex shapes but still follows the same principle of representing valid solutions within given constraints.

Review Questions

• How does the shape and location of a feasible region impact potential solutions to a system of inequalities?
  • The shape and location of a feasible region are crucial as they define which combinations of variables can satisfy all constraints simultaneously. For instance, if the feasible region is small or constrained by many inequalities, there may be fewer viable solutions. Conversely, a larger feasible region suggests more potential solutions that meet the criteria set by the inequalities. Understanding this relationship helps in determining where optimal solutions may lie.
• Discuss the significance of vertices within a feasible region when solving optimization problems.
  • Vertices hold particular importance in optimization because they are potential candidates for optimal solutions. In linear programming, it is proven that if an optimal solution exists, it will occur at one of the vertices of the feasible region. Therefore, evaluating these points allows us to efficiently determine maximum or minimum values of an objective function without having to examine every point within the entire feasible region.
• Evaluate how changing one constraint affects the feasible region and the implications for finding optimal solutions.
  • Altering a single constraint can significantly change the shape and size of the feasible region, potentially creating new vertices or eliminating existing ones. For example, tightening a constraint may shrink the feasible region, reducing possible solutions, while loosening it might expand it, allowing for more options. These changes can shift where optimal solutions are located, making it essential to analyze how each constraint interacts with others when optimizing.

Related terms

Linear Inequalities: Mathematical expressions that involve a linear function and an inequality symbol, used to define constraints in a system.

Vertices: The corner points of the feasible region, which are critical for determining optimal solutions in linear programming.

Objective Function:

A mathematical expression that defines the goal of optimization, usually to maximize or minimize a particular value within the feasible region.