Corner-point theorem

Definition

The corner-point theorem states that in a linear programming problem, if there exists an optimal solution, it will occur at one of the vertices (corner points) of the feasible region defined by the constraints. This theorem highlights the importance of examining these corner points when determining the maximum or minimum values of a linear objective function.

5 Must Know Facts For Your Next Test

1. The feasible region is formed by the intersection of the constraints and can be represented graphically as a polygon or polytope.
2. Every corner point of the feasible region corresponds to a specific combination of decision variable values, making them essential to finding optimal solutions.
3. To find the optimal solution, you evaluate the objective function at each corner point and identify which gives the highest or lowest value, depending on whether you are maximizing or minimizing.
4. If a linear programming problem is unbounded, it may not have an optimal solution, as the feasible region extends infinitely in a certain direction.
5. In some cases, there can be multiple optimal solutions if two or more corner points yield the same maximum or minimum value for the objective function.

Review Questions

• How does the corner-point theorem apply to determining optimal solutions in linear programming?
  • The corner-point theorem is crucial for identifying optimal solutions in linear programming because it asserts that if an optimal solution exists, it will be found at one of the vertices of the feasible region. By evaluating the objective function at each corner point, you can pinpoint which one provides the maximum or minimum value. This simplifies the process of finding solutions since you only need to focus on a finite number of points rather than every possible combination of variable values.
• Discuss how constraints influence the shape and size of the feasible region in linear programming and its implications for corner-point analysis.
  • Constraints play a critical role in shaping the feasible region in linear programming by determining where feasible solutions exist. Each constraint forms a boundary, and their intersections define the vertices or corner points. The arrangement and number of constraints can lead to various types of feasible regions—such as bounded or unbounded—which directly impacts how many corner points must be analyzed to find an optimal solution. In scenarios where constraints are tight, corner-point analysis becomes essential for navigating limited options.
• Evaluate how understanding the corner-point theorem can improve decision-making processes in real-world business scenarios involving resource allocation.
  • Understanding the corner-point theorem enhances decision-making in business by providing a structured approach to optimizing resource allocation. In real-world scenarios where companies face various constraints—like budget limits or capacity restrictions—analyzing corner points allows decision-makers to identify optimal combinations of resources that maximize profits or minimize costs. This systematic evaluation not only aids in achieving better outcomes but also helps businesses effectively visualize trade-offs between different variables, leading to more informed and strategic decisions.

Related terms

Feasible Region:

The set of all possible points that satisfy the constraints of a linear programming problem, typically represented as a polygon in graphical methods.

Linear Objective Function: A mathematical function that aims to be maximized or minimized, expressed as a linear combination of decision variables in a linear programming problem.

Constraints: Inequalities or equations that limit the values of decision variables in a linear programming problem, defining the feasible region.