Pivoting

Definition

Pivoting refers to the process of exchanging one basic variable for a non-basic variable in a linear programming solution to improve the overall objective function. This technique is essential in optimizing solutions by systematically moving from one vertex of the feasible region to another until the best possible outcome is achieved. It helps determine the most efficient allocation of resources and is a fundamental concept in the simplex method for solving linear programming problems.

5 Must Know Facts For Your Next Test

1. During pivoting, one basic variable leaves the basis while one non-basic variable enters, allowing for an updated solution.
2. The goal of pivoting is to improve the objective function value, which could be maximizing profit or minimizing costs.
3. Pivoting is crucial in the simplex method, as it systematically explores adjacent feasible solutions.
4. The process continues until no further improvements can be made, indicating that an optimal solution has been reached.
5. Each pivot operation is based on determining which non-basic variable will enter and which basic variable will leave, typically assessed using tableau methods.

Review Questions

• How does pivoting facilitate movement within the feasible region of a linear programming problem?
  • Pivoting allows for systematic transitions between feasible solutions by exchanging basic and non-basic variables. When a basic variable is removed from the solution, a non-basic variable enters, which alters the vertex of the feasible region being considered. This process helps explore all possible solutions efficiently, ensuring that the search for the optimal outcome is thorough and structured.
• Discuss how the choice of which variable to pivot on can impact the efficiency of finding an optimal solution.
  • The efficiency of finding an optimal solution through pivoting largely depends on selecting the right entering and leaving variables. If a non-basic variable that significantly improves the objective function is chosen to enter, and a basic variable that constrains progress is selected to leave, this can speed up convergence to an optimal solution. Conversely, poor choices can lead to unnecessary iterations and increased computational time, making strategic selection critical for optimization success.
• Evaluate the implications of pivoting on the overall strategy for solving linear programming problems and its relevance to real-world applications.
  • Pivoting plays a vital role in optimizing linear programming problems by enabling precise adjustments within constraints and improving objective function values. Its relevance extends beyond theoretical exercises; industries like logistics, finance, and manufacturing employ these techniques to optimize resource allocation, production schedules, and supply chain decisions. As such, understanding pivoting equips practitioners with powerful tools for making informed, efficient decisions in complex operational environments.

Related terms

Basic Variables:

Variables in a linear programming problem that are included in the solution basis and are assigned positive values at optimality.

Non-Basic Variables: Variables that are not part of the solution basis and are typically set to zero in a linear programming problem.

Simplex Method: An algorithm used for solving linear programming problems that involves moving along the edges of the feasible region to find the optimal solution.