Pivoting

Definition

Pivoting is a fundamental operation in linear programming that involves selecting a non-basic variable to enter the basis and a basic variable to leave the basis. This process allows for the improvement of the objective function by transforming the tableau, ultimately leading towards an optimal solution. It is crucial in both the iteration process of optimization and in methods like the revised simplex method, where efficient calculations are necessary to navigate through feasible solutions.

5 Must Know Facts For Your Next Test

1. In pivoting, the entering variable is chosen based on which non-basic variable can improve the objective function most significantly.
2. The leaving variable is determined by applying the minimum ratio test, which ensures that all variables remain non-negative after the pivot operation.
3. Pivoting operations are repeated iteratively until no further improvements can be made, indicating that an optimal solution has been reached.
4. In the revised simplex method, pivoting is performed using updated matrices to enhance computational efficiency, reducing memory usage and speeding up calculations.
5. The selection of pivot elements can greatly affect the performance and efficiency of reaching an optimal solution; poor choices may lead to cycling or slower convergence.

Review Questions

• How does pivoting contribute to improving the objective function in linear programming?
  • Pivoting contributes to improving the objective function by systematically selecting which non-basic variable to introduce into the basis and which basic variable to remove. This selection process aims to maximize or minimize the objective function effectively by navigating through feasible solutions. Each pivot operation alters the tableau, ideally driving the solution closer to optimality while adhering to constraints.
• What role does pivoting play in enhancing the efficiency of the revised simplex method compared to traditional methods?
  • In the revised simplex method, pivoting plays a crucial role in enhancing efficiency by utilizing updated matrix forms instead of recalculating entire tableaux from scratch. This approach allows for quicker adjustments during each iteration, as only relevant parts of the data are manipulated. Consequently, pivoting in this context helps reduce computational overhead and speeds up convergence towards an optimal solution.
• Evaluate how poor pivot selection can impact the overall process of solving linear programming problems and suggest strategies to mitigate these issues.
  • Poor pivot selection can significantly slow down the process of finding an optimal solution by leading to cycling or unnecessary iterations. This not only increases computation time but may also cause convergence issues. To mitigate these problems, techniques such as Bland's Rule or implementing anti-cycling measures can be used to ensure that pivot selections do not repeat previous states. By making informed choices on entering and leaving variables based on specific criteria, one can maintain a smooth progression towards optimality.

Related terms

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

Basis: A set of linearly independent vectors in a vector space that can be used to represent other vectors; in linear programming, it refers to the current set of basic variables.

Tableau: A structured format that organizes the coefficients of a linear programming problem for use in the simplex method, showing the relationships among variables and constraints.