Pivoting

Definition

Pivoting refers to the process of changing the basis of a linear programming solution by replacing one of the basic variables with a non-basic variable in order to move to an adjacent vertex of the feasible region. This technique is crucial in optimizing a linear objective function and is a key operation in the simplex method. It enables finding an improved solution by navigating through feasible solutions until the optimal one is reached.

5 Must Know Facts For Your Next Test

1. In pivoting, the goal is to improve the objective function's value while maintaining feasibility within the constraints.
2. Each pivot operation modifies the tableau representation in the simplex method, facilitating transitions from one vertex to another in the feasible region.
3. The selection of entering and leaving variables is critical during pivoting, as it determines which variable will increase and which will decrease.
4. The process can continue iteratively until no further improvements can be made, indicating that an optimal solution has been found.
5. Pivoting can be performed using various rules, such as the largest coefficient rule or the smallest ratio rule, which affect efficiency and computational performance.

Review Questions

• How does pivoting affect the feasible region during the optimization process?
  • Pivoting impacts the feasible region by moving from one vertex to an adjacent vertex while remaining within the bounds defined by the constraints. Each pivot operation effectively shifts the solution along the edges of the feasible region, allowing for adjustments that may lead to improved objective function values. This movement is essential for exploring potential optimal solutions without violating any constraints.
• What factors influence the choice of entering and leaving variables during a pivot operation in linear programming?
  • The choice of entering and leaving variables during a pivot operation is influenced by several factors, including which variable can improve the objective function value most effectively and which basic variable can be replaced without violating feasibility. Common strategies include using rules like the largest coefficient rule to maximize improvement in each iteration or employing ratios to ensure that basic variables remain non-negative. This selection process is crucial for ensuring efficient convergence towards an optimal solution.
• Evaluate how different pivot selection strategies can impact the efficiency of solving linear programming problems using the simplex method.
  • Different pivot selection strategies can significantly impact the efficiency and speed at which optimal solutions are reached in linear programming problems. For example, using the largest coefficient rule may lead to faster improvements in some cases but could also result in cycling or suboptimal paths. On the other hand, adopting more conservative strategies like Bland's rule can enhance stability but may slow down convergence. Analyzing these strategies helps optimize performance based on specific problem characteristics, ensuring more effective resource allocation during computational processes.

Related terms

Basic Variable: A variable that is included in the current solution of a linear programming problem and has a value greater than zero.

Non-Basic Variable: A variable that is not part of the current solution, typically set to zero in the context of a linear programming problem.

Feasible Region:

The set of all possible solutions that satisfy the constraints of a linear programming problem.