Pivot operations

Definition

Pivot operations are techniques used in linear programming to systematically change the basic feasible solution in order to improve the objective function. They are crucial in the Simplex method, allowing for the movement between vertices of the feasible region defined by constraints. By selectively choosing a pivot element, these operations facilitate a structured approach to finding the optimal solution to linear programming problems.

5 Must Know Facts For Your Next Test

1. Pivot operations can be categorized into two main types: entering and leaving variable operations, where an entering variable replaces a leaving variable in the basic feasible solution.
2. The choice of pivot element significantly affects the efficiency of the Simplex method, as poor choices can lead to more iterations and longer computation times.
3. Pivoting is done by performing row operations on the tableau, which transforms it in such a way that one variable increases while another decreases.
4. The concept of feasibility is essential during pivot operations, ensuring that each new solution remains within the defined constraints.
5. When executed correctly, pivot operations guide the algorithm towards an optimal solution while maintaining all constraints of the linear program.

Review Questions

• How do pivot operations facilitate the transition between basic feasible solutions in the context of linear programming?
  • Pivot operations facilitate transitions between basic feasible solutions by altering which variables are considered 'basic' or 'non-basic'. When a pivot operation is performed, an entering variable takes the place of a leaving variable, allowing the algorithm to explore a different vertex of the feasible region. This systematic change helps improve the objective function incrementally until an optimal solution is reached.
• Discuss how the choice of pivot element impacts the efficiency of solving a linear programming problem using the Simplex method.
  • The choice of pivot element directly influences how quickly the Simplex method converges to an optimal solution. Selecting a good pivot can minimize the number of iterations needed, while poor choices may lead to unnecessary steps and longer computation times. Efficient pivoting strategies aim to select elements that maximize or minimize certain aspects, thereby enhancing overall performance and reducing computational complexity.
• Evaluate the role of pivot operations in maintaining feasibility during the optimization process of linear programming problems.
  • Pivot operations play a critical role in maintaining feasibility throughout the optimization process by ensuring that each new basic feasible solution adheres to all constraints of the problem. As variables enter and leave the basis during these operations, it's crucial that the resulting solution remains within the defined feasible region. This balance allows for an effective search for an optimal solution without violating any constraints, showcasing how pivotal these operations are to successful linear programming.

Related terms

Basic feasible solution: A solution that satisfies all the constraints of a linear programming problem and is formed by setting some variables to zero while solving for others.

Simplex method: An algorithm for solving linear programming problems that utilizes pivot operations to navigate through vertices of the feasible region in search of the optimal solution.

Objective function: The function that needs to be maximized or minimized in a linear programming problem, usually represented as a linear equation involving decision variables.