Non-basic variables

Definition

Non-basic variables are those variables in an optimization problem that are not included in the current solution or basis of the solution. They typically take a value of zero in basic feasible solutions and do not directly influence the objective function at that solution point. Their role is essential in determining the potential for improving the objective function by adjusting the values of basic variables.

5 Must Know Facts For Your Next Test

1. Non-basic variables are often identified during the simplex method as those that do not occupy a position in the current basic solution.
2. In a tableau, non-basic variables correspond to columns that do not contain leading 1s, indicating their zero value in basic feasible solutions.
3. They can be altered to explore other potential solutions, which may lead to improvements in the objective function.
4. Understanding non-basic variables is crucial for sensitivity analysis, as changes in their values can indicate how robust a solution is to perturbations.
5. In equality constrained optimization, non-basic variables may become basic if a pivot operation occurs, changing the nature of the solution.

Review Questions

• How do non-basic variables impact the process of finding optimal solutions in equality constrained optimization?
  • Non-basic variables play a crucial role in identifying potential improvements during optimization. They are set to zero in basic feasible solutions, which allows for focus on basic variables that drive the current solution. By analyzing how changes to non-basic variables can affect the objective function, one can explore whether moving to a new basic feasible solution would yield better results.
• What is the significance of recognizing non-basic variables when applying the simplex method to solve an equality constrained optimization problem?
  • Recognizing non-basic variables is essential when using the simplex method because they indicate which aspects of the solution can be adjusted to improve outcomes. In this method, operations such as pivoting allow for transitioning non-basic variables into basic status, potentially leading to an improved objective function. This understanding facilitates effective exploration of the feasible region and helps identify optimal solutions.
• Evaluate how non-basic variables relate to sensitivity analysis in equality constrained optimization and why this connection is important.
  • Non-basic variables are integral to sensitivity analysis because they help determine how changes in parameters or constraints might affect optimal solutions. By evaluating the potential shifts in non-basic variables, one can assess how sensitive an optimal solution is to variations in inputs or resource availability. This connection is important as it enables decision-makers to understand risks and adjust strategies accordingly when conditions change.

Related terms

basic variables: Basic variables are the variables in an optimization problem that are included in the current solution or basis, taking on non-zero values to satisfy the constraints.

feasible region: The feasible region is the set of all possible solutions that satisfy the constraints of an optimization problem.

slack variables: Slack variables are additional variables added to convert inequalities into equalities, allowing for a standard form representation of the optimization problem.