Non-basic variable

Definition

A non-basic variable is a variable in a linear programming problem that is not part of the solution at a particular vertex of the feasible region. These variables typically take a value of zero in the optimal solution, contrasting with basic variables that are active and determine the solution's coordinates. Understanding the distinction between basic and non-basic variables is essential for interpreting the results of linear programming models and for the simplex algorithm's operations.

5 Must Know Facts For Your Next Test

1. Non-basic variables are typically set to zero in the optimal solution of a linear programming model, which helps simplify calculations and interpretations.
2. The number of non-basic variables in a linear programming problem is equal to the total number of variables minus the number of basic variables.
3. In the simplex algorithm, non-basic variables can be changed from zero to positive values when pivoting to a new basic feasible solution.
4. Understanding which variables are non-basic helps identify which constraints are binding or non-binding at optimality.
5. Non-basic variables provide insight into potential changes in the objective function if they were to take on non-zero values.

Review Questions

• How does understanding non-basic variables contribute to solving linear programming problems?
  • Understanding non-basic variables is crucial because they represent dimensions of potential change in a linear programming model. Knowing that these variables are typically set to zero at optimal solutions allows for easier identification of which constraints are active and which can be adjusted. This comprehension helps in navigating the simplex algorithm more effectively by highlighting which variables can be brought into play without violating constraints.
• Explain how non-basic variables interact with basic variables during the simplex algorithm process.
  • During the simplex algorithm, non-basic variables remain at zero while basic variables assume positive values. As the algorithm iterates through feasible solutions, it identifies entering and leaving variables where a non-basic variable can increase from zero while a basic variable decreases. This interaction allows for pivoting, which effectively moves toward optimal solutions while ensuring that all constraints are satisfied. Understanding this relationship aids in strategizing which variable adjustments could yield better objective function values.
• Analyze how changing non-basic variables affects the overall solution in linear programming problems.
  • Changing non-basic variables can lead to shifts in the feasible region and impact the overall solution significantly. When a non-basic variable transitions from zero to a positive value, it may alter binding constraints and redefine which basic variables remain active. This interplay may result in different optimal solutions or improve existing ones, showcasing how sensitivity analysis can be conducted to assess potential outcomes based on varying conditions within a model.

Related terms

basic variable: A basic variable is a variable in a linear programming problem that is included in the solution at a specific vertex of the feasible region, contributing to the value of the objective function.

feasible region: The feasible region is the set of all possible solutions to a linear programming problem that satisfy all given constraints, typically represented graphically.

simplex algorithm:

The simplex algorithm is a widely used method for solving linear programming problems by moving along the edges of the feasible region to find the optimal solution.