Basic Variables

Definition

Basic variables are the variables in a linear programming problem that have non-zero values in a given basic feasible solution. In the context of the simplex algorithm, they correspond to the columns of the tableau that contain a leading one (pivot) in their rows. These variables represent the active constraints of the system and play a critical role in determining the optimal solution of the objective function.

5 Must Know Facts For Your Next Test

1. In a standard linear programming problem, there are typically more variables than constraints, which means some variables will be non-basic in any given solution.
2. Basic variables are associated with the constraints of the system and provide a means of expressing other variables through them.
3. Each basic feasible solution corresponds to a vertex of the feasible region defined by the constraints of the linear programming problem.
4. The number of basic variables is equal to the number of constraints in the system, assuming all constraints are independent and there are no redundant constraints.
5. In each iteration of the simplex algorithm, basic variables may change as new ones enter and leave the basis through pivoting.

Review Questions

• How do basic variables influence the solution space in linear programming?
  • Basic variables define a specific corner point or vertex of the feasible region in linear programming. Each combination of basic variables corresponds to a unique feasible solution, illustrating how they constrain the possible values of non-basic variables. By examining these combinations, we can identify potential optimal solutions as we navigate through different vertices using the simplex algorithm.
• Discuss how changing basic variables affects the tableau during simplex iterations.
  • When basic variables are changed during simplex iterations, it directly affects the tableau structure and thus influences how solutions progress towards optimality. The pivot operation introduces new basic variables while removing others, which alters their values and impacts overall feasibility. These changes help refine the search for an optimal solution as we move through adjacent vertices of the feasible region.
• Evaluate the importance of basic variables in relation to finding optimal solutions using the simplex algorithm.
  • Basic variables are crucial for identifying optimal solutions with the simplex algorithm because they determine which constraints actively shape feasible solutions. Each step taken within the tableau modifies these basic variables, leading to new potential solutions. The iterative process relies on systematically adjusting these basic variables while tracking performance metrics until an optimal configuration is achieved, ultimately guiding us toward maximizing or minimizing our objective function.

Related terms

Non-basic Variables: Non-basic variables are those variables in a linear programming problem that are set to zero in a basic feasible solution. They do not contribute to the solution but can be adjusted during the optimization process.

Slack Variables: Slack variables are added to linear inequalities to convert them into equalities, allowing for the representation of surplus resources in the simplex tableau.

Pivot Operation: The pivot operation is the process used in the simplex algorithm to switch basic and non-basic variables, aimed at improving the objective function value by moving towards an optimal solution.