Artificial variable

Definition

An artificial variable is a temporary addition to a linear programming problem that helps in finding an initial feasible solution when the standard constraints do not naturally allow for one. These variables are used specifically in the simplex method to aid in the process of optimization by ensuring that basic feasible solutions can be reached, even when certain constraints are not satisfied initially.

5 Must Know Facts For Your Next Test

1. Artificial variables are usually introduced when using methods like the Big M method or two-phase simplex method to solve linear programming problems.
2. In the two-phase simplex method, artificial variables are used during the first phase to find a basic feasible solution without violating any constraints.
3. Once a feasible solution is identified, artificial variables should ideally have zero value in the optimal solution, indicating they are not needed.
4. In practical terms, if an artificial variable remains in the final solution with a non-zero value, it indicates that no feasible solution exists for the original problem.
5. The coefficients of artificial variables in the objective function are typically set to a very large negative value (in the Big M method) to drive them out of the solution.

Review Questions

• How do artificial variables assist in finding a basic feasible solution in the simplex method?
  • Artificial variables help in finding a basic feasible solution by providing a way to include constraints that do not allow for an immediate feasible point. They are added to the problem when initial solutions are not evident. By using these variables, the simplex method can begin optimization from a point where all constraints are satisfied, thus allowing for further progression toward an optimal solution.
• What is the process and significance of removing artificial variables during optimization in the two-phase simplex method?
  • In the two-phase simplex method, artificial variables play a crucial role during the first phase by helping identify a basic feasible solution. In phase two, the goal is to eliminate these artificial variables from the solution entirely, which signifies that we have moved toward an actual feasible region defined by the original problem. The successful removal of these variables indicates that we can optimize without their presence, reflecting a valid solution to the original constraints.
• Evaluate the implications if an artificial variable remains with a positive value in the final simplex tableau after optimization is complete.
  • If an artificial variable retains a positive value in the final simplex tableau after optimization, it indicates that no feasible solution exists for the original linear programming problem. This situation highlights that the constraints imposed by the problem are incompatible, preventing any combination of decision variables from satisfying them. Recognizing this outcome is critical for understanding the limitations of modeling and guides further exploration of alternative formulations or problem adjustments.

Related terms

basic feasible solution: A solution to a linear programming problem that satisfies all the constraints and has a non-negative value for all variables, including slack and artificial variables.

slack variable:

A variable added to a linear programming constraint to convert an inequality into an equality, representing unused resources in a resource allocation problem.

simplex tableau: A tabular representation of the linear programming problem used in the simplex method that organizes coefficients, variables, and constraints to facilitate the optimization process.