Artificial variables

Definition

Artificial variables are additional variables introduced into a linear programming problem to facilitate the finding of an initial feasible solution, particularly in cases where the original constraints do not allow for a straightforward solution. They play a critical role in methods like the Big M method and the two-phase method, helping to navigate situations where there are no obvious basic feasible solutions, such as when dealing with inequality constraints or surplus variables. Their introduction helps maintain the integrity of the optimization process while allowing for the exploration of potential solutions.

5 Must Know Facts For Your Next Test

1. Artificial variables are typically added when there is a need to convert an inequality constraint into an equality, enabling the use of simplex methods.
2. In the Big M method, a large positive constant (M) is assigned to artificial variables in the objective function to ensure they are not part of the final optimal solution.
3. The two-phase method initially solves a simplified version of the problem that includes only artificial variables to find a feasible starting point before addressing the original objective function.
4. Once an optimal solution is found in a linear program with artificial variables, they should ideally have zero value in the final solution to indicate that they were not needed.
5. Artificial variables are crucial for resolving issues associated with degenerate solutions, which can occur when multiple solutions exist at a vertex of the feasible region.

Review Questions

• How do artificial variables help in finding initial feasible solutions in linear programming problems?
  • Artificial variables are introduced into linear programming problems to create an initial feasible solution when standard methods cannot identify one directly. They act as placeholders to satisfy equality constraints that arise from inequalities or surplus variables. By doing this, they allow algorithms like simplex or two-phase methods to start exploring potential solutions, even in challenging situations where feasible points aren't readily apparent.
• What are the differences between using artificial variables in the Big M method versus the Two-Phase Method?
  • In the Big M method, artificial variables are included in the objective function with a large penalty term (M), which discourages their inclusion in any optimal solution. This approach allows for immediate optimization but can lead to numerical issues if M is not appropriately chosen. In contrast, the Two-Phase Method separates finding feasibility and optimization into two distinct steps: Phase One focuses solely on minimizing the sum of artificial variables to find a feasible solution, while Phase Two then optimizes the original objective function without these variables.
• Evaluate how artificial variables influence the outcomes of optimization problems and their role in addressing multiple optimal solutions.
  • Artificial variables significantly impact optimization outcomes by providing necessary structure for solving linear programs that lack clear feasible starting points. They ensure that algorithms can explore all potential solutions effectively. In scenarios with multiple optimal solutions, the presence of artificial variables can help clarify which solutions meet constraints without biasing results. Once optimization is complete, having artificial variables with zero values confirms that an efficient solution was found without reliance on these auxiliary elements, reinforcing the integrity of identified optimal solutions.

Related terms

Big M Method: A technique used in linear programming to solve problems with artificial variables by adding a large penalty (M) to the objective function, discouraging their use in the final solution.

Two-Phase Method: A linear programming algorithm that solves optimization problems in two distinct phases: the first phase focuses on finding a feasible solution using artificial variables, and the second phase optimizes the original objective function.

Feasible Region: The set of all possible points that satisfy the constraints of a linear programming problem, representing potential solutions to be evaluated.