Slack Variable

Definition

A slack variable is an additional variable introduced in a linear programming problem to transform an inequality constraint into an equality constraint. By adding a slack variable, the unused resources or capacities in a constraint are explicitly accounted for, allowing for a clearer representation of the feasible region. This concept is crucial for identifying optimal solutions as it helps define the boundaries of the solution space.

5 Must Know Facts For Your Next Test

1. Slack variables are always non-negative, meaning they cannot take on negative values as they represent unused capacity in the system.
2. When a slack variable is added to an inequality constraint, it essentially represents how much 'slack' or room there is before the constraint is fully utilized.
3. In standard form linear programming problems, all constraints must be equalities, hence slack variables play a critical role in converting inequalities.
4. Each inequality constraint in a maximization problem typically has one corresponding slack variable to ensure that solutions remain feasible.
5. The value of a slack variable at optimality indicates how much of that resource remains unused in the solution.

Review Questions

• How do slack variables contribute to defining the feasible region in linear programming?
  • Slack variables help define the feasible region by converting inequality constraints into equalities. This ensures that all constraints are represented as boundaries in the solution space. When these boundaries are plotted, they help to shape the feasible region where potential optimal solutions can be found. The presence of slack variables allows for a clear visualization of resource usage and availability within that region.
• Discuss how slack variables are treated differently from basic and non-basic variables in a linear programming model.
  • In a linear programming model, slack variables are classified as basic variables when they are included in the solution basis, while non-basic variables are set to zero. Slack variables represent unused resources and directly contribute to maintaining feasibility within constraints. Their values indicate how much of each resource remains after satisfying other constraints. In contrast, non-basic variables do not contribute to the solution until they enter the basis, affecting the overall optimization process.
• Evaluate the impact of introducing slack variables on the overall efficiency of solving a linear programming problem.
  • Introducing slack variables significantly enhances the efficiency of solving linear programming problems by allowing constraints to be expressed as equalities, which simplifies the mathematical formulation. This transformation facilitates the use of optimization algorithms like the Simplex method, leading to faster convergence toward an optimal solution. Additionally, by accounting for unused capacities explicitly, slack variables provide valuable insights into resource allocation and help identify areas for improvement or adjustment within operational processes.

Related terms

Feasible Region: The set of all possible points that satisfy the constraints of a linear programming problem, including both equality and inequality constraints.

Basic Feasible Solution: A solution to a linear programming problem that corresponds to a vertex of the feasible region, formed by setting non-basic variables to zero and solving for basic variables.

Artificial Variable: A variable added to a linear programming model to facilitate the initial solution process, particularly when dealing with equality constraints or when no obvious basic feasible solution exists.