Slack variable

Definition

A slack variable is a non-negative variable added to a linear programming constraint to convert an inequality into an equality. It represents the difference between the left-hand side and the right-hand side of a less-than-or-equal-to constraint, allowing for the model to satisfy constraints while optimizing the objective function. This makes it possible to include constraints in a linear programming problem without altering its fundamental structure.

5 Must Know Facts For Your Next Test

1. Slack variables allow for the transformation of inequalities into equalities, making it easier to apply the simplex method for solving linear programming problems.
2. They are always non-negative, meaning that if a constraint is not tight, the slack variable will take on a positive value, indicating unused resources.
3. In a graphical representation of a linear programming problem, slack variables can be visualized as the distance from the boundary of the feasible region to the constraint line.
4. The introduction of slack variables can help identify how much a resource can be increased without affecting the optimal solution.
5. Slack variables play a critical role in sensitivity analysis by showing how changes in constraints affect the optimal solution.

Review Questions

• How do slack variables help in converting inequalities into equalities in linear programming problems?
  • Slack variables serve to convert inequalities into equalities by quantifying the difference between the actual usage of resources and their maximum available limits. When an inequality constraint is less than or equal to something, adding a slack variable allows us to express this as an equality. For example, if we have a constraint like `x + y <= 10`, introducing a slack variable `s` lets us rewrite it as `x + y + s = 10`, where `s` is non-negative.
• Discuss how slack variables contribute to understanding resource utilization in linear programming solutions.
  • Slack variables provide insight into how efficiently resources are utilized in linear programming solutions. When we solve for these variables, any positive value indicates that there is unused capacity within that resource. This helps decision-makers understand where they can potentially adjust resource allocation without negatively impacting the overall outcome, allowing for better strategic planning and optimization of available resources.
• Evaluate the role of slack variables in sensitivity analysis within linear programming models.
  • In sensitivity analysis, slack variables are crucial because they reveal how changes in constraints can impact the optimal solution. By analyzing slack values, one can determine how much a constraint can be altered before it affects feasibility or optimality. For instance, if a slack variable is zero, it means that resource utilization is at its limit and any increase in demand would require adjustments elsewhere in the model. This evaluation helps assess risk and inform decisions about resource management under varying conditions.

Related terms

Linear Programming: A mathematical method used for optimizing a linear objective function, subject to linear equality and inequality constraints.

Objective Function: The function that is being maximized or minimized in a linear programming problem, which depends on the decision variables.

Feasible Region: The set of all possible points that satisfy the constraints of a linear programming problem, typically represented graphically.