Britannica Sources used in this answer

What does a slack variable show about unused resources in a linear programming problem?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

optimization problem — optimization Constraint set bounded by the five lines x₁ = 0, x₂ = 0, x₁ = 8, x₂ = 5, and x₁ + x₂ = 10. These enclose an infinite number of points that represent feasible solutions.

In linear programming, a slack variable is a non-negative variable introduced to convert a linear inequality constraint into an equality constraint.12

These variables quantify the difference between the limit imposed by a constraint and the actual amount used. For example, if a constraint is expressed as $x_{1} + x_{3} = 8$ , where $x_{1}$ is a production variable and $x_{3}$ is the slack variable, the value of $x_{3}$ represents the amount of the resource that remains unused.12

When a slack variable is greater than zero, it indicates that the corresponding constraint is not "binding," meaning there is leftover capacity or resource. If a slack variable equals zero, it indicates that the resource is fully utilized, meaning the constraint is binding at that specific extreme point. By incorporating these variables, the simplex method can algebraically analyze the boundaries of the feasible region to determine the optimal solution.12