What is a surplus variable in linear programming?

A surplus variable is utilized in linear programming to convert a "greater than or equal to" (≥) constraint into an equality. In linear programming, constraints are often expressed in terms of inequalities, and to apply methods like the Simplex algorithm, these inequalities need to be transformed.

When you have a constraint that states one quantity must be greater than or equal to another (e.g., (x_1 + x_2 \geq 10)), a surplus variable is introduced to adjust this constraint to reflect an equality. By subtracting the surplus variable from the left-hand side of the inequality, you can rewrite it as (x_1 + x_2 - s = 10), where (s) represents the surplus variable. This represents the amount by which the left-hand side exceeds the minimum required value.

The concept of surplus variables is essential for correctly setting up linear programming problems, particularly when formulating constraints that involve limits or minimum requirements. Each surplus variable takes on a non-negative value, indicating how much the actual solution exceeds the required constraint.