Introduction to Surplus Variables in Linear Programming

Linear programming is a mathematical optimization technique used to determine the optimal solution for a set of linear equations. In linear programming, we have to maximize or minimize an objective function subject to a set of constraints. There are different types of constraints, such as equality constraints, inequality constraints, and non-negativity constraints. Surplus variables are introduced in linear programming problems to convert inequality constraints into equality constraints.

Explanation of Surplus Variables

Surplus variables are used to convert inequality constraints into equality constraints. In the standard form of linear programming problems, all constraints are written as equality constraints. However, in some cases, it is necessary to use inequality constraints. When we encounter inequality constraints in a linear programming problem, we can introduce a slack variable or a surplus variable to convert the inequality constraint into an equality constraint.

If we have an inequality constraint of the form Ax ≤ b, we can introduce a surplus variable s to convert it into an equality constraint of the form Ax + s = b. The surplus variable represents the amount by which the left-hand side of the inequality constraint exceeds the right-hand side of the constraint.

Example of Surplus Variables

Consider the following linear programming problem:

Maximize Z = 5x1 + 4x2
Subject to:
2x1 + 3x2 ≤ 6
4x1 + 2x2 ≤ 8
x1, x2 ≥ 0

To convert the first constraint into an equality constraint, we introduce a surplus variable s1. The constraint becomes:

2x1 + 3x2 + s1 = 6

Similarly, we introduce a surplus variable s2 for the second constraint. The constraint becomes:

4x1 + 2x2 + s2 = 8

The linear programming problem now becomes:

Maximize Z = 5x1 + 4x2
Subject to:
2x1 + 3x2 + s1 = 6
4x1 + 2x2 + s2 = 8
x1, x2, s1, s2 ≥ 0

Conclusion

Surplus variables are introduced in linear programming problems to convert inequality constraints into equality constraints. They represent the amount by which the left-hand side of the inequality constraint exceeds the right-hand side of the constraint. Surplus variables are used to convert the problem into standard form, where all constraints are written as equality constraints.