What is the Big M Method?

The Big M Method is a specialized version of the Simplex algorithm used in linear programming to solve optimization problems where the constraints include "greater-than-or-equal-to" (≥) or "equal-to" (=) signs. Unlike standard maximization problems that only use slack variables, the Big M method introduces artificial variables to find an initial feasible basis.

When to use the Big M Method?

In linear programming, if a model does not have a natural starting point at the origin (0,0), we need a way to force the artificial variables to zero to reach the actual feasible region. The "M" represents an arbitrarily large positive constant. By assigning a penalty of -M (for maximization) or +M (for minimization) to these artificial variables in the objective function, the algorithm naturally seeks to drive these variables to zero during the iteration process.

How the Solver Works

1. Standardization: The solver first converts all inequalities into equalities by adding slack variables (for ≤), surplus variables (for ≥), and artificial variables (for ≥ and =).
2. Objective Function Modification: It appends the artificial variables to the objective function with a high penalty coefficient (M).
3. Tableau Construction: It builds the initial Simplex tableau including all coefficients.
4. Iteration: The solver performs row operations (pivoting) to improve the objective value until no further improvement is possible or the problem is found to be unbounded or infeasible.

Frequently Asked Questions

Q: Why is it called "Big M"?
A: Because "M" is used to represent an infinitely large value that ensures artificial variables are prioritized for removal from the basis.

Q: What happens if an artificial variable remains in the final solution?
A: If at the optimal stage an artificial variable is still positive, it indicates that the original linear programming problem has no feasible solution.