What is the theoretical basis for the Big M method in operations research?

For linear programming problems of general form, reduced to standard type, both the Big M method and the two-stage method can be solved. If solved by hand, there is no difference in the application of the two algorithms.

In case of computer programming, the two-stage algorithm is preferred. The reason for this is that the large M method may be subject to computational errors due to the value of the large M taken.

In the maximization problem, an artificial variable is assigned an M as its coefficient; in the minimization problem, an artificial variable is assigned an M as its coefficient, with M being an arbitrarily large (not infinite) positive number. Consider M as an algebraic symbol involved in the operation and solve it by the simplex method.

Expanded:

In the process of improving the objective function by the simplex method, if there is an optimal solution to the original problem, it is inevitable that the artificial variables will be gradually changed to non-basic variables, or their values will be zero. It will be impossible to minimize or maximize the objective function value.

In the iterative process, if all the artificial variables become non-basic variables, the artificial variables in the column can be deleted from the simplex table, this time to find an initial base feasible solution to the original problem. If this base feasible solution is not the optimal solution of the original problem, then continue to iterate until all the tests are less than or equal to 0, the optimal solution.