A basis-deficiency-allowing variation of the simplex method for linear programming

As one of the most important and fundamental concepts in the simplex methodology, basis is restricted to being a square matrix of the order exactly equal to the number of rows of the coefficient matrix. Such inflexibility might have been the source of too many zero steps taken by the simplex method in solving real-world linear programming problems, which are usually highly degenerate. To dodge this difficulty, we first generalize the basis to allow the deficient case, characterized as one that has columns fewer than rows of the coefficient matrix. Variations of the primal and dual simplex procedures are then made, and used to form a two-phase method based on such a basis, the number of whose columns varies dynamically in the solution process. Generally speaking, the more degenerate a problem to be handled is, the fewer columns the basis will have; so, this renders the possibility of efficiently solving highly degenerate problems. We also provide a valuable insight into the distinctive and favorable behavior of the proposed method by reporting our computational experiments.