Implementation of a Double-Basis Simplex Method for the General Linear Programming Problem

Abstract

The basis handling procedures of the simplex method are formulated in terms of a “double basis.” The basis is expressed as a matrix product, one of the factors being the basis matrix of the last refactorization. Forward and backward transformations and update are presented for each of two implementations of the double-basis method. The double-basis update is restricted to a matrix of dimension limited by the refactorization frequency and two permutation matrices. This can lead to a saving in storage space and updating time. The cost is that the time for the forward and backward transformations is about double. Computational comparisons of storage and speed are made with the standard simplex method on problems of up to 1,480 constraints. Generally, the double-basis method performs best on larger, denser problems. Density seems to be the more important factor, and the problems with large nonzero growth between refactorizations are the better ones for the double-basis method. Storage saving in the basis inverse representation versus the standard method is as high as 36%, whereas the double-basis run times are 1.2 or more times greater.