A General Pricing Scheme for the Simplex Method

Abstract

In the simplex method for linear programming the algorithmic step of checking the reduced costs of nonbasic variables is called the “pricing” step. If these reduced costs are all of the “right sign” the current basis (and solution) is optimal, if not, this procedure selects a candidate vector that looks profitable for inclusion in the basis. While theoretically the choice of any profitable vector will lead to a finite termination (provided degeneracy is handled properly) but the number of iterations until termination depends very heavily on the actual choice (which is defined by the selection rule applied). Pricing has long been an area of heuristics to help make better selection. As a result, many different and sophisticated pricing strategies have been developed, implemented and tested. So far none of them is known to be dominating all others in all cases. Therefore, advanced simplex solvers need to be equipped with many strategies so that the most suitable one can be activated for each individual problem instance. In this paper we present a general pricing scheme. It creates a large flexibility in pricing. It is controlled by three parameters. With different settings of the parameters many of the known strategies can be reproduced as special cases. At the same time, the framework makes it possible to define new strategies or variants of them. The scheme is equally applicable to general and network simplex algorithms.