Bregman's balancing method

Abstract

In this paper we observe that most of the independently discovered balancing methods, used in transportation planning and in other fields, are in fact special cases of a method of Bregman. Examples include the usual Kruithof or Furness method, the Evans-Kirby three dimensional balancing procedure, the Murchland multiproportional balancing procedure, the Osborne or Grad method for preconditioning matrices, the Jefferson-Scott procedure for gravity models with inequality constraints, and the method considered by Macgill for partially constrained gravity models. The convergence of all of these methods follows from Bregman's general result.