A Treatment of Transportation Problems by Primal Partition Programming

J. B. Rosen's Primal Partition Programming method [Rosen, J. B. 1964. Primal partition programming for block diagonal matrices. Numerische Mathematik 6.] is applied to the classical transportation problem and to some related generalizations or extensions. Observations on the special structure of the constraint matrix result in a subproblem which may be solved by inspection, a unimodular master inverse of order m and in a considerable reduction in the number of vectors spanning the dual space of the master problem. This inverse and master problem may be replaced by a directed network of at most m(m - 1) arcs and m nodes, where m is the number of sources. Dual feasibility is maintained throughout the optimization procedure. The generalizations and extensions treated are: (i) The upper bounded transportation problem, (ii) the transportation problem in which the source availabilities are subject to some general linear constraints and, (iii) the parametric transportation problem. Programming of the method for a digital computer is quite simple and requires only integer arithmetic (except for (ii)). Computing efficiency compares quite well with the specialization of the Dantzig-Wolfe Decomposition method to transportation problems [Williams, A. C. 1962. A treatment of transportation problems by decomposition. J. Soc. Indust. Appl. Math. 10(1, March).] and other methods of solution.