Dual Algorithms for Pure Network Problems

This paper reports the development of a new algorithmic implementation of the dual algorithm for the capacitated minimum cost network flow problem. Furthermore, it introduces dual reoptimization procedures and compares primal and dual algorithms for the optimization and reoptimization of network problems. The implementation makes use of cut-sets for the efficient execution of the entering variable selection and the selection of the leaving variable is tied to the basis structure at an iteration. Empirical testing of the dual algorithm for optimization shows that it dominates the primal procedure for assignment problems with 400 nodes, transportation problems with more than 600 nodes, and transshipment problems with more than 1500 nodes. Computational testing with typical changes found in decomposition and relaxation procedures indicates the superiority of dual reoptimization over primal reoptimization. For a sequence of parametric changes, as would be typical in large-scale programming techniques, on the average, dual reoptimization is found to require between 75 and 93% fewer pivots and between 20 and 50% less time than primal reoptimization depending on the type of change.