One-Pass Algorithms for Some Generalized Network Problems

The generalized network problem and the closely related restricted dyadic problem are two special model types that occur frequently in applications of linear programming. Although they are next in order after pure network or distribution problems with respect to ease of computation, the jump in degree of difficulty is such that, in the most general problem, there exist no algorithms for them comparable in speed or efficiency to those for pure network or distribution problems. There are, however, numerous examples in which some additional special structure leads one to anticipate the existence of algorithms that compare favorably with the efficiency of those for the corresponding pure cases. Also, these more special structures may be encountered as part of larger or more complicated models. In this paper we designate by topological properties two special structures that permit evolution of efficient algorithms. These follow by extensions of methods of Charnes and Cooper and of Dijkstra for the corresponding pure network problems. We obtain easily implemented algorithms that provide an optimum in one “pass” through the network. The proofs provided for these extended theorems differ in character from those provided (or not provided) in the more special “pure” problem algorithms published.