Twelve: Maximal Flow, Shortest Path, Multicommodity Flow, and Network Synthesis Problems

Two special and important network flow problems are the maximal flow problem and the shortest path problem. Both of these problems can be solved by either the network simplex method of Chapter 9 or the out-of-kilter algorithm of Chapter 11. However, their frequent occurrence in practice and the specialized, more efficient procedures that can be developed for handling these two problems provide a strong case for considering them separately. We also include in this chapter an introduction to the class of network flow problems called multicommodity network flows. In Chapters 9, 10, and 11 we have considered network flow problems in which it was not necessary to distinguish among the units flowing in the networks. There was essentially a single commodity or type of unit. There are network flow problems in which different types of units must be treated. In these instances, supplies and demands are by commodity type, and the distinction among the commodities must be maintained. We shall examine this multicommodity flow problem, consider the difficulty in dealing with it, and present a decomposition-based procedure for solving it. Another topic that we introduce is network synthesis. The network flow problems considered thus far are analysis problems. A network is given, and some analysis is performed on it. On the other hand, the network synthesis problem requires one to construct an optimal network (in some defined sense) that satisfies certain specifications. Several of the tools developed in this book, including network analysis and decomposition methods, can be used to study such problems.