Optimal flows in networks with multiple sources and sinks

In their famous book, Ford and Fulkerson [ 3 ] state that the situation in which there are multiple sources and sinks, with flow permitted from any source to any sink, presents nothing new. They claim that a multiple source, multiple sink network reduces to a single source, single sink network by the adjunction of a supersource, a supersink, and several arcs. That is of course true if only a maximization of the total flow through the network is desired. However, under certain circumstances, especially in economical applications, one wishes not only to maximize the total flow but also to distribute it "fairly" among the sinks or the sources. For instance, it might be desirable to maximize the minimum amount supplied to individual sinks ' or, alternatively, delivered from individual sources. On the other hand, the labelling method for solving maximal flow problems [21 very often generates "unfair" flows, in the sense that most of the flow comes from one source, or goes to one sink. In fact, this is the nature of the labelling method.