Heuristic Methods for Estimating the Generalized Vertex Median of a Weighted Graph

The generalized vertex median of a weighted graph may be found by complete enumeration or by some heuristic method. This paper investigates alternatives and proposes a method that seems to perform well in comparison with others found in the literature. THE PROBLEM of supplying some number of destinations, n, from a number of sources, p, has been attacked with a variety of assumptions and methods. If both sources and destinations are at fixed locations with given quantities available and required, the standard transportation problem of linear programming appears. With given locations for destinations, the determination of locations of sources in a euclidean plane may be called the generalized Weber problem, after the nineteenth century student of industrial location who examined this problem for the special case of the single location. Operational generalizations of this form of the problem have been investigated by COOPER 1 2, 31 using iterative approximation methods that are appropriate where the locations for sources may be treated as continuous in the plane. If destinations consist of fixed nodes on a network, but sources may lie anywhere on network links and destination demands are fixed while source capacities are unconstrained, it has been shown by HAKIMI14, 6 that the problem resolves itself into finding the generalized absolute median of the weighted shortest path graph corresponding to the network. Hakimi also demonstrated that there will exist such a generalized p-median that includes only vertices of the graph, that is, nodes on the network. Thus a solution to this problem will correspond to the case where both destinations and sources lie on nodes of a network, a situation like that investigated for fixed destination demands and unconstrained source capacity by MARANZANA.161 In this paper we deal with the problem of choice of location of p sources of unconstrained capacity from among n destinations having fixed demands and located at nodes of a network. The problem is essentially the same as