Heuristic Solution of the MultisourceWeber Problem as a p - Median

Good heuristic solutions for large Multisource Weber problems can be obtained by solving related p-median problems in which potential locations of the facilities are users locations and then solving Weber problems for the sets of users of each facility. Weber's problem is to locate a facility in the Euclidean plane in order to minimize the sum of its (weighted) distances to the locations of a given set of users. It is central to continuous location theory, and has been extended in many ways, see Wesolowsky (1993) for a recent survey. Weiszfeld's (1937) iterative algorithm, extended to ensure convergence even if an iterate coincides with a user's location (e.g. Brimberg and Love (1993)), is the standard way to solve Weber's problem. Rosen and Xue (1991) suggest accelerations. Considering simultaneous location of several facilities leads to two generalizations: (i) the Multifacility Weber problem in which several facilities each producing a diierent product (or rendering a diierent service) are to be located in order to minimize the sum of weighted distances between all facilities and all users as well as between the facilities, and (ii) the Multisource Weber problem in which several facilities producing the same product are to be located in order to minimize the sum of weighted distances from all users to their closest facility. While the former problem is one of convex optimization for which practically eecient algorithms (e.g.