A Note on Spanning Trees for Network Location Problems

A p-facility location problems on a network N consist of locating p new facilities on N such that some function of distances among them and vertices of N is minimum. We consider a class of such problems where objective function is nondecreasing in distance. The median, the center and the centdian problems belong to this class. We prove that the optimal solution on the network and on the corresponding spanning tree are equal. Since location problems on tree network are easier to solve than on general network, we propose a descent local search heuristic that solve optimaly the problem on spanning tree in each iteration. Facility location analysis deals with the problem of locating one or several new facilities with respect to existing facilities (clients, users or demand points) in order to optimize some economic criteria (for introduction in location analysis see for example [9]). Examples of facilities are plants, warehouses, schools, hospital, administrative buildings, departure stores, waste material dumps, ambulance or fire engine depots, etc. The economic criteria is usually obtained by a objective function that decreases when the distance from the facilities to the clients decrease. Network location problems occur when new facilities are to be located on a network (see [5]). The network of interest may be a road network, an air transport network, a river network, or a network of computers. For a given network location problem, the network is represented by a graph and the new and existing facilities are often idealized as points. The demand points are generally taken to be at the vertices of the graph. A special case arises when the underlying network has a tree structure (a connected graph without a circuit). Most of the problems are easier to solve on tree than on general graph (see [12] for example. The efficiency of the algorithms is due primarily to the general convexity properties of tree networks [4]. The algorithms for tree networks may be applied to a general networks if the problem can be decomposed into subproblems on trees or if the graph has few cycles and the algorithms can be modified for nearly acyclic networks (many rural networks in a bounded region are nearly acyclic). ∗The research has been partially supported by the DGICYT project PB95-1237-C03-02. †D.E.I.O.C. University of La Laguna, Tenerife, Spain ‡G.E.R.A.D. University of Montreal. Montreal, Canada §D.E.I.O.C. University of La Laguna, Tenerife, Spain