The maximum utilization subtree problem

A number of network design problems can be built on the following premise: Given a tree network, T, containing node set, V, identify a single subtree, t, containing nodes, v, so that the subtree is located optimally with respect to the remaining, unconnected nodes {V-v}. Distances between unconnected nodes and nodes in the subtree t can be defined on travel paths that are restricted to lie in the larger tree T (the travel-restricted case), or can be defined on paths in an auxiliary complete graph G (the travel-unrestricted case). This paper presents the Maximum Utilization Subtree Problem( MUSP), a bicriterion problem that trades off the cost of a subtree, t, against the utilization of the subtree by the sum of the populations at nodes connected to the subtree, plus the distance-attenuated population that must travel to the subtree from unconnected nodes. The restricted and unrestricted cases are formulated as a two objective integer programs where the objectives are to maximize utilization of the subtree and minimize the cost of the subtree. The programs are tested using linear programming and branch and bound to resolve fractions. The types of problems presented in this paper have been characterized in the existing literature as "structure location" or "extensive facility location" problems. This paper adds two significant contributions to the general body of location literature. First, it draws explicit attention to the travel-restricted and travel-unrestricted cases, which may also be called "limited-access" and "general-access" cases, respectively. Second, the distance-attenuated demands represent a new objective function concept that does not appear in the location literature.