The bi-criteria doubly weighted center-median path problem on a tree

Given a tree network T with n nodes, let P _L be the subset of all discrete paths whose length is bounded above by a prespecified value L. We consider the location of a path-shaped facility P ε P _L, where customers are represented by the nodes of the tree. We use a bicriteria model to represent the total transportation cost of the customers to the facility. Each node is associated with a pair of nonnegative weights: the center-weight and the median-weight. In this doubly weighted model, a path P is assigned a pair of values (MAX (P), SUM (P)), which are, respectively, the maximum center-weighted distance and the sum of the median-weighted distances from P to the nodes of the tree. Viewing P _L and the planar set {(MAX(P), SUM(P)) : P ε P _L} as the decision space and the bi-criteria or outcome space respectively, we focus on finding all the nondominated points of the bi-criteria space. We prove that there are at most 2n nondominated outcomes, even though the total number of efficient paths can be Ω(n ²), and they can all be generated in O(n log n) optimal time. We apply this result to solve the cent-dian model, whose objective is a convex combination of the weighted center and weighted median functions. We also solve the restricted models, where the goal is to minimize one of the two functions MAX or SUM, subject to an upper bound on the other one, both with and without a constraint on the length of the path. All these problems are solved in linear time, once the set of nondominated outcomes has been obtained, which in turn, results in an overall complexity of O(n log n). The latter bounds improve upon the best known results by a factor of O(log n).