In this paper we propose a general approach solution method for the single facility ordered median problem in the plane. All types of weights (non-negative, non-positive, and mixed) are considered. The big triangle small triangle approach is used for the solution. Rigorous and heuristic algorithms are proposed and extensively tested on eight different problems with excellent results.

We consider a variant of the classical k-median problem, introduced by Anthony et al. [1]. In the Robust k-Median problem, we are given an n-vertex metric space (V, d) and m client sets {Si ⊆ V }mi=1. We want to open a set F ⊆ V of k facilities such that the worst case connection cost over all client sets is minimized; that is, minimize maxi ∑ v∈Si d(F, v). Anthony et al. showed an O(logm) appr...

The optimization problem that arises out of the least median of squared residuals method in linear regression is analyzed. To simplify the analysis, the problem is replaced by an equivalent one of minimizing the median of absolute residuals. A useful representation of the last problem is given to examine properties of the objective function and estimate the number of its local minima. It is sho...

The polyhedral structure of the K-median problem on a tree is examined. Even for very small connected graphs, we show that additional constraints are needed to describe the integer polytope. A complete description is given of those trees for which an optimal integer LP solution is guaranteed to exist. We present a new and simpler demonstration that an LP characterization of the 2-median problem...

Where to locate one or several facilities on a network so as to minimize the expected users-closest facility transportation cost is a problem well studied in the OR literature under the name of median problem. In the median problem users are usually identified with nodes of the network. In many situations, however, such assumption is unrealistic, since users should be better considered to be di...

A set of vertices S in a graph is called geodetic if every vertex of this graph lies on some shortest path between two vertices from S. In this paper, minimum geodetic sets in median graphs are studied with respect to the operation of peripheral expansion. Along the way geodetic sets of median prisms are considered and median graphs that possess a geodetic set of size two are characterized. © 2...

The diff-convex (DC) problem of perturbed spatial median and the Weiszfeld algorithm in a framework for incomplete data is studied, and some level set theorems for general DC problems are provided. These results are then applied to study certain multiobjective formulations of clustering problems, and to yield a new algorithm for solving the multisource Weber problem.