Ranking Decomposition for the Discrete Ordered Median Problem

Given a set $\mathcal{N}$ of size n, a nonnegative, integer-valued distance matrix D of dimensions $n\times n$, an integer $p\in \mathbb{N}$ and an integer-valued weight vector $\mathit{\lambda }\in {\mathbb{Z}}^n$, the discrete ordered median problem (DOMP) consists of selecting a subset $\mathcal{C}$ of exactly p points from $\mathcal{N}$ (also referred to as the centers) so as to: 1) assign each point in $\mathcal{N}$ to its closest center in $\mathcal{C}$; 2) rank the resulting distances (between every point and its center) from smallest to largest in a sorted vector that we denote $d^{*}$; 3) minimize the scalar product $〈\mathit{\lambda },d^{*}〉$. The DOMP generalizes several classical location problems such as the p-center, the p-median and the obnoxious median problem. We introduce an exact branch-and-bound algorithm to solve the DOMP. This branch-and-bound decouples the ranking attribute of the problem to form a series of simpler subproblems which are solved using innovative binary search methods. We consider several acceleration techniques such as warm-starts, primal heuristics, variable fixing, and symmetry breaking. We perform a thorough computational analysis and show that the proposed method is competitive against several MIP models from the scientific literature. We also comment on the limitations of our method and propose avenues of future research.

History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete.

Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grants 2017-06106, 2020-06311, and 2021-03327].

Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0059) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2023.0059). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/.