In this paper, we propose the Ordered Median Tree Location Problem (OMT). The OMT is a single-allocation facility location problem where p facilities must be placed on a network connected by a non-directed tree. The objective is to minimize the sum of the ordered weighted averaged allocation costs plus the sum of the costs of connecting the facilities in the tree. We present different MILP formulations for the OMT based on properties of the minimum spanning tree problem and the ordered median optimization. Given that ordered median hub location problems are rather difficult to solve we have improved the OMT solution performance by introducing covering variables in a valid reformulation plus developing two pre-processing phases to reduce the size of this formulations. In addition, we propose a Benders decomposition algorithm to approach the OMT. We establish an empirical comparison between these new formulations and we also provide enhancements that together with a proper formulation allow to solve medium size instances on general random graphs.

翻译：本文提出有序中位数树选址问题（OMT）。OMT是一种单分配设施选址问题，需将p个设施部署于由无向树连接的网络中。目标是最小化有序加权平均分配成本与设施间树形连接成本之和。我们基于最小生成树问题和有序中位数优化的特性，提出了OMT的多种混合整数线性规划（MILP）模型。鉴于有序中位数枢纽选址问题的求解难度较高，我们通过引入覆盖变量进行有效重构，并开发了两种预处理阶段以缩小模型规模，从而提升了OMT的求解性能。此外，我们提出了一种Benders分解算法用于求解OMT。通过对这些新模型进行实证对比，并结合适当的建模优化，我们实现了在一般随机图上对中等规模实例的有效求解。