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：本文提出了有序中位数树选址问题（OMT）。该问题是一种单分配设施选址问题，需在由无向树连接的网络上部署p个设施，目标是最小化有序加权平均分配成本与设施间连接成本之和。我们基于最小生成树问题与有序中位数优化的性质，提出了OMT的多种混合整数线性规划（MILP）公式。由于有序中位数枢纽选址问题的求解难度较大，我们通过引入覆盖变量构建有效重构公式，并开发两个预处理阶段以缩减公式规模，从而提升OMT的求解性能。此外，我们提出了一种Benders分解算法来求解OMT。通过对比实验验证了新公式的有效性，并提出了增强策略，结合恰当的公式化表达，可实现对一般随机图的中等规模实例的求解。