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。该问题属于单分配设施选址问题，要求在由无向树连接的网络上布置p个设施，目标函数是最小化有序加权平均分配成本与设施间树状连接成本的总和。基于最小生成树问题与有序中位数优化的性质，我们提出了多种混合整数线性规划模型。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构建有效重构模型，并开发两个预处理阶段以缩减模型规模，从而提升了有序中位数树选址问题的求解性能。此外，我们设计了Benders分解算法来求解该问题。本文通过实证研究对比了这些新模型，并提出多种改进策略；结合适当的模型构建，这些方法能够求解一般随机图上的中等规模算例。