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）。OMT是一种单分配设施选址问题，要求将p个设施置于由无向树连接的网络中。其目标是最小化有序加权平均分配成本之和与树中设施连接成本之和。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OMT的多种混合整数线性规划（MILP）公式。考虑到有序中位数枢纽选址问题求解难度较大，我们通过引入覆盖变量构建有效重构公式，并开发了两个预处理阶段以缩减该公式的规模，从而提升OMT求解性能。此外，我们提出了一种Benders分解算法来求解OMT。我们通过实证比较了这些新公式的性能，并提出了增强策略，结合适当的公式可求解一般随机图上的中等规模实例。