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）。该问题为单分配设施选址问题，需将p个设施布置于由无向树连接的网络上。目标是最小化有序加权平均分配成本与树中设施连接成本之和。我们基于最小生成树问题与有序中位数优化的性质，提出该问题的多种混合整数线性规划（MILP）模型。针对有序中位数枢纽选址问题求解困难的特点，通过在有效重构中引入覆盖变量并开发两阶段预处理缩减模型规模，从而提高OMT求解性能。此外，我们提出邦德斯分解算法逼近OMT解。通过对新模型开展实证比较，并结合增强策略与恰当模型形式，能够在通用随机图上求解中等规模实例。