The ordered median tree of hubs location problem

Abstract

In this paper, we propose the Ordered Median Tree of Hub Location Problem (OMTHL). The OMTHL is a single-allocation hub location problem where p hubs must be placed on a network and connected by a non-directed tree. Each non-hub node is assigned to a single hub and all the flow between origin-destination pairs must cross at least one hub. The objective is to minimize the sum of the ordered weighted averaged collection and distribution costs plus the sum of the interhub flow costs. We present different MILP formulations for the OMTHL based on the properties of the Minimum Spanning Tree Problem, the ordered median optimization and on the different ways of modeling flow within the network. Given that ordered median hub location problems are rather difficult to solve, we have improved the OMTHL solution performance by introducing covering variables in two valid reformulations. In addition, we have developed two pre-processing phases to reduce the size of these formulations. 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.

An extended version of the Ordered Median Tree Location Problem including appendices and detailed computational results

“…These costs are assumed to be symmetrical between each pair of nodes, = for , ∈ , and nodes can be allocated to themselves with no cost, = 0 for each ∈ , that is the so called free self-service assumption. In addition, the allocation costs of the | | clients to their corresponding facilities are compensated using scaling factor parameters = ( 1 , … , | | ) (see Nickel and Puerto, 2006;Boland et al, 2006;Kalcsics et al, 2010b;Marín et al, 2009Marín et al, , 2010Puerto et al, 2011Puerto et al, , 2013Puerto et al, , 2016Pozo et al, 2021 for different ordered median location models). If client is allocated to facility at a cost and this cost is non-decreasing ordered in the -th position among this type of costs, then this term would be scaled by , i.e., the corresponding compensated cost would be .…”

“…The reader is referred to Figure 2 for an overview of the different considered problems and their relationships. In addition, the OMT can be understood as a subproblem of the Ordered Median Tree of Hubs Location Problem (OMTHL) (Pozo et al, 2021), where flow exchange between origin-destination pairs is considered for an OMT. The interested reader may also check the tree of hubs location problem (Contreras et al, 2010) and the ordered median hub location problem (Puerto et al, 2011).…”

“…Several papers and surveys cover different aspects of the ConnFl and new cases of study based on this problem (see, e.g, Fortz, 2015;Ljubić, 2021). As detailed in Gollowitzer and Ljubić (2011), ConnFl is also related to other problems in the literature, as for example rent-or-buy problems (Kumar et al, 2002), Steiner tree-star problem (Lee et al, 1993;Khuller and Zhu, 2002), general connected facility location problem (Bardossy and Raghavan, 2010), prize collecting capacitated connected facility location problem (Leitner and Raidl, 2011) or the tree of hubs location problem (Contreras et al, 2010;Pozo et al, 2021).…”

“…DOMP objective function have been successfully applied in the field of location analysis and distribution models (see, e.g., Puerto and Tamir, 2005;Boland et al, 2006;Puerto, 2008;Marín et al, 2009;Kalcsics et al, 2010a;Puerto and Rodríguez-Chía, 2015). DOMP structure has also been embedded within other well-known problems giving rise to, e.g., ordered weighted average problems (see Galand and Spanjaard, 2012;Fernández et al, 2014Fernández et al, , 2017 or ordered hub location problems (see Puerto et al, 2011Puerto et al, , 2013Puerto et al, , 2016Pozo et al, 2021).…”

On Multistage Multiscale Stochastic Capacitated Multiple Allocation Hub Network Expansion Planning

“…The HLP-Ordered Median Tree problem (HLP-OMTP) is presented in [29] in its singleallocation unconstrained version, where each network node is only allocated to one hub, and a given number of hubs should be located. The objective function to minimize is the weighted average cost of the ordered flow collection and distribution, as well as the hub edges flow.…”