## 2 Background  
The H&S design problem is widely known as the hub location problem (HLP) and refers to determining the location of hub facilities and allocating the radial spoke nodes within a network. Depending on the problem formulation, H&S models can be classified in different respects as follows. Initially, the problem can be classed as either single-hub, where only one hub node is to be allocated, or multiple-hub, if more than one hub node exists. Another classification refers to the cases in which a spoke is connected to only one hub (single-allocation) or many hubs (multiple-allocation). An additional categorization is made on how the number of hubs is set, either as a predetermined planning decision (exogenous model) or as a result of the analysis process (endogenous model). In the former case, the problem is known as the p-hub location problem, where p is the predetermined number of hubs. A particular case is the p-hub median problem, in which the hub nodes are fully interconnected, and the traveling cost across the interhub link is assumed to be zero to drop out the quadratic term. Another parameter that differentiates the analysis is related to the capacity level of hub nodes serving incoming and outgoing flows. If a capacity limitation is in effect, the H&S problem is referred to as capacitated; otherwise, it is called uncapacitated. Finally, the H&S formulation can be classified based on the analysis objectives. In particular, the general p-hub location problem aims to minimize the cumulative cost of the transport carrier among all origin-destination pairs. A typical competing objective, which leads to a denser cluster, is to minimize the maximum distance (or cost) among all individual origin-destination pairs, and, in this case, the problem falls within the hub-center location category. An extension to the cumulative cost objective formulation is the hub-covering location problem, in which a constraint for the maximum allowable distance/cost is set for any (or specific) origin-destination pair. In the early research development phase for this problem, researchers have primarily focused on developing mathematical model formulations for the basic H&S problem. Within the next phase, studies have emphasized enriching and optimizing the models to reduce the complexity and reflect some realistic aspects of the scheme. As computational capabilities grew, several algorithms and techniques were proposed for solving large-scale H&S optimization problems. Nowadays, research goes on unabated considering additional parameters and directions, such as uncertainty, congestion limitations, and environmental aspects. A detailed presentation of existing studies is provided below. [PERSON] was the pioneer in hub-location network structure research [3; 4; 5], proposing a quadratic integer programming approach for the hub-median problem along with introducing fixed costs for defining the number of hubs as one of the decision parameters [6]. Since then, the topic has attracted the scientific community's interest because of its high added value in real-life implementation, and numerous studies have been conducted to tackle different aspects of the problem. [PERSON] [7] presented an early linear integer programming formulation for the single allocation p-hub median problem and later [8] proposed integer programming methodologies for four different hub location problems. [PERSON] et al. [9] tested [PERSON]'s model and stated that the LP relaxation results in highly fractional solutions. [PERSON] and [PERSON] [10] presented a linear programming formulation for the single-allocation p-hub median problem and a heuristic algorithm based on simulated annealing for solving the problem, and later [11], they extended their work on the scheme by proposing several mixed-integer linear programming (MILP) formulations based on the shortest paths. The same researchers [12] also introduced a heuristic algorithm based on simulated annealing (SA) and random descent (RDH) to solve the capacitated single-allocation hub location problem.