ABSTRACT This paper studies a two‐stage adaptive robust hub location problem with multiple assignments under demand uncertainty. In our setting, the capacitated hubs are strategically located in the first stage to minimize the worst‐case scenario cost over a budgeted uncertainty set, and the routing decisions, which are adaptive to uncertainty realizations, are made in the second stage to transport all commodities. For the large‐scale instances of the problem, we develop a novel column‐and‐constraint generation approach that integrates Benders decomposition. In this approach, we design a customized Benders decomposition to efficiently solve the master problem involving a subset of uncertain scenarios, in which a tailored cutting plane algorithm is developed to solve Benders dual subproblems and a cut refinement strategy is proposed to generate strong Benders cuts. Besides, to quickly identify possible uncertain scenarios, we reformulate the second‐stage problem into a more tractable form, which is further simplified by significantly reducing the number of redundant variables and constraints. Extensive computational experiments on the well‐known instances with up to 200 nodes are conducted to evaluate the effectiveness of proposed model and the performance of the solution method. The computational results demonstrate that our developed solution method outperforms the conventional column‐and‐constraint generation or Benders decomposition. Compared with the two‐stage stochastic programming model, our proposed model can provide more reliable and robust solutions with superior out‐of‐sample performance. The results also illustrate the effect of uncertainty budgets and highlight the advantages of incorporating features such as hub capacity and multiple assignments into our model. Moreover, we make extensions by applying our framework to handle more general polyhedral uncertainty sets.

Containers account for 16% of all tonnage transported by sea. Therefore, the efficient repositioning of empty containers is a necessary, albeit often unprofitable, activity. This process supports both the local and global efficiency of container transport, especially given the significant imbalance in container flows from Asia to Europe. The transport sector is responsible for 21.2% of total CO2 emissions, making Empty Container Repositioning (ECR) crucial from an environmental perspective. Efficient repositioning and the identification of an optimal location for an empty container depot are crucial problems that can be effectively solved within a hub system. The aim is to find solutions that minimize the overall impact on the environment while reducing holding and travel costs. This paper proposes a robust and feasible method for determining the optimal location for an inland empty container depot, demonstrating its effectiveness on a large-scale logistics hub location problem in Slovenia. The proposed method takes into account both the economic costs and the emissions generated during the repositioning process and aims to select a hub location that minimizes these factors. The optimization is performed using a heuristic approach, Particle Swarm Optimization (PSO), which requires fewer constraints and assumptions compared to traditional linear optimization techniques. This method takes advantage of the behavior of swarms and their ability to find optimal solutions to complex problems. In contrast to classical approaches such as linear programming, which require considerable effort to formulate and solve, this method simplifies the process and efficiently identifies optimal solutions.

This paper introduces an innovative approach for constructing large-scale, realistic benchmark geospatial datasets specifically designed for the single-allocation hierarchical hub median problem, a key variant of the hierarchical hub location problem. Addressing the lack of publicly available large-scale datasets in location science, we have provided a detailed step-by-step methodology to generate benchmark datasets. Also, we have provided datasets of various granularities based on Kolkata and Mumbai, two of India’s most populous metropolitan cities as a ready reference. The datasets are built using actual building-level geographic data acquired from OpenStreetMap and QGIS. Benchmark datasets are solved for the single-allocation hierarchical hub median problem formulation using IBM ILOG CPLEX to provide exact solutions, supporting robust empirical validation. The datasets are fully reproducible, scalable, and adaptable to various hierarchical hub location problem formulations and urban planning scenarios. By making these resources publicly available, this work fills a significant research gap and offers a foundational platform for algorithmic benchmarking and future methodological innovations in hierarchical hub location studies.

Exact solution method for stochastic single-allocation hub location problems

An approximation algorithm for multi-allocation hub location problems

This paper studies a two-stage decision-making framework for the profit-maximising hub location problem under uncertain transportation costs, wherein the first stage, the optimal location of hubs, is determined, and in the second stage, flow routeing decisions are made. We assume that while the true probability distribution of uncertain parameters might be unknown, the decision maker has access to a set of historical data from which a prior belief is formed about the underlying distribution. We propose to construct a set of probability distributions around this belief in the sense of Wasserstein distance. Then, we study a data-driven distributionally robust optimisation (DRO) approach to the studied problem to maximise the worst-case profit over the set of candidate probability distributions. We propose the Benders' decomposition and L-shaped algorithms to solve the studied problem. Numerical results show that the L-shaped algorithm reaches optimal solutions with less computation effort when compared to the Benders' decomposition algorithm and an off-the-shelf solver. Moreover, the Wasserstein DRO approach effectively enhances the out-of-sample performance compared to deterministic and stochastic programming approaches. Finally, the Wasserstein DRO approach leads to a more resilient network by establishing more hubs and links in the worst-case scenario compared to other approaches.

Single allocation hub location problems with congestion: Mixed-integer second-order cone programming and Benders decomposition

Stochastic Profit Maximization and Pricing in Hub Location Problems with Elastic Demand: Mathematical Formulations and Exact Algorithms

Aiming for convenience and the low cost of goods transfer between towns, this paper proposes a trade hub location and allocation method based on a novel artificial eagle-inspired optimization algorithm. Firstly, the trade hub location and allocation model is established, taking the total cost consisting of construction and transportation costs as the objective function. Then, to solve the nonlinear model, a novel artificial eagle optimization algorithm (AEOA) is proposed by simulating the collective migration behaviors of artificial eagles when facing a severe living environment. Three main strategies are designed to help the algorithm effectively explore the decision space: the situational awareness and analysis stage, the free exploration stage, and the flight formation integration stage. In the first stage, artificial eagles are endowed with intelligent thinking, thus generating new positions closer to the optimum by perceiving the current situation and updating their positions. In the free exploration stage, artificial eagles update their positions by drawing on the current optimal position, ensuring more suitable habitats can be found. Meanwhile, inspired by the consciousness of teamwork, a formation flying method based on distance information is introduced in the last stage to improve stability and success rate. Test results from the CEC2022 suite indicate that the AEOA can obtain better solutions for 11 functions out of all 12 functions compared with 8 other popular algorithms. Faster convergence speed and stronger stability of the AEOA are also proved by quantitative analysis. Finally, the trade hub location and allocation method is proposed by combining the optimization model and the AEOA. By solving two typical simulated cases, this method can select suitable hubs with lower construction costs and achieve reasonable allocation between hubs and the rest of the towns to reduce transportation costs. Thus, it is used to solve the trade hub location and allocation problem of Henan province in China to help the government make sound decisions.

This paper presents an enhanced hub location model tailored to port–hinterland logistics planning, grounded in the Capacitated Multiple-Allocation Hub Location Problem (CMAHLP). The formulation incorporates nonlinear cost structures, hub-specific operating costs, adaptive capacity constraints, and a feasibility condition based on the Social Net Present Value (NPVsocial) to support the design of intermodal freight networks under asymmetric spatial and socio-environmental conditions. The empirical case focuses on Spain, leveraging its strategic position between Asia, North Africa, and Europe. The model includes four major ports—Barcelona, Valencia, Málaga, and Algeciras—as intermodal gateways connected to the 47 provinces of peninsular Spain through calibrated cost matrices based on real distances and mode-specific road and rail costs. A Genetic Algorithm is applied to evaluate 120 scenarios, varying the number of active hubs (4, 6, 8, 10, 12), transshipment discounts (α = 0.2 and 1.0), and internal parameters. The most efficient configuration involved 300 generations, 150 individuals, a crossover rate of 0.85, and a mutation rate of 0.40. The algorithm integrates guided mutation, elitist reinsertion, and local search on the top 15% of individuals. Results confirm the central role of Madrid, Valencia, and Barcelona, frequently accompanied by high-performance inland hubs such as Málaga, Córdoba, Jaén, Palencia, León, and Zaragoza. Cities with active ports such as Cartagena, Seville, and Alicante appear in several of the most efficient network configurations. Their recurring presence underscores the strategic role of inland hubs located near seaports in supporting logistical cohesion and operational resilience across the system. The COVID-19 crisis, the Suez Canal incident, and the persistent tensions in the Red Sea have made clear the fragility of traditional freight corridors linking Asia and Europe. These shocks have brought renewed strategic attention to southern Spain—particularly the Mediterranean and Andalusian axes—as viable alternatives that offer both geographic and intermodal advantages. In this evolving context, the contribution of southern hubs gains further support through strong system-wide performance indicators such as entropy, cluster diversity, and Pareto efficiency, which allow for the assessment of spatial balance, structural robustness, and optimal trade-offs in intermodal freight planning. Southern hubs, particularly in coordination with North African partners, are poised to gain prominence in an emerging Euro–Maghreb logistics interface that demands a territorial balance and resilient port–hinterland integration.

Abstract The tree‐of‐hubs location problem (THLP) is a variant of the classical hub location problem, in which the set of hubs must be connected in a tree topology. The key decision variables involve the selection of hub locations, the allocation of spokes to hubs, and the design of a tree‐structured interhub network. In this paper, we introduce a new extension of the THLP that incorporates stopover nodes—intermediate locations situated along the paths between hubs. The inclusion of stopovers helps reduce transportation costs by minimizing unnecessary back‐and‐forth trips and limiting transshipments at hubs. However, this benefit comes at the expense of potential detours in the interhub connections. We propose a mixed‐integer linear programming (MILP) formulation to model this problem, with an objective function that minimizes the total cost of transporting commodities between multiple origins and destinations. Computational experiments are conducted using adapted instances from the AP‐200 dataset. We also perform a sensitivity analysis on the discount factors associated with the use of stopovers. The results show that stopovers can lead to logistics cost savings of up to 15%. In addition, we provide managerial insights and quantify the impact of stopovers on network design costs by comparing solutions with and without their inclusion. Finally, we discuss the conditions under which stopovers can be effectively leveraged in practice.

The intermodal express network of high-speed railways and highways can fully utilize the flexibility of highways and the advantages of high-speed railways, such as low cost, high efficiency, and low carbon emission. This paper studies the hub location and flow assignment problem in the intermodal express network of high-speed railways and highways, which can not only increase the transportation efficiency but also provide door-to-door service. Considering the characteristics of multiple modes, flow balance, carbon emission, capacity constraints, and time constraints in the intermodal express network, a mixed-integer linear programming model is proposed with the objective of minimizing the total cost by determining the hub locations, allocations, mode selections, and flow assignments. Owing to the NP-hard computational complexity, an improved genetic algorithm with local search is designed by combining the genetic operators and two optimization strategies to solve the problem effectively. Lastly, numerical experiments are conducted to validate the feasibility of the model and the effectiveness of the algorithm.

In this paper we consider a capacitated hub location-routing problem (HLRP) which combines the hub location problem and multi-hub vehicle routing decisions. The HLRP not only determines the locations of the capacitated p-hubs within a set of potential hubs but also deals with the routes of the vehicles to meet the demands of customers. This problem is formulated as a 0-1 mixed integer programming model with the objective of the minimum total cost including routing cost, fixed hub cost and fixed vehicle cost. An optimal solution is obtained by using Xpress-MP for the small sized problems. The experimental results show that the proposed mathematical programming approach can be a viable solution method for the intelligent mobility in supply chain network.

Hub location problems: A meta review and ten disruptive research challenges

This paper studies a flow hub location problem (FHLP) stemming from recent trends in network design for e-commerce businesses. Specifically, e-commerce companies are flexible and agile in reoptimizing their logistics networks, including supplier (origin) and customer zone (destination) decisions. Furthermore, a large number of commodities (flows) and a relatively small sales volume for each product incentivize e-commerce retailers to lease warehouse spaces as hubs, yielding a large number of hub location candidates. As such, the proposed FHLP determines the origin and destination of each flow simultaneously with the hub location and flow routing decisions in contrast to the classical hub location problems, where the origins and destinations of all flows are predetermined. To solve this large-scale optimization problem, we propose an optimization algorithm that combines Lagrangian relaxation and Benders decomposition. Novel acceleration techniques, such as a clustering-empowered multicommodity Benders reformulation, learning-empowered elimination tests, and variable reduction techniques, are further developed to improve the performance and convergence of the algorithm. The efficiency of the proposed algorithm is evaluated via extensive computational experiments. The numerical results show that when compared with five other benchmark methods, the proposed algorithm can achieve optimal solutions faster for small-sized test instances and reduce optimality gaps for large-sized ones. For example, the proposed method achieves optimal solutions for a set of 10 test instances, with node sizes ranging from 225 to 450, within 20 minutes on average. In comparison, the automatic Benders decomposition method implemented in the commercial CPLEX solver achieves an average optimality gap of 2% within one hour. History: Accepted by Russell Bent, Area Editor for Network Optimization: Algorithms & Applications. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0367 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0367 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Redesigning urban distribution networks contributes to reducing the negative impacts on the environment and people’s well-being caused by urban freight delivery operations. One solution is to locate a set of hubs in strategic sites in the city to make the distribution more efficient and environmentally friendly. The objective of this paper is to study the hub location problem and vehicle allocation, and to design a distribution network for freight deliveries in urban areas. Data from a case study in the city of Bogota, Colombia is used as an example of the proposed solution. Firstly, a mathematical programming model is proposed to minimize transportation costs, hub location costs, and CO2 emissions. Given the multi-objective nature of the problem, the solution is obtained following a lexicographic approach. Moreover, multiple scenarios are considered to analyse which might be the most suitable according to the decision-makers’ objectives or requirements. The results obtained through the proposed approach provide a tool for decision-makers to be strategic when choosing the solutions to implement for the case study. These insights and the modelling and solution approach can be generalized for implementation in other cities with similar concerns regarding urban freight distribution.

The hierarchical multimodal hub location problem for cross-border logistics networks considering multiple capacity levels, congestion and economies of scale

The properties of spatial information have been shown to aid in identifying optimal solutions for location–allocation problems. Little effort, though, has been made to develop a spatially informed approach to solving hub location problems, as this class of problems entails a more complex model structure and greater challenges in terms of solving capability. To address this issue, this research proposes the spatially informed hub location problem (SI-HLP), derived from investigating the behavior of hub location problems in determining hubs and their allocations to nonhubs to achieve optimal solutions leveraged by underlying spatial characteristics among nodes, links, and routes. The performance of SI-HLP is achieved with two strategies to distinguish essential and nonessential decision variables for location and allocation decision variables, using an innovative convex-hull-based method, HUBI-COV, to capture nodes with high positive network autocorrelations and their allocated links. Simulation experiments under robustly designed settings were conducted to generalize the findings and assess the effectiveness of SI-HLP, indicating that SI-HLPs provide a novel avenue for advancing the solution of large-scale hub location problems.

Logistic hub location problem under fuzzy Extended Z-numbers to consider the uncertainty and reliable group decision-making

Profit maximization in congested hub location problems: Demand models and service level constraints