Hub location problems are among the fundamental and widely studied problems in the fields of operations research, transportation network design, and supply chain management. Hubs function as intermediate nodes within a network, and their primary role is to consolidate, process, and distribute flows among different nodes. The use of hubs reduces the number of direct routes, decreases transportation costs, improves efficiency, and facilitates the management of goods and information flows. Over the past two decades, hub location has attracted considerable attention from researchers, as the optimal selection of hub locations can have a significant impact on overall network performance. In this context, various models have been proposed to address the problem from different perspectives, such as cost minimization, transportation time reduction, service level improvement, or a combination of these objectives. Despite significant advances in this field, several challenges remain unresolved. One of the most important challenges is uncertainty in key parameters, such as demand, transportation costs, and facility capacities. Ignoring such uncertainties may lead to unrealistic and impractical decisions. Therefore, stochastic models and scenario-based approaches have gained particular importance as effective tools for incorporating uncertain conditions into decision-making processes. Unlike direct transportation from origins to destinations, hub networks concentrate flows at central nodes and benefit from economies of scale. In this process, flows originating from different origins but sharing the same destination are consolidated at hubs, while flows from a single origin to multiple destinations are efficiently routed through the network. This mechanism reduces the number of direct routes between origin and destination nodes and enables demand satisfaction under limited resources [1].

The first study in this area was conducted by Hakimi [2], who investigated concepts similar to hub location problems. Later, in 1985, Toh et al. [3] examined the application of hub networks in airlines and the aviation industry, which is recognized as the first direct research contribution in this field. Subsequently, Campbell proposed several mathematical formulations for these problems and modeled them analogously to classical facility location problems [4]. Serious research in this area began with the pioneering work of O’Kelly [5], who introduced the first well-known mathematical model for minimizing total transportation costs, including time and distance, in air passenger transportation networks. In 1987, O’Kelly [6] introduced a dataset based on airline passenger flows among 25 U.S. cities, which was later validated by the Civil Aeronautics Board (CAB). Over time, this dataset became a primary benchmark in hub location research and has been widely used in the literature under the title of the CAB dataset. Hub networks can generally be classified into two main structures: single-allocation and multiple-allocation configurations. The distinction between these two structures lies in the way demand centers are connected to hubs. In the single-allocation model, each demand node is assigned to only one hub, and all its inbound and outbound flows pass through that hub. In contrast, in the multiple-allocation model, a node can send or receive portions of its flow through more than one hub [7]. O'Kelly et al. [8] demonstrated that the single-allocation problem is in fact a special case of the multiple-allocation model, obtained by adding a specific constraint. The development of multiple-allocation problems was largely shaped by Campbell’s binary linear programming model, which laid the foundation for the idea of distributing flows among several hubs. In this class of models, each non-hub origin–destination pair can exchange flows through multiple hubs; a feature that, in practice, often results in allocation based on the nearest hub. In decision-making contexts, uncertainty is generally classified into two categories. In the first case, the source of instability cannot be described by probabilistic laws, making it difficult to model. In the second case, although the exact values of parameters are unknown, their behavior can be predicted using probability distributions. The latter category is commonly analyzed and controlled using stochastic methods [9].

In this study, the objective is to develop a scenario-based, two-stage, bi-objective stochastic mathematical model that determines the optimal hub locations and allocates non-hub nodes to the selected hubs in the first objective, while the second objective allows for direct routing between nodes, thereby enabling the design of a hybrid transportation network. The proposed model is formulated as a Mixed-Integer Linear Programming (MILP) model and incorporates uncertainty through a set of probabilistically generated by scenarios that show hub conditions (operational and non-operational). In the first stage, strategic decisions, including hub location selection, are determined. In the second stage, tactical decisions, such as flow allocation and demand management under each scenario, as well as the possibility of direct shipment, are made. Furthermore, by employing a weighted aggregation of objective functions, the proposed approach enables risk assessment and performance comparison of the network under different conditions. On the one hand, it minimizes investment and operational costs, and on the other hand, it enhances the service level and network reliability. The remainder of this paper is organized as follows. Section 2 reviews the existing literature on the hub location problem, as well as studies addressing reliability considerations. Section 3 presents the mathematical formulation of the proposed model. Section 4 reports the computational results obtained from solving the model using standard benchmark datasets commonly employed in the hub location literature. Finally, the paper concludes with a summary of the findings and suggestions for future research.

Numerous studies have been conducted on hub location problems, a substantial portion of which has been devoted to the development of various models for analyzing different scenarios. Incorporating uncertainty into hub location optimization models can better reflect real-world conditions and improve the robustness and adaptability of decision-making schemes. For this reason, studies on hub networks have examined real-world constraints and attempted to incorporate such constraints into their models. In hub location problems, one of the fundamental challenges is the existence of uncertainty in key parameters, such as customer demand, transportation costs, facility capacities, and even the probability of network disruptions. Ignoring these uncertainties may lead to decisions that are not practically implementable under real conditions and may weaken network performance. Since many of these parameters are subject to variation in practice due to economic fluctuations, market volatility, climatic conditions, or other unpredictable factors, classical deterministic approaches are not capable of fully representing reality. In this regard, uncertainty-based models, including stochastic programming and scenario analysis, have been developed to evaluate network behavior under diverse and probable conditions. By considering probability distributions or a set of scenarios, these models enable risk analysis, the selection of more resilient hub locations, and the design of networks with higher flexibility. Accordingly, addressing uncertainty not only improves decision-making accuracy but also leads to the development of networks that operate more stably and efficiently against environmental changes. Shen et al. studied a reliable facility location problem in which facilities are subject to possible failures and customers assigned to a failed facility must be reassigned to other operational facilities. They formulated the problem as a two-stage stochastic programming model and proposed several heuristic approaches to obtain near-optimal solutions for this NP-hard problem [10]. Chaharsooghi et al. [11] investigated a reliable uncapacitated multiple-allocation hub location problem under hub disruptions. In their model, each hub may fail during operation and customers assigned to the disrupted hub are either reassigned to other operational hubs or incur a penalty if service cannot be provided. Momayezi et al. [12] proposed a hub location problem with modular capacitated single allocation under the probability of hub disruption, in which a limited number of capacitated transportation vehicles are used instead of fixed parameters. Soleimani et al. [13] developed a multi-objective mathematical programming model considering uncertainty in some parameters, particularly cost, represented as fuzzy numbers. In addition, backup hubs were selected for each primary hub to cope with disruptions and natural disasters and to prevent delays. Demir et al. [14] employed two metaheuristic approaches based on the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and the Archived Multi-Objective Simulated Annealing (AMOSA) method to solve the Multi-Objective Capacitated Multiple-Allocation Hub Location Problem (MOCMAHLP). Rahmati et al. [15] presented a two-stage robust optimization approach for the uncapacitated hub location problem, in which demand is uncertain. By introducing an expected aggregation function to adjust the uncertainty budget, they effectively improved the feasibility and reliability of the optimization strategy under uncertainty. In the proposed model, hub facility locations were determined in the first stage, while allocation decisions were made in the second stage. Ghaffarinasab et al. [16] addressed a single-allocation hub location problem using a risk-averse two-stage stochastic model, in which mean-CVaR was employed as the risk measure. They also proposed a new solution approach by combining Benders decomposition and scenario grouping. Wang et al. [17] introduced a two-stage mixed-integer programming model developed to address the proposed hub location-routing problem with hybrid logistics strategies. In the strategic stage, hub location and allocation decisions are made before the realization of uncertain demands in order to minimize total production and transportation costs. They then integrated third-party logistics into their self-built hub distribution network. Andaryan et al. [18] investigated a two-stage stochastic mathematical model for the capacitated single-allocation hub location problem under Bernoulli demand. Two strategies, namely facility outsourcing and customer outsourcing, were proposed. Guillot et al. [19] studied an extended version of this hub location problem with integrated fleet allocation decisions and proposed a two-stage stochastic integer programming hub location formulation for the problem. The first-stage variables represent hub location decisions, whereas the second-stage variables represent scenario-based transportation flows and fleet allocation decisions. Liu et al. [20] presented a two-stage stochastic optimization model aimed at intelligently deploying hub capacity to achieve timely delivery, high consolidation, and network flexibility, while minimizing the hub establishment budget and truck fleet cost.

Let G=(N,E) denote the graph representing the network, where N is the set of nodes and E\subseteq(N\times N) is the set of edges. In the hub network, J nodes out of the N nodes are selected as hubs, while the remaining N-J nodes are treated as non hub nodes (spokes) that must be assigned to the hub nodes. The variables and parameters used in the model are summarized in Table 1.

Eq. (1) use for calculate the shipping cost.

cijkm = cik + αckm + cmj. (1)

Based on the variables and parameters in Table 1, the two-stage stochastic mathematical model is as follows.

To evaluate the efficiency of the mathematical model, computational experiments were conducted using widely used datasets available in the literature on the hub location problem, and the results are presented in this section. The dataset used for the computations is the CAB dataset, which is based on air travel data among 25 different cities in the United States. Since the CAB dataset does not include fixed costs for establishing hubs, three values were considered for the fixed-cost parameter, namely 100, 150, and 200. The discount factor parameter was set at three levels, 0.2, 0.6, and 1, in accordance with commonly used values in the related literature. The mathematical model was solved using the CPLEX solver in GAMS 24.7. All computations were performed on a computer equipped with an Intel Core i7-12700H CPU with a clock speed ranging from 2.3 to 4.7 GHz and 16 GB of RAM, under the Windows 11 operating system. The obtained results are reported in Table 3.

In Table 3, the term penalty means that, instead of shipping through direct transportation at its corresponding direct cost, a fixed penalty value ( θ = 1000 ) is considered. The results presented in the table indicate that changes in the weights of the objectives have a direct effect on the network structure and the objective function value. When the weight associated with the fixed costs of establishing hubs (w1) increases, the model tends to select fewer hubs in order to avoid a sharp increase in capital costs. This issue is particularly observable at higher weights, where the combination of selected hubs is reduced to smaller sets. In contrast, when the weight of operational costs (w2) is higher, the model prefers to keep more hubs active in order to reduce ongoing costs and cover flows through shorter routes, even if this decision leads to a relative increase in capital costs. For example, in the case where the two objectives are considered with equal weights ( w1 = 0.5 and w2 = 0.5 ), the objective function value increases noticeably because both cost components are simultaneously influential. On the other hand, increasing the fixed cost of hub establishment (fk) leads to changes in the selected hub configuration and an increase in the objective function value. By increasing this cost level from 100 to 150 and 200, the model becomes less inclined to keep all initial hubs open, and the ( = . , = , = . , = . ).

According to Figs. 1, 3, and 5, as (w2) increases, the amount of direct shipment also increases, because the model shows a greater tendency to cover flows through shorter routes. It can also be stated that the direct allocations in the penalty case, shown in Figs. 2, 4, and 6, differ from those in the direct-shipment case. This is due to the nature of the model, which compares the transportation cost through hubs with the penalty value. Overall, it can be concluded that, in the design of real-world networks, direct shipment can play an effective role as a complementary strategy in improving network efficiency and resilience.

In this study, a bi-objective single-allocation hub location model was proposed by considering uncertainty in hub availability and the possibility of direct shipment. The proposed approach demonstrated that combining flows routed through hubs with direct shipments between some origin–destination pairs can lead to the design of a more efficient and resilient network. Accordingly, the network not only benefits from the advantages of centralization and economies of scale arising from flow consolidation at hubs, but also achieves greater flexibility in dealing with demand variations and potential disruptions through the use of direct routes. The obtained results indicate that direct shipment can play an important complementary role in network design and can contribute to cost reduction and improved stability under different conditions. This capability becomes particularly important when there are limitations on the number of hubs and flows need to be managed through alternative routes. Therefore, the proposed hybrid approach can provide an effective decision-making framework for the design of real-world networks, especially in transportation and distribution industries.

This research received no external funding.

Sufficient data is available upon request.

No potential conflict of interest was reported by the authors.

All authors have provided their consent for the publication of this manuscript.