The reliable huband-spoke design problem : Models and algorithms

Hub-and-spoke structure is widely adopted in industry, especially in transportation and telecommunications applications. Although hub-and-spoke paradigm demonstrates significant advantages in improving network connectivity with less number of routes and saving operating cost, the failure of hubs and reactive disruption management could lead to substantial recovery cost to the operators. Thus, we propose a set of reliable huband-spoke network design models, where the selection of backup hubs and alternative routes are taken into consideration to proactively handle hub disruptions. To solve these nonlinear mixed integer formulations for reliable network design problems, Lagrangian relaxation and Branch-and-Bound methods are developed to efficiently obtain optimal solutions. Numerical experiments are conducted with respect to real data to demonstrate algorithm performance and to show that the resulting hub-and-spoke networks are more resilient to hub unavailability. 2015 Elsevier Ltd. All rights reserved. 1. Background and motivation The hub-and-spoke system has been widely employed in various industrial applications, such as transportation and telecommunications system designs. It is a fully interconnected network with material/information flow between any two nodes being processed at a small number of critical nodes (i.e., hubs) so that the operators can benefit from the economies of scale by consolidating flows from and to spoke nodes and increasing the utilization of equipment and staff at those critical nodes. Clearly, a hub-and-spoke network heavily relies on hubs to make the whole system functional, and therefore it is vulnerable to any disruptions and degradations of hubs. Traditional hub-and-spoke network design solves the problem of hub location and allocations of spoke nodes to hubs, assuming network components work properly. In practice, nevertheless, operators have to face various disruptions and apply disruption management techniques to recover the system. Such an issue is most prominently demonstrated in air transportation where severe weather, labor strikes, terrorism threats, and runway incursions disrupt regular operations and make airports partially or completely unavailable (Palpant et al., 2009; Løve and Sørensen, 2001). To deal with the vulnerability issue of the hub-and-spoke system, several mitigation strategies have been proposed and implemented, such as delaying, canceling, and rerouting in air transportation (Janic, 2005; Ball et al., 2006) and network peering in telecommunications systems (O’Kelly et al., 2006). However, most of mitigation strategies are reactive, which are often costly to implement and inefficient, given that the initial network is designed for perfect conditions. For example, 104 Y. An et al. / Transportation Research Part B 77 (2015) 103–122 it is observed in Bratu and Barnhart (2006) that, although the disrupted passengers were only three percent of the total passengers, they suffered 39 percent of the total passenger transportation delays with much lower customer satisfaction. Clearly, the initial network design affects the selections of backup hubs and alternative routes, which affects the cost of mitigation operations. Therefore, to achieve both economic advantage and system reliability, the network design problem should consider both the hub locations and regular route designs as well as the backup hubs and alternative route designs under disruptions in a holistic modeling framework. Therefore, in this paper, we propose a reliable hub-and-spoke network design strategy by explicitly considering the hub unavailability, i.e., backup hub and alternative route decisions will be considered in the design stage and related cost will be included in the objective function of the design problem. With this strategy, we aim to develop a new type of optimization models to minimize the operating cost considering both the normal situation, which is disruption free, and disrupted situations where survived hubs serve as backup hubs for rerouting disrupted flights due to unavailable hubs. As illustrated in Fig. 1, where the solid line denotes a regular route for the flight from Tampa to San Francisco and the dotted line denotes an alternative route using Dallas as a backup hub if the Miami hub is unavailable. This strategy will not only benefit airlines but also other industries who adopted hub-and-spoke distribution paradigm with which they can build and operate their networks with both reliability and economic advantages. Compared to classical models, the introduction of backup hubs and alternative routes drastically increases the complexity of the network design problem. As the choice of backup hubs and alternative routes depends on the hubs in regular routes, a large number of nonlinear terms are introduced to capture the dependency. As a result, nonlinear mixed integer formulations are constructed. Their structures are further investigated and solution methods developed. To the best of our knowledge, our study is the first analytical work on the reliable hub-and-spoke design with consideration of backup hubs and alternative routes. The developed algorithm is easy to implement and can solve practical instances in a reasonable amount of time. Numerical study demonstrates that our reliable models can serve more passengers under the disruption situations and sensitivity analysis shows that the resulting designs are robust to hub unavailability. The proposed reliable hub-and-spoke network design also yields a set of useful tools for practitioners, such as airlines, to re-structure their networks or to identify strategic partners to hedge against various disruptions and achieve better performance. The rest of the paper is organized as follows. In Section 2, literature review on hub-and-spoke design is presented as well as recent research on reliable facility location models. In Section 3, the reliable single allocation hub-and-spoke model is formulated and the solution methods are elaborated. In Section 4, the study is extended to the reliable multiple allocation model. Section 5 demonstrates computational performance of the developed algorithms using the CAB data set from airline operations as the case study and provides comparisons between our reliable hub-and-spoke design models and classical models. In addition, system design and performance with proposed model are analyzed and discussed, including sensitivity Fig. 1. Regular and alternative routes. Y. An et al. / Transportation Research Part B 77 (2015) 103–122 105 analysis and the demonstration of applying proposed model to a recent airlines merger. Section 6 concludes this paper with some discussions on future research directions. 2. Literature review The hub-and-spoke design problem is conventionally called hub location problem (HLP), which is concerned with locating hub facilities and allocating spoke nodes to hubs. There are generally two basic structures: single allocation (SA) and multiple allocation (MA). In SA hub-and-spoke model, all outbound/inbound flows of any node must travel directly from/to a specific hub. In MA model, flows of a given node can go directly from/to different hubs. When the number of hubs, denoted by p, is given, the problem is called the p-hub median problem (HMP). In the remainder of this paper, we use SA-HMP or MAHMP to denote the corresponding design problem. O’Kelly (1987) proposes the first mathematical formulation for HMP and presented the first quantitative analysis on this type of network structure using the Civil Aeronautics Board (CAB) data set. Since then, as hub-and-spoke structures are of significant theoretical and practical values, a large number of studies have been conducted on developing models with more practical features and on designing efficient algorithms. We first briefly describe a few important results on formulation and algorithm design. Ernst and Krishnamoorthy (1996, 1998a) formulate SA-HMP and MA-HMP, respectively, based on the idea of ‘‘multicommodity flow’’. Skorin-Kapov et al. (1996) propose mixed integer formulations for both SA-HMP and MA-HMP that yield tight linear relaxations. As for the customized algorithm development, Branch-and-Bound process and Lagrangian relaxation have been widely used to obtain exact solutions (Ernst and Krishnamoorthy, 1998b and Pirkul and Schilling, 1998). Different from the p-hub median problem, the hub location problem with fixed costs treats the number of hubs as a decision variable and seeks to minimize the transportation cost and the construction cost where a fixed construction cost is associated with a decision of hub location. O’Kelly (1992) and Campell (1994) study a few formulations of HLP with fixed costs. There are also extensive literature in search of effective solution algorithms for these problems, see Cunha and Silva (2007), Chen (2007), Cánovas et al. (2007) and Contreras et al. (2011a) for examples. One may refer to Alumur and Kara (2008) and Campbell and O’Kelly (2012) for a comprehensive review of modeling techniques and solution methods of HLP. In the remainder of this paper, unless we explicitly mention, the hub-and-spoke network design problem indicates p-hub median problem. Recent studies focused on extending classical SA and MA models by incorporating practical factors, such as hub congestion (Grove and O’Kelly, 1986; Elhedhli and Wu, 2010), hub capacity (Contreras et al., 2012), nonlinear economies of scale (de Camargo et al., 2009), and dynamic/stochastic nature of demand and cost (Contreras et al., 2011b,c). Nearly all studies on HLP assumed that the chosen hubs would always operate functionally as planned. Nevertheless, in practice, hubs could fail due to different reasons. As the typical cases in air transportation industry, adverse weather often significantly deteriorates the availability of a hub airport and results in huge disruption costs. Similar situations have been observed in facility-and-clie