Abstract
The hub location problem (HLP) has a pivotal role in designing many-to-many transportation networks with increasing applications in logistics, postal services, passenger air transportation, etc. While developing their hub networks, firms need to take account of their competitors’ network structure to determine the optimal location of their hubs, so that the highest market share is captured. However, due to potential natural or man-made disruptions on the located hubs, decision-makers need to consider backup paths for routing the origin–destination (OD) flows. This paper introduces the competitive multiple allocation hub location problem considering primary and backup routes in a duopoly market consisting of a leader–follower pair of firms that locate their hubs sequentially in the spirit of Stackelberg competition. The firms also try to hedge the risk of inadequate customer service by considering backup routes for the OD flows. The backup routes differ from the primary ones based on one of the two strategies called the partial and complete backup strategies. The problem is modeled as a bilevel MILP formulation, and an enumeration algorithm, as well as a simulated annealing heuristic, are proposed for solving instances from two well-known existing data sets. A large set of computational experiments is performed to test the efficiency of two solution algorithms and analyze the impact of various input parameters on the optimal solutions. The results indicate the necessity of paying a special attention to the possibility of disruptions, particularly for the leader firm.
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Appendix
Appendix
This appendix provides further information regarding the calculation of \(\lambda \) for real case studies and actual circumstances. As observed in the previous sections, a fixed value has been assigned to \(\lambda \), representing the probability of using primary routes. This value is assigned to the whole network independent of the path used. However, this assumption can be relaxed as it is more practical to assign a unique probability for each path (i.e., path-dependent probabilities). To this end, \(\lambda \) can be redefined in the following manner, which has previously been introduced in Kim and O’Kelly (2009) as the probability of the successful delivery of flows for the path \(i \rightarrow k \rightarrow m \rightarrow j\):
In this equation, \(R_{ij}\) indicates the reliability of link (i, j), and \(\eta \in (0,1)\) shows the fixed rate used to represent the enhanced reliability on the inter-hub links. According to this definition, a primary path can only be used when all its links are active; otherwise, it is required to use the backup route. Therefore, the probability of using the corresponding backup route can be obtained as \((1 - \lambda _{ijkm})\). In other words, the backup route is used when the primary route is disrupted, and as \(\lambda _{ijkm}\) shows the percentage of times the primary route is used, \(1 - \lambda _{ijkm}\) is defined to show the percentage of times that the backup route is utilized.
Considering this redefinition, we solve some instances from the CAB data set using the proposed enumeration algorithm, and the results are presented in Table 29. The results show the flows captured by the follower and the optimal hub locations for both the players obtained from the HMed-PBR and the HCen-PBR problems using the complete backup strategy. In this table, it is assumed that \(\alpha =0.6\), \(\eta =0.5\), \(R_{ik}=R_{mj}=0.9\), and \(R_{mk}=0.8\). With these values, \(\lambda _{ijkm}\) can be calculated as: \(\lambda _{ijkm} = 0.9 \times 0.8^{(1-0.5)} \times 0.9 = 0.7245\). As it can be observed, the value of \(\lambda _{ijkm}\) is almost between the ones that we used in the computational experiments section. Therefore, as it is expected, the results represent the same trends that have been manifested before.
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Khodaee, S., Roghanian, E. & Ghaffarinasab, N. Competitive multiple allocation hub location problem considering primary and backup routes. Comp. Appl. Math. 41, 143 (2022). https://doi.org/10.1007/s40314-022-01849-8
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DOI: https://doi.org/10.1007/s40314-022-01849-8