Intermodal Hub Network Design with Probabilistic Service-Level Constraints

In this paper, we study the intermodal hub network design problem with probabilistic service-level constraints ensuring that total service time requirements of customers’ orders are satisfied with a minimum probability. The intermodal network is modeled as a Jackson queueing network with $M/M/s$ queues for the hubs and $M/GI/\infty$ queues for transport operations. We characterize the total service time distribution and propose a cutting-plane algorithm that exploits the characteristics of this distribution. We show that the α-level sets of the total sojourn time distribution for a transport path including two or more hubs are homothetic with some homothetic center. This characteristic allows for the derivation of valid inequalities leading to significant reductions in the solution time. We propose a worst-case renewal approximation considering $GI/PH/1$ queues to extend our analysis to non-Jackson networks. We prove that the properties of the total sojourn time distribution derived for Jackson networks also hold for this renewal approximation, allowing the application of the derived cutting-plane approach to the general case. Extensive computational experiments are performed on the Australian Post and Colombian data sets to assess the performance of the proposed formulations and solution algorithms.

Funding: The authors acknowledge support from the Natural Sciences and Engineering Research Council of Canada.

Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0657.