Multimodal Hub-and-Spoke Network Design Problems to Integrate Urban Air Mobility and High-Speed Rail Services: A Semi-Quadratic Assignment Formulation and its Linearization

Abstract Urban air mobility (UAM) leverages advanced air transportation systems to facilitate efficient and reliable air travel within and around urban areas. It involves the use of electric aircraft capable of vertical take-off and landing (VTOL), such as air taxis, which navigate urban airspace to transport passengers quickly between locations. High-speed rail (HSR) service, on the other hand, is a rail transport system that utilizes trains operating at higher speeds than traditional rail services, providing rapid connectivity between major metropolitan areas. In this study, we aim to develop a hub-and-spoke network design (HSND) problem that integrates UAM and HSR (UAMHSR–HSND) to improve the efficiency of multimodal transportation systems. The hub-and-spoke model captures the interfacility interactions between passenger flows and vertiport–station assignments via a semi-quadratic assignment model, which poses significant computational challenges. To address this challenge, we compare different linearization models that leverage the special network configuration of UAMHSR–HSND and allow the problem to be solved using off-the-shelf solvers. We further propose a hybrid solving strategy that uses NET-QAP to generate warm-start solutions and then refine them with stronger linearization models, thereby enhancing scalability for large instances. Numerical results show that the NET-QAP model consistently solves large-scale cases with reasonable optimality gaps, and that integrating it with the multicommodity-flow linearization can obtain optimality (GAP = 0%) within reasonable time limit on large-scale instances. Case studies conducted along the Beijing–Shanghai high-speed rail corridor, with spoke vertiports scattered throughout both Beijing and the Changjiang Delta, demonstrate the practicality of the proposed method.