The bi-level line-planning model of Li et al. (2020) simultaneously optimizes train frequencies, train routing and passenger assignment for high-speed railway networks, but merges all stations of the same city into a single network node and represents stopping with an average stop time and an average stop rate, so it can neither route trains through different stations of one metropolitan area nor let passengers transfer between those stations at their actual inter-station travel times. The open problem is to reformulate and solve this joint optimization at station-level resolution: every station of a city is a distinct node, trains may be routed via any of them, and the simultaneous passenger assignment uses the actual transfer time between distinct stations of the same city. Later work optimizes train routing across the multiple stations of a single hub but explicitly excludes transfer passenger flow, so the network-level joint model with intra-city station choice and inter-station transfer times remains unresolved.

Background

A railway line plan (LP) specifies which train lines are operated on a network, at what frequency each line runs, and which stations the trains stop at; train routing decides the path (and, within a city, the station sequence) each train follows, and passenger assignment allocates the origin-destination (OD) travel demand to the resulting services. These decisions are usually taken sequentially, which degrades plan quality because assignment consequences feed back into routing and frequency choices. Li, Li, Hu, Yan and Wang (2020) proposed a bi-level multi-objective mixed-integer nonlinear program for the high-speed railway (HSR) case: the upper level decides the train operation routings and train frequencies (an integer program whose objectives are to minimize the total train travel time and the average capacity utilization rate over the used network sections), while the lower level assigns passengers so as to minimize total passenger travel time and maximize the average train attendance rate over all sections, with the number of trains on each routing derived from the assigned passenger flows. A particle swarm optimization (PSO) heuristic — a population-based metaheuristic — solves the coupled model on a large Chinese HSR network. To keep the network tractable, that model merges all stations located in the same city into a single node: a passenger's city is one aggregate stop, trains implicitly 'serve the city', and stopping is described only by an average stop time tstt_{st} and an average stop rate σk\sigma_k (the fraction of trains stopping at a node). Many metropolises, however, contain several HSR stations (for example Beijing's Beijing South, Beijing West and Qinghe stations) connected by urban transport, and trains may be routed through different ones. A separate literature line optimizes the train routing problem (TRP) for such multi-station hubs: Wang, Song, He and Song (2022) formulate a mixed-integer node-arc model (a network formulation whose decision variables represent flows on individual arcs) minimizing the total cost of train and passenger routes for the Zhengzhou multi-station hub, and Wang, Song, He, Song and Chi (2022) solve the hub TRP by Lagrangian relaxation (dualizing hard constraints so the problem decomposes into train and passenger shortest-path subproblems). These hub models treat the stations of one urban area as distinct nodes, but they restrict attention to departure passenger flow: Wang et al. (2022, Sustainability) state that transfer passenger flow is not considered and is deferred to future work, and Wang (2026) likewise restricts its hub division-of-labor model to the outward travel demand of passengers, stating that the transfer needs of transfer passengers are not accounted for and will become a critical optimization direction for future work. Network-level passenger assignment work such as Wu, Shi, Zhao, Xu and Yang (2022), a Dantzig-Wolfe decomposition algorithm (an iterative method that decomposes a large optimization model into a master problem and smaller subproblems) for capacitated assignment with time-varying demand, likewise uses station-level networks without modeling choice among, and transfer between, the distinct stations of one city. The gap is therefore the union of the two ingredients inside one joint line-planning model: station-level resolution of same-city stations and the actual transfer time between them in the simultaneous passenger assignment.

Problem Statement

Reformulate the bi-level joint optimization framework of Li et al. (2020) for high-speed railway line planning — upper level deciding train routing and frequencies, lower level performing simultaneous passenger assignment — so that (i) every station of a city is a distinct node of the network rather than one merged city-node, allowing trains to be routed through different stations of the same metropolitan area; and (ii) the passenger assignment carries the actual transfer time between distinct stations of the same city, so that a passenger whose access, exit or transfer involves different stations of one city incurs the travel time of the corresponding inter-station leg instead of that movement being absorbed in the merged city node. The transfer times used under (ii) must be pair-specific values reflecting the travel times actually afforded by the urban transport connections between the stations concerned; a degenerate choice — every intra-city entry set to zero, or to one pair-independent constant — would leave intra-city movement free, would make the passenger side indistinguishable from the merged-node treatment, and does not satisfy (ii). The formulation must also state explicitly how stopping is represented once the stations of a city are separate nodes (whether the average stop time tstt_{st} and the average stop rate σk\sigma_k are retained per station, replaced by station-specific dwell times and stop decisions, or a combination of the two), and it keeps the four original objective terms — total train travel time, average capacity utilization rate, total passenger travel time and average train attendance rate — re-derived per station rather than per city. The task is to give the complete mathematical formulation (sets, parameters, decision variables, objectives and constraints, including how passengers choose among and transfer between the stations of one city), to solve the resulting model on a realistic multi-station metropolitan high-speed railway network, and to quantify how the station-level resolution with inter-station transfer times changes the optimized line plan (train routing, frequencies, passenger paths, total passenger travel time and capacity utilization) relative to the merged-node baseline of the original model, reporting each objective term and the corresponding decisions separately, since the bi-level model is multi-objective and has no single scalar optimum.

References

  1. Li X, Li D, Hu X, Yan Z, Wang Y (2020) Optimizing train frequencies and train routing with simultaneous passenger assignment in high-speed railway network. Computers & Industrial Engineering 148:106650. DOI: 10.1016/j.cie.2020.106650. https://doi.org/10.1016/j.cie.2020.106650
  2. Wang Y, Song R, He S, Song Z (2022) Train Routing and Track Allocation Optimization Model of Multi-Station High-Speed Railway Hub. Sustainability 14(12):7292. DOI: 10.3390/su14127292. https://doi.org/10.3390/su14127292
  3. Wang Y, Song R, He S, Song Z, Chi J (2022) Optimizing Train Routing Problem in a Multistation High-Speed Railway Hub by a Lagrangian Relaxation Approach. IEEE Access 10:61992-62010. DOI: 10.1109/ACCESS.2022.3181815. https://doi.org/10.1109/ACCESS.2022.3181815
  4. Wang Y (2026) Research on optimization of train routing problem for railway hubs considering passenger demand. PLoS One 21(6):e0351229. DOI: 10.1371/journal.pone.0351229. https://doi.org/10.1371/journal.pone.0351229
  5. Wu R, Shi F, Zhao S, Xu G, Yang H (2022) A Dantzig-Wolfe decomposition-based algorithm for capacitated passenger assignment problem with time-varying demand in high-speed railway networks. Transportation Research Part C: Emerging Technologies 145:103909. DOI: 10.1016/j.trc.2022.103909. https://doi.org/10.1016/j.trc.2022.103909