Shared passenger–freight transportation in high-speed rail (HSR) systems provides an effective way to improve railway capacity utilization and operational revenue while offering a more sustainable alternative for time-sensitive freight services. This study investigates the train capacity planning problem for shared passenger–freight HSR services under uncertain passenger and freight demand. The problem explicitly considers heterogeneous service requirements, including passengers’ departure-time preferences, freight delivery time windows, train capacity restrictions, transfer feasibility, and the different operational characteristics of passenger train units (PTUs) and freight train units (FTUs). To describe these operations, we construct an extended space–time network that captures passenger movements, freight transportation, loading and unloading, transshipment, and the potential reallocation of underutilized PTUs for freight service. Based on this network, a two-stage stochastic programming model is formulated to determine train composition decisions in the first stage and passenger–freight flow assignment decisions in the second stage, with the objective of maximizing expected operational profit while preserving passenger service quality. The stochastic model is approximated through the Sample Average Approximation (SAA) method and reformulated as a mixed-integer linear programming (MILP) model. To solve the resulting large-scale problem, we develop an improved integer L-shaped (ILS) algorithm with strengthened optimality cuts and acceleration strategies. Numerical experiments based on the Beijing–Shanghai HSR corridor demonstrate the computational efficiency of the proposed ILS algorithm and its applicability to practical-scale instances. The results also show that accounting for demand uncertainty and jointly using PTUs and FTUs under flexible train composition improve solution robustness and operational efficiency.

This study aims to design a high-quality customised feeder bus (CFB) service for high-speed railway (HSR) passengers while controlling operating costs. First, a CFB scheduling optimisation model considering travel time uncertainty is proposed to optimise the timetable with the design of the skip-stop pattern which would influence the selection of service paths, enabling a reduction in operation time. Second, to improve computation efficiency, this study suggests an iterative heuristic with a two-stage approach to address the travel time uncertainty. Numerical results demonstrated that compared with the commonly used algorithm, the genetic algorithm, the proposed algorithm could reduce the objective by 62.38% within a shorter time. Furthermore, compared with the traditional bus service, the CFB service generated by the proposed model could averagely reduce passenger travel time by 48.46%. These results indicated that the proposed model and algorithm could help to provide a more convenient access/egress service to HSR passengers.

Recurrent hazardous-material shipments may concentrate traffic on a limited number of corridors when transportation cost dominates route choice, yet absolute flow does not provide a consistent basis for comparing heterogeneous road segments. This study develops a stochastic path-flow allocation framework for controlling localized hazardous-material flow concentration across a road network under uncertain OD demand while accounting for heterogeneity in arc-specific safety-informed reference capacities. Hazardous-material flows are expressed as normalized flow burdens, and a common normalized-flow-burden criterion is used to coordinate path-flow allocation across arcs and demand scenarios. The model jointly determines scenario-specific path-flow allocations and a normalized-burden threshold shared across all arcs and modeled demand scenarios, balancing maximum normalized flow burden against expected transportation cost according to the decision maker’s preference. A Lagrangian relaxation decomposition algorithm is developed to solve the model. Numerical experiments show that, relative to the cost-minimizing benchmark, the proposed framework reallocates flows, lowers the maximum normalized flow burden, and mitigates excessive concentration of hazardous-material flows on a limited number of arcs, albeit at additional transportation cost. The framework provides regulators with a quantitative basis for coordinating network-wide path-flow allocation and evaluating the trade-off between localized-burden control and transportation efficiency.

Integrating freight transportation into metro systems can improve network utilization and reduce urban freight emissions. This study develops a dynamic train carriage allocation strategy for coordinated passenger-freight metro operations under time-varying demand and loading conditions. A bi-objective optimization model is formulated to jointly consider passenger waiting time and freight transportation profit, which is transformed into a unified framework maximizing overall system benefit. A hybrid VNS-based matheuristic framework is designed for efficient solution. Numerical experiments on a unidirectional metro line demonstrate that the proposed approach enhances system utilization and revenue without requiring additional train services. The results demonstrate the potential of coordinated passenger-freight metro operations for sustainable urban logistics.

In railway systems, the fare-free reward scheme (FFRS) provides passengers the chance to redeem reward points for free trips on specific trains. Passengers have to determine an optimal strategy for accumulating and redeeming points across multiple trips, involving various origin-destination (O-D) pairs and scheduled serving trains. This paper focuses on the issue of passenger assignment in the FFRS system, considering the intricate interplay of reward points on passengers' multiple itinerary choices. To define passenger classes, the demand proportions for each O-D pair are adopted to characterize heterogeneous travel demand structure. Then a multi-class user equilibrium model is formulated to capture passengers' multi-trip choices about serving train and payment approaches, where the class-specific reward point value can be determined endogenously at an equilibrium solution. Meanwhile, we turn to formulate an equivalent mathematical programming (MP) model, where the solution is proved to satisfy the proposed passenger equilibrium assignment conditions and the Lagrange multipliers associated with the reward points constraint are related with reward point value. Furthermore, an Augmented Lagrangian Multiplier (ALM) passenger assignment algorithm is developed. Two numerical examples are designed to demonstrate the effectiveness of the proposed model and algorithm. The results also show that different FFRS rules can regulate the passenger choice behavior, providing a practical demand management strategy.

The potential spread of infectious diseases, e.g. Influenza A, SARS, and COVID-19, has brought challenges to railway operations. One issue is how to improve railway utilization while implementing social distancing strategies. This study proposes to jointly optimize train stop planning, scheduling, and seat allocation when incorporating social distancing strategies (TSPS-SA-SD). The proposed approach incorporates heterogeneous social distancing requirements for passengers from areas associated with different levels of risks in epidemics/pandemics. To tackle the TSPS-SA-SD problem, an integer nonlinear programming (INLP) model is developed, which minimizes travel time and maximizes rail revenue while ensuring that social distancing strategies are respected. The proposed INLP is reformulated into an integer linear programming (ILP) model. Then, a decomposition-based heuristics solution (DHS) algorithm, which leverages the adaptive large neighborhood search (ALNS) and GUROBI solver, has been introduced to solve large instances. The proposed approach is evaluated on a small network and two real instances.

Passenger demand in high-speed railway (HSR) networks often exhibits significant temporal and spatial variability, posing challenges for HSR service planning. To address these issues, this study introduces an integrated optimization framework that jointly considers train scheduling, stopping patterns, flexible train compositions with extra-long trains, and seat allocation strategies, which are interdependent decisions that collectively govern system efficiency. Notably, the proposed modeling framework allows for the deployment of extra-long trains exceeding standard platform lengths to increase capacity during peak demand periods. Operating extra-long trains raises safety and efficiency challenges in passenger boarding and alighting. A carriage-specific seat allocation method and a station-specific train docking position control strategy are utilized to avoid cross-carriage walking during passenger boarding or alighting (for extra-long trains). The studied problem can be formulated as a bi-level optimization model. The upper-level model determines the operator's decisions regarding scheduling, stopping patterns, train compositions, train docking position control and seat allocation, while the lower-level model reflects passenger travel choices in response to the provided services. The objective jointly considers operator profit and the penalty for unserved passenger demand. To ease the solution process, the proposed bi-level model is reformulated into a single-level mixed-integer linear programming (MILP) model by applying Karush-Kuhn-Tucker conditions and appropriate linearization techniques. To solve the proposed MILP of realistic problem sizes, we develop a temporal decomposition-based algorithm enhanced with local search, which partitions the problem by time periods, incorporates local search for solution refinement, and utilizes acceleration cuts to improve computational efficiency. Numerical studies for the Hong Kong-Shenzhen-Guangzhou and Guangzhou-Zhuhai-Xinhui corridors indicate that the proposed integrated optimization framework with extralong trains shows potential to improve operator profit and passenger service performance. It is also shown that the advantages of extra-long trains can be better leveraged when train scheduling, flexible train composition, and seat allocation are jointly optimized.

Urban rail transit (URT) systems face operational challenges from inherently unbalanced passenger demand-spatially (across line sections and directions) and temporally (peak/off-peak fluctuations). We address this by integrating short-turning and modular train strategies into the train operational plan (TOP) via a novel mixed-integer linear programming (MILP) model. The approach simultaneously optimises the number of train services, the non-cyclic train schedule (departure/arrival times, routes and compositions) and rolling stock circulation, while explicitly incorporating passenger route choice under tight capacity constraints. Formulated as a time-discrete service network design problem, the model strikes a balance between passenger travel costs and operating costs. Numerical experiments on large, realistic instances validate the approach: embedding these strategies in the TOP substantially reduces both operating and passenger costs and remains effective across varied demand-supply conditions and scales. The results provide operators with a practical decision-support tool for designing flexible URT operations and optimising future TOPs.

Saving energy has both significant environmental benefits and economic advantages. As the urban rail transit network and its consumed energy continue to expand, it is crucial to optimize the energy-saving operation scheme of trains. Energy-saving train operation often requires longer section running times, which is obviously not conducive to the quality of passenger service. In order to ensure passenger service quality when pursuing the decrease of the train's energy consumption and rolling stocks' operation cost, this paper proposes an integrated optimization model of the demand-oriented and energy-efficiency train timetable and rolling stock circulation plan for urban rail transit. Its objective is to minimize train net energy consumption, rolling stock utilization cost, as well as passenger waiting time and travel time. Specifically, the net energy consumption is defined as the difference between train required traction energy consumption and regenerative braking energy utilization. To efficiently solve this large-scale mixed integer nonlinear model, we design a solution algorithm combining Variable Neighborhood Search (VNS) and CPLEX, in which seven different neighborhood structures are constructed. Based on the data of Guangzhou Metro Line 13, we have verified the effectiveness and performance of the model and algorithm through numerical experiments of various scales, as well as through comparisons with other algorithms and models. The results demonstrate that the timetable and rolling stock circulation plan obtained by VNS can reduce net energy consumption by 9.55 %, rolling stock utilization cost by 9 %, and passenger waiting time by 4.77 %, and their travel time by 0.87 % compared to the current timetable.

Understanding the passenger distribution within the metro system is a prerequisite for metro network planning and operation. However, as automatic fare collection (AFC) data records only entry and exit information, directly obtaining passenger distribution through AFC data and established timetables remains challenging. Although many studies have explored passenger distribution in metro systems based on accurate timetable inputs, parameter collection and calibration are challenging due to the spatiotemporal dynamics of both passenger demand and headway. This study proposes a data-driven passenger-to-train assignment model (PTAM). The posterior probability of passengers boarding the train is computed using a two-stage Gaussian mixture model (GMM). This method does not require precise timetable inputs, and both the initial parameter collection and final estimation processes are automated, eliminating the need for manual calibration. Using the Nanjing metro as a case study, the effectiveness of the PTAM is demonstrated. Additionally, the study computes in-vehicle passengers, left-behind passengers, and passengers' willingness to pay (WTP) using PTAM. The results demonstrate significant differences in crowding level and left-behind at different stations on the same line. During the evening peak, passengers bear about 50% of welfare costs. The findings can provide managers with a basis for passenger flow organization and guidance for passenger's travel decision.

To mitigate the transmission risk of infectious diseases (e.g., COVID-19) on trains while maintaining a certain level of railway service and improving railway operation revenue, this study proposes a passenger group-based railway seat allocation policy. First, the shedding rate is used to estimate the risk of virus transmission among passenger groups (different infectious diseases can be treated in a similar manner). Then, a mixed integer programming model is established, which minimizes the summation of shedding rates and maximizes the railway operation revenue. In the proposed policy, both virus diffusion between cities associated with different risk levels and virus transmission among different passenger groups in train carriages are considered. A tailored heuristic algorithm based on the multi-start variable neighborhood search (M-VNS) algorithm is then designed to quickly produce a high-quality solution for practical large-scale applications. A series of numerical studies are conducted, demonstrating the efficacy of the proposed policy in enhancing railway revenue and mitigating the (estimated) transmission risk of infectious disease.

This paper investigates a railway seat allocation problem with a focus on equity. We aim to distribute the railway capacity more fairly among passengers from different Origin-Destination (OD) pairs while enhancing profitability. We first develop a Mixed Integer Linear Programming (MILP) model for scenarios with deterministic demand. We then further extend our study by formulating Stochastic Programming (SP) and Distributionally Robust Optimization (DRO) models for scenarios with demand uncertainty. Additionally, we derive the deterministic equivalent of the DRO model using a box ambiguity set. Furthermore, we explore the relationships between the proposed DRO and SP models, both of which can be efficiently solved by common MILP solvers like GUROBI. To validate our approach, we perform numerical studies on a small-scale example and the Zhengzhou-Xi'an high-speed railway corridor. The results demonstrate that the proposed optimization methods improve equity across OD pairs, where the DRO model can yield high-quality solutions.

In railway systems, the transport capacity of trains depends on the number of train compositions, which are then allocated to different OD pairs. Therefore, train composition planning is a critical capacity inventory control technique in railway revenue management. However, passenger demand for high-speed railway (HSR) exhibits a strong spatial-temporal imbalance, with significant differences between peak and off-peak periods, and also entails considerable uncertainty, directly impacting enterprise revenue. To address these challenges, this study accounts for the stochastic nature of passenger demand and proposes a flexible train composition (FLTC) strategy that integrates extra-long trains (XLTs), allowing train lengths to exceed platform lengths. This strategy allows for the flexible adaptation of train compositions to accommodate variations in passenger demand. By deploying XLTs without necessitating alterations to the current infrastructure, it enhances transport capacity during peak hours, thereby boosting operational revenue for the enterprise. Additionally, for XLTs, we propose a new operational mechanism to fully utilize their transport capacity while ensuring passenger boarding and alighting experiences, i.e., docking position control and seat allocation methods. Then, the problem studied can be formulated as a two-stage stochastic programming (SP) model, where the first stage determines the number of train compositions and the docking positions of train composition units (TCUs), and the second stage determines the seat allocation scheme based on demand. To enhance the tractability of the model, we reformulate it as a mixed-integer linear programming (MILP) model using appropriate linearization techniques. To solve the proposed model efficiently, we develop a column generation (CG)-based solution method, thereby enabling the generation of near-optimal solutions. To evaluate the effectiveness of the proposed method, numerical experiments are conducted using both a small-scale instance and a real-world case study based on the Beijing-Shanghai HSR corridor. The computational results demonstrate that the proposed approach can significantly improve transport capacity during peak periods, thereby increasing the overall revenue of the HSR system, while also effectively accommodating the stochastic nature of passenger demand.

The accumulation of large passenger flows at metro stations often poses congestion risks to urban rail systems, including an increased likelihood of accidents (e.g., slips, trips, and falls) and train departure delays. However, during peak hours, the priority boarding rights of passengers at upstream stations often lead to congestion and overcrowding at downstream popular stations. We propose a new passenger flow control strategy to address this issue, namely destination- to-gate assignment. This approach assigns specific gates to destinations served by the station, enabling passengers to board the appropriate train carriages. This strategic allocation facilitates a more even distribution of passengers, reducing congestion and enhancing spatial equity in passenger travel, thereby mitigating operational risks associated with overcrowding. For the problem of interest, we propose a nonlinear integer programming model to optimize the destination-to-gate assignment, aiming to simultaneously minimize risks related to passenger crowding and waiting times. The model adopts a first-come, first-served (FCFS) boarding rule to accurately capture the dynamic nature of passenger flow while considering the capacity limitations of train carriages. Leveraging the model's characteristics, we employ a set of linearization methods to equivalently transform it into a mixed-integer linear programming (MILP) model. To address the computational challenges posed by real-world scale, we develop a customized heuristic algorithm that uses Variable Neighborhood Search (VNS) combined with passenger flow simulation to efficiently generate high-quality solutions. Finally, we conduct a series of numerical experiments using data from Guangzhou Metro Line 9 to demonstrate the effectiveness of our proposed approach. The results show that the proposed destination- to-gate assignment strategy effectively alleviates congestion-related risks across all stations and promotes spatial equity in passenger travel, even under varying levels of passenger compliance, demand, and train delays. It can thus be recommended as a self-organizing and easily implementable passenger flow control method.

High-speed railways (HSR) and airlines can complement each other in long-distance journeys served by connecting flights through a hub. To encourage airline-HSR cooperation and promote intermodal transport services, we examine two capacity agreements under the capacity limitations of HSR and/or airlines and their effects on ticket pricing, passenger volume, travel time, frequency, consumer surplus, social welfare, and profitability. In a capacity sell agreement, HSR sells its capacity to airline (HSA), and airline provides the bundling service. In a capacity purchase agreement, HSR purchases airline capacity (HPA) and offers the bundled service. We first analytically compare two capacity agreements with no-cooperation cases, and find that the joint ticket price under the HSA agreement offers a price advantage over individual ticket purchases in non-cooperation scenarios only when HSR is capacity-constrained. Although the HPA agreement also exhibits a similar joint price advantage as the HSA agreement, it requires the stringent condition that both transport modes have limited capacities. Regarding consumer surplus, it tends to increase under the HSA agreement when both transport modes have capacity limitations or both do not have capacity limitations. However, the increasing trend can be relaxed to different capacity limitation conditions under the HPA agreement. We then directly compare the HSA and HPA agreements through simulations. The results indicate that the HSA agreement yields higher consumer surplus and profits than the HPA agreement, while the HPA agreement achieves greater social welfare.

Given the dynamic and unevenly distributed metro or urban rail passenger demand, this paper investigates the train timetabling optimization that is responsive to demand fluctuations while also improving energy efficiency with a flexible train composition mode. For the studied problem, a mixed-integer nonlinear programming (MINLP) model is first developed to simultaneously optimize the number of train compositions (e.g., train carriages), the train headways, and the optimal speed profile selection decisions over the planning time horizon to minimize passenger waiting time and energy consumption. The nonlinear model is then reconstructed through a series of linearization techniques into an equivalent mixed-integer linear programming (MILP) model that can be solved by commercial MILP solvers. Furthermore, a customized heuristic algorithm employing Variable Neighborhood Search (VNS) is developed to produce high-quality solutions for large-scale problems. To demonstrate the effectiveness of the proposed model and algorithm, two numerical examples are presented, i.e., a small example and a real-world example based on the Yizhuang metro line. Computational findings demonstrate that our method significantly reduces energy consumption while maintaining service quality, thus contributing to the advancement of sustainable urban transportation systems, in contrast to existing methods reliant on fixed train compositions.

To accommodate the uneven spatio-temporal distribution of passenger demand and improve the maximum transportation capacity for high-speed railway (HSR) corridors, this study proposes a demand-oriented flexible train composition strategy. The proposed strategy allows flexibility in selecting the number of train composition units (e.g., train carriages) for each train to accommodate the demand variations, where extra-long trains (longer than the length of the train station platform) might be utilized. Under such a strategy, a seat allocation method coupled with a train stopping position control strategy is proposed to avoid the need for cross-carriage walking during passenger boarding or alighting (for those extra-long trains). The train stopping position control strategy specifies the carriages that dock at or extend beyond the station platform, while the seat allocation method allocates seats within each carriage to specific origin-destination pairs. The studied problem can be formulated as an integer nonlinear programming model, where the weighted sum of the HSR operating cost and passenger travel time cost is minimized. The proposed model is then reformulated into an integer linear programming model via a series of linearization techniques. A tailored heuristic algorithm based on the variable neighborhood search (VNS) and GUROBI solver is designed to produce high-quality solutions for large-scale problems. Two numerical examples, i.e., a small example and a real-world Shanghai-Hangzhou HSR line example, are presented to illustrate the effectiveness of the proposed approach. The computation results show that, in comparison to traditional fixed and flexible train composition strategies without extra-long trains, the proposed strategy can significantly reduce the total operating cost and total travel time cost.

Time-varying demand distribution (TDD) is a critical input data for operation and management in HSR systems. This paper proposed a bi-level model to estimate the TDD with the ticket booking date and using the schedule-based User Equilibrium (UE) assignment. The up-level aims to determine the TDD with maximum entropy value and minimal error between the path flow (ticket booking volumes) and the corresponding equilibrium flows (determined from lower-level); the lower-level is a schedule-based UE assignment with rigid capacity constraints to reflect the interactions of ticket booking choices behaviors between different OD pairs in the HSR networks, and further, the advance booking cost is considered endogenously as a part of passenger choice equilibrium. The bi-level model is converted into a single-level model through equivalent complementary constraints. Then, based on linear relaxation, the single-level model is transformed into a mixed-integer quadratic program (MIQP). Furthermore, in order to improve the computational efficiency of the MIQP, the approach of reducing the calculation size of our problem is proposed. By solving the MIQP we get the information about the upper and lower bounds of our original problem, and then a global optimal solution algorithm with four piecewise interval strategies is proposed. The effectiveness and applicability of the proposed algorithm are illustrated with a simple case and three real-world cases.

Subsidization is vital for supporting the full-life-cycle operation and sustainable development of urban rail transit (URT) systems. It is crucial for ensuring the financial viability of transit agencies, boosting ridership, and facilitating sustainable access to public transportation for urban populations. This paper analyzes three different subsidy schemes (namely, fixed subsidy, fare-based subsidy, and distance-based subsidy) for a URT system to maximize social welfare. Firstly, a benchmark model is formulated with a fixed subsidy scheme and the optimal conditions for fares, operating frequency, and subsidy are derived and determined by the Lagrange multipliers method. Two additional subsidy models are then developed based on fare and trip distance, respectively. Compared with the fixed subsidy scheme, the results show that (i) the effects and system performance of the fixed subsidy scheme are worse than for the other two subsidy schemes; (ii) the fare-based subsidy scheme has the highest social welfare and passenger utility among the three subsidy schemes; and (iii) the distance-based subsidy scheme is the most profitable and requires the least subsidy. In this paper, we explicitly derive the Karush-Kuhn-Tucker (KKT) Conditions for the three different subsidy models, providing valuable insights into the precise subsidization of URT systems. Additionally, computational experiments demonstrate the significance of adopting an appropriate subsidy scheme to maximize social welfare in public transportation.