Integrating emerging shared mobility with traditional fixed-line public transport is a promising solution to the mismatch between supply and demand in urban transportation systems. The advent of modular vehicles (MVs) provides opportunities for more flexible and seamless intermodal transit. The MVs, which have been implemented, are comprised of automated modular units (MUs), and can dynamically change the number of MUs comprising them at different times and stops. However, this innovative intermodal urban transit brings with it a new level of dynamism and uncertainty. In this paper, we study the problem of jointly optimizing the timetable and the vehicle schedule within an intermodal urban transit network utilizing MVs within the context of distributionally robust optimization (DRO), which allows MVs to dynamically (de)couple at each stop and permits flexible circulations of MUs across different transportation modes. We propose a DRO formulation to explore the trade-off between operators and passengers, with the objective of minimizing the worst-case expectation of the weighted sum of passengers’ and operating costs. Furthermore, to address the computational intractability of the proposed DRO model, we design a discrepancy-based ambiguity set to reformulate it into a mixed-integer linear programming model. In order to obtain high-quality solutionss of realistic instances, we develop a customized decomposition-based algorithm. Extensive numerical experiments demonstrate the effectiveness of the proposed approach. The computational results of real-world case studies based on the operational data of Beijing Bus Line illustrate that the proposed integrated timetabling and vehicle scheduling method reduces the expected value of passengers’ and operating costs by about 6% in comparison with the practical timetable and fixed-capacity vehicles typically used in the Beijing bus system.

With the rapid development and expansion of the urban rail transit network in China, the coordination between different lines brings great challenges to the operation due to its high complexity. In order to minimize the unsuccessful transfer-passenger flows of last-train under the worse case with the given level of tolerance, a distributionally robust chance-constrained programming model is proposed for the last-train connection planning problem in urban rail transit networks under uncertain transfer-passenger demand. In particular, the probability distribution of uncertain parameters is only partially known. By analyzing the relationship between the distributionally robust optimization model and the corresponding robust optimization model, it is proved that the former is an extension of the latter. Furthermore, the original model can be reformulated into a second-order mixed-integer conic programming form under the Gaussian perturbations ambiguity set based on the limited information of expectation and variance, which can be solved by CPLEX. The results of numerical examples indicate that the proposed model can be solved to optimality quickly by CPLEX on a small network, and can effectively avoid over-conservative solutions compared to the robust optimization model and reduce unsuccessful transfer-passenger flows in the extreme situation compared with the stochastic programming model, which exhibits more robust performance.

The collaborative design of the timetable and dynamic-capacity allocation plan of emerging modular vehicles (MVs) is a promising solution to the mismatch between supply and demand in public transportation studies; however, such efforts are subject to high-level dynamics and uncertainty inherent in operating environments. In this study, we focus on the timetabling and dynamic-capacity allocation problem of MVs within the context of distributionally robust optimization under time-dependent demand uncertainty. The dynamic capacity refers to the number of modular units (MUs) comprising an MV can be potentially changed at different times and stops. A Wasserstein distance-based ambiguity set with a time-dependent and station-wise perturbation parameter is adopted to incorporate all potential distributions within a 1-Wasserstein distance for addressing the uncertainty of passenger demand. Further, a data-driven distributionally robust optimization model that considers time-varying capacity is formulated to minimize passenger waiting costs and dispatching costs of operators over all possible demand distributions within the ambiguity set. Subsequently, an expansion that allows for flexible formations of MVs assigned to each trip at each stop is proposed, and this results in more customized operational plans driven by the passenger demand. To improve the computational efficiency of realistic problems, we design a customized integer L-shaped method to exactly solve the models, which incorporates a class of valid equalities to further speed up the computation. The effectiveness of the proposed approaches in reducing the costs for both passengers and operators compared with the practical fixed-capacity operations is verified by real-world case studies based on the operating data of Beijing Bus Line 468. Furthermore, the superiority of the distributionally robust optimization method in comparison to the stochastic programming and the robust optimization approaches is demonstrated.

The passenger demand and operating environment of the urban public transport system are highly uncertain in time and space due to external disturbance, bringing great challenges to the operating organization. To enhance the ability of the bus system to deal with the impact of the two-fold uncertainties rooted in the passenger demand and the operating scenarios, a distributionally robust optimization method of the single-line bus timetabling problem is proposed in this paper. A discrete set of scenarios is used to describe the uncertain demand, and a multi-scenario distributionally robust optimization (DRO) model is established to minimize the excepted number of detained passengers and conditional-value at risk (CVaR) by taking account of wide-ranging constraints. For the convenience of computing, a fuzzy set of uncertain quantities is constructed with the limited known distribution information. On this basis, dual theory and conventional linearized approaches are then employed to transform the original model into a mixed-integer linear programming form. Finally, a case study of a bus line in Beijing is conducted to demonstrate the effectiveness and efficiency of the proposed model. The results show that the linear model obtained from equivalent transformation can be quickly solved to optimality by the GUROBI optimization soft package, and the timetable obtained based on the DRO model can effectively deal with the double uncertainties. In addition, compared to the SO (stochastic optimization) model, with the increase of uncertainty, the distributionally robust optimization approach is insensitive to various possible uncertain scenarios, which is expected to improve the stability of the public transport system.

Bus timetables play an important role in improving the level of service and reducing operations costs of a bus transit system. Without dedicated bus lanes, bus travel times, which are important input data for bus timetabling, are usually time-dependent due to recurrent traffic congestion. However, few studies on bus timetabling have explicitly considered such travel time time-dependency in creating timetables. This paper addresses the problem of how to optimally modify an existing single-line bus timetable by slightly shifting vehicle departure times at the departure terminal and holding vehicles at other stops taking into account time-dependent travel times. The problem is mathematically formulated as a nonlinear programming model. According to the special structure and properties of the model, a derivative-free constrained compass search algorithm with revised step-size updating rule is applied to solve it. A case study of a bus line in Beijing, China is conducted to demonstrate the effectiveness and efficiency of the proposed model and solution algorithm. The case study results show that by utilizing the proposed methodology the optimized bus timetable can significantly reduce the total passenger travel time and improve ridership comfort, while rarely increasing the average vehicle cycle time. This study offers a promising and practical methodology for optimizing single-line bus service taking into account time-dependent travel times.

Considering the continuous arrival characteristics of outside arrival passenger flow and the impulsive feature of transferring passengers, this paper investigates the metro train timetabling and passenger flow control strategy under the influence of transferring passengers. This paper formulates an integer nonlinear collaborative optimization model for the metro train timetabling and passenger flow control strategy, which aims to minimize the number of detained passengers. The proposed model is reformulated into an integer linear programming model by introducing the 0-1 decision variables. A case study of a real-world urban rail transit line is performed to verify the effectiveness of the proposed approach, which is solved by the CPLEX software. The results reveal that, the proposed approach has good optimization quality and computational efficiency. Compared to the plan only optimize timetables, the obtained plan reduced the number of detained passengers by 17.69% and the service level was significantly improved. This study provides theoretical support for the high-quality operation of the urban rail transit system.