When urban rail transit (URT) does not provide 24-hour services, passengers who travel at late night may not be able to reach their destinations with only URT trains. As a result, passengers have to find alternative transport means, or combine URT trains with other transport services to fulfill their journeys. This paper investigates the integrated optimization of last train timetabling and bridging service design with consideration of passenger path choices. Two bridging services are considered: taxis and buses. Based on pre-constructed path sets, a bi-objective mixed-integer nonlinear programming (MINLP) model is developed, aiming at minimizing total passenger travel time and total passenger travel cost. To reduce the model scale and improve solution efficiency, three path dominance principles are proposed to remove redundant passenger paths without loss of optimality. An adaptive iterative algorithm is designed to obtain the Pareto frontier curve. The proposed model and solution methods are demonstrated on the Chengdu URT network. Results indicate that passenger travel costs and travel times can be significantly reduced by the integrated optimization. It also provides passengers with a safer night travel environment due to the reduction in passenger travel times in taxis.

With the emergence of innovations associated with public transport (PT) services, such as Mobility-as-a-Service, demand responsive transit, and autonomous vehicles, the door-to-door PT journey is achievable via multiple transfers between and within different PT modes. As such, seamless transfers between different modes of public transportation become an increasingly important factor for the attractiveness of PT services. At the same time, recent developments in travel time prediction methodologies offer new, reliable data sources for the optimization of PT operations. This work, with the consideration of these two elements, develops a mixed integer linear programming model for the PT schedule synchronization problem. The novelty is threefold. First, a novel concept of path-oriented scheduling is proposed. The path transfer time is explicitly formulated and minimized to provide a seamless travel experience considering that the emerging multimodal mobility inevitably induces multiple transfers. Second, time-dependent travel time data is also utilized in the model, which allows us to harness new and more representative data sources for improving PT services. Third, in order to complement the increase in computational complexity as a result of the utilization of time-dependent travel time data, three novel valid inequalities (VIs) are derived. Numerical studies show that the use of time-dependent travel time data is beneficial in terms of reducing path transfer times, when compared to using the mean historical travel times. The numerical study also reveals a tradeoff between the maximum allowable path transfer time and trip time. Using simulation studies on three bus lines in Copenhagen, we demonstrate that the valid inequalities could reduce the computation time by 8.5% on average, where the maximum reduction of computation time could reach 54.0%. The proposed valid inequalities are benchmarked against two classes of valid inequalities in the literature. It is found that the proposed valid inequalities could outperform those in the literature. We also found that further improvement in computational performance can be attained by using a combination of the proposed valid inequalities.

In cities where the urban rail transit (URT) systems do not provide 24-h services, passengers may not be able to reach their destinations if the last train services have closed by the time they arrive at the transfer stations. This paper aims to seek a well-coordinated last train timetable that can transport as many passengers as possible to their destinations (referred to as reachable passengers) and also transport those passengers who cannot reach their destinations (referred to as unreachable passengers) to the stations as close as possible to their destinations. A bi-objective mixed-integer linear programming (MILP) model is developed to maximize the number of reachable passengers and minimize the total remaining travel distance of all passengers. The augmented ε-constraint method is applied to generate all Pareto optimal solutions of the bi-objective MILP model. Numerical experiments were implemented in the Chengdu URT network. Results indicate that compared to the current-in-use timetable, the optimized timetable by our methods significantly increased the number of reachable passengers and meanwhile reduced the average remaining travel distance of unreachable passengers. In addition, we discussed two possible strategies to improve passengers’ destination reachability, which are encouraging passengers to arrive early at their origin stations, and optimizing the timetable of last trains and non-last trains at the same time.