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.@en

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.@en