A distributionally robust optimization model for the bus timetabling problem under two-fold uncertainties

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. © 2023 Northeast University. All rights reserved.