Scheduling zonal-based flexible bus service under dynamic stochastic demand and Time-dependent travel time

•A zonal-based flexible bus service (ZBFBS) is scheduled under dynamic stochastic demand and time-dependent travel time.•Zonal-based time-space network is proposed for flexible bus scheduling with flexibility in arrival and departure times.•A reliability-based gradient solution method with relaxations is developed to solve this inherently non-convex problem. This paper schedules the zonal-based flexible bus service (ZBFBS) considering elastic stochastic demand, stochastic location, time-dependent travel time, and passenger time window constraints based on a scheduled-based formulation. Unlike a traditional time–space network that stipulates the precise arrival and departure times on specific nodes, a zonal-based time–space network is proposed to define the routes in terms of zonal visits of the flexible buses while allowing for flexibility in their arrival and departure times to cater for randomness. The ZBFBS scheduling problem is formulated as a two-stage decision-dependent stochastic problem with recourse. The first stage schedules the zonal visits of flexible buses and the second stage matches each passenger with either flexible bus or ad hoc service, with the latter incurring extra cost to carry the unmatched passengers. To effectively solve the problem, a state-augmented network, that integrates time and zone, is proposed to reduce the number of variables. Moreover, relaxation formulations based on vehicle types and routes are introduced, with an insertion heuristic implemented for vehicle scheduling. The problem is solved by a gradient-based solution approach. Numerical studies demonstrate the efficiency and quality of the solution methods under a variety of ride requests, as well as its advantage over the frequency-based approach in substantially reducing the ad hoc service cost. The applicability of the model is validated by solving an instance of Chengdu, China, with real data.