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In this paper, we develop a zonal-based flexible bus services (ZBFBS) by considering both passenger demands' spatial (origin-destination or OD) and volume stochastic variations. Service requests are grouped by zonal OD pairs and number of passengers per request, and aggregated into demand categories which follow certain probability distributions. A two-stage stochastic program is formulated to minimize the expected operating cost of ZBFBS, in which the zonal visit sequences of vehicles are determined in stage 1, whereas in stage 2, service requests are assigned to either regular routes determined in stage 1 or ad hoc services that incur additional costs. Demand volume reliability and detour time reliability are introduced to ensure quality of the services and separate the problem into two phases for efficient solutions. In phase 1, given the reliability requirements, we minimize the cost of operating the regular services. In phase 2, we optimize the passenger assignment to vehicles to minimize the expected ad hoc service cost. The reliabilities are then optimized by a gradient-based approach to minimize the sum of the regular service operating cost and expected ad hoc service cost. We conduct numerical studies on vehicle capacity, detour time limit and demand volume to demonstrate the potential of ZBFBS, and apply the model to Chengdu, China, based on real data to illustrate its applicability. [ABSTRACT FROM AUTHOR]

• A novel framework was proposed to jointly optimize demand-responsive flexible bus dispatching, pricing, and passenger assignment. • Stochastic elastic demand volume, stochastic location, and random cancellation behaviour due to competition with taxis are jointly captured in the model. • The problem is formulated as a two-stage stochastic problem with recourse and solved by a reliability-based solution method. • Sensitivity analyses are conducted on cancellation penalty, taxi price, and zone division, and the model is applied to a scenario based on real data. Order cancellation is ubiquitous in on-demand mobility pre-booking systems, which affects the pricing and operational strategies of on-demand mobility services. This paper investigates the order cancellation behaviour in a frequency-based flexible bus (FB) system, which provides customized door-to-door service, considering stochastic elastic demand volume and stochastic spatial demand distribution. A two-stage stochastic model that simultaneously optimises the FB service price, FB dispatching, and passenger assignment is developed while capturing the effect of pre-booked order cancellation probability. This cancellation probability is calculated based on the pricing and availability of street-hailing taxi and FB services. The availability of FBs is derived from the underlying stochastic demand and order assignment process, which makes the model nonlinear. To solve the mixed integer nonlinear model, a two-phase decomposition scheme is employed, and then approximation techniques are implemented to make each decomposed problem solvable by commercial solvers. Finally, a gradient decent solution algorithm is applied. Numerical studies are conducted to show the necessity of capturing the impact of order cancellation and illustrate the influence of cancellation penalty, taxi price, and zone division. Finally, a case study with Chengdu data is implemented to examine the performance of our solution algorithm for large-scale optimisation. [ABSTRACT FROM AUTHOR]

• 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. [ABSTRACT FROM AUTHOR]