Vehicle scheduling optimization for demand responsive transit with flexible stops

Flexible stops demand responsive transit is a public transportation mode that combines fixed service stops with the ability for passengers to book at flexible stops. Traditional research focuses on the planning of individual trips, while ignoring the connectivity of vehicles performing multiple trips throughout the entire operating period. This paper establishes a joint model for timetabling and vehicle scheduling within a single period, with the objective of minimizing total cost. Our model utilizes a time-space network to represent the service logic and constraints of flexible stops demand responsive transit service. Furthermore, based on the problem characteristics, an adaptive variable neighborhood search algorithm (AVNS) that integrates 8 types of neighborhood operators and an improved maximum network algorithm is designed to efficiently solve real-world large-scale complex problems. Finally, numerical experiments conducted in Guangzhou's Huangpu District demonstrate that bus utilization rate can enhance 8.7% and total cost can reduce 20% to 70%, respectively, compared with traditional two-stage method. Besides, the AVNS provides better solution quality for the vast majority of order sizes, saving over 30% of total cost for an order size of 60, compared with traditional algorithms.