## 1 Introduction
Numerous studies have demonstrated the NP-hard nature of the bus rescheduling problem, which involves timetabling and vehicle scheduling, in different scenarios, such as single-line rescheduling [3; 5; 6], single- depot rescheduling [4], and multi-depot rescheduling [7]. Some researchers [2; 8] have formulated the vehicle rescheduling problem (VRSP) based on information on the remaining trips and operating buses. The VRSP is a combinatorial optimization problem that focuses on minimizing trip cancellations and deviations from the original bus schedules when recovering operations. With consideration of trip delays, studies have derived practical schemes through cancellations and reassignments to guarantee the successful operation of as many remaining trips as possible. However, trip delays are predefined based on the idle time of a single bus and remain unmodified. Individually adjusting each trip's departure time would overlook the overall trip departure distribution, potentially resulting in irregular departure intervals and destabilizing the bus service. Therefore, it is necessary to modify departure times with consideration of neighboring trips. The hysteresis effects of disruptions also make it difficult to determine a long-term viable solution for the remaining transit operations. When disruptions occur, their impacts propagate backward along schedules, turning rescheduling into a long-term decision-making challenge for the entire day. Manual rescheduling is the most common method during actual operation. According to controllers' experience, the impacts of disruptions can be mitigated through local schedule adjustments. However, this approach only provides limited optimization for trips departing within a short time frame, thereby giving rise to subsequent operational challenges. For instance, inadequate consideration of the overall schedule may lead to numerous cancellations due to the inappropriate allocation of bus resources, resulting in significantly divergent plans from the original schedule. Moreover, the rescheduling problem becomes more intricate and difficult to solve manually as the number of buses and trips increases. And, the NP-hard nature of the problem also necessitates the introduction of more effective solution approaches. To address the above challenges in the VRSP problem, we propose an integrated service capability and regularity (ISCR) model, and an improved large neighborhood search (LNS) algorithm based on the dynamic programming (DP) framework, named DP-LNS, is introduced to solve the problem effectively. Our main contributions are as follows:  
1. Service regularity is innovatively combined with service capability in constructing the ISCR model. Rather than separate adjustments in the timetable before rescheduling, we directly introduce the overall departure regularity into the formulation of the objective function, which simultaneously achieves more sufficient and regular services. Figure 1: Schematic illustration of rescheduling scenarios. (**a**) Diagram of the actual travel processes of two bus lines. (**b**) Original schedule. (**c**) Free operation. (**d**) Intuitive rescheduling. (**e**) Comprehensive rescheduling. (**f**) Labels of Figure 1b–e. The rescheduling problem is formulated with the dynamic programming framework, and an improved large neighborhood search (LNS) algorithm is applied based on it. To achieve long-term optimization for the subsequent operations of the day, dynamic programming decomposes the problem into a series of stages over time and considers the aftereffects of previous periods. In addition, by incorporating operational rules, we improve the LNS algorithm with scheduling-based heuristic operators to solve the problem effectively. 3. Several experiments are conducted under a real public transit operation scenario in Shenzhen, China. The method offers an effective and robust rescheduling scheme for bus services, facilitating the recovery of operations and ensuring stable service provision.