## 1 Introduction  
A stable and reliable bus service system is crucial for urban mobility, providing convenient travel services for residents. However, buses are susceptible to disruptions due to the heterogeneous traffic conditions. Unforeseen events such as vehicle breakdowns frequently disrupt bus schedules, resulting in severe delays and even service interruptions. These disruptions prevent transit services from successful implementation as planned, and running a bus bridging service using an experience-based response strategy is a common solution. However, delaying or canceling services is often necessary when there are insufficient spare buses or widespread disruptions occur. The delays and cancellations of services increase passenger waiting times, making services irregular and negatively impacting both bus service quality and passenger satisfaction. Although several predictive-based offline or scroll generation methods have been developed to enhance schedule robustness [1], it is important to note that the real-world events leading to delays and cancellations mentioned above have not been properly considered due to their inherently random nature of occurrence. Therefore, bus rescheduling, which adjusts initial fleet plans such as timetables and bus schedules in response to the current conditions of a public transportation system, is imperative to restore transit functionality and ensure stable operation for the remainder of the day. Bus rescheduling aims to refine the original schedule to minimize the impacts of disruptions on both service capability and regularity [2; 3]. A small case is illustrated in Figure 1, where Figure 1a shows a schematic diagram of the actual travel processes of two bus lines, and Figure 1b-f shows the corresponding operational schedule. \(trip_{2}\), \(trip_{4}\), \(trip_{6}\), and \(trip_{7}\) are trips along line 1, whereas \(trip_{1}\), \(trip_{3}\), and \(trip_{5}\) are trips along line 2. The departure time is indicated above the top-left corner of each trip node, whereas the arrival time is indicated above the top-right corner. Each bus serves multiple trips; for example, bus 1 operates \(trip_{1}\to trip_{4}\), bus 2 manages \(trip_{3}\to trip_{6}\), and bus 3 is assigned \(trip_{2}\to trip_{5}\to trip_{7}\). And, a deadhead is the movement of an empty bus to a destination without any passengers, which helps relocate bus resources [4]. In this case, bus 2 experiences a breakdown while starting \(trip_{3}\) at 7:50, impacting the remaining trips (i.e., \(trip_{3}\) and \(trip_{6}\)). At that moment, bus 1 is preparing to start \(trip_{4}\), while bus 3 is in the process of \(trip_{2}\). In scenario 1, no additional operations are applied, leading to the cancellation of both \(trip_{3}\) and \(trip_{6}\) and an increase in the maximum departure interval of line 1 by 35 min (100% of the original). Scenario 2 employs an intuitive rescheduling method, which completes \(trip_{6}\) by assigning it to an available vehicle (bus 1 in this case) without affecting other trips. However, the maximum departure interval still increases by 35 min (100% of the original). With overall consideration of the entire schedule and vehicle resources, all subsequent trips are completed following a regular departure interval in scenario 3, with only a 10-minute increase in the maximum departure interval (28.57% of the original), resulting in minor impacts on the original schedule. From this small case, two interesting conclusions can be drawn: (1) bus services can be recovered through rescheduling when disruptions occur, and (2) the comprehensive consideration of both subsequent schedules and vehicle resources is essential for ensuring both service capability and regularity. However, the rescheduling process involves the comprehensive consideration of various complex factors, including trip departure times, the number of trips, and bus assignments.