## 2 Related Works  
Rescheduling is an essential aspect of transportation systems, including buses, metros, railways, and aircraft [10; 11]. Many studies have been carried out on formulations, model establishment, and solution approaches. To better formulate the problem and design a mathematical model, researchers [6; 8] often construct an underlying network based on trip information and bus status. Time-space networks (TSNs) and connection networks (CN)s are the most popular basic structures. A TSN offers exact time and space information of buses, making it easy to describe the actual state of a public transportation system [12]. [PERSON] et al. [13] developed routines based on TSNs to reduce the wastage of bus resources. And, some studies [14; 15] have combined variable speed and TSNs to achieve a more practical rescheduled timetable. However, TSNs focus more on the details of the travel process, which makes rescheduling complicated. Compared with a TSN, a CN is constructed based on relationships between trips and vehicles and has fewer elements, offering a more intuitive and brief computational representation [16; 17]. It has been proven that CNs are more conducive to the development of effective methods when travel information is given [4; 12]. However, this approach limits the solution space to specific trip departure times. Recently, researchers [9; 18] have found that extending the connection network with time-delay windows can further improve the quality of solutions. In this paper, our main focus is on bus assignments and the precise modification of departure times, rather than the intricacies during a single trip. Therefore, we adopted the concept of a connection network with delayed trip copies to formulate the rescheduling problem in a clear and effective manner. The purpose of bus rescheduling is to minimize the impact of disruptions on the service capability and regularity of public transportation systems. Early works mainly considered recovering operations by canceling trips [2; 8], which significantly reduced bus service reliability. Thus, [PERSON] et al. [4] reconsidered real-world situations and expanded previous models with reassignments to ensure the executability of trips. However, trips are still prone to failure due to insufficient vehicles. Some studies [9; 18; 19] have introduced time-delay windows into models and obtained higher-quality rescheduled plans through reasonable time-sharing of vehicle resources. [PERSON] [20] additionally calculated the delays of each train to reduce the aftereffects of rescheduling. Based on the above aspects, researchers have constructed models with objectives concerning trip delays, trip cancellations, and deviations from bus schedules and made some achievements in guaranteeing service capability. Nevertheless, few studies have paid attention to service regularity after rescheduling, which is significant for public transit passengers. [PERSON] [21] perceived the regularities of train arrivals and departures as essential to measuring the performance of a transit system. [PERSON] [3; 22] raised the idea that service regularity is directly related to the excessive waiting times of passengers. Accordingly, researchers [23; 24] have built models considering passenger delays and waiting times. Moreover, [PERSON] [25] took the maximum train-loading rate into account. Researchers [26; 27; 28; 29] have also added deviations from departure intervals to models to satisfy service requirements. Service regularity has gradually become increasingly crucial during rescheduling, which can be represented by departure intervals. However, the cancellation and retiming of individual trips within transportation services can lead to significant service gaps and disrupt service regularity. There has been a noticeable lack of emphasis on addressing the issue of maintaining service regularity both before and after rescheduling, as well as its integration with service capability.