## 3 Mathematical Formulation
### Construction of the Integrated Model  
Based on the connection network, we utilize the decision variables \(e_{i,j}^{b}\) and \(T_{i}\) to construct our ISCR model. \(e_{i,j}^{b}=1\) if edge \((i,j)\in\hat{\mathbf{E}}^{b}\) is selected. \(\hat{\mathbf{E}}^{b}(i)\) contains the edges in \(\hat{G}^{b}\) pointing to trip node \(i\). Then, we propose the objective function of the ISCR model, considering both service capability and regularity. **Service capability:** Service capability focuses on the performance and successful operation of the transit schedule. The number of actually dispatched trips is a key indicator for measuring the capability of public transit services. During disruptions, it becomes crucial to ensure the completion of as many trips as possible, meaning that a sufficient number of nodes should be connected by edges within the connection network. Part of the  
Figure 2: Demonstration of the connection network. objective considering service capability is given in Formula (2), where \(R_{n}^{l}\) is the number of trips in \(\{i|S_{i}=l,i\in\mathbf{V}_{t}\}\) and \(R_{n}=\sum_{l}R_{n}^{l}\). \[z_{Q}=C^{Q}\sum_{l}(R_{n}^{l}-\sum_{b}\sum_{(i,j)\in\{(i,j)|(i,j)\in\hat{\mathbf{ E}}^{b},S_{j}=l\}}e_{i,j}^{b}) \tag{2}\]  
Furthermore, the rescheduling scheme should be easy to follow to mitigate additional risks. This entails minimizing the dispatch of vehicles and reducing changes in the assignment of vehicles to trips. The dispatch of empty vehicles, referred to as deadheading, introduces more instability into the schedule implementation and results in additional operational costs, such as idle time and fuel consumption. At the same time, extensive reassignments hinder the swift implementation of new schedules. To address these concerns, we include the cost of deadheads and reassignments in the service capability. The remaining objectives, which take service capability into account, are related to the deadheading time and the number of reassignments, as shown below:  
\[z_{C}=\sum_{b}\sum_{(i,j)\in\hat{\mathbf{E}}^{b}}C_{i,j}^{D}e_{i,j}^{b} \tag{3}\]  
\[z_{P}=C^{P}\sum_{b}\sum_{j}\biggl{[}\sum_{i_{c}(i,j)\in\hat{\mathbf{E}}^{b}(j )}e_{i,j}^{b}(e_{i,j}^{b}-p_{j}^{b})\biggr{]} \tag{4}\]  
**Service regularity:** Service regularity strives to provide a comparable level of transit service after rescheduling. Service frequency reflects the maximum waiting time of passengers, which is essential to evaluate the service. We introduce the changes in trip departure intervals to represent the variation in service frequency. The service regularity objective \(z_{H}\) is formulated as follows:  
\[z_{H}=C^{H}\sum_{l}\sum_{(i,j)\in\mathbf{E}_{sd}^{l}}\mid T_{j}-T_{i}-\hat{I}^ {l}\mid \tag{5}\]  
\[\mathbf{E}_{sel}^{l}=\{(i,j)|j=i+1,i\in\mathbf{V}_{sd}^{l}\} \tag{6}\]  
\[\mathbf{V}_{sel}^{l}=\{i|\sum_{b}\sum_{j:(j,i)\in\hat{\mathbf{E}}^{b}(i)}e_{ j,i}^{b}=1,T_{i+1}>T_{i},S_{i}=l,i\in\mathbf{V}\} \tag{7}\]  
where \(\mathbf{V}_{sel}^{l}\) represents the trip sequence in the solution that travels along line \(l\), and \(\hat{I}^{l}\) denotes the original average departure intervals of the corresponding trips. \(\mathbf{E}_{sel}^{l}\) contains consecutive trip pairs in \(\mathbf{V}_{sel}^{l}\). It is evident that \(z_{H}\) acts as the focal point for feedback regulation within the framework of a rescheduling scheme, rendering the model challenging to solve. We address the solution to this issue in Section 4. Finally, constraints are formulated for operational rules, except for the minimum bus idle time, which has already been satisfied during the construction of the connection network. The mathematical model is as follows:  
\[\min\quad z=z_{Q}+z_{C}+z_{P}+z_{H} \tag{8}\]  
s.t.