## 4 Solution Approach
### Large Neighborhood Search Algorithm  
The NP-hard characteristics of the VRSP problem and the nonlinear nature of the proposed ISCR model require a trade-off between solution quality and computational efficiency. Large neighborhood search (LNS) is an effective approach for solving bus operation problems, capable of rapidly converging toward an approximate optimal solution. The LNS algorithm commences with an initial solution and iteratively refines the current solution through a cycle of destroy and repair rules until a locally optimal solution is reached [37]. The primary content of LNS is illustrated in Algorithm 2. With flexible configurations of the initial solutions and destroy and repair rules, our study improves the LNS by incorporating bus operational rules relevant to service capability and regularity to effectively address the bus rescheduling problem. ```
Initialize parameters  Initial solution: Generate an initial solution \(f\) Until the costs of results remain stable within a small range  Calculate the selection probability of each position in solution \(f\) based on heuristic operators  Destroy: Apply destroy for the solution \(f\) based on the probability  Choose an allowed neighborhood to be repaired  Repair: Apply repair for the solution \(f\) based on the rule  Update the solution \(f\) with any improvements return\(f\)
```  
**Algorithm 2**LNS  
According to the above algorithm, we mainly focus on the design of the initial solution and the destroy and repair rules. The input parameters contain the connection network \(G\), \(C^{Q}\), \(C^{D}_{i,j}\), \(C^{P}\), \(C^{H}\), \(d_{max}\), \(i_{min}^{i\downarrow}\), \(h_{max}\), \(\varepsilon\), and \(\gamma\). **Initial solution:** In this study, an initial solution \(f\) is generated based on the connection network \(G\). The structure of the proposed model reveals the inherent nonlinearity in both the objective \(z_{H}\) and constraint (15), significantly increasing the complexity of the problem. In particular, \(z_{H}\) serves as an objective necessitating feedback regulation, which can be effectively managed through localized adjustments. Additionally, constraint (15) offers a higher level of flexibility. Taking these aspects into account, we opt to relax these elements, thus obtaining a feasible initial solution \(f_{k}\) by addressing the following problem:  
\[\min z^{\prime}=z_{Q}+z_{C}+z_{P}\] (23) s.t. (9) \[\sim \tag{14}\]  
Problem (23) primarily focuses on ensuring service capability, which constitutes a 0-1 linear programming problem that can be efficiently resolved using existing tools. In this paper, we utilize the Gurobi solver to address problem (23) and procure an initial solution for the subsequent recombination processes. **Destroy rules:** Service regularity is a primary challenge to be addressed during the recombination process, and it is closely associated with service frequency. Therefore, we devise a heuristic operator that considers service frequency and makes adjustments to departure times to assist in formulating recombination rules. The destroy operation is intended to identify a trip pair \((i,j)\) for adjustment. The heuristic operator is designed as the difference between the rescheduled service frequency and the baseline, as shown in Equation (24). Service frequency is characterized by trip departure intervals, with the original average value during the corresponding time periods, denoted as \(\bar{l}^{l}\), serving as the baseline. The probability of selecting a trip pair \((i,j)\) is determined by Equation (25). When a departure interval is less than the baseline, it indicates an excess of service supply, potentially leading to resource wastage, and thus necessitates an enlargement of the interval. Conversely, if the departure interval exceeds the baseline, it results in longer passenger waiting times, requiring a reduction. Similar to the expansion of the connection network, we only consider the approach of delaying trips. Therefore, \(cp_{(i,j)}\) plus or minus indicates which trip needs to be adjusted.