## 4 Model Solution Based on Improved Genetic Algorithm
### Algorithm Effectiveness Analysis  
The solution results are presented in Table 6. Upon analyzing the results, it is observed that when \(r=1\) or \(r=0\), the bi-objective model reduces to a single-objective model where either maintenance cost or connection time is the optimization target. The balance between the maintenance cost and connection time can be adjusted by the weight coefficients \(\omega_{1}\) and \(\omega_{2}\). When \(\omega_{1}\) is small and \(\omega_{2}\) is large, the objective function is significantly influenced by the train maintenance schedule, resulting in the lowest train cost but suboptimal train scheduling. Conversely, when \(\omega_{1}\) is large and \(\omega_{2}\) is small, the combined optimization model's total objective is dominated by the total connection time, which minimizes the total connection time but leads to less satisfactory maintenance outcomes. The preventive maintenance and train scheduling plans can be developed by the operations planners based on the actual maintenance requirements and the weight of each objective. The optimization calculation was performed using MATLAB R2021b on a PC with 8 GB of RAM, an AMD Ryzen7 5800H processor with Radeon Graphics, and Windows 11. The improved genetic algorithm (described in Section 4.2) was used to solve the model. During the computation, the data from Tables 1-3 were incorporated into the model. The crossover probability \(p_{c}\) was set to 0.9, mutation probability \(p_{m}\) to 0.09, population size \(p\) to 120, and iteration count \(N\) to 600. Additionally, the improved genetic algorithm (IIGA) was compared with the traditional genetic algorithm (GA), differential-based optimization  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Route** & **Number of** & **Departure** & **Arrival** & **Start** & **End Station** \\ \hline R1 & 12 & 5:30 & 21:37 & A & C \\ \hline R2 & 4 & 5:35 & 12:14 & B & C \\ \hline R3 & 5 & 5:40 & 12:19 & C & B \\ \hline R4 & 8 & 5:55 & 21:32 & A & C \\ \hline R5 & 7 & 6:10 & 19:29 & B & \\ \hline R6 & 7 & 6:25 & 19:47 & C & A \\ \hline R7 & 11 & 6:40(DBO) algorithm, and chaotic genetic algorithm (IGA). The DBO algorithm is an optimization algorithm that explores the solution space through differential operations and effectively guides the search process to avoid falling into local optimality. The IGA is an optimization method that combines chaotic mapping with the GA. The IGA improves the initialization, selection, crossover, and mutation operations in the genetic algorithm by introducing a chaotic sequence, aiming to improve the global search ability of the algorithm and avoid falling into the local optimal solution, thereby improving the efficiency and accuracy of the optimization process. As shown in Figure 10, the improved genetic algorithm converges faster than the traditional genetic algorithm, chaotic genetic algorithm, and DBO algorithm, reaching a  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Weight \(\omega_{1}\)** & **Weight \(\omega_{2}\)** & **Objective Value \(f\)** & \begin{tabular}{c} **Total Maintenance** \\ **Count \(y\)** \\ \end{tabular}  & \ \begin{tabular}{c} **Maintenance Cost \(f_{1}\)** \\ **(10,000 CNY)** \\ \end{tabular}  & \begin{tabular}{c} **Connection Time \(f_{2}\)** \\ **(Minutes)** \\ \end{tabular}  \\ \hline
1.0 & 0.0 & 3975 & 27 & 21.36 & 3975 \\ \hline
0.9 & 0.1 & 3752.369 & 27 & 20.69 & 4167 \\ \hline
0.8 & 0.2 & 3533.594 & 28 & 19.97  
satisfactory solution set by the 458 th generation. In contrast, the DBO algorithm, chaotic genetic algorithm, and traditional genetic algorithm converged at the 479 th, 546 th, and 554 th generations, respectively. The improved genetic algorithm utilizes a tent chaotic mapping strategy for population initialization, which enhances both the convergence speed and solution efficiency due to its randomness, ergodicity, and initial value sensitivity.