## 4 Numerical Experiments and Results
### Example and Parameter Setting
#### 4.2.4 Comparison with Other Algorithms  
The results of 14 examples of the two-stage VNS-TA algorithm are compared with those of the IPSO algorithm [32] and IVNS algorithm [42]. Among the 14 examples, the two-stage VNS-TA algorithm optimizes 12 solutions. Specifically, the maximum reduction value is 37.712, and the maximum optimization percentage is 2.61%. The optimal values of the other two examples are consistent with the objective function values of the current optimal solutions of the other two algorithms. In addition, the standard variance of the objective function values of the 14 examples solved by the two-stage VNS-TA algorithm is mainly between 0.3625 and 9.8541. Only the vrpnc11-S2 example has a large standard variance of 45.5913. This result indicates that the designed algorithm has good solution  
Figure 4: Comparison of the objective function value and running time for different maximum intervals of path traversal. (**a**) Comparison of optimal value, worst value and average value of objective function value respectively. (**b**) Comparison of optimal value, worst value and average value of running time respectively. stability overall. It can be observed that the two-stage VNS-TA algorithm has a strong optimization ability (Table 2). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Example of** **Calculation**} & \multirow{2}{*}{**IPSO**} & \multirow{2}{*}{**IVNS**} & \multicolumn{5}{c}{**Two-Stage VNS-TA**} \\ \cline{3-8}  & & & **Optimal** & **Value of** & **Percentage of** & **Average** & **Standard** \\  & & & **Value** & **Reduction** & **Optimization** & **Value** & **Variance** \\ \hline vrpnc1-S2 & 552.19 & 550.6958 & 550.6958 & 0 & 0.00\% & 551.5908 & 0.7308 \\ vrpnc1-S3 & 549.69 & 547.9608 & 547.9608 & 0 & 0.00\% & 548.3637 & 0.3625 \\ vrpnc2-S2 & 933.72 & **874.2201** & **868.606** & \(-\)5.6141 & \(-\)0.64\% & 872.4336 & 2.3628 \\ vrpnc2-S3 & 980.56 & **883.5585** & **878.328** & \(-\)5.2305 & \(-\)0.59\% & 883.9418 & 4.1727 \\ vrpnc3-S2 & 941.71 & **873.992** & **864.8485** & \(-\)9.1435 & \(-\)1.05\% & 870.6048 & 5.0803 \\ vrpnc3-S3 & 942.17 & **872.7455** & **872.2247** & \(-\)0.5208 & \(-\)0.06\% & 879.6754 & 6.7235 \\ vrpnc4-S2 & 1236.16 & **1091.991** & **1084.4192** & \(-\)7.5718 & \(-\)0.69\% & 1090.7992 & 3.274 \\ vrpnc4-S3 & 1272.13 & **117.096** & **1102.6068** & \(-\)14.4892 & \(-\)1.30\% & 1112.9597 & 9.6698 \\ vrpnc5-S2 & 1632.95 & **1393.437** & **1364.9778** & \(-\)28.4592 & \(-\)2.04\% & 1374.8837 & 6.147 \\ vrpnc5-S3 & 1658.33 & **1446.556** & **1408.844** & \(-\)37.712 & \(-\)2.61\% & 1418.3487 & 9.4022 \\ vrpnc11-S2 & 1317.87 & **1094.415** & **1092.8736** & \(-\)1.5414 & \(-\)0.14\% & 1150.4099 & 45.5913 \\ vrpnc11-S3 & 1327.51 & **1214.479** & **1204.437** & \(-\)10.042 & \(-\)0.83\% & 1215.6311 & 8.1814 \\ vrpnc12-S2 & 916.37 & **828.0884** & **820.5216** & \(-\)7.5668 & \(-\)0.91\% & 822.2101 & 1.2296 \\ vrpnc12-S3 & 993.57 & **924.0961** & **923.737** & \(-\)0.3591 & \(-\)0.04\% & 931.5986 & 9.8541 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the objective function values of the results of the two-stage VNS-TA algorithm and other algorithms. Figure 5: Comparison of the objective function value and running time under the different solution acceptance criteria. (**a**) Comparison of optimal value, worst value and average value of objective function value respectively. (**b**) Comparison of standard variance, range, and range percentage of objective function value respectively. (**c**) Comparison of optimal value, worst value and average value of running time respectively. (**d**) Comparison of standard variance, range, and range percentage of running time respectively. In general, the two-stage VNS-TA algorithm is effective and has a strong optimization ability in solving the MCVRP. Scenario Simulation Analysis of the Two-Stage VNS-TA Algorithm to Solve the MCVRP Considering Traffic Congestion under the Mixed Carbon Policy  
4.3.1.