## 6 Example Solution and Analysis
### Sensitivity Analysis
#### 6.3.1 Effectiveness of Improved Grey Wolf Algorithm
A comparison between the initial solution by GWO and its three variant algorithm is shown below in Table 7.  
Examining the upper right iteration diagram of Figure 9, we find that the large-scale neighborhood search algorithm falls into the local optimal solution three times in the process of solving, and the third stagnation time is very long, which affects the solving speed of the algorithm. The ant colony algorithm is slow in the early iteration, and the grey wolf algorithm is obviously better than the other two algorithms. However, our improved grey wolf algorithm still has an advantage over all the other algorithms: compared with them, the calculation time is reduced by 16.88%, 3.39%, and 18.83%, respectively. In terms of the optimization results, the improved grey wolf algorithm also has the highest solution quality. Compared with the other three algorithms, the total delivery route length is reduced by 8.11%, 10.61%, and 11.03%, respectively. In addition, to verify the superiority of our algorithm in more general cases, we used a new set of artificial data with 202 customer nodes and compared the convergence speed between the traditional GWO and our algorithm (see the bottom charts in Figure 9). This distribution pattern is four times the size of the previous one, and it is larger than any current pharmaceutical cold chain logistics in China (2020). Owing to the sharp increase in customer nodes, it took more than 50 iterations for our algorithm to reach the optimal solution, while the traditional GWO took more than 150 iterations. Furthermore, we compared the GAP values of the four algorithms by continuously increasing the number of client nodes. Table 8 compares the GAP values, defined as (mean value of 20 runs of a certain algorithm--optimal solution obtained by all algorithms)/optimal solution obtained by all algorithms * 100%. The results indicate that the IGWO algorithm has a smaller dispersion, signifying its superior stability and performance when compared to the other algorithms. When the number of target nodes is less than 160, there is little difference in the optimization time among the four algorithms, with the IGWO algorithm exhibiting a slight advantage. However, when the number of target nodes exceeds 160, the GWO algorithm's slope increases, while the slopes of the LNS, AGO, and GWO algorithms also increase when the number of target nodes surpasses 240. This is because the three algorithms converge too early during the later iterations, resulting in local optimal solutions instead of the global optimal solution. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Results** & **Initial Solution (GWO)** & **IGWO** & **AGO** & **LNS** \\ \hline Region 1 path length/km & 1223.09 & 1191.965 & 1632.375 & 1771.365 \\ Region 2 path length/km & 910.7994 & 1212.071 & 911.97 & 995.585 \\ Region 3 path length/km & 1024.642 & 987.709 & 1197.331 & 1172.693 \\ Region 4 path length/km & 1554.866 & 896.957 & 1621.225 & 1193.375 \\ Region 5 path length/km & 940.924 & 970.818 & 1092.455 & 973.78 \\ Region 6 path length/km & 1306.518 & 995.585 & 966.274 & 1371.156 \\ Region 7 path length/km & 1459.393 & 1482.041 & 1233.797 & 1218.066 \\ Algorithm running time/s & 11.8979 & 9.889 & 10.34 & 12.1818 \\ Total delivery distance/km & 8420.227 & 7737.146 & 8655.427 & 8696.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison table of four algorithms. The comparison reveals that the IGWO algorithm outperforms the others when optimizing more target nodes, demonstrating higher efficiency and a better path planning ability in complex environments. It provides superior path optimization compared to the other comparative algorithms, verifying the feasibility and effectiveness of the proposed improvement strategy. Figure 10 presents a comparison of the running times for the four algorithms across various numbers of nodes.