## 5 Numerical Examples
### Comparative Experiments
#### 5.2.1 Experiment 1  
To verify the feasibility and effectiveness of the algorithm, computational experiments are conducted by six modified Solomon instances. The tabu search (TS) algorithm is a heuristic algorithm based on neighborhood search proposed by [PERSON] [39]. It can solve a class of combinatorial optimization problems including variants of the VRP that involve \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(N\) & \(m\) & \(\sum d_{ij}\) & \(f(x)\) & **Time (s)** \\ \hline
800 & 3 & 1100 & 101.00 & 2.17 \\
1000 & 2 & 1230 & 72.30 & 2.36 \\
2000 & 2 & 1220 & 72.20 & 2.68 \\
4000 & 2 & 1190 & 71.90 & 2.93 \\
5000 & 2 & 1190 & 71.90 & 3.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The results of the VNS algorithm. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Request** & **Demand** & **Time Window** \\ \hline
1 & 4 & [8:10, 8:20] \\
2 & 4 & [8:10, 8:20] \\
4 & 4 & [8:10, 8:20] \\
5 & 4 & [8:40, 8:50] \\
6 & 4 & [8:40, 8:50] \\
7 & 4 & [10:10, 11:00] \\
8 & 4 & [10:10, 11:00] \\
9 & 4 & [10:40, 11:00] \\
10 & 4 & [10:40, 11:00] \\
11 & 4 & [10:40, 11:00] \\
12 & 4 & [10:40, 11:00] \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Demand and time window of the 12 medical requests. stochastic factors. For small instances, the performance of the VNS algorithm is compared with CPLEX. For large instances, we employed a tabu search algorithm (TS) and VNS algorithm for experimental comparison. In the case of small instances, we select the first 15 nodes from these Solomon instances. The coordinates, demands, and time windows of the nodes remain unchanged. Each node is assigned a floor information where the first five requests are located on the first floor, the sixth to tenth requests on the second floor, and so on. The AMR's capacity is the original vehicle capacity of the Solomon instance, while the other parameters remain unchanged, as shown in Table 2.  
Table 7 presents a comparative analysis of the CPLEX optimizer and the VNS algorithm to solve six modified small Solomon instances. We set the maximum runtime per event to 1800 s for CPLEX. The iteration number of the VNS algorithm is 5000, and we run ten times for each instance. Here, \(m\) represents the total number of AMRs, \(\sum d_{ij}\) denotes the sum of travel distance, \(Time\) indicates the total running time (unit: second), and \(f\) is the objective value. For the VNS, we present both the best (_Best_) and average (_Avg_) values for the following metrics: the number of AMRs, travel distance, running time, and objective value. We define \(Gap=(f-f^{*})/f*100\) and \(G^{\prime}=\left(\overline{f}-f^{*}\right)/\overline{f}*100\), where \(f^{*}\) is the best objective function value obtained by VNS, \(\overline{f}\) is the average objective function value obtained by VNS, and \(f\) is the best lower bound of the objective function of the final solution obtained by the CPLEX. Figure 8: Optimal service routes of medical AMRs with 12 requests. Table 7 shows that although the objective function value obtained through VNS does not reach the optimality achieved by CPLEX, the difference was less than 1% and the average gap (%) is \(-0.96\). This indicates the effectiveness of the VNS algorithm in finding near-optimal solutions. The VNS algorithm achieved competitive performance compared to CPLEX in terms of optimizing the objective function. We compare the VNS algorithm with ten repeated runs on different instances, and the results are shown in Table 7, where \(f^{*}\) and \(\overline{f}\) are the optimal and average values of the ten runs, respectively. The smaller the absolute value of \(G^{\prime}\), the better the robustness of the VNS algorithm. Table 7 shows that the average value of \(G^{\prime}\) is about 10%. Overall, these results highlight the effectiveness and robustness of the VNS algorithm. Table 8 provides a comparative analysis between the TS algorithm and the VNS algorithm for solving six modified large Solomon instances.