## 5 Numerical Examples
### Parameter Tuning  
Suppose that the daily fixed cost of each AMR is \(\zeta_{1}=30\) (unit: $) and the travel cost per unit distance of each AMR is \(\zeta_{2}=0.01\) (unit: \(\xi/m\)). Let \(\alpha=0\), \(\beta=0.8\). The variables \(S_{i}\) (service time) and \(T_{ij}\) (travel time) follow normal distributions and \(S_{i}\sim N\big{(}600,\sigma_{s}^{2}\big{)}\), \(T_{ij}\sim N\big{(}\mu_{t},\sigma_{t}^{2}\big{)}\), where \(\mu_{t}=\,d_{ij}/v_{r}+6+51.25\cdot I_{ij}\), \(\sigma_{t}^{2}=\sigma_{0}^{2}+\sigma_{f}^{2}\cdot I_{ij}\), and \(I_{ij}\) is an indicator function of \(\big{|}f_{ij}\big{|}\), i.e.,  
\[I_{ij}= \begin{cases}1,&\big{|}f_{ij}\big{|}\
eq 0,\\ 0,&\big{|}f_{ij}\big{|}=0.\end{cases} \]  
To guarantee the service quality, the probabilistic constraint in (10) is satisfied with a confidence level of 0.95, i.e., \(\varepsilon=0.05\). The specifications of AMRs are shown in Table 2, where \(r=\frac{1}{216}percent/s\) and \(v_{q}=\frac{1}{162}percent/s\). The AMR can work for 6 h continuously, and the full charging time is 4.5 h. Without being specified, the parameters are the same in what follows. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Specification** \\ \hline Running Speed \(v_{r}\) & \(1\)\(m/s\) \\ \hline Full Battery Capacity Level & \(1\) \\ \hline Electricity Consumption Rate \(r\) & \(1/216\ percent/s\) \\ \hline Charging Rate \(v_{q}\) & \(1/162\ percent/s\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of AMRs. All the following computation is run by MATLAB 2014a in the experimental environment as Inter(R) Core (TM)ij9-9900K CPU @ 3.60 GHz, 16.00 GB and the operating system is Windows 10. It is well known that CPLEX as a solver for optimization problems is suitable for solving optimization problems with small data. In this study, we compare the performance of the VNS algorithm with CPLEX. Specifically, we aim to determine the appropriate value for the iteration count \(N\) in the VNS algorithm. We conducted experiments using a dataset provided by Singapore Changi General Hospital, which consists of 12 medical requests. There is a depot \(d\) and a charging station \(c\), and the AMR capacity \(Q=20\) kg. The distance and floor difference between the 12 medical requests is presented in Tables 3 and 4, respectively. The detailed information about the demand and time window of each request can be found in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**Floor** & **Depot** & & & & & & & & & & & & **Charging** \\  & \(d\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & \(c\) \\ \(d\) & 0 & 1 & 6 & 2 & 5 & 3 & 4 & 4 & 3 & 5 & 2 & 6 & 1 & 0 \\ \(1\) & 1 & 0 & 5 & 1 & 4 & 2 & 3 & 3 & 2 & 4 & 1 & 5 & 0 & 1 \\ \(2\) & 6 & 5 & 0 & 4 & 1 & 3 & 2 & 2 & 3 & 1 & 4 & 0 & 5 & 6 \\ \(3\) & 2 & 1 & 4 & 0 & 3 & 1 & 2 & 2 & 1 & 3 & 0 & 4 & 1 & 2 \\ \(4\) & 5 & 4 & 1 & 3 & 0 & 2 & 1 & 1 & 2 & 0 & 3 & 1 & 4 & 5 \\ \(5\) & 3 & 2 & 3 & 1 & 2 & 0 & 1 & 1 & 0 & 2 & 1 & 3 & 2 & 3 \\ \(6\) & 4 & 3 & 2 & 2 & 1 & 1 & 0 & 0 & 1 & 1 & 2 & 2 & 3 & 4 \\ \(7\) & 4 & 3 & 2 & 2 & 1 & 1 & 0 & 0 & 1 & 1 & 2 & 2 & 3 & 4 \\ \(8\) & 3 & 2 & 3 & 1 & 2 & 0 & 1 & 1 & 0 & 2 & 1 & 3 & 2 & 3 \\ \(9\) & 5 & 4 & 1 & 3 & 0 & 2 & 1 & 1 & 2 & 0 & 3 & 1 & 4 & 5 \\ \(10\) & 2 & 1 & 4 & 0 & 3 & 1 & 2 & 2 & 1 & 3 & 0 & 4 & 1 & 2 \\ \(11\) & 6 & 5 & 0 & 4 & 1 & 3 & 2 & 2 & 3 & 1 & 4 & 0 & 5 & 6 \\ \(12\) & 1 & 0 & 5 & 1 & 4 & 2 & 3 & 3 & 2 & 4 & 1 & 5 & 0 & 1 \\ \(c\) & 0 & 1 & 6 & 2 & 5 & 3 & 4 & 4 & 3 & 5 & 2 & 6 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Floor difference between the 12 medical requests.