## 5 Numerical Examples
### Case Study  
As mentioned in the introduction, this study was provoked by the actual demands of Singapore Changi General Hospital using AMRs to perform some service. This section addresses the realistic problem encountered at the hospital with the approach proposed in this paper.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Distance** & \(d_{ij}\) & \(2d_{ij}\) & \(4d_{ij}\) & \(6d_{ij}\) & \(8d_{ij}\) \\ \hline Number of AMRs & 2 & 2 & 4 & 4 & 8 \\ Charging Times & 0 & 0 & 0 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 10: The influence of distance on charging times and total number of AMR.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(n\) & \(m\) & \(\sum d_{ij}\) & \(f\) \\ \hline
1 & 2 & 1190 & 71.9 \\
10 & 2 & 1190 & 71.9 \\
100 & 4 & 1180 & 131.8 \\ \hline \hline \end{tabular}
\end{table}
Table 9: The effect of variance of variables on the AMR scheduling problem.  
A set of homogeneous AMRs is utilized to fulfill specific medical requests at the hospital. These requests involve delivering drugs from a pharmacy to 32 wards twice daily. The 64 recorded requests are denoted by 1, 2, \(\cdots\), 64, where requests \(i\) and \(32+i\) belong to the same ward. The depot, denoted by 0, serves as both the starting point and the location for charging the AMRs. Details regarding the demand, mean service time, and time window of these 64 requests can be found in Table A1 of the Appendix A. Additionally, Tables A2 and A3 are the distance and floor difference between the 32 wards, respectively. Let \(\sigma_{t}^{2}=40+160l_{ij}\), \(\sigma_{s}^{2}=360\) and the initial battery level of the AMRs be 50%, while the remaining parameters are the same as those in Section 5.1.  
Table 11 presents the optimized service routes, where the first AMR runs three routes, the second AMR runs three routes, and the third AMR runs two routes. The computed metrics for each route include travel distance, total load, electricity consumption, and the mean arrival time of requests. For instance, the first route of the third AMR is 0\(\rightarrow\)45\(\rightarrow\)13\(\rightarrow\)53\(\rightarrow\)21\(\rightarrow\)17\(\rightarrow\)49\(\rightarrow\)41\(\rightarrow\)31\(\rightarrow\)6  
Note that in some practical scenarios, demands may arise dynamically, such as garbage collection in wards, and stochastic load demands may arise during the service process. In such cases, AMRs need to adjust their service routes based on new arrival requests. Future research can extend the AMR scheduling problem to study the optimization of routes with stochastic load demands and dynamic requests.  
Conceptualization, L.C., N.Z., K.W. and Z.C.; methodology, L.C., N.Z., K.W. and Z.C.; software, L.C.; validation, L.C.; writing--original draft preparation, L.C.; writing--review and editing, L.C., N.Z. and K.W.; visualization, L.C.; supervision, L.C. and N.Z. All authors have read and agreed to the published version of the manuscript.  
This research was supported in part by State Key Laboratory of Industrial Control Technology (Grant No. ICT2021B51).  
Not applicable.  
The data used to support the findings of this paper are available from the corresponding author upon request.  
The authors declare no conflict of interest.