## 3 Model Formulation  
From the perspective of hospital resource allocation, it is desirable to minimize the number of AMRs required to fulfill all medical requests. This necessitates enhancing the utilization of AMRs and devising optimal service routes. In this section, we formulate a mathematical programming model aimed at minimizing the economic cost incurred by the hospital. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Description** \\ \hline \(D_{1}=\{d\}\) & the set of starting points of AMRs \\ \hline \(D_{2}=\{d\}\) & the set of ending points of AMRs \\ \hline \(R\) & the set of request points \\ \hline \(C\) & the set of charging stations \\ \hline \(V_{1}=D_{1}\cup R\cup C\) \\ \hline \(V_{2}=D_{2}\cup R\cup C\) \\ \hline \(V=D_{1}\cup R\cup C\cup D_{2}\) \\ \hline \(m\) & number of AMRs participating in medical request services \\ \hline \(q_{i}\) & demand of request \(i\) \\ \hline \([e_{i}h_{i}]\) & the time window of request \(i\) \\ \hline \(Q\) & capacity of AMRs \\ \hline \(S_{i}\) & service time of request \(i\) \\ \hline \(d_{ij}\) & the distance between any two medical requests \(i\) and \(j\) \\ \hline \(\left|f_{ij}\right|\) & their floor difference between any two medical requests \(i\) and \(j\) \\ \hline \(T_{ij}\) & the travel time from request \(i\) to request \(j\) \\ \hline \(X_{pi}^{k}\) & the time when the \(k\)th AMR arrives at the request \(i\) on the \(p\)th route \\ \hline \(\gamma_{pi}^{k}\) & the time when the \(k\)th AMR starts service request \(i\) on the \(p\)th route \\ \hline \(\alpha\) & the lower bound of the AMR battery level \\ \hline \(\beta\) & the upper bound of the AMR battery level \\ \hline \(r\) & electricity consumption rate, i.e., the AMR consumes \(r\) percent of its fully charged battery level if it runs a unit time \\ \hline \(v_{q}\) & the charging rate of AMRs, i.e., the AMR increases \(v_{q}\) percent of its fully charged battery level if it is charged for a second \\ \hline \(\gamma_{pi}^{k}\) & the remaining battery level of the \(k\)th AMR when it arrives at request (or depot) \(i\) on its \(p\)th route \\ \hline \(u_{pi}^{k}\) & the remaining capacity of the \(k\)th AMR when it arrives at request \(i\) on its \(p\)th route \\ \hline \(h_{0}\) & any infinite large number \\ \hline \(\xi_{1}\) & the daily fixed cost of each AMR \\ \hline \(\xi_{2}\) & the unit travel cost of the AMR \\ \hline \(\xi_{3}\) & the penalty cost for violating the time window \\ \hline \(\epsilon\) & the probability that the time window is violated \\ \hline \(x_{pi}^{k}\) & binary decision variable, i.e., \\ \(x_{pi}^{k}=\begin{cases}1,&\text{the $k$th AMR serves requests $i$ and $j$ successively on its $p$th  route;}\\ 0,&\text{otherwise.}\end{cases}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nomenclature list. Assume that the number of AMRs participating in medical services is \(m\), where \(m\) is a decision variable. Each AMR may drive multiple routes every day. Let the set of traveling routes of the \(k\)th AMR be \(L_{k}=\left\{1,2,\cdots\right\},\)\(k=1,\ldots,m\). Assume that \(T_{ij}\) and \(S_{i}\) follow normal distributions. The \(k\)th AMR arrives at request \(i\) on its \(p\)th route at \(X_{pi}^{k}\), and it starts to serve request \(i\) at \(Y_{pi}^{k}\). Moreover, \(Y_{pi}^{k}=\max\left\{X_{pi}^{k},e_{i}\right\}\) since the AMR needs to wait until \(e_{i}\) if \(X_{pi}^{k}<e_{i}\). If the \(k\)th AMR serves requests \(i\) and \(j\) successively on its \(p\)th route, the arrival time \(X_{pi}^{k}\) is  
\[X_{pi}^{k}=Y_{pi}^{k}+S_{i}+T_{ij},\forall i,j\in V,k=1,2,\ldots,m,p\in L_{k} \tag{1}\]  
Let the full battery capacity be 1, and the electricity consumption rate is a percentage of the battery capacity. The remaining battery level of the AMRs is denoted by a percentage of the fully charged battery level. Suppose that when the \(k\)th AMR reaches request \(i\) on the \(p\)th route (\(i\in R\)), the remaining battery level is \(\gamma_{pi}^{k}\), and the remaining capacity is \(u_{pi}^{k}\).