## 3 Problem Formulation
### Problem Formulation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Description** \\ \hline \(0\) & Index of the rescue center \\ \(N\) & Set of all nodes, including the rescue sites and the rescue center \\ \(R\) & Set of all rescue locations \\ \(H\) & Set of all helicopters \\ \(TH\) & Set of transport helicopters \\ \(MH\) & Set of medical helicopters \\ \(a_{i}\) & Earliest possible rescue time of location \(i\), \(i\in N\) \\ \(b_{i}\) & Latest possible rescue time of location \(i\), \(i\in N\) \\ \(d_{ij}\) & Distance between locations \(i\) and \(j\), \(ij\in N\), \(i\
eq j\) \\ \(dm_{i}\) & Demand for the supplies of location \(i\), \(i\in N\) \\ \(dp_{i}\) & The number of survivors waiting to be rescued at the rescue site \(i\), \(i\in N\) \\ \(lt_{i}\) & Life strength for survivors at the rescue site \(i\), \(i\in N\) \\ \(cm_{k}\) & Maximum capacity for the material of transport helicopter \(k\), \(k\in TH\) \\ \(tt_{k}\) & Minimum life strength threshold of transport helicopter \(k\), \(k\in TH\) \\ \(cp_{k}\) & Maximum capacity for casualty care of medical helicopter \(k\), \(k\in MH\) \\ \(mt_{k}\) & Minimum life strength threshold of medical helicopter \(k\), \(k\in MH\) \\ \(t_{i}\) & Service duration for rescue site \(i\), \(i\in N\) \\ \(u_{ik}\) & Start time of service for rescue site \(i\), \(i\in N\), \(k\in H\) \\ \hline \hline \end{tabular} Decision variable:  
\(x_{ijk}\): if a helicopter \(k\) visits and serves the node \(i\) and travels directly to the node \(j\), \(x_{ijk}=1\); otherwise, \(x_{ijk}=0\);  
Objective:  
\[Min\sum\limits_{i\in N}\sum\limits_{j\in N}\sum\limits_{k\in H}d_{ij}x_{ijk} \tag{1}\]  
Constraints:  
\[\sum\limits_{k\in TH}\sum\limits_{j\in R}x_{ijk}=1\quad\forall i\in N \tag{2}\]  
\[\sum\limits_{k\in MH}\sum\limits_{j\in R}x_{ijk}\leq 1\quad\forall i\in N \tag{3}\]  
\[\sum\limits_{j\in R}x_{0 jk}=1\quad\forall k\in H \tag{4}\]  
\[\sum\limits_{i\in R}x_{i0k}=1\quad\forall k\in H \tag{5}\]  
\[\sum\limits_{i\in N}\sum\limits_{j\in N}dm_{i}x_{ijk}\leq cm_{k}\quad\forall k \in TH \tag{6}\]  
\[\sum\limits_{i\in N}\sum\limits_{j\in N}dp_{i}x_{ijk}\leq cp_{k}\quad\forall k \in TH \tag{7}\]  
\[a_{i}\leq u_{ik}\leq b_{i}\quad\forall i\in N,\forall k\in H \tag{8}\]  
\[lt_{i}e^{-0.037u_{ik}}\geq tt_{k},\forall i\in N,\forall k\in TH \tag{9}\]  
\[lt_{i}e^{-0.037u_{ik}}\geq mt_{k},\forall i\in N,\forall k\in MH \tag{10}\]  
\end{table} 
Table 1: Notation used in the R-VRPTWLST model. \[x_{ijk}\in\{0,1\}\quad\forall i,j\in N,\forall k\in H \tag{11}\]  
Equations (1)-(11) outlines the constraints and objectives of the model as follows: Equation (1) aims to minimize the total driving distance of all helicopters. Equation (2) guarantees that each rescue site is serviced exactly once by transport helicopters. Equation (3) specifies that each rescue site is serviced exactly once by medical helicopters. Equations (4) and (5) ensure that all helicopters depart from and ultimately return to the rescue center. Equations (6) and (7) enforce the constraint that helicopters do not exceed their maximum carrying capacity when transporting supplies and rescuing injured individuals along their route. Equation (8) requires that the service start time of the transport helicopter at each rescue site falls within the specified time window. Equations (9) and (10) guarantee that the life strength of survivors remains above the threshold. Finally, Equation (11) restricts the decision variables.