## 2 Problem Formulation
### Interdependency between Sub-Problems
\begin{table}
\begin{tabular}{c l} \hline \hline
**Sets and Sequences** & \\ \hline \(G=(N,E)\) & The graph representing the transportation network \\ \(N=\{0\ldots n\}\) & The set of nodes representing the base and customer locations \\ \(E=\{[i,j]\ |i_{j}\ j\in N,i\ \
eq j\}\) & The set of edges defined between each pair of nodes \\ \(CL_{m,l}=\left(N_{m,l},E_{m,l}\right)\) & The subgraph of \(G\) representing \(m^{\text{th}}\) cluster in \(l^{\text{th}}\) flying-time window; \(o(l)\)—number of clusters in \(l^{\text{th}}\) flying-time window \\ \(N_{m,l}\) & The set of nodes in the \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window, \(N_{m,l}\subseteq N\) \\ \(E_{m,l}\) & The set of edges in \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window, \(E_{m,l}\subseteq E\) \\ \hline \hline \end{tabular} Parameters Unmanned Aerial Vehicle (UAV) Technical Parameters  
\begin{tabular}{c l} \hline \(Q\) & The maximum loading capacity of a UAV [Kg] \\ \(cp\) & The empty weight of a UAV [Kg] \\ \(vg_{i,j}\) & The ground speed of a UAV from node \(i\) to \(j\) [m/s] \\ \(\delta_{i,j}\) & The angle of the vector of ground speed \(vg_{i,j}\) [degrees] \\ \(w_{i,j}^{\text{th}}\) & The airspeed of a UAV from node \(i\) to \(j\) in the \(l^{\text{th}}\) flying-time window [m/s] \\ \(\overline{v_{i,j}^{\text{th}}}\) & The maximum range of \(v_{i,j}^{\text{th}}\) [m/s] \\ \(w_{i,j}^{\text{th}}\) & The minimum range of \(v_{i,j}^{\text{th}}\) [m/s] \\ \(\overline{P_{max}}\) & The maximum energy capacity of a UAV [J] \\ \(P_{i,j}^{\text{th}}\) & The amount of energy consumed per time unit from node \(i\) to \(j\) by \(l^{\text{th}}\) UAV [J/s] \\ \(w\) & The time spent for take-offs and landings of UAVs [s] \\ \(K\) & The size of the fleet of UAVs \\ \hline \hline \end{tabular} Network Parameters  
\begin{tabular}{c l} \hline \(d_{i,j}\) & The travel distance from node \(i\) to \(j\) [m] \\ \(t_{i,j}\) & The travel time from node \(i\) to \(j\) [s] \\ \(D_{i}\) & The demand at node \(i\in N_{0}=N\cup\{0\}\) [Kg] \\  & The binary variable of crossing edges \\ \(b_{[i,j]\lambda(a,\beta]}\) & \(b_{[i,j]\lambda(a,\beta)}=\begin{cases}1&\text{when an edge }[i,j]\text{ and }[\alpha,\beta]\text{ is utilized}\\ 0&\text{otherwise}\end{cases}\) \\ \hline \hline \end{tabular} Environmental Parameters  
\begin{tabular}{c l} \hline \(H\) & The time period \(H=[0,t_{max}]\) \\ \(STW_{T}\) & The sub-mission time window \(T\): \(STW_{T}=[STW_{T},STW_{T}]\), \(STW_{T}\) / \(STWE_{T}\) is a start/end time of \(STW_{T}\) \\ \(FTW_{l}\) & The flying-time window \(l\): \(FTW_{l}=[FTWS_{l},FTWE_{l}]\), \(FTWS_{l}\)/\(FTWE_{l}\) is the start time of \\ \(FTW_{l}\) & The number of flying-time windows \\ \(vw_{l}\) & The Wind Speed in the \(l^{\text{th}}\) flying-time window \\ \(\overline{vw_{l}}\) & The Maximum range of \(vw_{l}\) \\ \(\overline{\theta_{l}}\) & The Wind direction in the \(l^{\text{th}}\) flying-time window \\ \(WUV\) & The Weather utility value \(WUV=\alpha\) (the standard deviation of the wind direction) + \(\beta\) (the standard deviation the wind speeds) \\ \(\alpha\), \(\beta\) & The Weighted parameters corresponding to wind speed and wind direction \\ \hline \hline \end{tabular} Variables Decision Variables  
\begin{tabular}{c l} \hline \hline \(x_{i,j}^{k}\) & The binary variable used to indicate if \(l^{\text{th}}\) UAV travels from node \(i\) to node \(j\) \\ \(x_{i,j}^{k}\) & \(x_{i,j}^{k}=\begin{cases}1&\text{if }k^{\text{th}}\text{UAV travels along from node }i\text{ to node }j\\ 0&\text{otherwise}\end{cases}\) \\ \(y_{i}^{k}\) & The time that kth UAV arrives at the node \(i\) \\ \(s_{0}^{k}\) & The time that kth UAV starts to fly from node 0 \\ \(c_{i}^{k}\) & The payload weight amount delivered to node \(i\) by \(k^{\text{th}}\) UAV \\ \(f_{i,j}^{k}\) & The payload weight carried by a UAV from node \(i\) to \(j\) by \(k^{\text{th}}\) UAV \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation table.