## 3 Problem Modeling  
The objective function  
\[\textit{maximize}:\ \ \ \underset{i\in N_{0}}{\text{min}}\ \{\sum_{n=1}^{\psi(m,l)} \ \sum_{m=1}^{\sigma(l)}\ \sum_{l=1}^{\Phi}\ \text{Cs}_{i,n,m,l}\}. \tag{1}\]  
The objective is to maximize the customer satisfaction levels for all the customers by delivering as much as demanded as possible. _The constraints_  
_Arrival time at nodes:_ Relationship between \(x_{i,j}^{k}\) and \(y_{i}^{k}\). \[(x_{i,j}^{k}=1)\Rightarrow(y_{j}^{k}=y_{i}^{k}+t_{i,j}+w),\ \ \forall(i,j)\in N,\ \forall k\in K. \tag{2}\]  
If an edge \((i,j)\) is utilized by UAV \(k\) in a given flying-time window then the arrival time \(y_{j}^{k}\) to node \(j\) is equal to the sum of travel time \(t_{i,j}\) between node \(i\) to \(j\), time spent for take up landing \(w\) and the arrival time \(y_{i}^{k}\) to node \(i\) (2). _Collision avoidance:_ When two edges are crossing (\(b_{[i,j];[\alpha,\beta]}=1\)), UAVs occupied in those two edges (\(x_{i,j}^{k}=1\) and \(x_{\alpha,\beta}^{v}=1\)) must not be occupied at the same time. \[\left[(b_{[i,j];[\alpha,\beta]}=1)\wedge\left(x_{i,j}^{k}=1\right)\wedge\left( x_{\alpha,\beta}^{v}=1\right)\right]\Rightarrow\left[(y_{j}^{k}\leq y_{ \alpha}^{v})\vee(y_{i}^{k}\leq y_{\beta}^{v})\right],k,v=1\ldots K,\ k\
eq v\ \textit{and}\ \{i,j\},[\alpha,\beta ]\in E_{m,j} \tag{3}\] \begin{table}
\begin{tabular}{l l} \hline \hline Output Variables & \\ \hline \(S_{n,m,l}\) & The \(n^{\text{th}}\) sub-mission in the \(m\)th cluster in the \(l^{\text{th}}\) flying-time window, \\ \(S_{n,m,l}\) & \(S_{n,m,l}=(\ R_{n,m,l},P_{n,m,l},C_{n,m,l});\psi(m,l)\) – the number of sub-missions in the \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window \\  & the \(l^{\text{th}}\) flying-time window \\ \(C_{n,m,l}\) & The customer satisfaction levels of \(n^{\text{th}}\) sub-mission in \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window \\ \(R_{n,m,l}\) & The routes of the \(n^{\text{th}}\) sub-mission in the \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window \\ \(P_{n,m,l}\) & The schedules of \(n^{\text{th}}\) sub-mission in \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window \\  & The \(s^{\text{th}}\) scenario in the \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window, \\ \(RL_{s,m,l}\) & \(RL_{s,m,l}=(R_{0,m,l},P_{n,m,l},C_{s,m,l});\psi(m,l)\) – the number of scenarios in the \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window \\  & the \(l^{\text{th}}\) flying-time window \\ \(Ro_{r,m,l}\) & The \(r^{\text{th}}\) route in the \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window \\ \(Pl_{r,m,l}\) & The schedule of the \(r^{\text{th}}\) route in the \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window \\  & The customer satisfaction levels of the \(r^{\text{th}}\) route in the \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window \\ \(CS_{r,m,l}\) & window \\  & The customer satisfaction level of the \(i^{\text{th}}\) node of the \(n^{\text{th}}\) sub-mission in the \(m^{\text{th}}\) cluster in the \(l^{\text{th}}\) flying-time window \\  & in the \(l^{\text{th}}\) flying-time window \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)_Capacity:_ The demand assigned to a UAV should not exceed its capacity. \[\sum_{i\in N_{m\downarrow}}\sum_{j\in N_{m\downarrow}}x_{i,j}^{k}c_{j}^{k}\leqslant Q,\qquad k=1\ldots K. \tag{4}\]  
Sum of all the carried weights \(c_{j}^{k}\) by UAV \(k\) should not exceed the maximum carrying payload \(Q\)  
Figure 3: Problem modeling illustration. _The flow of UAVs:_ When a UAV arrives at a node that UAV must leave from that node. \[\sum_{j\in N_{m,l}}\ x_{i,j}^{k}-\sum_{j\in N_{m,l}}\ x_{j,i}^{k}=0,\ k=1\ldots K,\ \forall i\in N_{0}=N_{m,l}\cup\{0\}\enspace. \tag{5}\]  
The sum of all the occupied edges which goes with node \(i\ (\sum_{j\in N_{m,l}}\ x_{j,i}^{k})\) should be equal to the sum of all the edges which leave from node \(i\ (\sum_{j\in N_{m,l}}\ x_{i,j}^{k})\).