# Multi-UAV Mission Allocation under Constraint
## 2 Problem Formulation
### Mathematical Model  
Parameters and decision variables are shown in Appendix A Table 1.  
The improved model of the multi-UAV mission allocation problem is stated as follows:  
\[\max f=\sum\limits_{i=0}^{N}\sum\limits_{j=0}^{N}\sum\limits_{k=1}^{M}R\left( t_{j}\right)\cdot x_{ijk}\cdot c_{i} \tag{3}\]  
where \(R\left(t_{j}\right)=C\cdot\left(R_{p}+R_{q}\left(1-e^{\left(-\beta t_{j}\right) }\right)\right)\)  
Subject to:  
\[\sum\limits_{i=0}^{N}\sum\limits_{j=0}^{N}t_{ijk}\cdot x_{ijk}+\sum\limits_{i =0}^{N}\sum\limits_{j=0}^{N}t_{j}\cdot x_{ijk}\leq T_{f} \tag{4}\]  
\[\sum\limits_{i=0}^{N}\sum\limits_{j=0}^{N}t_{j}\cdot x_{ijk}\leq T_{r} \tag{5}\]  
\[\left\{ \begin{array}{ll}\sum\limits_{j=0}^{N}\sum\limits_{k=1}^{M}x_{ijk}=1, &i=1,2,3,\cdots,N\\ \sum\limits_{j=0}^{N}\sum\limits_{k=1}^{M}x_{ijk}=M,&i=0\end{array} \right. \tag{6}\]  
Figure 1: Multi-unmanned aerial vehicle (UAVs) reconnaissance mission planning schematic diagram. \[\left\{ \begin{array}{ll}\sum\limits_{i=0}^{N}\sum\limits_{k=1}^{M}x_{ijk}=1,&j=1,2,3,\cdots,N\\ \sum\limits_{i=0}^{N}\sum\limits_{k=1}^{M}x_{ijk}=M,&j=0\end{array} \right. \tag{7}\]  
\[\sum\limits_{j=0}^{N}x_{0 jk}=1,\hskip 42.679134 ptk=1,2,3,\cdots,M \tag{8}\]  
\[\sum\limits_{i=0}^{N}x_{i0k}=1,\hskip 42.679134 ptk=1,2,3,\cdots,M \tag{9}\]  
\[\sum\limits_{i=0}^{N}\sum\limits_{k=1}^{M}x_{ijk}-\sum\limits_{i=0}^{N}\sum \limits_{k=1}^{M}x_{ijk}=0,\hskip 28.452756 ptj=1,2,3,\cdots,N \tag{10}\]  
\[a_{i}\leq T_{i}\leq b_{i} \tag{11}\]  
\[f\;x_{ijk}=1,\;then\;T_{i}+t_{i}+t_{ijk}+W_{j}=T_{j} \tag{12}\]  
\[W_{j}=\max\!\left(0,a_{j}-\left(T_{i}+t_{i}+t_{ijk}\right)\right) \tag{13}\]  
\[\sum\limits_{i=0}^{N}\sum\limits_{j=0}^{N}x_{ijk}\cdot T_{j}+\sum\limits_{i=0 }^{N}\sum\limits_{j=0}^{N}x_{ijk}\cdot t_{j}+t_{j0k}+W_{j}\leq T_{f} \tag{14}\]  
The objective function is to maximize the total UAV reconnaissance reward, where the time and resource are subject to many constraints. Equation (3) is an objective function, which indicates that each target area is maximized by multiplying the reconnaissance reward after the reconnaissance time by the value of the target area. Where \(x_{ijk}\) is the decision variable and \(c_{i}\) is the fixed coefficient that measures the strategic value of each target area. \(R(t_{j})\) is the reconnaissance reward of each target area determined by the decision variable \(t_{t}\). Equation (4) specifies that the total flight time for each UAV to perform missions is less than the maximum flight time of the UAV. Where \(t_{ijk}\) is the flight time required for the UAV \(k\) to fly from the target area \(i\) to the target area \(j\), \(t_{j}\) is the reconnaissance time allocated by the UAV \(k\) to the target area \(j\), and also is the flight time of UAV \(k\) flying in the target area \(j\). \(T_{f}\) is the maximum flight time of the UAV \(k\). Equation (5) specifies that the total reconnaissance time of each UAV is less than the maximum opening time of the reconnaissance sensor. Where \(T_{r}\) represents the maximum opening time of the reconnaissance sensor carried by the UAV \(k\). Equations (6) and (7) ensure that each target area can be reconnoitered by one and only one UAV. Where \(M\) is number of UAV performing reconnaissance missions. The starting point constraint and ending point constraint are defined in Equations (8) and (9). Where \(x_{0 jk}\) indicates that the UAV \(k\) flies from the base to the target area \(j\), and \(x_{i0k}\) indicates that the UAV \(k\) flies back to the base from the target area \(i\). Time window feasibility constraints are defined in Equations (11)-(14). Where \(a_{i}\) is earliest arrival time, \(b_{i}\) is latest arrival time, \(T_{i}\) is arrival time when the UAV reaches the target area \(i\). Equation (11) defines that the time of UAV reached the target area \(i\) must be within the time window of target area \(i\).