## 2 Problem Description and Modelling
\[trans(t_{g},t_{h})=\left\{ \begin{matrix}transA(\Delta),&\text{if $g$ and $h$ are in the same orbit}\\ 0,&\text{otherwise}\end{matrix} \right. \tag{4}\]  
where \(transA(\Delta)\) is the following piecewise linear function [29]:  
\[transA(\Delta)=\left\{ \begin{matrix}11.66,\Delta\leq 10\\ 5+\frac{\Delta}{15},10<\Delta\leq 30\\ 10+\frac{\Delta}{2},30<\Delta\leq 60\\ 16+\frac{\Delta}{25},60<\Delta\leq 90\\ 22+\frac{\Delta}{3},90<\Delta\end{matrix} \right. \tag{5}\]  
The look angle of an AEOS during VTW varies over time, as it is influenced by the satellite parameters and the relative positions of the satellite and target. Thus, according to (4) and (5), the transition time \(trans(t_{g},t_{h})\) is dependent on time, and hence the AEOS observation schedule considered in this paper is time-dependent. Given a target allocation \(\mathbf{a}_{s}\) for AEOS \(s\), we define the (partial) fitness value \(F(\mathbf{a}_{s})\) as follows:  
\[F(\mathbf{a}_{s})=\sum_{g\in\Gamma_{s}}\phi_{g}-\frac{1}{\mathbf{e}^{\tau}}- \frac{\sum_{g\in\Gamma_{s}}d_{g}}{\text{Ave}(s)\times|\Gamma_{s}|} \tag{6}\]  
where \(\Gamma_{s}\) collects all observable targets in \(\mathbf{a}_{s}\), and \(\tau=1/\left(|\Gamma_{s}|+1\right)\), \(|\Gamma_{s}|\) is the number of targets in \(\Gamma_{s}\), and \(\text{Ave}(s)=\sum_{k\in R}\Sigma_{g\in G}(e_{sg}k-st_{sg}k)/n\) is the average sum of VTM length for each satellite. The proposed fitness Function (6) considers three factors: task execution benefits, satellite task load, and satellite observation capabilities towards the targets. Clearly, the higher the profit sum of the observable targets, the bigger the fitness value \(F(\mathbf{a}_{s})\). To enforce the system load balance, when two satellites observe the same set of targets, the one with the longer average VTW is assigned a higher fitness value according to equation (6). The global fitness value for the solution \(\mathbf{a}=[\mathbf{a}_{1},\,\mathbf{a}_{2},\,\ldots,\,\mathbf{a}_{m}]\) is defined as \(\text{Y}(\mathbf{a})=\Sigma_{g\in S}F(\mathbf{a}_{s})\). Let \(x_{skg}\) be the decision variable such that \(x_{skg}=1\) if target \(g\) is observed by AEOS \(s\) in the orbit \(k\in R\) and \(x_{skg}=0\) otherwise. Another variable \(y_{skgh}=1\) if satellite \(s\) selects target \(h\) after observing \(g\) in the orbit \(k\), and \(y_{skgh}=0\) otherwise. In addition, let \(l_{sk}\) and \(e_{sk}\) be the dummy source and sink targets in orbit \(k\) associated with \(s\), respectively. It is assumed that the observations of \(s\) in orbit \(k\) is start from \(l_{sk}\) but end with \(e_{sk}\). The mathematical model of the observation scheduling of the AEOS constellation can be expressed as follows:  
\[J=\text{Max}\sum_{s\in S}\left\{\sum_{k\in R}\sum_{g\in G}\left\{x_{skg}\phi_{ g}-\frac{x_{skg}d_{g}}{\text{Ave}(s)}\right\}-\frac{1}{e^{\sum_{k\in R}\sum_{g\in G }x_{skg}+1}}\right\} \tag{7}\]  
s.t. \[\sum_{s\in S}\sum_{k\in R}x_{skg}\leq 1,\forall g\in G\] (8)  
\[\sum_{h\in G\cup\{e_{sk}\}\setminus\{g\}}y_{skgh}=\sum_{h\in G\cup\{l_{sk}\} \setminus\{g\}}y_{skhg}=x_{skg},\forall s\in S,\forall k\in R,\forall g\in G \tag{9}\]  
\[\sum_{g\in G\cup\{l_{sk}\}}y_{skge_{sk}}=\sum_{g\in G\cup\{e_{sk}\}}y_{sklg}= 1,\forall s\in S,\forall k\in R \tag{10}\]  
\[st_{sg}^{k}*x_{skg}\ \leq t_{g},\forall s\in S,\forall k\in R,\forall g\in G \tag{11}\]\[t_{g}+d_{g}\leq e_{sg}^{k}+K(1-x_{skg}),\forall s\in S,\forall k\in R,\forall g\in G \tag{12}\]  
\[t_{g}+d_{g}+trans(t_{g},\ t_{h})-\ t_{h}\leq K(1-y_{ggh}),\forall s\in S,\forall k \in R,\forall g\in G \tag{13}\]  
\[x_{skg}\in\{0,\ 1\},\ y_{skgh}\in\{0,\ 1\},\forall s\in S,\forall k\in R,\forall g,h\in G \tag{14}\]  
\[t_{g}\geq 0,\forall g\in G \tag{15}\]  
where \(K\) is an integer that is large enough. Equation (7) implies that the objective function is to maximize the global fitness value. Equation (8) implies that each target is observed at most once.