## 5 Genetic Algorithm
Figure 13 shows the convergence history of the population mean fitness and the fitness of the best individual, respectively. If the constraint gain is too small (0.001), the constraint violation is discounted, and the resultant solution is sub-optimal. If the gain is too high (10 and 1000), the solution is also sub-optimal, so the most optimal solution is found to be 0.1. Tables 16 and 17 show similar GA trends, as already shown in SA, reaching either 3-plane or 5-plane optima in multiple runs. Overall GA achieves better score (lower \(J\)) by maintaining fine balances between performance and cost (mass). For example, the 3-plane GA solution has a negative _delta_alt_, resulting in a lower launch mass compared to the 3-plane SA solution with a positive _delta_alt_. The 5-plane GA solution has three satellites per plane, which also reduces the constellation launch mass compared to the 5-plane SA solution with four satellites per plane. In both cases, degradation in coverage or revisit performance is justified by savings in launch mass. The two GA solutions have nearly identical \(J\) score although the 5-plane solution is marginally better than the 3-plane solution. In either case, the individual satellite weighs 2 ton and the aperture has a diameter of 1.2 m, not significantly exceeding the ranges in the Earth-observation satellite database (Table A1 and \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variable** & **Range** & **Bits** & **Type** \\ \hline RGT ratio & [13/2, 14/1] & 4 & Discrete \\ Altitude difference & [\(-\)100, 100] & 12 & Continuous \\ Number of orbital planes & [2; 7] & 4 & Integer \\ Number of satellites per plane & [1; 7] & 4 & Integer \\ Field of regard (FoR) & [5; 60] & 12 & Continuous \\ Propellant Type & - & - & Fixed \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Range and initial value of design variables in GA.  
Figure 12: Genetic algorithm (GA) for ReCon optimization. Figure 11). The physical feasibility is thus verified, and the performance-to-cost (mass) analysis is conducted in the next section. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Symbol** & **Description** & **Optimum** \\ \hline \multirow{4}{*}{Design variable} & _nk_ & RGT ratio & \(15/1\) \\  & _delta_alt_ & Altitude difference & \(-53.9\) km \\  & _n_planes_ & Number of orbital planes & \(3\) \\  & _n_sats_ & Number of satellites per plane & \(4\) \\  & _regard_ & Field of regard & \(46.8^{\circ}\) \\ \hline \multirow{4}{*}{Performance metrics} & \(F_{1}\) & GOM area coverage & \(1.95\) \% \\  & \(F_{2}\) & ROM revisit time & \(1346\) sec \\ \cline{1-1}  & \(F_{3}\) & Constellation mass & \(25\),\(187\) kg \\ \cline{1-1}  & \(F_{4}\) & Reconfiguration time & \(2.92\) days \\ \cline{1-1}  & \(J\) & Objective function & \(1.385\) \\ \hline \hline \end{tabular}
\end{table} 
Table 16: GA optimal solution (3 planes). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Symbol** & **Description** & **Optimum** \\ \hline \multirow{4}{*}{Design variable} & _nk_ & RGT ratio & \(15/1\) \\  & _delta_alt_ & Altitude difference & \(-54.7\) km \\  & _n_planes_ & Number of orbital planes & \(5\) \\  & _n_sats_ & Number of satellites per plane & \(2\) \\  & _regard_ & Field of regard & \(47.1^{\circ}\) \\ \hline \multirow{4}{*}{Performance metrics} & \(F_{1}\) & GOM area coverage & \(1.95\) \% \\  & \(F_{2}\) & ROM revisit time & \(1602\) sec \\ \cline{1-1}  & \(F_{3}\) & Constellation mass & \(21\),\(318\) kg \\ \cline{1-1}  & \(F_{4}\) & Reconfiguration time & \(2.93\) days \\ \cline{1-1}  & \(J\) & Objective function & \(1.382\) \\ \hline \hline \end{tabular}
\end{table} 
Table 17: GA optimal solution (5 planes). Figure 13: GA convergence history: (**a**) Population mean; (**b**) best individual.