## 4 Simulated Annealing
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Design** & **Description** & **Range** &
\begin{tabular}{c} **Initial** \\ **Value** \\ \end{tabular} & **Type** \\ \hline _nk_ & RGT ratio & \([13/1,31/2]\) & \(31/2\) & Discrete \\ _delta\_alt_ & Altitude difference & \([-200,200]\) & \(-200\) km & Continuous \\ _n\_planes_ & Number of orbital planes & \([2,9]\) & \(2\) & Integer \\ _n\_sats_ & Number of satellites per plane & \([1,5]\) & \(1\) & Integer \\ _regard_ & Field of regard (FoR) & \([5,50]\) & \(50^{\circ}\) & Continuous \\ _prop_ & Propellant Type & Monopprellant & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Range and initial value of design variables in SA. Figure 10: Simulated Annealing (SA) algorithm. and a weight (\(w_{i}\)). Scaling factors are used to prevent any figure of merit from dominating the others, and weights balance the relative importance of the four objectives. Similarly, each constraint term is multiplied by a relative scale factor (\(c_{j}\)), and the sum of their linear combination is again multiplied by a global gain \(g\), as summarized in Table 12.  
Figure 11 reports the time history of fitness where several SA runs were run with different penalty gains. As the gain grows, the optimizer tries to avoid constraint violation because any nonzero penalty will be amplified significantly. The lowest fitness was achieved with \(g=0.001\), but a penalty from constraint violation was non-zero. Therefore, \(g=1000\) was chosen for optimization which had the second lowest fitness and incurred no penalty. Tables 13 and 14 summarize two types of optimal ReCon configuration obtained using SA. They differ in the number of orbital planes and the sign of _delta_alt_, but the magnitude of _delta_alt_ and the field of regard are similar. The 3-plane solution has superior GOM coverage and ROM revisit time to the 5-plane solution, but its requirement of heavier constellation launch mass has resulted in a lower score (higher value of \(J\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Figure of Merit** & **Definition** & **Typical Value** & **Scaling (\(s_{i}\))** & **Weighting (\(w_{i}\))** \\ \hline \(F_{1}\) & \((-1)\times\) GOM coverage (\%) & \(-5\) & \(0.5\) & \(0.25\) \\ \(F_{2}\) & ROM revisit time (s) & \(1000\) & \(0.001\) & \(0.25\) \\ \(F_{3}\) & Constellation mass (kg) & \(10\),\(000\) & \(0.0001\) & \(0.30\) \\ \(F_{4}\) & Reconfiguration time (day) & \(2\) & \(1\) & \(0.20\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Figures of merit, scaling and weighting factors in single-objective optimization. Figure 11: SA convergence history. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Constraint** & **Definition** & **Typical Value** & **Scaling (\(s_{i}\))** & **Gain (\(w_{i}\))** \\ \hline \(h_{1}\) & Minimum altitude (km) & \(350\) & \(0.5\) & \\ \(h_{2}\) & Maximum altitude (km) & \(1200\) & \(0.001\) & \\ \(h_{3}\) & Maximum aperture (m) & \(1.8\) & \(0.0001\) & \\ \(h_{4}\) & Max propellant mass fraction & \(0.3\) & \(1\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Constraints, scaling factors, and gains in single-objective optimization.