## 5 Genetic Algorithm  
Evolutionary optimization mimics natural selection processes where individuals compete for survival in the population. Only the fittest can survive and reproduce, improving the entire population over generations. There are many algorithms falling into this category, depending on the forms which individuals may take: Genetic algorithm (gene sequences), genetic programming (solver programs), differential evolution (numeric vectors), neuroevolution (neural net weights), learning classifier system (rules or conditions) and so forth [55]. This study employs a genetic algorithm (GA) which is the most widely used amongst evolutionary optimization methods. Figure 12 explains the GA steps where randomized initial design variables of each individual are encoded into Boolean alleles for genetic operations (selection, crossover, mutation, and insertion) and decoded back for fitness evaluation prior to the next step or termination. Note that an initially diverse population reaches an equilibrium where the fittest (lowest fitness score) individuals with homogeneous characteristics constitute the final population. In genotypes, each individual design is represented by a schema which is a template consisting of 0's and 1's [56]. The schema is a concatenation of binary forms of design variables, which are given a varying number of bits according to their values and required accuracy. Table 15 summarizes design variables and the number of bits to quantities them. Discrete variables don't incur quantization errors in encoding or decoding, whereas continuous variables have quantization errors of (_maximum \(-\) minimum_)/2_bits_ each time. Note also that the range of each variable has been modified from that used in SA such that GA can find an optimal solution faster. The genetic operation parameters are set up and tuned as follows. The selection process uses a roulette wheel selection scheme. A crossover rate of 0.95 and a mutation rate of 0.001 are used along with a population of 50. Experimentation has shown that higher mutation rates often lead to poor convergence because new, mutated species are continually injected into the population; on the other hand, lower mutation rates with a small \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 & \(-42.9\) km \\  & _n\_planes_ & Number of orbital planes & 3 \\  & _n\_sats_ & Number of satellites per plane & 5 \\  & _regard_ & Field of regard & \(47.8^{\circ}\) \\ \hline \multirow{4}{*}{Performance metrics} & \(F_{1}\) & GOM area coverage & \(3.32\) \% \\  & \(F_{2}\) & ROM revisit time & \(1018\) sec \\  & \(F_{3}\) & Constellation mass & \(32\),\(796\) kg \\  & \(F_{4}\) & Reconfiguration time & \(3.13\) day \\  & \(J\) & Objective function & \(1.570\) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: SA 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 & \(49.6\) km \\  & _n\_planes_ & Number of orbital planes & 5 \\  & _n\_sats_ & Number of satellites per plane & 2 \\  & _regard_ & Field of regard & \(46.8^{\circ}\) \\ \hline \multirow{4}{*}{Performance metrics} & \(F_{1}\) & GOM area coverage & \(2.89\) \% \\  & \(F_{2}\) & ROM revisit time & \(1609\) sec \\  & \(F_{3}\) & Constellation mass & \(26\),\(276\) kg \\  & \(F_{4}\) & Reconfiguration time & \(3.17\) days \\  & \(J\) & Objective function & \(1.463\) \\ \hline \hline \end{tabular}
\end{table} 
Table 14: SA optimal solution (5 planes). population yield sub-optimal solutions as well because the whole population prematurely converges homogeneously to a sub-optimal solution. Using the settings in Tables 11 and 12, the same objective function as in SA (Equation 15) is to be minimized.