## 4 Illustrative Example
### Experiments and Results  
This case study employs Python 3.6 for the experiment in an environment with AMD Ryzen Threadripper 3970X CPU, 128 GB 2200 MHZ memory, and a Linux operating system. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**System Type**} & \multicolumn{6}{c}{**Activities**} \\ \cline{2-7}  & **F\_S1** & **F\_S2** & **F\_S3** & **F\_D1** & **F\_D2** & **F\_II** & **F\_I2** \\ \hline Destroyer & 0 & 1 & 0 & 0 & 1 & 1 & 1 \\ EW Aircraft & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ Stealth Fighter & 1 & 1 & 1 & 1 & 0 & 1 & 1 \\ UAV & 1 & 1 & 1 & 1 & 0 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Functions provided by different types of system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**System Type**} & **Class** & **System Number** & \
\begin{tabular}{c} **Acquisition Cost** \\ **(m5)** \\ \end{tabular} & **Function Capacity** &
\begin{tabular}{c} **Communication** \\ **Bandwidth** \\ \end{tabular} \\ \hline Destroyer & Class A & Sys1 & 200 & 70 & 50 \\  & Class B & Sys2 & 250 & 80 & 75 \\ Destroyer & Class C & Sys3 & 300 & 95 & 100 \\  & Class D & Sys4 & 350 & 105 & 125 \\  & Class E & Sys5 & 400 & 120 & 150 \\ \hline \multirow{4}{*}{EW Aircraft} & Class A & Sys6 & 100 & 40 & 45 \\  & Class B & Sys7 & 125 & 50 & 60 \\  & Class C & Sys8 & 150 & 60 & 75 \\  & Class D & Sys9 & 175 & 70 & 90 \\  & Class E & Sys10 & 200 & 80 & 105 \\ \hline \multirow{4}{*}{Fighter} & Class A & Sys11 & 80 & 25 & 45 \\  & Class B & Sys12 & 100 & 35 & 55 \\  & Class C & Sys13 & 120 & 40 & 65 \\  & Class D & Sys14 & 140 & 45 & 75 \\  & Class E & Sys15 & 160 & 55 & 85 \\ \hline \multirow{4}{*}{UAV} & Class A & Sys16 & 50 & 25 & 30 \\  & Class B & Sys17 & 70 & 30 & 35 \\ \cline{1-1}  & Class C & Sys18 & 90 & 35 & 45 \\ \cline{1-1}  & Class D & Sys19 & 110 & 45 & 50 \\ \cline{1-1}  & Class E & Sys20 & 130 & 50 & 55 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Candidate systems and some of the synthetic input data. Regarding the NSGA-II based Algorithm 1, we run the experiments with a crossover rate between (0.5, 1), a mutation rate between (0.01, 0.2), and a population size between (50, 200). With regard to the inner-loop Algorithm 2, we conduct experiments with a mutation rate between (0.01, 0.5), crossover rate between (0.1, 0.5), and population size between (50, 100). The selected parameters are listed in Table 4.  
The algorithm takes around 33 h to achieve convergence. Since this method aims to solve the architecture design problem, it is still acceptable. Figure 8 illustrates the obtained Pareto sets at different generations when a system randomly receives attacks and falls to a complete failure. The plot shows clearly how the output reaches the Pareto front from the 25 th generation to 100 th generation. Since only one system is destroyed, a decent amount of capability value survives. In the 100 th generation, if the surviving capability is small (e.g., if the attacked system significantly affects SoS effectiveness), the recovered capability is relatively large on the Pareto front. This means that the optimal reconfiguration strategy can help recover SoS capability as much as it can. If multiple systems receive attacks, the surviving and recovered capabilities at the 100 th generation are shown in Figure 9. The overall trend is not so different from the results in Figure 8, except for the lower capability values. Hence, the discussion of the results will focus on the situation with one system disruption. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Parameters** & **Values** \\ \hline \multirow{4}{*}{Algorithm 1 (NSGA-II)} & Crossover rate & 0.9 \\  & Mutation rate & 0.05 \\  & Population size & 100 \\  & Maximum generation & 100 \\ \hline \multirow{4}{*}{Algorithm 2 (GA)} & Mutation rate & 0.05 \\  & Crossover rate & 0.3 \\ \cline{1-1}  & Population size & 80 \\ \cline{1-1}  & Maximum generation & 400 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Experiment parameters for the algorithms.