## 3 Numerical Experiments and Results
### Simulation Experiment and Results  
The NS is determined from the DFS model, assuming that all parameters are certain, and the robust solution (RS) is determined from the RO model, assuming that enemy's threat level and threat time parameters are uncertain; NS and RS are compared in the experiment. Experiments were conducted considering four different number of targets: n = 5, 10, 15, and 20. Here, the uncertain threat, \(\tilde{f_{j}}\), of enemies belongs to the uncertain set \(UF_{j}=\left\{\tilde{f_{j}}:\;f_{j}-f_{j}\theta_{j}\leq\tilde{f_{j}}\leq f_{j} +f_{j}\theta_{j}\right\}\), in which \(\theta_{j}\) refers to a uncertainty level selected from {0.2, 0.4, 0.6, 0.8}. The uncertain threat time, \(\tilde{a_{j}}\), of enemies belongs to the uncertain set \(UA_{j}=\left\{\tilde{a_{j}}:\;a_{j}-a_{j}\theta_{j}\leq\tilde{a_{j}}\leq a_{j} +a_{j}\theta_{j}\right\}\), which is selected from {0.2, 0.4, 0.6, 0.8}. The Monte Carlo simulation was conducted to compare the NS and RS; 15 scenarios were created, and 10,000 sets of uncertain data were sampled for each scenario to compare the objective value. The uncertain threat \(\tilde{f_{j}}\) of enemies is randomly selected from section \(\left(f_{j}-f_{j}\theta_{j},f_{j}+f_{j}\theta_{j}\right)\), and the uncertain threat occurrence time \(\tilde{a_{j}}\) of enemies is randomly selected from \(\left(a_{j}-a_{j}\theta_{j},\;a_{j}+a_{j}\theta_{j}\right)\). Table 3 presents the computation time required to search for the optimal robust solution in the CPLEX software. As presented in Table 3, as the number of targets (n) increases, the corresponding computation time increases exponentially. In particular, when the number of targets (n) is eight or larger, the solution cannot be found within three hours by using the CPLEX software. However, firing scheduling should be determined within three hours in a real-world operation. Thus, the optimization method could not be used when the number of targets (n) was eight or larger, and it was verified that the methodology proposed in this study had to be used. Table 4 presents the results of 10,000 simulations for each target number (n) and an uncertainty level (\(\theta\)) in 15 cases. In Table 4, N refers to the number of better objective values between RS and NS among 150,000 simulation results. For example, when n = 5 and \(\theta\) = 0.2, the simulation execution result for 150,000 iterations exhibited that 78,556 iterations yielded better results using RS and that 71,444 iterations yielded better results using NS. As shown in Figure 2, RS had a higher number of better results than that of NS for all target sizes. Also, the average values of the objective values obtained using RS were smaller than those of NS. The minimum value of the objective value obtained using RS was smaller than that of NS overall, although NS had smaller minimum values than RS in some cases. For example, when n = 10 and \(\theta\) = 0.4, the minimum value of the objective value using RS was 129.8, whereas that using NS was 116.2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Best** & **Worst** & **ST.DEV** & **Computation Times (s)** \\ \hline
1 & 873 & 873 & 0 & 00:02.202 \\
2 & 757.5 & 757.5 & 0 & 00:02.052 \\
3 & 585 & 585 & 0 & 00:01.967 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter verification test results of genetic algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Size (n)** & **5** & **6** & **7** & **8** \\ \hline Computation time (s) & 3.25 & 16.63 & 317.92 & 18330.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: CPU times of RO approach using scenario concept with IBM ILOG CPLEX Optimization software package. The maximum value of the objective value exhibited that RS was better than NS in all problem sizes and uncertainty levels. That is, the maximum value of the objective value using NS was always larger than that using RS. Here, the objective value means the total threat of the enemy inflicted on friendly forces. The smaller the objective value, the better.