# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 3 Simulation Results and Statistical Analysis
The defense station loss rate (36%) is moderate compared with the other cases because the weighting of defense stations is raised to the level of a minor post. The 80% confidence interval is the difference of defense costs at the 10 th percentile and the 90 th percentile of the CDF curve. Similarly, the 95% confidence interval is the difference of defense costs at the percentiles of 2.5 and 97.5 of the CDF curve. A small confidence interval implies a more predictable defense cost in any game, and a large confidence interval implies more uncertainty in the resulting defense cost. Table 5 lists the statistics of loss against pincer attack formation. With the optimal deployment plan based on \(\zeta_{1}\), the average hit on the north post (0.09) is the lowest among all the four cases. The average hit on the south post (1.23) is higher than that on the north post, because the only defense station within 40 km of the former is placed on the flank \begin{table}
\begin{tabular}{c c c c c c} \hline
**Defense Cost Function** & \(\xi_{1}\) & \(\xi_{2}\) & \(\xi_{3}\) & \(\xi_{4}\) & **Empirical** \\ \hline average hits on major post & 0.73 & 1.35 & 0.13 & 0.77 & 1.57 \\ \hline average hits on north post & 0.61 & 1.87 & 2.25 & 1.15 & 1.11 \\ \hline average hits on south post & 0.21 & 0.66 & 0.72 & 0 & 0.33 \\ \hline defense stations lost (\%) & 43.3 & 30.2 & 40.1 & 36 & 56.1 \\ \hline interceptors launched (\%) & 81.925 & 81.738 & 81.613 & 86.388 & 82.513 \\ \hline attackers lost (\%) & 82.8 & 82.75 & 77.125 & 85.5 & 78.125 \\ \hline average defense cost & 3.317 & 1.668 & 5.208 & 4.527 & - \\ \hline median of defense cost & 3.3 & 1.34 & 5.1 & 4.39 & - \\ \hline
80\% confidence interval & 2.1995 & 1.9986 & 2.2796 & 2.5582 & - \\ \hline
95\% confidence interval & 3.579 & 3.0912 & 3.2401 & 3.0803 & - \\ \hline \end{tabular}
\end{table} 
Table 4: Statistics of loss against slant attack formation. instead of up front to guard off or attract the approaching attackers. Indeed, one of the attackers aiming for the south post is almost certain to hit the latter. The average hit on the major post (1.81) is the highest among all the four cases, implying that three defense stations within 40 km of the major post are not enough for protection. The attacker loss rate (79.95%) is comparable to the other cases, which is related to the hits on the major post and the south post. With the optimal deployment plan based on \(\zeta_{2}\), the average hit on the major post (0.93) is relatively low since five defense stations are placed within 40 km of the former. The average hit on the north post (1.85) is the highest among all the four cases. The only defense station within 40 km of the north post fires all the equipped interceptors, and the latter becomes a sitting duck for the attackers approaching later. The average hit on the south post (1.8) is the highest among all the four cases since no defense station is placed within 40 km of the south post. The only defense station placed between the south post and the south attack stations fires all the interceptors in the early stage, leaving the south post defenseless against the attackers approaching later. The defense stations loss rate (29.2%) is the lowest among all the four cases. The southernmost defense station is easily attacked since no other defense station is placed within 40 km of the former. All the other defense stations have some cohort stations within 40 km to cover them and are occasionally hit by detouring attackers originally aiming for another defense asset. The attacker loss rate (78.275%) is comparable to the other cases, which is related to the high hits on the north and south posts. With the optimal deployment plan based on \(\zeta_{3}\), the average hit on the major post (1.03) is relatively low since five defense stations are placed within 40 km of the former.