# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 3 Simulation Results and Statistical Analysis  
Given a formation of attack stations and a defense cost function chosen among Equations (11)-(14), the PSO algorithm is applied to search for a global best particle, which prescribes an optimal deployment plan of defense stations. Then, 100 games are played according to this deployment plan, and the resulting defense costs out of these games are sorted to derive a cumulative distribution function (CDF) for statistical analysis. In this section, the general statistical features on these 100 games will be reviewed and compared under two different attack formations and four different types of defense cost function. Some interesting outlier cases will be inspected in the next section to learn more about the nature of the game. For each type of defense cost function, under slant or pincer attack formation, the PSO algorithm is applied four times, acquiring an optimal deploment plan each time. Each optimal deployment plan is then used to play 100 games and the resulting defense costs are presented in the form of a CDF curve. Figures 4 and 5 show the CDF curves on four types of defense cost function, against slant attack formation and pincer attack formation, respectively. For example, the four curves in Figure 4a are the results played with four optimal deployment plans, respectively, under cost function type of \(\zeta_{1}\), and their values lie in [1.7, 6.1], [2.8, 12], [2, 12.4] and [3.9, 9.6], respectively. Figure 4: CDF of defense cost function against slant formation, (a) \(\zeta_{1}\); (b) \(\zeta_{2}\); (c) \(\zeta_{3}\); (d) \(\zeta_{4}\), with four PSO runs. One out of the four optimal deployment plans in Figure 4a that leads to the lowest median defense cost is selected to check how the defense cost of its associated global best particle converges with iterations. The black curve in Figure 6 shows such convergence with iterations for defense cost function of type \(\zeta_{1}\). Similarly, the other three curves in Figure 6 show the convergence of cost functions for types \(\xi_{2}\), \(\xi_{3}\), and \(\zeta_{4}\), associated with the curve with lowest median defense cost in Figure 4b-d, respectively. The convergent values of \(\zeta_{1},\zeta_{2},\zeta_{3}\) and \(\zeta_{4}\) are 1.92, 0.24, 3.9, and 3.38, respectively. Due to the uncertainty aroused by imperfect kill probability, the defense cost may remain constant over many iterations before dropping a lower value. The value of \(\zeta_{2}\) is relatively lower than the other three because the weighting on minor posts, defense stations and interceptors are reduced by tenfold compared with their counterparts in the benchmark case (\(\zeta_{1}\)). On the other hand, the values of \(\zeta_{3}\) and \(\zeta_{4}\) are relatively higher due to the increased weighting on minor posts and defense stations, respectively. Similarly, Figure 7 shows the convergence of defense costs associated with four optimal deployment plans, one selected from each of the four subfigures in Figure 5, under pincer attack formation. The convergent values of \(\zeta_{1},\zeta_{2},\zeta_{3}\) and \(\zeta_{4}\) are 2.72, 0.322, 4.24, and 4.26, respectively. Figure 8a shows the optimal deployment plan of defense stations against slant attack formation, based on defense cost \(\zeta_{1}\). It is observed that three defense stations are placed within 40 km of the major post, and one defense station is placed within 40 km of the north and the south minor posts, respectively, Three defense stations are placed between the  
Figure 5: CDF of defense cost function against pincer formation, (**a**) \(\zeta_{1}\); (**b**) \(\zeta_{2}\); (**c**) \(\zeta_{3}\); (**d**) \(\zeta_{4}\), with four PSO runs. major post and the south post to cover both, and three defense stations are placed in the upper left area to guard both the major post and the north post. More defense stations are placed within 40 km of the major post than within 40 km of the other two minor posts.