# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 5 Scenarios with Perfect Kill Probability  
The imperfect kill probability of interceptors arouses uncertainty in every episode that an attacker falls within the impact radius of an interceptor. Hence, the outcomes of two games with the same attacker dispatch pattern and the same deployment plan of defense stations will not be the same. On the other hand, if the kill probability is set to 1, the outcome of a game will be predictable once the attacker dispatch pattern and the deployment plan of defense stations are specified. Theoretically, the PSO will converge to a steady-state global best position, based on the given type of defense cost function, which is then mapped to the optimal deployment plan. Figure 16 shows the convergence of defense costs against slant attack formation. The convergent values of \(\zeta_{1}\), \(\zeta_{2}\), \(\zeta_{3}\), and \(\zeta_{4}\) are 2, 0.204, 1.74, and 2.24, respectively. Figure 17 shows the convergence of defense costs against pincer attack formation, all within 30 iterations. The convergent values of \(\zeta_{1}\), \(\zeta_{2}\), \(\zeta_{3}\), and \(\zeta_{4}\) are 2.5, 0.222, 3.4, and 2.4, respectively, which are higher than their counterparts against slant attack formation. Figure 18a shows an optimal deployment plan of defense stations against slant attack formation, with kill probability set to 1, based on defense cost \(\zeta_{1}\). It is observed that two defense stations are placed within 40 km of the major post, four defense stations are placed near the north post, with 2 within 40 km of the latter, three defense stations are placed near the south post, with 1 within 40 km of the latter. The four defense stations around the north post are closer to the attack stations and will also intercept some attackers aiming for the major post. Figure 16: Convergence of defense cost against slant formation with kill probability set to 1, black: \(\zeta_{1}\), orange: \(\zeta_{2}\), green: \(\zeta_{3}\), purple: \(\zeta_{4}\). Figure 18b shows an optimal deployment plan of defense stations based on defense cost \(\zeta_{2}\). It is observed that three defense stations are placed within 40 km of the major post, more than that on \(\zeta_{1}\), possibly because the weighting of one hit on the major post is relatively higher. Two defense stations are placed southwest to the north post within 40 km of the latter, and one defense station is placed west to the south post within 40 km of the latter, two defense stations are placed between the major post and the north post to cover both, and two defense stations are placed between the major post and the south post to cover both. Figure 18c shows an optimal deployment plan of defense stations based on defense cost \(\zeta_{3}\). There are four defense stations placed near the major post, with three within 40 km of the latter, three defense stations are placed near the north post, with one within 40 km of the latter, two defense stations are placed near the south post, with one to the west and within 40 km of the latter. The major post is covered by more defense stations within 40 km of the former since it is targeted by many more attackers. The south post is the farthest from the attack stations, hence is covered with fewer defense stations. Figure 18d shows an optimal deployment plan of defense stations based on defense cost \(\zeta_{4}\). There are four defense stations placed near the major post, with one to the southeast within 40 km of the latter, three defense stations are placed near the north post, but none is within 40 km of the latter, and three defense stations are placed near the south post, with one to the northwest and within 40 km of the latter. Compared to the other three types of defense cost function, only the north post under \(\zeta_{4}\) has no defense station within 40 km of it, possibly because the weighting on each defense station is on par with one hit on the north post.