# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 2 Game Design
### Default Parameters in Simulations  
Tables 1 and 2 list the parameters of the defense force and the attack force, respectively. The major post is located at \((T_{x},T_{y})\), and the center of the deployment area shown in Figure 2. The particle position is confined by \(x_{\text{max}}=T_{x}+50\), \(x_{\text{min}}=T_{x}-50\), \(y_{\text{max}}=T_{y}+200\) and \(y_{\text{min}}=T_{y}-200\). Table 3 lists the parameters of optimization. The particle velocity is confined by \(V_{x,\text{max}}=0.2(x_{\text{max}}-x_{\text{min}})\), \(V_{x,\text{min}}=-V_{x,\text{max}}\), \(V_{y,\text{max}}=0.2(y_{\text{max}}-y_{\text{min}})\) and \(V_{y,\text{min}}=-V_{y,\text{max}}\). A total of \(M\) defense stations are deployed to protect the assets from the attack force, and each defense station is armed with \(N_{br}\) interceptors. Each defense station is assigned a radius of coverage (\(R_{r}\)), within which any attacker can be marked for interception. The attacker impact radius (\(D_{a}\)) and interceptor impact radius (\(D_{b}\)) are effective ranges within which the attacker or interceptor will hit its target, with a given kill probability (\(p_{k}\)). The radius of coverage (\(R_{r}\)), interceptor speed (\(v_{b}\)), attacker speed (\(v_{a}\)), interceptor impact radius (\(D_{b}\)), and attacker impact radius (\(D_{a}\)) are roughly estimated from available data collected from public sources. The maximum flight time of interceptor (\(B_{\tau}\)) and attacker (\(A_{\tau}\)) are estimated from their respective maximum flight range. Empirically, neither interceptor nor attacker can guarantee to hit its intended target, and such uncertainty is characterized with a kill probability (\(p_{k}\)) derived from available records. When an aimed target falls within the impact radius of an interceptor or an attacker, the target will survive with probability of \(1-p_{k}\). The use of kill probability makes the game versatile and unpredictable, more or less reflecting the mist of wars. The total number of attack stations (\(N_{sa}\)) and the number of attackers (\(N_{am}\)) on each attack station determine the type and intensity of an assault mission, be it provocative, repressive, or fatal. The efficacy of \(N_{sa}\) and \(N_{am}\) also depends on the number of defense stations (\(M\)), the number of interceptors (\(N_{br}\)) on each defense station, and how these stations are deployed. Likewise, the efficacy of \(M\) and \(N_{br}\) depends on \(N_{sa}\), \(N_{am}\), the intention of attack force, and how much defense cost will be tolerated. The deployment plan of defense stations is optimized under the worst condition that the attack force can achieve its best performance. The attack force puts the highest priority on the major post and the second priority on two minor posts. Thus, two diverse paths are planned from each attack station to the major post, one from left flank and the other from right flank. Both paths take off at a departure angle (\(\theta\)) about a straight line between the major post and the attack station. Similarly, the path planned to attack a minor post or a defense station also takes off at departure angle \(\theta\). An attacker is endowed certain self-defense capability. When an attacker finds itself being marked by an interceptor within alert radius (\(R_{a}\)), it will adjust its evading direction, \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline major post coordinates & \((T_{x},T_{y})\) & (381,250) km \\ minor post coordinates & - & (421,400) km, (421,100) km \\ number of defense stations & \(M\) & 10 \\ number of interceptors per defense station & \(N_{br}\) & 8 \\ radius of coverage & \(R_{r}\) & 200 km \\ interceptor impact radius & \(D_{b}\) & 50 m \\ interceptor speed & \(v_{b}\) & 1 mach \\ interceptor flight time & \(B_{\tau}\) & 900 s \\ interceptor kill probability & \(p_{k}\) & 0.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of defense force.