# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 2 Game Design
### Engagement of Attacker and Interceptor  
If an attacker \(A_{m}\) flies within the coverage area of some defense station, the latter will fire an interceptor \(B_{n}\) on \(A_{m}\). If \(A_{m}\) is marked by only one interceptor and is within 50 km of its target, a defense station within 40 km of the target will fire another interceptor on \(A_{m}\). A targeted defense station may fire another interceptor on \(A_{m}\) by itself. At each simulation time step, the defense force computes an achievable intercept time, \(t_{h}\), for \(B_{n}\) (at \(Q_{n}\)) to hit \(A_{m}\) (at \(P_{m}\)) by solving  
\[|P_{m}+\bar{U}_{am}t_{h}-Q_{n}|-v_{b}t_{h}=0,\ \ 0<t_{h}\leq 1000 \tag{1}\]  
where \(\bar{U}_{am}\) is the velocity vector of \(A_{m}\) and \(v_{b}\) is the interceptor speed. If no intercept time is found, the interceptor will tailgate \(A_{m}\). If \(A_{m}\) no longer exists, \(B_{n}\) will detour for the nearest attacker.  
At each time step, each attacker will check if it is marked by any interceptor. If an interceptor \(B_{n}\) (at \(Q_{n}\)) appears within alert radius \(R_{a}\) of attacker \(A_{m}\) (at \(P_{m}\)), the attack force computes an anticipated intercept time, \(t_{h}\), by solving  
\[|(P_{m}+\bar{U}_{am}t_{h})-(Q_{n}+\bar{U}_{bn}t_{h})|=0,\ \ 0<t_{h}\leq 1000 \tag{2}\]  
where \(\bar{U}_{bn}\) is the velocity vector of \(B_{n}\).  
An attacker vanishes if it runs out of fuel or gets hit by an interceptor. An attacker hits its target if their separation distance is shorter than the impact radius \(D_{a}\) of the attacker. An interceptor is lost if it runs out of fuel or hits an attacker. An interceptor hits an attacker with kill probability \(p_{k}\) if the latter falls within the impact radius \(D_{b}\) of the interceptor.  
In this work, assume all the attackers and interceptors fly at the same speed, and an attacker has certain maneuverability to evade an interceptor. When interceptor \(B_{n}\) (at \(Q_{n}\)) appears within an alert radius \(R_{a}\) of attacker \(A_{m}\) (at \(P_{m}\)), the attack force will compute an  
Figure 2: Exemplary scenario of route planning, square with cross: major post, square with slash: minor post, blue square: defense station, red square: attack station, \(\underline{\  
intercept time by solving (2). If an intercept time is found, \(A_{m}\) will evade \(B_{n}\) by adjusting its velocity vector as  
\[\vec{U}_{am}=v_{a}\frac{\alpha_{e}\hat{a}\{Q_{n}X_{m}\}+(1-\alpha_{e})\hat{a}\{P _{m}X_{m}\}}{|\alpha_{e}\hat{a}\{Q_{n}X_{m}\}+(1-\alpha_{e})\hat{a}\{P_{m}X_{m} \}|} \tag{3}\]  
where \(v_{a}\) is the attacker speed, \(\alpha_{e}\) is the attacker maneuverability, with \(0\leq\alpha_{e}\leq 1\), \(X_{m}\) is the anticipated intercept point, \(\hat{a}\{Q_{n}X_{m}\}\) is the unit vector of \(\overline{Q_{n}X_{m}}\), and \(\hat{a}\{P_{m}X_{m}\}\) is the unit vector of \(\overline{P_{m}X_{m}}\). Attacker \(A_{m}\) will regularly adjust its velocity vector as long as \(B_{n}\) remains within its alert radius and an intercept point is found. If intercept point no longer exists, \(B_{n}\) will tailgate \(A_{m}\), and \(A_{m}\) will fly for its assigned target via a straight path. Interceptor \(B_{n}\) will track \(A_{m}\) via a straight path if an intercept point emerges again. Otherwise, \(B_{n}\) will tailgate \(A_{m}\) and may not be able to catch the latter since they fly at the same speed.  
If \(A_{m}\) is marked by more than one interceptors, it will detour for the nearest defense station or post via a straight path. If the defense station originally targeted by \(A_{m}\) has been hit, \(A_{m}\) will detour for the nearest defense station or post via a straight path.  
Each defense station is composed of a radar unit and a few interceptor launching units at safe distance from the radar. Assume that the attackers only aim for the radar unit; hence, a defense station being hit means that the radar unit is lost and no more interceptors can be fired. The remaining interceptors will not be counted in the defense cost at the end of the game.