# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 2 Game Design
### Deployment of Defense Stations  
Multiple attackers are dispatched from attack stations to strike the commanding posts and the defense stations via planned routes. Imperfect kill probability of interceptors, maneuverability of attackers for evading interception, and the maximum flight range of attackers and interceptors are implemented in the game. Due to the complexity of the game, exacerbated by the imperfect kill probability, it is difficult to apply analytical approaches to find the optimal deployment plan of defense stations. In this work, a particle swarm optimization (PSO) algorithm [21], reputed for fast convergence, is applied to find the optimal deployment plan of defense stations, by minimizing a specific defense cost function \(\xi_{a}\). A set of \(N\) particles are used to represent the candidate coordinates of \(M\) defense stations, with the position of the \(n\)th particle given by \(\bar{S}_{n}=(S_{n1x},S_{n1y},S_{n2x},S_{n2y},\cdots,S_{nMx},S_{nMy})\), where \((S_{nmx},S_{nmy})\) are the coordinates of the \(n\)th defense station. Similarly, the velocity of the \(n\)th particle is given by \(\bar{V}_{n}=(V_{n1x},V_{n1y},V_{n2x},V_{n2y},\cdots,V_{nMx},V_{nMy})\). In the beginning, the position and velocity of each particle are randomly selected from given intervals as  
\[x_{\min}\leq S_{nmx}\leq x_{\max} \tag{5}\] \[y_{\min}\leq S_{nmy}\leq y_{\max}\] (6) \[V_{x,\min}\leq V_{nmx}\leq V_{x,\max}\] (7) \[V_{y,\min}\leq V_{nmy}\leq V_{y,\max} \tag{8}\]  
with \(1\leq n\leq N\) and \(1\leq m\leq M\). The position of the \(n\)th particle is used to deploy the \(M\) defense stations and a game is played accordingly, resulting in a defense cost function \(\zeta_{\alpha n}\). If \(\zeta_{\alpha n}\) is lower than the lowest defense cost in previous iterations, the best defense cost is updated as \(\zeta_{\alpha n}^{\prime}=\zeta_{\alpha n}\), and the best position is updated as \(\bar{S}_{bn}=\bar{S}_{n}\). The lowest defense cost among all the \(N\) particles iterated so far is recorded as the global best defense cost, \(\zeta_{\alpha g}\), and the associated particle position is recorded as the global best position, \(\bar{S}_{g}\). In the next iteration, the velocity of the \(n\)th particle is updated as  
\[\bar{V}_{n}\gets w_{v}\bar{V}_{n}+c_{1}r_{1}(\bar{S}_{bn}-\bar{S}_{n})+c _{2}r_{2}(\bar{S}_{g}-\bar{S}_{n}) \tag{9}\]  
where \(w_{v}\) is a weight on particle velocity, \(c_{1}\) and \(c_{2}\) are acceleration constants, \(r_{1}\) and \(r_{2}\) are uniform random numbers over \([0,1]\). The updated velocity components are constrained within the given intervals. If \(V_{nmx}>V_{\alpha,\max}\), then set \(V_{mma}=V_{\alpha,\max}\), with \(\alpha=x,y\). Similarly, if \(V_{mma}<V_{\alpha,\min}\), then set \(V_{nma}=V_{\alpha,\min}\), with \(\alpha=x,y\). Next, the position of the \(n\)th particle is updated as  
\[\bar{S}_{n}\leftarrow\bar{S}_{n}+\bar{V}_{n} \tag{10}\]  
The updated position is constrained by reflection boundary conditions. If \(S_{nmx}>\alpha_{\max}\), then \(S_{nmx}\gets 2\alpha_{\max}-S_{nmx}\), with \(\alpha=x,y\). Similarly, if \(S_{nmx}<\alpha_{\min}\), then \(S_{nmx}\gets 2\alpha_{\min}-S_{nmx}\), with \(\alpha=x,y\). The updated position \(\bar{S}_{n}\) is then used to play another game. The global best position \(\bar{S}_{g}\) after a given number of iterations will be used as the optimal deployment plan of all the defense stations. In [22], a multi-agent decision support system (DSS), including missile, radar, and command center, was established for missile defense. The objective function of DSS counted in the constraints of radar, offensive missile, intercept missile, and commander. A variable neighborhood negative selection particle swarm optimization (VNNSPSO) algorithm was proposed to obtain an optimal interception plan dynamically. The proposed algorithm was verified by comparing with basic PSO and several variants in terms of convergence rate, iteration number, average fitness value, and standard deviation.