# Deployment Optimization of Defense Stations in an Attack-Defense Aerial War Game
## 1 Introduction
In [16], a reinforcement learning (RL) algorithm through amplification of imitation effect (AIE) was proposed to enhance the capability of single or multiple combat unmanned aerial vehicles (CUAVs) in pursuing a target while evading missiles fired from the latter. A self-imitation learning (SIL) algorithm based on the best previous decision and a random network distillation (RND) algorithm were used to guide the exploration. However, the kill probability of UAV or missile was not considered in these works. In [17], an all-domain simulation (ACS) was proposed to dispatch a group of UAVs (attackers), without path planning or map navigation, to attack multiple static targets while bypassing the area covered by radar stations (defenders). Each target was marked by only one attacker, the targets were located outside of the radar coverage area, and no interceptors were fired against the attackers. In [18], a virtual target approach was proposed to guide a missile to hit a static target or a target moving at constant speed. A missile was first planned to hit a virtual target at a given impact angle by following a nonsingular terminal sliding-mode guidance law. Then, the missile flew at a fixed angle to the real target by following a proportional navigation guidance (PNG) law. An optimization algorithm was applied to simultaneously control the impact time and angle on the real target. In comparison, our work aims to search an optimal deployment plan of defense stations, and the interceptors in the air maneuver by specific rules to cope with the attackers on the move. Chasing episodes between pursuer(s) and evader(s) are indispensable ingredients in a complicated game scenario. In [19], a pursuit-evasion differential game was studied, with multiple attackers trying to capture a moving target armed with no interceptor. Time-varying optimal control algorithms were developed for both attackers and target, with limited on-site information. Some attackers might lure the target to a region for the other attackers to capture it off guard. In [20], a pursuit-evasion differential game was played between three pursuers and an evader. The strategies for all the agents were derived with a geometric method, which could be extended to involve more pursuers. The chasing episodes together with imperfect kill probability sometimes may turn the tide of a game, which will be demonstrated with some interesting cases in this work. Most works in the literature highlighted a few factors of interest to assess their effects. By combining more practical factors into the scenario, the game may become closer to real-life operations, manifesting some nuances that will otherwise be missing in the lack of interaction among different factors. In this work, an aerial war game between an attack force and a defense force is developed, aiming to find an optimal deployment plan of defense stations for the protection of the defense force. The attack force is composed of multiple attack stations in slant formation or pincer formation, with each station equipped with multiple attackers. The defense force is composed of three commanding posts and multiple defense stations, with each station armed with multiple interceptors against the approaching attackers. A particle swarm optimization (PSO) algorithm is applied to find the optimal deployment plan of defense stations, based on four possible options of defense cost. Multiple games are then played on the optimal deployment plan, and the feasibility of the plan is evaluated by the statistics on the simulation results. The rest of this work is organized as follows: The game design is presented in Section 2, the simulation results and statistical analysis are presented in Section 3, some interesting outlier cases are reviewed in Section 4, simulation results with perfect kill probability are discussed in Section 5, comparisons and further discussions are presented in Section 6, empirical deployment plans are proposed and simulated in Section 7, and some conclusions are drawn in Section 8.