Heterogeneous surface-to-air missile defense battery location: a game theoretic approach

In the context of an air defense missile-and-interceptor engagement, a challenge for the defender is that surface-to-air missile batteries often must be located to protect high-value targets dispersed over a vast area, subject to which an attacker may observe the disposition of batteries and subsequently develop and implement an attack plan. To model this scenario, we formulate a two-player, extensive form, three-stage, perfect information, zero-sum game that accounts for, respectively, a defender’s location of batteries, an attacker’s launch of missiles against targets, and a defender’s assignment of interceptor missiles from batteries to incoming attacker missiles. The resulting trilevel math programming formulation cannot be solved via direct optimization, and it is not suitable to solve via full enumeration for realistically-sized instances. We instead adapt the game tree search technique Double Oracle, within which we embed either of two alternative heuristics to solve an important subproblem for the attacker. We test and compare these solution methods to solve a designed set of 52 instances having parametric variations, from which we derive insights regarding the nature of the underlying problem. Enhancing the solution methods with alternative initialization strategies, our superlative methodology attains the optimal solution for over 75% of the instances tested and solutions within 3% of optimal, on average, for the remaining 25% of the instances, and it is promising for realistically-sized instances, scaling well with regard to computational effort.