A Stackelberg equilibrium for a missile procurement problem

This paper deals with a procurement problem of missiles involving the efficient assignment of the missiles to some targets. Within a fixed amount of budget, a leader purchases several types of missiles, by which he aims to damage as much value as possible a follower hides into some facilities later. The effectiveness of the missile depends on the type of missile and facility. A payoff of the game is the expected amount of destroyed value. The problem is generalized as a two-person zero-sum game of distributing discrete resources with a leader and a follower. Our problem is to derive a Stackelberg equilibrium for the game. This type of game has an abundance of applications. The problem is first formulated into an integer programming problem with a non-separable objective function of variables and it is further equivalently transformed into a maximin integer knapsack problem. We propose three exacts methods and an approximation method for an optimal solution.