Generalized Bomber and Fighter Problems: Offline Optimal Allocation of a Discrete Asset

Abstract

The classical bomber problem concerns properties of the optimal allocation policy of a given number, n, of anti-aircraft missiles, with which an airplane is equipped. The airplane begins at a distance t >0 from its destination and uses some of the anti-aircraft missiles when intercepted by enemy planes that appear according to a homogeneous Poisson process. The goal is to maximize the probability of reaching its destination. The fighter problem deals with a similar situation, but the goal is to shoot down as many enemy planes as possible. The optimal allocation policies are dynamic, depending upon both the number of missiles and the time which remains to reach the destination when the enemy is met. The present paper generalizes these problems by allowing the number of enemy planes to have any distribution, not just Poisson. This implies that the optimal strategies can no longer be dynamic, and are, in our terminology, offline. We show that properties similar to those holding for the classical problems hold also in the present case. Whether certain properties hold that remain open questions in the dynamic version are resolved in the offline version. Since ‘time’ is no longer a meaningful way to parametrize the distributions for the number of encounters, other more general orderings of distributions are needed. Numerical comparisons between the dynamic and offline approaches are given.