An asymptotic result for the multi-stage weapon-target allocation problem

A multi-stage version of the weapon-target allocation problem is considered. This problem models the following battle scenario. The offense launches a number of weapons (the targets) which are aimed at assets of the defense. These targets are assigned values by the defense. The defense has a number of (nonreusable) defensive weapons each of which can engage at most one target. The outcome of such an engagement is stochastic. In each stage of the engagement the defense observes the outcomes of the assignments made in the previous stage before assigning a subset of the remaining weapons in the present stage. The objective is to assign weapons to targets so as to minimize the total expected value of the targets which survive all stages. In this study it is assumed that all targets have a value of unity and that the engagement of a target by a weapon depends only on the stage number. If the number of weapons used in each stage is linearly dependent on the number of targets, then it is shown that as the number of targets approaches infinity, the solution to this stochastic problem can be obtained by solving a related deterministic one