Optimal Adaptive Target Shooting with Imperfect Feedback Systems

Abstract

In military combats, a common task is to destroy multiple targets with limited number of missiles. The weapons are not perfect and each can eliminate its target with a certain probability. Suppose after each shot we may receive some information on the state of the target. Realistically these feedbacks are subject to error, thus we will not know for sure whether the target is really destroyed. This research focuses on finding the optimal allocation of missiles so that maximum number of targets can be destroyed, or the probability of destroying all targets is maximized.

Formally, we consider the following sequential target shooting problem. Given a fixed number of targets and missiles, the objective is to find the optimal strategy, where the missiles are fired sequentially at the targets in the attempt to destroy as many targets as possible. The probability of destroying a target at each shot is known and after each shot, a report becomes available on the state of the target: either destroyed or intact. However, the reports are subject to two types of errors and the probabilities of making these errors are also known. We call the report system imperfect if the probability of getting wrong reports is positive.

Note that after each shot, the probability that a target is intact can be evaluated based on the imperfect reports. We have shown that the myopic decision strategy, which always shoots the target with the highest intact probability, is optimal when all the missiles have the same hitting probability and the targets are homogeneous. In addition, techniques for comparing imperfect feedback systems are developed. A partial ordering of the systems is provided when the optimal strategy is carried out.