Optimal Threshold Policy for Sequential Weapon Target Assignment

We investigate a variant of the classical Weapon Target Assignment (WTA) problem, wherein N targets are sequentially visited by a bomber carrying M homogenous weapons. We are interested in the optimal assignment of weapons to targets. A weapon launched by the bomber destroys the jth target with probability pj and upon successful elimination, the bomber earns a positive reward rj. There is feedback in that the bomber, upon deploying a weapon, is notified whether or not it successfully destroyed the target. Whereupon, it decides whether to move on to the next target or allocate an additional weapon to the current target. We provide a tractable solution method for computing the optimal closed loop control policy that results in maximal expected total reward. Moreover, we show that a thresholding policy is optimal, wherein a weapon from an inventory of k weapons is dropped on the jth target iff k exceeds a stage dependent threshold value, κ(j).