Optimal Search of a Moving Target

There are n boxes. A target is initially in box i with a given probability p_ı, where p_ı ≧ 0, ∑p_ı = 1. Then at discrete time t = 1, 2, …, it moves from box to box. If at time t the target is in box i, then it will be in box i with probability p_ij at time t + 1, where p_ij ≧ 0, ∑_jp_ij = 1. A search of box i costs C_ı(c_ı > 0) and finds the target with probability α_ı if the target is in that box. A sequential search is made. The objective is either to maximize the probability of finding the target in a given number of searches or to minimize the expected searching cost before finding the target. Optimal strategies are characterized for special types of probability matrices [p_ij] for the n-box model.