A STOCHASTIC SEQUENTIAL ALLOCATION PROBLEM WHERE THE RESOURCES CAN BE REPLENISHED

Abstract

Suppose a hunter starts hunting over certain given t periods with i bullets in hand. A distribution of the value of each appearing target and the hitting probability of a bullet are known. For shooting, he takes a strategy of shoot-look-shoot scheme, implying that if a bullet just fired does not hit the target, then the hunter must decide whether or not to shoot an additional one. At the end of each period, it is allowed to replenish a given number of bullets by paying a certain cost. The objective here is to examine the properties of the optimal policy which maximizes the total expected net reward. We get the following main results: (1) the optimal policy for shooting is monotone in the number of bullets in hand if it is always optimal either to replenish a certain number of bullets every period or not to replenish them at all, (2) if only one bullet can be replenished per period, then both the optimal policies for shooting and replenishment are monotone in the number of bullets in hand, (3) if more than one can be replenished per period, then there exist examples where the optimal policies for shooting are not monotone in the number of remaining bullets.