Monotone stopping-allocation problems

Abstract

Stopping-allocation problems are concerned with how best to allocate observations among some K competing stochastic populations and when to stop the observation process. The goal of the decision-maker is to choose a stopping–allocation rule to maximize the expected value of a payoff function. First the stopping rule is fixed, and the local and global optimality of the myopic allocation rule are derived under some monotonicity conditions. An application is considered, namely the inspection problem and its use in solving a computer scheduling problem. Next, optimization is done with respect to both the allocation rule and the stopping rule. For any given stopping-allocation rule, it is shown that under some monotonicity conditions, the decision-maker can improve on it by using a ‘partial' myopic allocation rule and a generalized one-stage-look-ahead stopping rule; this result is then extended, under the same conditions and other monotonicity requirements, to derive the joint optimality of the myopic allocation rule and the one-stage-look-ahead stopping rule. Finally this latter result is applied to the inspection problem.