Simple Policies for Joint Replenishment Can Perform Badly

We consider the stochastic joint replenishment problem in which several items must be ordered in the face of stochastic demand. Previous authors proposed multiple heuristic policies for this economically-important problem. We show that several such policies are not good approximations to an optimal policy, since as some items grow more expensive than others, the cost rate of the heuristic policy can grow arbitrarily larger than that of an optimal policy. These policies include the well-known RT policy, the P (s, S) policy, the Q(s, S) policy and the recently-proposed (Q,S, T ) policy. To compensate for this problem, we propose a QI(s, S) policy, which is a generalization of the Q(s, S) policy, and in which items are ordered if an expensive item is demanded or if demand for other items reaches Q. Our numerical results demonstrate that QI(s, S) policies do indeed overcome the weakness of the other heuristics, and can cost less than the Q(s, S) heuristic even when the ratio of the cost of expensive items to other items is only a factor of three.