An analytical study of the Q(s,S) policy applied to the joint replenishment problem

We evaluate the performance of a joint replenishment inventory control policy, called the Q(s,S) policy, which was first suggested and characterized by Viswanathan [Manage. Sci. 43 (1997) 1447]. We argue why joint replenishment problems can be seen as one product, multiple location problems, and hence why they should be considered supply chain management problems. The paper uses Markov decision theory to work out an analytical solution procedure to evaluate the costs of a particular Q(s,S) policy, and thereby a method for computing the optimal Q(s,S) policy, under the assumption that demands follow a Poisson process. For a fixed order size Q, the problem can be decomposed for each item to find an optimal (s,S) policy with a stochastic review period. These subproblems can be solved by the algorithm of Zheng and Federgruen [Oper. Res. 39 (4) (1991) 654]. Thereafter it remains to find the optimal value of Q, which is done by systematic search over all relevant values of Q. The performance of the Q(s,S) policy is compared with the performance of other known joint ordering policies as in Pantumsinchai [Decis. Sci. 23 (1992) 111] and Viswanathan [Manage. Sci. 43 (1997) 1447]. It is shown that the Q(s,S) performs best among the considered policies.