The joint replenishment problem: Optimal policy and exact evaluation methody

Abstract

We propose a new method to evaluate any stationary joint replenishment policy under compound Poisson demand. The method makes use of an embedded Markov chain that only considers the state of the system after an order is placed. The resulting state space reduction allows exact analysis of instances that until now could only be evaluated using approximation procedures. In addition, the size of the state space is not affected if we include nonzero lead times, backlog, and lost sales. We characterize the optimal joint replenishment policy, and use these characteristics to develop a greedy-optimal algorithm that generalizes the can-order policy, a well-known family in the class of joint replenishment policies. We numerically show that this generalized can-order policy only marginally improves the best conventional can-order policy. For sizeable systems with multiple items, the latter can now be found using our exact embedded Markov-chain method. Finally, we use our method to improve and extend the well-known decomposition approach.