Stochastic Joint Replenishment Problem with Limited Demand Distribution Information

We consider a stochastic joint replenishment problem (JRP) of coordinating the replenishment of a sequence of products. The base stock level and replenishment cycle of each product are determined to minimize the average total cost rate resulting from placing orders, holding, and backlogging products. Under the assumption that partial demand distribution information is known, we study this problem in the paradigm of robust optimization using a minimax approach. An attractive feature of using the minimax approach is that the approach allows us to present a brief structural characterization for the stochastic JRP. In particular, we show that the minimax base stock level is a linear function of the replenishment cycle. With this result, the stochastic JRP can be converted into a deterministic one. Consequently, a power-of-two heuristic is proposed to generate minimax replenishment cycles with guaranteed 98% effectiveness. A case study based on an actual supply chain setting is provided to examine the performance of the minimax solutions. The numerical results are promising, and show that using minimax solutions instead of optimal solutions under full demand distribution information result in very small expense. Two extensions of the stochastic JRP are also considered