Lower Bounds and Heuristics for Supply Chain Stock Allocation

Assume that m periods with stochastic demand remain until the next replenishment arrives at a central warehouse. How should the available inventory be allocated among N retailers? This paper presents a new policy and a new lower bound for the expected cost of this problem. The lower bound becomes tight as N → ∞ . The infinite horizon problem then decomposes into N independent m-period problems with optimal retailer ship-up-to levels that decrease over the m periods, and the warehouse is optimally replenished by an order-up-to level that renders zero (local) warehouse safety stock at the end of each replenishment cycle. Based on the lower bound solution, we suggest a heuristic for finite N. In a numerical study it outperforms the heuristic by Jackson [Jackson, P. L. 1988. Stock allocation in a two-echelon distribution system or what to do until your ship comes in. Management Sci. 34(7) 880–895], and the new lower bound improves on Clark and Scarf's [Clark, A. J., H. Scarf. 1960. Optimal policies for a multi-echelon inventory problem. Management Sci. 6(4) 475–490] bound when N is not too small. Moreover, the warehouse zero-safety-stock heuristic is comparable to Clark and Scarf's warehouse policy for lead times that are not too long. The suggested approach is quite general and may be applied to other logistical problems. In the present application it retains some of the risk-pooling benefits of holding central warehouse stock.