Modified Echelon ( r, Q ) Policies with Guaranteed Performance Bounds for Stochastic Serial Inventory Systems

We consider the classic continuous-review N stage serial inventory system with a homogeneous Poisson demand arrival process at the most downstream stage (Stage 1). Any shipment to each stage, regardless of its size, incurs a positive fixed setup cost and takes a positive constant lead time. The optimal policy for this system under the long-run average cost criterion is unknown. Finding a good worst-case performance guarantee remains an open problem. We tackle this problem by introducing a class of modified echelon (r, Q) policies that do not require Q i + 1 / Q i to be a positive integer: Stage i + 1 ships to Stage i based on its observation of the echelon inventory position at Stage i ; if it is at or below r i and Stage i + 1 has positive on-hand inventory, then a shipment is sent to Stage i to raise its echelon inventory position to r i + Q i as close as possible . We construct a heuristic policy within this class of policies, which has the following features: First, it has provably primitive-dependent performance bounds. In a two-stage system, the performance of the heuristic policy is guaranteed to be within (1 + K 1 / K 2 ) times the optimal cost, where K 1 is the downstream fixed cost and K 2 is the upstream fixed cost. We also provide an alternative performance bound, which depends on efficiently computable optimal ( r, Q ) solutions to N single-stage systems but tends to be tighter. Second, the heuristic is simple, it is efficiently computable and it performs well numerically; it is even likely to outperform the optimal integer-ratio echelon ( r,Q ) policies when K 1 is dominated by K 2 . Third, the heuristic is asymptotically optimal when we take some dominant relationships between the setup or holding cost primitives at an upstream stage and its immediate downstream stage to the extreme, for example, when h 2 / h 1 (rightarrow) 0, where h 1 is the downstream holding cost parameter and h 2 is the upstream holding cost parameter.