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 4r1Q5 policies that do not require Qi+1/Qi 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 ri and Stage i+ 1 has positive on-hand inventory, then a shipment is sent to Stage i to raise its echelon inventory position to ri +Qi 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 41 +K1/K25 times the optimal cost, where K1 is the downstream fixed cost and K2 is the upstream fixed cost. We also provide an alternative performance bound, which depends on efficiently computable optimal 4r1Q5 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 4r1Q5 policies when K1 is dominated by K2. 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 h2/h1 → 0, where h1 is the downstream holding cost parameter and h2 is the upstream holding cost parameter.