1.73-Optimal Echelon Stock (R,nQ) Policies in Two-Stage Stochastic Serial Systems

Abstract

We consider a classical two-stage continuous-review serial inventory system with unit-sized Poisson demand of rate lambda, backlog cost rate p, lead times L_1, L_2, echelon holding-cost rates h_1, h_2, and fixed shipment costs K_1, K_2. The classical echelon stock (R,nQ) policy offers a simple operating rule: Stage 1 orders a fixed Q_1-lot, Stage 2 orders n >= 1 such lots at a time, and a Stage-1 request waits until a complete lot is available. Existing uniform guarantees restrict lead times or induced lot sizes, and the known all-instance guarantee is primitive-dependent, leaving open whether this restrictive integer-ratio class admits any uniform guarantee over the full primitive space. We develop a setup-preserving cost-allocation lower bound and show that the infimum cost within the classical class is at most 1.73 times the optimal cost over a rate-balanced admissible comparison class, for every lambda, p, L_1, L_2, h_1, h_2, K_1, K_2 >= 0. The result holds for exact integer lot sizes and under both setup conventions, one charging each positive dispatch and the other every complete lot. On the boundaries the guarantee is one, as an equality of infima, when lambda=0, p=0, or h_2=0, and improves to 5/3 when K_2=0. Numerically, across three broad parameter grids, the best-found (R,nQ) policy costs at most 1.098 times the evaluated lower bound, with a median ratio of 1.007; a deliberately adversarial stress test reports 1.6004 under the conservative lot-cost convention. At that point, finite searches under shipment-cost accounting find no benefit from allowing Stage 2 to ship an incomplete Q_1-lot, which is consistent with looseness in the evaluated lower bound. The simple (R,nQ) policy, therefore, combines a straightforward fixed-lot implementation with a uniform guarantee over the full nonnegative primitive space.