A 1.932 Guarantee for (r,Q) Policies in Continuous-Review Lost-Sales Inventory Systems
Abstract
We provide a uniform guarantee for a continuous-review lost-sales inventory system with unit Poisson demand, deterministic lead time, and fixed ordering costs: a simple algorithm computes an (r,Q) policy and chooses between that policy and no replenishment, with cost at most (√2+√6)/2<1.932 times the unrestricted nonanticipative optimum. The algorithm starts by generating an optimal integer base-stock policy, which at zero setup cost achieves the tight worst-case factor √2 relative to the optimal system cost. For positive setup costs, it uses the reorder point of this integer base-stock policy and an integer lot at the economic order quantity scale. The inclusion of ordinary base stock in the continuous-review capped base-stock family also improves that family's earlier asymptotic $1.79$ upper guarantee to a uniform √2 guarantee valid at every finite parameter vector. To establish these guarantees, we construct a computable lower bound on the zero-setup optimal cost by tracking units already on hand or committed for delivery. We then combine it with a bound on holding and setup costs required by batching to obtain the general-setup guarantee and instance-specific performance bounds for any admissible replenishment policy. In our numerical study, the largest estimated policy-cost-to-lower-bound ratios are about $1.59$ for our heuristic $(r, Q)$ policy and $1.47$ after joint optimization of r and Q. The heuristic also performs well against the optimized policy: over the sampled instances, its median extra cost ranges from 2.33% to 3.91%, depending on the setup cost.
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BibTeX
@misc{hu20261932,
title = {{A 1.932 Guarantee for (r,Q) Policies in Continuous-Review Lost-Sales Inventory Systems}},
author = {Ming Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34576},
eprint = {2610.34576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34576},
}
RIS
TY - UNPB
TI - A 1.932 Guarantee for (r,Q) Policies in Continuous-Review Lost-Sales Inventory Systems
AU - Ming Hu
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34576
UR - https://arxiv.org/abs/2610.34576
AB - We provide a uniform guarantee for a continuous-review lost-sales inventory system with unit Poisson demand, deterministic lead time, and fixed ordering costs: a simple algorithm computes an (r,Q) policy and chooses between that policy and no replenishment, with cost at most (√2+√6)/2<1.932 times the unrestricted nonanticipative optimum. The algorithm starts by generating an optimal integer base-stock policy, which at zero setup cost achieves the tight worst-case factor √2 relative to the optimal system cost. For positive setup costs, it uses the reorder point of this integer base-stock policy and an integer lot at the economic order quantity scale. The inclusion of ordinary base stock in the continuous-review capped base-stock family also improves that family's earlier asymptotic $1.79$ upper guarantee to a uniform √2 guarantee valid at every finite parameter vector. To establish these guarantees, we construct a computable lower bound on the zero-setup optimal cost by tracking units already on hand or committed for delivery. We then combine it with a bound on holding and setup costs required by batching to obtain the general-setup guarantee and instance-specific performance bounds for any admissible replenishment policy. In our numerical study, the largest estimated policy-cost-to-lower-bound ratios are about $1.59$ for our heuristic $(r, Q)$ policy and $1.47$ after joint optimization of r and Q. The heuristic also performs well against the optimized policy: over the sampled instances, its median extra cost ranges from 2.33% to 3.91%, depending on the setup cost.
ER -