In periodic-review stochastic inventory control, replenishment policies must balance holding and backlogging costs while coping with uncertain demand that may be arbitrarily correlated and non‑stationary, as well as evolving forecasts. Levi, Pál, Roundy, and Shmoys (2007) showed that the fully adaptive optimum can be approximated within a factor $2$, a guarantee that has stood unchanged for years.
We break this long‑standing barrier by achieving a $\frac{e}{e-1}\approx1.582$ approximation against the fully adaptive benchmark, while preserving general demand correlations and dynamic forecasts. The key is a structural link between marginal inventory accounting and weighted stopping problems: each unit’s conditional holding charge forms a supermartingale, and postponing its order incurs a history‑dependent backlog charge. Leveraging the random‑threshold technique of Ma and Tzamos (2023), we develop a potential‑function analysis that yields a $\frac{e}{e-1}$ approximation for any unit’s stopping problem. A common‑threshold policy then aggregates these marginal decisions into a global replenishment policy without additional loss.