Bounds and asymptotics for planning critical safety stocks

We develop bounds and approximations for setting base-stock levels in production–inventory systems with limited production capacity. Our approximations become exact as inventories become critical, meaning either that the target service level is very high or the backorder penalty is very large. Our bounds apply even without this requirement. We consider both single-stage and multi-stage systems. For single-stage systems, we find tight bounds and asymptotically exact approximations for optimal base-stock levels: for multistage systems, our results give partial characterizations of the optimal levels. Part of our analysis is a precise connection, in the critical regime, between a multistage system and an associated single-stage system consisting solely of the bottleneck facility.