444. (s, S)

Continuous review, order-up-to . Generalizes (Q, r): instead of always ordering the same Q, top up to a target level.

Decision rule: monitor inventory continuously. When inventory position drops to (or below) the reorder point , order enough to bring inventory position back up to .

Two parameters:

• = reorder point (trigger)
• = order-up-to level

The actual order quantity varies: . If inventory drops exactly to before triggering, . If demand causes a drop below (overshoot), .

444.0.1. Setup

Same as (Q, r): demand rate with std , lead time , costs , service level .

444.0.2. Difference from (Q, r)

(Q, r) commits to a fixed batch every order — doesn’t matter if inventory dropped to exactly or much below. (s, S) compensates: if inventory overshoots downward, the order is larger to make up.

In practice, with continuous review and small per-period demand (smooth depletion), the two are nearly identical. Differences emerge under lumpy demand — large discrete withdrawals can drop inventory well below in one go.

444.0.3. Set

Same logic as in (Q, r):

444.0.4. Set

For large fixed cost (most cases), should approximately equal the EOQ:

Why: when inventory drops exactly to (no overshoot), this gives the same order quantity as (Q, r). When it overshoots, the order grows automatically.

For low/zero fixed cost: (order one unit at a time — the “base stock” / (S-1, S) policy in the limit; see [base_stock.typ](base_stock.typ)).

444.0.5. Final formulas

Average inventory under (s, S) is slightly higher than (Q, r) because of the overshoot correction — but only by an amount proportional to expected overshoot per cycle, usually small.