448. (s, S)

Continuous review, order-up-to $𝑆$. 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 $𝑠$, order enough to bring inventory position back up to $𝑆$.

Two parameters:

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

The actual order quantity varies: ${𝑄}_{\text{actual}}=𝑆-\text{(inventory position when triggered)}$. If inventory drops exactly to $𝑠$ before triggering, ${𝑄}_{\text{actual}}=𝑆-𝑠$. If demand causes a drop below $𝑠$ (overshoot), ${𝑄}_{\text{actual}}>𝑆-𝑠$.

448.0.1. Setup

Same as (Q, r): demand rate $𝑑$ with std ${𝜎}_{𝑑}$, lead time $𝐿$, costs ${𝑆}_{\text{setup}},ℎ$, service level $𝑧$.

448.0.2. Difference from (Q, r)

(Q, r) commits to a fixed batch $𝑄$ every order — doesn’t matter if inventory dropped to exactly $𝑟$ 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 $𝑠$ in one go.

448.0.3. Set $𝑠$

Same logic as $𝑟$ in (Q, r):

448.0.4. Set $𝑆$

For large fixed cost (most cases), $𝑆-𝑠$ should approximately equal the EOQ:

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

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

448.0.5. 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.