451. (R, S)

Periodic review, order-up-to $𝑆$. The simplest periodic-review policy.

Decision rule: at every review point (every $𝑅$ time units), observe the inventory position and order just enough to bring it up to $𝑆$. No reorder point — order at every review whether or not inventory is depleted.

Two parameters:

• $𝑅$ = review interval (often fixed by external factors — a weekly truck, a monthly billing cycle)
• $𝑆$ = order-up-to level

451.0.1. When to use

Periodic review — either because continuous monitoring is expensive (manual stock-taking) or because deliveries arrive on a fixed cadence (the supplier’s truck comes every Monday whether you want it to or not). (R, S) is the workhorse for this regime.

451.0.2. Inventory profile

Sawtooth, with reviews as the cycle anchor:

• At review $𝑡$: observe position ${𝐼}_{𝑡}$. Order $𝑆-{𝐼}_{𝑡}$. Order arrives at $𝑡+𝐿$.
• Between reviews: inventory drains stochastically. No intervention even if it runs out.
• Stockout window: from order placed until the next order arrives — that’s $𝑅+𝐿$ (review interval plus lead time).

451.0.3. Set $𝑆$

The protection window is $𝑅+𝐿$ (not just $𝐿$ as in continuous review!). Demand during this window has:

• Mean: ${𝜇}_{𝑅+𝐿}=𝑑\left(𝑅+𝐿\right)$
• Std: ${𝜎}_{𝑅+𝐿}={𝜎}_{𝑑}\sqrt{𝑅+𝐿}$

For service level $1-𝛼$:

The safety stock $𝑧{𝜎}_{𝑅+𝐿}$ is larger than continuous-review safety stock by factor $\sqrt{\left(𝑅+𝐿\right)/𝐿}$.

451.0.4. No $𝑟$ in (R, S)

Unlike continuous review, there’s no reorder point — every review triggers an order, even if inventory is still high. Order quantity varies cycle to cycle:

Sometimes ${𝑄}_{\text{actual}}\approx 0$ (just received last order); sometimes large (after a busy period). Average: ${𝑄}_{\text{actual}}=𝑅𝑑$ (you replace what was consumed since last review).

451.0.5. Final formulas