(R, s, S)

452. (R, s, S)

Periodic review with a reorder point. Generalizes (R, S) by adding a “do nothing” option when inventory is high.

Decision rule: at every review point (every $𝑅$ time units), observe the inventory position $𝐼_{𝑡}$ .

If $𝐼_{𝑡} \leq 𝑠$ (inventory low): order $𝑆 - 𝐼_{𝑡}$ (bring up to $𝑆$ ).
Else ( $𝐼_{𝑡} > 𝑠$ , inventory still healthy): don’t order.

Three parameters:

$𝑅$ = review interval
$𝑠$ = reorder point
$𝑆$ = order-up-to level

452.0.1. When to use

Use (R, s, S) when:

Reviews are periodic (truck arrives weekly), but
You don’t want to order every review when inventory is still healthy (saves order cost when consumption was light).

Compared to (R, S): saves order cost in low-consumption periods. Compared to (s, S): adds the periodic-review constraint (simpler operations, less precise).

(R, s, S) is widely considered the most general periodic policy. It generalizes both (R, S) (set $𝑠 = 𝑆$ , always order) and zero ordering (set $𝑠$ very low, never order).

452.0.2. Inventory profile

Sawtooth, with order skipping:

Some reviews: $𝐼_{𝑡} > 𝑠$ , no order. Inventory continues to drain.
Other reviews: $𝐼_{𝑡} \leq 𝑠$ , order up to $𝑆$ .
Order arrives at next review or before, depending on $𝐿$ vs $𝑅$ .

If $𝑅$ is short relative to mean cycle, most reviews do not trigger orders. If $𝑅$ is long, almost every review triggers.

452.0.3. Set $𝑠$ — protection-window logic

Same as (R, S): the protection window is $𝑅 + 𝐿$ , because an order placed at review covers demand until the next review’s order arrives. So:

𝑠 = 𝜇_{𝑅 + 𝐿} + 𝑧 𝜎_{𝑅 + 𝐿}

Identical to the $𝑆$ in (R, S)! That’s because $𝑠$ in (R, s, S) plays the same protection role as $𝑆$ in (R, S) — it’s the level above which we still have enough buffer.

452.0.4. Set $𝑆$

$𝑆 - 𝑠$ should be the EOQ-style batch size. Heuristic:

𝑆 = 𝑠 + 𝑄_{EOQ} = 𝑠 + \sqrt{\frac{2 𝐷 𝑆_{setup}}{ℎ}}

When inventory drops to exactly $𝑠$ between reviews, the next order is for $𝑆 - 𝑠 = 𝑄_{EOQ}$ . With overshoot (inventory below $𝑠$ at review), the order is bigger — same as (s, S).

452.0.5. Computing optimal $𝑠, 𝑆$ exactly

Closed forms exist for special cases (Wagner’s lower-bound formula); in general, $(𝑠^{*}, 𝑆^{*})$ require dynamic programming or simulation. The heuristic above is within a few % of optimal for typical industrial parameter ranges.

452.0.6. Final formulas

𝑠 = 𝜇_{𝑅 + 𝐿} + 𝑧 𝜎_{𝑅 + 𝐿} 𝑆 = 𝑠 + \sqrt{\frac{2 𝐷 𝑆_{setup}}{ℎ}} SS = 𝑧 𝜎_{𝑅 + 𝐿}

Example

Given (same policy-comparison params):

$𝐷 = 12000$ /yr, $𝑑 = 33$ /day, $𝜎_{𝑑} = 5$ , $𝐿 = 14$ days, $𝑅 = 30$ days
$𝑆_{setup}$ = $50, $ℎ$ = $2/unit/yr, $𝑧 = 1.645$

Step 1 — reorder point (same as $𝑆$ from (R, S))

𝑠 = 𝜇_{𝑅 + 𝐿} + 𝑧 𝜎_{𝑅 + 𝐿} = 1452 + 54.6 \approx 1507

Step 2 — order-up-to level

𝑄_{EOQ} = \sqrt{\frac{2 \cdot 12000 \cdot 50}{2}} \approx 775

𝑆 = 𝑠 + 𝑄_{EOQ} = 1507 + 775 \approx 2282

Step 3 — typical reviews

Steady-state inventory between reviews drops by about $𝑅 𝑑 = 990$ units per review interval.

Review 1: position is 2282 (just received order). Drains to $\approx 2282 - 990 = 1292$ by review 2.
Review 2: position 1292. Below $𝑠 = 1507$ ✓ → order $2282 - 1292 = 990$ units.
Continues this pattern.

Almost every review triggers an order because $𝑅$ is comparable to the average cycle time. Behaviorally similar to (R, S).

Step 4 — when “don’t order” actually fires

Suppose demand is unusually light over a review period (e.g., only 200 units consumed instead of 990). Then inventory at the next review is $2282 - 200 = 2082 > 1507 = 𝑠$ . No order placed → save $50.

Then next review: another 200 consumed → 1882, still above $𝑠$ → no order. Then another: 1682 → still above $𝑠$ → no order. Eventually drops below $𝑠$ , then big catch-up order.

(R, s, S) sleeps through low-consumption stretches; (R, S) would have ordered every review regardless.

Compare to (R, S)

(R, S): always order every review. $𝑆$ at protection-window level (1507). Number of orders = $\frac{365}{𝑅} = 12.2$ /year.
(R, s, S): may skip some reviews. Same $𝑠$ (1507), $𝑆 = 𝑠 + 𝑄_{EOQ}$ (2282). Order frequency varies; expected $\approx \frac{365}{\frac{𝑄_{EOQ}}{𝑑}} = \frac{365}{\frac{775}{33}} = 15.5$ /year but with variability.

Use (R, s, S) when order setup cost is significant and consumption variability is high — the “skip a review” behavior amortizes the setup cost across larger batches.