(R, nQ, s)

453. (R, nQ, s)

Periodic review, fixed pack size, with reorder point. Combines the periodic-review structure of (R, s, S) with the fixed-pack constraint of (nQ, r).

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

If $𝐼_{𝑡} \leq 𝑠$ (inventory low): order the smallest integer multiple $𝑛 𝑄$ that brings position above $𝑠$ (or above some target).
Else: don’t order.

Three parameters:

$𝑅$ = review interval
$𝑄$ = fixed pack size (supplier-mandated)
$𝑠$ = reorder point

(The “S” of (R, s, S) is replaced by implicit $𝑆 = 𝑠 + 𝑛 𝑄$ — the resulting position depends on how many packs are needed.)

453.0.1. When to use

The pragmatic real-world combination:

Reviews are periodic (cycle counts done weekly or monthly).
Pack size is fixed by the supplier (full case, dozen, pallet).
Want to skip the order when inventory is still healthy.

Common in retail (Each store reorders periodically, pack sizes are case-of-N) and pharmacy (DEA-mandated periodic counts of controlled substances, fixed pack size from manufacturer).

453.0.2. Inventory profile

Sawtooth, with two layers of discreteness:

Some reviews: skip (inventory above $𝑠$ ).
Other reviews: order $𝑛 𝑄$ where $𝑛$ is the integer chosen at trigger time.

Average inventory is higher than (R, s, S) because the integer-pack rounding inflates each order. Difference is small if $𝑄 ≪ 𝑠$ , large if $𝑄$ is comparable to $𝑠$ .

453.0.3. Set $𝑠$

Same protection-window logic as (R, s, S) and (R, S):

𝑠 = 𝜇_{𝑅 + 𝐿} + 𝑧 𝜎_{𝑅 + 𝐿}

Pack-size constraint affects the order quantity, not the trigger.

453.0.4. Choose $𝑛$ at trigger time

When triggered with position $𝐼_{𝑡} \leq 𝑠$ :

𝑛 = ⌈ \frac{𝑠 - 𝐼_{𝑡} + 𝑄}{𝑄} ⌉ (smallest 𝑛 s.t. 𝐼_{𝑡} + 𝑛 𝑄 > 𝑠)

A simpler rule: pick $𝑛$ to make post-order position $\approx 𝑠 + 𝑄 / 2$ (midpoint of typical $𝑠$ -to- $𝑠 + 𝑄$ range). For the smooth-demand case, $𝑛 = 1$ usually suffices.

453.0.5. Set $𝑄$

$𝑄$ is given by the supplier. If you have flexibility (multiple pack sizes available), pick the one closest to $𝑄_{EOQ} = \sqrt{2 𝐷 𝑆_{setup} / ℎ}$ .

453.0.6. Final formulas

𝑠 = 𝜇_{𝑅 + 𝐿} + 𝑧 𝜎_{𝑅 + 𝐿} 𝑄 = supplier-fixed pack size 𝑛 = ⌈ \frac{𝑠 - 𝐼_{𝑡} + 𝑄}{𝑄} ⌉

Example

Given (same policy-comparison params + a fixed pack):

$𝐷 = 12000$ /yr, $𝑑 = 33$ /day, $𝜎_{𝑑} = 5$ , $𝐿 = 14$ days, $𝑅 = 30$ days
$𝑧 = 1.645$ , $ℎ$ = $2/unit/yr
Pack size: $𝑄 = 1000$ units (case)

Step 1 — reorder point

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

Step 2 — typical review with healthy inventory

Inventory at review = 1800 (above $𝑠 = 1507$ ). Skip — no order.

Step 3 — typical review with depleted inventory

Inventory at review = 1200 (below $𝑠$ ). Compute $𝑛$ :

𝑛 = ⌈ \frac{1507 - 1200 + 1000}{1000} ⌉ = ⌈ \frac{1307}{1000} ⌉ = ⌈ 1.307 ⌉ = 2 packs

Order $𝑛 𝑄 = 2000$ units. Post-order position: $1200 + 2000 = 3200$ . Well above $𝑠$ .

Step 4 — typical review with slight depletion

Inventory at review = 1450 (just below $𝑠$ ). Compute $𝑛$ :

𝑛 = ⌈ \frac{1507 - 1450 + 1000}{1000} ⌉ = ⌈ \frac{1057}{1000} ⌉ = ⌈ 1.057 ⌉ = 2 packs

Hmm — even a tiny dip below $𝑠$ orders 2 packs. The formula always rounds up to ensure the post-order position is above $𝑠 + 𝑄$ . A more economical choice would be 1 pack (post-order = 2450, above $𝑠$ = 1507 ✓), but then over time you’d accumulate small deficits.

Refinement: a common variant orders the smallest $𝑛$ such that post-order position exceeds just $𝑠$ (not $𝑠 + 𝑄$ ):

𝑛_{loose} = ⌈ \frac{𝑠 - 𝐼_{𝑡}}{𝑄} ⌉ = ⌈ \frac{1507 - 1450}{1000} ⌉ = ⌈ 0.057 ⌉ = 1 pack

Choice of rule depends on tolerance for overshoot vs subsequent shortages. The tighter rule (always above $𝑠 + 𝑄$ ) is safer; the looser rule (just above $𝑠$ ) is leaner.

Step 5 — compare to (R, s, S)

(R, s, S) on same params: $𝑠 = 1507$ , $𝑆 = 𝑠 + 𝑄_{EOQ} = 2282$ .

At trigger with position 1450, order $𝑆 - 1450 = 832$ . Post-order: 2282.
Compare to (R, nQ, s) with $𝑄 = 1000$ : order $1 \cdot 1000 = 1000$ . Post-order: 2450.

(R, nQ, s) carries about 10% more inventory (2450 vs 2282) because integer packs round up. Acceptable when supplier inflexibility forces the constraint.

Use (R, nQ, s) when: periodic review + fixed pack size + want skip-when-healthy logic. The most operationally realistic of the periodic policies.