(nQ, r)

450. (nQ, r)

Continuous review, fixed pack size. Like (Q, r), but the supplier requires orders in multiples of a fixed batch $𝑄$ (cases, dozens, pallet quantities).

Decision rule: monitor inventory continuously. When inventory position drops to (or below) reorder point $𝑟$ , order the smallest integer multiple $𝑛 𝑄$ that brings inventory position back above $𝑟$ .

Three parameters:

$𝑄$ = fixed pack size (set by supplier, e.g., case of 24)
$𝑛$ = number of packs to order this cycle (a function of how far inventory dropped, not a free parameter)
$𝑟$ = reorder point

450.0.1. When does this matter?

(Q, r) assumes you can order any integer quantity $𝑄$ . (nQ, r) handles the practical case where:

Supplier ships only full pallets / cases / drums.
Pack size is fixed by packaging or logistics; you choose how many.

If $𝑄$ is small relative to lead-time demand, (nQ, r) is essentially (Q, r) with rounding noise. If $𝑄$ is large (e.g., a pallet contains a year’s worth of demand), the round-up effect inflates inventory significantly.

450.0.2. Decision logic when triggered

When inventory position falls to $𝑟 - 𝛿$ (for some overshoot $𝛿 \geq 0$ ):

𝑛 = ⌈ \frac{𝑠 - position + 𝑄}{𝑄} ⌉

Or simpler in practice: order the smallest $𝑛 𝑄$ such that the post-order position exceeds some target (e.g., $𝑟 + 𝑄$ ).

In the small overshoot case (smooth demand), $𝑛 = 1$ almost always — order one pack of $𝑄$ . (nQ, r) reduces to (Q, r).

450.0.3. Set $𝑟$

Same as (Q, r):

𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿}

The pack-size constraint affects $𝑄$ , not $𝑟$ . The reorder trigger logic doesn’t change.

450.0.4. Set $𝑄$

If $𝑄$ is given (supplier-mandated pack size), no decision. If you can choose pack-size from a discrete set (dozen, gross, case, pallet), pick the closest to $𝑄_{EOQ} = \sqrt{2 𝐷 𝑆_{setup} / ℎ}$ .

450.0.5. Average inventory

Slightly higher than (Q, r): $\approx 𝑄 / 2 + SS + \frac{expected overshoot}{2}$ . Overshoot is small under continuous review of smooth demand but grows with demand lumpiness.

450.0.6. Final formulas

𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿} 𝑄 = supplier-fixed pack size 𝑛 = ⌈ \frac{triggered position deficit + 𝑄}{𝑄} ⌉

Example

Given (same policy-comparison params):

$𝐷 = 12000$ /yr, $𝑑 = 33$ /day, $𝜎_{𝑑} = 5$ , $𝐿 = 14$ days
$𝑆_{setup}$ = $50, $ℎ$ = $2/unit/yr, $𝑧 = 1.645$
Pack size: $𝑄 = 800$ (one pallet contains 800 units; supplier won’t break pallets)

Step 1 — reorder point

𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿} = 462 + 30.8 \approx 493

Step 2 — pack-size sanity check

Unconstrained EOQ would be $\approx 775$ . Forced pack of 800 is very close — pack-size penalty is negligible.

Step 3 — typical order

Inventory position drops to roughly 493 over time (smooth demand). Order $𝑛 𝑄$ where $𝑛$ is the smallest integer such that the new position exceeds the next reorder cycle’s needs.

At trigger, position $\approx 493$ .
One pack: $493 + 800 = 1293$ . Above $𝑟$ , well above $𝑟 + 𝑄 / 2 = 893$ . ✓
Order 1 pack (800 units).

Step 4 — when overshoot matters

If a sudden burst drops inventory from 600 to 200 (a 400-unit lumpy withdrawal that crosses below $𝑟$ ):

One pack brings position to $200 + 800 = 1000$ . Above $𝑟$ ✓.
But average inventory just after order is $1000$ , which is near $𝑆_{target} \approx 1293$ for the previous cycle — still in normal range.

Now consider $𝑄 = 4000$ (much larger pack):

One pack: $200 + 4000 = 4200$ . Way above $𝑟$ , but we just bought 10 cycles of inventory in one go.
Average inventory: $\approx \frac{𝑄}{2} = 2000$ (vs $\frac{𝑄_{EOQ}}{2} = 388$ optimal).
Holding cost more than 5x optimal.

Compare to (Q, r)

Under matched pack size ( $𝑄 = 800 \approx 𝑄_{EOQ}$ ), (nQ, r) and (Q, r) are nearly identical — same trigger, same order quantity. The difference grows when pack size diverges from the unconstrained optimum.

Use (nQ, r) when: supplier mandates pack sizes; you want predictable shipping units; pack size happens to be close to EOQ. Switch to (Q, r) only if you can break packs.