446. (nQ, r)

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

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

Three parameters:

446.0.1. When does this matter?

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

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

446.0.2. Decision logic when triggered

When inventory position falls to (for some overshoot ):

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

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

446.0.3. Set

Same as (Q, r):

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

446.0.4. Set

If 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 .

446.0.5. Average inventory

Slightly higher than (Q, r): . Overshoot is small under continuous review of smooth demand but grows with demand lumpiness.

446.0.6. Final formulas

Example

Given (same policy-comparison params):

  • /yr, /day, , days
  • = $50, = $2/unit/yr,
  • Pack size: (one pallet contains 800 units; supplier won’t break pallets)

Step 1 — reorder point

Step 2 — pack-size sanity check

Unconstrained EOQ would be . 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 where is the smallest integer such that the new position exceeds the next reorder cycle’s needs.

  • At trigger, position .
  • One pack: . Above , well above . ✓
  • 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 ):

  • One pack brings position to . Above ✓.
  • But average inventory just after order is , which is near for the previous cycle — still in normal range.

Now consider (much larger pack):

  • One pack: . Way above , but we just bought  10 cycles of inventory in one go.
  • Average inventory: (vs   optimal).
  • Holding cost more than 5x optimal.

Compare to (Q, r)

Under matched pack size (), (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.