449. (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 time units), observe inventory position .
- If (inventory low): order the smallest integer multiple that brings position above (or above some target).
- Else: don’t order.
Three parameters:
- = review interval
- = fixed pack size (supplier-mandated)
- = reorder point
(The “S” of (R, s, S) is replaced by implicit — the resulting position depends on how many packs are needed.)
449.0.1. 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).
449.0.2. Inventory profile
Sawtooth, with two layers of discreteness:
- Some reviews: skip (inventory above ).
- Other reviews: order where 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 , large if is comparable to .
449.0.3. Set
Same protection-window logic as (R, s, S) and (R, S):
Pack-size constraint affects the order quantity, not the trigger.
449.0.4. Choose at trigger time
When triggered with position :
A simpler rule: pick to make post-order position (midpoint of typical -to- range). For the smooth-demand case, usually suffices.
449.0.5. Set
is given by the supplier. If you have flexibility (multiple pack sizes available), pick the one closest to .
449.0.6. Final formulas
Example
Given (same policy-comparison params + a fixed pack):
- /yr, /day, , days, days
- , = $2/unit/yr
- Pack size: units (case)
Step 1 — reorder point
Step 2 — typical review with healthy inventory
Inventory at review = 1800 (above ). Skip — no order.
Step 3 — typical review with depleted inventory
Inventory at review = 1200 (below ). Compute :
Order units. Post-order position: . Well above .
Step 4 — typical review with slight depletion
Inventory at review = 1450 (just below ). Compute :
Hmm — even a tiny dip below orders 2 packs. The formula always rounds up to ensure the post-order position is above . A more economical choice would be 1 pack (post-order = 2450, above = 1507 ✓), but then over time you’d accumulate small deficits.
Refinement: a common variant orders the smallest such that post-order position exceeds just (not ):
Choice of rule depends on tolerance for overshoot vs subsequent shortages. The tighter rule (always above ) is safer; the looser rule (just above ) is leaner.
Step 5 — compare to (R, s, S)
(R, s, S) on same params: , .
- At trigger with position 1450, order . Post-order: 2282.
- Compare to (R, nQ, s) with : order . 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.