(Q, r)

449. (Q, r)

Continuous review, fixed order quantity. The most common stochastic-demand policy in textbooks and practice.

Decision rule: monitor inventory position continuously. When it drops to (or below) the reorder point $𝑟$ , order a fixed quantity $𝑄$ .

Two parameters:

$𝑄$ = order quantity (a fixed batch every time)
$𝑟$ = reorder point (the trigger level)

449.0.1. Setup

Random demand with rate $𝑑$ per unit time and standard deviation $𝜎_{𝑑}$ . Lead time $𝐿$ (constant for now). Holding cost $ℎ$ , fixed order cost $𝑆$ , plus a service-level requirement $𝑧$ (e.g., $𝑧 = 1.645$ for 95% cycle service level).

Inventory position $= on-hand + on-order - backorders$ . Continuous review means we always know it.

449.0.2. Inventory profile

Sawtooth shape, but with random consumption rate:

Inventory drains stochastically.
When position hits $𝑟$ , a Q-unit order is placed; arrives $𝐿$ time later.
During lead time, demand consumes $𝐷_{𝐿} =$ random draw from a distribution with mean $𝜇_{𝐿} = 𝑑 𝐿$ and std $𝜎_{𝐿} = 𝜎_{𝑑} \sqrt{𝐿}$ .
Service level (probability of no stockout per cycle) = $𝑃 (𝐷_{𝐿} \leq 𝑟)$ .

449.0.3. Set $𝑟$ — the reorder point

To meet cycle service level $1 - 𝛼$ :

𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿}

where:

$𝜇_{𝐿} = 𝑑 𝐿$ (expected demand during lead time)
$𝜎_{𝐿} = 𝜎_{𝑑} \sqrt{𝐿}$ (std of lead-time demand, assuming i.i.d. periods)
$𝑧 = Φ^{- 1} (1 - 𝛼)$ (standard-normal quantile for the desired service level)

The term $𝑧 𝜎_{𝐿}$ is the safety stock — buffer for demand variability.

449.0.4. Set $𝑄$ — the order quantity

When fixed cost $𝑆$ is significant, use the EOQ formula:

𝑄 = \sqrt{\frac{2 𝐷 𝑆}{ℎ}}

(Approximate — exact joint optimization of $𝑄$ and $𝑟$ exists but EOQ is within a few % of optimal.)

449.0.5. Final formulas

𝑄 \approx \sqrt{\frac{2 𝐷 𝑆}{ℎ}} 𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿} SS = 𝑧 𝜎_{𝐿}

Average inventory (approximation): $𝑄 / 2 + SS$ , so total holding cost $\approx ℎ (𝑄 / 2 + 𝑧 𝜎_{𝐿})$ .

Example

Given (shared policy-comparison params):

Annual demand: $𝐷$ = 12,000 units/year ( $𝑑 = 33$ units/day)
Daily demand std: $𝜎_{𝑑} = 5$ units/day
Lead time: $𝐿 = 14$ days (constant)
Order cost: $𝑆$ = $50 / order
Holding cost: $ℎ$ = $2 / unit / year
Service level: 95% → $𝑧 = 1.645$

Step 1 — order quantity (EOQ)

𝑄 = \sqrt{\frac{2 \cdot 12000 \cdot 50}{2}} = \sqrt{600000} \approx 775 units

Step 2 — lead-time demand statistics

𝜇_{𝐿} = 𝑑 𝐿 = 33 \cdot 14 = 462 units

𝜎_{𝐿} = 𝜎_{𝑑} \sqrt{𝐿} = 5 \cdot \sqrt{14} \approx 18.7 units

Step 3 — reorder point

𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿} = 462 + 1.645 \cdot 18.7 \approx 462 + 30.8 \approx 493 units

Safety stock: $SS = 30.8$ units.

Step 4 — interpret

Whenever inventory position drops to 493, place an order for 775. The order arrives 14 days later. Expected demand during those 14 days is 462; the extra 31 units of safety stock cover up-to-95th-percentile demand fluctuations.

Average ordering rate: $\frac{𝐷}{𝑄} = \frac{12000}{775} \approx 15.5$ orders / year. Average cycle length: $\frac{𝑄}{𝐷} = \frac{775}{12000} \approx 24$ days.

Average inventory: $\frac{𝑄}{2} + SS = 388 + 31 \approx 419$ units. Annual holding cost: $ℎ \cdot 419 \approx$ $838 / year. Annual ordering cost: $𝑆 \cdot \frac{𝐷}{𝑄} = 50 \cdot 15.5 \approx$ $775 / year. Total: $\approx$ $1613 / year (vs $1549 for deterministic EOQ — the gap is the cost of safety stock).