Base Stock

447. Base Stock

447.1. (S−1, S) Policy — base stock / one-for-one

The simplest continuous-review policy: every consumed unit triggers an immediate replenishment of one unit. Sometimes called the base-stock policy.

Decision rule: maintain inventory position at exactly $𝑆$ . As soon as one unit is sold, place an order for one unit.

Equivalently: trigger an order at $𝑠 = 𝑆 - 1$ (whenever inventory dips below $𝑆$ ). Hence the name.

Single parameter:

$𝑆$ = base-stock level (the only decision)

447.1.1. When does this make sense?

Use base-stock when fixed ordering cost is negligible (or items are too critical to risk by batching):

Spare parts and critical components: stockouts are catastrophic, individual items are cheap to order.
High-value, low-volume items: holding cost dominates; ordering one-at-a-time is fine.
Vendor-managed inventory with continuous data exchange: ordering happens automatically per consumption.

Don’t use base-stock if you have meaningful per-order setup costs — the EOQ logic says batch up.

447.1.2. Inventory position vs. on-hand

In (S−1, S):

Inventory position = on-hand + on-order = $𝑆$ always (just before each consumption).
On-hand = $𝑆 - outstanding orders$ .
An outstanding order is one consumed within the past $𝐿$ time units that hasn’t yet arrived.

If demand during lead time is $𝐷_{𝐿}$ , outstanding orders just before the next consumption is exactly $𝐷_{𝐿}$ . So:

On-hand = {(𝑆 - 𝐷_{𝐿})}^{+}

Stockouts occur when $𝐷_{𝐿} > 𝑆$ .

447.1.3. Set $𝑆$ — Newsvendor style

Choose $𝑆$ to balance holding cost vs. shortage cost.

For Poisson lead-time demand $𝐷_{𝐿} \sim Poisson (𝜇_{𝐿})$ :

𝑆^{*} = smallest integer with 𝑃 (𝐷_{𝐿} \leq 𝑆) \geq CR

Critical ratio: $CR = 𝐶_{𝑢} / (𝐶_{𝑢} + 𝐶_{𝑜})$ where $𝐶_{𝑢}$ is shortage cost per unit, $𝐶_{𝑜}$ is holding cost per unit per cycle.

For a cycle service-level requirement (probability of no stockout in lead time = $1 - 𝛼$ ):

𝑆 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿} (normal approximation)

For Poisson: $𝜎_{𝐿} = \sqrt{𝜇_{𝐿}}$ , so $𝑆 \approx 𝜇_{𝐿} + 𝑧 \sqrt{𝜇_{𝐿}}$ .

447.1.4. Average inventory

Steady-state expected on-hand:

𝐸 [on-hand] = 𝐸 [{(𝑆 - 𝐷_{𝐿})}^{+}] = 𝑆 - 𝜇_{𝐿} + 𝜎_{𝐿} 𝐿 (𝑧)

where $𝐿 (𝑧) = 𝜑 (𝑧) - 𝑧 (1 - Φ (𝑧))$ is the standard normal loss function (see [expected_profit.typ](../newsvendor/expected_profit.typ)).

Annual holding cost: $ℎ \cdot 𝐸 [on-hand]$ .

447.1.5. Final formulas

𝑆 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿} SS = 𝑧 𝜎_{𝐿} 𝐸 [on-hand] \approx 𝑆 - 𝜇_{𝐿} + 𝜎_{𝐿} 𝐿 (𝑧)

Example

Given (same policy-comparison params, but interpret as low-volume spare-part regime):

Demand: $𝑑 = 1$ unit / day (Poisson) — instead of $33$ /day to make the spare-part case realistic
Lead time: $𝐿 = 14$ days
Service level: 95% → $𝑧 = 1.645$

Lead-time demand: $𝐷_{𝐿} \sim Poisson (𝜇_{𝐿} = 14)$ , $𝜎_{𝐿} = \sqrt{14} \approx 3.74$ .

Step 1 — base-stock level

Normal approximation:

𝑆 = 𝜇_{𝐿} + 𝑧 𝜎_{𝐿} = 14 + 1.645 \cdot 3.74 \approx 20 units

Exact (Poisson lookup): smallest $𝑆$ with $𝑃 (𝐷_{𝐿} \leq 𝑆) \geq 0.95$ .

$𝑃 (𝐷_{𝐿} \leq 19) \approx 0.923$
$𝑃 (𝐷_{𝐿} \leq 20) \approx 0.952$ ✓
$𝑆^{*} = 20$ .

Step 2 — interpret

Maintain inventory position at 20 at all times. Every time a unit is consumed, immediately reorder one. Average backlog of in-transit orders is $𝜇_{𝐿} = 14$ . Average on-hand:

𝐸 [on-hand] \approx 20 - 14 = 6 units (plus loss-function correction)

Step 3 — what if demand drops to 1 unit permonthinstead of per day?

$𝜇_{𝐿} = 14 \cdot (\frac{1}{30}) \approx 0.47$ , $𝜎_{𝐿} = \sqrt{0.47} \approx 0.68$ .

$𝑆 = 0.47 + 1.645 \cdot 0.68 \approx 1.6 \to 2$ (round up).
Most of the time, on-hand = 2; very occasional stockouts.

Compare to (Q, r)

In the high-volume case ( $𝑑 = 33$ /day): (Q, r) gives $𝑟 = 493$ , $𝑄 = 775$ . Base-stock would give $𝑆 = 493$ , but with $𝑆 - 1 = 492$ as the trigger — meaning 15 orders per day on average. Per-order cost dominates → use (Q, r).

In the low-volume spare-part case ( $𝑑 = 1$ /day): per-order cost is moot (warehouse already has the receiving infrastructure), and base-stock’s responsiveness wins. (S−1, S) is the right choice.

Rule of thumb: base-stock is optimal when $𝑆_{setup} \to 0$ and converges to $𝑟$ (no batching). For any non-trivial setup cost, batched policies (Q, r) or (s, S) reduce per-unit ordering cost.