Safety Stock

413. Safety Stock

The portion of inventory held as a buffer against uncertainty — in demand, in lead time, or both. Distinguished from cycle stock (which exists due to batching, not uncertainty).

SS = 𝑧 \cdot 𝜎_{LD}

where $𝜎_{LD}$ is the standard deviation of lead-time demand and $𝑧$ is the quantile of the standard normal at the desired service level.

413.0.1. Why this form

The reorder point under (Q, r) is set such that the probability of running out during lead time equals the target stockout probability $𝛼$ :

𝑃 (𝐷_{𝐿} > 𝑟) = 𝛼

For normally distributed lead-time demand $𝐷_{𝐿} \sim 𝒩︀ (𝜇_{𝐿}, 𝜎_{LD}^{2})$ :

𝑟 = 𝜇_{𝐿} + 𝑧 𝜎_{LD}, 𝑧 = Φ^{- 1} (1 - 𝛼)

Decompose:

$𝜇_{𝐿}$ = expected demand during lead time. Just covers the average — no safety.
$𝑧 𝜎_{LD}$ = the safety stock. Buffer above the average for the variability.

413.0.2. Lead-time demand variance — the four cases

The form of $𝜎_{LD}^{2}$ depends on which of demand and lead time are random. Combining via the law of total variance:

𝜎_{LD}^{2} = 𝐸 [𝐿] 𝜎_{𝑑}^{2} + {(𝐸 [𝑑])}^{2} 𝜎_{𝐿}^{2}

where $𝜎_{𝑑}$ = std of per-period demand, $𝜎_{𝐿}$ = std of lead time, $𝐸 [𝑑] = | 𝑑 |$ and $𝐸 [𝐿] = | 𝐿 |$ .

The four classical cases:

Case	$𝜎_{LD}$	Where the variance comes from
Constant $𝑑$ , constant $𝐿$	$0$	No variance — basic EOQ regime, no safety stock needed
Variable $𝑑$ , constant $𝐿$	$\sqrt{𝐿} \cdot 𝜎_{𝑑}$	Demand variability over $𝐿$ periods
Constant $𝑑$ , variable $𝐿$	$\| 𝑑 \| \cdot 𝜎_{𝐿}$	Lead time variability scales the deterministic demand
Variable $𝑑$ , variable $𝐿$	$\sqrt{𝐿 𝜎_{𝑑}^{2} + {\| 𝑑 \|}^{2} 𝜎_{𝐿}^{2}}$	Both contribute

The combined formula reduces to the special cases when one variance is zero. The ${| 𝑑 |}^{2}$ factor in the lead-time term often dominates when both are random — small percentage variance in lead time can produce huge swings in lead-time demand.

413.0.3. Safety stock = $𝑧 𝜎_{LD}$

Once you know $𝜎_{LD}$ , the safety stock formula is:

SS = 𝑧 \cdot 𝜎_{LD}

Choose $𝑧$ from the desired service level (see [cycle_service_level.typ](../service_levels/cycle_service_level.typ) and [fill_rate.typ](../service_levels/fill_rate.typ)).

413.0.4. Risk pooling reduces safety stock

When inventory is held centrally rather than distributed across $𝑛$ independent locations:

Decentralized: each location holds $𝑧 𝜎_{𝑑} \sqrt{𝐿}$ , total = $𝑛 \times 𝑧 𝜎_{𝑑} \sqrt{𝐿}$ .
Centralized: one location holds $𝑧 𝜎_{𝑑} \sqrt{𝐿} \sqrt{𝑛}$ (because aggregate variance scales with $𝑛$ , std with $\sqrt{𝑛}$ ).

Total safety stock falls by factor $\sqrt{𝑛}$ under centralization (assuming independent demands across locations). This is the square-root law of inventory aggregation.

Example

Given (compare four scenarios):

$| 𝑑 | = 33$ units / day
$| 𝐿 | = 14$ days
$𝑧 = 1.65$ (95% CSL)

Case 1 — constant demand, constant lead time

No variance. Safety stock = 0. Just hold $| 𝑑 | | 𝐿 | = 462$ units of cycle/pipeline; no buffer.

Case 2 — variable demand, constant lead time ( $𝜎_{𝑑} = 5$ , $𝜎_{𝐿} = 0$ )

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

SS = 𝑧 𝜎_{LD} = 1.65 \cdot 18.7 \approx 31 units

Case 3 — constant demand, variable lead time ( $𝜎_{𝑑} = 0$ , $𝜎_{𝐿} = 3$ days)

𝜎_{LD} = | 𝑑 | \cdot 𝜎_{𝐿} = 33 \cdot 3 = 99

SS = 1.65 \cdot 99 \approx 163 units

Lead time variability dominates demand variability in this case — same $𝑧$ , same demand, but 5× higher safety stock. Because $| 𝑑 | = 33$ is much larger than $𝜎_{𝑑} = 5$ , the multiplier $| 𝑑 |$ amplifies $𝜎_{𝐿}$ .

Case 4 — variable demand AND variable lead time

𝜎_{LD} = \sqrt{𝐿 𝜎_{𝑑}^{2} + {| 𝑑 |}^{2} 𝜎_{𝐿}^{2}} = \sqrt{14 \cdot 25 + 33^{2} \cdot 9} = \sqrt{350 + 9801} = \sqrt{10151} \approx 100.8

SS = 1.65 \cdot 100.8 \approx 166 units

Combined std is barely larger than the lead-time-only case — when one source of variance dominates, the other contributes little. Reducing the bigger one (lead-time variability here) gives the biggest safety-stock savings.

Practical takeaway

Reduce demand-side uncertainty → modest safety-stock savings.
Reduce lead-time uncertainty → often dramatic savings (because $| 𝑑 |$ multiplies $𝜎_{𝐿}$ ).
Centralize inventory → safety stock shrinks by $\sqrt{𝑛}$ (risk pooling).

413. Safety Stock

413.0.1. Why this form

413.0.2. Lead-time demand variance — the four cases

413.0.3. Safety stock = 𝑧𝜎LD

413.0.4. Risk pooling reduces safety stock

413.0.3. Safety stock = $𝑧 𝜎_{LD}$