409. 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).
where is the standard deviation of lead-time demand and is the quantile of the standard normal at the desired service level.
409.0.1. 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 :
For normally distributed lead-time demand :
Decompose:
- = expected demand during lead time. Just covers the average — no safety.
- = the safety stock. Buffer above the average for the variability.
409.0.2. Lead-time demand variance — the four cases
The form of depends on which of demand and lead time are random. Combining via the law of total variance:
where = std of per-period demand, = std of lead time, and .
The four classical cases:
| Case | Where the variance comes from | |
| Constant , constant | No variance — basic EOQ regime, no safety stock needed | |
| Variable , constant | Demand variability over periods | |
| Constant , variable | Lead time variability scales the deterministic demand | |
| Variable , variable | Both contribute |
The combined formula reduces to the special cases when one variance is zero. The 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.
409.0.3. Safety stock =
Once you know , the safety stock formula is:
Choose 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)).
409.0.4. Risk pooling reduces safety stock
When inventory is held centrally rather than distributed across independent locations:
- Decentralized: each location holds , total = .
- Centralized: one location holds (because aggregate variance scales with , std with ).
Total safety stock falls by factor under centralization (assuming independent demands across locations). This is the square-root law of inventory aggregation.
Example
Given (compare four scenarios):
- units / day
- days
- (95% CSL)
Case 1 — constant demand, constant lead time
No variance. Safety stock = 0. Just hold units of cycle/pipeline; no buffer.
Case 2 — variable demand, constant lead time (, )
Case 3 — constant demand, variable lead time (, days)
Lead time variability dominates demand variability in this case — same , same demand, but 5× higher safety stock. Because is much larger than , the multiplier amplifies .
Case 4 — variable demand AND variable lead time
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 multiplies ).
- Centralize inventory → safety stock shrinks by (risk pooling).