Reorder Points and Safety Stock
The Economic Order Quantity
answers how much to order. It leaves the other half of the job untouched: when to
order. Orders do not arrive by magic the instant the shelf empties — there is a lead time
while the supplier ships. And demand during that lead time is uncertain. Order too late and you
stock out before the delivery lands; order too early and you carry needless stock. The tools that solve
this are the reorder point and its cushion, the safety stock.
The continuous-review (Q, r) policy
The simplest and most common rule watches inventory continuously and works on two numbers:
- Q — the order quantity (typically the EOQ): how much
to buy each time.
- r — the reorder point: the stock level that, when
reached, triggers a fresh order of size Q.
Between deliveries the stock draws down as a familiar sawtooth; the moment it crosses
r, an order is placed, and it arrives one lead time later — ideally just as
the stock nears the floor. Watch a few cycles:
The reorder point must cover the demand expected during the lead time, plus a buffer for the
days demand runs high. That buffer is the safety stock, the shaded band along the bottom that the plan
tries never to dip into.
Splitting the reorder point in two
Let demand during the lead time have mean \mu_{LT} and standard deviation
\sigma_{LT}. The reorder point is simply the expected lead-time demand plus a
cushion:
r = \underbrace{\mu_{LT}}_{\text{cover the average}} \;+\; \underbrace{z\,\sigma_{LT}}_{\text{safety stock}}.
The safety stock \text{SS} = z\,\sigma_{LT} is measured in
standard deviations of lead-time demand. The multiplier
z
is a z-score chosen from your target service level — the probability of not
running out during a cycle. Want to be safe 95% of the time? Use the z with 95% of the normal curve
below it, z = 1.645. If demand never varied
(\sigma_{LT} = 0) you would need no safety stock at all: uncertainty is the
only thing you are buffering against.
Worked example
A warehouse sells a part at a steady average, and demand over the two-week supplier lead time has mean
\mu_{LT} = 200 units and standard deviation
\sigma_{LT} = 30 units. Management wants a 95% service level,
so z = 1.645. Then
\text{SS} = z\,\sigma_{LT} = 1.645 \times 30 \approx 49 \text{ units}, \qquad r = \mu_{LT} + \text{SS} = 200 + 49 = 249 \approx 250.
So the rule is: when stock falls to 250, order a fresh batch. On an average cycle the 50 units
of safety stock go untouched and arrive as the new delivery lands; on an unlucky, high-demand cycle they
are the margin that keeps the shelf from going empty. Raise the target to 99%
(z = 2.33) and safety stock jumps to
2.33 \times 30 \approx 70 — an extra 21 units bought for the last four points
of service.
The price of certainty
Safety stock rises with the z-score, and the z-score rises ever more steeply as the service
level nears 100%. The table below (all with \sigma_{LT} = 30) shows the
cruel geometry: the jump from 90% to 95% costs about 11 units; from 99% to 99.9%, over 20 more.
| Service level | z-score | Safety stock = z·σLT |
|---|
| 90% | 1.28 | ≈ 38 units |
| 95% | 1.645 | ≈ 49 units |
| 98% | 2.05 | ≈ 62 units |
| 99% | 2.33 | ≈ 70 units |
| 99.9% | 3.09 | ≈ 93 units |
Each extra "nine" of reliability buys a smaller reduction in stockouts for a larger pile of inventory.
This is why almost no one targets 100%: it would demand infinite safety stock, because the normal
distribution has no upper bound.
A flat percentage padding ignores how variable demand actually is. Two products with the same
average lead-time demand can behave completely differently: one steady as a metronome
(\sigma_{LT} small), one wildly erratic
(\sigma_{LT} large). The steady one needs almost no buffer; the erratic one
needs a big one. Scaling the safety stock by \sigma_{LT} — the actual spread
of demand — puts the cushion exactly where uncertainty lives. Longer lead times hurt too: a longer
window means more demand can accumulate and more variability, so
\sigma_{LT} grows with the lead time, and so must the safety stock.
Chasing a very high service level is a game of sharply diminishing returns. Because the z-score
explodes toward infinity as the service level approaches 100%, going from 95% to 99% might add half as
much safety stock again, and reaching 99.9% can more than double it — all to eliminate rare stockouts
that may cost far less than the inventory tied up to prevent them. The right target is an
economic choice: weigh the cost of a stockout against the cost of holding the buffer. "Never run
out" sounds like good service; it is usually a very expensive way to gold-plate the shelf.