Pooling warehouses can halve your safety stock — if demand actually cooperates

The square-root pooling benefit assumes independent demand; positive correlation erodes it.

Published 26 Aug 2026 Updated 04 Sep 2026 8 min read Studied in Logistics & Supply Chain Analytics, Imperial College London

You only need Variance of a sum of random variables, and what safety stock is for.

On this page

The square root law of inventory centralisation says that consolidating $n$ stocking locations into one leaves you holding $1/\sqrt{n}$ of the stock those $n$ sites held between them 1. Maister’s own illustration is that three depots hold 1.732 times what one depot would 3. It is a real theorem with a clean proof — and it arrives with a list of stated assumptions, one of which is that demand at the separate locations is uncorrelated. That assumption is not a technicality at the edge of the result. It is the entire mechanism.

Where the square root comes from

Safety stock at a single location, over a fixed lead time, is $z\sigma$: a service-level multiple $z$ times the standard deviation $\sigma$ of demand over that lead time, in units of stock. Run $n$ locations, each sized the same way, and the decentralised total is

$$ \text{decentralised} = n \, z\sigma \quad (\text{units}). $$

Consolidate into one location serving the same total demand. Its safety stock depends on the standard deviation of the pooled demand, and that is where the assumption enters. For $n$ locations each with variance $\sigma^2$ and a common pairwise correlation $\rho$,

$$ \operatorname{Var}(\text{total}) = n\sigma^2 + n(n-1)\rho\sigma^2 = n\sigma^2\bigl(1 + (n-1)\rho\bigr), $$

because there are $n$ variance terms and $n(n-1)$ ordered covariance terms, each equal to $\rho\sigma^2$. Taking the square root and multiplying by $z$,

$$ \text{pooled} = z\sigma\sqrt{n\bigl(1 + (n-1)\rho\bigr)}, \qquad \frac{\text{pooled}}{\text{decentralised}} = \sqrt{\frac{1 + (n-1)\rho}{n}}. $$

Set $\rho = 0$ and the covariance terms vanish, leaving $\sqrt{1/n}$ — the square root law. Nothing about consolidation itself produced that; the zero did.

What else the theorem assumes Boylan sets out the assumptions behind Maister’s safety-stock result: equal demand variance at every location, the same safety-stock multiple everywhere (which implies a fixed lead time), uncorrelated demand between locations, and unchanged total system demand before and after consolidation, with everything else held constant 3. Equal variance is doing real work too: Das proved in 1978 that the decentralised-to-centralised ratio is strictly below $\sqrt{n}$ unless demand variability is the same at all locations, in which case it attains its maximum of $\sqrt{n}$ 3. The square root law is the best case, not the typical one.

Four warehouses at a 95% service level

Four regional warehouses. Weekly demand at each has standard deviation $\sigma = 100$ units, the service target is 95%, and the one-sided normal quantile for 95% is 1.6449, rounded here to $z = 1.65$:

$$ \text{decentralised} = 4 \times 1.65 \times 100 = 660 \text{ units}, \qquad \text{pooled at } \rho = 0 = 1.65 \times 100 \times \sqrt{4} = 330 \text{ units}. $$

Exactly half, and the halving does not depend on $z$: the ratio $\sqrt{n}\,z\sigma / (n z\sigma)$ cancels $z$ and $\sigma$ entirely, so it would be 0.5 at a 90% or a 99% service level too. Using the unrounded 1.6449 gives 657.94 and 328.97 units — the same ratio, and a reminder that the 660 and 330 are rounded quantities of stock, not exact ones.

Turning the correlation dial

Now move $\rho$ and hold everything else. With $n = 4$ the ratio is $\sqrt{(1+3\rho)/4}$:

Pairwise correlation ρ	Pooled ÷ decentralised (unitless)	Pooled safety stock (units)	Reduction
−1/3	0.000	0	100%
0	0.500	330	50%
0.25	0.661	437	34%
0.5	0.791	522	21%
0.75	0.901	595	10%
1	1.000	660	0%

A third of the headline benefit is gone by $\rho = 0.25$ and three-fifths of it by $\rho = 0.5$. At $\rho = 1$ every location moves in lockstep, the combined standard deviation is $n\sigma$, and consolidation buys nothing at all. This is Eppen’s result in the form used here: the magnitude of the saving depends on the correlation of demand, and the square-root case is the special case in which demands are identical and uncorrelated 2 3.

Why the correlation floor is −1/3 and not −12 lines

Combined variance is $n\sigma^2(1 + (n-1)\rho)$, and a variance cannot be negative, so $1 + (n-1)\rho \ge 0$, giving $\rho \ge -1/(n-1)$. At $n = 4$ that floor is $-1/3$, where the combined variance is exactly zero and pooled safety stock is zero.

Two locations can reach $\rho = -1$, since $-1/(n-1) = -1$ at $n = 2$; that is the equal-weight version of the perfect hedge in two-asset portfolio theory. Four locations cannot all be perfectly opposed to one another simultaneously, which is what the $-1/3$ floor records.

What a consolidation case has to estimate

The gap between $\rho = 0$ and the real $\rho$ is not a rounding error in a business case; at $\rho = 0.5$ it is the difference between a 50% and a 21% reduction, or 192 units of stock in the worked example above. So the question a consolidation proposal has to answer is not whether pooling helps — Eppen showed the centralised system’s expected holding and penalty costs never exceed the decentralised system’s 2 3 — but by how much, and that answer is set by a correlation the proposal has usually not measured.

Any shared driver pushes $\rho$ upward: a national promotion, a common seasonal pattern, a macroeconomic cycle, an upstream disruption that hits every region at once. None of those are exotic, and each one moves the answer toward the right-hand end of the table.

Estimating ρ is itself the hard part The honest version of the pitch estimates pairwise correlation from historical demand data rather than assuming zero. That estimate is not free: Boylan notes that covariance estimates carry large sampling errors, particularly for slow-moving items where demand data is sparse, and that the models requiring a correlation for every pair of depots can be difficult to apply for exactly that reason 3. Reporting a range of outcomes across a plausible band of ρ is more defensible than reporting a single √n figure — and considerably more defensible than reporting one without saying which ρ produced it.

Which step in the derivation is the independence assumption actually used?Recall

The variance of the pooled demand. In general $\operatorname{Var}(\text{total}) = n\sigma^2(1 + (n-1)\rho)$, with $n$ variance terms and $n(n-1)$ covariance terms. Independence sets every covariance term to zero, leaving $n\sigma^2$, a combined standard deviation of $\sigma\sqrt{n}$, and a ratio of $1/\sqrt{n}$. Everything else in the derivation — the service multiple, the demand scale, the number of locations — survives unchanged when $\rho \ne 0$.

A four-site consolidation is pitched on a 50% safety-stock saving. Historical demand shows ρ ≈ 0.5. What is the saving?Recall

About 21%. The ratio is $\sqrt{(1 + 3 \times 0.5)/4} = \sqrt{0.625} = 0.791$, so 660 units of decentralised safety stock becomes about 522 rather than 330 — 192 units more than the pitch claimed, and less than half the promised reduction.

Why can't four warehouses reach the zero-variance case that two can?Recall

Because a common pairwise correlation across $n$ locations is bounded below by $-1/(n-1)$, or the combined variance would be negative. Two locations can be perfectly opposed at $\rho = -1$; four can only reach $-1/3$, which is still enough to drive the combined variance to zero, but it is not “exact opposition” and no arrangement of four locations achieves that.

Sources and further reading

Centralisation of Inventories and the ‘Square Root Law’ (external source) David H. Maister International Journal of Physical Distribution 6(3), 124–134; publisher copy is subscription-gated 1976
Used for: The original statement that total inventory in a system is proportional to the square root of the number of locations at which a product is stocked.
Note—Effects of Centralization on Expected Costs in a Multi-Location Newsboy Problem (external source) Gary D. Eppen Management Science 25(5), 498–501; publisher copy is subscription-gated 1979
Used for: The result that the saving from centralisation depends on the correlation of demand between locations, and that the square-root case is the special case of identical, uncorrelated demands.
The Centralisation of Inventory and the Modelling of Demand, §2.2–2.4 and §14.3.2 (external source) John E. Boylan PhD thesis, University of Warwick; free full text from the Warwick Research Archive Portal 1997
Used for: Maister’s claim quoted verbatim with its √3 worked example, the ceteris paribus condition and the eight assumptions behind it, Das’s (1978) proof that the decentralised-to-centralised ratio is below √n unless demand variability is equal at every location, Eppen’s three results and his assumptions, the Zinn–Levy–Bowersox two-depot correlation formula, and the difficulty of estimating pairwise demand covariances in practice.