Risk pooling & centralization

Imagine a shampoo brand with four regional warehouses, in Mumbai, Pune, Delhi and Chennai. Each one holds safety stock against its own uncertain demand, say 500 units each, 2,000 in total.

Now suppose all four regions are served from one central warehouse. On any given week Mumbai might sell more than forecast while Chennai sells less, so the surprises partly cancel out. The combined demand is less jumpy than the four pieces added up, and the central warehouse needs much less buffer to hit the same service level.

If the four regions are similar and independent, the total safety stock falls by a factor of $\sqrt{n}$, where $n$ is the number of locations pooled. Four locations become one, so the buffer drops from 2,000 to about 1,000 units, half as much. This is often called the square-root law.

Two catches keep it honest:

• Correlation. If all regions rise and fall together (a national heatwave, a festival everywhere), the surprises don’t cancel and the saving shrinks towards zero.
• Distance. One central site is further from customers, so delivery takes longer and outbound freight usually costs more. Pooling trades inventory against speed and transport.

You can get some of the benefit without closing warehouses: share stock between nearby sites (transshipment), use common components across products, or postpone the final customization.

Why surprises cancel

Let each of $n$ regions have weekly demand with standard deviation $\sigma$, independent of the others. Variances add, so combined demand has:

Safety stock is proportional to the standard deviation, so pooling $n$ equal, independent locations cuts total safety stock by a factor of $\sqrt{n}$.

Correlation shrinks the benefit

If demands move together, with correlation $\rho$ between every pair of regions:

At $\rho = 0$ this is $\sqrt{n}\,\sigma$. At $\rho = 1$ it is $n\,\sigma$, and pooling saves nothing. Negative correlation, where one region gains what another loses (customers moving between nearby stores, say), makes pooling even more powerful.

Using it for network changes

The same logic gives the square-root law of network design: total safety stock scales with the square root of the number of stocking points. Going from 10 warehouses to 5 multiplies safety stock by $\sqrt{5/10} \approx 0.71$, a cut of about 29%, if demand is similar and independent across them. Cycle stock falls too when each site orders its own EOQ, because EOQ also grows with the square root of demand.

For unequal regions, use the general rule: add the variances (and covariances), take the square root, and compare it with the sum of the separate buffers.

Centralize the right items, not the whole range

Pooling saves the most where uncertainty is high relative to volume: slow movers, expensive parts and erratic items, where each region’s buffer is large compared with its sales. For fast, steady movers the uncertainty is already small relative to demand, and the extra transport cost and delivery time of a central site usually outweigh the saving. Many companies therefore run a hybrid: fast movers stocked in every region, slow and expensive items held centrally and shipped on demand. Spare-parts networks are the classic case.

Pooling without moving warehouses

• Product pooling: design products to share components, so one common part replaces several specific ones.
• Postponement: keep stock generic and customize late, pooling the uncertainty of the variants.
• Virtual pooling: keep stock in separate places but share it, through transfers between stores or ship-from-store for online orders. This needs real-time visibility of stock everywhere.
• Capacity pooling: make plants able to produce several products. Jordan and Graves (1995) showed that limited flexibility, arranged as a closed chain linking products and plants, captures most of the benefit of full flexibility.

The costs on the other side

A central warehouse is further from most customers, so outbound transport per unit usually rises (smaller shipments over longer distances) and delivery takes longer. Inbound transport often falls, since full truckloads go to one site. The right number of warehouses balances inventory, inbound and outbound freight, facility costs and delivery time, which is the job of network design.

The same effect in time

Safety stock scales with $\sqrt{L}$ for the same reason pooled demand scales with $\sqrt{n}$: independent days partly cancel, just as independent regions do. That is why halving a lead time cuts the buffer by only about 29%, not 50%.