Operations.

Pooling

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The question

Count the places you hold the same thing separately — stock split across three sites, cover split across four teams, a queue forming in front of each person rather than in front of all of them — and ask what it costs to keep them apart. Nobody invoices you for that number. Which is exactly why it has survived every budget you've ever run.

The idea

Every queue that feeds several counters at once — the bank, the airport, the passport hall — is a piece of arithmetic made physical, and it's the same arithmetic that sets how much stock you hold. Pool a set of uncertain demands and the average adds in full, but the variability adds far more slowly, because it's the variances that add rather than the swings themselves. So the cushion you need per unit of demand shrinks as the pool grows. Gary Eppen, a professor at the University of Chicago's business school, put a number on that in Management Science in 1979, in a paper with a title nobody quotes and a result everybody uses: Effects of Centralization on Expected Costs in a Multi-Location Newsboy Problem. Take a set of locations, each holding its own buffer against its own uncertain demand, and merge them into one. Where those demands are genuinely independent, the expected cost of holding and running short falls by a factor of the square root of the number of locations. Four depots become one and the buffer cost halves. Nine become one and it falls to a third. The size of the prize isn't set by how big your buffers are. It's set by how many separate piles you're keeping them in.

The framework

01
Find the same buffer held in more than one place.
Walk the business looking for one thing kept in several piles: safety stock at each site, a spare person on each team, a cash float per branch, an on-call rota per squad, a queue in front of each desk. The tell is a resource that all the piles could in principle serve, but that's held separately because of how the org chart was drawn rather than because any customer needs it that way. Write the list before you argue about a single item on it. In most businesses it's longer than anyone expects, and nobody owns it end to end.
02
The square root is a ceiling you can't argue with.
There's nothing here to negotiate. Unlike a setup cost, which you can attack until the batch follows it down, the square root is a ceiling on what pooling can ever give back — no amount of clever operating beats it. The only variable you control is how many pools you collapse, so run the arithmetic across the whole set before you reorganise any two of them. Nine into one cuts the buffer to a third. Three into two barely registers, and the merge still costs you freight, systems and a slower response at whichever site just lost its stock.
03
Do these demands actually move independently?
The square root only arrives if the pooled demands are genuinely unrelated, and plenty of them aren't. Four branches in one city share the weather, the public holiday and the same regional promotion, so their peaks land together and you don't get anything like the formula's promise. The reverse is the real prize: demands that peak at different times pool better than independent ones do. A business with a summer line and a winter line, or a weekday client base and a weekend one, is sitting on a gain most owners never think to go looking for.
04
Pool the commitment, not necessarily the building.
Centralising is the crude version and it's often the wrong one, because your customers still want the thing near them. The gain comes from pooling the uncertainty, and you can do that while leaving the physical arrangement alone: hold a shared reserve any site can draw on, delay the decision about where a unit goes until you know where it's needed, let sites ship to each other, keep a roster of people who float instead of a spare on every team. One queue feeding several servers rather than a queue per server is the same move, and it costs nothing to make.

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