Four regional depots against one central warehouse, worked on one part number: what pools, what does not, and where the real cost of the network sits.
The square root law says n depots need √n times the safety stock of one warehouse, so four depots should need double. That holds only for the demand part of the uncertainty. For part AF-7, where the supplier's lead time drives 90.5 per cent of the variance, four depots need 8,553 units against 7,547 centrally, 13.3 per cent more, not the 15,093 the law predicts. The large penalty of the network is in cycle stock, which does double.
Worked in full in Supply Chain Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Pooling is one of the most quoted results in inventory theory and one of the most misapplied. It is used to justify closing depots, and to estimate the stock a new regional site will need. Both uses go wrong when the law is applied to the whole buffer instead of to the part of the buffer it actually describes.
| Input | One warehouse | Each of four depots |
|---|---|---|
| Mean weekly demand | 1,840 | 460 |
| Standard deviation of weekly demand | 462 | 231 |
| Mean lead time, weeks | 6 | 6 |
| Standard deviation of lead time, weeks | 1.9 | 1.9 |
| Cycle service level (z) | 98% (2.0537) | 98% (2.0537) |
| Unit cost, EUR / holding rate | 28.60 / 13.4% | 28.60 / 13.4% |
The depots' demand is independent, so each has a standard deviation of 462 ÷ √4 = 231. That is the condition under which the square root law is exact.
Split the buffer into its two sources and the reason is plain. On demand noise alone the warehouse needs 2,324 units and the four depots 4,648: a ratio of exactly 2.00, as the law says. On lead-time variability alone both need 7,180 units: a ratio of 1.00. A late ship from the supplier is late for every depot at once, and each depot's lead-time term, its demand times 1.9 weeks, adds up to exactly the warehouse's. Nothing diversifies.
The square root law is a statement about independent demand noise. Before applying it, split your variance. AF-7's is 9.5 per cent demand and 90.5 per cent lead time at the warehouse, which is why the law misses by a factor of nearly two.
The cycle stock behaves very differently. Each depot orders its own economic order quantity on a quarter of the demand. Because EOQ is itself a square root, each depot's batch is 2,192.25 units, half the warehouse's 4,384.50, and four of them carry twice the cycle stock.
| Item | One warehouse | Four depots | Extra |
|---|---|---|---|
| Safety stock, units | 7,546.70 | 8,553.23 | 1,006.53 |
| Cycle stock, units | 2,192.25 | 4,384.50 | 2,192.25 |
| Ordering cost, EUR a year | 8,401.59 | 16,803.18 | 8,401.59 |
| Extra stock, EUR | 91,485.19 | ||
| Extra holding plus ordering, EUR a year | 20,660.60 |
Of the 91,485.19 EUR of extra stock, 62,698.42 is cycle stock and only 28,786.77 is safety stock. A network review that budgets the depots' extra buffer with the square root law gets the wrong number in the wrong line: it doubles the safety stock, which barely moves, and ignores the batches, which do double.
| σL, weeks | One warehouse | Four depots | Ratio | Extra, EUR |
|---|---|---|---|---|
| 0.0 | 2,324 | 4,648 | 2.00 | 66,471 |
| 0.5 | 2,995 | 5,018 | 1.68 | 57,840 |
| 1.0 | 4,436 | 5,991 | 1.35 | 44,449 |
| 1.9 | 7,547 | 8,553 | 1.13 | 28,787 |
| 3.0 | 11,572 | 12,253 | 1.06 | 19,453 |
The pooling benefit grows as the supplier becomes more reliable. With a perfectly regular supplier the law holds exactly and the depots cost 66,471 EUR of extra safety stock; at today's 1.9 weeks they cost 28,787. A company that fixes its supplier first and then reviews its network will find the case for centralising stronger than it looked.
The number of sites matters in the same way. At a 1.9-week spread, two depots need 1.05 times the warehouse's buffer, eight need 1.29 and sixteen 1.56. With a regular supplier the same rows read 1.41, 2.83 and 4.00, the square roots of 2, 8 and 16.
Applying √n to the total buffer. It overstates the safety-stock penalty of a network whenever lead time is the main source of variance, and understates nothing else, so it biases every decision towards centralising. The companion files add a second correction: measured at equal fill rate rather than equal cycle service level, the depots need a cycle service level of 96.70 per cent and only 109.00 more units, or 65,815.76 EUR of extra inventory in all.
The free workbook for this case on the Supply Chain Management companion page runs the four depots against the warehouse with every input in a blue cell. The buffer formula itself is worked in how to calculate safety stock when lead time varies.
It states that total safety stock across n independent locations is the square root of n times the safety stock of one location serving the same demand. Four depots would need twice the stock of one warehouse. It assumes the uncertainty is demand noise that is independent between locations. On part AF-7, demand alone gives 4,648 units across four depots against 2,324 centrally, exactly the predicted ratio of 2.00.
Because lead-time variability from a common supplier does not diversify away. Each depot's lead-time term is its own demand times the lead-time spread, and those terms add up to the warehouse's. With a 1.9-week spread on AF-7, four depots need only 13.3 per cent more safety stock than one warehouse, a ratio of 1.13 rather than 2.00.
Yes, and more reliably than to safety stock, because the economic order quantity is itself a square root. Each AF-7 depot orders 2,192.25 units against 4,384.50 centrally, so four depots carry twice the cycle stock: 62,698.42 EUR more, plus 8,401.59 EUR a year of extra ordering cost.
This article is one calculation from Supply Chain Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
Get the book on Amazon →Free companion files
Also on Amazon UK · Amazon Germany · Amazon France · Amazon Canada
Reading guide: business operations: cost, contracts, pricing and supply chain → · All 681 articles →
If this book helped, or didn’t, a few lines on Amazon are worth more than they look: they are what the next reader goes on. Write a review. The workbook stays free either way.