Square root law of inventory

Got asked what would happen to inventory when the number of stocking locations change.  I thought for a minute and remembered a quick estimate.  The Square Root Law states that total safety stock can be approximated by multiplying the total inventory by the square root of the number of future warehouse locations divided by the current number.

X2 = (X1) * √ (n2/n1)

n1 = number of existing facilities
n2 = number of future facilities
X1 = existing inventory
X2 = future inventory

Here are two examples:

Current inventory is 4000 units, 2 facilities grow to 3.  Using the square root law the future inventory = (4000) * √ (3/2) = 4000 * 1.2247 = 4899 units.

Current inventory is 4000 units, 2 facilities grow to 8.  Using the square root law the future inventory = (4000) * √ (8/2) = 8000 units.

Related post: Square root law of inventory - application

There are 17 Comments

enorflis's picture

Thank you for the useful information! I look forward to future entries that will help remove the complication from critical concepts.

lloucka's picture

Ebony, you are most welcome. Come back often and feel free to submit a Lean Math topic of your own or make comments or ask questions.

caroline's picture

Sir can you please explain the meaning of square root law

Jamie Flinchbaugh's picture

This has always been a helpful process to at least think strategically about locations. Of course, when it comes to operationalizing the decision, we hope people think more deliberately about it, but it's good to be able to think about the consequences of a decision. Many large distributors, after vastly expanding their distribution footprint, ended up going back after consolidation because the benefits weren't worth the extra inventory. I recently talked a company out of adding another location (from 1 to 2) and this was part of the argument. Of course, the formula is only for inventory, and doesn't take into account the amount of extra MANAGEMENT resources needed to manage additional locations.

Also keep in mind the Taiichi Ohno rule of thumb: "The more inventory you have, the less likely you are to have what you need."

Jamie Flinchbaugh

lloucka's picture

Loucka's Corollary to Ohno's rule of thumb "The more locations you have, the less likely you will have the right product in the right place".

Haruhiko Matsumura (@harryconsult)'s picture

Thanks for the info. I’ve looked for useful site of square root law of inventory so far. This is informative site.

Ryan Orchard's picture

It is important to note that the benefits of aggregating stock only applies to the safety stock. You mention that in your description at the top, but the example implies that it applies to total inventory. The reality would be that the benefits would be only on a fraction of the the inventory (the safety stock, which in turn depends on replenishment lead times and demand variability.)

Ryan Orchard's picture

I would like to add to my previous comment - the square root law can apply to all types of inventory (i.e., cycle and safety stock), but has a number of assumptions, such as zero correllation in demand between the different regions. There is a very recent paper that is excellent on this topic (a bit techncial, but if you scroll through for key points it really nails it).

An empirical examination of the assumptions of the Square Root Law for inventory centralisation and decentralisation
by Gerald Oeser & Pietro Romano
International Journal of Production Research, Aug 2015

Anand Mantri's picture

Does the total inventory level change or only the safety stock changes as per square root law?

Kevin's picture


I know it's difficult to answer below without more details, but in general: does the square root law in general applies to Multi-echelon supply chains? So we have two central warehouses, 7 Regional DCs (the two central warehouses function also as a regional DC) and per regional DC there is a number of shops/miniwarehouses (end nodes) (varying from 3 to 8 per RDC)?

thanks in advance!

Kind regards,


lloucka's picture

The square root law is is a bit of a simplification. One key assumption is that each warehouse carries safety stock, and so more warehouses means more safety stock across the whole network. There are network inventory optimization modeling tools to look at such as Logility or Llamasoft.

John Davies's picture

There seems to be much confusion in the web literature on Maister's law. Surely it only applies to reduction of safety stock, and not to the cycle stock (stock held to meet the assumed mean of demand during each replenishment cycle). The Cycle stock for a given demand would remain unchanged no matter how many locations you held it in. However dispersing the stock would mean that at each location a mean demand level of stock for the area served would be accompanied by a safety stock for the variation experienced in that area. The "Law of Large Numbers" would seem to indicate that the proportion of variability would be lower in the central aggregated demand than in the dispersed local demands.
There is also much confusion as to the actual formula with some sources X2 = (X1)*(n2/n1)^.5 as above, and others suggesting for instance X2%= (1-(n2^.5/n1^.5))*100.

lloucka's picture

Small point - depending on packaging the cycle stock can increase with multiple stocking locations. Take as an example a liquid product shipped and stored in 55 gallon drums and dispensed in small quantities. With three warehouses the maximum inventory would be 165 gallons, minimum zero, average inventory would be half of 165. While with only one stocking location and the same total demand, the average inventory would be 27.5 gallons. Anyway, the 'law' is just a rule of thumb.

John Davies's picture

Point taken about the packaging issue. Also agreed that it is a rule of thumb, however it is a very useful rule of thumb.

CKR's picture

The real explanation is this:
If we have one system X (inventory) characterised by a variance Var(X) or sigma^2(X)
If we add another system Y to X, we have X+Y as our new system
(if and only if X and Y are independant is the Covariance=0)
Thus Var(X+Y)=Var(X)+Var(Y)
If Y has same variance as X -->

This is valid for 2 systems, and it can be demonstrated n independent systems of same variance.