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Square Root Law

The Problem Statement

The effect on maintaining the same item in more than one location on the overall levels of inventory escapes many practitioners. The principal problem is that an item stocked in multiple location increases the overall inventory levels. The key to the problem is the term location. Whilst we all understand physical warehouses being a location where inventory is stored, location can also be virtual locations or contracts where inventory is ring-fenced. Let us explore what this all means; we start with the relationship of inventory levels and the number of locations described in the square root law.

The Square Root Law

The square root law was first published by D.H. Maister in 1976 and describes the relationship levels of inventory levels and the number of locations an item in stored in. The relationship is as follows:

The variables of this equation are defined as follows, they apply to a discrete and unique item or material:

  • In1: Inventory Level in Current State
  • In2: Inventory Level in Future State
  • n1: number of locations in the Current State
  • n2: number of locations in the Future State

What this means in practice is that the inventory levels increase in a non-linear fashion. If we store an item instead of in a single location in two locations, then the increase is not the factor 2, but the square root of 2, which means 1,41. The below graph shows the increase of the inventory levels with increasing number of location where an item is stocked.

The reason inventory grows at the square root of the locations can be best explained as follows using the principles of statistics. The square root law is based on the determination of the standard error. The standard error measures the variability between sample means taken from the same population multiple times, whereas the standard deviation measures the variability within a single sample.

Applied to inventory control this can be seen as the accumulation of errors due to the variability across n locations.

Application of the Square Root Law and the Location Issue

Now a practitioner will say that such a high number of locations in a physical network is unrealistic. Well, that is true unless we better understand what a location really means. A location in the real supply chain world can be many things:

  • A Virtual Warehouse or Location where inventory is logically separated in a physical warehouse
  • A Contract which contains ring-fenced inventory, logically or sometimes physically separated in a physical warehouse acts similar to a location, in particular is not used for a single and committed customer demand
  • Allocated Inventory made through hard and soft reservations, acting similar to contracts if the allocation is used for a channel, customer or store grouping
  • A real Physical Warehouse

The square root law should be applied for all locations described about unless this location services a single customer demand with no safety stock. In practice, is it acceptable to take on more inventory risk when locations are primarily virtual locations in a single physical location because rebalancing inventory across multiple virtual inventory locations is straightforward and does not necessarily imply physical movements.

The Square Root Law and its Relationship to Inventory Strategy

Inventory represents a key element of the triangle: Cash, Service and Inventory. Increasing levels of inventory requires cash being spend on the value of inventory as well as the inventory carrying cost; the latter being somewhere between 20% to 25% of the inventory value.

A key question of the use of multiple location is how to set the targets for providing the right service levels to customer whilst controlling the cash outlay. Centralization versus distribution of inventory enables a company to provide different service levels. However:

  • some items are moving fast and some slow
  • some are highly predictable and some are not
  • some are sold in high volumes, some are not.

For each item, an ABC XYZ analysis has to be made. The ABC axis representing the volume and the XYZ a combination of predictability and velocity. The result will determine where to position the inventory, either centrally or distributed.

The XYZ dimension lead to an important view of centralizing inventory, which is risk pooling. Central inventory protects better against variability than distributed inventory. One example of that strategy can be found in retail where a 100% allocation to store lead to imbalances in the store network which cannot be easily compensated by store transfers, rather than reduce the initial allocation to 40% to 50% and the react with replenishment once the read of sales signals comes in.  Obviously, this method has consequences both in terms of logistics cost and lead-time, hence the level of centralized inventory versus distributed inventory must be evaluated.  This is also why forecasting and replenishment techniques are needed.

The Problem with a misunderstood Make to Order Business

One big contributor of inventory inefficiencies in a multi-location supply chain environment is the misunderstood nature of make to order. This phenomenon occurs in for example the fashion industry where a company is servicing major wholesale channels and have to place purchase orders to factories based on purchase orders from customers. The below depicts the decisions made along the cycle driven by the assumption that customers have actually committed 100% to their purchase orders placed.

One impact of using too many location, which leads to a micro-segmentation of inventory, is the loss of control over where inventory resides and how is it consumed.  Another aspect of the MTO businesses is that a company needs to have the right commercial terms in place.  If the customer buys it, they own it, and ideally would have strictly managed and limited returns opportunity. Selling a product and getting it returned is not “sales”.