Over the past 30 years, inventory management has been increasingly important to businesses, both within their own firms and across their supply chains. Accordingly, there has been extensive research on inventory management optimization aiming to cut costs. Mathematical models which address inventory management in flow systems have been the common approach.

To understand what this means, imagine you are looking at how things move through a large system, such as a factory or a transportation network. Such a system has clear boundaries, meaning one can tell its extent. Now, picture entities moving through this system. These entities arrive at the system from external sources, spend some time moving through it, and then depart from the system

This abstracted description encompasses many systems that include inventories, where entities, such as product units, raw material units and batches, arrive as replenishment, spend some time “on the shelf,” and then are shipped to customers. We can now focus on inventory systems as a specialized setting.

The ITR is a popular inventory management efficiency metric that measures the ratio of the number of entities that move through the inventory (or volume of entity flow) over a period to the time average of inventory held over that period. Loosely stated, ITR measures the number of times the average inventory is sold or used over a given time interval. In a financial context, ITR is alternatively stated in terms of the monetary value of the inventory items. A closely related companion metric is the Days Inventory Outstanding, or DIO, metric, which captures the average time spent by entities in the inventory system.

A very low ITR or high DIO metric serve as evidence of slow-moving inventory, suggesting inefficient inventory management. Conversely, a high ITR indicates efficient inventory management. This is why ITR is such a popular key performance indicator for executives, and managers are incentivised to improve it.

There are multiple computational methods currently used by practitioners to provide inventory-related insights. Companies often calculate ITR as a monetary formula by dividing the total monetary flow during a period by the average monetary value of inventory over the same period. However, there are various methods of measuring inventory flow through a company’s supply chain, as well as measuring the inventory level that a company holds in stock.

This causes several problems. For example, different ways of measuring and calculating metrics can make them hard to compare. This in turn can lead to misunderstandings about how efficient inventory management is. Furthermore, having too much flexibility in how metrics are calculated can lead to biased results. Companies might even use tricks in their accounting to make their performance look better than it is.

Given how important ITR metrics are, how can we make them more robust?

In recent research, Professor Benjamin Melamed of Rutgers University has presented a new computational approach to assess the ITR metric, based on firmer foundations. He proposes doing so using finite-horizon versions of Little’s Law.

Little’s Law is a fundamental equilibrium relation in queueing theory that describes the relationship between the mean number of customers in a system, the rate at which customers arrive, and the mean time a customer spends in the system. It is a powerful tool for analysing and optimizing systems as it helps in making decisions related to resource allocation, process improvements, and capacity planning to achieve better performance.

To adapt this to the ITR metric, Professor Melamed published a new formulation, which he refers to as Finite-Horizon Little’s Law. So, how has Professor Melamed adapted Little’s Law to create a more robust method of assessing inventory management?

The ITR metric can be measured in different ways depending on what you want to focus on. For instance, you might want to measure how efficiently you are managing your inventory, or you might be more interested in the financial efficiency.

The traditional Little’s Law is generally formulated in terms of long-run averages or steady-state settings. However, business decisions are generally based on finite time horizons and short-term statistics and metrics.

The standard use of Little’s Law also usually deals with individual items, where each item’s arrival increases the inventory count, and each departure decreases it. So, the count is always an integer number.

However, instead of just counting items, there are also circumstances where we need to focus on other characteristics associated with each item. For example, in finance, we might be more interested in the monetary value of items flowing through inventory rather than just the number of items.

In other cases, items do not arrive or leave all at once but rather move gradually over time. For instance, think about batches of products arriving slowly or bulk material being continuously transferred. It is important to model these gradual movements too.

For meaningful results, it is essential to ensure that changes in inventory levels correspond to actual arrivals and departures of inventory items, not just fluctuations due to market conditions or other factors. Otherwise, the interpretation of inventory management metrics can be muddled.

A key feature of Professor Melamed’s Finite-Horizon’s Little Law formulations is that they can deal with all these variations that we have considered. They work for different types of inventory flows, including both synchronous flows (where things happen all at once) and asynchronous flows (where things happen gradually). They can also handle various attributes of inventory items, such as their monetary value.  Moreover, these formulas work in any situation, not just when the system is in steady state like the traditional Infinite Horizon Little’s Law requires.

Another useful feature is that there are two related methods of computing inventory-related metrics: counting entities or calculating the amount of time spent by entities in the system. This provides data processing departments with flexibility, and the option of selecting the method that has the lower computational complexity.

Professor Melamed’s research is relevant to businesses and executives across various sectors. Many important strategic decisions are taken based on inventory management performance, so it is vital that the way we measure it is robust and accurate. This work provides a more effective, flexible way to assess inventory management, which avoids the common pitfalls of more traditional methods.