Principle of Little's Law

The fundamental long-term relationship between Work-In-Process, throughput and flow time of a production system in steady state is:

Inventory =Throughput × Flow Time


Motivation
Little's law is both fundamental and simple. Because it relates three critical performance measures of any production system, it is a basic manufacturing principle. But, although it has deep mathematical roots, it is extremely intuitive. If we observe a milling machine that cuts 100 parts per hour with a queue of 800 parts in front if it, we say that it has "8 hours of WIP". In speaking of WIP in terms of time, we are making use of Little's law, which can be thought of as a conversion of units. Inventory is measured in pieces, flow time in hours, and throughput in pieces per hour. Hence, if we divide inventory by throughput we get flow time. So, to convert 800 pieces to the time it will take to process them, we divide by the throughput (100 pieces per hour) to get 8 hours. This conversion is useful when diagnosing a plant. If we see what physically looks like a large amount of inventory in the system, we cannot tell whether this is a signal of trouble until we know how much time is represented by the inventory. For instance, if we see 2000 pieces of WIP in a system that produces 10 per day, this is probably a disastrous system, while 2000 pieces in a system that produces 1000 per hour is probably extremely lean. (Note however that the efficiency of a system depends on more than how much WIP is present. To see how, go to diagnostic tools.)

But Little's law is actually much deeper and more general than a simple conversion of units. It applies to single stations, production lines, factories, and entire supply chains. It applies to systems with and without variability. It applies to single and multiple product systems. It even applies to non-production systems where inventory represents people, financial orders, or other entities. Generally speaking, there are only two requirements for Little's law to hold:


1.inventory, throughput and flow time must represent long-term averages of a stable system, and


2.inventory, throughput and flow time must be measured in consistent units.

To appreciate the first requirement, suppose we have a production system that has just started up, so that it has inventory but no throughput yet, then inventory will be positive, throughput will be zero and flow time will be undefined. Clearly Little's law will not hold. But over time, as the system stabilizes and we get averages for the three quantities, it will hold.

The requirement of consistent units is simple for the single product case. For instance, if throughput is measured in pieces per hour, then flow time must be in hours (not weeks or months). But it is more subtle if we want to apply Little's law to a multi-product system. For instance, we might have a workstation that processes several part types, each with different inventory, throughput and flow time. While it is perfectly consistent to apply Little's law to the parts one at a time, we can also speak of them in aggregate terms if we use the proper units. For instance, we could measure all inventories in units of dollars and speak of throughput in terms of dollars per hour. Here, however, we must be careful to maintain consistency by measuring throughput in terms of cost of goods sold (instead of in terms of revenue), so that the dollars used match those used to measure inventory. Measuring inventories and flows in dollars makes Little's law applicable to a vast number of situations.

Examples

1.
Estimating Waiting Times: If are in a grocery queue behind 10 persons and estimate that the clerk is taking around 5 minutes/per customer, we can calculate that it will take us 50 minutes (10 persons x 5 minutes/person) to start service. This is essentially Little's law. We take the number of persons in the queue (10) as the "inventory". The inverse of the average time per customer (1/5 customers/minute) provides us the rate of service or the throughput throughput. Finally, we obtain the waiting time as equal to number of persons in the queue divided by the processing rate (10/(1/5) = 50 minutes).



2.

Planned Inventory Time: Suppose a product is scheduled so that we expect it to wait for 2 days in finished goods inventory before shipping to the customer. This two days is called planned inventory time and is sometimes used as protection against system variability to ensure high delivery service. Using Little's law the total amount of inventory in finished goods can be computed as

FGI = throughput × planned inventory time

3.

Tracking Flow Time: What is the flow time of an automobile? The answer, of course, is "it depends". Does flow time include only the time on the final assembly line? Does it include time for engine assembly? What about components? Although we could easily clock the time it takes an automobile to travel down the final assembly line, it is much less clear how one could directly measure the flow time when other levels of the production process are included. However, we can apply Little's law by noting that

Flow Time = Inventory/Throughput

If we want to estimate flow time for a complex assembly, like an automobile, then we simply measure the inventory (in dollars) of all the segments of the process we wish to consider, measure throughput in units of cost of goods sold, and take the ratio. This will provide a perfectly consistent measure of flow time.

4.

WIP Reduction: Since Little's law implies that for a line Flow Time = WIP/Throughput, it is clear that reducing WIP while holding throughput constant will reduce flow time. One might be tempted to conclude that WIP reduction will always reduce cycle time. However, we must be careful. Reducing WIP in a line without making any other changes will also reduce throughput (see the discussion of Closed Asynchrounous Lines for an explanation of the throughput vs. WIP relationship). For this reason, simply reducing inventory is not enough to achieve a lean manufacturing system. An integral part of any lean manufacturing implementation is a variability reduction effort, to enable a line to achieve the same (or greater) throughput with less WIP.