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Happy Gold Anniversary Little’s Law

A VERY quick way to get faster production and more profits from a factory or any other system.  By: Henry Fitzhugh Camp, November 14, 2011



John D. C. Little, PhDFifty years ago John D. C. Little, a Boston native and now a professor at his alma mater MIT, proved what is now referred to as Little’s Law.  The Law is the soul of simplicity, yet broadly applicable.  It states that the average number of items (L) in a system is equal to the average time an item spends (lambda) in the system multiplied by the average rate at which these items enter (W) the system (or leave, since the system is expected to be neither forever increasing nor decreasing to 0).

Little's Law

Now, it is not necessary to be a mathematician to appreciate Little’s Law, particularly because of the things you don’t have to know to apply it.  You don’t have to bother with pesky statistics that describe the variations in the actual rate of arrivals into the system.  Nor must you worry with the definition of the different routes various items take while in the process of passing through the system.  We don’t even have to know what happens within the system or how fast these things happen.  Not a single Normal Distribution curve must we plot.

I have been in conversation with David and Carroll over the last week or so and expect another week or so of conversation before they embrace the simplicity of Little’s Law. 

They run a factory that, while small, is remarkably complex.  They make a broad range of power equipment used by a specific industry.  They buy thousands of parts and raw materials which they process through numerous steps until they leave the plant later, aggregated into a specific piece of equipment.  They too often miss an order because one of their distributors needs a piece of equipment faster than they are able to deliver.  Ergo, they came up with the simple solution of stocking the popular equipment so that they can immediately deliver to distributors with an immediate need.  The problem they have is that they are struggling to produce enough to even catch up with their existing orders, much less build the needed inventory.

David and Carroll are contemplating adding significantly to their labor pool so as to have the capacity to produce the extra equipment they need.  Of course, adding people means undertaking more risk.  There is a chance that the amount of additional sales they capture will produce less gross profit than the certain added cost of labor, its benefits and management, hurting rather than helping their profits.  But is this risk necessary at all?

Let’s define the “system” as their factory.  The products that leave the factory during a month (W) are determined by the orders the company gets from its distributors.  The company's problem: its customers have to wait too long for their equipment and, therefore, they are too often forced to buy from a competitor.  In other words, the average processing time through the factory (lambda) is too high.  Little’s Law to the rescue!  If we apply a little (no pun intended) algebra, we get this modified version of the Law:

Little's Law 2

So, if we want to produce finished equipment in ½ the time, we just have to change reality by reducing the amount of stuff in the factory (L) to be half of what it currently is. 

Wait a minute, is that possible at all?  Everything in the factory is being worked on, otherwise it wouldn’t be there, right?  I asked Carroll that same question.  His estimation was that the piece of equipment that automatically drills a pattern of holes in sheet metal has about 50 pieces of sheared to size sheet metal waiting in a queue to be processed at any given time.  This is exactly the situation of almost all manufacturers.  The total of all the queue time for a part is far in excess of its actual processing time.  So, yes, it is possible to cut the number of items in the factory (L) in half without impacting productivity measured by the rate equipment leaving the plant (W).   The outcome will be that the time to produce each piece of equipment (lambda) likewise drops to half of what it was before.  The only real change is that the queues are much shorter than before.

Under this scenario, David and Carroll would have capacity to build inventory and to capture sales that were lost before.  Better still, they do it without increasing labor, other consequent costs and complication. 

How do they start?  They know the average time that the manufacturing process currently takes.  They have a dated work order that starts the production process by releasing raw materials and purchased parts onto the production floor.  They also know exactly when each work order is closed.  Knowing this average duration, when they get an order, they add some extra time to protect against Murphy’s attacks to come up with a quoted lead time. 

I suggested that they hold each work order until half the quoted lead time has already passed before releasing the work order to the production floor.  Simple but scary!

There are two erroneous but common sense assumptions that keep most manufacturers from doing this: 1) the sooner we start the sooner we finish and 2) a worker standing idle reduces profits.  Little’s Law refutes the first nicely.  The second has to be thought through a bit.  Although many queues are long, like the one before the relatively slower hole drilling machine, others are short.  The short ones are in front of the faster processes.  If we cut the amount of stuff in the plant in half, as I suggested, there is a high probability that some of the faster operations will burn through their queues altogether, from time to time, and their operators will sometimes be standing idle. 

A quick check allows us to verify that David and Carroll should support this situation.  The cost of operating the plant in dollars remains the same as before.  They neither hire new people nor buy more raw materials.  However, since production is much faster, they can now build inventory.  Once they have inventory, they capture the sales they formerly missed.  So sales go up, bringing more money in.  Meanwhile, work-in-process inventory shrank in half, so their investment decreased.  Profits must increase and with less investment, therefore, return on investment increases (global efficiency) because we encourage workers with nothing to do to do nothing (local inefficiency).  God bless you Dr. Little.

Productivity

 
10 Comments
I love it. Thanks for keeping me in the loop and especially for making your blog iPad friendly. I hope your well. I would like to forward it to a few manufacturers I know.
Henry,

Nice blog post. I am a big fan of Prof. Little and appreciated reading your article.

We teach Little’s Law during our Pull Design Workshops, and it is always eye-opening for participants. We show how even small reductions in cycle time, say just 1 day, can lead to huge savings in inventory. It is a great way to show the compelling business case for cycle time and inventory reductions.
Hum? If the flow time is cut in half, then they respond quicker to the actual demand of the customer. Hum? It seems that they may not have to make as much finished goods inventory as they thought! They can protect the customer demand by SPEED! Just think of the excess capacity they will discover! I hope they are not giving away their new found effectiveness for nothing. SPEED has real value connected to it.
Hum? If the flow time is cut in half, then they respond quicker to the actual demand of the customer. Hum? It seems that they may not have to make as much finished goods inventory as they thought! They can protect the customer demand by SPEED! Just think of the excess capacity they will discover! I hope they are not giving away their new found effectiveness for nothing. SPEED has real value connected to it.
Hum? If the flow time is cut in half, then they respond quicker to the actual demand of the customer. Hum? It seems that they may not have to make as much finished goods inventory as they thought! They can protect the customer demand by SPEED! Just think of the excess capacity they will discover! I hope they are not giving away their new found effectiveness for nothing. SPEED has real value connected to it.
"Simple but scary!" [An actual LOL. So true. So hard to sell.]

That sounds good to me because the idle workers (after having a coke and a smile), will become nervous about their job security and look for ways to help their neighbors. Unlike building throughput-blocking queues, the "idle workers" could actually help the plants throughput. I now put "idle workers" in quotes because their boots-on-the-ground insight may be more valuable than their managers'. Rapture!
Jim C -
Thanks for the blog. I wanted to alert you to a (very) little known fact (no pun intended, as well) about Little's Law. You probably know this already but here's the real story: this formula was around for a long time before JDC Little gave a "formal" proof, which cemented his name to this identity. Per Little: "We attribute the explicit formula "L = AW' to Philip Morse and his book, Queues, Inventories and Maintenance, (Morse, 1958)." He also attributes Cobham's mention in 1954.
I think you did an awesome job with this! Great job. -cap
I think Little's Law would apply to almost any business involving process flow. Consider a direct mail business and fulfillment operation for example. It all comes down to balancing the amount of staff to take orders with order inflow, then matching the work force and inventory levels necessary to fulfill customers orders. This would hold true for many other business.
Wayne H -
Very nice example, Henry!
stewartwitt -



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