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Harish Jose


Where Ohno and VUT Intersect

Insights into Kingman’s formula

Published: Wednesday, September 25, 2019 - 12:03

One of my favorite equations from Factory Physics, by Wallace Hopp and Mark Spearman (Waveland Press, third edition, 2011) is Kingman’s formula, usually represented as “VUT.”

The VUT equation is named after Sir John Kingman, a British mathematician:

The first factor represents variability and is a combination of variability factors representing arrival and service times (e.g., flow variability and process variability). The second factor represents utilization of the workstation or the assembly line. The third factor represents the average processing time in the workstation or the assembly line. The VUT equation shows that the average cycle time or wait time is proportional to the product of variability, utilization, and process time.

The most important lesson from VUT is: If a station increases utilization without making any other change, average work in process (WIP) and cycle time will increase in a highly nonlinear fashion.

The influence of variability on cycle time is shown below. The red line shows that with high variability, any increase in utilization will result in an exponentially higher cycle time. If the variability is low (indicated by the green line), then the increase in the cycle time happens at a slower rate. If there is no variability, then the cycle time will be a constant. In other words, an increase in variability always degrades the performance of a production system.

Some of the lessons that we can learn from the VUT equation are:
1. To maintain a steady cycle time, reduce utilization if variability cannot be reduced. Reducing utilization means increasing capacity. As demand goes up, do not try to run the line at 100-percent utilization.
2. The VUT equation can be used in conjunction with Little’s Law, which states that WIP is proportional to the product of throughput rate and cycle time. In other words, WIP is proportional to the product of throughput and VUT. If you try to reduce WIP without trying to reduce variability, the throughput will go down. Thus, implementing one-piece flow without trying to reduce variability will result in a reduction in throughput.
3. Reducing process variability will reduce cycle time variability.
4. Adding buffer space at bottlenecks will improve throughput. Adding buffers elsewhere will not have a positive impact on throughput.
5. Variability will always be buffered in the form of inventory, capacity, or time. If variability is not reduced, you pay in terms of high WIP, underutilized capacity, and reduced customer service. This is further explained here.
6. Utilization effects are not linear but are highly nonlinear. Thus, the effect of variability at 40-percent utilization is not half of the effect of variability at 80-percent utilization.
7. Reducing variability reduces uncertainty regarding cycle time or project lead times.
8. First reduce variability, and then go for increasing throughput.
9. The rule of thumb is to run a line at or near 80-percent utilization. You should experiment to learn more about your production system.
10. In lean, the variability factor can be viewed as mura (unevenness), and the burden from pushing for 100-percent utilization can be viewed as muri (overburdening). Both result in muda (waste).

VUT and the Toyota Production System

Taiichi Ohno, the father of the Toyota Production System (TPS), learned by trial and error and by actively learning from the gemba. Ohno realized early on that the first step in increasing throughput is by achieving stability. The idea of variability is closely tied to the idea of mura (unevenness) in TPS. Ohno pushed for the idea of standard work for kaizen. He taught that kaizen is not possible without standard work. Standard work is aimed at reduction of variability in the process. In addition, Ohno came up with kanban to minimize variability in the process flow. He further pushed for reduction in WIP once process stability was achieved. Ohno constantly pushed to remove “waste” from the production system through kaizen. This continuous improvement cycle helped to maintain process stability.

As lean thinker Art Smalley puts it, “What Toyota (Ohno) learned the hard way is that in the beginning of a transformation, you need lots of basic stability before you can succeed with the more sophisticated elements of lean.... Veterans of Toyota comment that certain preconditions are needed for a lean implementation to proceed smoothly. These include relatively few problems in equipment uptime, available materials with few defects, and strong supervision at the production-line level.”

Smalley poses four questions to evaluate stability:
1. Do you have enough machine uptime to produce customer demand?
2. Do you have enough material on hand every day to meet your production needs?
3. Do you have enough trained employees available to handle the current processes?
4. Do you have work methods, such as basic work instructions, defined, or standards in place?

If the answer is emphatically “no” to any of these questions, stop and fix the problem before proceeding. Attempting to flow product exactly to customer demand with untrained employees, poor supervision, or little inventory in place is a recipe for disaster.

Ohno’s first go-to method to train the production team to start thinking in terms of improvement was to ask the line to maintain current throughput with one less operator. In many regards, this can be viewed as reducing capacity or increasing utilization. As we learned from VUT, increasing utilization is a bad thing. Why would Ohno do that? He firmly believed that doing is the main way to learn something. He explained that, “Knowledge is something you buy with money; wisdom is something you acquire by doing.”

Ohno was able to “see” wastes in a process that hindered flow. He had to train others to see these wastes like he did. This could be because they were able to meet production demand with their current process, and thus were unaware of hidden wastes. The only way that Ohno could make them improve further was by asking them to do the same with one less operator. This challenged the team to look at their standard work and the process to see where waste was. This challenge is part of the “respect for people” pillar of the Toyota Way. It is said that TPS also stands for “thinking production system.” Toyota develops its people to think so they can see problems and fix them.

Fujio Cho, the former president of Toyota and a student of Ohno, has said that the TPS pioneered by Ohno is not just a method of production; it is a different way of looking and thinking about things. Ohno developed the management team by giving genchi genbutsu-based practical tasks through which the team members were matched in a “competition of wits” against him. Cho called Ohno’s methods “hands-on human resources nurturing.” Ohno believed that if he was in a position to give orders, he could not do so unless he was confident about what he was asking. Ohno saw that the current condition could be improved, and he challenged teams to do that by deliberately pushing utilization up.

I encourage everyone to learn more about VUT here and here. Always keep on learning....

First published June 23, 2019, on Harish’s Notebook.


About The Author

Harish Jose’s picture

Harish Jose

Harish Jose has more than seven years experience in the medical device field. He is a graduate of the University of Missouri-Rolla, where he obtained a master’s degree in manufacturing engineering and published two articles. Harish is an ASQ member with multiple ASQ certifications, including Quality Engineer, Six Sigma Black Belt, and Reliability Engineer. He is a subject-matter expert in lean, data science, database programming, and industrial experiments, and publishes frequently on his blog Harish’s Notebook.


Variation impacts production management

This article is extremely important from not only a quality but also an operations management perspective. Goldratt's and Cox's The Goal showed why, in a balanced factory without excess capacity (or even if there is some excess capacity), favorable variation in processing and material transfer times does not offset unfavorable variation, with the result that inventory piles up.

A good takeaway from this article is that we need to think about variation in processing and material transfer times  (service and arrival times in the VUT equation shown here) as well as variation in part dimensions, which is what quality practitioners usually think about when variation is mentioned. I never thought of the kind of variation mentioned in this article until I read The Goal, in a factory that was implementing Goldratt's methods.