Which Way You Should Go Depends on Where You Are

The dangers of misapplying linearity

Published: Thursday, December 14, 2017 - 12:02

I recently read Jordan Ellenberg’s wonderful book, How Not To Be Wrong: The Power of Mathematical Thinking (Penguin Books, 2014). I found the book to be enlightening and a great read. Ellenberg has the rare combination of being knowledgeable and capable of teaching in a humorous and engaging way. One of the gems in the book is, “Which way you should go depends on where you are.”

This lesson is about the dangers of misapplying linearity. When we are thinking in terms of abstract concepts, the path from point A to point B may appear to be linear. After all, the shortest path between two points is a straight line. This type of thinking is linear thinking.

To illustrate this, let’s take the example of poor quality issues on the line. The first instinct to improve quality is to increase inspection. In this case, point A = poor quality, and point B = higher quality. If we plot this incorrect relationship between quality and inspection, we might assume it is a linear relationship—i.e., increasing inspection results in better quality.

However, increasing inspection will not result in better quality in the long run and will result in higher costs of production. We must build quality in as part of the normal process at the source and not rely on inspection. In the Toyota Production System (TPS), there are several ways to do this, including poka yoke and jidoka.

In a similar fashion, we may look at increasing the number of operators in the hopes of increasing productivity. This might work initially. However, increasing production at the wrong points in the assembly chain can hinder the overall production and decrease overall productivity. Taiichi Ohno, the father of TPS, always asked to reduce the number of operators to improve the flow. TPS relies on the thinking of the people to improve the overall system.

The two cases discussed above are nonlinear in nature. Thus, increasing one factor may increase the response factor initially. However, continually increasing the factor can yield negative results. One example of a nonlinear relationship is shown below:

The actual curve may of course vary depending on the particularities of the example. In nonlinear relationships, which way you should go depends on where you are. In the productivity example, if you are at the yellow star location on the curve, increasing the operators will only decrease productivity. You should reduce the number of operators to increase productivity. However, if you are at the red star, you should look into increasing the operators. This will increase productivity up to a point, after which the productivity will decrease. Which way you should go depends on where you are!

In order to know where you are, you need to understand your process. As part of this, you need to understand the significant factors in the process. You also need to understand the boundaries of the process, where things will start to break down. The only way you can truly learn your process is through experimentation and constant monitoring. It is likely that you did not consider all of the factors or the interactions. Everything is in flux, and the only constant thing is change. You should be open for input from the operators and allow improvements to happen from the bottom up.

I will finish off with the anecdote of the “Laffer curve,” which Ellenberg used to illustrate the concept of nonlinearity. One political party in America has been pushing for lowering taxes on the wealthy. The conservatives made this concept popular using the Laffer curve. Arthur Laffer was an economics professor at the University of Chicago. The story goes that Laffer drew the curve on the back of a napkin during dinner in 1974 with senior members of then President Gerald Ford’s administration. The Laffer curve is shown below:

The horizontal axis shows the tax rate, and the vertical axis shows the revenue that is generated from taxation. If there is no taxation, then there is no revenue. If there is 100-percent taxation, there is also no revenue because nobody would want to work and make money if they cannot hold on to it. The argument that was raised was that America was on the right-hand side of the curve, and thus reducing taxation would increase revenue.

It has been challenged whether this assumption was correct. Ellenberg used the following passage from Greg Manikiw, a Harvard economist and a Republican, who chaired the Council of Economic Advisors under the second President Bush: “Subsequent history failed to confirm Laffer’s conjecture that lower tax rates would raise tax revenue. When Reagan cut taxes after he was elected, the result was less tax revenue, not more. Revenue from personal income taxes fell by 9 percent from 1980 to 1984, even though average income grew by 4 percent over this period. Yet once the policy was in place, it was hard to reverse.”

The Laffer curve may not be symmetric as shown above. The curve may not be smooth and even, as shown above, and could be a completely different curve altogether. Ellenberg states in the book, “All the Laffer curve says is that lower taxes could, under some circumstances, increase tax revenue; but figuring out what those circumstances are requires deep, difficult, empirical work, the kind of work that doesn’t fit on a napkin."

Always keep on learning....

About The Author

Harish Jose

Harish Jose has more than seven years of 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.