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Columnist: A. Blanton Godfrey

Photo: A. Blanton Godfrey


Brave New Universes

A. Blanton Godfrey


Recently, my son gave me a copy of Ian Stewart’s wonderful book, Flatterland (Perseus Books, 2001). The title reflects Stewart’s appreciation of a century-old classic, Flatland (1880), written by "A. Square," the pseudonym of Edwin Abbott Abbott, a headmaster and Shakespeare scholar in Victorian England. In Flatland, Abbott describes a world of two dimensions, a flat Euclidean plane. The inhabitants are geometric figures: lines, squares, triangles, pentagons and other shapes. One day a character arrives from another world, one with three dimensions. This, of course, shakes up the Flatland residents to no end. But what Abbott was really leading his readers to consider was something even more outlandish: four dimensions.

Flatland has been in print for more than 100 years, and during this time several other derivative books, including Sphereland and The Planiverse, have appeared. Stewart’s Flatterland is another derivative with a serious purpose. In these early days of the 21st century, he says, mathematics and science have come a long way from where they were at the end of the 19th. The notion of four dimensions, after all, is rather commonplace when compared to the "mind-boggling inventions of geometers and physicists—spaces with many dimensions, spaces with none, spaces with fractional dimension, spaces with infinitely many points, curved spaces, spaces that get mixed up with time and spaces that aren’t really there at all."

If nothing else, Stewart’s book is a great read for bending one’s mind in a few new dimensions. Just when you think he’s introduced a new mathematical theory that must be truly useless, he follows it up with surprising and practical examples of how it has led to a technological breakthrough. For example, he relates how thinking in multidimensional space led to creating the error-correcting codes that made the Internet possible.

But one line in the book really stopped me: Stewart casually mentions that economists are working in million-dimensional space. With tens of thousands of products, prices, costs, consumer needs and so forth, it’s easy to see how complex the economic world can quickly become. And, of course, this is the world we all live in when we work in quality. Each product or process includes many quality dimensions. Each customer has many different needs, wants and weightings of each of these. Our competitors, regulators, governments and other factors further complicate the space. Yet quality tools are, by and large, designed for looking at only one dimension at a time.

Even when using some of the more advanced methods taught in Six Sigma Black Belt or Master Black Belt courses, we’re usually focused on one output variable. We may have a number of inputs, the X’s in the basic equation, Y = f(X), but we rarely attempt to deal with more than one output at a time. We identify many that are critical to quality outputs—the CTQs—but we rarely treat these outputs as an interdependent group. When we look at two simple outputs of one variable, for example, the mean and the variance, we find out how complicated life can get. Sometimes with response surface methods we can find an optimum, but more often we compromise. And when we begin to optimize several variables, we find life gets even more complicated.

A few years ago, I taught a design for Six Sigma course during which a Black Belt related his difficulty with maximizing two strengths of a critical part and minimizing the weight. Thanks to some timely guidance by J. Stuart Hunter, my co-instructor that week, the Black Belt developed a well-planned experiment that allowed him to do the analysis. By comparing multiple response surfaces, he could create a new design that met all his objectives in a few minutes. Needless to say, he became a champion of experimental design in his company.

Too often, though, companies and quality professionals ignore the complexities of multiple dimensions and simply try to manage things one dimension at a time. We often end up suboptimizing or in many cases actually creating destructive conflicts between critical variables. Multivariate methods exist, but most are quite difficult and not often applied in business. Graphical statistics offer one alternative to this problem; using them, we can compare many outputs graphically and make reasonable decisions. In many cases the outputs are independent, and we can use standard methods for each output. However, sometimes the inputs for different outputs may not only be interdependent but actually the same, and this introduces another complexity. The optimum levels for one set of input variables for our output dimensions may be far different from the optimum levels we need for another output variable.

Perhaps I’m wrong and many useful tools are being developed and used every day for these complex-space problems. I look forward to the flood of mail and examples describing them.

About the author

A. Blanton Godfrey is dean and Joseph D. Moore Professor in the College of Textiles, North Carolina State University. He is also the founding editor of Six Sigma Forum Magazine and the co-author of the recently published second edition of Modern Methods for Quality Control and Improvement. Letters to the editor regarding this column can be e-mailed to letters@qualitydigest.com.