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
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.
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