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As many of you are aware, our mentor and friend, Dave Doehlert, passed away last month. You may also be aware that Dave was the inspiration behind Gosset, and a big believer in computer generated optimal designs, particularly I-optimal designs.
It is particularly interesting that this article should come to light now, so shortly after the publication of Ray Meyers article in the January issue of the Journal of Quality Technology. This article certainly presents a perspective that is in sharp contrast to the perspective presented by Dr. Meyers and his learned associates. We are indebted to Dave's estate for allowing this publication.
Our apologies to Selden Crary. It was our intention to run his article on this general subject this month. Dr. Crary has graciously consented to allow us to present some of his related views in our next issue.
By the way, Dave supported both the use of Gosset and Selden Crary's I-OPT design engine. Links to both are available on our DOE software web page. (See Free Software Article in this issue)
http://www.processbuilder.com/doe/Links/doe_software.htm
The foundation no lounger generates designs since Dave's death, but we would be glad to direct you to someone who can. We also can provide training in the use of Gosset. Drop me a line at abcorwin@processbuilder.com, and I will see that you get the information you require.
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When you use optimal experiments designed on a computer, you get to choose the number of runs
Computer generated optimal experiment designs can put you way ahead of your competition. It's easy now to be on the leading edge in R & D. A short history of DOE will show how times have changed: The well known factorial designs you've probably seen were new in the 1920s (Fisher). They are still good for the simplest R & D challenges.
Here's the factorial design for the times you are varying 3 factors each from a low to a
high limit.
| Trials | X1 Temp |
X2 Time |
X3 Speed |
| 1 | low | low | low |
| 2 | high | low | low |
| 3 | low | high | low |
| 4 | high | high | low |
| 5 | low | low | high |
| 6 | high | low | high |
| 7 | low | high | high |
| 8 | high | high | high |
The results from these 8 trials can be used to discover synergy and to predict the performances of all other combinations of X1, X2 and X 3 between the high and low limits. Use the following model for interpolations that predict all performance of combinations of X1, X2, and X3:
Y= b0+ b1X1 + b2X2 + b3X3+ b12X1X2 + b13X1X3 + b23X2X3 + b123X1X2X3
This model includes terms for measuring synergy: the last four terms. This model often interpolates predictions very well in industrial R & D ... but not always.
The factorial designs are excellent because the b's you get in the analysis do not cross contaminate each other. And these designs measure all the interaction terms (synergies).
However, for more factors the number of runs gets too big: 5 factors 32 runs 8 factors 256 runs k factors 2 k runs
That's too many for k>5 for most R & D budgets. So in the 1940s Finney began leaving out some of the interactions by using fractional factorials. Only 1/2 of a five factor factorial is needed to measure all bi Xi main effects and all bij Xi Xj two factor interactions when you feel safe in assuming all higher order interactions are zero.
Many experimenters are willing to give up measuring the higher order interactions to save money, and that often works well. To help experimenters, National Bureau of Standards (now NIST) put out three "blue books" of fractional factorials in which the user could find a list of combinations to run. There were not many choices of designs, and many of them were too expensive. But this was a big improvement over the 19th Century. As the years went by the selection improved.
In the 1950s Box suggested adding Xi^2 terms to the model and fitting dome and basin shaped response surfaces to show how a process or product was behaving. His "composite design" looks like this:
He added the star points (*), to the factorial points (f) to get sufficient data, well placed to allow the terms X1^2 and to be X2^2 to be added to the model.
Now there were more models to select from. But some users were finding that these arrays from Box and others called for more runs than they could afford. Kennard and Stone introduced CADEX, in the 1960s, an algorithm that helps you generate a design with the number of trials you want. CADEX spread out the trials rather well.
CADEX had a further benefit: You could make your design fit within the boundaries of your problem, not just square or circular boundaries. (Or cubical or spherical for 3 or more factors).
In 1959 Kiefer and Wolfowitz came up with several ways to measure optimality. Some of the factorials were already optimal; some other designs were not. Now the quality
of a design was defined. But we did not then know how to write down the combinations of factor levels in a design that would be optimal. In the 1990s with increased computing power, we at TESF now generate the optimal design for your problem to meet the challenge that you are facing in industrial R & D. Examples of challenges:
Computer aided optimal Design of Experiments can handle all these at once. You list your needs; then the computer program plans the runs that meet your needs.
And now to answer the question: How many runs for how many factors? The answer is that you control that decision.
Computer generated designs can now be optimal. In the past some were optimal, some were not. And you can go back later to the computer to ask, what runs should I now make beyond the ones I have already done to get still better results.
The control of the size your experiment design is in your hands. This is the good news in the 1990s for the experimenter.
Computer generated DOE is a tool to add to your kit for solving industrial problems and improving products.
In the past you had to settle for general purpose designs. Now a computer run generates designs for your special situations
References
Fisher, R.A., Design of Experiments, Reprinted by Oxford Press (1990)
Finney, D.J., Annals of Eugenics 12 pp 291-301 (1945)
Box, G.E.P. and Wilson, K.B. J. Roy, Stat. Soc.B, pp 1-45 (1951).
Kennard, R.W. and Stone, L.A. Technometrics, pp 137-148 (1969).
Kiefer, J. and Wolfowitz, J., Ann. Inst. Stat. Math, pp 79-108 and 295-303(1964).
Doehlert, D.H., 1. Roy, Stat. Soc.C, pp 23 1 239 (1970)
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There are freely downloadable versions of the two premier design engines that we are aware of. Both Selden Crary's I-OPT design engine and Gosset from Neil Sloane and Ron Hardin at AT&T Bell Labs. As far as we know, these are the only two design engines that can generate I-optimal designs. (Both of these applications may require commercial to use them in your regular work; the downloads are for evaluation and academic use only.)
Links to these and other interesting DOE sites can be found on our Miscellaneous DOE links page < http://www.processbuilder.com/doe/Links/miscellaneous.htm>.
Both of these programs generate excellent designs. The I-OPT program is very easy to master. Gosset requires some training, but it is extremely powerful and flexible.
You need a UNIX box to run either of these programs. Selden has a Windows version under construction, but no firm release date. Look for the Windows version to be significantly more powerful than the current UNIX offerings.
By the way, we do offer training in Gosset as well as software capable of analyzing designs generated by either of these programs. Contact abcorwin@processbuilder.com for details.
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Apr 20, 21, 22 -- San Jose, CA
A Modern DOE Workshop
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A Modern DOE Workshop
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Thank you all for the many kind notes about Dave.
I am trying to put together an on-line biography of Dave, and the records are sketchy at best. I've learned a number of things about Dave in the last few days that I never knew before. Is there anywhere that he did not teach?
Bottom Line: I need your help. The longer ago you took one of Dave's classes, the more we need your help.
I would like to know when Dave came to your company, what changed as a result of his classes, and what accomplishments, if any, came as a direct result of Dave's teaching.
If you feel that Dave changed your life, 1 would really appreciate if you could say how he changed it. I would like this work to emphasize how Dave touched people.
My very thin start at this can be found at: http://www.processbuilder.com/doe/Dave_Doehlert/default.htm
Thanks in advance for your help.
Sincerely yours,
Al Corwin
President
Process Builder