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By William D. Kappele
Statistical Tolerance limits provide you with a range in
which some percentage of all your products will lie. Here is an example:
Suppose you are making ball bearings. You know that you
have response variation in your process, so you know that the ball bearings
produced over time will vary in diameter. You calculate the average and
standard deviation from the diameter measurements of ten randomly
selected ball bearings from your process. You find an average diameter of
0.125 inches with a standard deviation of 0.004 inches. You would like to know
lower and upper limits on the diameters of ball bearings produced by your
process.
As is often the case in the real world, you cannot define
these limits for ALL of the ball bearings made. You can, though, determine
limits on 99% of all ball bearings made. These limits are called
"Tolerance Limits on 99% of the Population." As is also often
the case in the real world, you cannot be 100% confident that these limits are
correct. However, you can be 95% confident that they are correct.
These limits are then called "95% Tolerance limits on 99% of the
Population."
If your diameter measurements fall in bell-shaped piles
(are "Normally Distributed") then you can calculate 95% Tolerance
Limits on 99% of the Population using the simple formula,
Lower Tolerance Limit = Average - Ks
Upper Tolerance Limit = Average + Ks
s is the standard deviation and K is a number from a table
of K values. You can find a table at http://www.MathOptions.com/tolerance.htm.
For the ball bearing example,
Lower Tolerance Limit = 0.125 - 4.433 * 0.004 = 0.107
Upper Tolerance Limit = 0.125 + 4.433 * 0.004 = 0.143
So, over the course of ball bearing production we would
expect 99% of all of the ball bearings produced to have diameters between 0.107
inches and 0.143 inches. We are also 95% confident that these limits are
correct.
Tolerance limits for different percentages of the
population are also easy to calculate. You can also change the level of
confidence. You use the same formula, but a different value for K.
You can find a thorough table of K values at http://www.MathOptions.com/tolerance.htm.
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Bill Kappele is the president of Math Options. He
would be delighted to receive your letter at Bill@MathOptions.com
Visit him at his web site and sign up for his free
newsletter, E-Math News.
http://www.MathOptions.com
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A maintenance upgrade of STRATEGY (version 11.4.1) has just
been released. It clears up the problem where STRATEGY refuses to accept
certain valid linear constraints when defining a GridSearch. This upgrade
is free to all current STRATEGY subscribers. For your convenience, here is
a link to our download information form.
http://www.processbuilder.com/doe/Software/strategy/downloads.htm
If you are not using the linear constraints in GridSearch,
you may want to start. Let's say that you have run an experiment that
contains both a mixture and process factors. Use a constraint to tell
GridSearch which factors are parts of the mixture.
There are two ways to do this.
Note that both of these methods assume that the regression analysis did
not include one of the mixture factors. The entry point for each option is
after you have successfully run your regression analysis and defined your other
GridSearch criteria.
Option 1 -- Using an Inequality Constraint
Go to the Linear Constraints Tab of the GridSearch
Definition Form. The main grid on the page shows one column for each
factor starting just to the right of a lower limit and upper limit. Each
row of the grid is a different constraint.
Remember that one mixture factor is not showing. If
the percentage of that missing factor can vary from zero to one hundred percent,
the constraint is quite easy to describe. Set the constraint's lower limit
to zero and the upper limit to 1. Place a one in that row under each of
the remaining mixture factors. GridSearch will then know to only consider
results where the total percentage of the indicated components is somewhere
between zero and one.
If the unseen mixture component must be at least 10 percent
of the mixture, but can't be any more than 50 percent, everything would be the
same except the upper and lower limit. The lower limit for the other
components would then be 50 (100 - 50). The upper limit would be 90
percent. The range of permitted totals is thus anywhere between 50 percent
and 90 percent.
Note that you can define constraints for any of the named
individual factors either on this Tab or the Factor Limits Tab. This
approach is only necessary for the omitted factor.
Option 2 -- All Mixture Factors Appear in the Design File
If you have included all of the factors in the design file,
you may have the GridSearch report show them all. This is more explicit
and clearer, but it is a little more work.
First, you must go the Additional Factors tab of the
GridSearch Definition Form. When the form is loaded, GridSearch compares
your model file to your design file. If you one or more columns that are
not listed in the model file (a factor or a response) and not called RUNORD,
GridSearch will list those column names here. These won’t be just factor
names. If you have transformed columns or unused response columns, they
will show up also.
Select the mixture factor or factors that have been left
out. Move to the Linear Constraints form. Note that the last columns
in the worksheet are the additional factors you just specified. For each
mixture, set the upper and lower limits to one hundred percent. Mark each
mixture component, including the omitted factor, by placing a 1 in the column
for that factor for that row. GridSearch will then only consider mixture
combinations that add up to one hundred percent.
We recommend this second approach. It is particularly
advantageous for experiments that include more than one mixture.
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In-house classes and related activities are impacting.
We are aggressively working (see below) to make a few more classes available to
the public in the fourth quarter, but this is the only public class that we can
commit to now.
Availability for in-house classes is also limited. Drop
me a line as soon as possible at abcorwin@processbuilder.com
if you need to bring either our introductory or advanced workshop to your
company before the end of the year.
Sept 9, 10
Advanced DOE Workshop
Boulder, CO
http://www.processbuilder.com/doe/Classes/mixtures.htm
Note that this is our advanced class covering mixtures,
discrete factors, and optimal designs. Prior completion of a qualifying
introductory class is required.
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We are looking for a primary Design of Experiments training
partner. Our preference is to find an organization that is already
teaching a set of courses that matches our educational objectives and
philosophy, but finding the right partnership is far more important to us than
finding the right courses (assuming that you are willing to customize the
courses to our needs). We also will give serious consideration to a
motivated entrepreneurial individual, but time is a critical consideration.
Our selection process is thus strongly biased in favor of those who already have
something in place
Our primary goal is to present STRATEGY and the Nine-Step
Method for Experiment Design to working scientists and engineers, but we do not
intend to make them use our software. The courses should be flexible
enough so that they can be taught in whatever software the customer is already
using at their location. The ideal organization or individual will be
familiar with a variety of DOE software including Gosset and I-OPT.
The chosen training partner will have the opportunity to
influence the development of STRATEGY and other products for the professional
scientist. We want to stay on the leading edge of design of experiments,
but we can only do that with partners and associates who have a similar
commitment.
Process Builder currently sponsors and markets eight public
classes a year. Over three thousand people a month visit our
highly-regarded design of experiments web site, and the subscriber list for DOE
Perspectives (this newsletter) has grown by over ten times in the past year.
STRATEGY is a unique product that is constantly evolving. We have
technical alliances that will bring exciting new products to the market in the
near future.
We currently get over a hundred requests for information
about design of experiments every month, and send out over twelve hundred
notices of upcoming classes and in-house training opportunities. Those
numbers grow significantly with every passing month. The ideal candidate
will work with Process Builder to provide training, custom designs, and guidance
in response to these requests.
Biases:
We have inherited our DOE prejudices from Dave Doehlert and
thus favor classic and custom-computed optimal designs. We are at least
the most vocal supporters of I-Optimal designs, and we explicitly do not support
Taguchi methodology.
We like our current curriculum but would look at expanding
it or even replacing parts of it. Our customers are working scientists and
engineers with real problems, and they are looking for practical solutions.
The courses should be hands-on, designed to formal instructional objectives, and
not require any math skills more advanced than basic algebra. We are
flexible about the length and shape of these courses, but we are unwilling to
consider any classes that are less than fifty percent hands-on.
If this sounds like the kind of alliance that you are
looking for, drop me a line (abcorwin@processbuilder.com)
or give me a call (206-525-9093).
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Thanks for your attention.
Sincerely yours,
Al Corwin
President
Process Builder
<http://www.processbuilder.com/doe/>
abcorwin@processbuilder.com