Table Of ContentA UNIFIED APPROACH TO PROCESS OPTIMIZATION
By
PHILIP JOHN MCGOFF
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2000
This dissertation is dedicated to my family, especially my to father, who is not here
to see me become the first in the family with a doctorate.
ACKNOWLEDGMENTS
I would like to thank my family and friends for supporting me in my return
to graduate school. I would also like to thank all of my teachers-from elementary
school, high school, undergraduate, and graduate school-for spending time with me
and challenging me to do my best. I would especially like to thank Mr. Kristo and
the rest of the math department at Owatonna High School for giving me an excellent
foundation in basic mathematics. Finally I would like to thank the University of
Florida Department of Statistics, the faculty, staff, and fellow students, for making
my stay in Gainesville an excellent experience.
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TABLE OF CONTENTS
gage
ACKNOWLEDGMENTS
iii
ABSTRACT
vi
CHAPTERS
INTRODUCTION
1 1
METHODS OF OPTIMIZATION
2 6
2.1 The Taguchi Method 6
2.2 Dual Response Surface Optimization 11
2.3 Alternative Methods of Dual Response Surface Optimization 16
2.4 Criticisms of the Various Optimization Schemes 22
2.5 Multiple Response Optimization 25
2.6 Criticisms of Multiple Response Optimization Methods ... 34
2.7 Robust Multiple Response Optimization 36
2.8 Optimizing a Function 38
3 ROBUST PARAMETER DESIGN 41
3.1 A Description of the Proposed Method 41
3.2 Printing Process Example 47
3.3 Example with Lifetime Data 54
4 MULTIPLE RESPONSE OPTIMIZATION 59
4.1 A Description of the Proposed Method 59
4.2 Tire Tread Compound Example 63
4.3 Repeatability of a Multiresponse Optimization Experiment 68
.
4.4 Albumin Nanospheres Example 72
5 MULTIPLE RESPONSE ROBUST PARAMETER DESIGN .... 79
5.1 A Description of the Proposed Method 79
5.2 Oxygen Plasma Anodization Example 85
5.3 Repeatability of Anodization Example 90
6 CONCLUSION AND FURTHER RESEARCH 93
IV
APPENDICES
A SOLVER COMMAND IN MICROSOFT EXCEL 95
B FORTRAN CODE FOR MULTIVARIATE NORMAL PROBABILITY 99
C BASIC OUTLINE OF AN OPTIMIZATION EXPERIMENT ... 128
REFERENCES 133
BIOGRAPHICAL SKETCH 136
V
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A UNIFIED APPROACH TO PROCESS OPTIMIZATION
By
Philip John McGoff
May 2000
Chairman: G. Geoffrey Vining
Major Department: Statistics
The goal of any optimization experiment is to find the settings of the factors
that can be controlled which results in optimal levels of the responses of interest. In
robust parameter design, the two responses of interest are the mean and variance of
a quality characteristic. In multiple response optimization, the responses of interest
are the quality characteristics of the product. In both of these cases, a quantity that
is a function of the estimates of the responses of interest is either maximized or
minimized. A variety of quantities have been proposed for robust parameter design
and multiple response optimization, but all of the proposed quantities are lacking in
some respect-they may lack intuitive appeal, depend too heavily on the definition
of subjective parameters, or fail altogether in certain situations. In addition, most
of the quantities proposed for robust parameter design cannot be adapted easily
to multiple response optimization. The probability that all of the responses are
simultaneously within their upper and lower specification limits is a quantity which
vi
can be used for robust parameter design and multiple response optimization. The
probability method also has an intuitive appeal that will make it easy to explain to
people in fields outside of statistics. This method does not depend on the definition
of subjective parameters, and it works in all of the situations that have been
addressed. It may also be extended to multiple response robust parameter design,
which none of the other methods has attempted.
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CHAPTER
1
INTRODUCTION
It is the goal ofany industry to produce high quality products as inexpensively
as possible. The quality of a product is usually measured by physical characteristics,
such as diameter, purity, taste, or hardness. When these characteristics are at
specific target levels or values, the product is thought to be of high quality. As the
characteristics deviate from the target levels, the quality of the product decreases.
Therefore producing a high quality product translates into producing the product
with quality characteristics at specific levels. There are three possible situations for
an individual characteristic:
• Target is best: the quality characteristic has a target level that is most desirable.
For instance, the diameter of a roller bearing or viscosity of a fluid.
• Larger is better: the quality characteristic should be made as large as possible.
For instance, the purity of a chemical or a car’s gas mileage.
• Smaller is better: the quality characteristic should be made as small as possible.
For instance, impurities in a chemical or number of bubbles in a paint job.
Optimizing a product or process is a situation found in many areas of applied
statistics. A set of control factors affects a response of interest, which is to be
maximized (or minimized). The goal of the experimenter is to find the combination
of the control factors that does maximize (or minimize) the response of interest.
Box and Wilson (1951) introduced the topic of response surface methods in an
attempt to answer this problem. They recommended that a series of experiments
be performed, first to find out which control factors actually affect the response
of interest, and secondly to find the optimal settings of the control factors. The
1
2
optimal settings are found by running a response surface experiment capable of
fitting the response to a second order polynomial in the control factors. Using some
basic calculus, the maximum (or minimum) of the second order polynomial can be
found. The situation can become much more complicated if the response of interest
does not have constant variance in the region in question, or if there is more than
one response of interest. Robust parameter design deals with the situation where
there is a single response that does not have a constant variance in the region of
interest. Multiple response optimization deals with the situation where there is more
than one response of interest.
Robust Parameter Design
In the manufacturing and use of the product, there will exist factors that will
affect the values ofthe quality characteristic, not all ofwhich can be easily controlled.
Those that can be easily controlled are referred to as control factors. These could be
the amount of material going into a chemical process, the temperature of a chemical
bath, or the pressure in a reactor. Those factors that cannot be easily controlled, or
are too costly to control, are referred to as noise factors. These could be ambient
humidity and temperature, the speed at which a customer drives his car, or the
temperature at which a customer bakes his cake.
Optimizing the industrial process entails finding the levels of the control
factors that will produce a product that has the desired quality characteristics. Due
to natural variability, it is impossible to find settings that will always give the same
values of the quality characteristics, and so all we ask is that it give those values
a high percentage of the time on average. This variability will lead to decreased
quality, as the product will not always be at its target value. One approach that has
been traditionally taken is a two step process. The first step gets the characteristic
at its desired level, on average. The second step is to find the sources of variability
3
and either control them or eliminate them. Control or elimination of sources of
variability can be difficult and/or costly.
Another approach is to find settings of the control factors that make the
product insensitive to the sources of variability. This is the approach used in the
area of parameter design or robust parameter design. This approach has the appeal
that it is typically more cost effective than controlling or trying to totally eliminate
the sources of variation. If levels of the control factors can be found for which
the product is insensitive to the sources of variation, and the product’s quality
characteristics can be made consistently close to their desired levels, this will result
in a consistently high quality product.
Experiments can be run in order to find the settings of the control factors
that will result in a consistently high quality product. The experiment will have
the quality characteristics as the responses of interest, and the control and noise
factors as the variables of interest. Notice that some noise factors can be controlled
during experimentation, even though they cannot be easily controlled otherwise. An
example of this would be the speed a car is driven or the temperature at which a
cake is baked. Even if no noise factors exist or are known, an experiment can still be
run to find the settings of the control factors that produce the smallest variability
while still achieving a desired level of the response.
Multiple Response Optimization
Most products and processes have more than one response of interest, and all
of these responses may depend on a set of control factors. These responses may have
either an ideal target value that is desired, or a range of values that will result in a
product that is of satisfactory quality. These targets and ranges are typically given
in the form of targets and upper and lower specification limits. Often the product
cannot be shipped to a customer unless all of the individual responses are within
their upper and lower specifications. Ideally, there would exist a combination of the
