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Electronic Theses, Treatises and Dissertations
The Graduate School
2006
Efficient Mixed-Level Fractional Factorial
Designs: Evaluation, Augmentation and
Application
Yong Guo
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THE FLORIDA STATE UNIVERSITY
FAMU-FSU COLLEGE OF ENGINEERING
EFFICIENT MIXED-LEVEL FRACTIONAL FACTORIAL DESIGNS:
EVALUATION, AUGMENTATION AND APPLICATION
by
YONG GUO
A Dissertation submitted to the
Department of Industrial and Manufacturing Engineering
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Degree Awarded:
Spring Semester, 2006
Copyright © 2006
Yong Guo
All Rights Reserved
The members of the Committee approve the dissertation of Yong Guo defended on April
7, 2006.
James R. Simpson
Professor Directing Dissertation
Xufeng Niu
Outside Committee Member
Samuel A. Awoniyi
Committee Member
Joseph J. Pignatiello, Jr.
Committee Member
Approved:
Chuck Zhang, Chair, Department of Industrial Engineering
Ching-Jen Chen, Dean, FAMU-FSU College of Engineering
The Office of Graduate Studies has verified and approved the above named committee
members.
ii
ACKNOWLEDGEMENTS
I would like to express my gratitude to my advisor, Dr. James Simpson, for his
support, patience, guidance and encouragement throughout my graduate studies. I am so
proud of him as my advisor, professor, boss and friend. It is not often that one finds an
advisor that always finds the time for listening to the little problems and roadblocks that
unavoidably crop up in the course of performing research. His technical and editorial
advice was essential to the completion of this dissertation and has taught me innumerable
lessons and insights on the workings of academic research in general.
My thanks also go to the member of my committee member, Professor Samuel
Awoniyi for supporting and helping me during my time in FAMU-FSU College of
Engineering. I would like to thank Joseph Pignatiello for reading previous drafts of this
dissertation and providing many valuable comments that improved the presentation and
contents of this dissertation. I also thank Dr. Niu from Statistics Department for his
advisement and comments for this dissertation.
Micelle Zeisset is much appreciated for her truly friendship during my graduate
studies. I am also grateful to my friend Wayne Wesley for staying with me in prelim
exam and dissertation writing. Lisa, Francisco and Rupert in Quality Lab are appreciated
and have led to many interesting and good-spirited discussions relating to this research.
I would like to thank Irene, Charlie, Marcus, Noah, David, Bernie, Faqing,
Fangyu for their friendship along this journey. Their encouragement was in the end what
made this dissertation possible. Last, but not least, I thank other friends in the Department
of Industrial Engineering and all the friends at the Rogers Hall basketball court.
My parents receive my deepest gratitude and love for their supporting and
understanding during my study on abroad that provided the foundation for this work.
iii
TABLE OF CONTENTS
LIST OF TABLES ........................................................................................................... vi
LIST OF FIGURES ....................................................................................................... viii
ABSTRACT...................................................................................................................... ix
CHAPTER 1 INTRODUCTION ..................................................................................... 1
1.1 Motivation................................................................................................................. 5
1.2 Problem Statement .................................................................................................... 5
1.3 Research Objective ................................................................................................... 6
CHAPTER 2 A REVIEW OF SOME MINIMUM ABERRATION CRITERIA FOR
EVALUATING FRACTIONAL FACTORIAL DESIGNS .......................................... 8
2.1 Introduction............................................................................................................... 8
2.2 The Usual Minimum Aberration Criterion for Two-Level Designs......................... 9
2.3 Minimum Aberration Criterion for Designs with Two- and Four-Level Factors... 10
2.4 Other Minimum Aberration Criterion Definitions.................................................. 12
2.4.1 Minimum G-Aberration Criterion.................................................................... 15
2.4.2 Generalized Minimum Aberration Criterion ................................................... 19
2.4.3 Minimum Moment Aberration Criterion ......................................................... 22
2.4.4 Moment Aberration Projection Criterion......................................................... 24
2.4.5 Additional Minimum Aberration Definitions .................................................. 26
2.5 Conclusion .............................................................................................................. 28
CHAPTER 3 THE GENERAL BALANCE METRIC FOR FRACTIONAL
FACTORIAL DESIGNS ................................................................................................ 30
3.1 Introduction............................................................................................................. 30
3.2 General Near-balanced Designs.............................................................................. 31
3.3 Features of General Balance Metric ....................................................................... 34
3.4 More Examples ....................................................................................................... 38
3.5 Conclusion .............................................................................................................. 41
iv
CHAPTER 4 OPTIMAL FOLDOVER PLANS FOR MIXED-LEVEL
FRACTIONAL FACTORIAL DESIGNS .................................................................... 43
4.1 Introduction............................................................................................................. 43
4.2 Foldover Strategy for Multi-Level Factors ............................................................. 44
4.3 General Balance Metric .......................................................................................... 45
4.4 Optimal Foldovers of Mixed-Level Designs .......................................................... 47
4.5 Conclusion .............................................................................................................. 53
CHAPTER 5 ANALYSIS OF MIXED-LEVEL EXPERIMENTAL DESIGNS
INCLUDING QUALITATIVE FACTORS.................................................................. 54
5.1 Introduction............................................................................................................. 54
5.2 Indicator Variables for Qualitative Factors............................................................. 54
5.3 Contrast Coefficients for Qualitative Factors ......................................................... 56
5.4 Classic Regression Models for Mixed-Level Designs............................................ 58
5.4.1 First-Order Models........................................................................................... 58
5.4.2 First-Order Models with Interactions............................................................... 59
5.4.3 Second-Order Models ...................................................................................... 59
5.5 Interactions between Qualitative Factors................................................................ 59
5.6 Regression Analysis on Fire Fighting Data ............................................................ 61
5.7 Conclusion .............................................................................................................. 67
CHAPTER 6 GENERAL CONCLUSIONS AND FUTURE RESEARCH .............. 68
APPENDICES ................................................................................................................. 72
A. Efficient Mixed-Level Designs................................................................................ 72
B. MATLAB Codes ...................................................................................................... 82
C. Glossary.................................................................................................................... 93
REFERENCES................................................................................................................ 94
BIOGRAPHICAL SKETCH ......................................................................................... 98
v
LIST OF TABLES
Table 1 Full Factorial Design - 213151 ................................................................................ 3
Table 2 Treatments Combinations for 2 Factors with 2 Levels Each................................. 4
Table 3 A 8-Run 2441 Mixed-Level Design........................................................................ 5
Table 4 Three 27-2 design options...................................................................................... 10
Table 5 A Mixed-Level Design with One Four-Level Factor and Four Two-Level Factors
................................................................................................................................... 11
Table 6 Three 4124 Design Options .................................................................................. 12
Table 7 The 12-Run Plackett-Burman Design.................................................................. 14
Table 8 Plus and Minus Signs for Calculating J k ( d ) ....................................................... 17
Table 9 Comparison of Two Designs Using MGA ........................................................... 18
Table 10 Comparison of Two Designs Using MG2A........................................................ 19
Table 11 Calculation of GMA for d3 ................................................................................. 21
Table 12 Comparison of Designs in Example 2 Using GMA ........................................... 22
Table 13 Comparison of Designs in Example 1 Using MMA........................................... 23
Table 14 Comparison of Designs in Example 2 Using MMA........................................... 24
Table 15 All k-Factor Projections and Their K-Values for d3 .......................................... 25
Table 16 All k-Factor Projections and Their K-Values for d4 .......................................... 25
Table 17 Frequency Distribution of Kk -Values of Factor Projections for Example 2 ..... 25
Table 18 Frequency Distribution of Kk -Values of Factor Projections for Example 1 ..... 26
Table 19 Features of Minimum Aberration Criteria ......................................................... 28
Table 20 Relationship Summery between Minimum Aberration Criteria........................ 29
Table 21 An Example of Coding Mixed-Level Factor Interactions ................................. 31
Table 22 A Mixed-Level Design (6, 2231) ........................................................................ 34
Table 23 Three 27-2 Design Options.................................................................................. 38
Table 24 Comparison of Three 27-2 Design Options ......................................................... 39
Table 25 Comparison of Two OA(12, 243) Designs ........................................................ 40
Table 26 Comparison of Three OA(12, 244) designs ....................................................... 41
Table 27 A Mixed-Level Design (6, 3122) ........................................................................ 47
Table 28 All Foldover Alternatives for Design (6, 3122) .................................................. 48
vi
Table 29 Statistics of EA(15, 315171) ................................................................................ 49
Table 30 Comparison of the Combined Design with EA(30, 315171) ............................... 50
Table 31 Optimal Foldover Plans for EAs in terms of GBM and B.................................. 52
Table 32 Decomposition of Factor A Using Contrast Coefficients.................................. 56
Table 33 Two Orthogonal Decomposition Options.......................................................... 56
Table 34 Four Fire Fighting System Methods .................................................................. 61
Table 35 Descriptive Statistics of Factors and Responses................................................ 62
Table 36 Contrast Coefficients for Surface and Method .................................................. 62
Table 37 Analysis of Variance of the Fire Data (All Terms) ........................................... 65
Table 38 Analysis of Variance of the Fire Data (Significant Terms) ............................... 65
Table 39 Model Terms Estimation ................................................................................... 66
vii
LIST OF FIGURES
Figure 1 The components of a process. .............................................................................. 2
Figure 2 Application of different minimum aberration criteria. ....................................... 13
Figure 3 Two (12, 25) designs. .......................................................................................... 14
Figure 4 Two OA(12, 243) designs. .................................................................................. 15
Figure 5 Coincidence matrices for Example 1.................................................................. 23
Figure 6 Balance relationship among types of columns. .................................................. 32
Figure 7 Two OA(12, 243) designs. .................................................................................. 40
Figure 8 Three OA(12, 244) designs. ................................................................................ 41
Figure 9 Rotate a five-level factor. ................................................................................... 44
Figure 10 EA(15, 315171)................................................................................................... 49
Figure 11 EA(30, 315171)................................................................................................... 51
Figure 12 Interaction of qualitative factors A and B. ....................................................... 60
Figure 13 Plot of extinguishment time verse method. ...................................................... 63
Figure 14 Plot of extinguishment time verse surface........................................................ 63
Figure 15 Probability plot of extinguishment time. .......................................................... 64
Figure 16 Probability plot of transformed extinguishment time....................................... 65
viii
ABSTRACT
In general, a minimum aberration criterion is used to evaluate fractional factorial
designs. This dissertation begins with a comprehensive review and comparison of
minimum aberration criteria definitions regarding their applications, relationships,
advantages, limitations and drawbacks. A new criterion called the general balance metric,
is proposed to evaluate and compare mixed-level fractional factorial designs. The general
balance metric measures the degree of balance for both main effects and interaction
effects. This criterion is related to, and dominates orthogonality criteria as well as
traditional minimum aberration criteria. Besides, the proposed criterion provides
immediate feedback and comprehensively assesses designs and has practical
interpretations. The metric can also be used for the purpose of design augmentation to
improve model fit. Based upon the proposed criterion, a method is proposed to identify
the optimal foldover strategies for efficient mixed-level designs. The analysis of mixedlevel designs involving qualitative factors can be achieved through indicator variables or
contrast coefficients. A regression model is developed to include qualitative factor
interactions which have been previously ignored.
ix
CHAPTER 1
INTRODUCTION
An experiment is a test or series of tests conducted under controlled conditions
made to demonstrate a known truth, examine the validity of a hypothesis, or determine
the efficacy of action previously untried. In an experiment, one or more input process
variables are changed deliberately in order to observe the effect that changes have on one
or more response variables. Experiments are performed a number of times in order to
evaluate the output response variables under the different input process variable
conditions. The design of experiments is an efficient method for planning experiments so
that the data obtained can be analyzed to yield valid and objective conclusions. The
method for conducting designed experiments begins with determining the objectives of
an experiment and selecting the process factors for the study. A designed experiment
requires establishing a detailed experimental plan in advance of conducting the
experiment, which results in a streamlined approach in the data collection stage.
Appropriately choosing experimental designs maximizes the amount of information that
can be obtained for a given amount of experimental effort.
Experimental designs are used to investigate industrial systems or processes. A
typical process model is given in Figure 1. Purposeful changes are made to the
controllable input factors of a process so as to observe and identify the reasons for
changes that may be observed in the output responses. The noise factors are considered as
random effects that cannot be controlled. Experimental data are used to derive a statistical
empirical model linking the outputs and inputs. These empirical models generally contain
first and second-order terms. For more information regarding the statistical empirical
model, see Montgomery (2005).
1
……
Controllable ……
Input Factors
Noise Factors
Process
……
Output Responses
Figure 1 The components of a process.
Many experiments involve the study of the effects of two or more factors on one
or more output responses. Full factorial designs are test matrices that contain all possible
combinations of the levels of the factors. For example, if factor A has a levels and factor
B has b levels, then the two-factor full factorial design contains ab combinations. Table.
1 shows another example, a full factorial design with three factors: one with two levels,
one with three and one with five. The shorthand notation for this design is (213151), which
displays the factor levels as the base numbers and the number of factors with that many
levels as the exponent.
One purpose of factorial designs is to study the effects of these factors on the
response. The main effect of a factor is defined to be the change in response produced by
a change in the level of the factor. The term main effect is used because it refers to the
primary factors of interest in the experiment. A main effect reflects the individual impact
of each factor.
One-factor-at-a-time testing is an extensively used experimentation strategy. This
method consists of selecting a starting point setting for each factor, then successively
varying the settings of each factor over its range, with the other factors held constant
(Montgomery, 2005). Compared with one-factor-at-a-time experiments, factorial designs
are more efficient. Factorial designs allow the effects of a factor to be estimated at several
levels of the other factors because the difference in response between the levels of one
factor may not be the same at all levels of the other factor. Therefore, factorial designs
are necessary when interactions may be present.
2
Table. 1Full Factorial Design - 213151
Run Factor A Factor B Factor C
1
1
1
1
2
2
1
1
3
1
2
1
4
2
2
1
5
1
3
1
6
2
3
1
7
1
1
2
8
2
1
2
9
1
2
2
10
2
2
2
11
1
3
2
12
2
3
2
13
1
1
3
14
2
1
3
15
1
2
3
16
2
2
3
17
1
3
3
18
2
3
3
19
1
1
4
20
2
1
4
21
1
2
4
22
2
2
4
23
1
3
4
24
2
3
4
25
1
1
5
26
2
1
5
27
1
2
5
28
2
2
5
29
1
3
5
30
2
3
5
A specific case of general factorial designs is the 2k factorial design. That is, these
designs have k factors, each at only two levels. These levels may be either quantitative or
qualitative. Normally “+” is used to represent the high level and “–” is used to represent
the low level in the 2-level factorial designs. A complete replicate of such a design
3
requires 2k observations and is called a 2k factorial design. Table 2 gives an example for
k=2 in three replicates.
Table 2. Treatments Combinations for 2 Factors with 2 Levels Each
Factor Treatment
A B Combination
– – A low, B low
+ – A high, B low
– + A low, B high
+ + A high, B high
Replicate
I II III
28 25 27
36 32 32
18 19 23
31 30 29
The interaction effect AB is defined as the average change in response between
the effect of A at the high level of B and the effect of A at the low level of B. The
methods used for generating 2k factorial designs are straightforward. Each column
represents a factor. The levels for the first column follow “– + – + …– +”. The levels for
the second column follow the pattern of “– – + +…– – + +”. For the nth column, the
pattern will be “–…– +…+” and the number of minus signs and plus signs is n for each.
Many experimental design textbooks and software packages emphasize the use of
factorial and fractional factorial designs in which all factors in the experiment have two
levels, often called 2k-p designs, where k is the number of factors, p is the degree of
fractionation, and 2k-p is the number of runs. It is true that technological experiments
often have only quantitative factors; however, it is not uncommon for technological
experiments to also include factors that are qualitative in nature. There are often more
than two levels of such factors. In order to include factors that have more than two levels,
mixed-level designs are used, which have become more practically used in the field of
design of experiments. For example, an experimental design (Table 3) considers five
factors: four factors with two levels and one factor with four levels.
4
Table 3. A 8-Run 2441 Mixed-Level Design
Two-level factors Four-level factor
Run A B C D
E
1
-1 -1 -1 -1
1
2
1
1 1 1
1
3
-1 -1 1 1
2
4
1
1 -1 -1
2
5
-1
1 -1 1
3
6
1 -1 1 -1
3
7
-1
1 1 -1
4
8
1 -1 -1 1
4
1.1 Motivation
The full factorial mixed-level design could be very large in terms of run number,
depending on the number of factors and the factor levels. For example, a mixed-level
design considers three factors: a three-level factor, a five-level factor, and a seven-level
factor. The full factorial design contains a total of 105 runs. Therefore, it may be
desirable to use fractional factorial mixed-level designs instead of the full factorial. Some
mixed-level designs are available in the literature. Orthogonal and near orthogonal
mixed-level designs are discussed by Sloane (2006) and Xu (2002). In the case that
balanced designs are not available, a good solution then is to use near-balanced efficient
mixed-level designs (Guo, Simpson, and Pignatiello 2005).
However, different fractions from a full factorial may have the same balance and
orthogonality property. An important consideration is how to further select the “best”
fractional factorial mixed-level designs. In situations where we have little or no
knowledge about the effects that are potentially important, it is appropriate to use the
minimum aberration criterion.
1.2 Problem Statement
The “usual” definition of minimum aberration (MA) criterion for regular two-level
designs was introduced by Fries and Hunter (1980). A type of mixed-level design, 4m2n-p ,
can be developed by using the usual MA criterion (Wu and Hamada 2000, Ankenman
1999, and Montgomery 2005). Even though the usual definition of minimum aberration
5
works well for designs with two-level factors and four-level factors, it is not possible to
extend this usual definition to other applications, such as two-level non-regular designs,
multi-level designs and mixed-level designs, since the principle of the usual MA is based
upon design generators. Therefore, new MA definitions have been developed to meet
these requirements. Some definitions include the minimum G-aberration criterion (Deng
and Tang 1999), the minimum G2-aberration criterion (Tang and Deng 1999, Ingram and
Tang 2005), the generalized minimum aberration criterion (Xu and Wu 2001), the
minimum moment aberration criterion (Xu 2003), the moment aberration projection (Xu
and Deng 2005), the minimum generalized aberration criterion (Ma and Fang 2001), and
a general criterion of minimum aberration (Cheng and Tang 2005). In general, the
currently existing minimum aberration criteria are complicated and not easy to apply in
industrial situations.
1.3 Research Objectives
The first objective of this dissertation is to review the existing minimum
aberration criteria. Examples including non-regular two-level designs and mixed-level
designs are used for comparing these criteria. The goal is to introduce these minimum
aberration criteria to practitioners with practical examples so that the practitioners can
know the relationships, advantages and drawbacks of these minimum aberration criteria.
The second objective is to develop a new minimum aberration criterion, the
general balance metric, for mixed-level fractional factorial designs. The performance of
this criterion will be compared with other criteria.
The third objective is to fold over efficient mixed-level designs using the general
balance metric. The purpose is to decompose aliased model terms. With this method, find
the optimal foldover plans for given mixed-level designs via algorithms. Finally, provide
optimal foldover plans for existing efficient mixed-level designs.
The fourth objective is to analyze mixed-level designs involving qualitative factor
interactions via contrast coefficients. The goal is to analyze qualitative factor interactions
from the point of view of model building and to propose a regression model that includes
qualitative factor interactions.
6
This dissertation is structured as follows. Chapter 2 reviews the existing minimum
aberration criteria. Chapter 3 develops a new criterion, called general balance metric.
Chapter 4 proposes a method to fold over mixed-level fractional factorial designs.
Chapter 4 identifies qualitative factor interactions and develops a regression model to
incorporate qualitative factor interactions. Finally, general conclusions from these
research topics will be discusses in Chapter 6.
7
CHAPTER 2
A REVIEW OF MINIMUM ABERRATION
CRITERIA FOR EVALUATING FRACTIONAL
FACTORIAL DESIGNS
2.1 Introduction
In the last 20 years, significant attention has been paid on developing new
minimum aberration criteria and constructing minimum aberration designs using those
criteria. The concept of minimum aberration was first introduced by Fries and Hunter
(1980) as a way of selecting the best two-level fractional factorial designs from those
with equal maximum resolution. The resolution for two-level fractional factorial designs
was proposed by Box and Hunter (1961). Minimum aberration designs have the best alias
structure and possess robust properties (Cheng, Steinberg and Sun 1999 and Tang and
Deng 1999).
Regular two-level designs indicate those designs who are constructed by design
generators (Motgomery 2005). Regular two-level designs are denoted by 2m-q, and have
simple alias structures. Non-regular designs have more flexible design sizes than regular
designs, but their alias structures are more complicated. Examples of non-regular designs
are Plackett-Burman designs (Deng and Tang, 1999) and supersaturated designs (Xu
2003, Xu and Wu 2005). Since the usual minimum aberration criterion definition can not
be applied directly to non-regular designs, it was necessary to develop new minimum
aberration definitions for evaluating non-regular designs. Extensive work on non-regular
two-level fractional factorial designs was developed by Chen and Hedayat (1996), Chen
(1992), Bingham and Sitter (1999), Sitter, Chen and Feder (1997), Huang, Chen and
Voelkel (1998), Tang and Wu (1996), Ma and Fang (2001), Wu and Zhu (2003), Cheng
and Tang (2005) and Xu and Deng (2005).
8
As an extension of two-level fractional factorial designs, Franklin (1984) and
Suen, Chen and Wu (1997) discuss the construction of multi-level minimum aberration
designs. Xu and Wu (2001) proposed a generalized minimum aberration for mixed-level
(asymmetrical) fractional factorial designs. Wu and Zhang (1993) and Ankenman (1999)
used minimum aberration designs in two-level and four-level mixed-level designs.
Mukerjee and Wu (2001) developed minimum aberration designs for mixed-level
fractional factorial designs involving factors with two or three distinct levels.
In the following section, the usual minimum aberration criterion for two-level
designs is briefly reviewed. Then the application of this criterion to special cases of
mixed-level designs with two- and four-level factors is discussed. In the subsequent
section, other proposed minimum aberration criteria are reviewed and compared via
examples. The last section discusses the conclusions and suggestions. For the reader’s
convenience, we use consistent notation throughout the paper. A complete glossary can
be found in Appendix C.
2. 2 The Usual Minimum Aberration Criterion for Two-Level Designs
A two-level 2m-q design is defined to be a fractional factorial design with m factors,
each at two levels, consisting of 2m-q runs. Therefore, it is a 2-q fraction of the 2m full
factorial design in which the fraction is determined by q generators, where a generator
consists of letters which are the names of the factors denoted by A, B, …. The number of
letters in a word is its word length and the word formed by the q defining words is called
the defining relation.
For a 2m-q design, let Ak ( d ) be the number of words of length k in the defining
contrast subgroup. The vector
W ( d ) = ( A1 ( d ) , A2 ( d ) ,
, Am ( d ) )
is called the word length pattern of the design d (Fries and Hunter, 1980). The resolution
of a 2m-q design, R, is defined to be the smallest r such that Ar ( d ) ≥ 1, that is, the length
of the shortest word in the defining contrast subgroup. For any two 2m-q designs d1 and d2,
let r be the smallest integer such that Ar ( d1 ) ≠ Ar ( d 2 ) . Then d1 is said to have less
9
aberration than d2 if Ar ( d1 ) < Ar ( d 2 ) . If no design exists with less aberration than d1,
then d1 has minimum aberration.
Consider a 27-2 experiment, with three design options. Table 4 provides the design
generators for three designs along with their defining relations. In this example, d3 has
less aberration than d1 or d2 because the first unequal number in word length pattern is in
the fourth position and d3 has the smallest number in that position. Design d3 is the
minimum aberration 27-2 design. Other 2m-q minimum aberration designs and their design
generators are presented in Montgomery (2005). Montgomery (2005) gives a slightly
different formatted word length pattern from Wu and Zhang (1993)’s. Instead of using
numbers of words of length k in the defining contrast subgroup, Montgomery (2005)
directly shows the length of each word in the defining contrast group (Table 4).
Table 4. Three 27-2 design options
Design
d1
d2
d3
Options
Generators
F=ABC, G=BCD
F=ABC,G=ADE
F=ABCD, G=ABDE
Defining
I=ABCF=BCDG=ADFG I=ABCF=ADEG=BCDEFG I=ABCDF=ABDEG=CEFG
Relations
WLP
Wu and Zhang
(0, 0, 0, 3, 0, 0)
(0, 0, 0, 2, 0, 1)
(0, 0, 0, 1, 2, 0)
(1993)
WLP
Montgomery
{4, 4, 4}
{4, 4, 6}
{4, 5, 5}
(2005)
2.3 Minimum Aberration Criterion for Designs with Two- and FourLevel Factors
The application of the usual minimum aberration (MA) criterion can be expanded
for designs other than the regular two-level designs using certain schemes. The literature
(Addelman (1962), Wu and Hamada (2000), Ankenman (1999) and Montgomery (2005))
10
shows that multi-level factors can be replaced by two-level factors, thereby taking
advantage of two-level fractional factorial design alternatives. This method was
successfully used for four-level factors. Wu and Hamada (2000) proposed a formal
procedure for replacing four-level factors with two-level factors. The idea is to replace
one four-level factor with three two-level factors by the following rule.
X
A
1
−
2 Æ +
3
−
4
+
B AB
− +
− −
+ −
+ +
Now consider an experiment with five factors, one with four levels, and four with
two levels. The full factorial contains 64 runs, but an 8-run fractional is of interest. This
design (Table 5) illustrates that although only two, two-level factors are used to replace a
four-level factor, the interaction of these two two-level factors is also used. Thus, this
four-level factor is replaced by three single degree-of-freedom two-level factors.
Table 5. A Mixed-Level Design with One Four-Level Factor and Four Two-Level Factors
Run X1 X2 X1X2
1
− −
+
2
− −
+
3
+ −
−
4
+ −
−
5
− +
−
6
− +
−
7
+ +
+
8
+ +
+
B
−
+
−
+
−
+
−
+
C
+
−
+
−
−
+
−
+
D
+
−
−
+
+
−
−
+
E
Run X B
−
1
1 −
+
2
1 +
+
3
2 −
− Æ 4
2 +
+
5
3 −
−
6
3 +
−
7
4 −
+
8
4 +
C
+
−
+
−
−
+
−
+
D
+
−
−
+
+
−
−
+
E
−
+
+
−
+
−
−
+
Wu and Zhang (1993) present three 4124 designs (one four-level factor A mixed
with four two-level factors, B, C, D, and E). Let 1, 2, 3, 4 be four columns of the 24 full
factorial design. Let (1, 2, 1×2) be the four-level factor A. Factor B uses column 3 and C
is column 4. Then factors D and E are formed from combinations of columns 1, through 4
11
according to three different schemes (Table 6). Those schemes result in different
aberration scores for each design. Among all three designs, design d1 has minimum
aberration because design A3 ( d1 ) =1 but A3 ( d 2 ) = A3 ( d3 ) =2. Furthermore, d2 has less
aberration than d3 since A4 ( d 2 ) = 0 < A4 ( d 2 ) = 1 .
Table 6. Three 4124 Design Options
1
d1
d2
d3
A
1
1
1
A
A2
2
2
2
B
A3
12 3
12 3
12 3
C
D
E
Defining Relations
WLP
4 134 23 I = A1BCD = A2BE = A3CDE (0, 0, 1, 2, 0)
4 14 23 I = A1CD = A2BE = A3BCDE (0, 0, 2, 0, 1)
4 124 34 I = A3BDE = BCE = A3CD (0, 0, 2, 1, 0)
2.4 Other Minimum Aberration Criterion Definitions
Even though the usual definition of minimum aberration (MA) works well for
designs with two-level factors and four-level factors, it is difficult to extend this usual
definition to other applications, such as two-level non-regular designs, multi-level
designs and mixed-level designs, since the principle of the usual MA is based upon design
generators. Therefore, new MA definitions have been developed to meet these
requirements. Some definitions include the minimum G-aberration criterion (Deng and
Tang (1999)), the minimum G2-aberration criterion (Tang and Deng (1999), Ingram and
Tang (2005)), the generalized minimum aberration criterion (Xu and Wu (2001)), the
minimum moment aberration criterion (Xu (2003)), the moment aberration projection
(Xu and Deng (2005)), the minimum generalized aberration criterion (Ma and Fang
(2001)), and a general criterion of minimum aberration (Cheng and Tang (2005)). The
application of these MA criteria is shown in Figure 2.
12
Mixed-level
designs
Multi-level
designs
Non-regular
two-level
designs
Regular
two-level
designs
9
Usual
minimum
aberration
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
Minimum
Gaberration
Minimum
G2aberration
Generalized
minimum
aberration
Minimum
moment
aberration
Moment
aberration
projection
Minimum
generalized
aberration
criterion
General
criterion of
minimum
aberration
Figure 2 Application of different minimum aberration criteria.
To facilitate explanation, it is useful to work with example designs that cannot be
evaluated nor compared using the usual MA. Two examples are considered for the
reminder of this discussion.
Example 1
Consider the 12-run Plackett-Burman design given in Table 7. Two sub-designs
can be formed from this base design. The first design d1 contains columns 1-4 and 10,
second design d2 uses only columns 1-5 (Xu and Deng (2005)).
13
Table 7. The 12-Run Plackett-Burman Design
Run
1
2
3
4
5
6
7
8
9
10
11
12
1
+
−
+
−
−
−
+
+
+
−
+
−
2
+
+
−
+
−
−
−
+
+
+
−
−
3
−
+
+
−
+
−
−
−
+
+
+
−
4
+
−
+
+
−
+
−
−
−
+
+
−
5
+
+
−
+
+
−
+
−
−
−
+
−
6
+
+
+
−
+
+
−
+
−
−
−
−
7
−
+
+
+
−
+
+
−
+
−
−
−
8
−
−
+
+
+
−
+
+
−
+
−
−
9 10 11
− + −
− − +
− − −
+ − −
+ + −
+ + +
− + +
+ − +
+ + −
− + +
+ − +
− − −
Figure 3 gives these two sub-designs (12, 25), representing 5 two-level factors in 12 runs.
In general, ( n, l1k1 l2k2
lTkT ) denotes an n-run fractional factorial design, involving k1
factors with l1 levels, k2 factors with l2 levels so and so on.
A
+
−
+
−
−
−
+
+
+
−
+
−
B
+
+
−
+
−
−
−
+
+
+
−
−
d1
C
−
+
+
−
+
−
−
−
+
+
+
−
D
+
−
+
+
−
+
−
−
−
+
+
−
E
+
−
−
−
+
+
+
−
+
+
−
−
A
+
−
+
−
−
−
+
+
+
−
+
−
d2
B C
+ −
+ +
− +
+ −
− +
− −
− −
+ −
+ +
+ +
− +
− −
Figure 3. Two (12, 25) designs.
14
D
+
−
+
+
−
+
−
−
−
+
+
−
E
+
+
−
+
+
−
+
−
−
−
+
−
Example 2
A second example considers two mixed-level designs (Figure 4). Each design has
5 factors in a total of 12 runs. Factor A has three levels and factors B, C, D, and E has
two levels each. The factor levels are coded as “1, 2, 3,… ”, representing the first-, second-,
third-… level of that factor.
A
B
C
D
d3
E
A
Figure 4 Two OA(12, 243) designs.
B
C
D
E
d4
2.4.1 Minimum G-Aberration Criterion
The first MA criterion is the minimum G-aberration (MGA) criterion for two-level
designs, proposed by Deng and Tang (1999). Ingram and Tang (2005) used this criterion
to construct two-level designs with 24 runs. For an experimental design matrix, d, let n
represent the number of rows (runs) and m be the number of columns (factors). Let
s = [ c1 , c2 ,
, ck ] represent a k-column subset matrix from d and cij is the ith element of
column cj. Define
Jk (d ) =
n
∑c c
i =1
i1 i 2
cik for k = 1, … , m .
15
J k ( d ) corresponds to all k-factor interactions, 1 through m.
Let Fk ( d ) be the vector that contains the frequencies of the different J k ( d )
values
for
design
d.
as F ( d ) = ⎡⎣ F1 ( d ) , F2 ( d ) ,
Define
the
confounding
frequency
vector
of
d
, Fm ( d ) ⎤⎦ . Let r be the smallest number for which
Fr ( d1 ) ≠ Fr ( d 2 ) . If Fr ( d1 ) < Fr ( d 2 ) , d1 is said to has less G-aberration than d2. This
criterion was developed strictly for two-level designs.
MGA can be used to compare the designs in Example 1. First, calculate plus and
minus signs for interactions by multiplying the appropriate preceding columns, row by
row. The calculation of J k ( d1 ) is shown in Table 8. It is found that J k ( d1 ) has three
possible values, (8, 4, 0), listed in a decreasing order. So Fk ( d1 ) contains the frequencies
of J k ( d1 ) = 8 , followed by frequencies of J k ( d1 ) = 4 , and J k ( d1 ) = 0 . For the main
effects
and
two-factor
interaction
effects,
J1 ( d1 ) = J 2 ( d1 ) = 0 .
Therefore,
F1 ( d1 ) = ( 0, 0,5) and F2 ( d1 ) = ( 0, 0,10 ) . That is all five J1 ( d1 ) are equal to 0 for main
effects and all ten J 2 ( d1 ) are equal to 0 for two-factor interactions. For three-factor and
four-factor interactions, all J 3 ( d1 ) and J 4 ( d1 ) equal 4, so F3 ( d1 ) = ( 0,10, 0 ) and
F4 ( d1 ) = ( 0,5, 0 ) . That is all ten J 3 ( d1 ) in three-factor interactions equal to 4 and all five
J 4 ( d1 ) in four-factor interactions is 4. For five-factor interactions, the only J 5 ( d1 ) is 8,
so F5 ( d1 ) = (1, 0, 0 ) .
In a similar way, J k ( d 2 ) is calculated and there are the same possible numbers (8,
4, 0). The frequencies of the different J k ( d 2 ) are
F ( d 2 ) = ⎡⎣( 0, 0,5 )1 , ( 0, 0,10 )2 , ( 0,10, 0 )3 , ( 0,5, 0 )4 , ( 0, 0, 0 )5 ⎤⎦ .
The comparison of two designs is given in Table 9. It can be seen that design d2 is better
since J 5 ( d 2 ) =0, but J 5 ( d1 ) =1.
16
Table 8. Plus and Minus Signs for Calculating J k ( d )
Run
1
2
3
4
5
6
7
8
9
10
11
12
17
A
B
C
D
E AB AC AD AE BC BD BE CD CE DE ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE ABCD ABCE ABDE ACDE BCDE ABCDE
1
-1
1
-1
-1
-1
1
1
1
-1
1
-1
1
1
-1
1
-1
-1
-1
1
1
1
-1
-1
-1
1
1
-1
1
-1
-1
-1
1
1
1
-1
1
-1
1
1
-1
1
-1
-1
-1
1
1
-1
1
-1
-1
-1
1
1
1
-1
1
1
-1
-1
1
-1
-1
-1
1
1
-1
1
1
-1
-1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
1
1
1
-1
1
-1
-1
-1
-1
-1
1
1
1
1
-1
1
-1
-1
1
-1
1
-1
-1
1
-1
1
-1
-1
-1
1
1
-1
1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
-1
-1
-1
1
1
1
1
-1
-1
1
-1
-1
-1
1
1
-1
1
1
1
-1
-1
-1
1
1
-1
-1
1
1
1
-1
1
0 0 0 0 0
0
0
0
0
0
0
0
0
0 0 0 0 0
0
0
0
0
0
0
0
0
1
1
-1
-1
-1
1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
-1
1
-1
1
-1
-1
-1
1
1
-1
-1
-1
1
1
-1
-1
-1
-1
-1
1
1
1
1
1
1
-1
-1
1
-1
1
-1
0
0 -4
-4
0
0
4
-1
1
1
1
1
1
1
1
-1
-1
1
-1
-1
1
-1
-1
-1
1
-1
1
1
-1
-1
-1
1
-1
-1
1
1
-1
-1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
-1
1
-1
1
-1
-1
-1
-1
1
1
-1
1
1
1
1
1
1
-1
4
4 -4 -4 -4
4
4
1
1
1
-1
1
-1
1
1
-1
1
1
-1
-1
1
-1
1
-1
-1
1
-1
-1
1
-1
-1
-1
1
-1
1
-1
-1
-1
1
-1
-1
-1
1
-1
1
1
-1
1
-1
1
1
1
-1
1
1
1
-1
1
1
-1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
1
-1
-1
-1
-1
1
-1
1
1
1
1
1
-1
-1
-1
1
1
1
-1
-1
1
-1
-1
-1
-1
-1
-1
-1
1
-1
4
4 -4
-4
4
4
-4
4
-8
4
4
4
4
4
4
4
8
12
∑c c
i =1
Jk (d ) =
i1 i 2
cik
12
∑c c
i =1
i1 i 2
cik
4
4
4
4
4
Table 9. Comparison of Two Designs Using MGA
MGA
d1
d2
Frequencies of J-values (8, 4, 0)
[(0, 0, 5)1, (0, 0, 10)2, (0, 10, 0)3, (0, 5, 0)4, (1, 0, 0)5]
[(0, 0, 5)1, (0, 0, 10)2, (0, 10, 0)3, (0, 5, 0)4, (0, 0, 0)5]
The use of a frequency vector may not be convenient when large computations
are involved. The minimum G2-aberration (MG2A) criterion is proposed by Tang and
Deng (1999) as a relaxed version of MGA. Define
⎛ J (d ) ⎞
Bk ( d ) = ∑ ⎜ k
⎟ , for k=1, …, m,
n ⎠
s =k ⎝
2
where s = k indicates that the summation covers all subset matrices with k columns.
The criterion of MG2A sums the square of the normalized J k ( d ) for k =1, …, m.
For two designs d1 and d2, suppose that r is the smallest integer such that
Br ( d1 ) ≠ Br ( d 2 ) . Then d1 is said to have less G2-aberration than d2 if Br ( d1 ) < Br ( d 2 ) .
Now, apply MG2A for the designs in Example 1. For both d1 and d2,
2
2
⎛ 0⎞
⎛ 0⎞
B1 ( d ) = 5 × ⎜ ⎟ = 0 , B2 ( d ) = 10 × ⎜ ⎟ = 0
⎝ 12 ⎠
⎝ 12 ⎠
2
2
5
⎛ 4 ⎞ 10
⎛ 4⎞
B3 ( d ) = 10 × ⎜ ⎟ = =1.111 and B4 ( d ) = 5 × ⎜ ⎟ = =0.556.
9
⎝ 12 ⎠ 9
⎝ 12 ⎠
However, for 5-factor interactions, the two designs diverge with different B5 -values,
which are,
2
2
4
⎛ 8⎞
⎛ 0⎞
B5 ( d1 ) = ⎜ ⎟ = =0.444 and B5 ( d 2 ) = ⎜ ⎟ = 0 .
9
⎝ 12 ⎠
⎝ 12 ⎠
A summary of the results (Table 10) shows that d2 is better than d1 by MG2A.
18
Table 10. Comparison of Two Designs Using MG2A
MG2A ( B1 ( d ) , B2 ( d ) , B3 ( d ) , B4 ( d ) , B5 ( d ) )
d1
(0, 0, 1.111, 0.556, 0.444)
d2
(0, 0, 1.111, 0.556, 0)
The MG2A criterion is equivalent to the usual MA for evaluating regular designs.
The MG2A is computationally much easier than the MGA because it uses a single number
for each interaction, instead of a frequency vector. As a result, MG2A can be helpful for
comparing or evaluating large designs.
The MG2A criterion can be further generalized into the minimum Ge-aberration
criterion (Ingram and Tang (2005)). That is
⎛ J (s) ⎞
Bk ( d ) = ∑ ⎜ k
⎟ .
n ⎠
s =k ⎝
e
With values of e larger than 2, more emphasis is put on lower order interactions.
2.4.2 Generalized Minimum Aberration Criterion
Xu and Wu (2001) proposed a generalized minimum aberration (GMA) criterion
for multi-level and mixed-level designs. For a design d, the ANOVA model has the
following form
Y = X 0α 0 + X 1α1 +
+ X mα m + ε ,
where Y is the response, α k is the vector of all k-factor interactions and Xk= ⎡⎣ xij( k ) ⎤⎦ is the
matrix of contrast coefficients for α k . Let
Ak (d ) = n
−2
n
∑ ∑x
j
i =1
(k )
ij
2
.
The Ak ( d ) are invariant with respect to the choice of orthogonal contrasts. The vector
( A (d ), A (d ),
1
2
, Am ( d ) ) is called the generalized word length pattern. Then the
generalized minimum aberration criterion is to sequentially minimize Ak ( d ) for k=1, …,
m.
19
For two-level fractional factorial designs, the criterion of GMA is equivalent to
MG2A in mathematical form. Since the two designs in Example 1 are two-level designs,
GMA will give the exact same results as MG2A.
For the mixed-level designs from Example 2. Normalized orthogonal polynomials
are used as the contrast coefficients
Factor Level
1
2
Contrast Coefficient
0.7071
−0.7071
Factor Level
1
2
3
Contrast Coefficient
0.4082
−0.7071
0 −0.8165
0.7071
0.4082
Table 11 shows the model matrix of X1 and X2, and the calculation of the corresponding
A1 and A2 for design d3. A comparison of d3 versus d4 using GMA is given in Table 12.
Since A1 ( d3 ) = A1 ( d 4 ) = 0 but A2 ( d3 ) > A2 ( d 4 ) , design d4 has less aberration than d3.
Therefore, d4 is better than d3 by the GMA.
20
Table 11. Calculation of GMA for d3
___________________________________________________________________________________________________________
X1
1
2
A
A
B
C
D
E
Sum
0
0
0
2
2
2
2
2
2
2
A1(d3) ( 0 + 0 + 0 + 0 + 0 + 0 ) / 12 = 0
0
0
0
21
__________________________________________________________________________________________________________________________________________________
1
AB
Sum
0
1
1
AC
1
AD
0
2
AE
0
AB
0
0
2
AC
0
X2
A2D
A2E
0
BC
0
1.9994
BD
0
BE
CD
0
CE
0
DE
0
3.9988
A2(d3) ( 02 + 02+ 02 + 02+ 02+ 02+ 02+ 02 +1.99942 + + 02+ 02+ 02+ 02 + 3.99882 ) / 122 = 0.1388
___________________________________________________________________________________________________________________________________________________
Table 12. Comparison of Designs in Example 2 Using GMA
GMA
( A ( d ) , A ( d ) , A ( d ) , A ( d ) , A ( d ))
d3
(0, 0.1388, 0.0833, 0.0000)
d4
(0, 0.0000, 0.1249, 0.0208)
1
2
3
4
5
2.4.3 Minimum Moment Aberration Criterion
The MGA, MG2A, and GMA criteria all require contrast coefficients of factors. Xu
(2003) developed a minimum moment aberration criterion (MMA), which does not need
contrast coefficients. For a design matrix d , let dij be the elements of ith row and jth column.
The coincidence between two elements dij and dlj is defined by δ ( d ij , d lj ) , where
δ (d ij , d lj ) = 1 if dij = dlj and 0 otherwise. The value of
∑δ ( d
m
j =1
ij
, dlj ) measures the
coincidence between the ith and lth rows of d. The kth power moment is defined by Xu (2003)
as
k
K k (d ) = [ n(n − 1) / 2]
−1
⎡m
⎤
∑
⎢ ∑ δ ( dij , dlj ) ⎥ .
1≤i ≤l ≤ n ⎣ j =1
⎦
For two designs d1 and d2, d1 is said to have less moment aberration than d2 if there
exists an r such that K r (d1 ) < K r (d 2 ) and K t (d1 ) = K t (d 2 ) for all t=1, …, r-1. Therefore,
d1 is said to have minimum moment aberration if there is no other design with less moment
aberration than d1.
Figure 5 is the coincidence matrix for the designs in Example 1, where the (ith, jth)
element indicating the coincidence between the ith row and jth row. Since the lower
triangular matrix is symmetric to the upper triangular matrix, only the elements in upper
triangular matrix are used for calculation. Table 13 gives the calculation results, which
shows that design d2 is better than d1.
22
Run
1
2
3
4
5
6
7
8
9
10
11
12
1
0
0
0
0
0
0
0
0
0
0
0
0
2
1
0
0
0
0
0
0
0
0
0
0
0
3
2
2
0
0
0
0
0
0
0
0
0
0
4
3
3
2
0
0
0
0
0
0
0
0
0
5
1
3
2
1
0
0
0
0
0
0
0
0
6
3
1
2
3
3
0
0
0
0
0
0
0
7
3
1
2
1
3
3
0
0
0
0
0
0
8
3
3
2
3
1
1
3
0
0
0
0
0
9
3
3
2
1
3
1
3
3
0
0
0
0
10 11
3 2
3 2
2 5
3 2
3 2
3 2
1 2
1 2
3 2
0 2
0 0
0 0
12
1
3
2
3
3
3
3
3
1
1
2
0
9
2
3
3
1
2
1
2
4
0
0
0
0
10 11
2 3
3 2
3 4
3 2
2 3
3 2
0 3
2 1
3 2
0 2
0 0
0 0
12
1
2
2
2
3
4
3
3
2
2
1
0
(a) Coincidence Matrix for d1
Run
1
2
3
4
5
6
7
8
9
10
11
12
1
0
0
0
0
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
3
2
1
0
0
0
0
0
0
0
0
0
0
4
4
3
1
0
0
0
0
0
0
0
0
0
5
1
4
2
2
0
0
0
0
0
0
0
0
6
2
1
3
3
2
0
0
0
0
0
0
0
7
3
2
2
2
3
2
0
0
0
0
0
0
8
3
2
2
2
1
2
3
0
0
0
0
0
(b) Coincidence Matrix for d2
Figure 5 Coincidence matrices for Example 1.
Table 13. Comparison of Designs in Example 1 Using MMA
MMA
( K ( d ) , K ( d ) , K ( d ) , K ( d ) , K ( d ))
d1
d2
(2.3, 5.9, 16.8, 51.4, 167.7)
(2.3, 5.9, 16.8, 51.4, 165.9)
1
2
3
23
4
5
The application of MMA can also cover multi-level or mixed-level designs. For
designs in Example 2, design d4 is better than d3 with criterion MMA (Table 14). MMA
investigates the relationship between runs (rows) instead of factors (columns). Therefore,
MMA can be used for any design and it is computationally quick. Although MMA has the
capability to discriminate among designs, the k-th power moment in the MMA definition is
not related to k-factor interactions.
Table 14. Comparison of Designs in Example 2 Using MMA
MMA
( K ( d ) , K ( d ) , K ( d ) , K ( d ) , K ( d ))
d3
d4
(2.1, 5.3, 14.8, 43.9, 134.8, 427.1)
(2.1, 5.0, 13.0, 35.9, 103.9, 312.3)
1
2
3
4
5
2.4.4 Moment Aberration Projection Criterion
In order to address the drawback that k-th power moment is not corresponding to kfactor interactions, Xu and Deng (2005) proposed a criterion, called the moment aberration
projection (MAP). MAP uses the coincidence matrix for all factor projections. For a given k
⎛m⎞
( 1 ≤ k ≤ m ), there are ⎜ ⎟ k-factor projections. The frequency distribution of Kk-values of
⎝k⎠
these projections is called the k-dimensional K-value distribution and is denoted by Fk(d).
For two designs d1 and d2, suppose that r is the smallest integer such that the r-dimensional
K-value distributions are different, that is, Fr(d1) ≠ Fr(d2). Hence, d1 is said to have less
MAP than d2 if Fr(d1) < Fr(d2). Moreover, the criterion of MAP was developed for twolevel non-regular designs and it also can be used in multi-level and mixed-level designs.
For the designs in Example 2, the one-, two-, three-, four-, five-factor projections
and their K-values are shown in Table 15 and Table 16. The summarized K-value
frequency distributions are given in Table 17. According to F2, design d4 is better than d3.
MAP values are also generated for Example 1 (Table 18) to compare designs d1 and d2.
24
Table 15. All k-Factor Projections and Their K-Values for d3
k=1
K1
k=2
K2
k=3
K3
k=4
K4
k=5
K5
A
18
AB
60
ABC
270
ABCD
1252
ABCDE
8898
k-Factor Projections
K-Value
B
C
D
E
30
30
30
30
AC
AD
AE
BC
BD
60
60
60
88
84
ABD ABE
ACD
ACE
ADE
222
222
222
222
294
ABCE ABDE ACDE BCDE
1252
1408
1408
2180
BE
84
BCD
342
CD
84
BCE
342
CE
84
BDE
402
DE
100
CDE
402
CE
84
BDE
330
DE
84
CDE
330
Table 16. All k-Factor Projections and Their K-Values for d4
k=1
K1
k=2
K2
k=3
K3
k=4
K4
k=5
K5
A
18
AB
60
ABC
246
ABCD
0052
ABCDE
6858
k-Factor Projections
K-Value
B
C
D
E
30
30
30
30
AC
AD
AE
BC
BD
60
60
60
84
84
ABD ABE
ACD
ACE
ADE
222
222
222
222
246
ABCE ABDE ACDE BCDE
1152
1152
1152
1728
BE
84
BCD
330
CD
84
BCE
330
Table 17. Frequency Distribution of Kk -Values of Factor Projections for Example 2
Frequency Distribution
d3
d4
F1: (30, 18)
(4, 1)
(4, 1)
F2: (100, 88, 84, 60)
(1, 1, 4, 4)
(0, 0, 6, 4)
F3: (402, 342, 330, 294, 270, 246, 222) (2, 2, 0, 1, 1, 0, 4) (0, 0, 4, 0, 0, 2, 4)
F4: (2180, 1728, 1408, 1252, 1152)
(1, 0, 2, 2, 0)
(0, 1, 0, 0, 4)
F5: (8898, 6858)
(1, 0)
(0, 1)
k-Factor Projections: (K-Values)
25
Table 18 Frequency Distribution of Kk -Values of Factor Projections for Example 1
k-Factor Projections: (K-Values)
F1: 30
F2: 84
F3: 330
F4: 1728
F5: (11070, 10950)
Frequency Distribution
d2
d1
5
5
10
10
10
10
5
5
(1, 0)
(0, 1)
Unlike MMA which uses all moments for the whole design, MAP uses k moments
for k-factor projections. As a result, k-factor projections correspond to k-factor interactions.
Therefore, MAP reflects the interaction alias structure better than MMA. However, MAP
uses a distribution vector instead of a single value, making itself more cumbersome in
terms of application.
2.4.5 Additional Minimum Aberration Definitions
In addition to the definitions already discussed, there are two other minimum
aberration criteria proposed in literature, which will be briefly introduced here. The first
criterion is the minimum generalized aberration (MGA) of Ma and Fang (2001), and Fang,
Ge, Liu and Qin (2003), which is based upon code theory. This criterion can be used for
multi-level designs. For a p-level design d, let
Ek ( d ) = n −1 {( c, d ) : d H (c, d ) = k , c, d ∈ D} , for k =0, …, m,
where dH(c, d) is the hamming distance between two runs c and d, which is the number of
places where they differ. The vector (E0(d), …, Em(d)) is called the distance distribution of
d. In algebraic coding theory, hamming distance can be calculated by n− δ ij , where δ ij is
the coincidence between two rows in the criterion of MMA.
The vector ( A1g ( d ), A2g ( d ),
, Amg ( d ) ) is called the generalized word length pattern,
where
26
A (d ) = [ n(q − 1)]
−1
g
i
m
∑ P ( j; m) E (d ) , i=1, …, m
j =0
i
j
⎛ k ⎞⎛ m − k ⎞
j
and Pj (k ; m) = ∑ r =0 (−1) r (q − 1) j − r ⎜ ⎟ ⎜
⎟ is the Krawtchouk polynomial. The MGA
⎝ r ⎠⎝ j − r ⎠
criterion is to sequentially minimize Aig ( d ) for i=1, …, m. The MGA was proposed for
multi-level designs. The mathematical principles behind MGA are theoretical and its
application can be covered by the other criteria. However, for readers who are interested in
combining criteria of minimum aberration and uniformity, Ma and Fang (2001) is a good
paper to review.
One final measure to mention is the general criterion of minimum aberration
(GCMA) of Cheng and Tang (2005). For two-level regular fractional factorial designs, γ 0
is the overall mean, and γ 1 is a set of effects to be estimated. The fitted model is then
Y = γ 0 I + W1γ 1 + ε ,
where Y is the vector of responses, W1 is the model matrix corresponding to γ 1 and ε is
the vector of uncorrelated random errors. Besides γ 1 , the remaining effects may not be
negligible. Suppose that these remaining effects can be divided into J-1 groups, γ 2 , … , γ J ,
via previous experience. However, these effects groups have to be ordered in such a way
that the effects in γ j are more important than those in γ j +1 for j = 2, … , J − 1 . The true
model is
Y = γ 0 I + W1γ 1 + W2γ 2 +
+ WJ γ J + ε ,
where W j is the model matrix corresponding to γ j for j = 1,
,J .
The general criterion of minimum aberration is defined as sequentially minimizing
a vector of
( N 2 ,…, N J ) ,
those in γ 1 , for j = 2,
where N j is the number of effects in γ j that are aliased with
, J . If γ j are used for the j-factor interactions, the relationship of
general criterion of minimum aberration and the usual minimum aberration can be
established as
N j = ( j + 1) Aj +1 + ( m − j + 1) Aj −1
for j = 2, … , m − 1 , where m is the number of factors and N m = Am −1 .
27
This criterion is established aiming at unifying different versions of MA criteria. It
is true that GCMA can be reduced to a more practical criterion such as GMA. However, it
is not convenient to make use of a criterion, which needs to satisfy model assumptions. For
preliminary readers whose goal is to find appropriate criterion to evaluate two-level nonregular designs, GCMA is not recommended. However, for advanced readers, we suggest
them referring the original paper for more details.
2.5 Conclusions
Minimum aberration criteria have been used to compare two-level regular
fractional factorial designs. However, it is theoretically difficult to extend the usual
minimum aberration definition to two-level non-regular, multi-level or mixed-level design
situations. Therefore, many statisticians have placed their efforts in developing new
minimum aberration criteria for handling these situations. This chapter reviews and
compares existing definitions with the intent to introduce these definitions to engineers and
industrial scientists. Table 19 provides a summary of some features for applying these
minimum aberration criteria.
Table 19. Features of Minimum Aberration Criteria
Contrast
Features
Coefficient
Uses frequency vector of J k ( d ) for each k
Yes
MA
Notation
MGA
F
MG2A
B
Yes
GMA
A
Yes
MMA
K
No
MAP
F
No
MGAC
Ag
No
GCMA
N
Yes
Uses sum of squares of the normalized J k ( d )
Needs orthonormal contrast coefficients for factors
Equals MG2A when evaluating two-level designs
Uses coincidence relations between rows
But K k (d ) does not reflect factor interactions
Uses coincidence matrix for each k-factor subset
Corresponds to k-factor interactions
Uses frequency vector of K-value for each k
Complicated mathematical principles
Connected to uniformity
A general MA that can derive some other MA criteria
Assume prior knowledge of effects to rank
γ 1, γ 2 , γ J
28
In terms of evaluating two-level designs, the MGA and related the MG2A criteria
are appropriate for both regular and non-regular designs. The MG2A is simpler and easier
to compute than MGA. The GMA is equivalent to MG2A for two-level designs. However,
the GMA criterion also applies to multi-level or mixed-level designs, if contrast
coefficients are used. The MMA is another criterion that can be used to evaluate multi- or
mixed-level designs. The MMA criterion does not require contrast coefficients. The MAP
criterion provides more detailed information than the MMA criterion but the MAP criterion
is more complicated. The relationships between MA criteria are shown in Table 20.
Table 20 Relationship Summery between Minimum Aberration Criteria
MA Criteria
MGA
MG2A
MG2A
GMA
MMA
MAP
MAP
MGA
MGAC
MMA
Relationships
Both use J characteristics of k factor interaction. MGA provides
frequency but MG2A is sum of squares of normalized J
Exact same for two-level designs
Both depend on coincidence matrix. MMP counts frequency of kfactor interactions but MMA sums k-power
Both are represent as frequency vectors
MGAC uses Hamming distance d H and MMA uses coincidence δ ,
where d H = m − δ
In conclusion, minimum aberration discriminates many types of fractional factorial
designs based upon their alias relations. The proposition of different MA criteria satisfies
different design situations. Once better understood, these MA criteria can be widely used to
help practitioner for the real problem solving.
29
CHAPTER 3
THE GENERAL BALANCE METRIC FOR
FRACTIONAL FACTORIAL DESIGNS
3.1 Introduction
Many industrial experiments involve factors with more than two levels, especially
when some factors are qualitative in nature. Factorial designs containing factors with
different numbers of levels are called mixed-level factorial designs. For mixed-level
factorial designs, the number of runs increases quickly as the number of factors and/or
number of factor levels increases. Mixed-level fractional factorial designs should be
considered when a full factorial is not affordable. One common property for traditional
mixed-level fractional factorial designs is balance, which indicates that in each column all
factor levels appear equally often. Balanced designs are also called U-type designs in the
literature (Fang et al. (2000), Fang et al. (2003a), Fang et al. (2003b), Fang et al. (2004)).
In balanced designs, there is consistency in the variances of the difference of observations
at pairs of treatment combinations.
Most of the mixed-level design literature considers only balanced designs.
Orthogonal or near-orthogonal balanced designs are constructed by Nguyen (1996),
DeCock and Stufken (2000), Wang and Wu (1991, 1992), Wang (1996), Xu (2002) and
Hedayat, Solane and Stufken (1999). Xu and Wu (2005) discuss optimal supersaturated
designs that are strictly balanced designs. Minimum aberration mixed-level designs are
also balanced (Xu and Wu (2001), Cheng, Steinberg and Sun (1999), Deng and Tang
(1999), Franklin (1984), Wu and Zhang (1993), Ankenman (1999) and Mukerjee and Wu
(2001)).
For unbalanced mixed-level fractional factorial designs, the degree of balance was
evaluated using a balance coefficient, (Guo (2003)). However, this balance coefficient was
proposed only for model main effects. In next section, we generalize this balance
coefficient and propose a new metric, which will measure the degree of balance of mixedlevel factorial designs beyond the main effects and can be used to compare different
30
mixed-level fractional factorial designs for modeling quality. The discussion then focuses
on the features of this generalized balance measurement and provides a comparison with
existing criteria. We demonstrate that a relationship exists between the new criterion and
the word length pattern often used in aberration assessment for two-level fractional
factorials. Finally, some examples are provided to illustrate the application of this new
definition.
3.2 General Near-balanced Designs
For traditional mixed-level factorial designs, the concept of balance only pertains to
the main effect factor columns. We consider generalizing the definition of balance to also
include interaction effects. A t-factor interaction is said to be balanced if all level
combinations associated with those factors appear equally often. Levels of t-factor
interactions can be coded according to factor level combinations.
Coding t-factor interactions
In mixed-level designs, the factor levels are commonly coded as “1, 2, 3…”,
representing the first-, second-, third-… level of that factor. In general, the levels for factor
interactions can be defined by the sequence of combinations in standard order of all factors.
t
For t factors with l1, l2, …, lt levels, then all possible t factor level combinations is
∏l .
i =1
i
The standard order arrangement is also called Yate’s order (Montgomery (2001)). For
example, suppose factor A has two levels and factor B has three levels. By definition, the
column of interaction AB has six levels. The combinations of these two factors in standard
order are shown in Table 21.
Table 21. An Example of Coding Mixed-Level Factor Interactions
Standard
A B AB
Order
1
1 1 1
2
1 2 2
3
1 3 3
4
2 1 4
5
2 2 5
6
2 3 6
31
For an n × k design matrix d, n is the number of rows and k is the number of
factors. Let d t ( t = 1,
, k ) denote matrices including all t-factor interaction columns, and
d 1 is the one-factor-interaction matrix for the main effects. Note that d 1 is equivalent to d.
Therefore, the whole interaction matrix involves all t-factor interaction matrices d t . That
is,
D = ⎡⎣ d 1
d2
dt
d k ⎤⎦ .
Columns in D are called interaction columns, which can be assessed in terms of balance.
Balanced columns contain all the levels equally often. Columns whose levels do not
appear equally often are called unbalanced. Among unbalanced columns, the concept of
near-balance denotes that while not all levels appear equally often, due to design size
limitations, all levels appear as equally often as possible. Therefore, both balance and nearbalance designs are considered to have optimal balance status given the constraint on the
number of runs. An unbalanced column is considered not near-balanced when it is neither
balanced nor near-balanced. Figure 6 shows the relationship of these concepts.
Balance
Optimal
Near-balance
Unbalance
Not near-balance
Figure 6 Balance relationship among types of columns.
A design matrix d can be classified by the degree of balance of columns in D.
General balanced designs are designs in which every column in D is balanced. Only the
full factorial design is general balanced. In fractional factorial designs, not all columns can
reach balance simultaneously. If some columns are near-balanced, the designs are called
general near-balanced designs (GNBD). The degree of general balance for mixed-level
fractional factorial designs can be evaluated by a new criterion, which is called general
balance metric.
32
General Balance Metric
Let l tj be the number of levels of the jth column in d t ( 1 ≤ t ≤ k ). Let crjt be the
T
number of times the r th level appears in the j th column of d t . Let c tj = ⎡⎢c1t j , c2t j ,..., cltt j ⎤⎥ be
j ⎦
⎣
the counts for each level for the j th column of d t .
The notation H t is used for the balance coefficients of d t . We employ a distance
function to reflect the degree of balance and define the jth column balance coefficient as
l tj
H = ∑ ( crjt − T jt ) 2
t
j
r =1
for the k-factor interaction matrix, where T jt =
n
n
is fixed. Substituting T jt = t , then H tj
t
lj
lj
becomes
2
⎛
n⎞
H = ∑ ⎜ crjt − t ⎟ .
⎜
l j ⎟⎠
r =1 ⎝
l tj
t
j
This balance coefficient measures two aspects of the degree of balance for interaction
columns: 1) equality of frequencies of the levels and 2) completeness of the levels.
The balance coefficients H t for d t just sum the H tj and are defined as
⎛k ⎞
⎜ ⎟
⎝t⎠
⎛k⎞
⎜ ⎟ lt
⎝t⎠ j
2
⎛
n⎞
H = ∑ H = ∑∑ ⎜ crjt − t ⎟ .
⎜
l j ⎟⎠
j =1
j =1 r =1 ⎝
t
t
j
Then, the general balance metric (GBM) can be defined as
GBM = ( H 1 , H 2 ,
,Ht,
,Hk ).
For two designs d1 and d2, suppose r is the smallest value such that H r (d1) ≠ H r (d2). We
say that d1 is more general balanced than d2 if H r (d1) < H r (d2). If no design is more
general balanced than d1, then d1 is said to be the most general balanced design. Therefore,
a GNBD is a mixed-level design in which H t are sequentially optimized.
As an example, consider a mixed-level design (Table 22) involving two factors
with 2 levels and one factor with three levels in a total of 6 runs, denoted by (6, 2231). The
level counts c tj , for d 1 , d 2 and d 3 are also given. This design is optimal under the
33
criterion of the general balance metric by observing c tj . This design is a GNBD because
the BC and ABC interactions are near-balanced and A, B, C, AB and AC are balanced. In
this example, the GBM is (0, 1, 3).
Table 22. A Mixed-Level Design (6, 2231)
Interactions
dt
d1
A B C
1 1 2
1 2 1
2 1 2
2 2 1
3 1 1
3 2 2
⎡ 2 3 3⎤
⎢ 2 3 3⎥
⎢
⎥
⎢⎣ 2
⎥⎦
d2
AB AC BC
1 4 3
4 1 2
2 5 3
5 2 2
3 3 1
6 6 4
1
1 1⎤
⎡
⎢1 1 2 ⎥
⎢
⎥
⎢1 1 2 ⎥
⎢
⎥
⎢1 1 1 ⎥
⎢1 1
⎥
⎢
⎥
⎣1 1
⎦
c tj
H tj
Ht
GBM
0
0
0
0
0
0 1
1
(0, 1, 3)
d3
ABC
7
4
8
5
3
12
⎡0 ⎤
⎢0 ⎥
⎢ ⎥
⎢1 ⎥
⎢ ⎥
⎢1 ⎥
⎢1 ⎥
⎢ ⎥
⎢0 ⎥
⎢1 ⎥
⎢ ⎥
⎢1 ⎥
⎢0 ⎥
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎢0 ⎥
⎢⎣1 ⎥⎦
3
3
3.3 Features of General Balance Metric
The GBM criterion provides a measurement of orthogonality for mixed-level
designs. An orthogonal array of strength t denotes a matrix where, for any t columns, all of
34
the level combinations appear equally often (Rao (1947)). In a design matrix, if some level
combinations do not appear due to the small size of the design, the matrix is called a nearorthogonal array ((Taguchi (1959), Wang and Wu (1992), Nguyen (1996), Xu (2002)). If
the model matrix is specified, criteria such as D optimality (Wang and Wu (1992)) and
ave(s2) (Booth and Cox (1962)) can be used to assess the orthogonality of mixed-level
designs via contrast coefficients. The proposed GBM, however, does not need the contrast
coefficients. Other orthogonality criteria for mixed-level designs include J2 optimality (Xu
(2002)) and f-related statistics (Fang et al. (2000)). These two criteria were defined for
balanced designs and can be used to evaluate design othgonality of strength 2. However,
the GBM criterion reveals a more complete assessment of the orthogonality property. Thus,
the GBM also establishes a connection between orthogonality and aberration for mixedlevel designs.
Because general near-balanced designs are optimal for each d t sub-matrix, these
GNBDs have minimum aberration. Therefore, GBM can be used as a minimum aberration
criterion for mixed-level fractional factorial designs. Thus, GNBDs enjoy all of the robust
properties of minimum aberration designs (Cheng, Steinberg and Sun (1999) and Tang and
Deng (1999)). After Fries and Hunter (1980) proposed the concept of minimum aberration
as a way of selecting the best fractional factorial designs, much work has been done on
two-level fractional factorial designs. Besides Fries and Hunter (1980), others who
contributed to developing two-level fractional factorial designs are Chen and Cheng (1999),
Chen and Hedayat (1996), Chen (1992), Cheng, Steinberg and Sun (1999), Sitter, Chen
and Wu (1997), Tang and Wu (1996) and Xu and Deng (2005). Franklin (1984) and Suen,
Chen and Wu (1997) extended this criterion of minimum aberration to p-level fractional
factorial designs. Xu and Wu (2001) proposed a generalized minimum aberration (GMA)
criterion for asymmetrical fractional factorial designs.
The concept of minimum aberration has a wide application in all types of
experimental designs. Huang, Chen and Voelkel (1998) and Bingham and Sitter (1999)
have proposed minimum aberration two-level fractional factorial split-plot designs. Deng
and Tang (1999) proposed generalized resolution and minimum aberration criteria for
Plackett-Burman and other nonregular factorial designs. Wu and Zhu (2003) examined the
use of a minimum aberration criterion for design selection in robust parameter design. Xu
35
(2003) proposed minimum aberration for supersaturated designs. In terms of the
application of minimum aberration in mixed-level designs, Wu and Zhang (1993) and
Ankenman (1999) used minimum aberration designs in two-level and four-level mixedlevel designs. Mukerjee and Wu (2001) developed minimum aberration designs for two
types of mixed-level fractional factorials: (sr)×sn, and (sr1)×(sr2)×sn factorial.
Generalized minimum aberration has several definitions: minimum G-aberration
criterion (Deng and Tang (1999)), minimum G2 aberration criterion (Tang and Deng
(1999), Ingram and Tang (2005)), minimum generalized aberration criterion (Ma and Fang
(2001)), generalized minimum aberration (GMA) criterion (Xu and Wu (2001)), minimum
moment aberration (MMA) criterion (Xu (2003)), and a general criterion of minimum
aberration (Cheng and Tang (2005)). By contrast, the GBM criterion is easy to use and has
a practical interpretation. The GBM works with the design matrix and thus contrast
coefficients are not required. The GBM is easy to compute. In addition, the GBM is defined
based upon coded interactions, which allows for clever development and augmentation of
mixed-level designs.
The word length pattern (Fries and Hunter (1980), Wu and Zhang (1993)) is an
important concept in the definition of minimum aberration two-level fractional factorial
designs. Therefore, it is useful to show the relationship between GBM and the word length
patterns for two-level fractional factorial designs.
Relationship with Word Length Pattern
There are two versions of word length pattern described in the literature (Fries and
Hunter (1980) and Wu and Zhang (1993)). Wu and Zhang (1993) define the vector
WLP ( d ) = ( A 1 ( d ) , A 2 ( d ) ,
, A t (d ),
, A k ( d ))
as the word length pattern (WLP) of the design d, where At ( d ) is the number of words of
length t in the defining contrast subgroup. The resolution of the design is defined as the
smallest t with non-zero At ( d ) in its word length pattern. For two designs d1 and d2,
suppose r is the smallest value such that Ar ( d1 ) ≠ Ar ( d 2 ) . Then d1 has less aberration than
d2 if Ar ( d1 ) < Ar ( d 2 ) . If no design has less aberration than d1, then d1 is said to have
minimum aberration.
36
The relationship between the GBM and the word length pattern in two-level
fractional factorial designs can be found by defining the number of unbalanced columns as
(
B ( d ) = M1 ( d 1 ) , M 2 ( d 2 ) ,
, Mt (dt ),
)
, Mk (d k ) ,
where M t ( d t ) is the number of unbalanced columns in d t . The generalized resolution
(GR) of mixed-level fractional factorial designs is defined to be the smallest t such that
M t ( d t ) > 0 . It turns out that the relationship of B and WLP for two-level fractional
factorial designs can be expressed with the following equation:
⎛
⎛ k − ( t − 1) ⎞
At = M t − ⎜⎜ M t −1 × ⎜
⎟−
1
⎝
⎠
⎝
⎞
p ⎟⎟
⎠
(1)
⎛ k − ( t − 1) ⎞
where p is the number of terms that are repeated in all the M t −1 × ⎜
⎟ terms. This
1
⎝
⎠
equation is established based upon the fact that any term (interaction or main effect) joined
with an unbalanced term will generate another unbalanced term. In two-level fractional
factorial designs, both WLP and GBM can be used to find designs with same aberration as
the usual word length pattern.
To demonstrate the use of GBM and WLP, consider three 27-2 designs with different
design generators (Table 23). Both WLP definitions are shown in this example but only the
Wu and Zhang version will be used in subsequent examples. In this example, WLP shows
that d1 has three 4-letter terms in generating relations, d2 has two 4-letter terms and one 6letter term in generating relations, d3 has one 4-letter term and two 5-letter terms in
generating relations. Therefore, d3 has less aberration than d2, and d2 has the less aberration
than d1. With B , we compare the number of unbalanced columns in a 4-factor interaction
matrix for the three design options because M 4 ( d 4 ) is the first nonzero number in B . It is
found that for d 4 , d3 has one unbalanced column, which is fewer than the other two
designs, (since there are two unbalanced columns in d2, and three unbalanced columns in
d1). Therefore, d3 is the most general balanced design among three designs. GBM also
shows d3 is the most general near-balanced design because the balance coefficient for d 4
is 64, which is smallest one among three design options. All three designs are resolution IV.
37
Table 23. Three 27-2 Design Options
Design
d1
d2
d3
Generators
F=ABC, G=BCD
F=ABC,G=ADE
F=ABCD, G=ABDE
Generating
I=ABCF=BCDG=ADFG I=ABCF=ADEG=BCDEFG I=ABCDF=ABDEG=CEFG
Relations
WLP*
(4, 4, 4)
(4, 4, 6)
(4, 5, 5)
**
WLP
(0, 0, 0, 3, 0, 0)
(0, 0, 0, 2, 0, 1)
(0, 0, 0, 1, 2, 0)
(0, 0, 0, 3, 9, 7)
(0, 0, 0, 2, 6, 7)
(0, 0, 0, 1, 5, 7)
B
(0, 0, 0, 192, 228, 144)
(0, 0, 0, 128, 192, 112)
(0, 0, 0, 64, 160, 112)
GBM
*
Fries and Hunter, 1980
**
Wu and Zhang, 1993
For design d2 in this example, the two unbalanced columns (terms) of d 24 are ABCF and
ADEG, which here are used to generate the six unbalanced terms of d 25 : ABCDF, ABCEF,
ABCFG, ABDFG, ACDEG, and ADEFG. These t=5 unbalanced interaction terms are
obtained by multiplying the t=4 term ABCF by D, E and G respectively, and the ADEG
term by B, C, and F respectively. This behavior is captured in (1) such that there are
⎛ 7 − ( 5 − 1) ⎞
2×⎜
⎟ = 6 unbalanced columns. In this case p=0, since no terms are replicated.
1
⎝
⎠
Accordingly
⎛ ⎛ 3⎞ ⎞
A5 = 6 − ⎜ 2 × ⎜ ⎟ − 0 ⎟ = 0
⎝ ⎝1⎠ ⎠
.
Similarly
it
can
be
shown
that
⎛ ⎛ 2⎞ ⎞
A6 = 7 − ⎜ 6 × ⎜ ⎟ − 6 ⎟ = 1 .
⎝ ⎝1⎠ ⎠
3.4 More Examples
The relationship among GBM, aberration, and WLP is further illustrated in the
following examples. In each example two more criteria, the generalized minimum
aberration and the minimum moment aberration, are used along with GBM for the purpose
38
of comparison. For the previous example (Table 23), the values of GMA and MMA are
calculated and displayed in Table 24. The first nonzero number in GMA is in the fourth
position. Because design d3 has a smaller fourth number for GMA than the other two
designs, d3 has the minimum aberration among three designs. With MMA, the first unequal
number is also the fourth one and d3 has the minimum aberration among three designs.
This example shows GBM working consistently with both GMA and MMA in two-level
fractional factorial designs.
Table 24. Comparison of Three 27-2 Design Options
Design
WLP
B
GBM
GMA
d1
(0, 0, 0, 3, 0, 0)
(0, 0, 0, 3, 9, 7)
(0, 0, 0, 192, 228, 144)
(0, 0, 0, 0.19)
(3.4, 12.9, 52.2, 223.5,
1006.0, 4735.5)
MMA
d2
(0, 0, 0, 2, 0, 1)
(0, 0, 0, 2, 6, 7)
(0, 0, 0, 128, 192, 112)
(0, 0, 0, 0.12)
(3.4, 12.9, 52.2, 221.9,
978.9, 4437.4)
d3
(0, 0, 0, 1, 2, 0)
(0, 0, 0, 1, 5, 7)
(0, 0, 0, 64, 160, 112)
(0, 0, 0, 0.06)
(3.4, 12.9, 52.2, 220.4,
959.5, 4278.7)
The next example considers two orthogonal array designs OA(12, 243), shown in
Figure 7. These two designs are compared under B , GBM, GMA, and MMA, in Table 25.
It can be seen from B that d1 has two unbalanced 2-factor interactions while all 2-factor
interactions
of
d2
are
balanced.
In
this
case,
H 1 ( d1 ) = H 1 ( d 2 ) = 0
but
H 2 ( d1 ) = 20 > H 2 ( d 2 ) = 0 , so d2 is better than d1 under B and GBM. Both GMA and
MMA show that d2 has less aberration than d1.
39
d1
d2
Figure 7. Two OA(12, 243) designs.
Table 25 Comparison of Two OA(12, 243) Designs
Design
GR
B
GBM
GMA
MMA
d1
II
(0, 2, 6, 5, 1)
(0, 20, 44, 31, 9)
(0, 0.14, 0.12, 0.03)
(2.1, 5.3, 14.8, 43.9, 134.8, 427.1)
d2
III
(0, 0, 6, 5, 1)
(0, 0, 24, 29, 9)
(0, 0, 0.17, 0.06)
(2.1, 5.0, 13.0, 35.9, 103.9, 312.3)
The last example considers three OA(12, 244) designs in Figure 8. These three
designs are constructed by the method of replacement (Wu and Zhang (1993)). Table 26
compares these three designs using several criteria. The first criterion used is WLP.
Because A1 and A2 are zero for all three designs, but A3 ( d1 ) = 1 < A3 ( d 2 ) = A3 ( d3 ) = 2 , d1
has less aberration than both d2 and d3. Design d2 has less aberration than d3
because A4 ( d 2 ) = 0 < A4 ( d3 ) = 1 . The same conclusion can be drawn by using GMA and
MMA. Under criterion of B and GBM, d1 is the most general near-balanced design among
the
three
designs
because
H1 = H 2 = 0
for
all
three
designs,
but
H 3 ( d1 ) = 1 < H 3 ( d 2 ) = H 3 ( d3 ) = 2 . Design d2 is more general near-balanced than d3
because H 4 ( d 2 ) = 4 < H 4 ( d3 ) = 5 . All criteria rank the three designs consistently.
40
d1
d2
4
d3
Figure 8. Three OA(12, 2 4) designs.
Table 26 Comparison of Three OA(12, 244) designs
Design
WLP*
GR
B
GBM
GMA
d1
d2
(0, 0, 1, 2, 0)
(0, 0, 2, 0, 1)
III
III
(0, 0, 1, 4, 1)
(0, 0, 2, 4, 1)
(0, 0, 16, 32, 12)
(0, 0, 32, 32, 12)
(0, 0, 0.0681, 0.0781)
(0, 0, 0.1305, 0.0312)
(2.1, 5.0, 12.9,
(2.1, 5.0, 13.3,
MMA
34.6, 96.1, 273.0)
37.0, 106.1, 309.0)
* Method of replacement, Wu and Zhang (1993).
d3
(0, 0, 2, 1, 0)
III
(0, 0, 2, 5, 1)
(0, 0, 48, 48, 12)
(0, 0, 0.1930, 0.0368)
(2.1, 5.0, 13.7,
41.0, 132.1, 449.0)
3.5 Conclusion
Based upon a mechanism of coding factor interactions in mixed-level designs, the
general balance metric measures the degree of balance for both main effects and
interaction effects. The GBM also serves as an orthogonality criterion, and goes beyond
order two to provide a complete orthogonality assessment. In addition, the GBM can be
used as a minimum aberration criterion, and performs consistently with other minimum
aberration criteria definitions, such as the GMA and MMA. The B metric, derived from the
GBM, is used as a supplementary criterion primarily to show the relationship with WLP for
two-level fractional factorial designs. With B , the concept of resolution is generalized
41
because the whole interaction matrix D corresponding to all potential model terms is
considered. Because GBM not only incorporates the number of unbalanced columns shown
in B , but also reflects the degree of unbalance within each column, it is suggest that the
GBM be used to compare designs of similar quality.
Although more study is necessary, the GBM has a potential capability to dominate
all orthogonality and minimum aberration criteria in terms of comparing and evaluating
mixed-level fractional factorial designs. The GBM can also be applied if augmentation of
mixed-level designs is desired. The GBM does not require contrast coefficients and hence
is straightforward to use in practice.
42
CHAPTER 4
OPTIMAL FOLDOVER PLANS FOR MIXED-LEVEL
FRACTIONAL FACTORIAL DESIGNS
4.1 Introduction
Fractional factorial designs are widely used by practitioners for screening
experiments. However, one consequence of using fractional factorial designs is the alias
structure of main effects or interactions. A common method to dealias is to add more runs.
Foldover of a fractional factorial design is a quick technique to create a design with twice
as many runs, which typically releases aliased factors or interactions.
A standard approach to fold over two-level fractional factorial designs is to reverse
the plus minus signs of one or more columns of the original design (Box et al. (1978),
Montgomery (2005), Wu and Hamada (2000)). Montgomery and Runger (1996) discussed
foldovers of two-level resolution IV designs considering switching the signs of one or two
factors. Li and Mee (2002) emphasized foldovers of two-level resolution III designs. Li,
Lin and Ye (2003) gave a complete discussion of regular two-level designs and provided
optimal foldover plans using an exhaustive search method. The criterion they used was the
aberration (Fries and Hunter (1980)) of the combined designs, which included the original
design and its foldover. Li and Lin (2003) extended their method to nonregular two-level
designs. Miller and Sitter (2001) also studied foldovers of Plackett-Burman designs, a type
of nonregular designs. Recently, Miller and Sitter (2005) examined the use of
nonorthogonal foldover designs.
In some situations however, investigating foldovers for mixed-level designs are
required. Mixed-level factorial designs contain factors with different numbers of levels.
Therefore, the foldover technique for two-level designs cannot be directly applied to
mixed-level designs. There is a rich literature on the construction of mixed-level designs.
Wang and Wu (1991), and Wang (1996) proposed an approach for constructing orthogonal
mixed-level designs based upon difference matrices. DeCock and Stufken (2000) proposed
an algorithm for constructing orthogonal mixed-level designs using existing two-level
43
orthogonal designs. Wang and Wu (1992) and Nguyen (1996) constructed near-orthogonal
mixed-level designs. Xu (2002) proposed an algorithm to construct orthogonal and nearorthogonal designs based on the concept of J2-optimality (Xu 2002).
The theory of
orthogonal designs was systematically discussed by Hedayat, Sloane and Stufken (1999).
Guo (2003) developed genetic algorithms to construct efficient fractional factorial mixedlevel designs.
In this chapter, a method is proposed to fold over mixed-level designs. This chapter
begins with a discussion of a strategy of extending the “reversing signs” method to multilevel factors. Then a general method is developed to find optimal foldovers for mixeddesigns. The method is used to present optimal foldover plans for commonly used mixedlevel designs. The last section provides some concluding remarks and suggestions for
follow on research.
4.2 Foldover Strategy for Multi-Level Factors
The method of “reversing signs” loses its meaning when the original designs
involving factors with more than two levels. For an n × k design matrix d, let dij be the
elements of ith row and jth column. Let l j represent factor levels. The factor levels are
commonly coded as “1, 2, 3…”, representing the first-, second-, third-… level of that
factor.
One method for folding over factors with more than two levels is to rotate the
factor levels. For a column with l levels, factor levels are rotated by replacing the i-th level
by i + p if i ≤ l − p and by i + p − l if i > l − p , where 1 ≤ p ≤ l − 1 . For example, Figure 9
shows the rotation of a five-level factor column using different p values.
p
Factor Levels
1
2
3
4
5
Rotate
→
1
2
3
4
5
1
Figure 9 Rotate a five-level factor.
44
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
The rotation method changes every factor level to a different level. For the original
fractional factorial design d, let d' represent its foldover, in which factor levels in one or
more columns are rotated. The design combining the original design d and its foldover d'
is denoted by [ d ; d' ] .
4.3 General Balance Metric
The general balance metric (Guo, Simpson, and Pignatiello (2005)) can be
employed to evaluate combined designs. In mixed-level designs, the factor levels are
commonly coded as “1, 2, 3…”, representing the first-, second-, third-… level of that
factor. The levels for factor interactions can be defined by the sequence of combinations in
standard order of all factors. Therefore, the interaction of t factors (with l1, l2, …, lt levels)
t
has
∏l
i =1
i
levels.
Suppose a design d has n rows and k columns. Let d t ( t = 1,
, k ) denote
matrices including all t-factor interaction columns, so d 1 is the one-factor-interaction
matrix for the main effects. Note that d 1 is equivalent to d. Therefore, the whole
interaction matrix involves all t-factor interaction matrices d t . That is,
D = ⎡⎣ d 1
d2
dt
d k ⎤⎦ .
Let l tj be the number of levels of the jth column in d t ( 1 ≤ t ≤ k ). Let crjt be the number of
T
times the r th level appears in the j th column of d t . Let c tj = ⎡⎢c1t j , c2t j , ..., cltt j ⎤⎥ be the
j ⎦
⎣
counts for each level for the j th column of d t . The j th column of d t can be described as
⎧= 0 Balanced
⎪
max ( c ) − min ( c ) ⎨ = 1 Near - balanced .
⎪≥ 2 Non - balanced
⎩
t
j
t
j
The notation H t is used for the balance coefficients of d t . The balance
coefficients for t-factor interactions are defined as
45
⎛k ⎞
⎜ ⎟
⎝t⎠
⎛k⎞
⎜ ⎟ lt
⎝t⎠ j
2
⎛
n⎞
H = ∑ H = ∑∑ ⎜ crjt − t ⎟ .
⎜
l j ⎟⎠
j =1
j =1 r =1 ⎝
t
t
j
For fixed t, the values of H t decreases as n increases. Then, the general balance metric
(GBM) can be defined as
GBM = ( H 1 , H 2 ,
,Ht,
,Hk ).
For two designs d1 and d2, suppose r is the smallest value such that H r (d1) ≠ H r (d2).
Design d1 is more general balanced than d2 if H r (d1) < H r (d2). If no design is more
general balanced than d1, then d1 is said to be the most general balanced design. Therefore,
a general near-balanced design is a design in which H t are sequentially minimized. As a
supplementary criterion, the number of non-balanced columns is defined as
(
B ( d ) = M1 ( d 1 ) , M 2 ( d 2 ) ,
, Mt (dt ),
)
, Mk (d k ) ,
where M t ( d t ) is the number of non-balanced columns in d t .
To illustrate the use of GBM, consider a mixed-level design (Table 27) involving
two factors with 2 levels and one factor with three levels in a total of 6 runs, denoted by (6,
3122). The level counts c tj , for d 1 , d 2 and d 3 are also given in Table 22. This design is
optimal under the criterion of the general balance metric by observing c tj . In this example,
the GBM is (0, 0.0093, 0.0833) and B = ( 0, 0, 0 ) .
46
Table 27. A Mixed-Level Design (6, 3122)
Interactions
dt
d1
A B C
1 1 2
1 2 1
2 1 2
2 2 1
3 1 1
3 2 2
⎡ 2 3 3⎤
⎢ 2 3 3⎥
⎢
⎥
⎢⎣ 2
⎥⎦
d2
AB AC BC
1 4 3
4 1 2
2 5 3
5 2 2
3 3 1
6 6 4
⎡1 1 1 ⎤
⎢1 1 2 ⎥
⎢
⎥
⎢1 1 2 ⎥
⎢
⎥
⎢1 1 1 ⎥
⎢1 1
⎥
⎢
⎥
⎣1 1
⎦
ctj
H tj
t
H
Mt
GBM
B
0
0
0
0
0
0
d3
ABC
7
4
8
5
3
12
⎡0 ⎤
⎢0 ⎥
⎢ ⎥
⎢1 ⎥
⎢ ⎥
⎢1 ⎥
⎢1 ⎥
⎢ ⎥
⎢0 ⎥
⎢1 ⎥
⎢ ⎥
⎢1 ⎥
⎢0 ⎥
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎢0 ⎥
⎢⎣1 ⎥⎦
0 0.0093 0.0833
0.0093
0.0833
0
0
(0, 0.0093, 0.0833)
(0, 0, 0)
4.4 Optimal Foldovers of Mixed-Level Designs
With the criterion of GBM, the optimal foldover plans can be found by searching
all foldover alternatives. The foldover is a more computationally efficient technique for
augmenting fractional factorial designs compared to searching for additional runs from the
full factorial. Searching the full factorial guarantees the optimal augmentation, but it may
not be practical. For example, suppose one is interested in augmenting a relatively small
design (15, 315171) using additional 15 runs. Since the full factorial contains 105 runs,
47
⎛ 105 − 15 ⎞
16
there are ⎜
⎟ ≈ 4.58 × 10 possible alternatives for the complete exhaustive search.
⎝ 15 ⎠
However, if the rotate method is used to foldover the original design, there are 104
alternatives. Therefore, the rotation foldover strategy is computationally feasible while
searching the full factorial may be prohibitive.
The search algorithm used in this chapter covers any column combinations with
any p. Therefore, it is an exhaustive search associated with rotation method. Consider
again the (6, 3122) mixed-level design example and let A, B, and C be the factor names.
The superscript represents p, which is omitted if factor levels equal 2. Table 28 provides all
foldover alternatives and the corresponding evaluation of the combined designs. It turns
out that foldovers with p=1 for A are equivalent to foldovers with p=2. p is the degree of
rotation. The design (6, 3122) is a half fractional factorial design. Therefore, the best
combined design should be the full factorial design, and the optimal foldover is to rotate
factor B or C only (Table 28).
Table 28 All Foldover Alternatives for Design (6, 3122)
A1
Foldover Alternatives
B
C
GBM
B
A1B
(0.000, 0.009, 0.028)
(0.000, 0.000, 0.000)
(0.000, 0.000, 0.000)
(0, 1, 1)
(0, 0, 0)
(0, 0, 0)
1
AC
BC
A1BC
(0.000,0.0000, 0.0556)
(0, 0, 1)
(0.000, 0.000, 0.056)
(0, 0, 1)
(0.000, 0.009, 0.083)
(0, 1, 1)
(0.0000,0.009, 0.028)
(0, 1, 1)
A2
A2B
A2C
A1BC
(0.000, 0.009, 0.028)
(0, 1, 1)
(0.000,0.0000, 0.0556)
(0, 0, 1)
(0.000, 0.000, 0.056)
(0, 0, 1)
(0.0000,0.009, 0.028)
(0, 1, 1)
One advantage of the method proposed in this chapter is that it can be used for
mixed-level designs. Efficient mixed-level designs (efficient array (EA)) are fractional
factorial designs whose factors have different numbers of runs (Guo, Simpson, and
Pignatiello 2004). Most EAs contain unbalanced columns due to the limitation of design
48
sizes. However, all EAs are generated to be optimal in terms of the balance property and
the orthogonality property. As an example, Figure 10 is an EA consisting of three factors,
one with three levels, one with five levels and one with seven levels. The smallest balanced
design would require all 105 runs. In this design, both of three-level and five-level factors
are balanced. The seven-level factor is not balanced but it is near-balanced, because it
contains three of second level and two of the other levels (Guo, Simpson, and Pignatiello
2004).
⎡1
⎢1
⎢
⎢1
⎢
⎢1
⎢1
⎢
⎢2
⎢2
⎢
d = ⎢2
⎢2
⎢
⎢2
⎢
⎢3
⎢3
⎢
⎢3
⎢3
⎢
⎣⎢ 3
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
7⎤
2 ⎥⎥
5⎥
⎥
4⎥
3⎥
⎥
2⎥
5⎥
⎥
6⎥
7⎥
⎥
1⎥
⎥
4⎥
1⎥
⎥
2⎥
3⎥
⎥
6 ⎦⎥
Figure 10 EA(15, 315171).
This EA is evaluated by GBM criterion (Table 29). It is a minimum aberration
design since B ( d ) has all zero values, which indicates all main effects, two-factor
interactions, and three-factor interactions are balanced or near-balanced.
Table 29. Statistics of EA(15, 315171)
GBM ( d )
(0.0013, 0.0180, 0.0568)
B (d )
(0, 0, 0)
49
The optimal foldover plan for this design is to rotate the first column (three-level) with p=2,
and the third column (seven-level factor) with p=1. For the combined design,
GBM ( d ; d' ) = (0.0005, 0.0057, 0.0238) and B ( d ; d' ) = (0, 1, 0). This combined design can
be compared with another design, EA(30, 315171) (Figure 11). The comparison results
(Table 30) shows that the combined design is not optimal, because the combined design
uses and is limited by the existing resource (original design). This conclusion can be
verified with foldover of regular two-level designs (Li and Lin (2003)).
Table 30. Comparison of the Combined Design with EA(30, 315171)
[ d ; d' ]
GBM ( d )
EA(30, 315171)
(0.0005, 0.0057, 0.0238) (0.0005, 0.0035, 0.0238)
B (d )
(0, 1, 0)
(0, 0, 0)
Non-balanced
Columns
BC
none
The rotation method can be applied for the optimal foldovers for EAs developed in
Guo, Simpson, and Pignatiello (2005). These EAs can be found in Appendix I. Since all
designs have less than ten factors, the exhaustive search method is employed to identify the
best foldover plans in terms of the general balance property of the combined designs. Table
31 shows the best foldovers and the general balance property of the original designs and
the combined designs. The general balance property of efficient designs is improved by
combining original designs with the best foldovers using the rotation strategy.
Foldover is a simple way to augment a fractional factorial design for the purpose of
releasing some aliased main effects or interactions. For those EAs, the alias structures are
complex, resulting in partial aliases. Additional details regarding partial alias structures are
discussed by Hamada and Wu (2000). The GBM and B criteria can assess designs in terms
of aberration properties, but neither criterion can identify the alias structure directly.
50
⎡1
⎢1
⎢
⎢1
⎢
⎢1
⎢1
⎢
⎢1
⎢1
⎢
⎢1
⎢1
⎢
⎢1
⎢
⎢2
⎢2
⎢
⎢2
⎢2
⎢
⎢2
D=⎢
2
⎢
⎢2
⎢
⎢2
⎢2
⎢
⎢2
⎢3
⎢
⎢3
⎢3
⎢
⎢3
⎢3
⎢
⎢3
⎢
⎢3
⎢3
⎢
⎢3
⎢3
⎣
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1⎤
7 ⎥⎥
4⎥
⎥
2⎥
6⎥
⎥
3⎥
6⎥
⎥
1⎥
5⎥
⎥
2⎥
⎥
4⎥
5⎥
⎥
2⎥
7⎥
⎥
1⎥
7⎥
⎥
3⎥
⎥
6⎥
3⎥
⎥
4⎥
2⎥
⎥
2⎥
5⎥
⎥
1⎥
7 ⎥⎥
6⎥
⎥
1⎥
3⎥
⎥
4⎥
5 ⎥⎦
Figure 11 EA(30, 315171).
51
Table 31. Optimal Foldover Plans for EAs in terms of GBM and B
d
EA (20, 243141)
EA (20, 243151)
EA (21, 26315171)
EA (30, 26315171)
EA (21, 325171)
EA (21, 314171)
EA (21, 324171)
EA (20, 235171)
EA (20, 245171)
EA (20, 314151)
EA (20, 23314151)
EA (20, 24314151)
EA (28, 236171)
EA (28, 246171)
EA (21, 316171)
EA (24, 416171)
EA (24, 516171)
EA (24, 31516171)
EA (20, 41516171)
GBM ( d )
B (d )
Optimal Foldovers
(0.0003, 0.0027, 0.0144)
(0, 0, 12)
(0.0003, 0.0014, 0.013)
(0, 0, 10)
(0.0010, 0.0039, 0.0165)
(0, 2, 48)
(0.0002, 0.0012, 0.0095)
(0, 1, 46)
(0.0005, 0.0067, 0.0356 )
(0, 0, 1)
(0.0006, 0.0057, 0.0357)
(0, 0, 0)
(0.0004, 0.0044, 0.0327)
(0, 0, 1)
(0.0004, 0.0044, 0.0221)
(0, 0, 4)
(0.0004, 0.0037, 0.0199)
(0, 0, 9)
(0.0006, 0.0050, 0.0328)
(0, 0, 0)
(0.0003, 0.0027, 0.0182)
(0, 0, 9)
(0.0002, 0.0023, 0.0162)
(0, 0, 17)
(0.0003, 0.0023, 0.0144)
(0, 1, 6)
(0.0003, 0.0018, 0.0126)
(0, 1, 11)
(0.0011, 0.0098, 0.0397)
(0, 0, 0)
(0.0010, 0.0075, 0.0357)
(0, 0, 0)
(0.0015, 0.0124, 0.0365)
(0, 0, 0)
(0.0011, 0.0090, 0.0329)
(0, 0, 0)
(0.0014, 0.0139, 0.0432)
(0, 0, 0)
E2F1
E2F4
ABCDEFH3
CEFG1H4I3
A2B1C2D3, A1B2C3D4
B1, B2, B3
A1C3
D4E3
DF6
A1
C1D2E2
AE1F2
AB1C1D3
D1E3F2
A2B3C3, A1B3C4
A3B1C3
A5B2C3
A1B5C2D4
B4C3D1
52
GBM ( d ; d' )
B ( d ; d' )
(0.0001,0.0001,0.0004)
(0, 1, 10)
(0.0001, 0.0004, 0.0058)
(0, 0, 15)
(0.0001, 0.0009, 0.0074)
(0, 3, 52)
(0.0001, 0.0005, 0.0043)
(0, 8, 69)
(0.0002, 0.0019, 0.0140)
(0, 1, 4)
(0.0002, 0.0019, 0.0119)
(0, 0, 0)
(0.0001, 0.0014, 0.0131)
(0, 0, 2)
(0.0002, 0.0015, 0.0122)
(0, 3, 8)
(0.0001, 0.0012, 0.0100)
(0, 5, 14)
(0.0001, 0.0013, 0.0081)
(0, 0, 0)
(0.0001, 0.0007, 0.0062)
(0, 2, 12)
(0.0001, 0.0005, 0.0050)
(0, 2, 18)
(0.0001, 0.0006, 0.0094)
(0, 0, 7)
(0.0001, 0.0006, 0.0056)
(0, 1, 15)
(0.0000, 0.0034, 0.0159)
(0, 1, 0)
(0.0001, 0.0023, 0.0148)
(0, 1, 0)
(0.0003, 0.0044, 0.0176)
(0, 2, 1)
(0.0002, 0.0035, 0.0147)
(0, 2, 4)
(0.0004, 0.0052, 0.0205)
(0, 3, 4)
4.5 Conclusions
In this chapter, the strategy is developed to fold over mixed-level fractional
factorial designs by rotating factor levels, which can be regarded as an extension of the
“reversing signs” methods for two-level designs. An exhaustive search technique is
employed to find the optimal foldover plans. Computationally, the search time is quite
reasonable. For designs contain ten factors, the average search time is about 5 minutes
under the MATLAB environment with a standard Pentium 4 PC. The minimum aberration
criterion used to evaluate the combined designs is the general balance metric. We also use
B , the number of non-balanced columns to assist in the evaluation.
The quality of the combined designs is determined by the original designs. Hence,
using optimal designs as the original designs is highly recommended. Examples of optimal
designs include orthogonal designs and minimum aberration designs. The combined
designs may not be optimal even though the original designs are optimal. This phenomena
is due to the limitation that half the combined designs runs are predetermined or restricted
to the original design.
The proposed approach can be applied to finding optimal foldovers for all types of
mixed-level designs. Optimal foldovers of EAs are provided as the preliminary results. All
the optimal foldover plans and their minimum aberration properties are tabulated and ready
for use. Foldovers based on unique mixed-level needs can be built in a matter of minutes.
53
CHAPTER 5
ANALYSIS OF MIXED-LEVEL EXPERIMENTAL
DESIGNS INCLUDING QUALITATIVE FACTORS
5.1 Introduction
In practice, some experimental designs may involve one or more qualitative factors.
For example, in a fire extinguishing experiment engineers are interested to model fire
extinguishing time against equipment flow rate and the fire size. The objective is to
establish the relationship between the extinguish time and the flow rate and fire size.
However, this experiment also considers two qualitative factors. The first qualitative factor
is surface, on which the fires are fought. There are three different types of surfaces: water,
gravel, soil/sod. Another qualitative factor is the fire system. There are four different fire
fighting systems involved in this experiment. Therefore, the influence of those two
qualitative factors and their interactions with flow rate and fire size on extinguish time
becomes of interest. The current literature suggests considering the interactions between
quantitative factors and interactions between quantitative and qualitative factors. However,
interactions between qualitative factors are always ignored.
In the next section, a common method is reviewed to analyze qualitative factors
through indicator variables. The contrast coefficients are then introduced for decomposing
qualitative factors. With the decomposed components, different regression models
involving both quantitative factors and qualitative factors are discussed in the following
section. The interactions between qualitative factors are analyzed and highlighted. An
example is given in the last section to illustrate the practical application.
5.2 Indicator Variables for Qualitative Factors
One way to model qualitative factors is to use indicator variables (Myers and
Montgomery (2001)). For example, a two-level qualitative factor A can be defined as
follows:
54
Indicator Variable of A
0
1
1st level of two-level qualitative factor
2nd level of two-level qualitative factor
If another qualitative factor B has three levels, the different levels would be accounted for
by two indicator variables defined as follows:
Indicator Variables of B
0
0
0
1
1
0
1st level of three-level qualitative factor
2nd level of three-level qualitative factor
3rd level of three-level qualitative factor
Note that this arrangement of settings for indicator variables is not unique.
In general, a qualitative variable with l levels is represented by l-1 indicator
variables, which are assigned the values either 0 or 1. Indicator variables can be centered to
improve the orthogonality of designs (Myers and Montgomery (2001)). All indicator
variables after centering will sum into zero. Therefore, the analysis of two-level and threelevel factor can be performed by the following indicator variables.
A
1
2
Centered Indicator Variable of A
−1
1
B
1
2
3
Centered Indicator Variable of B
1
0
0
1
−1
−1
where these settings for indicator variables reflect centering, but are also not unique. These
types of indicator variables provide a convenient interpretation of the data analysis. For
example, the first indicator variable compares the influence of the first level with that of
the last level; the second indicator variable compares the influence of the second level with
the last level, so and so on. Centered indicator variables are also called contrast coefficients,
which will be discussed in the next section.
55
5.3 Contrast Coefficients for Qualitative Factors
Let
l
be
the
number
of
levels
for
qualitative
factors.
The
matrix
C= [ c1 c2 … cl −1 ] is a contrast coefficients matrix if and only if c'1 = 0. That is,
contrast coefficients sum to zero. Contrast coefficients are used accomplishing the
decomposition of qualitative factors. Normally, the factor with l levels will be partitioned
into l −1 single degree of freedom components. Table 32 shows the decomposition of the
factor A (l −level) into l−1 components, where Ai represents the ith component of A.
Table 32 Decomposition of Factor A Using Contrast Coefficients
A A1 A2 … Al-1
1
2
3 c1 c2 … cl-1
…
l
If c'c
i j = 0 for any i ≠ j, the contrast coefficients are orthogonal contrast
coefficients. The orthogonal decomposition of qualitative factors is not unique. For
example Table 33 shows how a six-level factor can be decomposed into two different sets
of orthogonal components through two different sets of orthogonal contrast coefficients.
Table 33 Two Orthogonal Decomposition Options
Orthogonal Decomposition I Orthogonal Decomposition II
A A1 A2 A3 A4 A5 A1 A2 A3 A4
A5
1
-1
-1
0
-1
0
-5
5
-5
1
-1
2
-1
2
0
0
0
-3
-1
7
-3
5
3
-1
-1
0
1
0
-1
-4
4
2 -10
4
1
0
-1
0
-1
1
-4
-4
2
10
5
1
0
2
0
0
3
-1
-7
-3
-5
6
1
0
-1
0
1
5
5
5
1
1
Orthogonal Decomposition II (Table 33) uses orthogonal polynomials. Orthogonal
polynomials can be introduced using formulas or tables. A brief table of the numerical
values of these orthogonal polynomials is given in Montgomery (2005). More extensive
56
tables of coefficients of orthogonal polynomials are available in Delury (1960). Also, the
formulas used to calculate these coefficients are given in Montgomery, Peck and Vining
(2001).
Using orthogonal polynomials as contrast coefficients has many benefits. First,
they are easy to obtain. Second, these contrast coefficients are orthogonal, resulting in
independent hypothesis tests. Lastly, using them makes for practical interpretation. The
first column is used to test whether there is a linear relation between the factor and
responses. The second column is used to test quadratic relation so and so on.
If c'c
i i = 1 for any i, the contrast coefficients are called normalized contrast
coefficients. Normalized orthogonal contrast coefficients are called orthonormal contrast
coefficients (Xu and Wu (2001)). If a factor have two levels, l=2, this factor has a single
degree of freedom. For the single degree of freedom contrast, the sum of squares can be
calculated by
SS =
(c ' Y ) 2
,
n (c ' c )
where n represent the replicates (Montgomery, 2005). If n=1 and c is orthonormal, then
SS= ( c 'Y ) .
2
When factor A is decomposed (Table 32) using orthnormal contrast coefficients,
the sum of squares of A can be represented by the total of sums of square of all
components, which is
l −1
SS A = ∑ SS Ai = Y ' CC ' Y .
i =1
The orthonormal contrasts coefficients will guarantee that the hypothesis tests performed
are independent. In addition, different decompositions will produce the same total sum of
squares if the contrast coefficients are orthonormal. This can be showed by following
derivation:
(
)(
)
SS = Y 'C1C1'Y = Y 'C1 IC1'Y = Y 'C1 ( C 2C 2 ) C1'Y = Y ' C1C 2 ' C1C 2 ' Y = Y 'C 3C 3'Y ,
'
where C1 and C2 represent two sets of orthonormal contrast coefficients, and C3 = C1C 2 is
a new orthonormal contrast coefficient matrix.
57
The analysis of mixed-level designs can be performed with contrast coefficients.
The analysis objective are performing hypothesis tests, checking factors/components
significance, fitting regression models and interpreting results.
5.4 Classic Regression Models for Mixed-Level Designs
For experimental designs involving both quantitative and qualitative factors, we use
xi to represent two-level quantitative factors and zi to represent qualitative factors. For
qualitative factors with more than two levels, we use zij represent the jth component of ith
qualitative factor. Generally, the linear model is
y = β 0 + ∑ β i xi + ∑∑ β ij xi x j + ∑ ∑ γ ij z ij + ∑∑∑ δ kij x k z ij + ∑∑∑∑ ρ klij x k xl z ij + ε
i
i≤ j
i
j
k
i
j
k
l
i
j
Since the full model may require a large number of experiment runs (to fit all the
model parameters), the experimenter will often consider reduced models. Efficient mixedlevel designs (Guo, 2003) are appropriate choices to fit the reduced models.
Conversely, we recommend testing as many terms as possible, provided there are
enough degrees of freedom available to fit all these model term parameters. The priority
for the model terms from high to low are: qualitative factor and quantitative factor main
effects, interactions among quantitative factors, interactions between quantitative factors
and qualitative factors and second order model terms.
5.4.1 First-Order Models
First-order models are used for the purpose of factor main effects screening. The
standard model for a first-order model including main effects for both quantitative factors
and qualitative factors is
y = β 0 + ∑ β i xi + ∑∑ γ ij z ij + ε .
i
i
j
If 0-1 indicator variables are used, the parameters in this model may be easily interpreted.
If zij = 1, then βij goes into intercept; if 0, just remove this parameter from model.
58
5.4.2 First-Order Models with Interactions
Suppose interactions between quantitative factors are also of interest. The model is
as follows:
k
y = β 0 + ∑ β i xi + ∑ ∑ β ij xi x j + ∑∑ γ ij z ij + ε .
i =1
i≤ j
i
j
Then we can further incorporate interactions between quantitative factors and qualitative
factors. The general model for first-order models with interactions is as follows:
y = β 0 + ∑ β i xi + ∑∑ β ij xi x j + ∑ ∑ γ ij z ij + ∑∑∑ δ kij x k z ij + ∑∑∑∑ ρ klij x k xl z ij + ε
i≤ j
i
i
j
k
i
j
k
l
i
j
The presence of those kinds of interactions implies that certain coefficients change as one
changes levels of the qualitative factors.
5.4.3 Second-Order Models
The situation can be considerably more complicated in the case of a second-order
model when qualitative factors are present. A second-order model can be given as follows:
y = β 0 + ∑ β i xi + ∑∑ β ij xi x j + ∑ β ii xi2 + ∑∑ γ ij z ij
i≤ j
i
i
i
j
+ ∑∑∑ δ kij x k z ij + ∑∑∑∑ ρ klij x k xl z ij + ∑∑∑η kij x k2 z ij + ε
k
i
j
k
l
i
j
k
i
j
This model contains all first- and second-order terms as well as interactions between
quantitative factors, quantitative by qualitative factors. However, fitting such a
complicated regression model may be extremely costly. The highly number of model terms
require adequate degree of freedoms. In other words, the design size can be very large in
this case. On the assumption of that there are enough degrees of freedoms to estimate those
model parameters. See Guo (2003) to see how to build an efficient mixed-level design with
specified number of runs.
5.5 Interactions between Qualitative Factors
Much attention has been put on interactions between quantitative factors and
qualitative factors. The interpretation for interactions between quantitative factors and
59
qualitative factors is quite meaningful: different level of qualitative factors will affect the
influence of quantitative factors on responses. That is, the coefficients of quantitative
factors vary across levels of the qualitative factors. If there is no interaction between
quantitative factors and qualitative factors, that means changing the levels of qualitative
factors only changes the intercept.
Due to the lack of degree of freedom, the interactions between qualitative factors
are always assumed ignorable, which could not be true. For example, there are only two
two-level qualitative factors A and B involved in an experiment. The interaction between
these two factors can be illustrated in Figure 12.
B−
B+
Response
B−
B+
Factor A
Figure 12 Interaction of qualitative factors A and B.
Suppose factor A is a significant. Factor B is not significant, but changing the
levels of B will affect the influence of A on the response. This indicates the significance of
the AB interaction. The regression model for this experiment
y = β 0 + γ 1 z1 + γ 12 z1 z 2 + ε .
We use 0 to represent the first level of z2 and 1 for the second level. If z2=0, the model is
y = β 0 + γ 1 z1 + ε ,
if z2=1, the model is
y = β 0 + (γ 1 + γ 12 )z1 + ε .
Therefore, the coefficient of A (not intercept) will vary across the different level of B.
In general, the linear model including qualitative factor interactions can be given as
60
y = β 0 + ∑ β i xi + ∑∑ β ij xi x j + ∑∑ γ ij zij
i
i≤ j
i
j
+ ∑∑∑ δ kij xk zij + ∑∑∑∑ λipjq zij z pq + ∑∑∑∑ ρ klij xk xl zij + ε
k
i
j
i
p
j
q
k
l
i
.
j
The contrast coefficients for qualitative factor interactions can be obtained by multiplying
the corresponding elements of contrast coefficients of qualitative factor components.
5.6 Regression Analysis on Fire Fighting Data
Consider the example mentioned in the introduction section. The objective of this
experiment is to test new technologies for large scale fire fighting versus the current
techniques. The current used fire fighting method, P19, is the base, with which the other
three new methods are compared. The detailed information regarding four fire fighting
system methods are showed in Table 34.
Table 34 Four Fire Fighting System Methods
Fire Fighting System
P19
Compressed Air
Foam (CAF)
Combined Agent
Fire Fighting
System (CAFFS)
Ultra High Pressure
System (UHPS)
Information
The P-19 uses the Aqueous Film Forming Form (ARFF) fire
fighting technology.
The CAF system functions by injecting compressed air into the
pressurized line between the pump and the nozzle. This results in a
higher expansion ratio AFFF solution at the nozzle inlet. The
resulting foam on the fire is less dense than foam from
conventional systems (P-19) and provides better cooling and
insulation between the fuel and the fire.
The CAFFS system functions similarly to the CAF by injecting
compressed air foam, but added the benefits of dry chemical. A
special nozzle is used that discharges the dry chemical through a
central orifice. The compressed air foam is discharged through an
annular opening around the dry chemical orifice.
The UHPS system delivered AFFF solution at approximately 1500
psi. Operating at this pressure significantly changed the
characteristics of the solution and its effect on the fire.
61
Besides fire fighting method, different surfaces the fighting experiments performed
on are also of interest. There are four surfaces: water, gravel, soil/sod. Both fire fighting
method and surface are qualitative factors. In addition, there are two quantitative
controllable factors involved: flow rate and fire area. The response variable for this
problem is extinguishment time in the unit of second. As a summary, Table 35 shows
descriptive statistics for predictor variables and response variables.
Table 35 Descriptive Statistics of Factors and Responses
Factors
Surface
Method
Flow Rate
Area
Extinguishment Time
Unit
Notation
z1
z2
gpm
x1
Sq.ft.
x2
Second y
Classification
Qualitative
Qualitative
Quantitative
Quantitative
Quantitative
Factor Levels
Gravel, Soil/Sod, Water
CAF, CAFFS, UHPS, P19
47~597
877~6600
6~203
The qualitative factors z1 and z2 are decomposed into single degree of freedom
components using the contrast coefficients showed in Table 36. We choose to compare
surface gravel and soilsod with water because water is the most stable surface (Figure 13).
We choose to compare CAF, CAFFS, UHPS with P19 because P19 is the current used
method (Figure 14).
Table 36 Contrast Coefficients for Surface and Method
z1
z11 z12
Gravel
1
0
Surface
SoilSod
0
1
Water
−1
−1
z2
z21 z22 z23
CAF
1 0 0
Methods CAFFS
0 1 0
UHPS
0 0 1
P19
−1 −1 −1
62
Extinguishment Time vs Method
Ext inguishment Time
200
150
100
50
0
CAF
CAFFS
P19
UHPS
Met hod
Figure 13 Plot of extinguishment time verse method.
Extinguishment Time vs Surface
Ext inguishment Time
200
150
100
50
0
Gravel
Soil/ Sod
Surface
Figure 14 Plot of extinguishment time verse surface.
63
Water
Therefore, the first-order regression model with factor interactions is
y = β 0 + β1 x1 + β 2 x2 + β12 x1 x2 + γ 11 z11 + γ 12 z12 + γ 21 z21 + γ 22 z22 + γ 23 z23
+δ111 x1 z11 + δ112 x1 z12 + δ121 x1 z21 + δ122 x1 z22 + δ123 x1 z23
+δ 211 x2 z11 + δ 212 x2 z12 + δ 221 x2 z21 + δ 222 x2 z22 + δ 223 x2 z23
+ ρ1121 z11 z21 + ρ1122 z11 z22 + ρ1123 z11 z23 + ρ1221 z12 z21 + ρ1222 z12 z22 + ρ1223 z12 z23 + ε
We use the normal probability plot (Figure 15) to check the normality assumption of the
responses and found that an appropriate transformation of the data is necessary. The
natural log transformation is applied (Figure 16).
The design was analyzed using SAS statistical software package. SAS uses a set of
procedural statements (PROC) to carry out regression analysis of data. Table 37 shows the
analysis of variance for the fire data including all terms and Table 38 only shows the
significant terms. We found the significant terms are x1(flow rate), x2(area), z1(surface),
z2(method), z1×z2(Surface×Method) and x2×z2(Method×Area).
Probability Plot of Extinguishment Time
Normal - 95% CI
99.9
Mean
StDev
N
AD
P-Value
99
95
Percent
90
80
70
60
50
40
30
20
10
5
1
0.1
-100
-50
0
50
100
150
200
Ext inguishment Time
Figure 15 Probability plot of extinguishment time.
64
52.83
36.04
235
7.474
< 0.005
Probability Plot of Transformed Extinguishment Time
Normal - 95% CI
99.9
Mean
StDev
N
AD
P-Value
99
95
Percent
90
80
70
60
50
40
30
20
10
5
1
0.1
1
2
3
4
5
6
Transformed Ext inguishment Time
Figure 16 Probability plot of transformed extinguishment time.
Table 37 Analysis of Variance of the Fire Data (All Terms)
Source
Surface
Method
Flowrate
Area
Surface×Method
Surface×Flowrate
Surface×Area
Method×Flowrate
Method×Area
Flowrate×Area
DF
2
3
1
1
6
2
2
3
3
1
SS
0.829
0.146
1.734
0.004
3.129
0.058
0.043
0.869
1.147
0.174
MS
0.414
0.049
1.734
0.004
0.521
0.029
0.021
0.290
0.382
0.174
F
3.72
0.44
15.55
0.03
4.68
0.26
0.19
2.6
3.43
1.56
pr > F
0.0259
0.727
0.0001
0.8548
0.0002
0.7707
0.826
0.0533
0.018
0.2124
Table 38 Analysis of Variance of the Fire Data (Significant Terms)
Source
Surface
Method
Flowrate
Area
Surface×Method
Method×Area
DF
2
3
1
1
6
3
SS
15.861
0.745
16.612
17.049
3.418
1.032
65
MS
7.930
0.262
16.612
17.049
0.570
0.344
F
70.46
2.32
147.59
151.47
5.06
3.06
pr > F
<0.0001
0.0759
<0.0001
<0.0001
<0.0001
0.0293
3.746
0.6915
235
0.552
0.153
The estimation of significant model terms is showed in Table 39. We also provide
the interpretation for all model terms especially qualitative factor interactions. For example,
interaction z11z21 actually compare the surface-method combination as the pattern of
(Gravel,CAF)−(Gravel,P19)−(Water,CAF)+(Water,P19).
Table 39 Model Terms Estimation
Parameter Interpretation
z11
z12
z21
z22
z23
x1
x2
z11z21
z11z22
z11z23
z12z21
z12z22
z12z23
x2z21
x2z22
x2z23
Estimate
Gravel-Water
0.422
SoilSod-Water
0.292
CAF−P19
-0.585
CAFFS−P19
-0.240
UHPS−P19
-0.679
Flow rate
0.0033
Area
0.0002
(Gravel,CAF) − (Gravel,P19)
0.147
− (Water,CAF)+(Water,P19)
(Gravel, CAFFS) − (Gravel,P19)
-0.018
− (Water,CAFFS)+(Water,P19)
(Gravel, UHPS) − (Gravel,P19)
0.679
− (Water,UHPS)+(Water,P19)
(SoilSod, CAF) − (SoilSod,P19)
0.162
− (Water,CAF)+(Water,P19)
(SoilSod CAFFS) − (SoilSod,P19)
-0.238
− (Water,CAFFS)+(Water,P19)
(SoilSod UHPS) − (SoilSod,P19)
0.374
− (Water,UHPS)+(Water,P19)
Area×(CAF−P19)
0.00009
Area×(CAFFS−P19)
-0.00003
Area×(UHPS−P19)
0.00006
Standard
Error
0.104
0.101
0.185
0.178
0.329
0.0003
0.00003
t
pr > |t|
Value
4.05 <0.0001
2.9 0.0041
-3.15 0.0018
-1.35 0.179
-2.06 0.0402
-12.15 <0.0001
7.69 <0.0001
0.143
1.03 0.3058
0.149
-0.12 0.9054
0.157
4.3 <0.0001
0.158
1.03 0.3063
0.158
-1.5
0.161
2.31 0.0216
0.00004
0.00004
0.00007
2.04 0.0429
-0.83 0.4102
0.85 0.3989
0.0138
With the estimated model term coefficients, the final regression model on the transformed
data is
y = 3.56 − 0.0033 x1 + 0.0002 x2 + 0.422 z11 + 0.292 z12 − 0.585 z21 − 0.240 z22 − 0.679 z23
+0.00009 x2 z21 − 0.00003 x2 z22 + 0.00006 x2 z23
+0.147 z11 z21 − 0.018 z11 z22 + 0.679 z11 z23 + 0.162 z12 z21 − 0.238 z12 z22 + 0.374 z12 z23 .
66
5.7 Conclusion
The analysis of variance of mixed-level designs including qualitative factors can be
done by using indicator variables or contrast coefficients. Indicator variables work well for
qualitative factors and they have nice practical interpretations. Contrast coefficients are the
extended general form of indicator variables and they can be more widely used to estimate
sum of squares for qualitative factors. The process of using contrast coefficients for
qualitative factors is called decomposition of qualitative factors. That is, via contrast
coefficients, qualitative factors are decomposed into single degree of freedom components.
The decomposition is not unique. However, if the used contrast coefficients are
orthonormal, different decomposition will give the same sum of square estimation.
There are several regression models can be fitted using indicator variables or
contrast coefficients. However, all these models assume there is no qualitative factor
interaction. The fire fighting example given in the chapter shows it is necessary to include
qualitative factor interactions in the model, because a qualitative factor interaction may be
significant even though both qualitative factors are not significant. Therefore, with
sufficient degree of freedom, we recommend analyzing mixed-level designs including
qualitative factor interactions.
67
CHAPTER 6
GENERAL CONCLUSIONS AND
FUTURE RESEARCH
The primary objective of this dissertation was to propose practical mixed-level
design solutions. The solutions include all aspects of mixed-level designs, such as
proposing evaluation criteria for mixed-level fractional factorial designs, constructing
optimal mixed-level designs, developing mixed-level design augmentation schemes, and
also providing analysis guidelines for mixed-level designs. Mixed-level factorial designs
are the necessary alternatives to the traditional two-level factorial designs when qualitative
factors are present. Full factorial mixed-level designs may contain too many runs and may
not be affordable. As a result, we recommend using fractional factorial designs.
Regular two-level fractional factorial designs are generated by design generators.
This type of designs has simple alias structure. Usually, the factors are all quantitative
factors. Non-regular two-level fractional factorial designs refer to designs that are not
generated by design generators. Examples of non-regular two-level designs are PlackettBurman designs. These designs have advantages of small sizes and flexible models.
However, the alias structure of non-regular designs is much more complicated than regular
designs.
Orthogonal two-level designs imply that the dot product of every two columns
equals zero. For designs involving factors with more than two levels, orthogonalilty of
strength 2 refers to all pairwise columns are orthogonal. Two columns are orthogonal if
each of their level combinations appears equally often. Usually, orthogonal designs are
balanced. Efficient mixed-level fractional factorial designs are appropriate when designs
can not reach perfect balance due to specified design sizes. The appendix A gives some
examples of efficient mixed-level designs. The design size is chosen to make most of
factors balanced and minimum number of factors near-balanced. Primarily, efficient
mixed-level designs are constructed associated with optimal balance and orthogonality
properties. For mixed-level designs with same size, same balance and orthogonality
68
properties, we can use other more discriminating criteria, such as minimum aberration
criteria, to assess them.
Chapter 2 reviews most of the newly proposed minimum aberration definitions. For
example, MGA and related MG2A were formulated for evaluating non-regular two-level
designs. Compared to MGA, MG2A is simpler and easier to compute. GMA and MG2A were
proposed for multi-level or mixed-level designs via contrast coefficients, which need not
necessarily be orthonormal. But orthogormal contrast coefficients help to generate
consistent MG2A values. The GMA is equivalent to MG2A for evaluating two-level designs.
MMA and MAP were proposed based upon coincidence relations between runs. Therefore,
they do not require contrast coefficients. MAP provides more detailed information than
MMA but MAP is more complicated. In terms of representation, MG2A, GMA, and MMA
use single value for each k-factor interaction. However, MGA and MAP use a frequency
vector for each k-factor interaction, which may not be a good idea when large number of
computations are involved.
All minimum aberration criteria have their own drawbacks. For example, simply
adding squared J-characteristic all k-factor interactions may not distinguish designs that
have the same sums but different individual values. GMA uses contrast coefficients to
calculate the generalized word length pattern. However, the calculated values are not
related to the alias structure of designs in any way. MMA uses k-powered summation for
each k-factor interaction, but k-powered summation does not reflect k-factor interaction.
MGAC and GCMA are established based upon many assumptions. In addition, the
mathematical principles behind them are complicated.
The general balance metric proposed in Chapter 3 is a good minimum aberration
criterion for non-regular two-level and mixed-level design situations. The general balance
metric is an extension of balance coefficients. It measures the balance property for both
main effects and interactions. In addition, the general balance metric can also detect the
orthogonality property of mixed-level designs. Compared to other minimum aberration
criteria in literature, this new criterion is easy to use and also is easy to interpret. The B
metric, derived from the GBM, is used as a supplementary criterion primarily to show the
relationship with WLP for two-level fractional factorial designs. With B , the concept of
resolution is generalized because the whole interaction matrix D corresponding to all
69
potential model terms is considered. Because GBM not only incorporates the number of
non-balanced columns shown in B , but also reflects the degree of non-balance within each
column, it is suggested that the GBM be used to compare designs of similar quality. One
area of future research would be integrating B with GBM into a single criterion, which
reflects the number of non-balanced columns but also the degree of non-balanced.
Chapter 4 uses the general balance metric to fold over mixed-level designs. The
GBM is standardized in order to compare designs with different number of runs. The
general balance metric can show the improvement of balance property of factor
interactions and also can reveal the reduction of non-balanced columns when the initial
designs combined with optimal foldovers. Since the quality of the foldovers is affected by
the quality of initial designs, we recommend using optimal efficient designs. Examples of
optimal designs are orthogonal designs and minimum aberration designs. The combined
designs may not be optimal even though the original designs are optimal. This is due to the
limitation of the predetermined runs of original designs. This situation also happens to
regular two-level designs.
Although the number of non-balanced columns reflects that some interaction levels
are missing, the general balance metric defined in this dissertation did not fully reveal the
alias structures. Reducing the aliased interaction and releasing aliased terms is the ultimate
goal of augmenting fractional designs. As a result, clarifying the alias structure of mixedlevel fractional factorial designs is one area for future research.
In Chapter 5, the analysis of variance of mixed-level designs can be conducted
through indicator variables or contrast coefficients. Indicator variables work well for
qualitative factors and they have simple practical interpretations. Contrast coefficients are
the extended general form of indicator variables and they can be more widely used to
estimate sum of squares for qualitative factors. The process of using contrast coefficients
for qualitative factors is called decomposition of qualitative factors. That is, via contrast
coefficients, qualitative factors are decomposed into single degree of freedom components.
Such a decomposition may not unique. However, if the used contrast coefficients are
orthonormal, different decomposition will give the same sum of square estimation. Chapter
5 also uses indicator variables or contrast coefficients to fit regression models. However,
commonly used regression models assume there is no qualitative factor interaction. The
70
fire fighting example given in the dissertation shows it is necessary to include qualitative
factor interactions in the model, because a qualitative factor interaction may be significant
even though both qualitative factors are not significant. Therefore, with sufficient degrees
of freedom, we recommend analyzing mixed-level designs including qualitative factor
interactions.
There are other aspects of this research that may be further investigated. First of all,
current construction methods do not guarantee optimal designs. Therefore, more efforts
need to be placed on construction methods of optimal mixed-level designs. Secondly,
foldovers of mixed-level designs are not absolutely optimal even though they are close to
the optimal. With more efficient algorithms and faster computation technologies, optimal
foldovers can be found by searching all possible candidates. This method can also be used
to augment designs with more specified number of runs. Finding foldovers can share the
same algorithms with constructing initial designs. The criteria to evaluate mixed-level
designs can be more improved in two aspects, enhancing current minimum aberration
criteria or developing better criteria. Finally, other future research includes construction
and evaluation of split-plot mixed-level designs, supersaturated mixed-level designs, and
second order mixed-level designs.
71
APPENDIX
A. Efficient Mixed-Level Designs
EA (20, 243141)
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1
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1
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1
EA (20, 243151)
1
3
1
1
3
2
1
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3
2
2
3
1
2
2
1
1
3
3
4⎤
1 ⎥⎥
4⎥
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2⎥
4⎥
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3⎥
3⎥
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3⎥
1⎥
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2⎥
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2⎥
4⎥
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3⎥
1⎥
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2⎥
4⎥
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1⎥
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2⎥
1⎥
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3⎦
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72
1
1
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1
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2
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1
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1
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1
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1
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1
2
1
1
2
1
2
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1
2
2
1
1
2
2
2
1
1
2
2
1
1
2
1
2
1
1
2
2
2
1
1
1
1
2
1
1
2
1
2
2
2
2
1
3
3
2
1
3
2
1
2
2
3
1
3
1
1
2
2
3
1
3⎤
1 ⎥⎥
1⎥
⎥
2⎥
2⎥
⎥
5⎥
4⎥
⎥
4⎥
3⎥
⎥
5⎥
⎥
1⎥
4⎥
⎥
2⎥
3⎥
⎥
4⎥
3⎥
⎥
5⎥
⎥
1⎥
5⎥
⎥
2⎦
EA (21, 26315171)
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73
1
1
1
2
1
2
2
1
2
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2
2
2
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1
2
1
1
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1
1
2
2
2
2
2
1
1
1
1
2
2
2
1
3
2
4
5
1
1
1
1
2
1
2
1
2
1
1
2
2
1
1
1
1
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1
2
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1
2
3
3
3
4
1
1
1
2
2
2
1
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1
1
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1
2
1
2
2
2
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1
1
3
5
2
4
2
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1
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1
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2
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1
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1
2
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2
3
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1
3
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1
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1
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1
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4
2
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1
1
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1
1
1
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1
2
1
1
1
2
3
1
5
5
3
4⎤
7 ⎥⎥
2⎥
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5⎥
1⎥
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4⎥
6⎥
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5⎥
3⎥
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2⎥
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3⎥
4⎥
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2⎥
3⎥
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6⎥
1⎥
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7⎥
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6⎥
5⎥
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7⎥
1 ⎥⎦
EA (30, 26315171)
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1
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1
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1
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1
EA (21, 325171)
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5
1
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3
4
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3
1⎤
3 ⎥⎥
4⎥
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7⎥
5⎥
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2⎥
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6⎥
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5 ⎥⎥
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74
1
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5
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1
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1
5
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1
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3
3
1
2
3
4
3
2
3
3
3
1
2
3
5
2
5
4⎤
5 ⎥⎥
2⎥
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7⎥
6⎥
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1⎥
3⎥
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4⎥
5⎥
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6⎥
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7⎥
3⎥
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1⎥
2⎥
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7⎥
5⎥
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1⎥
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6⎥
2⎥
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3⎥
4 ⎥⎦
EA (21, 314171)
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2
2
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1
2
3
3
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1
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1
EA (21, 324171)
6⎤
3 ⎥⎥
1⎥
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5⎥
2⎥
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7⎥
4⎥
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4⎥
3⎥
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5⎥
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2⎥
6⎥
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7⎥
1⎥
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5⎥
3⎥
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1⎥
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2⎥
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4 ⎥⎦
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75
1
1
1
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1
1
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3
1
2
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1
2
2
3
1
2
3
3
2
4
4
2
2
2
1
2
3
1
1
2
2
2
3
1
2
3
3
3
1
3
3
3
1
2
3
3
2
2
3
3
3
1
2
3
4
4
1
7⎤
4 ⎥⎥
6⎥
⎥
5⎥
1⎥
⎥
3⎥
2⎥
⎥
2⎥
4⎥
⎥
3⎥
⎥
6⎥
5⎥
⎥
1⎥
7⎥
⎥
1⎥
4⎥
⎥
3⎥
⎥
7⎥
6⎥
⎥
5⎥
2 ⎥⎦
EA (20, 235171)
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5
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1
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1
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1
1
1
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1
2
1
2
2
2
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1
1
4
5
3
2
5
4
4
3
1
1
2
EA (20, 245171)
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4⎤
3 ⎥⎥
2⎥
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5⎥
5⎥
⎥
1⎥
6⎥
⎥
4⎥
7⎥
⎥
6⎥
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4⎥
2⎥
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2⎥
5⎥
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1⎥
3⎥
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3⎥
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6⎥
1⎥
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7⎦
76
1
1
1
1
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1
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1
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2
2
2
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2
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2
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2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2
1
2
1
1
1
2
2
2
2
1
2
1
1
2
2
1
1
1
1
1
2
2
2
2
1
2
1
2
2
1
2
2
2
1
1
1
1
1
4
2
4
5
3
1
5
2
3
4
1
4
5
3
2
3
1
5
2
2⎤
3 ⎥⎥
1⎥
⎥
4⎥
5⎥
⎥
1⎥
3⎥
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7⎥
6⎥
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6⎥
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7⎥
1⎥
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5⎥
2⎥
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2⎥
3⎥
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4⎥
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4⎥
6⎥
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5⎦
EA (20, 314151)
EA (20, 23314151)
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3
2
1
1
2
1
3
2
1
2
2
3
1
2
3
2
1
3
3
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
1⎤
2 ⎥⎥
3⎥
⎥
4⎥
5⎥
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1⎥
2⎥
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3⎥
4⎥
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5⎥
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1⎥
2⎥
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3⎥
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5⎦
77
1
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1
2
1
2
2
1
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2
1
2
1
2
1
1
2
1
2
1
2
2
1
2
1
1
1
1
2
3
1
2
3
2
3
2
3
3
1
1
2
1
1
3
2
2
2
1
3
4
3
3
2
1
4
2
1
3
1
4
4
2
4
3
1
2
2⎤
2 ⎥⎥
5⎥
⎥
1⎥
1⎥
⎥
3⎥
4⎥
⎥
4⎥
3⎥
⎥
5⎥
⎥
5⎥
2⎥
⎥
1⎥
5⎥
⎥
2⎥
3⎥
⎥
4⎥
⎥
4⎥
3⎥
⎥
1⎦
EA (20, 24314151)
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2
1
2
1
2
1
1
2
2
1
2
2
1
2
1
2
1
2
1
1
2
2
1
2
1
1
2
1
2
1
2
1
1
1
2
2
1
2
1
2
EA (28, 236171)
2
1
3
3
1
2
2
3
1
1
2
1
3
2
3
3
1
2
2
1
3
1
1
4
4
3
4
2
2
2
2
3
2
4
3
3
1
1
1
4
3⎤
3 ⎥⎥
2⎥
⎥
4⎥
1⎥
⎥
2⎥
5⎥
⎥
5⎥
4⎥
⎥
1⎥
⎥
2⎥
4⎥
⎥
3⎥
3⎥
⎥
1⎥
5⎥
⎥
5⎥
⎥
1⎥
4⎥
⎥
2⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣
78
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
1
2
2
1
1
2
2
2
2
1
2
1
2
2
1
2
1
1
2
2
1
1
2
1
2
4
4
3
6
6
5
2
1
3
3
2
1
5
4
6
3
1
5
4
2
4
6
1
3
5
2
1
2⎤
3 ⎥⎥
6⎥
⎥
7⎥
1⎥
⎥
3⎥
5⎥
⎥
4⎥
7⎥
⎥
2⎥
⎥
4⎥
1⎥
⎥
5⎥
6⎥
⎥
2⎥
5⎥
⎥
1⎥
⎥
6⎥
4⎥
⎥
7⎥
3⎥
⎥
5⎥
6⎥
⎥
4⎥
3 ⎥⎥
2⎥
⎥
7⎥
1 ⎥⎦
EA (28, 246171)
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2
2
2
1
2
1
2
2
1
1
1
2
1
1
1
1
2
1
1
2
2
2
1
2
2
2
1
1
2
1
2
2
2
1
1
2
1
2
1
1
2
1
1
1
1
2
1
2
1
2
2
1
2
1
2
2
EA (21, 316171)
5
6
2
4
1
3
1
3
4
2
2
6
3
4
2
1
4
1
5
2
1
4
6
5
3
3
6
5
7⎤
3 ⎥⎥
6⎥
⎥
4⎥
3⎥
⎥
5⎥
5⎥
⎥
4⎥
1⎥
⎥
7⎥
⎥
2⎥
1⎥
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2⎥
6⎥
⎥
3⎥
7⎥
⎥
7⎥
⎥
1⎥
4⎥
⎥
5⎥
4⎥
⎥
2⎥
6⎥
⎥
2⎥
1 ⎥⎥
6⎥
⎥
5⎥
3 ⎥⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣
79
1
1
1
2
1
1
1
3
4
5
1
1
2
2
6
1
2
3
2
2
2
4
5
6
2
2
3
1
2
3
3
3
3
4
5
6
3
3
3
1
2
3
4⎤
5 ⎥⎥
1⎥
⎥
6⎥
3⎥
⎥
2⎥
7⎥
⎥
2⎥
7⎥
⎥
3⎥
⎥
5⎥
1⎥
⎥
6⎥
4⎥
⎥
2⎥
5⎥
⎥
4⎥
⎥
7⎥
3⎥
⎥
1⎥
6 ⎥⎦
EA (24, 416171)
EA (24, 516171)
⎡
⎢
⎢
⎢
⎢
⎢
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⎢
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⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1⎤
3 ⎥⎥
7⎥
⎥
5⎥
6⎥
⎥
2⎥
6⎥
⎥
1⎥
4⎥
⎥
7⎥
⎥
2⎥
3⎥
⎥
4⎥
2⎥
⎥
3⎥
1⎥
⎥
5⎥
⎥
7⎥
2⎥
⎥
6⎥
1⎥
⎥
4⎥
3⎥
⎥
5 ⎥⎦
80
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
6⎤
3 ⎥⎥
2⎥
⎥
7⎥
2⎥
⎥
1⎥
7⎥
⎥
3⎥
2⎥
⎥
6⎥
⎥
3⎥
5⎥
⎥
1⎥
5⎥
⎥
4⎥
2⎥
⎥
4⎥
⎥
5⎥
1⎥
⎥
3⎥
4⎥
⎥
1⎥
7⎥
⎥
6 ⎥⎦
EA (20, 41516171)
EA (24, 31516171)
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
4
1
5
3
1
4
2
3
1
4
2
3
1
4
2
5
3
1
5
5
3
2
4
2
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
3⎤
7 ⎥⎥
5⎥
⎥
6⎥
3⎥
⎥
4⎥
1⎥
⎥
2⎥
2⎥
⎥
2⎥
⎥
7⎥
1⎥
⎥
4⎥
5⎥
⎥
6⎥
3⎥
⎥
5⎥
⎥
6⎥
7⎥
⎥
1⎥
3⎥
⎥
4⎥
1⎥
⎥
2 ⎥⎦
81
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
2
5
6
3
1
1
6
4
2
3
4
2
1
5
6
5
3
2
1
4
3⎤
6 ⎥⎥
7⎥
⎥
5⎥
2⎥
⎥
4⎥
5⎥
⎥
6⎥
2⎥
⎥
1⎥
⎥
2⎥
1⎥
⎥
3⎥
4⎥
⎥
6⎥
7⎥
⎥
3⎥
⎥
4⎥
1⎥
⎥
5⎦
APPENDIX
B. MATLAB Codes
82
83
84
85
86
87
88
89
90
91
92
APPENDIX
C. Glossary
Notation
d
n
m
dij
OA(n, 2m)
A, B, C, D, E, …
A1, A2, …
Interpretation
Design matrix
Number of runs (rows) of d
Number of factors (columns) of d
Element of ith row and jth column of d
Orthogonal Array with m two-level factors in n runs
Factor identity
First and second column of contrast coefficient
matrix of factor A
2-level 2-qth fractional factorial designs
k-factor interactions
J characteristics of k factor interaction of d
2m-q
k
Jk (d )
s = [ c1 , c2 , , ck ]
cij
δ ij = δ ( d ij , d lj )
Subset matrix of d
( F (d ), F (d ),
, Fm ( d ) )
( B (d ), B (d ),
( A (d ), A (d ),
, Bm ( d ) )
1
2
1
2
1
2
, Am ( d ) )
The ith element of column cj
Coincidence between two elements dij and dlj, used
in minimum moment aberration and minimum
aberration projection
1. Minimum G-aberration
2. Minimum aberration projection
Minimum G2-aberration
αk
1. Word length pattern of the usual minimum
aberration definition
2. Generalized minimum aberration
Vector of all k-factor interactions
Xk= ⎡⎣ xij( k ) ⎤⎦
Matrix of contrast coefficients for α k
( K (d ), K (d ),
( A (d ), A (d ),
( E (d ), E (d ),
1
2
g
1
g
2
1
2
, Km ( d ))
, Amg ( d ) )
Minimum generalized aberration
, Nm ( d ) )
Distance distribution of d, used in minimum
generalized aberration
Krawtchouk polynomial, used in minimum
generalized aberration
General criteria of minimum aberration
, Em ( d ) )
Pj ( k ; m)
( N (d ), N (d ),
1
γ 1, γ 2 , γ J
Wj
2
Minimum moment aberration
Sets of effects to be estimated, used in general
criteria of minimum aberration
The model matrix corresponding to γ j
93
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BIOGRAPHICAL SKETCH
Education Background
Degree
Doctor of Philosophy in Industrial Engineering
Emphasis: Quality Engineering and Applied Statistics
Advisor: Dr. James R. Simpson
Awarded
Institution
04/2006
Florida State University
Tallahassee, Florida, USA
Master of Science in Industrial Engineering
Advisor: Dr. James R. Simpson
01/2004
Florida State University
Tallahassee, Florida, USA
Bachelor of Management in Industrial Management
Advisor: Dr. Qin Su
07/2000
Xi’an Jiao Tong University
Xi’an, China
Bachelor of Science in Mechanical Engineering
Advisor: Dr. Yi Pei
07/2000
Xi’an Jiao Tong University
Xi’an, China
Research Appointment
Position
Research
Assistant
Dates
May 2005 - May 2006
Projects
Advanced Fire Protection Deluge System
Advanced Data Analysis
Research
Assistant
April 2003 - May 2006
Improved Performance Research Integration Tool
Simulation Model Output Analysis
Research
Assistant
May 2005 - August 2005
Design of Graphical Interface for Optimization
Tools
Research
Assistant
April 2004 - August 2004
Statistical Data Analysis for Dean’s Office
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Teaching Appointment
Position
Dates
Employer
FAMU-FSU
College of Engineering
Courses
MAP3305 Engineering
Mathematics I
Instructor
Spring 2005
Program
Manager
Spring 2003 - Fall 2004
FAMU-FSU
College of Engineering
EGN1004L First Year
Engineering Lab
Teaching
Assistant
Spring 2005
Florida State
University
ESI4523/EIN5524 System
Modeling and Simulation
Teaching
Assistant
Spring 2003, Spring 2004
Florida State
University
ESI5451 Project Analysis and
Design
Teaching
Assistant
Spring 2003
Florida State
University
EIN4312 Tool and Process
Engineering
Teaching
Assistant
Fall 2002
Florida State
University
EIN 4395 Computer Integrated
Manufacturing
Teaching
Assistant
Fall 2002
Florida State
University
EIN4930 Manufacturing
Process & Material Engineering
Teaching
Assistant
Spring 2002
Florida State
University
EIN4611 Industrial Automation
& Robotics
Teaching
Assistant
Spring 2002
Florida State
University
EGN2123 Computer Graphics
for Engineering
Industrial Experiences
Position
Engineer
Dates
Employer
July 2000 - January 2002
BGRIMM Magnetic Materials & Technology
CO., LTD. Beijing, China
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