Citation
The design and analysis of split-plot experiments in industry

Material Information

Title:
The design and analysis of split-plot experiments in industry
Creator:
Kowalski, Scott M., 1969-
Publication Date:
Language:
English
Physical Description:
vi, 172 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Centroids ( jstor )
Degrees of freedom ( jstor )
Design analysis ( jstor )
Experiment design ( jstor )
Factorial design ( jstor )
Factorials ( jstor )
Industrial design ( jstor )
Mathematical variables ( jstor )
Ovens ( jstor )
Random allocation ( jstor )
Dissertations, Academic -- Statistics -- UF ( lcsh )
Statistics thesis, Ph. D ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 168-171).
General Note:
Printout.
General Note:
Vita.
Statement of Responsibility:
by Scott M. Kowalski.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
021563364 ( ALEPH )
43757288 ( OCLC )

Downloads

This item has the following downloads:


Full Text













THE DESIGN AND ANALYSIS OF SPLIT-PLOT
EXPERIMENTS IN INDUSTRY












By

SCOTT M. KOWALSKI


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS OF THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1999

















ACKNOWLEDGMENTS


I would like to express my sincere gratitude to Dr. G. Geoffrey Vining for serving

as my dissertation advisor. Many thanks go out to him for allowing me the oppor-

tunity to serve as an editorial assistant for the Journal of Quality Technology. His

generosity has given me experiences that most graduate students could only dream

of having. Also, not only as he been a mentor for me statistically, but he has also

been my friend.

I would also like to thank Dr. John Cornell for his extreme interest in the work

that I have done and for being a constant resource for me. In addition, I would like

to thank Drs. Jim Hobert, Richard Scheaffer, and Diane Schaub for serving on my

committee. Also, I would like to thank Dr. Frank Martin for sitting in on my defense

and for developing my interest in Design of Experiments through his course. I extend

a big thank you to Dr. Ronald Randles, chairman of the department of statistics, for

supporting me through my many years at the University of Florida.

I thank my parents for their endless love and support. They fully believed in me

when, at times, I wasn't so sure I would finish. Finally, I have to thank my wife for

her love and especially her patience. She has stood by me through e-verytliii g and I

owe her more than I could ever repay.

















TABLE OF CONTENTS


ACKNOW LEDGMIENTS ....................................................... ii

A B ST R A C T ................................................................... vi

CHAPTERS

1 INTRODUCTION ......................................................... 1

1.1 Response Surface Methodology ............................. ......... 1

1.2 Split-Plot Designs ................................................... 3

1.3 Dissertation Goals .................................................. 13

1.4 O verview ........................................................... 16

2 LITERATURE REVIEW ................................................. 17

2.1 Split-Plot Confounding ........................................ .... .. 17

2.2 Split-Plots in Robust Parameter Designs ............................ 19

2.3 Bi-Randomization Designs .......................................... 27

2.4 Split-Plots in Industrial Experiments ............................... 31

3 INCOMPLETE SPLIT-PLOT EXPERIMENTS .......................... 40

3.1 Fractional Factorials ................................................ 41

3.2 Confounding ....................................................... 44

3.3 Confounding in Fractional Factorials ................................ 46

3.4 Combining Fractional Factorials and Confounding in Split-Plot
Experim ents ....................................................... 47













3.5 Discussion of Minimum-Aberration in Split-Plot Designs ............ 55

3.6 Adding Runs to Improve Estimation ................................ 57

3.7 An Exam ple ........................................................ 72

3.8 Sum m ary ........................................................... 78

4 A NEW MODEL AND CLASS OF DESIGNS FOR MIXTURE
EXPERIMENTS WITH PROCESS VARIABLES ...................... 81

4.1 Experimental Situation ............................................. 84

4.2 The Combined Mixture Component-Process Variable Model ........ 85

4.3 Design Approach ................................................... 89

4.4 A analysis ........................................................... 104

4.5 Lack of Fit ........................................................ 111

4.6 Exam ple .......................................................... 114

4.7 Sum m ary ......................................................... 117

5 MIXTURE EXPERIMENTS WITH PROCESS VARIABLES IN A
SPLIT-PLOT SETTING .............................................. 119

5.1 First-Order Model for the Process Variables ....................... 120

5.2 Second-Order Model for the Process Variables ..................... 135

5.3 Sum m ary ......................................................... 139

6 SUMMARY AND CONCLUSIONS ...................................... 147

APPENDICES

A: TABLES FOR CHAPTER 3 DESIGNS ................................. 149

B: TABLES FOR CHAPTER 4 DESIGNS ................................. 161

C: SAS CODE FOR PROC MIXED ....................................... 167













R EFER EN C ES ............................................................... 168

BIOGRAPHICAL SKETCH .................................................. .. 172

















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements of the Degree of Doctor of Philosophy

THE DESIGN AND ANALYSIS OF SPLIT-PLOT
EXPERIMENTS IN INDUSTRY

By

Scott M. Kowalski

December, 1999


Chairman: G. Geoffrey Vining
Major Department: Statistics

Split-plot experiments where the whole plot treatments and the subplot treatments

are made up of combinations of two-level factors are considered. Due to cost and/or

time constraints, the size of the experiment needs to be kept small. Using fractional

factorials and confounding, a method for constructing sixteen run designs is presented.

Along with this, semifolding is used to add eight more runs. The resulting twenty-four

run design has better estimating properties and gives some degrees of freedom which

can be used for estimating the subplot error variance.

Experiments that involve the blending of several components to produce high

quality products are known as mixture experiments. In some mixture experiments,

the quality of the product depends not only on the relative proportions of the mixture













components but also on the processing conditions. A combined model is proposed

which is a compromise between the additive and completely crossed combined mixture

by process variable models. Also, a new class of designs that will accommodate the

fitting of the new model is considered.

The design and analysis of the mixture experiments with process variables is dis-

cussed for both a completely randomized structure and a split-plot structure. When

the structure is that of a split-plot experiment, the aiiialy.-i is more complicated since

ordinary least squares is no longer appropriate. With the process variables serving as

the whole plot factors, three methods for estimation are compared using a simulation

study. These are ordinary least squares (to see how inappropriate it is), restricted

maximum likelihood, and using replicate points to get an estimate of pure error. The

last method appears to be the best in terms of the increase in the size of the confi-

dence ellipsoid for the parameters and has the added feature of not depending on the

model.














CHAPTER 1
INTRODUCTION


A common exercise in the industrial world is that of designing experiments, explor-

ing complex regions, and optimizing processes. The setting usually consists of several

input factors that potentially influence some quality characteristic of the process,

which is called the response. Box and Wilson (1951) introduced statistical methods

to attain optimal settings on the design variables. These methods are commonly

known as response surface methodology (RSM), which continues to be an important

and active area of research for industrial statisticians.

Many times in industrial experiments, the factors consist of two types: some with

levels that are easy to change and one or more with levels that are difficult or costly

to change. Suppose for illustration that there is only one factor that is difficult to

change. When this is the case, the experimenter usually will fix the level of this factor

(ie., restrict the randomization scheme) and then run all combinations or a fraction of

all combinations of the other factors, which is known as a split-plot design. Too often,

the data obtained from this experiment are analyzed as if the treatment combinations

were completely randomized, which can lead to incorrect conclusions as well as a loss

of precision. Ainly.-i, of data obtained from experiments, such as the example above,

need to take the restricted randomization scheme into account.

1.1 Response Surface Methodology

In RSMI, the true response of interest, 7, can be expressed as a function of one or

more controllable factors (at least in the experiment being performed), x, by









2




q = .g(x) + c,

where the form of the function g is unknown and c is a random error term. The goal

is to find, in the smallest number of experiments, the settings among the levels of

x within the region of interest at which q is a maximum or minimum. Because the

form of g is unknown, it must be approximated. RSM uses Taylor series expansion to

approximate g(x) over some region of interest. Typically, first or second order models

are used to approximate g(x). The traditional RSM model would be

Yi= f(x)'/ + E,


where


yi is the ith response,

x, is the /th setting of the design factors,

f(x) is the appropriate polynomial expansion of x,

(3 is a vector of unknown coefficients, and

the ci's are assumed to be independent and identically (i.i.d.) distributed as

N(0, a2).


For a more detailed discussion on RSM see Khuri and Cornell (1996), Box and Draper

(1987), and Myers and Montgomery (1995).












1.2 Split-Plot Designs

A split-plot design often refers to a design with qualitative factors but can easily

handle quantitative factors. Also, a split-plot design usually has replication. However,

in the literature it has been common practice to refer to any design that uses one

level of restricted randomization regardless of replication as a "split-plot" design.

Therefore, in this dissertation, we will use the term split-plot design throughout.

When performing minultifactor experiments, there may be situations where com-

plete randomization might not be feasible. A common situation is when the nature

of the experiment or factor levels preclude the use of small experimental units. Often

a second factor can be studied by dividing the experimental units into sub-units. In

these situations, the split-plot experiment can be utilized. The experimental unit

is referred to as the whole plot while the sub-units are referred to as the subplots.

For every split-plot experiment there are two randomizations. Whole plot treatments

are randomly assigned to whole plots based on the whole plot design. Within each

whole plot, subplot treatments are randomly assigned to subplots with a separate

randomization for each whole plot. This leads to two error terms, one for the whole

plot treatments and one for subplot treatments as well as the interaction between

whole plot treatments and subplot treatments. Split-plot experiments have been used

extensively in agricultural settings. Even so, the following example from Montgomery

(1997) shows that there are applications for split-plot experiments in industrial set-

tings.

A paper manufacturer is interested in studying the tensile strength of paper based

on three different pulp preparation methods and four cooking temperatures for the












pulp. Each replicate of the full factorial experiment requires 12 observations, and the

experimenter will run three replicates. However, the pilot plant is only capable of

making 12 runs per day, so the experimenter decides to run one replicate on each of

three days. The days are considered blocks. On any day, he conducts the experiment

as follows. A batch of pulp is produced by one of the three methods. Then this

batch is divided into four samples, and each sample is cooked at one of the four

temperatures. Then a second batch of pulp is made using one of the remaining two

methods. This second batch is also divided into four samples that are tested at the

four temperatures. This is repeated for the remaining method. The data are given in

Table 1. This experiment differs from a factorial experiment because of the restriction

on the randomization. For the experiment to be considered a factorial experiment,

the 12 treatment combinations should be randomly run within each block or day. This

is not the case here. In each block a pulp preparation method is randomly chosen,

but then all four temperatures are run using this method. For example, suppose

method 2 is selected as the first method to be used, then it is impossible for any of

the first four runs of the experiment to be, say, method 1, temperature 200. This

restriction on the randomization leads to a split-plot experiment with the three pulp

preparation methods as the whole plot treatments and the four temperatures as the

subplot treatments. It should be noted that conducting a split-plot experiment, as

opposed to a completely randomized experiment, can be easier because it reduces

the number of times the whole plot treatment is changed. This usually will result

in a time savings which will lead to reduced costs. For example, suppose one is

interested in six subplot treatments and four whole plot treatments. Let the whole













Table 1: Data for Tensile Strength of Paper (from Montgomery (1997))


Block 1 Block 2 Block 3


Pulp Preparation Method 1 2 3 1 2 3 1 2 3
Temperature
200 30 34 29 28 31 31 31 35 32
225 35 41 26 32 36 30 37 40 34
250 37 38 33 40 42 32 41 39 39
275 36 42 36 41 40 40 40 44 45



plot treatments be comprised of a 22 factorial in time and temperature of a kiln. A

completely randomized experiment would require the kiln to be fired up quite possibly

24 times. With a split-plot experiment, the kiln only needs to be brought up to the

correct temperature 4 times per replicate. This leads to a savings of time and possibly

money.

A split-plot experiment can be run inside of many standard designs, such as the

completely randomized design (CRD) and the randomized complete block (RCB) de-

sign. As in the example from Montgomery (1997), suppose the split-plot experiment

is performed using a RCB design. Let Yijk denote the observation for subplot treat-

ment k receiving whole plot treatment i in block j. Kemnipthorne (1952) uses as his

model


Yijk = / + Ti + f3j + 6ij + 7Yk + (T-7)ik + Cijk for i= 1,2,...,t

j = 1,2,..., b













where


t is the number of levels for the whole plot treatment,

b is the number of blocks or replicates of the basic whole plot experiment,

s is the number of levels for the subplot treatment,

yi is the overall mean,

T, is the effect of the ith whole plot treatment,

Oj is the effect of the jh block,


8ij is the whole plot error term,


Yk is the effect of the kth subplot treatment,

(TY)ik is the whole plot treatment by subplot treatment interaction, and

Eijk is the subplot error.


He uses randomization theory to derive the expected mean squares summarized in

Table 2. In this table, ,6 is the experimental error variance for the whole plot treat-

ments, and a2 is the experimental error variance for the subplot treatments.

Many ariiily-t-. assume that the blocks are random and use an unrestricted mixed

model to derive the appropriate mean squares. The most common model for this

approach is














Yijk = P + T, + 3j + (Tf),i + Yk + (TY)ik + 6tjk (1)

: = 1,2, j = 1,2,.. b k = 1,2,...,s,

where


it is the overall mean,

T-: is the effect of whole plot treatment i,

Oj is the effect of block j,

(TOf3)ij is the block x whole plot treatment interaction,

-Yk is the effect of subplot treatment k,

(T77)ik is the whole plot treatment x subplot treatment interaction, and

Eijk is the subplot error.

The (T/3)ij term will be the whole plot error term for the case of an RCB design under

the usual assumption of no block x whole plot treatment interaction. The analysis

of variance table associated with the model in Equation (1), assuming whole plot

treatments and subplot treatments are fixed and blocks are random, is given in Table

3. If the block by whole plot interaction is called the whole plot error, then Tables 2

and 3 suggest the same basic testing procedures. The following additional constraints

and assumptions are needed for hypothesis testing:


T= 0, >k= 0,
i k












0, N(0 2) V(0, Or2) 6^~N(0, ar2)
N (0,Oo ),bi N (o), N

and where bij and Cijk are independent.
Montgomery (1997) uses a restricted mixed model as the basis for his analysis of

the following form


Yijkh = A1 + Ti + O3j + (TfO)i + fYk + (7-r)'4k + (0),)j, + (T[3-)i3k + iikh,

where

h = 1, 2,. ., r is the number of replicates,

(TOf)j is the random block by whole plot treatment interaction,

is the whole plot treatment x subplot treatment interaction,

(/30Y)jk is the random block by subplot treatment interaction,

(TO3-y)ijk is the random block by whole plot treatment by subplot treatment
interaction.

Under this restricted mixed model, the random interactions involving a fixed factor
are assumed subject to the constraint that the sum of that interaction's effects over
the levels of the fixed factor is zero. Table 4 gives the resulting expected mean squares,
which suggests that there are three distinct error terms. The block by whole plot by
subplot interaction is used to test the whole plot by subplot interaction; the block
by subplot interaction is used to test the main effect of the subplot treatment; and
the block by whole plot interaction is used to test the main effect of the whole plot.













Table 2: Expected Mean Squares Table Under Randomization Theory

Source df Expected Mean Square
Whole Plot Treatment t 1 ao, + I_ # T2


Blocks
Whole Plot Error


b-1
(t- 1)(b- 1)


8
Subplot Treatment s 1 a2 + b E 7y
ck=l

Whole Plot x Subplot (t 1)(s 1) a2 + (tb b E- i E=l (T)ik
(t1)(S 1) i Ek i(FYi

Subplot Error t(b 1)(s 1) a2



Note, if h = 1, the variance of Eijkh is not estimable. This restricted ailiuil.-is reduces

to the other two analyses only if the block by subplot interaction is unimportant. In

such a case, its contribution can be pooled with the block by whole plot by subplot

interaction to form the same error term as the randomization and unrestricted mixed

model analyses.

Whole plot treatments are applied to blocks of t units which can be divided

further into s subunits, where s is the number of levels of the subplot treatment. Any

differences among these blocks must be confounded with the whole plot treatment

comparisons. Consequently, comparisons among the subplot treatments are made

with greater precision, and this leads to the more important factor usually being

assigned to the subplot. Using the unrestricted model and Table 3, it is seen that the

null hypothesis of no whole plot treatment effect, H0 : rl = T2 .= Trt, versus at

least one not equal, is tested using the Block x Whole Plot Treatment interaction as


a2 + staj
2
or,













Table 3: Expected Mean Squares Table Under the Most Common Unrestricted Mixed
Model

Source df Expected Mean Square
Whole Plot Treatment t 1 a2 + saj + E ri2
'- =1

Blocks b 1 a2 + sa + sta

Block x Whole Plot Treatment (t 1)(b 1) a2 + sa6
Subplot Treatment s 1 a2 + k= k "
k=l
t s
(tC2+ b TY2
Whole Plot x Subplot (t- 1)(s- 1) a2 + b- E E (r')ik
i=1 k=1

Error t(b- 1)(s- 1) a2
N,,t.-: Whole Plot and Subplot Treatments are assumed fixed while Blocks
are assumed to be random.



Table 4: Expected Mean Squares Table Under the Restricted Mix:-. Mohdel


Source df Expected Mean Square
Whole Plot Treatment t -1 r2 + sr + Zi l T2

Blocks b -1 a2 + sa2 + sta
Block x Whole (t 1)(b 1) a2 + saU_
Subplot Treatment s- 1 ar2 + ta2 + ELI -Y

Block x Sub (b- 1)(s- 1) a2 + ta2

Whole Plot x Subplot (t 1)(s 1) a2 + or + b i l(T)2
0Block x Whole x Sub (-1)(-)(s-k1) a2+
Block x Whole x Sub (t 1)(b 1)(s 1) ar2 +F Or2.












the error term. The hypothesis of no subplot treatment effect, H0 : Y1 = 72 .. 'Y,

versus at least one not equal, is tested using Error which is also used to test the

significance of the whole plot x subplot treatment interaction.

Suppose the whole plot and subplot treatments are a factorial structure. In this

case, after the hypothesis tests above are performed, a more detailed investigation of

the individual factors and their interactions can be carried out. For example, consider

the situation discussed above with t = 4 whole plot treatments consisting of a 22

factorial in zl and z2 and s = 4 subplot treatments also consisting of a 22 factorial

in x, and x2. The t 1 = 3 degrees of freedom (df) for the whole plot treatments

can be partitioned into single df contrasts z1, z2, and z12z2. Likewise, the s 1 = 3

df for the subplot treatments can be partitioned into a single df contrasts X1, x2,

and xIx2. Also, the (t 1)(s 1) = 9 df for the whole plot x subplot treatment

interaction can be broken down into 9 single df effects involving z1, z2, xi, and x2

(see Table 5). Orthogonal contrasts should be calculated and tested for each factor

and the interactions using the appropriate error term from the original analysis. This

can be accomplished in SAS by using PROC GLM and the CONTRAST statement along

with the option E = error term after the model statement for the whole factors and

interactions. For the subplot factors, interactions among subplot factors, and whole

plot x subplot factor interactions, the aiialdy.is can be run a second time. In this

second imiilv'ik. the treatments in the model statement can be entered as factors and

interactions, similar to a regression model. The correct tests for the subplot factors

and whole plot x subplot factor interactions are given by SAS.

















Table 5: Analysis of Variance Table for a Split-Plot Experiment Run Using a RCB
Design With Factorial Structure and the Most Common Unrestricted Model

Source df
Whole Plot Treatment t 1 = 3
ZI 1
z2 1
ZlZ2 1

Blocks b- 1

Block x Whole Plot Treatment (t 1)(b- 1)
Subplot Treatment s 1 = 3
xl l+ 1
x2 1
XlX2 1

Whole Plot x Subplot (t 1)(s 1) = 9
zlxl 1t
z1X2 1
Z2X1 tt 1
Z2X2 i 1
zIx1x2 t 1
Z2XlX2 1
Z1Z2X1 1
Zl Z2X2 1t
Z1Z2X1X2 1

Error t(b- 1)(s- 1)


t These terms are tested using the Block x Whole Plot Treatment interaction.
tt These terms are tested using Error.












The concept of the split-plot design can be extended if further randomization

restrictions exist. For example, suppose there are two levels of randomization restric-

tions within a block in which case we might have a split-split-plot design. For a more

detailed discussion of split-plot designs and their extensions see Yates (1937), Cox

(1958), Wooding (1973) and .Mrito iiiery (1997).


1.3 Dissertation Goals


The focus of this dissertation is to enhance our understanding of the design and

anilyi, of split-plot experiments. The experiments considered will be industrial in

nature. As much as possible, the dissertation will focus on or discuss the types

of experiments that would be run in industry in terms of size and resources. An

important goal is to come up with methods that are clear, practical, and easy to

implement. In other words, this dissertation will address issues of real concern to

applied statisticians working in industry mand provide them some tools that can be

used with split-plot experiments. Below two industrial statisticians have been kind

enough to share real situations that help to show the relevance of the work in this

dissertation.

A Food Industry Example

Frozen heat-and-serve pastries, along with shelf-stable ready-to-eat pastries, rep-

resent a large segment of the convenience foods that today's consumers crave. Op-

timized proofing and baking operations are critical to the successful manufacture of

high quality baked goods such as these. However, as this market segment has grown,













so has the manufacturing capacity, which has necessitated the installation of new

proofers and ovens. Given the complexity of these operations, qualifying a new piece

of proofing or baking equipment poses a challenging experimental design problem:

how do you design an experiment to explore the operating profile for a new proofer

or oven?

As an example, consider a continuous oven, in which dough-based products move

through on a belt. The oven has two zones, which are controlled independently. In

each zone you can atju-t the Temperature, the Relative Humidity (RH), the Air Flow

Speed (AF), and the Residence Time. In general, the conditions in each zone will be

different, as each zone is used to impart different characteristics to the product. All

of these variables will impact the quality of the finished product.

Experimenting with this type of oven requires a restricted randomization. You

can easily reset the air flow and residence time in each zone on the fly, but changes in

the temperature and relative humidity require a waiting period to allow the oven to

return to steady state. Thus, oven experiments are typically conducted as a split-plot

design with four whole plot treatments, namely


Zone 1 Temp, Zone 1 RH, Zone 2 Temp, and Zone 2 RH,


and four split plot treatments, namely


Zone 1 AF, Zone 1 Res Time, Zone 2 AF, and Zone 2 Res Time.


In addition, we typically want to evaluate the effect of the oven on at least two prod-

ucts (,











product that is baked is evaluated in a variety of ways, including sensory characteri-

zations and analytical and physical testing.

What makes this experimental setup so difficult, is that exploring the profile of

a new oven must typically occur on prototype equipment at the oven manufacturer.

This means the experiment must be conducted in a very short period of time, often

two days or less. This makes it imperative that the experiment have as few runs

as possible usually between 12 and 20 runs. The research described in this thesis is

directly applicable to problems like this, and will be very useful for teams of process

engineers charged with gathering the data they need to fully evaluate candidate ovens

and proofers.


An Integrated Circuits Example


Integrated microflex circuits are manufactured over several, very complicated pro-

cess steps. Circuit plating is a key step in this process. It involves depositing a

uniform layer of copper on the microflex circuits. Copper thickness is a key quality

characteristic due to functionality issues. High variability in copper thickness results

in poor bonding of chips to these circuits. Some of the variables that effects circuit

thickness are the circuit geometry, the line speed, the current in amperes, the copper

concentration in the chemistry bath, and the concentration of both sulfuric acid and

hydrogen peroxide.

Designing experiments to optimize copper thickness is a challenge because of the

presence of hard-to-change variables. In particular, restricted randomization occurs

with circuit geometry, line speed, and current. Randomization is not restricted for












the remaining variables. Data from such experiments are analyzed by assuming that

it arose from a completely randomized experiment. Research in the area of split-plot

experiments with multiple whole plot and subplot factors is lacking and the work in

this dissertation should be of real help to industrial statisticians.


1.4 Overview


The literature review that follows in Chapter 2 is intended to familiarize the reader

with other work that discusses split-plot designs in RSI. Chapter 3 begins with some

background information onl 2k factorial experiments and the remainder of Chapter

3 is devoted to a more in-depth look at confounding in split-plot experiments. In

Chapter 4, a new model and class of designs for mixture experiments with process

variables will be developed in a completely randomized setting. Finally, the last

chapter will assume a split-plot structure for the mixture experiments with process

variables described in Chapter 4.
















CHAPTER 2
LITERATURE REVIEW


The split-plot error structure has been underutilized in RSM. Most RSM ex-

periments assume a completely randomized error structure. Letsinger, Myers, and

Lentner (1996, pg. 382) point out, "Unfortunately, while this completely randomized

assumption simplifies analysis and research, independent resetting of variable levels

for each design run may not be feasible due not only to equipment and resource con-

straints, but also budget restrictions." This chapter focuses on the literature involving

restricted randomization within RSM.


2.1 Split-Plot Confounding

When the whole plot and/or subplot treatments are of a factorial nature, it is

possible to reduce the number of whole plots and/or subplots needed through frac-

tionating. This is important in industrial experiments where constraints limit the size

of the experiment. Bartlett (1935) suggested the possibility of confounding higher-

order subplot interactions to reduce the number of subplots needed within each whole

plot. Later, split-plot confounding was studied by Addelman (1964). He provided

a table containing factorial and fractional-factorial arrangements that involve split-

plot confounding. However, he did not consider confounding within the whole plots.

Letsinger, Myers, and Lentner (1996) discuss the possibility of split-plot confounding












with the use of their noncrossed bi-randomization designs. Box and Jones (1992)

illustrate split-plot confounding using a cake mix example.

In some experiments, there are constraints on the number of subplots within each

whole plot. When the whole plots are arranged in a CRD, Robinson (1967) discussed

situations where the number of subplots per whole plot is less than the number

of subplot treatments. The whole plots are treated as blocks and then a balanced

incomplete block (BIB) design is used to allocate the subplot treatments to the whole

plots. If the whole plots are arranged in an RCB design, the same procedure can be

applied. If the number of whole plots per block is less than the number of whole plot

treatments, then an incomplete block design can be used there as well. Robinson

(1970) gave details on the case when both whole plot and subplot treatments are

arranged in incomplete block designs. Essentially, the procedure amounts to arranging

the whole plot treatments in blocks using a BIB design and then considering the whole

plots as blocks and arranging the subplot treatments in another BIB design. Robinson

(1970) provided formulas for the estimates of the main effects and interactions for

three cases: within whole plot, between whole plot within block, and between blocks.

Formulas are also given for the variance of the differences of these estimates for each

case.

Huang, Chen and Voelkel (1998) also investigate frictiiiiiitliiig, two-level split-plot

designs at both the whole plots and the subplots. They consider 2(n"l+2)-(kl+k2)

split-plot designs which are associated with a subset of the 2n-k fractional factorial

designs where n = n7 + n2 and k = k, + k2. The criterion used to select the optimal

design is that of minimum-aberration which is the design that has smallest number of












words in the defining contrast with the fewest letters. Two methods are presented for

constructing minimum-aberration split-plot designs. The first method decomposes

the 2 "-k design into the 2(n-+n2)-(kl+k2) split-plot design. This method is used to

derive extensive, though incomplete, tables of the designs. The second and more

complicated method which involves linear integer programming is used when the first

method fails.

Minimum-aberration two-level split-plot designs are also discussed in Binghamn

and Sitter (1999). A combined search and sequential algorithm is presented for con-

structing all non-isomorphic minimum-aberration split-plot designs which include the

tables of Huang, Chen and Voelkel (1998). Bingham and Sitter (1999) catalog designs

for 16 and 32 runs containing up to 10 factors. Included in this catalog are the second

and third best minimum-aberration designs since sometimes it may be desirable to

use these designs.


2.2 Split-Plots in Robust Parameter Designs


Genichi Taguchi proposed methods for designing experiments for product design

that are robust to environmental variables. The goal of robust design is to design an

experiment that identifies the settings of the design factors that make the product

robust to the effects of the noise variables. The design factors, which are factors

controlled during manufacturing, make up the inner array while the environmental

factors, or noise factors, make up the outer array. Environmental factors are fac-

tors that are difficult to control and can cause variation in the use or performance

of products. The experimental design or "crossed arrive" consists of crossing each












experimental design setting of the inner array with each experimental design setting

of the outer array. Unless the number of factors in these arrays is small, Taguchi's

designs become large and expensive.

An alternative to Taguchi's crossed array is the "combined" array. The combined

array utilizes a single experimental design in both the design and environmental fac-

tors. Therefore, the response is modeled directly as a function of the design factors

and the environmental factors using a single linear model. More details on the com-

bined array can be found in Welch, Kang, and Sacks (1990); Shoemaker, Tsui, and

Wu (1991); and O'Donnell and Vining (1997).

Bisgaard (1999) discusses split-plot designs in association with inner and outer

array designs. He focuses on screening experiments that use restricted randomization.

The paper gives a nice overview of defining relations and confounding structures for

the 2k-p x 2q-, split-plot designs. In addition to split-plot confounding, Bisgaard

(1999) points out that the same fraction of the subplot factors can be run in each

whole plot. The appropriate standard errors for testing effects when using split-plot

confounding are also given.

Box and Jones (1992) investigate the use of split-plot designs as an alternative to

the crossed array. They consider three experimental arrangements where the robust

parameter design is set up as a split-plot design:


1. arrangement (a) thle whole plots contain the environmental factors and the

subplots contain the design factors;












2. arrangement (b) the whole plots contain the design factors and the subplots

contain the environmental factors;


3. arrangement (c) the subplot factors are assigned in "strips" across the whole

plot factors (commonly called a strip-block experiment).


These three arrangements are illustrated through an example seeking the best recipe

for a cake mix. Three design factors have been identified as affecting taste. They are

flour, shortening, and egg powder and are studied using a 23 factorial design. The

consumer may have an oven in which the temperature is biased up or down. Also,

the consumer may overcook or undercook the cake. Therefore, the recipe is to be

robust to two environmental factors, oven temperature and baking time, whose levels

are combined using a 22 factorial design.


Arrangement (a)


Under this arrangement, the whole plots contain the environmental factors and

the subplots contain the design factors. Suppose there are m levels of the envi-

ronmental factors, E1, E2,..., Ej,... I E,, applied to the whole plots, n levels of

the design factors, D1, D2,..., Di,..., Dn, applied to the subplots, and I replicates,

rl, r2,..., ru,... ,ri, with the whole plots in I randomized blocks. For the cake mix

example, mn = 4, n = 8, and 1 = 1. Arrangement (a) requires m x n x I subplots

and m x I whole plots. Thus for the cake mix example, 4 x 8 x 1 = 32 cake mix

batches are required, but only 4 x 1 =4 operations of the oven are necessary. By

comparison, a completely randomized cross-product array would require 32 cake mix












batches and 32 operations of the oven. Thus, the split-plot arrangement has saved
time by reducing the number of operations of the oven.

The model for arrangement (a) is


Yijk = IL + 'Yk + aj + 71jk + i + (a6)ij + Eijk ,

where Yijk is the response of the kth replicate of the ith level of factor D and the jth

level of factor E, p is the overall mean, yk is the random effect of the kth replicate

with -7k N( 0, o2), aj is the fixed effect of the jth level of factor E, 6i is the fixed

effect of the ith level of factor D, (a6)j is the interaction effect of the ith level of D

and the jth level of E, Tjk N(O, a2,,) is the whole plot error, jk N(0, ar') is the

subplot error, and qgk and fijk are independent.

Arrangement (b)

With this arrangement, the whole plots contain the design factors while the sub-

plots contain the environmental factors. Arrangement (b) requires only 8 x 1 = 8 cake

mix batches but requires 4 x 8 x 1 = 32 operations of the oven. Again, a completely

randomized cross-product array would use 32 cake mix batches and 32 operations

of the oven. Here, the savings of the split-plot design is not as great since only the

number of cake mix batches is reduced. This is not an ideal situation for indus-

trial experiments. First of all, the design factors are of greater interest. Therefore,

applying the design factors to the whole plots results in a loss of precision for the

design factors. Hence, it is possible to have large differences between the levels of

the design factors that are insignificant when tested. Also, from an economic point












of view, arrangement (b) is costly. It requires an inefficient use of the environmen-

tal factors which in industrial experiments are typically the difficult or expensive to

change factors.

The model for arrangement (b) is


Yijk "= :+ 7 k + bi + Trik + Cj + (a6)i + ijk ,

where yijk is the response of the kth replicate of the ith level of factor D and the jth

level of factor E, p is the overall mean, -y is the random effect of the kth replicate

with 7Yk N(O, a'), aj is the fixed effect of the jth level of factor E, 6i is the fixed

effect of the ith level of factor D, (a6b)ij is the interaction effect of the ith level of D

and the jth level of E, Oik N(O, o,,) is the whole plot error, fijk N(O, a) is the

subplot error, and Oik and Eijk are independent.

Arrangement (c)

Now, consider the arrangement where the subplot treatments are randomly as-

signed in strips across each block of whole plot treatments (see Table 6). For the

cake mix example, suppose each of the n = 8 batches of cake mix is subdivided into

m = 4 subgroups. One subgroup from each batch is then selected, and these eight

are baked in the same oven at the appropriate temperature for the appropriate time.

This arrangement requires only 8 cake mix batches and only 4 operations of the oven.

Therefore, the strip-block experiment is easier to run than the completely randomized

cross-product design, as well as both arrangements (a) and (b).













Table 6: Strip-Block Arrangement (Box and Jones (1992))

Block 1 Block 2 Block 3
a,1 a2 a3 a3 a2 a1 al a3 a2
bl b2 b1
b2 bi b2




The model for the strip-block arrangement is


Yijk = p + k + aj -+ '1jk + bi + Oik + (O)ij + Cijk ,

where Yijk is the response of the kth replicate of the ith level of factor D and the jth

level of factor E, it is the overall mean, Yk is the random effect of the kth replicate with

Yk N(O, a2), aj is the fixed effect of the jth level of factor E, 6i is the fixed effect
of the ith level of factor D, (a6)iy is the interaction effect of the ith level of D and the
jth level of E. In arrangement (c), rqjk N(O, ao), Oik N(O, aD), 'ijk N(Oa2)

and 'tljk, Oik and Eijk are independent.

ANOVA tables for all three arrangements are given in Box and Jones (1992).

These tables indicate the appropriate denominators for tests involving the design

factors, the environmental factors, and their interactions assuming replication. When

there is no replication, normal probability plots, one for the whole plot factors and

one for the subplot factors and whole plot x subplot interactions, can be used to

select significant effects. Also, if the design and environmental factors are factorial

combinations, it may be possible to pool negligible higher order interactions to get

estimates of the whole plot and subplot errors.












It is of great interest to the researcher to learn how and which environmental

factors influence the design variables. This information is contained in the subplot

x whole plot interactions. However, Taguchi's anih.lyni is commonly conducted in

terms of a performance statistic, such as the signal to noise ratio (SNR). The SNR

is calculated for each point in the inner array using data obtained from the outer

array about that point. Therefore, Taguchi ignores any information contained in the

interactions of the design and environmental factors. This is generally considered to

be a serious drawback to the Taguchi analysis.

Phadke (1989) presented an example involving a polysilicon deposition process

which he analyzed using Taguchi's SNR's. Polysilicon film is typically deposited

on top of the oxide layer of the wafers using a hot-wall, reduced pressure reactor.

The reactant gases are introduced into one end of a three-zone furnace tube and are

pumped into the other end. The wafers enter the low-pressure chemical vapor depo-

sition furnace in two quartz boats, each with 25 wafers, and polysilicon is deposited

simultaneously onil all 50 wafers. The desired output of this process is a wafer which

has a uniform layer of film of a specified thickness. Six design factors each at three

levels were identified: temperature, pressure, nitrogen flow, silane flow, setting time,

and cleaning method. Tube location and die location were considered noise factors.

Three responses, film thickness, particle counts, and deposition rate, were of interest.

The smaller the better SNR was used in the analysis for particles, the target is best

SNR was used for thickness, and a 20 logo10 transformation was used for deposition

rate. The data were analyzed using ANOVA techniques to determine the effect of












each design factor on the responses. A more detailed discussion of the selection of

fict.,r.,, design, and analysis of the SNR's is contained in Chapter 4 of Phadke (1989).

The actual structure of this experiment was a split-split-plot design because there

are three sizes of experimental units with different sources of variation. The design

factors are applied to the tube (run-to-run variability); the location in the tube affects

the wafer (wafer-to-wafer variability), whereas location in the wafer affects the die

(die-to-die variability). Therefore, using Taguchi's SNR's to analyze this experiment

will result in a complete loss of information in the design x noise factor interactions.

Cantell and Ramirez (1994) reanalyzed the data as if it were a split-split-plot design.

They pooled higher order interactions to get the necessary error terms in order to

perform hypothesis tests on the design factors and the design x noise factor inter-

actions. Interaction plots were used to determine the level of the design factor that

minimized the variation across the levels of the noise factors. Although the final

recommendations on the design factor levels by Cantell and Ramirez (1994) differed

from Phadke (1989) on only one of the six design factors, the use of the split-split-plot

design has allowed the process engineer to have a better understanding of the sources

of variation. This added information may lead to process improvement in the future.

Kempthorne (1952) and Box and Jones (1992) provide details on the relative

efficiency of these split-plot designs compared to the CRD and RCB. A summary of

their conclusions is provided here. Consider the split-plot experiment as a uniformity

trial. If the uniformity trial was run as a CRD or a RCB experiment, then, for

arrangements (a) and (b), the subplot factor effects and the subplot x whole plot

interactions are estimated more precisely than the whole plot factor effects. Compared












with arrangements (a) and (b), the strip-block design estimates the subplot x whole

plot interactions more precisely but the subplot factor effects with less precision.

However, the whole plot factor effects are estimated with equal precision. Based on

these results, arrangement (a) with the environmental factors applied to the whole

plots is generally preferred over arrangement (b). Both the strip-block design and

the split-plot design with the design factors applied to the subplots can be extremely

useful in robust parameter design.

2.3 Bi-Randomization Designs


Letsinger, Myers, and Lentner (1996) introduced bi-randomization designs (BRD's).

BRD's refer to designs with two randomizations similar to that of a split-plot design.

The whole plot variables are denoted by z = (z1, z2,..., zZ) while the sub-plot vari-

ables are denoted by x = (Xl, X2,..., xx). Hence, the ith design run is (zi, xi). BRD's

are broken into two classes, crossed and non-crossed. Crossed BRD's are constructed

as follows:


1. randomize the a unique combinations of z to the whole plot experimental units

(EU's), then


2. randomize the b levels of x to the smaller EU's within each whole plot (see

Table 7).


Thus every level of x is "crossed" with every level of z. These designs are the usual

split-plot designs.














Table 7: Crossed BRD From Letsinger,
Myers, and Lentner (1996)

ZI X1 ... Xb
Z2 XI ... Xb


Za X1 Xb



Table 8: Noncrossed BRD From Letsinger,
Myers, and Lentner (1996)

Zi X1 Xibj
Z2 X21 X2b2


Za Xal ... Xab,



The non-crossed BRD's differ from the crossed BRD's in that not all levels of x

are associated with zi. The whole plots have different levels of the sub-plots and need

not have the same number of levels. Non-crossed BRD's are constructed as follows:


1. randomize the a unique combinations of z to the whole plot EU's, then


2. randomize the bi levels of x to the smaller EU's within each whole plot (see

Table 8).


The distinction between these two can be thought of in terms of the sub-plot factors.

The crossed BRD might be represented by a 2k factorial in the sub-plot factors while

the non-crossed BRD might use a 2k-p fractional-factorial in the sub-plot factors but

not the same 2k-p set of treatments.












For both crossed and non-crossed BRD's, the two randomizations complicate the
error structure. The first randomization leads to the whole plot error variance, a,
while the second randomization leads to the sub-plot variance, ao. It is assumed that

the covariance between any two observations on the same whole plot is constant over
all whole plots and that observations on two sub-plots from different whole plots are
uncorrelated. The response surface model is

y=X/3 + 6 +E,

where 6 + E N(0, V) with V = aOJ + o2I, where J is a block diagonal matrix of

lb x 1' and where b, is the number of observations within the ith whole plot. Now
using generalized least squares (GLS), the maximum likelihood estimate (MILE) of
the response surface model is

(x'V' X' XV-1y (2)

with
Var() (X'V-1X)1. (3)

From Equation (2), it is seen that the model estimation depends on the matrix V
and thus both a2 and ao.
Suppose that the response surface model is partitioned into the whole plot and
sub-plot terms as
y = Z, + X*f3* + Z'AX*,

where A is a matrix of whole plot x sub-plot interaction parameters. The response
surface design should be large enough to test for general lack of fit as well as lack of












fit from the whole plots. Therefore, the number of whole plots available must exceed

the number of parameters in -f.

For the crossed BRD, there is an equivalence between ordinary least squares (OLS)

and GLS. This equivalence means that Equation (2) becomes


S= (X'X X'y

and the model estimation no longer depends on the error variance. However, for

testing purposes, the error variance must be estimated. Letsinger, Myers, and Lentner

(1996) suggest augmenting the response surface model with insignificant whole plot

terms, Z*p, to saturate the a 1 whole plot degrees of freedom. The whole plot

saturated model can be used to calculate lack of fit sums of squares for both the whole

plots and the sub-plots. Then approximate t-tests can be formed by substituting the

estimated error variances into Equation (3).

Non-crossed BRD's present a more complicated situation. The equivalency of OLS

and GLS is only true in the case of a first-order model. Although more complex, the

above method can be adapted for the first-order case. Letsinger, Myers, and Lentner

(1996) compare three methods for the second-order case. They are OLS, iterative

reweighted least squares (IRLS), and restricted maximum likelihood (REML). Though

IRLS and REML appear to be better methods, the "b-t." method depends on the

design, model, and any prior information.
Bi-randomization introduces the need for new definitions for design efficiency be-

cause efficient designs in the literature are based on a completely randomized error

structure. For example, for the BRD the D-optimality criterion (see, eg., Kiefer and












Wolfowitz (1959)) becomes

min N (X'V-'X)1

over all designs D. Letsinger, Myers, and Lentner (1996) provide comparisons of

several first- and second-order designs. For the second-order de.
central composite design (CCD) proves to be a good design.

2.4 Split-Plots in Industrial Experiments

Lucas and Ju (1992) investigated the use of split-plot designs in industrial experi-

ments where one factor was difficult to change and its levels served as the whole plot

treatments. They began their study with a simulation exercise using a four factor

face-centered cube design with four center points. They let x, correspond to the

hard-to-change factor, while x2, x3, and x4 were easy to vary. This design allowed for

the fitting of the quadratic model
4 4 3 4
Y +00 + x + iX2 + O3ijXiXj +.
i=1 i=1 i=1 j=i+l
However, since the error was the only term of interest, all the regression coefficients

can be zero. Therefore, data was generated using

Y = e = EU, + es ,

where ,, N(0, (,2,,) was the error term associated with changing the level of x, and

c, N(0, 0,) was the error associated with any new experimental run. Twentyeight

runs were generated using the following steps:


1. Generate E,,, N(O,a ,) and E, N(O, o).













2. Y = Ew+CS.


3. If the level of x, of the current run is different from that of the previous run, a

new value of both E,, and F, is generated. Otherwise, generate a new value for

E, only.


4. Go to step 2 until all 28 runs are completed.


The data were generated for completely randomized, completely restricted, and

partially restricted run orders. For a partially restricted run order, each level of

the hard-to-change factor was visited exactly twice and the runs at each level were

randomly divided into two equal groups. Each time a data set was generated, the

least squares estimates of the 13's were computed and the residual error was estimated.

The simulation procedure was repeated 1,000 times.

Lucas and Ju (1992) summarized their simulation results in a table with a listing

of the standard deviations of the regression coefficients for the three different ways

of running the experiment. The restricted randomization case has a much smaller

residual standard deviation and much smaller standard deviations for all the regres-

sion coefficients except those associated with the hard-to-change factor, 031 and I11.

These results correspond with the general result that split-plot designs will produce

increased precision on the subplot factors while sacrificing precision on the whole plot

factors. The magnitudes of the coefficients of the estimated standard deviations for

the partially restricted case were greater than those with the completely randomized

case but less than the corresponding estimates for the completely restricted case.












A similar simulation was conducted for two-level factorials (see Lucas and Ju

(1992)). They considered a 24 factorial with x, as the hard-to-change factor. This

allows the fitting of a regression model that includes the linear and interaction terms.

Again, a summary table is provided by Lucas and Ju (1992) showing s similar results to

the other experimental scenarios. The completely restricted experiment had smaller

standard deviations for all the regression coefficients except 01. Table 9 gives the

formula for the variance of the regression coefficients for a 2k factorial experiment

with one hard-to-change factor. Recall that in the partial restricted case, the blocking

was done at random. This can be improved on by blocking orthogonally. The 24

factorial can easily be blocked orthogonally in 4 blocks of size 4 or 8 blocks of size 2.

Both of these blocking schemes are an improvement over the partially restricted case

in that they have smaller standard deviations on the easy-to-vary factors.

Cornell (1988) discusses the analysis of data from mixture experiments with pro-

cess variables where the mixture blends are embedded in the process variable com-

binations as in "a split-plot design". The mixture process variables are factors that

are not mixture ingredients but whose levels could affect the blending properties of

the mixture components. To illustrate this situation, Cornell uses an example from

Cornell and Gorman (1984) involving fish patties. The mixture experiment involves

making fish patties from different blends of three fish species (mullet, sheepshead,

and croaker). The patties were subjected to factor level combinations of three process

variables (cooking temperature, cooking time, and deep-frying time). Each process

variable was studied at two levels. When process variables are included in a mixture












experiment, complete randomization tends to be impractical. This leads to a restric-

tion on randomization and lends itself to the split-plot design.

Cornell (1988) considers factor-level combinations of the process variables as the

whole plot treatments and the mixture component blends as the subplot treatments,

but points out that their roles can be switched. Hence, a combination of the levels

of the process variables is selected and all blends are run at this combination. An-

other combination of the process variable levels is chosen and all blends are run at

this combination. This procedure is continued until all combinations of the process

variables are performed. Following a replication of the complete design, the split-

plot nature of the experiment leads to two error terms which are used to assess the

significance of the effects of the whole plot treatments, the subplot treatments, and

their interaction. Several regression-type models are considered for estimating the

effects of the process variables, the blending properties of the mixture components,

and interactions between the two. The paper explains how to estimate the regression

coefficients as well as how to obtain variances and perform hypotheses tests. Both

balanced and unbalanced cases are considered. The hypothesis testing procedures are

illustrated with two completely worked-out numerical examples.

Santer and Pan (1997) discuss subset selection procedures for screening in two-

factor treatment designs. The paper deals mainly with split-plot designs run in com-

plete blocks; however, the strip-plot design is also discussed. One factor serves as the

whole plot factor while the other is the subplot factor. The goal is to select a subset

of the treatment combinations associated with the largest mean. Subset selection

procedures are given for additive and nonadditive factor cis,. where neither of the













Table 9: Variance of the Regression Coefficients For a 2k With
One Hard-To-Change Factor (from Lucas and Ju (1992))

Var(b) =- A, + Ba,

Hard To Change Variable Other Terms
A B A B

1-P P I1-P
i- -p-i

P = 1/(2k-2 + 1) for the completely randomized
design.
P = 1 for the completely restricted design.
P = (2k-l 2)/[2(2k-l 1)] for the partially
restricted design.


procedures depend onl the block variance.

Miller (1997) considers various fractional-factorial structures in strip-plot experi-

ments. These strip-plot experiments are identical in nature to the strip-block experi-

ments, arrangement (c), discussed in Box and Jones (1992). Strip-plot configurations

can be applied when the process being investigated is separated into two distinct

stages and it is possible to apply the second stage simultaneously to groups of the

first-stage product. Miller uses an example involving four washing machines and four

dryers in two blocks. Sets of cloth samples are run through the washing machines,

and then the samples are divided into groups such that each group contained exactly

one sample from each washer. Each group of samples would then be assigned to one

of the dryers. The response of interest was the extent of wrinkling.

It is convenient to represent strip-plot structures as rectangular arrays of experi-

mental units in which the levels of one treatment factor (or set of factors) are assigned












to the rows and the levels of a second treatment factor (or set of factors) are assigned

to the columns. Table 10 represents the laundry experiment in which each square

represents a cloth sample, rows represent sets of samples that were washed together,

and columns represent sets of samples that were dried together. The ANOVA table

for the laundry example, which is divided into "strata" corresponding to blocks, rows,

columns, and units, is given in Table 11. When making inferences about the effects

in a particular stratum, the estimate of variation must be based on the residual term

for that stratum.

Miller (1997) proposes a method for constructing strip-plot configurations for

fractional-factorial designs which consists of three steps:

1. Identify a suitable design for applying row treatments to rows ignoring columns;

2. Identify a suitable design for applying column treatments to columns ignoring

rows;

3. Select a suitable fraction of the product of the row and column designs.

The method is applied for two-level designs and then extended to m-level and mixed-

level designs. The procedure for two-level designs is presented here; for details on the

extended cases, see Miller (1997).

Consider the situation in which a proper fraction of a two-level factorial design is

to be run in a strip-plot arrangement using b = 2 blocks. Each block has r = 2M

rows and c = 2" columns. Let K and k represent the number of row and column

factors, respectively, and define Q = K (w + M) and q = k (w + mn). Then, the

procedure is as follows:


















Table 10: Strip-plot Configuration of the
Laundry Experiment (from Miller (1997))

Dryer Dryer
Washer 1 2 3 4 Washer 1 2 3 4
1 1
2 2
3 3
4 4
Block 1 Block 2


Table 11: ANOVA Table for the Laundry Example (from Miller (1997))

Strata Source df E(MS)
Block Blocks 1 a2 + 4a0 + 4a0, + 16a2
4
Row W-Washer 3 a2 + 4aR + (8/3) E W?
j=1
Row Residual 3 a2 + 4aR
4
Column D-Dryer 3 a2 + 474 + (8/3) Z Dk
j=l
Column Residual 3 a2 + 47C
4 4
Unit W x D 9 a2 + (1/9) E E [WD]?k
j=1 k=i
Unit Residual 9 a2












1. Select a row design that consists of a 2K-Q design in b blocks;

2. Select a column design that consists of a 2k-q design in b blocks;

3. Consider the product of the designs in steps 1 and 2 and select a Latin-Square

fraction of this product.

The selection of the design in steps 1 and 2 can be made on the basis that the analyses

for the row stratum and the column stratum will essentially be the analyses of these

designs. The Latin-Square fraction is selected so that the confounding array effects

in the unit stratum have desirable properties.

Mee and Bates (1998) consider split-lot experiments involving the etching of silicon

wafers. These experiments are performed in steps where a different factor is applied

at each step. Thus, there are an equal number of steps and factors. Specifying a

split-lot design involves determining the following:


1. the number of process steps with experimentation;

2. the number of factors and their levels at each processing step with experimen-

tation;

3. the subplot size at each processing step;

4. the number of wafers (experimental units) in the entire experiment;

5. a plan that details for each experimental wafer the process subplot at each step.

Mee and Bates emphasize symmetric designs, which are designs having the same

subplot size at each experimentation step.












The experimental plan in item 5 above will be determined as follows. First, to

define b subplots at each step, obtain b 1 contrasts for each experimental step. Then

assign factors to contrasts within the group intended for their respective processing

step. This is done in a way that gives the most information on the interaction effects

of interest. The approach is to determine a set of independent contrasts that can

be cycled to produce additional sets. The initial set of independent contrasts must

be chosen so that the groups of effects remain disjoint. This process and the result-

ing designs are illustrated for a variety of 64-wafer experiments (see .r.v and Bates

(1998)). Split-lot designs for three-level factors are also discussed. It should be noted

that if there are only two steps, the procedures by Miller (1997) can be applied with

one or with many factors at each step.
















CHAPTER 3
INCOMPLETE SPLIT-PLOT EXPERIMENTS


The focus of attention in factorial experiments centers on the effects of numerous

factors and their interactions. An important class of factorial experiments is the 2k

factorials where each of the k factors is assigned two levels. These experiments are

very useful in exploratory investigations as well as optimization problems because

they allow a large number of factors and their interactions to be examined.

Since there are only two levels of each factor, they will be denoted as low and high

for ease of reference. A treatment combination pertains to a level of each and every

factor and will be designated by lower case letters using the following conventions:

If a factor is at its low level, the corresponding letter is omitted from the treat-

ment designation. Conversely, if a factor is at its high level, the corresponding

letter is included.

When all factors are at their low levels, the treatment will be designated by the

symbol (1).


Under this notation, the treatments for a 22 factorial experiment in factors P and Q

are designated as (1), p, q, and pq. Factors and their effects will be designated by

capital letters.

Factorial experiments become large very rapidly so that often a single replicate

of the N = 2k runs requires more resources than are available, even with a moderate













number of factors, k. Even when resources are available, we may not want to estimate

all of the 2k 1 factorial effects. As an example, with k > 3, interactions involving

3 or more factors are generally considered to be negligible or of little importance.

Thus, a single replicate of a 2' requires 128 experimental units and provides a 64-fold

replication of each main effect. Of the 127 effects that can be estimated, only 28 may

be of major interest (seven main effects and 21 two-factor interactions).


3.1 Fractional Factorials


Finney (1945) proposed reducing the size of the experiment by using only a frac-

tion of the total number of possible treatment combinations. Such experiments are

called fractional factorials. He outlined methods of constructing fractions for 2k and
3k experiments. For screening purposes, Plackett and Burman (1946) gave designs

for the minimum possible number of experimental units, N = k + 1 where N is a

multiple of 4, and pointed out their utility in physical and industrial research. Since

then, these designs have found many applications, particularly in industrial research

and development. Their chief appeal is that they enable a large number of factors,

generally 5 or more, to be included in an experiment of practical size so that the

investigator can discover quickly which factors have an effect on the response. In this

chapter, the discussion will be limited to the case where every factor has only two

levels.

A 2k experiment that is reduced by a factor of 2-p will be called a 2k-p fractional

factorial experiment. These experiments have two major problems which can limit

their usefulness:












1. Every linear contrast of the treatments estimates more than one effect; hence,

each effect is aliased with one or more other effects. This can lead to the

misinterpretation of an effect which is not likely to happen with a complete

factorial experiment.

2. There is no independent estimate of experimental error.


Despite these limitations, fractional factorial experiments are used in exploratory

research and in situations that permit follow-up experiments to be performed. They

have been especially useful in industrial research and development where experimen-

tal errors tend to be small, the number of factors being investigated is large, and

experimentation is sequential. As a tool for exploratory research, fractional factorials

provide a means to efficiently evaluate a large number of factors using a relatively

small number of experimental units. This allows important factors to be detected

and unimportant factors to be screened or discarded rather than committing a large

amount of experimental resources on all of the factors.

Effects that are estimated by the same linear combination of treatments are called

aliases. Which effects are aliased depends on the factorial effects used to select the

treatments. The defining contrast is the effects) that is confounded with the constant

effect, I. It can be represented as an equation by setting the confounded effect equal

to I. The alias chain for an effect is found by forming the generalized interaction of

the effect with all terms in the defining contrast. For example, if a 23-1 fraction in

factors A, B, and C is run with defining contrast I = ABC, then the alias of the

main effect A is A(I) = A2BC which gives A = BC since A2 = I. Therefore, the













alias chains for the main effects, A, B and C are as follows:

A =BC
B = AC
C =AB.

For a 2k-P, there are 2P 1 effects in the defining contrast. The experimenter can

select any p factorial effects to be the defining contrast. The remaining 2P p 1

factorial effects are automatically determined as being the generalized interactions

among the p effects.

Box and Hunter (1961a, 1961b) classified fractional factorial plans by their degree

of aliasing of effects. This measure is called the resolution of the plan. The number

of letters in the shortest member of a set of defining contrasts determines the design's

resolution. Three important resolutions are


1. Resolution III- in which main effects are aliased with two-factor interactions;

2. Resolution IV in which main effects are aliased with three-factor interactions

and two-factor interactions are aliased with other two-factor intera;i tioii-:

3. Resolution V where two-factor interactions are aliased with three-factor in-

teractions.


Of course, if all three-factor and higher interactions are negligible, a design with

Resolution V is desired because it will allow the estimation of all main effects and

two-factor interactions since they are aliased with negligible effects.












3.2 Confounding

Suppose that a 2k factorial experiment is to be run in blocks. As noted earlier, the

main disadvantage of 2k factorial experiments is their size. Consequently, even for a

moderate number of factors, it may not be possible to find blocks with the required

number of homogeneous experimental units. When this occurs, it is necessary to use

smaller-sized blocks or incomplete block designs.

With an incomplete block design, there must be some loss of information. A

balanced incomplete block design, if it exists, distributes this loss equally to all treat-

ments. However, in factorial experiments, it is the main effects and interactions that

are important. For most factorial experiments with more than three factors, it is

highly unlikely that all effects, especially the higher-order interactions, are important.

If some effects can be assumed negligible prior to performing the experiment, then

a better procedure for constructing incomplete blocks, originally suggested by Fisher

(1926), would be finding arrangements which completely or partially sacrifice the in-

formation on these effects so that full information can be obtained on the rest. This

is done by forcing the comparisons among the blocks to be identical to the contrasts

for the negligible effects. Effects that are estimated by the same linear combination

of the treatments are said to be confounded. As a result, it is impossible to determine

if the observed difference is due to differences in blocks or due to the factorial effects

that are aliased woth the blocks.

Effects selected to be confounded with blocks are called the defining contrasts

since they determine which treatments will occur together in a block. These effects

are selected by the experimenter and should be effects thought to be negligible since












they are no longer separately estimable. Generally, these effects will be three-factor

interactions or higher so that all main effects and two-factor interactions can be

estimated.

When the block size of 2k is reduced by 2-p, each block will contain 2k-p experi-

mental units and each complete replicate will contain 2P blocks. In this case, it will

be necessary to confound 2P 1 effects in each replicate. The experimenter chooses p

of these effects with the remaining 2p p 1 effects being the generalized interactions

of the original p effects. When more than one replicate of the 2k-p fractional factorial

is performed, two types of confounding are possible:


1. Complete the same set of effects is confounded in each replicate;

2. Partial different sets of effects are confounded in different replicates.


Complete confounding is used whenever all information on the confounded effects

can be sacrificed. This should only be used when all confounded effects are believed

to be negligible. Complete confounding creates no problems with the analysis. It is

only necessary to find the effect totals for all unconfounded effects.

There are situations where effects believed to be important must be confounded,

for example, when available resources force the use of small block sizes. In these cases,

partial confounding is used. Partial confounding means confounding different effects

in different replicates so as to allow estimation of all effects. These estimates use only

the data from the replicates in which the effect is unconfounded. Thus, there will be

greater precision on effects that are unconfounded than on effects that are partially












confounded. While the amount of information is reduced, statistical significance of

each effect can be ascertained.


3.3 Confounding in Fractional Factorials


Although only a fraction of the treatments are included in a 2Ck-P experiment, this

number may still be too large for available blocks. As in any factorial experiment,

confounding is used to reduce the block size. Confounding an effect in a fractional

factorial experiment also confounds all of its aliases.

Consider a 26`1 fractional factorial experiment using the "be.r" defining contrast

for a half-replicate, I = ABCDEF. This requires 32 homogeneous experimental

units. If these are not available, then blocks of smaller size can be created by con-

founding additional effects. Suppose blocks of size 16 experimental units are available.

To create two blocks of size 16 for the 32 treatments it is necessary to confound one

effect. Since ABCDEF was used to define the half-fraction, it would appear logical

to select a five-factor interaction, say, ABODE. However, the alias of this interac-

tion or generalized interaction of the effect with ABCDEF is F and will also be

confounded with blocks. A better choice is to confound any three-factor interaction

since its alias will also be a three-factor interaction. As a result, no information is

lost on potentially important effects.

The word '"b(-,t." should be clarified. It is referring to the design which has the

least amount of aliasing among important effects which are usually thought to be

main effects and two-factor interactions. If important effects are not aliased with each

other, then 'Ljvt" refers to the design with highest Resolution. Therefore, "best" is













a criterion based onil estiinability. Throughout this chapter, wherever the phase 'best

design" is used it will be under the above setting.

3.4 Combining Fractional Factorials and
Confounding in Split-Plot Experiments

Splitting the plots or experimental units is possible with any experimental design.

The design refers to the assignment of the whole plot and subplot treatments and is

selected in order to control the known sources of extraneous variation. Regardless of

the choice of design, the subplot treatments can be thought of as being arranged in

blocks where the whole plots are the blocks. In each whole plot, if all the subplot

treatments can be run, then the situation resembles that of a complete block design

as far as the subplot treatments are concerned. However, there are situations where

in each whole plot not all of the subplot treatments can be performed so that some

form of an incomplete block design must be used. If the subplot treatments result

from a 2k factorial structure, then the methods discussed in the previous sections of

this chapter can be applied to reduce the number of subplot treatments in a whole

plot.

Consider the situation where both the whole plot treatments and the subplot

treatments have a 2k factorial structure. Assume that the design for the whole plot

treatments is a CRD. Suppose, only a fraction of the whole plot treatments are of

interest and only a fraction of the subplot treatments can be run for each whole plot.

We will consider the situation involving noise factors and design factors. The noise

factors will be the whole plot factors. Therefore, the goal of the experiment is to

estimate the following:












main effects for the whole plot factors;

main effects for the subplot factors;

two-factor interactions between the whole plot and subplot factors;

and if possible, two-factor interactions among the subplot factors.

Note that if there were sufficient resources to run all whole plot treatments and

subplot treatments, then all four goals would be automatically satisfied. However, in

most situations, this is not economically possible. Therefore, we shall try to estimate

as many effects as is possible within the restrictions on the resources available.

The idea of confounding effects in order to reduce the number of subplot treat-

ments per whole plot treatment and achieve the second goal has been around for some

time. Kempthorne (1952) has a section devoted to confounding in split-plot experi-

ments. Addelman (1964) also discusses ways of accomplishing this. Recently, the use

of split-plot experiments in industry has generated renewed interest in confounding.

Huang, Chen, and Voelkel (1998) and Bingham and Sitter (1999) discuss minimum-

aberration designs for factors with two-levels. This technique helps to improve the

estimation problem by raising the resolution concerning the subplot factors, but one

must be careful with the whole plot x subplot interactions. Bisgaard (1999) uses

inner and outer arrays, with factors at two-levels, as in robust parameter design and

provides the standard errors for various contrasts among the whole plot and subplot

factors.

We will use an example to compare the use of confounding in a split-plot exper-

iment. Consider a split-plot experiment with three whole plot factors, A, B, and C,












and three subplot factors, P, Q, and R where all factors have two levels. Suppose

only 16 runs are possible among the 64 total number of combinations. There are two

ways to allocate the whole plots and subplots for this experiment. We can use four

whole plots with each whole plot containing four subplots or we can use eight whole

plots with each whole plot containing two subplots. The goal of the experiment is to

estimate all six main effects and as many of the nine two-factor interactions between

the whole plot and subplot factors as is possible, although it is believed that some

two-factor interactions among the subplot factors might be significant. To conserve

space in the tables, the confounding structure or alias chains will be given only up to

order two. Therefore, if there is a blank space in the alias table, it means that the

effect is aliased with interactions of order higher than two.

First, suppose that the experimenter ignores the split-plot structure by consid-

ering the factors as a 26 factorial in a completely randomized design. Actually, the

experimenter would use a 26-2 fractional factorial design to obtain the 16 runs. The

best defining contrast is


I = ABCP = CPQR = ABQR,

which has Resolution IV. The layout is given in Table 12 and the alias chains are

given in Table 13. All main effects can be estimated, but two-factor interactions are

aliased with each other. Even if we assume that all two-factor interactions among A,

B, and C and all two-factor interactions among P, Q, and R are negligible, there is

still a problem since AQ is aliased with BR and AR is aliased with BQ. In other

words, some of the interactions we are interested in are aliased with each other.

















Table 12: Design Layout for 26-2 With Defining
Contrast I = ABCP = CPQR = ABQR


abcp ab cp
acr acq cpqr
bcq bcr bpr
apq qr apr


abcpqr
abqr
bpq
(1)


Table 13: Alias Structure for 26-2


A
B
C
P
Q
R
AB
AC
AP
AQ
AR
CQ
CR


CP+QR
BP
BC
BR
BQ
PR
PQ












Table 14: Design Layout for the Combined 23-1 x 23-1
With Defining Contrast I = ABC = PQR = ABCPQR

a b c abc
P P P P
pqr pqr pq pqr

q q q q
r r r r
pqr pqr pqr pqr


A second method, incorporating the split-plot nature and using four whole plots,
is to consider reducing the whole plot factors and subplot factors separately using
fractional factorials. A 23-1 fractional factorial with defining contrast I = ABC
will be used for selecting the whole plot treatments and combined with a 23-1 with
defining contrast I = PQR in selecting the subplot treatments (see Table 14). The
overall defining contrast for the experiment is

I = ABC = PQR = ABCPQR,

and the alias structure is shown in Table 15. Once we consider the split-plot structure,
the best we can do at the whole plot level is a Resolution III design. This method
provides a good design for estimating the two-factor interactions between the whole
plot and subplot factors. However, we must assume that the two-factor interactions
among the subplot factors are negligible in order to estimate the main effects for the
subplot factors.
Method three uses split-plot confounding and four whole plots. At the whole
plot level, a 23-1 fractional factorial with defining contrast I = ABC is used. Then,
the three-factor interaction, PQR, is confounded with factor C to reduce the eight













Table 15: Alias Structure for 23-1 x 23-1

A = BC
B = AC
C = AB
P = QR
Q = PR
R = PQ
AP =
AQ =
AR =
BP =
BQ =
BR =
CP =
CQ =
CR =



subplot treatments to four per whole plot (see Table 16). The idea is to put the

positive fraction of PQR wherever C is positive and the negative fraction wherever

C is negative. The overall defining contrast is given by


I = ABC = CPQR = ABPQR


with the alias structure provided in Table 17. This design is better than the second

design in terms of aliasing of the main effects for the subplot factors, but cannot

estimate all nine whole plot by subplot factor interactions without assuming that PQ,

PR, and QR are negligible. If, on the other hand, it is reasonable to assume that the

whole plot factor C will not interact with any of the subplot factors, then PQ, PR,

QR, the main effects for subplot factors and the remaining six whole plot by subplot













Table 16: Design Layout for Split-Plot Confounding
With Defining Contrast I = ABC = CPQR = ABPQR

a b c abc
(1) (1) p p
pq pq q q
pr pr r r
qr qr pqr pqr


factor interactions can be estimated using this design. Also, on a iiii.,iltini level,

some experimenters would feel more comfortable with this design since it uses all 8

subplot treatments.

The fourth method uses eight whole plots and split-plot confounding. Since there

are eight whole plots, the complete 23 factorial can be used for the whole plot fac-

tors. However, we must now reduce the number of subplots to two per whole plot.

This implies that we must confound two members in the defining contrast and their

generalized interaction completes the defining contrast. Using split-plot confounding,

the defining contrast is


I = ABPQ = ACQR = BCPR,


with the layout given in Table 18 and the alias structure given in Table 19. This design

is good for estimating main effects but has some serious deficiencies with interactions.

One possible problem with designs that use eight whole plots is cost. If the whole

plot factors are costly to change, then using eight whole plots as opposed to four

might be impractical. Another problem with designs using eight whole plots is the

breakdown of the degrees of freedom. There are 7 df for the whole plot design and














Table 17: Alias Structure for Split-Plot Confounding

A = BC
B = AC
C = AB
P=
Q =
R =
AP =
AQ=
AR=
BP=
BQ=
BR=
CP = QR
CQ = PR
CR = PQ


Table 18: Design Layout for Split-Plot Confounding
in 8 Whole plots With I = ABPQ = ACQR = BCPR

(1) a b ab c ac bc abc
AB+ AB- AB- AB+ AB+ AB- AB- AB+
AC+ AC- AC+ AC- AC- AC+ AC- AC+

PQ+ PQ- PQ- PQ+ PQ+ PQ- PQ- PQ+
QR+ QR- QR+ QR- QR- QR+ QR- QR+

pqr pr qr pq pq qr pr pqr
(1) q p r r p q (1)













Table 19: Alias Structure for Split-Plot Confounding
For 8 Whole Plots With I = ABPQ = ACQR = BCPR


A
B =
C
AB = PQ
AC = QR
BC = PR
P =
Q =
R=
AP = BQ
AQ = BP+ CR
AR = CQ
BR = CP


only 8 df left for the subplot factors and whole plot x subplot factor interactions.

Therefore, at the subplot level there are only enough df to estimate either three

main effects and five interactions or eight interactions. This may not be sufficient to

estimate all the effects of interest.


3.5 Discussion of Minimum-Aberration Split-Plot Designs


In split-plot designs using some sort of confounding, there is a concept of partial

resolution. The partial resolution of the whole plots refers to the resolution of terms

in the defining contrast involving only whole plot factors. The partial resolution of

the subplot factors refers to the resolution of terms in the defining contrast involving

either only subplot factors or both whole plot and subplot factors. Recall that the

definition of minimum-aberration is the design that has smallest number of words in













the defining contrast with the fewest letters. Therefore, it is looking at the overall

resolution of the design and not the partial resolution.

Huang, Chen and Voelkel (1998) and Bingham and Sitter (1999) have tabled

minimum-aberration (MA) designs for 16 and 32 runs for up to 10 factors. When a

design is needed that fits in these restrictions, one can simply look up the appropriate

design in these tables. However, the MA designs in these tables do not take into

account other design issues such as which effects are the most important to estimate.

This concept seems to be overlooked in the literature. For example, suppose the whole

plot factors are noise factors and only their main effects are of interest. Now, further

suppose that the two-factor whole plot by subplot interactions are the most important

effects to estimate (which is the case in many experiments). Then, it is better to

fractionate the whole plot treatments and subplot treatments separately since this

would alias the two-factor interactions of interest with higher order interactions. Note,

this design would not be the MA design since the partial resolution of the whole plot

factors would be too low.

Another concern with MA designs is the allocation of the runs. Consider the

MA designs for 16 runs involving combinations of 2, 3, and 4 whole plot and subplot

factors. With the exception of the cases involving 2 whole plot factors with 3 or 4

subplot factors, all the other MA designs use eight whole plots with two subplots per

whole plot. This raises several concerns.

1. Typically in industrial experiments, the whole plot factors are hard-to-change

or costly-to-change factors. If they are hard to change, then it would make more

sense to only change them four times as opposed to eight. Also, if changing













these factors is expensive, then again changing them four times seems more

reasonable.

2. It is not an efficient allocation of the degrees of freedom. Using eight whole

plots with two subplots per whole plot gives 7 df for whole plot factors and 8 df

for subplot factors and whole plot x subplot factor interactions. This allocates

a disproportionate number of degrees of freedom to the whole plot factors. In

contrast, using four whole plots with four subplots per whole plot gives 3 df for

whole plot factors and 12 df for terms involving subplot factors.

3. Using two subplots per whole plot is similar to using blocks of size two in a

block design which is not generally recommended.

MA designs are in general "good" designs, however, for split-plot experiments they

are based purely on the overall resolution of the design instead of partial resolution.

Also, they only use split-plot confounding to reduce the size of the experiment and

are not motivated by any other concerns such as those mentioned above.


3.6 Adding Runs to Improve Estimation


With the concerns of the previous section in mind, mainly the allocation of degrees

of freedom, we will focus our attention on 16 run designs that use four whole plots

with four subplots per whole plot. Within this allocation of the resources, the best

design is found for the two types of confounding discussed in the example in section

3.4. These are split-plot confounding and fractionating of the whole plot and subplot

factors separately (called the Cartesian product design in Bisgaard (1999)). The best













design is found using the "mininmum-aberration" (MA) criterion, but this differs from

just using MA because we are restricted to using four whole plot treatments with four

subplot treatments per whole plot. Therefore, the best resolution is desired within

this restricted setting. With the exception of the cases involving 2 whole plot factors

with 3 or 4 subplot factors, these designs will not be the overall MA design.

Once the sixteen run design is found, eight additional runs are considered in order

to break some of the alias chains. Along with breaking some of the alias chains, extra

degrees of freedom are now available in order to estimate additional effects. The

result is a 24 run design which we feel is a nice compromise between the 16 and 32

run designs presented in Huang, Chen and Voelkel (1998) and Bingham and Sitter

(1999). Which eight treatments should be added is the question to be answered next,

but first we briefly discuss foldover designs.

The concept of a foldover design was introduced in Box and Hunter (1961b).

Suppose an experiment involving k factors each at two levels is to be performed and

an initial Resolution III fractional factorial design is used. One way to do a the

foldover is to repeat the initial design and change the levels of one of the factors while

leaving the levels of the other factors unchanged. This allows the estimation of all

the interactions that contain the folded factor but doubles the size of the experiment.

A related idea is that of semifolding which folds only the points that are at the high

level of a factor (or the low level). The addition of the new points breaks certain alias

chains and allows estimates of interactions involving the factor that is semifolded to

be calculated while adding only half as many points as a complete foldover design.

In the rest of this chapter, we apply seminifolding to split-plot experiments.













In most of the cases studied here, the eight additional points are added to the

initial 16 run design by semifolding on either one or two subplot factors which results

in a 24 point design consisting of four whole plots with six subplots per whole plot.

he .,e designs will have 3 df for the whole plot treatments and 20 df for the subplot

treatments. The initial 16 point design is balanced over the subplot factors-each

factor has the same number of high and low levels present-which allows for the

effects to estimated with equal precision. It is desired to preserve this balance of the

subplot factors in the 24 point design as well as maintain the same number of subplot

treatments per whole plot. Therefore, in half of the whole plots the semifolding is on

the high level of a subplot factor while in the other half the semifolding is on the low

level of that factor.

In some cases, it is necessary to fold on a whole plot factor in order to estimate

the main effects of the whole plot factors. In these cases, two whole plots are added so

that the 24 point design consists of six whole plots with four subplots per whole plot.

These designs will have 5 df for the whole plot treatments and 18 df for the subplot

treatments. All nine cases involving 2, 3, and 4 whole plot and subplot factors are

considered. However, two cases do not need to be improved upon.


1. Two whole plot factors and two subplot factors: the 16 points represent the full

factorial. Since no fractionating or confounding is needed, there is nothing to

improve upon.

2. Two whole plot factors and three subplot factors: in this case, the MA design

presented in Huang, Chen and Voelkel (1998) is the best design possible and












allows estimates all of the main effects and all of the two-factor interactions.

For all situations involving less than four whole plot factors, a general method can

be used to construct 24 run designs. First, construct a 16 run design that uses four

whole plots with four subplots in each whole plot.


If there are two whole plot factors, then use the complete factorial in the whole

plot factors.

If there are three whole plot factors, then use one of the two half fractions found

using the defining contrast I = ABC.


To complete the 16 run design use either split-plot confounding or a separate fractional

factorial in the subplot factors to decide which subplot treatments will appear in each

whole plot. After the 16 run design is selected on, use semifolding to obtain two extra

subplot treatments for each whole plot treatment. The semifolding is, for the most

part, done on two subplot factors. In two of the whole plots, the subplot treatments

are folded on one factor (the high level of the factor in one whole plot and the low

level of the factor in the other whole plot). In the remaining two whole plots, the

subplot treatments are folded on a different subplot factor (again, on the high level

in one whole plot and the low level in the other whole plot). In the special case of

three subplot factors, the semifolding is done on just one factor since there is only

one alias chain in the defining contrast.

When there are four or more whole plot factors, using 4 whole plots results in

insufficient degrees of freedom to estimate the main effects of the whole plot factors.












Therefore, the additional eight runs will be added in the form of two extra whole

plots. This leads to a 24 run design with 6 whole plots with 4 subplots per whole

plot. The whole plot treatments used in the two additional whole plots are found

by semifolding on a whole plot factor. However, which subplot treatments should be

used in the two additional whole plots is case specific. To illustrate how to apply

these methods, the seven remaining cases involving 2, 3 and 4 whole plot and subplot

factors will be presented. For all the cases, tables which show the designs in highs

and lows for each factor are given in Appendix A.

For these designs, the aiily.-i.n could use one normal probability plot for the whole

plot effects and a separate plot for the effects involving the subplot factors and the

interactions between whole plot and subplot factors. One assumption of a normal

probability plot is that the effects are independent. This is not the case here. However,

it will be shown later in this chapter that the correlations are low (near zero) and

that a normal probability plot is therefore valid. In some cases there are degrees

of freedom left at the subplot level so that if desired they can be used to estimate

the an error variance. It should be reiterated that the goal of the experiment is to

estimate as well as test for significance the main effects for the whole plot factors, the

two-factor interactions between whole plot and subplot factors, the main effects of

the subplot factors, and if possible two-factor interactions among the subplot factors.

2 WP Factors (A, B) and 4 SP Factors (P, Q, R, S)

To obtain a 16 point design under this situation, only the subplot treatments need

to be fractionated or confounded. First, consider fractionating the subplot treatments.













The defining contrast is I = PQR = QRS = PS which is resolution II. Alias chains

involving both P and S need to be broken. Therefore, the additional eight points

are obtained by semifolding on high and low P in two whole plots and on high and

low S in the other two. The 24 point design is shown in Table 20. The chains are

almost completely broken. Only two of the three interactions. BP, BS, and PS

are estimable. If it can be assumed that PS is negligible, then everything else is

estimable. It is not unreasonable to believe that with four factors one of the two-

factor interactions is negligible and the experimenter should be able to help determine

which interaction is most likely to be negligible.

Next, consider split-plot confounding. The defining contrast is I = APQR =

BQRS = ABPS which is resolution IV. This design is the MA design given in Huang,

Chen and Voelkel (1998). Again, alias chains involving both P and S need to be

broken. Therefore, the additional eight points are obtained by semifolding on high

and low P in two whole plots and on high and low S in the other two. The 24

point design is shown in Table 21. All of the two-factor interactions between whole

plot and subplot factors can be estimated except BS which is aliased with QR. If

QR is assumed to be negligible, then BS can be estimated. Most of the two-factor

interactions among the subplot factors are aliased with each other. However, with

the 24 point design we can estimate three of the two-factor interactions between

the subplot factors without making any assumptions about negligibility, which is an

improvement over the MA design.
















Table 20: 24 Point Design for the Case of 2 WP Factors and 4 SP Factors Using
the Same Fraction [HP(HS)-denotes high P(S) and LP(LS)-denotes low P(S)]


a b ab (1)
q q q q
r r r r
ps ps ps ps
pqrs pqrs pqrs pqrs
Fold on
HP LP HS LS
s pq p qs
qrs pr pqr rs


Table 21: 24 Point Design for the Case of 2 WP Factors and 4 SP Factors Using
Split-Plot Confounding [HP(HS)-denotes high P(S) and LP(LS)-denotes low P(S)]


a b ab (1)
p s q qr
pqr pq r pqs
qs pr ps prs
rs qrs pqrs (1)
Fold on
HP LP HS LS
(1) ps p qrs
qr pqrs pqr s













Table 22: 16 Point Design for the Case of 3 WP Factors and 2 SP Factors
Using a Fraction Factorial of the Whole Plot Factors

a b c abc
P P P P
q q q q
pq pq pq pq
(1) (1) (1) (1)


3 WP Factors (A, B, C) and 2 SP Factors (P, Q)

In this case, only the whole plot factors need to be fractionated. Since nothing

needs to be done to the subplot factors, there is only one 16 point design. The defining

contrast is I = ABC which is Resolution III. Since the design estimates everything

set out in the goal of the experiment, no points need to be added to this design.

However, note that this is not the MA design which is run using eight whole plots

with 2 subplots per whole plot. The 16 point design in four whole plots with four

subplots per whole plot is shown in Table 22.

3 WP Factors (A, B, C) and 3 SP Factors (P, Q, R)

To obtain a 16 point design in this situation, both the whole plot and subplot

treatments need to be fractionated or confounded. First, consider fractionating the

whole plot and subplot treatments separately. The defining contrast is I = ABC =

PQR = ABCPQR which is resolution III. The two-factor interactions between whole

plot and subplot factors are already estimable. Therefore, there is only one alias

chain that needs to be broken, and that is associated with PQR. The additional













Table 23: 24 Point Design for the Case of 3 WP Factors and 3 SP Factors
Using the Same Fraction [HP-denotes high P and LP-denotes low P]

a b c abc
P P P P
q q q q
r r r r
pqr pqr pqr pqr
Fold on
HP LP LP HP
(1) pq pq (1)
qr pr pr qr


eight points are obtained by semifolding on factor P. The 24 points design is shown

in Table 23. The chain has been broken and now P, Q, R, PQ, PR, and QR are all

estimable. There are 5 df left over for a subplot error term.

Next, consider split-plot confounding. The defining contrast is I = ABC =

ABPQR = CPQR which is also resolution III. The two-factor interactions between

whole plot factors A and B and the subplot factors are already estimable. Therefore,

the only alias chain that needs to be broken is CPQR. The additional eight points

are obtained by semifolding on factor P while being careful to fold both high and low

P where C is high and where C is low. The 24 point design is shown in Table 24. The

chain has been broken and now all of the effects of interest including the two-factor

interactions among the subplot factors are estimable. There are 5 df left over for a

subplot error term.













Table 24: 24 Point Design for the Case of 3 WP
Using Split-Plot Confounding [HP-denotes high

a b c abc
Pq Pq P P
pr pr q q
qr qr r r
(1) (1) pqr pqr
Fold on
HP LP LP HP
q pqr pq (1)
r p pr qr


Factors and 3 SP Factors
P and LP-denotes low P]


3 WP Factors (A, B, C) and 4 SP Factors (P, Q, R, S)

To obtain a 16 point design in this situation, both the whole plot and subplot

treatments need to be fractionated or confounded. First, consider fractionating the

whole plot and subplot treatments separately. The defining contrast is I = ABC =

PQR = QRS = PS = ABCPQR = ABCQRS = ABCPS which is Resolution

II. In order to estimate the subplot factor main effects and possibly the two-factor

interactions among the subplot factors, the two chains PQR and QRS with resulting

chain PS need to be broken. The additional eight points are obtained by semnifolding

on both factors P and S. The 24 points design is shown in Table 25. The chains

are almost completely broken. Two resulting chains AP = PS and AS = PS are

left. The aliasing here means that the sum of AP and AS equals PS. Therefore, the

model can accommodate the fitting of any two of the three factors. So for example, if

PS is assumed negligible, then the effects of AS and AP are estimable. There are 4

df left that can be used as an error term or used to estimate PQ, PR, QS and RS.













Table 25: 24 Point Design for the Case of 3 WP Factors and 4 SP Factors Using
the Same Fraction [HP(HS)-denotes high P(S) and LP(LS)-denotes low P(S)]

a b c abc
q q q q
r7 r r r
ps ps ps ps
pqrs pqrs pqrs pqrs
Fold on
HP LP HS LS
s pq p qs
qrs pr pqr rs


Next, consider split-plot confounding. The defining contrast is I = ABC =

BCPQR = ACQRS = ABPS = APQR = BQRS = CPS which is resolution

III. Not much of anything is estimable free of two-factor interactions. Again, the

additional eight points are obtained by semifolding on factors P and S. The 24 point

design is shown in Table 26. Again, the chains are almost completely broken. Three

resulting chains C = PS, AP = PS and BS = PS are left. The aliasing here means

that the sum of C, AP, and BS equals PS. Therefore, the model can accommodate

the fitting of any three of the four factors. So for example, assuming PS is negligible

allows for C, AP and BS to be estimated. Also, any two of the remaining five

two-factor interactions among the subplot factors can be estimated.


4 WP Factors (A, B, C, D) and 2 SP Factors (P, Q)


In this case, only the whole plot treatments need to be fractionated. Note, with

four whole plot factors there are only 3 df for whole plot factor effects. Hence, whole













Table 26: 24 Point Design for the Case of 3 WP Factors and 4 SP Factors Using
Split-Plot Confounding [HP(HS)-denotes high P(S) and LP(LS)-denotes low P(S)]


a b c abc
p s qr q
pqr pq pqs r
qs pr prs ps
rs qrs (1) pqrs
Fold on
HP LP LS HS
(1) ps qrs p
qr pqrs s pqr


Table 27: 24 Point Design for the Case of 4 WP Factors and 2 SP
Factors Using a Fractional Factorial of the Whole Plot Factors

Fold on A
b c ad abcd d bcd
P P P P P P
q q q q q q
pq pq pq pq pq pq
(1) (1) (1) (1) (1) (1)


plots will need to be added for all cases involving four whole plot factors. Since

nothing needs to be done to the subplot factors, there is only one 16 point design. The

defining contrast is I = ABC = BCD = AD which is resolution II. The additional

whole plots are obtained by semifolding on factor A. The 24 point design is shown in

Table 27. The chains are broken and everything is estimable.












4 WP Factors (A, B, C, D) and 3 SP Factors (P, Q, R)

To obtain a 16 point design in this situation, both the whole plot and subplot

treatments need to be fractionated or confounded. First, consider fractionating the

whole plot and subplot treatments separately. The defining contrast is I = ABC =

BCD = AD = PQR = ABCPQR = BCDPQR = ADPQR which is resolution

II. The additional whole plot treatments are obtained by semifolding on factor A.

The positive fraction, I = PQR, is run in one whole plot while the negative fraction,
I = -PQR, is run in the other whole plot. The negative fraction can be thought of

as semifolding on any subplot factor and placing all of the points in one whole plot

instead of two as was done in all the cases up until now. The 24 point design is shown

in Table 28. The chains are broken and everything is estimable.

Next, consider split-plot confounding. The defining contrast is I = ABC =

BCD = AD = CPQR = ABPQR = BDPQR = ACDPQR which is resolution II.

Besides breaking chains among the whole plot factors, the chain, CPQR needs to be

broken. The additional whole plot treatments are obtained by semifolding on factor

A. Again, the positive fraction, I = PQR, is run in one whole plot with the negative

fraction, I = -PQR, is run in the other whole plot. Again, this can be thought of

as semifolding each fraction on any subplot factor and placing all four points in the

same whole plot. The 24 point design is shown in Table 29. The chains are broken

and everything is estimable.



















Table 28: 24 Point Design for the Case of 4 WP Factors
and 3 SP Factors Using the Same Fraction
Fold on A
b c ad abcd d bcd
P P P P P Pq
q q q q q pr
r r r r r qr
pqr pqr pqr pqr pqr (1)












Table 29: 24 Point Design for the Case of 4 WP Factors
and 3 SP Factors Using Split-Plot Confounding
Fold on A
b c ad abcd d bcd
Pq P pq P P Pq
pr q pr q q pr
qr r qr r r qr
(1) pqr (1) pqr pqr (1)












4 WP Factors (A, B, C, D) and 4 SP Factors (P, Q, R, S)

As the number of both whole plot factors and subplot factors increases, it becomes

impossible to break all of the relationships and estimate all of the important effects.

Therefore, some effects will need to be assumed negligible. Also, in the case of four

whole plot factors and four subplot factors, there is insufficient degrees of freedom to

estimate the four subplot factor main effects and the sixteen two-factor whole plot

by subplot interactions. Thus, some effects cannot be estimated anyway. Assuming

these effects to be negligible enables the estimation of the remaining effects.

To obtain a 16 point design in this situation, both the whole plot and subplot

treatments need to be fractionated. First, consider fractionating the whole plot and

subplot treatments separately. The defining contrast is I = ABC = BCD = AD =

PQR = QRS = PS = ABCPQR = ABCQRS = ABCPS = BCDPQR =

BCDQRS = BCDPS = ADPQR = ADQRS = ADPS which is resolution II. Be-

sides breaking chains among the whole plot factors, the chains, ADPS and PS need

to be broken. The additional whole plot treatments are obtained by semifolding on

factor A. The subplots are semifolded on factor P in one whole plot and factor S in

another whole plot. The 24 point design is shown in Table 30.

Next, consider split-plot confounding. The defining contrast is I = ABC =

BCD = AD = ACPQR = BDQRS = ABCDPS = BPQR = ABDPQR =

CDPQR = ACDQRS = CQRS = ABQRS = DPS = APS = BCPS which is

resolution II. The additional whole plot treatments are obtained by semifolding on

factor A. Again, the subplot factors are semifolded on P and S. Care must be taken












when choosing which whole plots the subplot factors are semifolded. Otherwise, the

same treatment combinations will occur in both additional whole plots. This occurs

when the semifolding uses the whole plots containing the subplot treatments defined

by PQR+, QRS+ and PQR-, QRS- or PQR+, QRS- and PQR-, QRS+. Any other

combination is fine. In this section, P is semnifolded in the whole plot containing

whole treatment c (PQR-, QRS+) and S is semifolded in the whole plot containing

whole plot treatment abcd (PQR+, QRS+). The 24 points design is shown in Table

31.

Most of the chains are broken but some of the two-factor interactions among the

subplot factors are aliased with each other. Also, four of the sixteen two-factor inter-

actions between whole plot and subplot factors must be assumed negligible. These

terms are AS, CS, DS, and DP. This is fairly nice since three of these terms involve

subplot factor S. Therefore, if it is believed that one of the subplot factors is unlikely

to interact with the whole plot factors, these terms or effects could be assumed negli-

gible. This does not seem unreasonable. Now the 18 subplot df are partitioned into 4

df for the subplot factor main effects, 12 df for the whole plot by subplot interactions,

and 2 for two-factor interactions among subplot factors (these two effects can be any

pair except PQ and QS or PR and RS).


3.7 An Example

To illustrate how an experiment could be carried out and analyzed, an example is

presented. The example, from Taguchi (1987), involves the study of a wool washing

and carding process. The original experiment used a 213-9 x 23-1 inner and outer





















Table 30: 24 Point Design for the Case of 4 WP Factors and
4 SP Factors Using the Same Fraction


b
q
r
I'
ps
prs
pqrs


4


C
q
r
PS
ps
pqrs


+


ad abed
q q
r r
ps ps
pqrs pqrs


Fold on A
d bed
pq qs
pr rs
s p
qrs pqr


Table 31: 24 Point Design for the Case of 4 WP Factors and
4 SP Factors Using Split-Plot Confounding


b c ad abed
p qrs (1) q
pqr s qr r
qs pq pqs ps
rs pr prs pqrs


Fold on A
d bcd
pqrs qs
ps rs
q P
r pqr












array. For our purposes, we shall only consider the first three factors in the inner

array along with the three factors in the outer array. The outer array will make up

the whole plot factors while the inner array will have the subplot factors. The original

experiment was not run using restricted randomization, but it will be assumed that

it was in order to present the analysis.

The 24 run designs discussed earlier can be utilized. For this example, the design

given in Table 23 with separate fractions at the whole plot and subplot levels is used.

The only difference is that the Taguchi example uses the negative fraction of the whole

plot factors instead of the positive fraction. To correspond with the factor names in

the Taguchi example, let X, Y and Z be the whole plot factors and A, B and C be

the subplot factors. The design along with the responses is shown in Table 32.

The analysis involves fitting the 18-term model involving the main effects of the

whole plot factors, the main effects of the subplot factors, the two-factor interactions

among subplot factors, and two-factor interactions between whole plot and subplot

factors. This leaves five degrees of freedom for a subplot error term. Table 33 gives

the estimated effects and t-tests. The tests for the three whole plot factor main effects

are not correct and should be ignored. From Table 33, there appears to be an effect

due to the interaction of factors B and C. Since there are only 5 df for error, one

might choose to use a normal probability plot to investigate at the subplot level. The

design is not completely balanced or orthogonal which leads to some effects having

one standard error and others having a different standard error. Therefore, instead

of just plotting the effects, the effects are divided by their standard error and then

plotted. The plot is shown in Figure 1 and gives BC and ZC as significant effects.




















Table 32: 24 Point Design for the Example

X Y Z A B C Response
+ 19.0
+ 22.5
+ 26.0
+ + + 21.5
20.0
+ + 18.5
+ + + 16.0
+ 21.0
+ 20.0
+ + + 16.0
+ + 24.0
+ -____ + 23.0
+ + + 17.5
+ 21.0
+ 22.0
+ + + 22.0
+ + 21.0
____ + + 19.0
+ + + 19.0
+ 22.5
+ 26.5
+ + + 23.0
23.0
+ + 26.5






















Table 33: Effects Table for the Example


Effect Coeff
21.266
-0.937 -0.469
1.406 0.703
-2.156 -1.078
1.000 0.500
-0.812 -0.406
0.937 0.469
0.031 0.016
-0.187 -0.094
-0.563 -0.281
-0.094 -0.047
0.312 0.156
0.875 0.438
0.344 0.172
0.500 0.250
-1.312 -0.656
1.375 0.687
1.250 0.625
-4.438 -2.219
d Tests


nt





















Vali


Term
Constai
X
Y
Z
y
z
A
B
C
X*A
X*B
X*C
Y*A
Y*B
Y*C
Z*A
Z*B
Z*C
A*B
A*C
B*C
t Not


Std Error
0.4288
0.4951
0.4288
0.4288
0.4951
0.4951
0.4951
0.4288
0.4951
0.4951
0.4288
0.4288
0.4288
0.4288
0.4288
0.4288
0.4951
0.4951
0.4951


t-value
49.60
-0.95
1.64
-2.51
1.01
-0.82
0.95
0.04
-0.19
-0.57
-0.11
0.36
1.02
0.40
0.58
-1.53
1.39
1.26
-4.48


P-value
0.000
0.3871
0.162t
0.054t
0.359
0.449
0.387
0.972
0.857
0.595
0.917
0.730
0.354
0.705
0.585
0.186
0.224
0.262
0.007

























1.0 ______
*
0.9 S

0.8
0.7 5
0
8 0.6
.X 0.5
' 0.4 -
0.3

0.2 -
0.1 ZC
0.0 BC
SI I I I
-5 -4 -3 -2 -1 0 1 2
Effect/Standard Error


Figure 1: Normal Probability Plot for the Example













3.8 Summary

The main goal of this chapter is to understand some of the complications involved

with using the two types of confounding in split-plot experiments. If the design is

chosen using the MA criterion, then designs constructed by combining the fractional

factorial of the whole plot treatments with a fractional factorial of the subplot treat-

ments are not considered. This is because their partial resolution is too low. However,

it has been shown that depending on what is to be estimated, it may not be wise

to eliminate such designs. Also, for 16 run designs the MA criterion tends to select

designs with eight whole plots and two subplots per whole plot. These designs do

not take into consideration the possible increase in the cost of experimentation or of

factors whose levels are hard to change.

We have presented a 24 run design which is a compromise between the 16 run

and 32 run designs. We begin with a 16 run design using four whole plots with four

subplots per whole plot. Such a strategy accommodates hard-to-change factors as

well as cost considerations involving the experiment. Then, eight additional runs are

added using semifolding of one or two factors. These runs preserve the balance of

high and low levels of each subplot factor as well as maintain the same number of

subplots for each whole plot. Except for the cases involving four whole plot factors,

the additional runs are at the subplot level. Thus, as long as adding the runs is

feasible, it should not be too costly. Also, the extra runs allow for additional effects to

be estimated and/or add degrees of freedom for estimating the subplot error variance.

Though designs using split-plot confounding and separate fractions differ in what

they can estimate in 16 runs, there is not much difference in estimability when using













Table 34: Condition Numbers for the Various Cases


# of WP Factors # of SP Factors
2 4
2 4


Type of Confounding
Same Fraction
Split-Plot Confounding


Condition Number, K
4.05
3.36


3 3 Same Fraction 2.00
3 3 Split-Plot Confounding 2.00

3 4 Same Fraction 4.05
3 4 Split-Plot Confounding 3.36
4 Neither 2.00

4 3 Same Fraction 5.83
4 3 Split-Plot Confounding 2.00

4 4 Same Fraction 5.83
4 4 Split-Plot Confounding 3.36



the 24 run designs.

The phrase "the chains are broken" used throughout this chapter does not mean

that all of the effects are no longer aliased. Sometimes the effects are aliased at

a lower degree than unity (completely aliased). Therefore, there is some degree of

collinearity between the effects. To measure the magnitude of this collinearity, the

condition number

Largest Eigenvalue of (X'X)
SSmallest Eigenvalue of (X'X)

is calculated for each case (see Table 34). Many textbooks declare collinearity to be

a problem if K > 30. It is seen from Table 34 that collinearity does not seem to be a

problem for the cases considered in this chapter. The variance inflation factors (VIF)













are also calculated. They are not reported here, but none of the VIF's exceeds 3.

This confirms that the collinearity is mild.

The cases used in this chapter are chosen because they cover a wide range of

industrial applications. The methods described in this chapter can be applied to

experiments involving even more than four factors at either or both the whole plot

and subplot level. However, as is seen with the case involving four whole plot factors

and four subplot factors, not all chains can be broken. However, the additional

points should result in a design which estimates more effects with less assumptions on

negligibility than a 16 point design. If the cost and time of adding the eight points is

acceptable and the goals of the experiment are those discussed in this chapter, then

the 24 run designs can improve the estimability of important effects.

















CHAPTER 4
A NEW MODEL AND CLASS OF DESIGNS FOR
MIXTURE EXPERIMENTS WITH PROCESS VARIABLES


Experiments that involve the blending of two or more components to produce high

quality products are known as mixture experiments. The quality of the end product

depends on the relative proportions of the components in the mixture. For example,

suppose we wish to study the flavor of a fruit punch consisting of juices from apples,

pineapples, and oranges. The flavor of the punch depends on the relative proportion

of the juices in the blend.

Consider a mixture experiment consisting of q components. Let xi, for i =

1, 2,..., q, represent the fractional proportion contributed by component i. Then the

proportions must satisfy the following constraints
q
0 i=1
and the experimental region is a (q 1)-dimensional simplex, Sq. For q =3, S3 is an

equilateral triangle and for q = 4, 54 is a tetrahedron. Typically, the blends used in

a mixture experiment are the vertices or single-component blends, the midpoints of

the edges, centroids of faces, etc., and the centroid of the simplex.

In some mixture experiments, the quality of the product depends not only on the

proportions of the components in the blend, but also on the processing conditions.

Process variables are factors that do not form any portion of the mixture but whose

levels, when changed, could affect the blending properties of the components. Cornell













(1990) discusses an experiment involving fish patties. The texture of the fish patties

depends not only on the proportions of three fish species that are blended but also

on three process variables which are cooking temperature, cooking time and deep fat

frying time.

A concern with mixture experiments involving process variables is that the size

of the experiment increases rapidly as the number of process variables, n increases.

In the fruit punch or fish patty examples above, it may not be necessary to limit the

size of the experiment. However, in most industrial experiments, cost and time do

impose restrictions on the number of runs permitted. Therefore, a design strategy

that uses fewer observations is preferred over a design that does not.

Cornell and Gorman (1984) presented combined mixture component-process vari-

able designs for n > 3 process variables that use only a fraction of the total number

of possible design points. They considered process variables each at two levels and

suggest fractions of the 2' factorial be considered. Two plans involving the frac-

tional factorial design in the process variables were discussed. The first plan, called

a matched fraction, places the same 23'-1 fractional replicate design at each mixture

composition point. The other plan, called a mixed fraction, uses different fractions at

the composition points. Each plan was applied to the situation involving three mnix-

ture components and three process variables with the total number of design points

ranging from 56 for the combined simplex-centroid by full 23 factorial, to only 16,

which relied on running the one-quarter fraction. It should be noted that if inter-

actions among the process variables are likely to be pre.-int. the use of a fractional

factorial will result in bias being present in the coefficient estimates. Cornell and












Gorman give recommendations regarding the choice of design which depend on the
form of the model to be fitted and whether or not there is prior knowledge on the

magnitude of the experimental error variance.

Czitrom (1988, 1989) considered the blocking of mixture experiments consisting

of three and four mixture components. She used two orthogonal blocks to construct

D-optimal designs. Draper et al. (1993) consider mixture experiments with four

mixture components. They treat a combination of the levels of the process variables

as defining blocks. Orthogonally blocked mixture designs constructed from Latin

Squares are presented. The optimal choice of a design using D-optimality is also given.

While the reduction in the number of observations required can be great, obtaining

D-optimality comes with a price. The D-optimal designs require very nonstandard

values for the component proportions.

We propose an alternative approach to reducing the size of the experiment which

borrows ideas from the above references. The concept of running only a subset of the

total number of mixture-process variable combinations is borrowed from Cornell and

Gorman (1984), although our fraction will involve the mixture component blends as

well. To evaluate the fraction, we shall make use of the D-criterion criterion (Czitrom

(1988, 1989)). The next section provides a type of experimental situation which led

to this research. In the section that follows, a combined model which is slightly

different in form from the combined mixture-process variable models ordinarily used

is presented. The method for constructing the design and comparing it's D-criterion

is discussed in the fourth section. The final section of this paper contains details on

the analysis of the experiments using the proposed designs and model forms.













4.1 Experimental Situation

Historically in the mixture literature, the interest in the blending properties of the

mixture components has been higher than that of studying the effects of the process

variables. Generally, the process variables have been treated as "'noike" factors. The

primary focus on the mixture by process variable interactions has been on the effects

of the process variables on the blending properties of the mixture components.

In many industrial situations, the interest in the process variables is at least equal

to that in the mixture components. Consider the production of a polymer which is

produced by reacting together three specific components. The research laboratory

proposes a specific formulation which is the result of a highly controlled environment

with reagent grade chemicals and laboratory glassware. The plant personnel use this

formulation during the pilot plant and initial start-up of the full production pro-

cess. During this period, the plant personnel are trying to find the proper processing

conditions to produce a useable product profitably.

At some point, plant personnel need to reevaluate the polymer's formulation in

light of the actual raw materials and the plant's full scale production capabilities.

Plant personnel need to find the "optimal" combination of the formulation and pro-

cessing conditions.

Traditionally, in response surface applications, the model assumed for process op-

timization is a second-order Taylor series. Such an assumption is based on background

knowledge in knowing the true surface over the experimental region can be approx-

imated by fitting a second-order model. Furthermore, in our polymer example, all

second-order terms involving mixture components, process variables, and the mixture













by process variable interactions are of equal importance. In fact, the specific mixture

component by process variable interaction terms may provide a significant amount of

insight into which operating conditions are optimal. For instance, the engineer truly

needs to know if a specific mixture component makes the reaction especially sensitive

to the reaction temperature.

This type of experimental situation leads us to propose a new model for extracting

information from a mixture experiment with process variables. The time and cost

constraints faced by plant personnel leads us to propose a new class of designs based

upon this model.


4.2 The Combined Mixture Component-Process Variable Model


In mixture experiments involving process variables, the form of the combined

model consisting of terms in the mixture proportions as well as in the process vari-

ables depends on the blending properties of the mixture components, the effects of

the process variables, and any interactions between the mixture components and pro-

cess variables. These models are typically second-order models that allow for pure

quadratic and two-factor interaction terms.

The general second-order polynomial in q mixture components is

q q q
71 = Ao + E Ox + E O.ix + E E Axxj (4)
i= i=1 i
Now using the constraints


xi = 1 and x. = xi 1- xj ,
i=1 j=1













Equation (4) becomes


(. Xi3-i ) + iE i + (Of3iiXi q q
i=1 i=1 i=1 j96i i q q q q
= E(Ao + 0i + 3i) x E ixi ELx + E1E ijxxj
i=1 i=1 j'i i q q
= E- x+ EE o >xx (5)
i=1 i where = /3o + A + Oij and /3 = Aj O3i ij for i, j = 1, 2,..., q, i < j.

Suppose that an experiment is to be performed with q mixture components,

x1, x2, ... Xq, and n process variables, z1, z2, ..., z,. In the process variables, let us

consider the model
n n
71pV = ao + E CkZk + E E CakZkZ (6)
k=l k Then there are two main types of combined models (see Cornell (1990)) that can be

used in this situation. The first type is a model which crosses the mixture model

terms in Equation (5) with each and every term of Equation (6). This produces the

combined model
q q q n q n
q(x, z) = 3,xi +E E3jXiXj + E E^xiZk + EEEYiklXiZkZ,
i=1 i q n q n
+ E E E ^ikXXjZk + E E E E ^>3j3kXiXZkZ (7)
i which includes parameters for three and four factor interactions. Depending on the

design, the model of Equation (7) provides a measure of the linear and nonlinear

blending properties of the mixture components averaged across the settings of the

process variables as well as the effects of the process variables on the linear and

nonlinear blending properties.













The second type of combined model is the additive model which combines the

models in Equations (5) and (6) without crossing any of the xi and zj terms. This

produces the model
q q
m/(x,z) + E
i=1 i n n
+ Eokzk+ EE akizkkZ. (8)
k=1 k Equation(8) provides a measure of the quadratic blending of the mixture components

on the response as well as up to two-factor interactions between the process variables

on the response. Since the model does not contain any crossproduct terms between

the mixture components and the process variables, when fitting Equation (8) the

user assumes the blending of the mixture components is the same at all factor-level

combinations of the process variables. This assumption is probably unrealistic in most

situations. Also, in some experiments like the one described in the previous section,

the mixture component by process variable interactions may be the most important

terms in the model.

A major concern with mixture experiments involving process variables is their

size. Many industrial situations require the use of small experiments due to time

and/or cost constraints. As the number of mixture components and/or process vari-

ables increases, the model in Equation (7) will require a design with a large number

of points. While the fitting of the model in Equation (8) permits the use of a smaller

design than the fitting of the model in Equation (7), it does not, as pointed out

earlier, address the estimation of the mixture components by process variable inter-

actions. If cost constraints limit the size of the experiment yet interactions between













mixture components and process variables are believed to be important, some sort of

compromise between these two models is needed.

Most of the model forms that have been proposed for response surface investi-

gations are based on a Taylor series approximation. In keeping with this tradition,

suppose that the true model for the n process variables is a second-order model.

Instead of Equation (6), such a model would be

71 n n
22
17Pv = ao + E Z kZk + E Z kkzk + E E klZkZ. (9)
k=l k=l k
Equation (9) is Equation (6) plus the n pure quadratic terms. Also, a Taylor series

approximation for a combined second-order model would include only up to two factor

interactions and would not be the model in Equation (7). Combining Equation (5)

with Equation (9), our proposed combined second-order model is

q q n
(xz) = Z)+ E+ 'z
i=l i n q n
+ E E kZkZ E E YikXiZk (10)
k
which includes the mixture model, plus pure quadratic as well as two-factor inter-

action effects among the process variables, and two-factor interactions between the

linear blending terms in the mixture components and the main effect terms in the

process variables. The minimum number of design points needed for the proposed

model (10) is less than what is needed for the completely crossed model (7) but is

more than is needed for the additive model (8). Also, the proposed model can be used

even if one does not feel the need for pure quadratic terms in the process variables

by simply omitting those n terms.












To support the fitting of Equation (10) we shall require a design that will support

nonlinear blending of the mixture components as well as the fitting of the full second

order model in the process variables. In the next section, we discuss a design approach

that will accommodate these terms.


4.3 Design Approach

In mixture experiments as in most response surface investigations, the design and

the form of the model to be fitted go hand in hand. For example, if a second order

model is suspected, it is necessary to select a design that will support the fit of this

model. The design chosen must have at least as many points as there are parameters

in the model. Therefore, a (q + n)(q + n + 1)/2 point design is needed to support the

fitting of the model in Equation (10).

A popular response surface design for fitting a second-order model of the form in

Equation (9) is the central composite design (ccd) which consists of a complete 2' (or

a Resolution V fraction of a 2n) factorial design, 2n axial points with levels a for

one factor and zero for the rest, and at least one center point. If a = 1 is selected,

the design region is a hypercube.

The approach to reducing the number of observations needed in a mixture exper-

iment begins with a ccd in the process variables. A simplex is then placed at each

point in the ccd with only a fraction of the mixture blends at each point. The mixture

blends at each design point are selected from the full simplex-centroid. A general

notion of balance among the mixture components across the process variables is de-

sired. First of all, let us insist on the same number of mixture blends to be present













at both the high and low levels of each process variable. Secondly, let us insist on all

of the mixture blends be present at each 1 factorial level for each process variable.

These ideas seem very intuitive and lead us to select some of the mixture blends to be

used at certain design points and different mixture blends to be used at other design

points.

Two designs are considered for the fitting of the model in Equation (10). With

both designs, the vertices of the simplex are run at one-half of the 2' factorial points

in the process variables with the midedge points of the simplex being run at the other

half. This is done in a such a way, that if the design is collapsed across the levels

of each process variable then one gets a simplex with vertices and midpoints at both

the low and high level of the remaining process variables. Hence, the information in

the mixture blends is spread evenly among the process variables. This is intuitively

appealing since if a process variable is deemed negligible then there is still complete

information on the mixture blends for the other process variables. Ntxt,. the axial

points in the process variables are paired with just the centroid of the simplex. This

allows for the centroid to also be present if the design is collapsed. The two designs

differ only in the number of points placed at the center of the process variables. With

one design the entire simplex-centroid is performed at the center while with the other

only the centroid mixture blend is performed at the center of the process variables.

Consider an example involving three mixture components and two process vari-

ables. The model for this example, using Equation (10), contains 15 terms. The two

designs are shown in Figures 2 and 3. For three mixture components, the design with

the full simplex-centroid at the center of the process variables consists of 23 points













while the second design with just the centroid consists of 17 points. Either could be

used to estimate the 15 terms in the model.

The designs in Figures 2 and 3 can be extended to experiments involving more

than 3 mixture components (MC) and/or more than 2 process variables (PV). The

extension is straightforward. Following the same general notion of balance described

earlier, one can generate the needed designs. In this paper, a total of five cases are

discussed: 3 MC, 2 PV; 3 MC, 3 PV; 4 MC, 2 PV; 4 MC, 3 PV; and 3 MC, 2 PV with

upper and lower bound constraints on the mixture component proportions. For four

mixture components, there are four vertices and six edge midpoints of the tetrahedron.

For three process variables, the layout is a cube with 23 = 8 factorial points, six axial

points, and a center point. Placing upper and lower bound constraints on the mixture

component proportions creates a more complicated mixture region than the simplex.

The constrained region is typically an irregular polygon. The example in this paper

(3 MC, 2 PV) uses the following constraints:

0.25 < x, < 0.40 0.25 < x2 < 0.40 0.25 < x3 < 0.40.

The resulting mixture region is a hexagon. Generally, the original components are
transformed to L-pseudocomponents, x (- )/(- =L) i = 1,2,...,, to

make the construction of the design and the fitting of the model easier. For the exam-

ple in this paper, the mixture components can be transformed to L-pseudocomponents

using
xi 0.25 =x 0.25
1 (0.25 + 0.25 + 0.25) 0.25 1,2,3.

Candidate points for the two designs in this case consist of the six vertices and







x_1 =1


x_2 = 1 xj


Figure 2: Proposed Design for the 3-2 Case With Full Simplex


z-2
A-


/


4 z1


=
]-=1


I ,'


A






x_1 =1

x_2 =1 x _3 = 1


z_2
A-


I z-


Figure 3: Proposed Design for the 3-2 Case With Just the Centroid


4\




Full Text
35
Table 9: Variance of the Regression Coefficients For a 2k With
One Hard-To-Change Factor (from Lucas and Ju (1992))
Var(6) = Act2 + Bct2
Hard To Change Variable
Other Terms
A
B
A B
l-p i p
^r + 2
i
Â¥
i-p i
Â¥
P = 1/(2*'
design.
P = 1 for
P = (2k~1
2 + 1) for the completely randomized
the completely restricted design.
2)/[2(2fc_1 1)] for the partially
restricted design.
procedures depend on the block variance.
Miller (1997) considers various fractional-factorial structures in strip-plot experi
ments. These strip-plot experiments are identical in nature to the strip-block experi
ments, arrangement (c), discussed in Box and Jones (1992). Strip-plot configurations
can be applied when the process being investigated is separated into two distinct
stages and it is possible to apply the second stage simultaneously to groups of the
first-stage product. Miller uses an example involving four washing machines and four
dryers in two blocks. Sets of cloth samples are run through the washing machines,
and then the samples are divided into groups such that each group contained exactly
one sample from each washer. Each group of samples would then be assigned to one
of the dryers. The response of interest was the extent of wrinkling.
It is convenient to represent strip-plot structures as rectangular arrays of experi
mental units in which the levels of one treatment factor (or set of factors) are assigned


THE DESIGN AND ANALYSIS OF SPLIT-PLOT
EXPERIMENTS IN INDUSTRY
By
SCOTT M. KOWALSKI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS OF THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999


169
[13]Cornell, J. A. (1988). "Analyzing Data from Mixture Experiments Contain
ing Process Variables: A Split-Plot Approach. Journal of Quality Technology 20,
pp. 2-23.
[14] Cornell, J. A. and Gorman, J. W. (1984). "Fractional Design Plans for
Process Variables in Mixture Experiments. Journal of Quality Technology 16,
pp. 20-38.
[15] Cox, D. R. (1958). Planning Experiments. John Wiley & Sons. New York, NY.
[16] Czitrom, V. (1988). Mixture Experiments with Process Variables: D-Optimal
Orthogonal Experimental Designs. Communications in StatisticsTheory and
Methods 17, pp. 105-121.
[17] Czitrom, V. (1989). "Experimental Designs for Four Mixture Components
with Process Variables. Communications in StatisticsTheory and Methods 18,
pp. 4561-4581.
[18] Draper, N. R. ; Prescott. P. ; Lewis, S. M. ; Dean, A. M. ; John, P.
W. M. ; and Tuck, M. G. (1993). Mixture Designs for Four Components in
Orthogonal Blocks. Technometrics 35, pp. 268-276.
[19] Finney, D. J. (1945). "The Fractional Replication of Factorial Arrangements.
Annals of Eugenics 12, pp. 291-301.
[20] Fisher, R. A. (1926). "The Arrangement of Field Experiments. Journal of
the Ministry of Agriculture 33. pp. 503-513.
[21] Huang, P. ; Chen, D. and Voelkel, J. O. (1998). Minimum-Aberration
Two-Level Split-Plot Designs. Technometrics 40, pp. 314-326.
[22] Kempthorne, O. (1952). The Design and Analysis of Experiments. John Wiley
& Sons, New York. NY.
[23] Khuri, A. I. and Cornell, J. A. (1996). Response Surfaces 2nd edition.
Marcel Dekker, New York. NY.
[24] Kiefer, J. and Wolfowitz, J. (1959). Optimum Designs in Regression
Problems. Annals of Mathematical Statistics 30, pp. 271-294.
[25] Letsinger, J. D. ; Myers. R. H. ; and Lentner, M. (1996). Response
Surface Methods for Bi-Randomization Structures. Journal of Quality Technol
ogy 28, pp. 381-397.
[26] Lucas, J. M. and Ju, H. L. (1992). Split Plotting and Randomization in In
dustrial Experiments. ASQC Quality Congress Transactions. American Society
for Quality Control, Nashville. TN. pp. 374-382.


24
Table 6: Strip-Block Arrangement (Box and Jones (1992))
Block 1 Block 2 Block 3
1
a2
3
03
02
Oi
Ol
0 3
2
61
02
bi
i>2
bi
62
The model for the strip-block arrangement is
Vijk V + Ik + Otj + 7)jk + + Oik + (a^)ij + eijk ,
where yljk is the response of the kth replicate of the 2th level of factor D and the jth
level of factor E, ¡i is the overall mean, 7*. is the random effect of the kth replicate with
7k ~ N(0, a2), ctj is the fixed effect of the jth level of factor E, 6l is the fixed effect
of the 2th level of factor D, (aS)^ is the interaction effect of the 2th level of D and the
jth level of E. In arrangement (c), rjjk ~ Ar(0,a|), 0ik ~ N(0,a2D), eijk ~ N(0, a2)
and r¡jkl 0ik and eijk are independent.
ANOVA tables for all three arrangements are given in Box and Jones (1992).
These tables indicate the appropriate denominators for tests involving the design
factors, the environmental factors, and their interactions assuming replication. When
there is no replication, normal probability plots, one for the whole plot factors and
one for the subplot factors and whole plot x subplot interactions, can be used to
select significant effects. Also, if the design and environmental factors are factorial
combinations, it may be possible to pool negligible higher order interactions to get
estimates of the whole plot and subplot errors.


30
fit from the whole plots. Therefore, the number of whole plots available must exceed
the number of parameters in 7.
For the crossed BRD, there is an equivalence between ordinary least squares (OLS)
and GLS. This equivalence means that Equation (2) becomes
3=(X'X)-1X'y
and the model estimation no longer depends on the error variance. However, for
testing purposes, the error variance must be estimated. Letsinger, Myers, and Lentner
(1996) suggest augmenting the response surface model with insignificant whole plot
terms, Z*p, to saturate the a 1 whole plot degrees of freedom. The whole plot
saturated model can be used to calculate lack of fit sums of squares for both the whole
plots and the sub-plots. Then approximate f-tests can be formed by substituting the
estimated error variances into Equation (3).
Non-crossed BRDs present a more complicated situation. The equivalency of OLS
and GLS is only true in the case of a first-order model. Although more complex, the
above method can be adapted for the first-order case. Letsinger, Myers, and Lentner
(1996) compare three methods for the second-order case. They are OLS, iterative
reweighted least squares (IRLS), and restricted maximum likelihood (REML). Though
IRLS and REML appear to be better methods, the best method depends on the
design, model, and any prior information.
Bi-randomization introduces the need for new definitions for design efficiency be
cause efficient designs in the literature are based on a completely randomized error
structure. For example, for the BRD the D-optimality criterion (see, eg., Kiefer and


x_1 = 1 z_2
Figure 3: Proposed Design for the 3-2 Case With Just the Centroid
CO
CO


39
The experimental plan in item 5 above will be determined as follows. First, to
define 6 subplots at each step, obtain 61 contrasts for each experimental step. Then
assign factors to contrasts within the group intended for their respective processing
step. This is done in a way that gives the most information on the interaction effects
of interest. The approach is to determine a set of independent contrasts that can
be cycled to produce additional sets. The initial set of independent contrasts must
be chosen so that the groups of effects remain disjoint. This process and the result
ing designs are illustrated for a variety of 64-wafer experiments (see Mee and Bates
(1998)). Split-lot designs for three-level factors are also discussed. It should be noted
that if there are only two steps, the procedures by Miller (1997) can be applied with
one or with many factors at each step.


10
Table 3: Expected Mean Squares Table Under the Most Common Unrestricted Mixed
Model
Source
df
Expected Mean Square
Whole Plot Treatment
t 1
*2 +
=1
Blocks
b- 1
a2 + sa2 + sta0
Block x Whole Plot Treatment
(t-m-
1)
a2 + so2
Subplot Treatment
s 1
*2 + £71
fc=i
Whole Plot x Subplot
(<-!)(-
1)
v /v i\ k\
Error
t(b- 1)(-
1)
a2
Note: Whole Plot and Subplot Treatments are assumed fixed while Blocks
are assumed to be random.
Table 4: Expected Mean Squares Table Under the Restricted Mixed Model
Source
df
Expected Mean Square
Whole Plot Treatment
t 1
a2 + sa20 + £ E=i Ti
Blocks
Block x Whole
1
o5 t1
1
**
a2 + sa20 + stop
a2 + sal0
Subplot Treatment
s 1
o'1 + to£ + & EU il
Block x Sub
(6 1)(* 1)
a2 + ta201
Whole Plot x Subplot
( 1)(* 1)
a2 + a2T01 + £-=i EUinOS
Block x Whole x Sub
(t- 1)(6- l)(a- 1)
g2 +


15
product that is baked is evaluated in a variety of ways, including sensory characteri
zations and analytical and physical testing.
What makes this experimental setup so difficult, is that exploring the profile of
a new oven must typically occur on prototype equipment at the oven manufacturer.
This means the experiment must be conducted in a very short period of time, often
two days or less. This makes it imperative that the experiment have as few runs
as possible usually between 12 and 20 runs. The research described in this thesis is
directly applicable to problems like this, and will be very useful for teams of process
engineers charged with gathering the data they need to fully evaluate candidate ovens
and proofers.
An Integrated Circuits Example
Integrated microflex circuits are manufactured over several, very complicated pro
cess steps. Circuit plating is a key step in this process. It involves depositing a
uniform layer of copper on the microflex circuits. Copper thickness is a key quality
characteristic due to functionality issues. High variability in copper thickness results
in poor bonding of chips to these circuits. Some of the variables that effects circuit
thickness are the circuit geometry, the line speed, the current in amperes, the copper
concentration in the chemistry bath, and the concentration of both sulfuric acid and
hydrogen peroxide.
Designing experiments to optimize copper thickness is a challenge because of the
presence of hard-to-change variables. In particular, restricted randomization occurs
with circuit geometry, line speed, and current. Randomization is not restricted for


140
Table 52: Relative Efficiencies for Comparing Methods of Estimating
V With 4 Mixture Components and 3 Process Variables
d
r = 2
r = 3
OLS
REML
PE
OLS
REML
PE
.11
1.06
1.05
1.04
1.06
1.05
1.06
.43
1.45
1.14
1.16
1.46
1.11
1.08
1.0
2.24
1.23
1.38
2.27
1.16
1.14
2.3
4.11
1.24
1.75
4.19
1.16
1.23
4.0
6.58
1.17
2.12
6.72
1.16
1.28
r 4
r 5
d
OLS
REML
PE
OLS
REML
PE
.11
1.06
1.05
1.10
1.06
1.05
1.15
.43
1.46
1.09
1.06
1.46
1.09
1.07
1.0
2.28
1.13
1.07
2.28
1.10
1.05
2.3
4.21
1.10
1.08
4.20
1.07
1.04
4.0
6.76
1.07
1.07
6.74
1.05
1.03
a level of some factors and then running combinations of the other factors leads to
an experiment with a split-plot structure. Designs are proposed which consider the
process variables as the whole plot factors and the mixture components as the subplot
factors. These designs are an extension to the designs given in Chapter 4.
The split-plot structure of the experiment complicates the estimation of the vari
ance components because OLS is no longer valid. Two alternative methods are pre
sented: REML and a pure error approach. A simulation is conducted to get estimates
of the variance components. The two methods along with OLS are compared using
the determinant of the variance of 3 and forming a relative efficiency in terms of the
asymptotic value. The relative efficiencies give the inflation factor of the size of the
confidence ellipsoid around (3, relative to the true size.


APPENDIX A
TABLES FOR CHAPTER 3 DESIGNS
The tables in this Appendix give the design points for the seven cases discussed
in Chapter 3. The coding convention is as follows:
1 is the low level of a factor
1 is the high level of a factor.
The points for 16 run designs are listed at the beginning of the table followed by the
additional 8 points.
149


21
2. arrangement (b) the whole plots contain the design factors and the subplots
contain the environmental factors;
3. arrangement (c) the subplot factors are assigned in strips across the whole
plot factors (commonly called a strip-block experiment).
These three arrangements are illustrated through an example seeking the best recipe
for a cake mix. Three design factors have been identified as affecting taste. They are
flour, shortening, and egg powder and are studied using a 23 factorial design. The
consumer may have an oven in which the temperature is biased up or down. Also,
the consumer may overcook or undercook the cake. Therefore, the recipe is to be
robust to two environmental factors, oven temperature and baking time, whose levels
are combined using a 22 factorial design.
Arrangement (a)
Under this arrangement, the whole plots contain the environmental factors and
the subplots contain the design factors. Suppose there are m levels of the envi
ronmental factors, Ei,E¡2, ... ,Ej,... ,Em, applied to the whole plots, n levels of
the design factors, Tfi, D2l..., D,..., Dn, applied to the subplots, and l replicates,
T\, r2,..., ru,..., r/, with the whole plots in l randomized blocks. For the cake mix
example, m = 4, n = 8, and 1 = 1. Arrangement (a) requires m x n x l subplots
and m x l whole plots. Thus for the cake mix example, 4 x 8 x 1 = 32 cake mix
batches are required, but only 4x1 = 4 operations of the oven are necessary. By
comparison, a completely randomized cross-product array would require 32 cake mix


5
Table 1: Data for Tensile Strength of Paper (from Montgomery (1997))
Pulp Preparation Method
Block 1
Block 2
Block 3
1
2
3
1
2
3
1
2
3
Temperature
200
30
34
29
28
31
31
31
35
32
225
35
41
26
32
36
30
37
40
34
250
37
38
33
40
42
32
41
39
39
275
36
42
36
41
40
40
40
44
45
plot treatments be comprised of a 22 factorial in time and temperature of a kiln. A
completely randomized experiment would require the kiln to be fired up quite possibly
24 times. With a split-plot experiment, the kiln only needs to be brought up to the
correct temperature 4 times per replicate. This leads to a savings of time and possibly
money.
A split-plot experiment can be run inside of many standard designs, such as the
completely randomized design (CRD) and the randomized complete block (RCB) de
sign. As in the example from Montgomery (1997), suppose the split-plot experiment
is performed using a RCB design. Let denote the observation for subplot treat
ment k receiving whole plot treatment i in block j. Kempthorne (1952) uses as his
model
Vijk /i + T /lj T $ij -(- yk T (T'y'jik T t-ijk
for
i = 1,2,... ,t
= 1,2,....6
k = 1,2, ...,s


19
words in the defining contrast with the fewest letters. Two methods are presented for
constructing minimum-aberration split-plot designs. The first method decomposes
the 2n~k design into the 2("1+n2)_(fcl+fc2) split-plot design. This method is used to
derive extensive, though incomplete, tables of the designs. The second and more
complicated method which involves linear integer programming is used when the first
method fails.
Minimum-aberration two-level split-plot designs are also discussed in Bingham
and Sitter (1999). A combined search and sequential algorithm is presented for con
structing all non-isomorphic minimum-aberration split-plot designs which include the
tables of Huang, Chen and Voelkel (1998). Bingham and Sitter (1999) catalog designs
for 16 and 32 runs containing up to 10 factors. Included in this catalog are the second
and third best minimum-aberration designs since sometimes it may be desirable to
use these designs.
2.2 Split-Plots in Robust Parameter Designs
Genichi Taguchi proposed methods for designing experiments for product design
that are robust to environmental variables. The goal of robust design is to design an
experiment that identifies the settings of the design factors that make the product
robust to the effects of the noise variables. The design factors, which are factors
controlled during manufacturing, make up the inner array while the environmental
factors, or noise factors, make up the outer array. Environmental factors are fac
tors that are difficult to control and can cause variation in the use or performance
of products. The experimental design or crossed array consists of crossing each


APPENDIX B
TABLES FOR CHAPTER 4 DESIGNS
The tables in this Appendix give the design points for the two proposed designs
and all cases discussed in Chapter 4. The coding convention is as follows:
the mixture components are on a scale of 0 to 1 and the decimal values refer to
proportions or fractions of the components
1 is the low level of a process variable
0 is the middle level of a process variable
1 is the high level of a process variable.
The additional design points necessary for the simplex-centroid at the center of the
process variable are given below the design using just the centroid at the center.
161


106
Table 39: Error Degrees of Freedom for the
Two Proposed Designs Under the 5 Cases
Error DF
Mixture
Process
Full
Just
Components
Variables
Simplex
Centroid
3
2
8
2f
3
3
16
10
4
2
14
4*
4
3
29
19
Upper and Lower Constraint
3
2
26
14
t The df may be insufficient for testing. It is
recommended that either the design with the
full simplex be used or replicates of the centroid
be taken at the center of the process variables.
In most cases, both of the proposed designs have sufficient df to estimate the
model. However, for testing the significance of the terms in the model, the design
using the full simplex-centroid at the center of the process variables has more df for
error. Table 39 lists the model and error df for the two proposed designs for the five
cases under consideration. For the unconstrained 3-MC, 2-PV and the 4-MC, 2-PV
cases, there may be insufficient df for error for the design with the single centroid
point, and tests of significance in these situations are not very powerful. In these
cases, it is recommended that either the design with the simplex-centroid at the center
of the process variables be used or replicates of the centroid blend be taken at the
center of the process variables to estimate the error variance.
The testing of the effects should begin with the mixture component by process
variable interactions. These interactions represent the effects of the process variables


131
Table 45: Values for the Variance of (3 (3 Mixture
Components and 3 Process Variables)
d
Asymptotic
OLS
REML
Pure Error
r = 2
.11
7.1 x 10-12
7.3 x 10~12
8.6 x 10_12(1.3 x 10~14)
7.5 x 10_12(7.3 x 10-15)
.43
7.7 x 10-10
9.2 x lo-10
8.7 x 10_lo(7.9 x 10~13)
8.7 x 10_10(7.5 x 10-13)
1.0
6.4 x 10~8
1.2 x 10-7
7.2 x 108 (1.3 x 10"10)
8.2 x 108 (1.8 x 10-10)
2.3
1.1 x 10~5
3.2 x 10-5
1.3 x 10-5 (5.3 x 10~8)
1.7 x 105 (7.4 x 108)
4.0
3.9 x 10~4
1.9 x 10~3
4.9 x 104 (2.6 x 10~6)
7.5 x 10"4 (4.8 x 10-6)
r = 3
.11
6.1 x HT12
6.3 x 10"12
7.2 x 10-12(1.0 x HT14)
6.7 x 10_12(8.7 x 1015)
.43
6.8 x 10-10
8.6 x 10-10
7.5 x 10_10(6.4 x 1013)
7.4 x 10_lo(6.0 x 10-13)
1.0
5.7 x 108
1.0 x 10-7
6.3 x 10~8 (1.0 x 10~10)
6.4 x 10-8 (9.6 x 10-10)
2.3
9.4 x 10"6
2.8 x 10~5
1.1 x 10-5 (3.2 x 108)
1.1 x 10-5 (3.7 x 10-8)
4.0
3.5 x 10~4
1.6 x 10~3
4.1 x 10-4 (1.8 x 10"6)
4.5 x 10-4 (2.2 x 10-6)
r = 4
.11
5.3 x 10-12
5.5 x 10-12
6.2 x 10 12(8.6 x HT15)
5.9 x 10_12(7.0 x 10-15)
.43
6.0 x 10-10
7.6 x 10~10
6.6 x 10_lo(5.4 x lO"13)
6.5 x 10_10(4.8 x lO13)
1.0
5.1 x 108
8.9 x lO"8
5.6 x 10"8 (7.9 x lO"11)
5.6 x 108 (6.4 x 10~n)
2.3
8.5 x 10"6
2.5 x lO"5
9.6 x 106 (2.7 x lO"8)
9.7 x lO6 (2.3 x lO-8)
4.0
3.2 x 104
1.5 x 10"3
3.5 x 10~4 (1.1 x 10"6)
3.7 x lO"4 (1.3 x 106)
r = 5
.11
4.7 x 10-12
4.9 x 1012
5.4 x 10 12(7.1 x 1015)
5.2 x 10_12(5.9 x lO"15)
.43
5.4 x lO"10
6.8 x lO"10
5.9 x 10_lo(4.8 x lO13)
5.8 x 10_10(3.9 x 10-13)
1.0
4.6 x 10~8
8.1 x HT8
5.0 x 10-8 (6.7 x 10"11)
4.9 x 108 (4.5 x lO11)
2.3
7.8 x 106
2.3 x 105
8.5 x 106 (1.7 x 10"8)
8.5 x 106 (1.5 x lO-8)
4.0
2.9 x 104
1.3 x lO3
3.2 x lO4 (7.2 x lO"7)
3.3 x HT4 (7.7 x lO7)
Standard Error for the Simulation are in Parentheses


BIOGRAPHICAL SKETCH
Scott Kowalski was born on February 5, 1969, in Kitchener, Ontario, Canada. He
received his bachelors degree in statistics at the University of Florida in 1993 and his
masters degree in staistics also from the University of Florida. Under the supervision
of Professor G. Geoffrey Vining, he received a Doctor of Philosophy degree in statistics
in December 1999.
He has been married to Kim since 1994. They have a beautiful daughter, Regan
Michelle.
172


145
Table 57: Relative Efficiencies for Comparing Methods of Estimating V With
4 Mixture Components and 3 Process Variables With a Quadratic Model
d
r
2
r =
3
REML
PE
REML
PE
.11
1.03
1.04
1.02
1.03
.43
1.05
1.09
1.05
1.05
1.0
1.05
1.20
1.04
1.09
2.3
1.04
1.39
1.03
1.13
4.0
1.02
1.52
1.02
1.18
r =
4
r =
5
d
REML
PE
REML
PE
.11
1.02
1.03
1.03
1.03
.43
1.04
1.03
1.04
1.04
1.0
1.03
1.05
1.03
1.04
2.3
1.02
1.07
1.02
1.07
4.0
1.01
1.07
1.01
1.06


I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Mtu. d so^
Diane A. Schaub
Assistant Professor of Industrial
and Systems Engineering
This dissertation was submitted to the Graduate Faculty of the Department of
Statistics in the College of Liberal Arts and Sciences and to the Graduate School and
was accepted as partial fulfillment of the requirements for the degree of Doctor of
Philosophy.
December 1999
Dean, Graduate School


162
Table 70: Design Points for 3-2 Case
X\
x3
Zl
z2
1
0
0
-1
1
1
0
0
1
-1
0
1
0
-1
1
0
1
0
1
-1
0
0
1
-1
1
0
0
1
1
-1
.5
.5
0
-1
-1
.5
.5
0
1
1
.5
0
.5
-1
-1
.5
0
.5
1
1
0
.5
.5
-1
-1
0
.5
.5
1
1
.33
.33
.33
-1
0
.33
.33
.33
1
0
.33
.33
.33
0
-1
.33
.33
.33
0
1
.33
.33
.33
0
0
Additional Points for
Simplex-Centroid
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
.5
.5
0
0
0
.5
0
.5
0
0
0
.5
.5
0
0


150
Table 59: Design Points for 2 WP Factors and 4 SP Factors Using Separate Fractions
A
B
P
Q
R
S
1
-1
-1
l
-1
-1
1
-1
-1
-l
1
-1
1
-1
1
-l
-1
1
1
-1
1
l
1
1
-1
1
-1
l
-1
-1
-1
1
-1
-l
1
-1
-1
1
1
-l
-1
1
-1
1
1
l
1
1
1
1
-1
l
-1
-1
1
1
-1
-l
1
-1
1
1
1
-l
-1
1
1
1
1
l
1
1
-1
-1
-1
l
-1
-1
-1
-1
-1
-l
1
-1
-1
-1
1
-l
-1
1
-1
-1
1
l
1
1
1
-1
-1
-l
-1
1
1
-1
-1
l
1
1
-1
1
1
l
-1
-1
-1
1
1
-l
1
-1
1
1
1
-l
-1
-1
1
1
1
l
1
-1
-1
-1
-1
l
-1
1
-1
-1
-1
-l
1
1


134
Table 48: Values for the Variance of /3 (3 Mixture Components and
2 Process Variables in a Constrained Region)
d Asymptotic OLS REML Pure Error
r = 2
.11
0.38
0.38
0.47(.00)
0.40(.00)
.43
16.27
17.69
18.77(.05)
17.61(.02)
1.0
404.89
502.15
494.88(12.13)
464.89(.61)
2.3
13200.28
21133.80
17230.56(31.84)
17048.69(28.36)
4.0
139796.42
290236.72
195563.56(558.13)
203788.49(536.60)
r 3
.11
0.25
0.26
0.27(.00)
0.28(.00)
.43
11.24
12.01
12.30(.03)
12.40(.02)
1.0
286.86
341.64
324.76(.53)
324.30(.60)
2.3
9737.79
14390.76
11437.38(17.68)
11380.29(16.95)
4.0
106541.24
197694.87
130885.49(257.89)
128693.51(235.56)
r = 4
.11
0.19
0.19
0.20(.00)
0.21(.00)
.43
8.58
9.09
9.14(.01)
9.69(.02)
1.0
222.11
258.87
243.24(.34)
249.69(.45)
2.3
7714.29
10908.96
8691.57(11.12)
8583.57(11.25)
4.0
86067.30
149886.77
99194.52(142.72)
96290.94(127.16)
r = 5
.11
0.15
0.15
0.16(.00)
0.18(.00)
.43
6.94
7.32
7.28(.01)
8.49(.02)
1.0
181.20
208.38
193.64(.17)
222.79(.58)
2.3
6387.07
8783.52
7032.91(75.21)
7754.71(15.89)
4.0
72193.86
120695.23
81590.13(105.11)
85671.18(135.44)
Standard Error for the Simulation are in Parentheses


x_1 = 1 z_2
Figure 2: Proposed Design for the 3-2 Case With Full Simplex
co
to


REFERENCES
[1] Addelman, S. (1964). Some Two-Level Factorial Plans With Split-Plot Con
founding. Technometrics 30. pp. 253-258.
[2] Anderson, R. L. (1952). Statistical Theory in Research. McGraw-Hill. New
York, NY.
;3] Bartlett, M. S. (1935). Discussion of Complex Experiments" by F. Yates.
Journal of the Royal Statistical Society. Ser. B, 2, pp. 224-226.
[4] Bingham, D. and Sitter, R. S. (1999). Minimum-Aberration Two-Level Frac
tional Factorial Split-Plot Designs. Technometrics 41, pp. 62-70.
[5] Bisgaard, S. (1999). The Design and Analysis of 2k~p x 2q~r Split Plot
Experiments. accepted by Journal of Quality Technology.
45] Box. G. E. P. and Draper. N. R. (1987). Empirical Model Building and
Response Surfaces. John Wiley : Sons, New York, NY.
[7] Box, G. E. P. and Hunter. J. S. (1961a). "The 2k~p Fractional Factorial
Designs, I. Technometrics 3. pp. 311-352.
[8] Box, G. E. P. and Hunter. J. S. (1961b). "The 2k~p Fractional Factorial
Designs, II. Technometrics 3. pp. 449-458.
[9] Box, G. E. P. and Jones, S. (1992). Split-Plot Designs for Robust Product
Experimentation. Journal of Applied Statistics 19. pp. 3-26.
[10] Box, G. E, P. and Wilson, K. B. (1951). On the Experimental Attainment
of Optimum Conditions. Journal of the Royal Statistical Society. Ser. B, 13,
pp. 1-45.
[11] Cantell, B. and Ramirez. J. G. (1994). Robust Design of a Polysilicon
Deposition Process Using a Split-Plot Analysis. Quality Reliability Engineering
International 10, pp. 123-132.
12] Cornell, J. A. (1990). Experiments With Mixtures: Designs, Models, and the
Analysis of Mixture Data 2nd ed. John Wiley & Sons, New York. NY.
168


104
Table 37: Relative Efficiencies of the Proposed Designs As Compared to
the Designs With the Same Number of Points Chosen by PROC OPTEX
Relative Efficiency
Mixture
Process
Full
Just
Components
Variables
Simplex
Centroid
3
2
73.5 f
75.1 ft
3
3
84.3 f
92.3
4
2
78.8 f
84.4 ft
4
3
84.3 f
91.8 f
Upper and Lower Constraint
3
2
83.0 f
83.2 f
f If relative efficiency is defined using the default N in
the denominator, relative efficiencies would increase.
ft If relative efficiency is defined using the default N in
the denominator, relative efficiencies would decrease.
4.4 Analysis
Cornell (1990) discusses the analysis for the additive and crossed models that use
standard designs. In this paper, the appropriate analysis upon fitting the compro
mised model given in Equation (10) using either of the two proposed designs is needed.
It will be assumed throughout this section that the design is run as a completely
randomized design.
For the proposed designs, there is a total of N 1 degrees of freedom. Estimating
the terms in the model requires p 1 df. Using effect sparsity, the remaining N p
df can be pooled to form an error source which can be used to test the significance
of the terms in the model. The ANOVA table is shown in Table 38.


160
Table 69: Design Points for 4 WP Factors and 4
SP Factors Using Split-Plot Confounding
A
B
C
D
P
Q
R
S
-1
1
-1
-1
1
-l
-1
-1
-1
1
-1
-1
1
i
1
-1
-1
1
-1
-1
-1
i
-1
1
-1
1
-1
-1
-1
-l
1
1
-1
-1
1
-1
-1
i
1
1
-1
-1
1
-1
-1
-l
-1
1
-1
-1
1
-1
1
l
-1
-1
-1
-1
1
-1
1
-l
1
-1
1
-1
-1
1
-1
-l
-1
-1
1
-1
-1
1
-1
i
1
-1
1
-1
-1
1
1
l
-1
1
1
-1
-1
1
1
-l
1
1
1
1
1
1
-1
l
-1
-1
1
1
1
1
-1
-l
1
-1
1
1
1
1
1
-l
-1
1
1
1
1
1
1
i
1
1
-1
-1
-1
1
1
i
1
1
-1
-1
-1
1
1
-l
-1
1
-1
-1
-1
1
-1
l
-1
-1
-1
-1
-1
1
-1
-l
1
-1
-1
1
1
1
-1
i
-1
1
-1
1
1
1
-1
-l
1
1
-1
1
1
1
1
-l
-1
-1
-1
1
1
1
1
i
1
-1


107
on the linear blending properties of the mixture components. Once these have been
investigated, the nonlinear blending properties of the mixture components and the
quadratic effects of the process variables can be tested and this may result in changing
the terms in the original model. The investigation of the interaction terms involves a
main question and the two subsequent questions:
Main: Is the effect of the process variable the same for all blends of the mixture
components?
1. If so, is this effect significantly different from 0?
2. If not, where and how is the effect different?
for each of the n process variables. For a given process variable, there are q interaction
terms of the form
l\kX\Zk, l2kX2zk , 7qkXqzk, for k = 1,2,... ,n.
The questions above partition the q df involving the q interaction terms into q 1
df for testing the main question and 1 df for testing sub-question number 1. The
main question above asks if the effect of Zk is the same on each of the mixture
components. Therefore, the hypotheses are
Ho : 7i k 72k = 4 = 7qk
: at least one not equal
and the tests are carried out for each process variable.
To illustrate how the tests on the interaction coefficients are carried out, consider
the case of three mixture components. Then for process variable, Z\, the hypotheses


44
3.2 Confounding
Suppose that a 2k factorial experiment is to be run in blocks. As noted earlier, the
main disadvantage of 2k factorial experiments is their size. Consequently, even for a
moderate number of factors, it may not be possible to find blocks with the required
number of homogeneous experimental units. When this occurs, it is necessary to use
smaller-sized blocks or incomplete block designs.
With an incomplete block design, there must be some loss of information. A
balanced incomplete block design, if it exists, distributes this loss equally to all treat
ments. However, in factorial experiments, it is the main effects and interactions that
are important. For most factorial experiments with more than three factors, it is
highly unlikely that all effects, especially the higher-order interactions, are important.
If some effects can be assumed negligible prior to performing the experiment, then
a better procedure for constructing incomplete blocks, originally suggested by Fisher
(1926), would be finding arrangements which completely or partially sacrifice the in
formation on these effects so that full information can be obtained on the rest. This
is done by forcing the comparisons among the blocks to be identical to the contrasts
for the negligible effects. Effects that are estimated by the same linear combination
of the treatments are said to be confounded. As a result, it is impossible to determine
if the observed difference is due to differences in blocks or due to the factorial effects
that are aliased woth the blocks.
Effects selected to be confounded with blocks are called the defining contrasts
since they determine which treatments will occur together in a block. These effects
are selected by the experimenter and should be effects thought to be negligible since


46
confounded. While the amount of information is reduced, statistical significance of
each effect can be ascertained.
3.3 Confounding in Fractional Factorials
Although only a fraction of the treatments are included in a 2k~p experiment, this
number may still be too large for available blocks. As in any factorial experiment,
confounding is used to reduce the block size. Confounding an effect in a fractional
factorial experiment also confounds all of its aliases.
Consider a 26_1 fractional factorial experiment using the best defining contrast
for a half-replicate, / = ABCDEF. This requires 32 homogeneous experimental
units. If these are not available, then blocks of smaller size can be created by con
founding additional effects. Suppose blocks of size 16 experimental units are available.
To create two blocks of size 16 for the 32 treatments it is necessary to confound one
effect. Since ABCDEF was used to define the half-fraction, it would appear logical
to select a five-factor interaction, say, ABODE. However, the alias of this interac
tion or generalized interaction of the effect with ABCDEF is F and will also be
confounded with blocks. A better choice is to confound any three-factor interaction
since its alias will also be a three-factor interaction. As a result, no information is
lost on potentially important effects.
The word best should be clarified. It is referring to the design which has the
least amount of aliasing among important effects which are usually thought to be
main effects and two-factor interactions. If important effects are not aliased with each
other, then "best refers to the design with highest Resolution. Therefore, best is


Table 46: Values for the Variance of (3 (4 Mixture
Components and 2 Process Variables)
d Asymptotic OLS REML
Pure Error
r = 2
.11
0.08
0.09
0.09(.00)
0.09(.00)
.43
2.23
3.25
2.65(.00)
2.54(.00)
1.0
35.69
80.65
46.40(.16)
46.60(.14)
2.3
751.67
320.39
1085.94(6.77)
1169.77(69.02)
4.0
6242.75
41467.20
8877.31(70.24)
11478.91(94.98)
r = 3
.11
0.06
0.07
0.07(.00)
0.07(.00)
.43
1.82
2.67
2.10(.00)
1.98(.00)
1.0
29.77
68.04
36.84(.12)
34.22(.07)
2.3
635.53
2679.31
820.09(45.48)
790.57(33.91)
4.0
5306.77
35863.68
6773.26(47.43)
6938.85(42.87)
r = 4
.11
0.05
0.05
0.05(.00)
0.05(.00)
.43
1.54
2.25
1.74(.00)
1.64(.00)
1.0
25.53
58.25
30.16(.08)
28.02(.04)
2.3
550.48
2315.20
644.11(27.95)
625.55(18.87)
4.0
4614.86
31104.49
5354.82(27.54)
5270.57(20.95)
r = 5
.11
0.04
0.05
0.05(.00)
0.05(.00)
.43
1.33
1.94
1.48(.00)
1.40(.00)
1.0
22.35
50.74
25.29(.05)
23.94(.03)
2.3
485.50
2028.05
547.27(19.38)
527.95(10.84)
4.0
4082.56
27307.82
4469.25(16.04)
4467.66(12.82)
Standard Error for the Simu
ation are in Parentheses


I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
G. Geoffrey Vining, Chairman
Associate Professor of Statistics
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
,^k.gA
John A. Cornell
Professor of Statistics
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
I certify that I have read this study and that in my opinion it conforms to accept
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Richard L. Scheaffer
Professor of Statistics


80
are also calculated. They are not reported here, but none of the VIFs exceeds 3.
This confirms that the collinearity is mild.
The cases used in this chapter are chosen because they cover a wide range of
industrial applications. The methods described in this chapter can be applied to
experiments involving even more than four factors at either or both the whole plot
and subplot level. However, as is seen with the case involving four whole plot factors
and four subplot factors, not all chains can be broken. However, the additional
points should result in a design which estimates more effects with less assumptions on
negligibility than a 16 point design. If the cost and time of adding the eight points is
acceptable and the goals of the experiment are those discussed in this chapter, then
the 24 run designs can improve the estimability of important effects.


130
Table 44: Values for the Variance of /3 (3 Mixture
Components and 2 Process Variables)
d Asymptotic OLS REML
Pure Error
r = 2
.11
0.01
0.01
0.01(.00)
O.Ol(.OO)
.43
0.20
0.25
0.22(.00)
0.22(.00)
1.0
2.54
4.51
3.18(.00)
3.04(.00)
2.3
46.98
142.17
64.98(.31)
65.89(.29)
4.0
372.51
1742.28
553.95(3.94)
606.82(3.96)
r =
= 3
.11
0.01
0.01
O.Ol(.OO)
O.Ol(.OO)
.43
0.16
0.20
0.18(.00)
0.01(.00)
1.0
2.09
3.72
2.53(.00)
2.33(.00)
2.3
39.51
121.40
48.90(.21)
46.91(.14)
4.0
315.56
1502.61
420.35(2.75)
393.39(1.96)
r =
= 4
.11
0.01
0.01
O.Ol(.OO)
O.Ol(.OO)
.43
0.13
0.17
0.15(.00)
0.14(.00)
1.0
1.78
3.21
2.02(.00)
1.92(.00)
2.3
34.08
104.63
40.24(.13)
38.25(.08)
4.0
273.71
1301.68
313.64(1.32)
311.67(.99)
r =
= 5
.11
0.01
0.01
O.OO(.OO)
O.Ol(.OO)
.43
0.11
0.14
0.12(.00)
0.12(.00)
1.0
1.55
2.79
1.76(.00)
1.65(.15)
2.3
29.97
91.52
33.35(.08)
32.40(.05)
4.0
241.66
1142.11
277.25(1.11)
263.17(.64)
Standard Error for the Simulation are in Parentheses


xml version 1.0 standalone yes
Volume_Errors
Errors
PageID P296
ErrorID 4
P299
4
P305
4
P314
4
P317
4
P386
4
P434
4


37
Table 10: Strip-plot Configuration of the
Laundry Experiment (from Miller (1997))
Dryer Dryer
Washer
1
2
3
4
Washer
1
2
3
4
1
1
2
2
3
3
4
4
Block 1
31ock 2
Table 11: ANOVA Table for the Laundry Example (from Miller (1997))
Strata
Source
df
E(MS)
Block
Blocks
1
a2 + 4cr^ + 4 cr£ + 16cr^
Row
W-Washer
3
a2+ 4^+ (8/3) W2
3=1
a2 + 4 o\
Row Residual
3
Column
D-Dryer
3
o2 + Aa2c + (8/3) D\
Column Residual
3
a2 + 4 Oq
Unit
W x D
Unit Residual
9
9
3=1k=1
a2


158
Table 67: Design Points for 4 WP Factors and 3
SP Factors Using Split-Plot Confounding
A
B
C
D
P
Q
R
-1
1
-1
-1
1
l
-1
-1
1
-1
-1
1
-l
1
-1
1
-1
-1
-1
l
1
-1
1
-1
-1
-1
-l
-1
-1
-1
1
-1
1
-l
-1
-1
-1
1
-1
-1
i
-1
-1
-1
1
-1
-1
-l
1
-1
-1
1
-1
1
i
1
1
-1
-1
1
1
i
-1
1
-1
-1
1
1
-l
1
1
-1
-1
1
-1
i
1
1
-1
-1
1
-1
-l
-1
1
1
1
1
1
-l
-1
1
1
1
1
-1
i
-1
1
1
1
1
-1
-l
1
1
1
1
1
1
i
1
-1
-1
-1
1
1
-l
-1
-1
-1
-1
1
-1
i
-1
-1
-1
-1
1
-1
-l
1
-1
-1
-1
1
1
i
1
-1
1
1
1
1
i
-1
-1
1
1
1
1
-l
1
-1
1
1
1
-1
l
1
-1
1
1
1
-1
-l
-1


59
In most of the cases studied here, the eight additional points are added to the
initial 16 run design by semifolding on either one or two subplot factors which results
in a 24 point design consisting of four whole plots with six subplots per whole plot.
These designs will have 3 df for the whole plot treatments and 20 df for the subplot
treatments. The initial 16 point design is balanced over the subplot factorseach
factor has the same number of high and low levels presentwhich allows for the
effects to estimated with equal precision. It is desired to preserve this balance of the
subplot factors in the 24 point design as well as maintain the same number of subplot
treatments per whole plot. Therefore, in half of the whole plots the semifolding is on
the high level of a subplot factor while in the other half the semifolding is on the low
level of that factor.
In some cases, it is necessary to fold on a whole plot factor in order to estimate
the main effects of the whole plot factors. In these cases, two whole plots are added so
that the 24 point design consists of six whole plots with four subplots per whole plot.
These designs will have 5 df for the whole plot treatments and 18 df for the subplot
treatments. All nine cases involving 2, 3, and 4 whole plot and subplot factors are
considered. However, two cases do not need to be improved upon.
1. Two whole plot factors and two subplot factors: the 16 points represent the full
factorial. Since no fractionating or confounding is needed, there is nothing to
improve upon.
2. Two whole plot factors and three subplot factors: in this case, the MA design
presented in Huang, Chen and Voelkel (1998) is the best design possible and


55
Table 19: Alias Structure for Split-Plot Confounding
For 8 Whole Plots With I = ABPQ = ACQR = BCPR
A
=
B
=
C
=
AB
=
PQ
AC

QR
BC
=
PR
P
=
Q

R
=
AP

BQ
AQ

BP + CR
AR

CQ
BR

CP
only 8 df left for the subplot factors and whole plot x subplot factor interactions.
Therefore, at the subplot level there are only enough df to estimate either three
main effects and five interactions or eight interactions. This may not be sufficient to
estimate all the effects of interest.
3.5 Discussion of Minimum-Aberration Split-Plot Designs
In split-plot designs using some sort of confounding, there is a concept of partial
resolution. The partial resolution of the whole plots refers to the resolution of terms
in the defining contrast involving only whole plot factors. The partial resolution of
the subplot factors refers to the resolution of terms in the defining contrast involving
either only subplot factors or both whole plot and subplot factors. Recall that the
definition of minimum-aberration is the design that has smallest number of words in


73
Table 30: 24 Point Design for the Case of 4 WP Factors and
4 SP Factors Using the Same Fraction
Fold on A
b c ad abed d bed
pq
qs
pr
rs
s
P
qrs
pqr
Q
q
Q
q
r
r
r
r
ps
ps
ps
ps
pqrs
pqrs
pqrs
pqrs
Table 31: 24 Point Design for the Case of 4 WP Factors and
4 SP Factors Using Split-Plot Confounding
b c ad abed
V
qrs
(1)
q
pqr
s
qr
r
qs
pq
pqs
ps
rs
pr
prs
pqrs
Fold on A
d bed
pqrs
qs
ps
rs
q
P
r
pqr


129
Comparison of Methods
The methods are compared for various values of d. Since it is assumed that a\ = 1,
then d a represents the whole plot error variance as well as the ratio of the two
error terms. Following Letsinger, Myers and Lentner (1996), five values are used for
d: .11, .43, 1.0, 2.3, and 4.0. This includes values where the whole plot error variance
is smaller than (.11, .43), equal to (1.0) and greater than (2.3, 4.0) the subplot error
variance. The designs consisted of r = 2, 3, 4 and 5 total replicates of the center of
the process variables (whole plot factors).
Tables 44 48 show the asymptotic value, the OLS value, the average determinant
values for REML, and the average determinant values for the pure error approach.
The standard errors for the simulated values are also given. It appears that in general
the standard errors are smaller for the pure error method especially for 4 and 5
replicates of the center of the process variables. Since it is difficult to compare the
average determinants directly, the notion of relative efficiency is used.
The results from the simulation are compared to the asymptotic values. This is
done through relative efficiency which for OLS is defined as
|(X'X)-X'VX(X'X)->|
ReL Eff- KX'V^X)-1!
where V is the true variance-covariance matrix. For REML and the pure error
method, relative efficiency is defined as
= avg. |(X/V~1X)~1X'V~1VV~1X(X/V;r1X)-1|
kx'v-x)-1!
where the average is over the 10,000 simulated determinants. Tables 49 53 show the
results for the five values of d and r = 2, 3, 4, and 5.


69
4 WP Factors (A, B, C, D) and 3 SP Factors (P, Q, R)
To obtain a 16 point design in this situation, both the whole plot and subplot
treatments need to be fractionated or confounded. First, consider fractionating the
whole plot and subplot treatments separately. The defining contrast is I = ABC =
BCD = AD = PQR = ABCPQR = BCDPQR = ADPQR which is resolution
II. The additional whole plot treatments are obtained by semifolding on factor A.
The positive fraction, I = PQR, is run in one whole plot while the negative fraction,
I = PQR, is run in the other whole plot. The negative fraction can be thought of
as semifolding on any subplot factor and placing all of the points in one whole plot
instead of two as was done in all the cases up until now. The 24 point design is shown
in Table 28. The chains are broken and everything is estimable.
Next, consider split-plot confounding. The defining contrast is I = ABC =
BCD = AD = CPQR = ABPQR BDPQR ACDPQR which is resolution II.
Besides breaking chains among the whole plot factors, the chain, CPQR needs to be
broken. The additional whole plot treatments are obtained by semifolding on factor
A. Again, the positive fraction, I = PQR, is run in one whole plot with the negative
fraction, I = PQR, is run in the other whole plot. Again, this can be thought of
as semifolding each fraction on any subplot factor and placing all four points in the
same whole plot. The 24 point design is shown in Table 29. The chains are broken
and everything is estimable.


CHAPTER 6
SUMMARY AND CONCLUSIONS
Many experiments that are performed in industry have cost and/or time con
straints which limit the size of the experiment. A concern with small experiments is
how much information can really be obtained in this setting. The global theme of
this dissertation is to propose methods for constructing designs for small experiments
that will give as much information as possible about the factors involved.
In Chapter 3, the experiments of interest are those that contain some factors that
are hard-to-change and some that are relatively easy-to-change. When this is the
case, it is more economical to run the experiment as a split-plot experiment. If all
of the factors are at two levels, the size of the experiment can be reduced by using
fractional factorials, split-plot confounding, or both. We have presented methods
for constructing sixteen run designs that will be economically feasible. Also, if the
experiment is in the framework of robust parameter design, then these designs handle
the subplot factors quite well. However, if possible, it is beneficial from an estimation
point of view to convince the experimenter to add eight more runs at the subplot
level. This will break alias chains in order to get better estimates of the parameters as
well as add some degrees of freedom for testing. Chapter 3 is concluded with several
examples of the proposed methods.
Mixture experiments are also widely used in industry. Often times the quality of
the product depends not only on the relative proportions of the mixture components
147


Table 5: Analysis of Variance Table for a Split-Plot Experiment Run Using a RCB
Design With Factorial Structure and the Most Common Unrestricted Model
Source
df
Whole Plot Treatment
t 1 = 3
V
1
22f
1
2l22f
1
Blocks
b- 1
Block x Whole Plot Treatment
(*-1X&-1)
Subplot Treatment
s- 1 = 3
Xi d
1
x2 tf
1
x\x2 d
1
Whole Plot x Subplot
(t-l)(s-l) = 9
zixi d
1
Z\X2 ft
1
22Z1 n
1
z2x2 ft
1
Z\Xix2 d
1
Z2X1X2 d
1
Z\Z2X\ d
1
Z\Z2X2 d
1
Z\Z2X\X2 d
1
Error
i(6- l)(s- 1)
f These terms are tested using the Block
x Whole Plot Treatment interaction.
ft These terms are tested using Error.


20
experimental design setting of the inner array with each experimental design setting
of the outer array. Unless the number of factors in these arrays is small, Taguchis
designs become large and expensive.
An alternative to Taguchis crossed array is the combined array. The combined
array utilizes a single experimental design in both the design and environmental fac
tors. Therefore, the response is modeled directly as a function of the design factors
and the environmental factors using a single linear model. More details on the com
bined array can be found in Welch, Kang, and Sacks (1990); Shoemaker, Tsui, and
Wu (1991); and ODonnell and Vining (1997).
Bisgaard (1999) discusses split-plot designs in association with inner and outer
array designs. He focuses on screening experiments that use restricted randomization.
The paper gives a nice overview of defining relations and confounding structures for
the 2k~p x 2q~r split-plot designs. In addition to split-plot confounding, Bisgaard
(1999) points out that the same fraction of the subplot factors can be run in each
whole plot. The appropriate standard errors for testing effects when using split-plot
confounding are also given.
Box and Jones (1992) investigate the use of split-plot designs as an alternative to
the crossed array. They consider three experimental arrangements where the robust
parameter design is set up as a split-plot design:
1. arrangement (a) the whole plots contain the environmental factors and the
subplots contain the design factors;


9
Table 2: Expected Mean Squares Table Under Randomization Theory
Source
df
Expected Mean Square
Whole Plot Treatment
t 1
'1 i bs Z2
*6 t1 1 'i
Blocks
Whole Plot Error
b- 1
a2 + SCTg
-I
Subplot Treatment
s 1
k= 1
Whole Plot x Subplot
(t l)(s 1)
Subplot Error
t(b-l)(s-l)
a2
Note, if h = 1, the variance of e^kh. is not estimable. This restricted analysis reduces
to the other two analyses only if the block by subplot interaction is unimportant. In
such a case, its contribution can be pooled with the block by whole plot by subplot
interaction to form the same error term as the randomization and unrestricted mixed
model analyses.
Whole plot treatments are applied to blocks of t units which can be divided
further into s subunits, where s is the number of levels of the subplot treatment. Any
differences among these blocks must be confounded with the whole plot treatment
comparisons. Consequently, comparisons among the subplot treatments are made
with greater precision, and this leads to the more important factor usually being
assigned to the subplot. Using the unrestricted model and Table 3, it is seen that the
null hypothesis of no whole plot treatment effect, H0 : T\ = r2 = = rt, versus at
least one not equal, is tested using the Block x Whole Plot Treatment interaction as


68
Table 26: 24 Point Design for the Case of 3 WP Factors and 4 SP Factors Using
Split-Plot Confounding [HP(HS)-denotes high P(S) and LP(LS)-denotes low P(S)]
abe abe
P
s
qr
Q
pqr
pq
pqs
r
qs
pr
prs
ps
rs
qrs
(1)
pqrs
Folc
HP LP
on
LS HS
(1)
ps
qrs
P
qr
pqrs
s
pqr
Table 27: 24 Point Design for the Case of 4 WP Factors and 2 SP
Factors Using a Fractional Factorial of the Whole Plot Factors
Fold on A
b c ad abed d bed
P
P
p
P
Q
q
Q
Q
pq
pq
pq
pq
(1)
(1)
(1)
(1)
plots will need to be added for all cases involving four whole plot factors. Since
nothing needs to be done to the subplot factors, there is only one 16 point design. The
defining contrast is I ABC BCD = AD which is resolution II. The additional
whole plots are obtained by semifolding on factor A. The 24 point design is shown in
Table 27. The chains are broken and everything is estimable.


26
each design factor on the responses. A more detailed discussion of the selection of
factors, design, and analysis of the SNRs is contained in Chapter 4 of Phadke (1989).
The actual structure of this experiment was a split-split-plot design because there
are three sizes of experimental units with different sources of variation. The design
factors are applied to the tube (run-to-run variability); the location in the tube affects
the wafer (wafer-to-wafer variability), whereas location in the wafer affects the die
(die-to-die variability). Therefore, using Taguchis SNRs to analyze this experiment
will result in a complete loss of information in the design x noise factor interactions.
Cantell and Ramirez (1994) reanalyzed the data as if it were a split-split-plot design.
They pooled higher order interactions to get the necessary error terms in order to
perform hypothesis tests on the design factors and the design x noise factor inter
actions. Interaction plots were used to determine the level of the design factor that
minimized the variation across the levels of the noise factors. Although the final
recommendations on the design factor levels by Cantell and Ramirez (1994) differed
from Phadke (1989) on only one of the six design factors, the use of the split-split-plot
design has allowed the process engineer to have a better understanding of the sources
of variation. This added information may lead to process improvement in the future.
Kempthorne (1952) and Box and Jones (1992) provide details on the relative
efficiency of these split-plot designs compared to the CRD and RCB. A summary of
their conclusions is provided here. Consider the split-plot experiment as a uniformity
trial. If the uniformity trial was run as a CRD or a RCB experiment, then, for
arrangements (a) and (b), the subplot factor effects and the subplot x whole plot
interactions are estimated more precisely than the whole plot factor effects. Compared


97
specify the number of points in the final design, N. Then the number of points in
the final design chosen by the computer is fixed at N or the default, which is the
number of terms in the model, p 15, plus ten additional points. A second option
is the DETMAX option. This option uses the detmax routine developed by Mitchell
(1974) to obtain a design with the maximum determinant of X'X, where X is a N x p
matrix containing the design runs. PROC optex returns the chosen design and its
D-criterion, which is defined as
D-criterion = 100 x
det (X'X)1/P
N
and is used to compare the designs.
Two designs generated by PROC OPTEX are shown in Figures 5 and 6. These
designs consist of 23 and 17 points, respectively, and are to be compared to our
designs in Figures 2 and 3, respectively. Note that unless PROC optex is run with
the same options and starting seed, the designs generated could be different. Tables
35 and 36 show the D-criterion for our proposed designs and the designs generated
by PROC optex for the cases where q 3 or 4 and n 2 or 3. The designs from
PROC OPTEX are with N restricted to the number of points from the proposed design
as well as the default N. All five cases under consideration are given.
The relative efficiencies of our two proposed designs are defined as
D-criterion for the proposed design
D-criterion for PROC optex designs with same number of points
and are given in Table 37. From Table 37, it is seen that in all cases, both of our
designs are at least 73% as efficient as the computer generated designs and excluding


Center
Figure 4: Proposed Design for the 3-3 Case
to
Cn


151
Table 60: Design Points for 2 WP Factors and 4 SP
Factors Using Split-Plot Confounding
A
B
P
Q
R
S
1
-1
1
-l
-1
-1
1
-1
1
l
1
-1
1
-1
-1
l
-1
1
1
-1
-1
-l
1
1
-1
1
-1
-l
-1
1
-1
1
1
i
-1
-1
-1
1
1
-l
1
-1
-1
1
-1
l
1
1
1
1
-1
i
-1
-1
1
1
-1
-l
1
-1
1
1
1
-l
-1
1
1
1
1
l
1
1
-1
-1
-1
l
1
-1
-1
-1
1
i
-1
1
-1
-1
1
-l
1
1
-1
-1
-1
-l
-1
-1
1
-1
-1
-l
-1
-1
1
-1
-1
l
1
-1
-1
1
1
-l
-1
1
-1
1
1
i
1
1
1
1
1
-l
-1
-1
1
1
1
l
1
-1
-1
-1
-1
i
1
1
-1
-1
-1
-l
-1
1


112
2. Collect additional observations at points other than the design points. These
are called check points.
3. Compare the observed data at the check points to the predicted response from
the fitted model at the check points.
The basic idea is that if the predictions are close to the observed values, then the
model does not suffer from lack of fit (see Cornell (1990) for more details).
If lack of fit is detected, then the proposed model needs to be upgraded by the
addition of higher-order terms. Significant terms in the fitted model might suggest
possible candidates for lack of fit. For the proposed model, some likely candidates
are
i. the special cubic terms involving the mixture components (ie. X\X2X3 ...)
ii. the interactions between binary blends and the linear effects of a process variable
(ie. XiX2zi ...)
iii. the interactions between the linear blending terms and the two-factor interactions
among the process variables (ie. X\Z\Z2 ...)
Once the necessary terms are added to the model, the new model should also be
checked for lack of fit.
One nice property of our proposed designs is that both designs can support the
fitting of higher order terms without adding more design points. The design with the
simplex-centroid at the center of the process variables is used since it has more degrees


D-Criterion D-Criterion
102
3-2 Case
15
o
LO
O
q
co
o
cvi
20 25 30
Design Size
3-3 Case
25 30 35 40
Design Size
35
45
Figure 7: Plot of D-Criterion Versus Design Size (C: Design With Just the Centroid,
F: Design With the Full Simplex-Centroid at Center of Process Variables)


148
but also on the processing conditions. We considered the process variables as two-
level factors. A new second order model that is intended to be a compromise between
the common mixture models used and standard response surface models is given in
Chapter 4. With this model in hand, a new class of designs is proposed that will
support the fit of this model. Keeping to the global theme, the designs are kept
relatively small. Assuming a completely randomized design, the analysis for these
experiments is provided with a small discussion of lack of fit.
Depending on the types of process variables being used, it may be too costly to
run a completely randomized experiment. In Chapter 5, we consider the experiment
as a split-plot. The process variables are assigned as the whole plot factors with the
mixture components serving as the subplot factors. The split-plot nature complicates
these experiments since ordinary least squares is no longer valid. An iterative ap
proach is needed. We investigate how ordinary least squares holds up to restricted
maximum likelihood under various ratios of the whole plot error variance to the sub
plot error variance. Also, another method for estimating these variance components
is presented which uses pure error from replicated points.
In consideration of the types of experiments that are being run in industry, the size
of the experiment has been of great concern in this dissertation. We have presented
methods for obtaining as much information as possible with a small experiment. Many
industrial experiments fall into the class of split-plot experiments either by nature or
by consequence of cost. We have shown how to design and analyze small experiments
which are in this class.


28
Table 7: Crossed BRD From Letsinger,
Myers, and Lentner (1996)
Zi
Xl
X&
z2
Xl
Xb
!
;
\
z a
Xl
X6
Table 8: Noncrossed BRD From Letsinger,
Myers, and Lentner (1996)
Zl
Xu
Xl6l
Z2
X21
X262

;
Za
Xal
Xa6a
The non-crossed BRDs differ from the crossed BRDs in that not all levels of x
are associated with z. The whole plots have different levels of the sub-plots and need
not have the same number of levels. Non-crossed BRDs are constructed as follows:
1. randomize the a unique combinations of z to the whole plot EUs, then
2. randomize the bl levels of x to the smaller EUs within each whole plot (see
Table 8).
The distinction between these two can be thought of in terms of the sub-plot factors.
The crossed BRD might be represented by a 2k factorial in the sub-plot factors while
the non-crossed BRD might use a 2k~p fractional-factorial in the sub-plot factors but
not the same 2k~p set of treatments.


22
batches and 32 operations of the oven. Thus, the split-plot arrangement has saved
time by reducing the number of operations of the oven.
The model for arrangement (a) is
Vijk H + 7fc + aj + Vjk + 6i + (Q)ij + eijk ,
where yijk is the response of the kth replicate of the zth level of factor D and the jth
level of factor E, // is the overall mean, 7*. is the random effect of the A:th replicate
with 7* ~ N(0, CTy), aj is the fixed effect of the jth level of factor E, is the fixed
effect of the th level of factor D, (a)^ is the interaction effect of the zth level of D
and the jth level of E, r¡jk ~ A^(0, is the whole plot error, etjk ~ N(0, ais the
subplot error, and 7)jk and are independent.
Arrangement (b)
With this arrangement, the whole plots contain the design factors while the sub
plots contain the environmental factors. Arrangement (b) requires only 8x1 = 8 cake
mix batches but requires 4 x 8 x 1 = 32 operations of the oven. Again, a completely
randomized cross-product array would use 32 cake mix batches and 32 operations
of the oven. Here, the savings of the split-plot design is not as great since only the
number of cake mix batches is reduced. This is not an ideal situation for indus
trial experiments. First of all, the design factors are of greater interest. Therefore,
applying the design factors to the whole plots results in a loss of precision for the
design factors. Hence, it is possible to have large differences between the levels of
the design factors that are insignificant when tested. Also, from an economic point


64
Table 22: 16 Point Design for the Case of 3 WP Factors and 2 SP Factors
Using a Fraction Factorial of the Whole Plot Factors
a b c abc
p
p
p
p
q
q
q
q
pq
pq
pq
pq
(1)
(i)
(i)
(i)
3 WP Factors (A, B, C) and 2 SP Factors (P, Q)
In this case, only the whole plot factors need to be fractionated. Since nothing
needs to be done to the subplot factors, there is only one 16 point design. The defining
contrast is I ABC which is Resolution III. Since the design estimates everything
set out in the goal of the experiment, no points need to be added to this design.
However, note that this is not the MA design which is run using eight whole plots
with 2 subplots per whole plot. The 16 point design in four whole plots with four
subplots per whole plot is shown in Table 22.
3 WP Factors (A, B, C) and 3 SP Factors (P, Q, R)
To obtain a 16 point design in this situation, both the whole plot and subplot
treatments need to be fractionated or confounded. First, consider fractionating the
whole plot and subplot treatments separately. The defining contrast is I = ABC =
PQR = ABCPQR which is resolution III. The two-factor interactions between whole
plot and subplot factors are already estimable. Therefore, there is only one alias
chain that needs to be broken, and that is associated with PQR. The additional


144
Table 56: Relative Efficiencies for Comparing Methods of Estimating V With
4 Mixture Components and 2 Process Variables With a Quadratic Model
d
r =
2
r =
3
REML
PE
REML
PE
.11
1.03
1.04
1.03
1.03
.43
1.07
1.09
1.07
1.05
1.0
1.09
1.20
1.08
1.10
2.3
1.10
1.38
1.06
1.15
4.0
1.08
1.41
1.05
1.17
r =
4
r
5
d
REML
PE
REML
PE
.11
1.02
1.01
1.03
1.03
.43
1.06
1.03
1.04
1.03
1.0
1.07
1.05
1.04
1.04
2.3
1.04
1.08
1.03
1.04
4.0
1.03
1.08
1.02
1.04


1
128
the process variables, the estimated subplot pure error term is
\ 2
/ in / \
, e e (y- y>)
q2 1 1=1 j= 1 ^
sp~ r 771-1
When there are three mixture components, the above term is the estimated subplot
error term. However, when there are four mixture components, two total replicates
of the centroid are added to the whole plots that contain the vertices. As was pointed
out earlier, this is done to have a balanced design. So in each of these whole plots,
one extra degree of freedom can be used to estimate the subplot error term. The
estimate of the subplot error term becomes a weighted error term and is given by
e2 (2 ~ + (2 1)5^2 + + (2 l)S^p V2h-i + SgpQ
sp ~ (2 1) + (2 1) + + (2 1) + r{m 1)
o2 Ssp,V 1 + Ssp,V2 ~l ^ S'sp.V^-l + ffap.O
2fc_1 + r(m 1)
where
V /
and SlpVk (Vi Y)2 + (Fj Y)2 is the estimate of subplot error from the two
replicated centroids in the whole plots that have the vertices, k = 1,2,..., 2k~1.
The variance of /3 is the same as given for the REML case. Once the pure er
ror estimates are obtained using the above formulas, they can be used to get Vc.
Again, the procedure is repeated 10,000 times and the det(Var(/3)) is calculated each
time. Then, the average of the determinants is compared to the determinant of the
asymptotic Var(¡3).


CHAPTER 1
INTRODUCTION
A common exercise in the industrial world is that of designing experiments, explor
ing complex regions, and optimizing processes. The setting usually consists of several
input factors that potentially influence some quality characteristic of the process,
which is called the response. Box and Wilson (1951) introduced statistical methods
to attain optimal settings on the design variables. These methods are commonly
known as response surface methodology (RSM), which continues to be an important
and active area of research for industrial statisticians.
Many times in industrial experiments, the factors consist of two types: some with
levels that are easy to change and one or more with levels that are difficult or costly
to change. Suppose for illustration that there is only one factor that is difficult to
change. When this is the case, the experimenter usually will fix the level of this factor
(ie., restrict the randomization scheme) and then run all combinations or a fraction of
all combinations of the other factors, which is known as a split-plot design. Too often,
the data obtained from this experiment are analyzed as if the treatment combinations
were completely randomized, which can lead to incorrect conclusions as well as a loss
of precision. Analysis of data obtained from experiments, such as the example above,
need to take the restricted randomization scheme into account.
1.1 Response Surface Methodology
In RSM, the true response of interest, r/, can be expressed as a function of one or
more controllable factors (at least in the experiment being performed), x, by
1


171
43] Taguchi, G. (9187). Systems of Experimental Design, Vol. 1 and 2. Kraus
International Publications, White Plains. NY.
[44] Welch, W. J. ; Kang, T. ; and Sacks, J. (1990). "Computer Experiments
for Quality Control by Parameter Design. Journal of Quality Technology 22.
pp. 15-22.
[45] Wooding, W. M. (1973). "The Split-Plot Design. Journal of Quality Tech
nology 5, pp. 16-33.
[46] Yates, F. (1937). The Design and Analysis of Factorial Experiments. Imperial
Bureau of Soil Sciences, Harpenden, England.


164
Table 72: Design Points for 4-2 Case
X\
Z2
X4
Zl
Z2
1
0
0
0
-1
1
1
0
0
0
1
-1
0
1
0
0
-1
1
0
1
0
0
1
-1
0
0
1
0
-1
1
0
0
1
0
1
-1
0
0
0
1
-1
1
0
0
0
1
1
-1
.5
.5
0
0
-1
-1
.5
.5
0
0
1
1
.5
0
.5
0
-1
-1
.5
0
.5
0
1
1
.5
0
0
.5
-1
-1
.5
0
0
.5
1
1
0
.5
.5
0
-1
-1
0
.5
.5
0
1
1
0
.5
0
.5
-1
-1
0
.5
0
.5
1
1
0
0
.5
.5
-1
-1
0
0
.5
.5
1
1
.25
.25
.25
.25
-1
0
.25
.25
.25
.25
1
0
.25
.25
.25
.25
0
-1
.25
.25
.25
.25
0
1
.25
.25
.25
.25
0
0
Additional Points for
Simplex-Centroid
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
.5
.5
0
0
0
0
.5
0
.5
0
0
0
.5
0
0
.5
0
0
0
.5
.5
0
0
0
0
.5
0
.5
0
0
0
0
.5
.5
0
0


101
the 3-MC, 2-PV case, the efficiencies are even higher. Also, if one simply turned
the computer loose to choose a design, the design efficiencies for almost all of our
designs increases. The design with just the centroid blend at the center setting of
the process variables is more efficient than when running the simplex-centroid design
at the center setting of the process variables, especially in the cases involving three
process variables. Hence, the addition of the mixture vertices and mid-edge points
at the center of the process variables does not seem to be beneficial in terms of D-
criterion. However, when considering the analysis, it is appealing to run the entire
simplex-centroid at some location. Therefore, as in most design situations, there are
trade-offs when deciding between designs. A plot of the D-criterion versus the size of
the design is shown in Figure 7. As the design size increases, the D-criterion increases
in value up to a certain point and then levels off. In other words, there is a point of
diminishing return when considering the model in Equation (10).
Suppose the model in Equation (6) is used for the process variables instead of
the model in Equation (9) (i.e. the model for the process variables is believed to
be first-order plus interactions). As mentioned earlier, the combined model can be
adjusted by omitting the n pure quadratic terms in the process variables. Then the
number of points in the two proposed designs can easily be reduced for this situation.
All that needs to be done is to eliminate the axial points which are no longer required
since the pure quadratic terms in the z/s are no longer in the model. This results
in a reduction in design size of 2n points for n process variables. This flexibility is
another appealing feature of the two proposed designs.


62
The defining contrast is I = PQR = QRS = PS which is resolution II. Alias chains
involving both P and S need to be broken. Therefore, the additional eight points
are obtained by semifolding on high and low P in two whole plots and on high and
low S in the other two. The 24 point design is shown in Table 20. The chains are
almost completely broken. Only two of the three interactions, BP, BS, and PS
are estimable. If it can be assumed that PS is negligible, then everything else is
estimable. It is not unreasonable to believe that with four factors one of the two-
factor interactions is negligible and the experimenter should be able to help determine
which interaction is most likely to be negligible.
Next, consider split-plot confounding. The defining contrast is I = APQR =
BQRS = ABPS which is resolution IV. This design is the MA design given in Huang,
Chen and Voelkel (1998). Again, alias chains involving both P and S need to be
broken. Therefore, the additional eight points axe obtained by semifolding on high
and low P in two whole plots and on high and low S in the other two. The 24
point design is shown in Table 21. All of the two-factor interactions between whole
plot and subplot factors can be estimated except BS which is aliased with QR. If
QR is assumed to be negligible, then BS can be estimated. Most of the two-factor
interactions among the subplot factors are aliased with each other. However, with
the 24 point design we can estimate three of the two-factor interactions between
the subplot factors without making any assumptions about negligibility, which is an
improvement over the MA design.


109
would produce the correct simultaneous test of the hypotheses. If H0 is rejected,
then there is an interaction present between the mixture components and the process
variable. Hence, the effect of the process variable is not the same for all mixture
components. A further investigation of the form of the interaction needs to be carried
out. This is easily accomplished with plots.
If H0 is not rejected, then for a fixed process variable, k, the 7*s are equal. In
other words, the process variable affects the mixture components equally when the
mixture region is a simplex. A follow-up test is needed since the equal magnitude
could be zero. As it turns out, this test will investigate whether or not there is an
additive effect due to the process variable.
Since H0 : 71* = 72k = 7,fc is not rejected, the terms in the model in Equation
(10) involving the interaction of the mixture components and process variable, k, can
be rewritten as
l\kX\Zk + ~(2kx2zk H 4- 7qkXqZk = 7*^1 zk + 7kx2zk -I h 7*****
= lkZk(xx + x2 + + xq)
- lkzk
Therefore, the follow-up test is a test of the significance of the additive effect of
process variable k. Note, this would have been the test for the additive model. All
that needs to be done for the analysis is to refit the model with zk replacing the
interactions of the mixture components with zk. This will provide the 1 df t-test that
combined with the q 1 df from above partitions the q df involving the interactions
of the mixture components and the process variables. This is the test for the additive
effect of the process variable zk and is done for each process variable that H0 is not


33
A similar simulation was conducted for two-level factorials (see Lucas and Ju
(1992)). They considered a 24 factorial with xx as the hard-to-change factor. This
allows the fitting of a regression model that includes the linear and interaction terms.
Again, a summary table is provided by Lucas and Ju (1992) showing s imilar results to
the other experimental scenarios. The completely restricted experiment had smaller
standard deviations for all the regression coefficients except ¡3\. Table 9 gives the
formula for the variance of the regression coefficients for a 2k factorial experiment
with one hard-to-change factor. Recall that in the partial restricted case, the blocking
was done at random. This can be improved on by blocking orthogonally. The 24
factorial can easily be blocked orthogonally in 4 blocks of size 4 or 8 blocks of size 2.
Both of these blocking schemes are an improvement over the partially restricted case
in that they have smaller standard deviations on the easy-to-vary factors.
Cornell (1988) discusses the analysis of data from mixture experiments with pro
cess variables where the mixture blends are embedded in the process variable com
binations as in a split-plot design. The mixture process variables are factors that
are not mixture ingredients but whose levels could affect the blending properties of
the mixture components. To illustrate this situation, Cornell uses an example from
Cornell and Gorman (1984) involving fish patties. The mixture experiment involves
making fish patties from different blends of three fish species (mullet, sheepshead,
and croaker). The patties were subjected to factor level combinations of three process
variables (cooking temperature, cooking time, and deep-frying time). Each process
variable was studied at two levels. When process variables are included in a mixture


83
Gorman give recommendations regarding the choice of design which depend on the
form of the model to be fitted and whether or not there is prior knowledge on the
magnitude of the experimental error variance.
Czitrom (1988, 1989) considered the blocking of mixture experiments consisting
of three and four mixture components. She used two orthogonal blocks to construct
D-optimal designs. Draper et al. (1993) consider mixture experiments with four
mixture components. They treat a combination of the levels of the process variables
as defining blocks. Orthogonally blocked mixture designs constructed from Latin
Squares are presented. The optimal choice of a design using D-optimality is also given.
While the reduction in the number of observations required can be great, obtaining
D-optimality comes with a price. The D-optimal designs require very nonstandard
values for the component proportions.
We propose an alternative approach to reducing the size of the experiment which
borrows ideas from the above references. The concept of running only a subset of the
total number of mixture-process variable combinations is borrowed from Cornell and
Gorman (1984), although our fraction will involve the mixture component blends as
well. To evaluate the fraction, we shall make use of the D-criterion criterion (Czitrom
(1988, 1989)). The next section provides a type of experimental situation which led
to this research. In the section that follows, a combined model which is slightly
different in form from the combined mixture-process variable models ordinarily used
is presented. The method for constructing the design and comparing its D-criterion
is discussed in the fourth section. The final section of this paper contains details on
the analysis of the experiments using the proposed designs and model forms.


135
OLS performs poorly except when d < 1. This is expected because when d is
small relative to one, the variance of /3 is dominated by the subplot error, at. So,
when d is small, the variance-covariance matrix is close to of times an identity matrix
which is what OLS is assuming.
The pure error approach and REML are quite comparable when three replicates
of the the center of the process variables are run. When only two replicates are run,
REML tends to perform better. With four or five replicates, the pure error method
performs very well. Therefore, if the experimenter has no prior knowledge of the error
variance values and can afford the extra center runs, the pure error method is a simple
and effective way to estimate the error variances. To get estimates of the coefficients,
once Vc is calculated, it can be inputted as the weight matrix and weighted least
squares can be performed. If REML is to be used, the replicates of the the center
of the process variables would most likely not be run. While the design without
these points cannot be directly compared to a design with these points, it is believed
that REML is helped by the inclusion of the replicate points. An added benefit of
including the replicate points and using the pure error approach is the ability to test
for lack of fit of the model.
5.2 Second-Order Model in the Process Variables
Suppose now that a second-order model is assumed for the process variables. The
combined model using a Taylor series approach is
Q Q n n q n
fi(x,z) = E Kxi + Y. E Pip&i + Z E otkiZkZi + a^zl + Y.Y.
i=l i

CHAPTER 2
LITERATURE REVIEW
The split-plot error structure has been underutilized in RSM. Most RSM ex
periments assume a completely randomized error structure. Letsinger, Myers, and
Lentner (1996, pg. 382) point out, Unfortunately, while this completely randomized
assumption simplifies analysis and research, independent resetting of variable levels
for each design run may not be feasible due not only to equipment and resource con
straints, but also budget restrictions. This chapter focuses on the literature involving
restricted randomization within RSM.
2.1 Split-Plot Confounding
When the whole plot and/or subplot treatments are of a factorial nature, it is
possible to reduce the number of whole plots and/or subplots needed through frac
tionating. This is important in industrial experiments where constraints limit the size
of the experiment. Bartlett (1935) suggested the possibility of confounding higher-
order subplot interactions to reduce the number of subplots needed within each whole
plot. Later, split-plot confounding was studied by Addelman (1964). He provided
a table containing factorial and fractional-factorial arrangements that involve split-
plot confounding. However, he did not consider confounding within the whole plots.
Letsinger, Myers, and Lentner (1996) discuss the possibility of split-plot confounding
17


6
where
t is the number of levels for the whole plot treatment,
b is the number of blocks or replicates of the basic whole plot experiment,
s is the number of levels for the subplot treatment,
/r is the overall mean,
is the effect of the ith whole plot treatment,
/3j is the effect of the jth block,
Sij is the whole plot error term,
7k is the effect of the kth subplot treatment,
(t7)ik is the whole plot treatment by subplot treatment interaction, and
ijk is the subplot error.
He uses randomization theory to derive the expected mean squares summarized in
Table 2. In this table, erf is the experimental error variance for the whole plot treat
ments, and a2 is the experimental error variance for the subplot treatments.
Many analysts assume that the blocks are random and use an unrestricted mixed
model to derive the appropriate mean squares. The most common model for this
approach is


Criterion D-Criterion
4-2 Case
25 30 35 40 45
Design Size
4-3 Case
Figure 7: Continued


components but also on the processing conditions. A combined model is proposed
which is a compromise between the additive and completely crossed combined mixture
by process variable models. Also, a new class of designs that will accomodate the
fitting of the new model is considered.
The design and analysis of the mixture experiments with process variables is dis
cussed for both a completely randomized structure and a split-plot structure. When
the structure is that of a split-plot experiment, the analysis is more complicated since
ordinary least squares is no longer appropriate. With the process variables serving as
the whole plot factors, three methods for estimation are compared using a simulation
study. These are ordinary least squares (to see how inappropriate it is), restricted
maximum likelihood, and using replicate points to get an estimate of pure error. The
last method appears to be the best in terms of the increase in the size of the confi
dence ellipsoid for the parameters and has the added feature of not depending on the
model.
Vll


4
pulp. Each replicate of the full factorial experiment requires 12 observations, and the
experimenter will run three replicates. However, the pilot plant is only capable of
making 12 runs per day, so the experimenter decides to run one replicate on each of
three days. The days are considered blocks. On any day, he conducts the experiment
as follows. A batch of pulp is produced by one of the three methods. Then this
batch is divided into four samples, and each sample is cooked at one of the four
temperatures. Then a second batch of pulp is made using one of the remaining two
methods. This second batch is also divided into four samples that are tested at the
four temperatures. This is repeated for the remaining method. The data are given in
Table 1. This experiment differs from a factorial experiment because of the restriction
on the randomization. For the experiment to be considered a factorial experiment,
the 12 treatment combinations should be randomly run within each block or day. This
is not the case here. In each block a pulp preparation method is randomly chosen,
but then all four temperatures are run using this method. For example, suppose
method 2 is selected as the first method to be used, then it is impossible for any of
the first four runs of the experiment to be, say, method 1, temperature 200. This
restriction on the randomization leads to a split-plot experiment with the three pulp
preparation methods as the whole plot treatments and the four temperatures as the
subplot treatments. It should be noted that conducting a split-plot experiment, as
opposed to a completely randomized experiment, can be easier because it reduces
the number of times the whole plot treatment is changed. This usually will result
in a time savings which will lead to reduced costs. For example, suppose one is
interested in six subplot treatments and four whole plot treatments. Let the whole


159
Table 68: Design Points for 4 WP Factors and 4 SP Factors Using Same Fraction
A
B
C
D
P
Q
R
S
-1
1
-1
-1
-1
i
-1
-1
-1
1
-1
-1
-1
-l
1
-1
-1
1
-1
-1
1
-l
-1
1
-1
1
-1
-1
1
l
1
1
-1
-1
1
-1
-1
l
-1
-1
-1
-1
1
-1
-1
-l
1
-1
-1
-1
1
-1
1
-l
-1
1
-1
-1
1
-1
1
i
1
1
1
-1
-1
1
-1
i
-1
-1
1
-1
-1
1
-1
-l
1
-1
1
-1
-1
1
1
-l
-1
1
1
-1
-1
1
1
i
1
1
1
1
1
1
-1
i
-1
-1
1
1
1
1
-1
-l
1
-1
1
1
1
1
1
-l
-1
1
1
1
1
1
1
i
1
1
-1
-1
-1
1
1
l
-1
-1
-1
-1
-1
1
1
-l
1
-1
-1
-1
-1
1
-1
-l
-1
1
-1
-1
-1
1
-1
i
1
1
-1
1
1
1
-1
i
-1
1
-1
1
1
1
-1
-l
1
1
-1
1
1
1
1
-l
-1
-1
-1
1
1
1
1
i
1
-1


43
alias chains for the main effects, A, B and C are as follows:
A = BC
B = AC
C =AB.
For a 2k~p, there are 2P 1 effects in the defining contrast. The experimenter can
select any p factorial effects to be the defining contrast. The remaining 2P p 1
factorial effects are automatically determined as being the generalized interactions
among the p effects.
Box and Hunter (1961a, 1961b) classified fractional factorial plans by their degree
of aliasing of effects. This measure is called the resolution of the plan. The number
of letters in the shortest member of a set of defining contrasts determines the designs
resolution. Three important resolutions are
1. Resolution Ilf in which main effects are aliased with two-factor interactions;
2. Resolution IV in which main effects are aliased with three-factor interactions
and two-factor interactions are aliased with other two-factor interactions;
3. Resolution V where two-factor interactions are aliased with three-factor in
teractions.
Of course, if all three-factor and higher interactions are negligible, a design with
Resolution V is desired because it will allow the estimation of all main effects and
two-factor interactions since they are aliased with negligible effects.


88
mixture components and process variables are believed to be important, some sort of
compromise between these two models is needed.
Most of the model forms that have been proposed for response surface investi
gations are based on a Taylor series approximation. In keeping with this tradition,
suppose that the true model for the n process variables is a second-order model.
Instead of Equation (6), such a model would be
n n n
Vpv = a0 + a^Zk + akkZk + £ akiZkZi (9)
fc=i fc=i k Equation (9) is Equation (6) plus the n pure quadratic terms. Also, a Taylor series
approximation for a combined second-order model would include only up to two factor
interactions and would not be the model in Equation (7). Combining Equation (5)
with Equation (9), our proposed combined second-order model is
q q n
i=l i n q n
+ EE &klZkZl + EEww (10)
k which includes the mixture model, plus pure quadratic as well as two-factor inter
action effects among the process variables, and two-factor interactions between the
linear blending terms in the mixture components and the main effect terms in the
process variables. The minimum number of design points needed for the proposed
model (10) is less than what is needed for the completely crossed model (7) but is
more than is needed for the additive model (8). Also, the proposed model can be used
even if one does not feel the need for pure quadratic terms in the process variables
by simply omitting those n terms.


105
Table 38: ANOVA Table for the Proposed
Designs Using the Compromised Model
Source
DF
mixture components:
linear
q 1
quadratic
q(q l)/2
process variables:
2-factor int.
n{n l)/2
quadratic
n
MC x PV int.:
2-factor
q n
Total Effects
i (g+2)(9-l)+n(n+l)+2qin
p J. 2
Error
N-p
Total
N l


84
4.1 Experimental Situation
Historically in the mixture literature, the interest in the blending properties of the
mixture components has been higher than that of studying the effects of the process
variables. Generally, the process variables have been treated as noise factors. The
primary focus on the mixture by process variable interactions has been on the effects
of the process variables on the blending properties of the mixture components.
In many industrial situations, the interest in the process variables is at least equal
to that in the mixture components. Consider the production of a polymer which is
produced by reacting together three specific components. The research laboratory
proposes a specific formulation which is the result of a highly controlled environment
with reagent grade chemicals and laboratory glassware. The plant personnel use this
formulation during the pilot plant and initial start-up of the full production pro
cess. During this period, the plant personnel are trying to find the proper processing
conditions to produce a useable product profitably.
At some point, plant personnel need to reevaluate the polymers formulation in
light of the actual raw materials and the plants full scale production capabilities.
Plant personnel need to find the optimal combination of the formulation and pro
cessing conditions.
Traditionally, in response surface applications, the model assumed for process op
timization is a second-order Taylor series. Such an assumption is based on background
knowledge in knowing the true surface over the experimental region can be approx
imated by fitting a second-order model. Furthermore, in our polymer example, all
second-order terms involving mixture components, process variables, and the mixture


70
Table 28: 24 Point Design for the Case of 4 WP Factors
and 3 SP Factors Using the Same Fraction
b c ad abed
p
P
P
P
q
q
q
q
r
r
r
r
pqr
pqr
pqr
pqr
Fold on A
d bed
P
pq
q
pr
r
qr
pqr
(1)
Table 29: 24 Point Design for the Case of 4 WP Factors
and 3 SP Factors Using Split-Plot Confounding
b c ad abed
pq
P
pq
P
pr
q
pr
q
qr
r
qr
r
(1)
pqr
(1)
pqr
Fold on A
d bed
P
pq
q
pr
r
qr
pqr
(1)


56
the defining contrast with the fewest letters. Therefore, it is looking at the overall
resolution of the design and not the partial resolution.
Huang, Chen and Voelkel (1998) and Bingham and Sitter (1999) have tabled
minimum-aberration (MA) designs for 16 and 32 runs for up to 10 factors. When a
design is needed that fits in these restrictions, one can simply look up the appropriate
design in these tables. However, the MA designs in these tables do not take into
account other design issues such as which effects are the most important to estimate.
This concept seems to be overlooked in the literature. For example, suppose the whole
plot factors are noise factors and only their main effects are of interest. Now, further
suppose that the two-factor whole plot by subplot interactions are the most important
effects to estimate (which is the case in many experiments). Then, it is better to
fractionate the whole plot treatments and subplot treatments separately since this
would alias the two-factor interactions of interest with higher order interactions. Note,
this design would not be the MA design since the partial resolution of the whole plot
factors would be too low.
Another concern with MA designs is the allocation of the runs. Consider the
MA designs for 16 runs involving combinations of 2, 3, and 4 whole plot and subplot
factors. With the exception of the cases involving 2 whole plot factors with 3 or 4
subplot factors, all the other MA designs use eight whole plots with two subplots per
whole plot. This raises several concerns.
1. Typically in industrial experiments, the whole plot factors are hard-to-change
or costly-to-change factors. If they are hard to change, then it would make more
sense to only change them four times as opposed to eight. Also, if changing


16
the remaining variables. Data from such experiments are analyzed by assuming that
it arose from a completely randomized experiment. Research in the area of split-plot
experiments with multiple whole plot and subplot factors is lacking and the work in
this dissertation should be of real help to industrial statisticians.
1.4 Overview
The literature review that follows in Chapter 2 is intended to familiarize the reader
with other work that discusses split-plot designs in RSM. Chapter 3 begins with some
background information on 2k factorial experiments and the remainder of Chapter
3 is devoted to a more in-depth look at confounding in split-plot experiments. In
Chapter 4, a new model and class of designs for mixture experiments with process
variables will be developed in a completely randomized setting. Finally, the last
chapter will assume a split-plot structure for the mixture experiments with process
variables described in Chapter 4.


60
allows estimates all of the main effects and all of the two-factor interactions.
For all situations involving less than four whole plot factors, a general method can
be used to construct 24 run designs. First, construct a 16 run design that uses four
whole plots with four subplots in each whole plot.
If there are two whole plot factors, then use the complete factorial in the whole
plot factors.
If there are three whole plot factors, then use one of the two half fractions found
using the defining contrast / = ABC.
To complete the 16 run design use either split-plot confounding or a separate fractional
factorial in the subplot factors to decide which subplot treatments will appear in each
whole plot. After the 16 run design is selected on, use semifolding to obtain two extra
subplot treatments for each whole plot treatment. The semifolding is, for the most
part, done on two subplot factors. In two of the whole plots, the subplot treatments
are folded on one factor (the high level of the factor in one whole plot and the low
level of the factor in the other whole plot). In the remaining two whole plots, the
subplot treatments are folded on a different subplot factor (again, on the high level
in one whole plot and the low level in the other whole plot). In the special case of
three subplot factors, the semifolding is done on just one factor since there is only
one alias chain in the defining contrast.
When there are four or more whole plot factors, using 4 whole plots results in
insufficient degrees of freedom to estimate the main effects of the whole plot factors.


65
Table 23: 24 Point Design for the Case of 3 WP Factors and 3 SP Factors
Using the Same Fraction [HP-denotes high P and LP-denotes low P]
a b c abc
P
P
P
P
q
q
q
q
r
r
r
r
pqr
pqr
pqr
pqr
Folc
HP LP
on
LP HP
(1)
pq
pq
(1)
qr
pr
pr
qr
eight points are obtained by semifolding on factor P. The 24 points design is shown
in Table 23. The chain has been broken and now P, Q, R, PQ, PR, and QR are all
estimable. There are 5 df left over for a subplot error term.
Next, consider split-plot confounding. The defining contrast is I = ABC =
ABPQR = CPQR which is also resolution III. The two-factor interactions between
whole plot factors A and B and the subplot factors are already estimable. Therefore,
the only alias chain that needs to be broken is CPQR. The additional eight points
are obtained by semifolding on factor P while being careful to fold both high and low
P where C is high and where C is low. The 24 point design is shown in Table 24. The
chain has been broken and now all of the effects of interest including the two-factor
interactions among the subplot factors are estimable. There are 5 df left over for a
subplot error term.


58
design is found using the minimum-aberration (MA) criterion, but this differs from
just using MA because we are restricted to using four whole plot treatments with four
subplot treatments per whole plot. Therefore, the best resolution is desired within
this restricted setting. With the exception of the cases involving 2 whole plot factors
with 3 or 4 subplot factors, these designs will not be the overall MA design.
Once the sixteen run design is found, eight additional runs are considered in order
to break some of the alias chains. Along with breaking some of the alias chains, extra
degrees of freedom are now available in order to estimate additional effects. The
result is a 24 run design which we feel is a nice compromise between the 16 and 32
run designs presented in Huang, Chen and Voelkel (1998) and Bingham and Sitter
(1999). Which eight treatments should be added is the question to be answered next,
but first we briefly discuss foldover designs.
The concept of a foldover design was introduced in Box and Hunter (1961b).
Suppose an experiment involving k factors each at two levels is to be performed and
an initial Resolution III fractional factorial design is used. One way to do a the
foldover is to repeat the initial design and change the levels of one of the factors while
leaving the levels of the other factors unchanged. This allows the estimation of all
the interactions that contain the folded factor but doubles the size of the experiment.
A related idea is that of semifolding which folds only the points that are at the high
level of a factor (or the low level). The addition of the new points breaks certain alias
chains and allows estimates of interactions involving the factor that is semifolded to
be calculated while adding only half as many points as a complete foldover design.
In the rest of this chapter, we apply semifolding to split-plot experiments.


51
Table 14: Design Layout for the Combined 23 1 x 23 1
With Defining Contrast / = ABC PQR = ABCPQR
a
b
c
abc
P
P
P
P
Q
q
q
q
r
r
r
r
pqr
pqr
pqr
pqr
A second method, incorporating the split-plot nature and using four whole plots,
is to consider reducing the whole plot factors and subplot factors separately using
fractional factorials. A 23_1 fractional factorial with defining contrast I = ABC
will be used for selecting the whole plot treatments and combined with a 23_1 with
defining contrast / = PQR in selecting the subplot treatments (see Table 14). The
overall defining contrast for the experiment is
I = ABC = PQR = ABCPQR,
and the alias structure is shown in Table 15. Once we consider the split-plot structure,
the best we can do at the whole plot level is a Resolution III design. This method
provides a good design for estimating the two-factor interactions between the whole
plot and subplot factors. However, we must assume that the two-factor interactions
among the subplot factors are negligible in order to estimate the main effects for the
subplot factors.
Method three uses split-plot confounding and four whole plots. At the whole
plot level, a 23-1 fractional factorial with defining contrast I = ABC is used. Then,
the three-factor interaction, PQR, is confounded with factor C to reduce the eight


146
Table 58: Relative Efficiencies for Comparing Methods of Estimating V With 3 Mix
ture Components and 2 Process Variables in a Constrained Region With a Quadratic
Model
d
r
2
r =
3
REML
PE
REML
PE
.11
1.06
1.14
1.06
1.10
.43
1.11
1.16
1.08
1.11
1.0
1.14
1.23
1.12
1.14
2.3
1.21
1.40
1.16
1.22
4.0
1.22
1.61
1.17
1.30
r =
4
r =
5
d
REML
PE
REML
PE
.11
1.07
1.09
1.05
1.06
.43
1.07
1.09
1.06
1.06
1.0
1.10
1.11
1.08
1.08
2.3
1.12
1.15
1.09
1.12
4.0
1.13
1.19
1.10
1.15


154
Table 63: Design Points for 3 WP Factors and 4 SP Factors Using Separate Fractions
A
B
C
P
Q
R
S
1
-1
-1
-1
l
-1
-1
1
-1
-1
-1
-l
1
-1
1
-1
-1
1
-l
-1
1
1
-1
-1
1
l
1
1
-1
1
-1
-1
l
-1
-1
-1
1
-1
-1
-l
1
-1
-1
1
-1
1
-l
-1
1
-1
1
-1
1
l
1
1
-1
-1
1
-1
l
-1
-1
-1
-1
1
-1
-l
1
-1
-1
-1
1
1
-l
-1
1
-1
-1
1
1
l
1
1
1
1
1
-1
l
-1
-1
1
1
1
-1
-l
1
-1
1
1
1
1
-l
-1
1
1
1
1
1
l
1
1
1
-1
-1
-1
-l
-1
1
1
-1
-1
-1
l
1
1
-1
1
-1
1
l
-1
-1
-1
1
-1
1
-l
1
-1
-1
-1
1
1
-l
-1
-1
-1
-1
1
1
l
1
-1
1
1
1
-1
l
-1
1
1
1
1
-1
-l
1
1


57
these factors is expensive, then again changing them four times seems more
reasonable.
2. It is not an efficient allocation of the degrees of freedom. Using eight whole
plots with two subplots per whole plot gives 7 df for whole plot factors and 8 df
for subplot factors and whole plot x subplot factor interactions. This allocates
a disproportionate number of degrees of freedom to the whole plot factors. In
contrast, using four whole plots with four subplots per whole plot gives 3 df for
whole plot factors and 12 df for terms involving subplot factors.
3. Using two subplots per whole plot is similar to using blocks of size two in a
block design which is not generally recommended.
MA designs are in general good designs, however, for split-plot experiments they
are based purely on the overall resolution of the design instead of partial resolution.
Also, they only use split-plot confounding to reduce the size of the experiment and
are not motivated by any other concerns such as those mentioned above.
3.6 Adding Runs to Improve Estimation
With the concerns of the previous section in mind, mainly the allocation of degrees
of freedom, we will focus our attention on 16 run designs that use four whole plots
with four subplots per whole plot. Within this allocation of the resources, the best
design is found for the two types of confounding discussed in the example in section
3.4. These are split-plot confounding and fractionating of the whole plot and subplot
factors separately (called the Cartesian product design in Bisgaard (1999)). The best


23
of view, arrangement (b) is costly. It requires an inefficient use of the environmen
tal factors which in industrial experiments are typically the difficult or expensive to
change factors.
The model for arrangement (b) is
Vijk = fJ- + 7*! + I + Vik + aj + + ijk ,
where is the response of the kth replicate of the ith level of factor D and the jth
level of factor E, y is the overall mean, 7*, is the random effect of the kth replicate
with 7*; ~ N(0, a^), aj is the fixed effect of the jth level of factor E, Si is the fixed
effect of the zth level of factor D, (a<5)^ is the interaction effect of the itil level of D
and the jth level of E, dik ~ V(0, t^,) is the whole plot error, e^k ~ iV(0, ) is the
subplot error, and 0tk and e^k are independent.
Arrangement (c)
Now, consider the arrangement where the subplot treatments are randomly as
signed in strips across each block of whole plot treatments (see Table 6). For the
cake mix example, suppose each of the n 8 batches of cake mix is subdivided into
m = 4 subgroups. One subgroup from each batch is then selected, and these eight
are baked in the same oven at the appropriate temperature for the appropriate time.
This arrangement requires only 8 cake mix batches and only 4 operations of the oven.
Therefore, the strip-block experiment is easier to run than the completely randomized
cross-product design, as well as both arrangements (a) and (b).


118
model allows the experimenter to see directly the interaction between specific mixture
components and specific process variables. These interactions can provide valuable
insights into the entire process being studied.


115
Table 42: Data for the Example
X\
X2
x3
Z\
z2
Response
1
0
0
-1
1
4
0
1
0
-1
1
9
0
0
1
-1
1
5
1
0
0
1
-1
9
0
1
0
1
-1
15
0
0
1
1
-1
8
.5
.5
0
-1
-1
12
.5
0
.5
-1
-1
7
0
.5
.5
-1
-1
8
.5
.5
0
1
1
15
.5
0
.5
1
1
12
0
.5
.5
1
1
12
.33
.33
.33
-1
0
7
.33
.33
.33
1
0
10
.33
.33
.33
0
-1
7
.33
.33
.33
0
1
8
.33
.33
.33
0
0
9


52
Table 15: Alias Structure for 23 1 x 23 1
A
=
BC
B

AC
C
=
AB
P
=
QR
Q
=
PR
R
=
PQ
AP
=
AQ
=
AR
=
BP
=
BQ
-
BR
=
CP
=
CQ
=
CR

subplot treatments to four per whole plot (see Table 16). The idea is to put the
positive fraction of PQR wherever C is positive and the negative fraction wherever
C is negative. The overall defining contrast is given by
I = ABC = CPQR = ABPQR
with the alias structure provided in Table 17. This design is better than the second
design in terms of aliasing of the main effects for the subplot factors, but cannot
estimate all nine whole plot by subplot factor interactions without assuming that PQ,
PR, and QR are negligible. If, on the other hand, it is reasonable to assume that the
whole plot factor C will not interact with any of the subplot factors, then PQ, PR,
QR, the main effects for subplot factors and the remaining six whole plot by subplot


63
Table 20: 24 Point Design for the Case of 2 WP Factors and 4 SP Factors Using
the Same Fraction [HP(HS)-denotes high P(S) and LP(LS)-denotes low P(S)}
a b ab (1)
Q
Q
Q
Q
r
r
r
r
ps
ps
ps
ps
pqrs
pqrs
pqrs
pqrs
Folc
HP LP
on
HS LS
s
pq
P
qs
qrs
pr
pqr
rs
Table 21: 24 Point Design for the Case of 2 WP Factors and 4 SP Factors Using
Split-Plot Confounding [HP(HS)-denotes high P(S) and LP(LS)-denotes low P(S)]
a b ab (1)
P
s
Q
qr
pqr
pq
r
pqs
qs
pr
ps
prs
rs
qrs
pqrs
(1)
Folc
HP LP
on
HS LS
(1)
ps
P
qrs
qr
pqrs
pqr
s


x 1 = 1 Z_2
Figure 9: Proposed Design for Split-Plot Structure With a Second-Order Model
CO
oo


27
with arrangements (a) and (b), the strip-block design estimates the subplot x whole
plot interactions more precisely but the subplot factor effects with less precision.
However, the whole plot factor effects are estimated with equal precision. Based on
these results, arrangement (a) with the environmental factors applied to the whole
plots is generally preferred over arrangement (b). Both the strip-block design and
the split-plot design with the design factors applied to the subplots can be extremely
useful in robust parameter design.
2.3 Bi-Randomization Designs
Letsinger, Myers, and Lentner (1996) introduced bi-randomization designs (BRDs).
BRDs refer to designs with two randomizations similar to that of a split-plot design.
The whole plot variables are denoted by z = (zi,z2, , zz) while the sub-plot vari
ables are denoted by x = (xi, x2,..., xx). Hence, the zth design run is (z, x). BRDs
are broken into two classes, crossed and non-crossed. Crossed BRDs are constructed
as follows:
1. randomize the a unique combinations of z to the whole plot experimental units
(EUs), then
2. randomize the b levels of x to the smaller EUs within each whole plot (see
Table 7).
Thus every level of x is crossed with every level of z. These designs are the usual
split-plot designs.


TABLE OF CONTENTS
ACKNOWLEDGMENTS
ABSTRACT vi
CHAPTERS
1 INTRODUCTION 1
1.1 Response Surface Methodology 1
1.2 Split-Plot Designs 3
1.3 Dissertation Goals 13
1.4 Overview 16
2 LITERATURE REVIEW 17
2.1 Split-Plot Confounding 17
2.2 Split-Plots in Robust Parameter Designs 19
2.3 Bi-Randomization Designs 27
2.4 Split-Plots in Industrial Experiments 31
3 INCOMPLETE SPLIT-PLOT EXPERIMENTS 40
3.1 Fractional Factorials 41
3.2 Confounding 44
3.3 Confounding in Fractional Factorials 46
3.4 Combining Fractional Factorials and Confounding in Split-Plot
Experiments 47
iii


142
Table 54: Relative Efficiencies for Comparing Methods of Estimating V With
3 Mixture Components and 2 Process Variables With a Quadratic Model
d
r =
2
r =
3
REML
PE
REML
PE
.11
1.03
1.04
1.02
1.04
.43
1.07
1.06
1.05
1.04
1.0
1.09
1.14
1.06
1.07
2.3
1.08
1.28
1.06
1.12
4.0
1.09
1.48
1.07
1.15
r =
4
r =
5
d
REML
PE
REML
PE
.11
1.02
1.04
1.02
1.04
.43
1.06
1.03
1.05
1.02
1.0
1.05
1.04
1.05
1.03
2.3
1.05
1.06
1.05
1.04
4.0
1.05
1.07
1.03
1.04


31
Wolfowitz (1959)) becomes
min
D
N (X'V_1X)
over all designs D. Letsinger, Myers, and Lentner (1996) provide comparisons of
several first- and second-order designs. For the second-order designs, the popular
central composite design (CCD) proves to be a good design.
2.4 Split-Plots in Industrial Experiments
Lucas and Ju (1992) investigated the use of split-plot designs in industrial experi
ments where one factor was difficult to change and its levels served as the whole plot
treatments. They began their study with a simulation exercise using a four factor
face-centered cube design with four center points. They let X\ correspond to the
hard-to-change factor, while x2, x3, and x4 were easy to vary. This design allowed for
the fitting of the quadratic model
4 4 3 4
y = fa + &iXi + S + E E Pijxixj +e
=1 =i =1 j=t+i
However, since the error was the only term of interest, all the regression coefficients
can be zero. Therefore, data was generated using
y + es ,
where e, ~ N(0, cr(,) was the error term associated with changing the level of X\ and
es ~ N(0, af) was the error associated with any new experimental run. Twentyeight
runs were generated using the following steps:
1. Generate e, ~ iV(0, a%,) and es ~ N(0,a^).


117
test statement is written in SAS and produces
F value: 0.0438 Prob>F: 0.9581
for process variable one and
F value: 1.6213 Prob>F: 0.5056 .
Therefore, none of the interactions between the mixture components and the process
variables are significant. The follow-up question asks if the process variable effects
are different from zero or additive effects. For process variable one, the p-value is
0.0012 and for process variable two it is 0.8611. This leads to an additive effect for
process variable one. It turns out that after eliminating the terms involving z2, there
is also an additive effect for z\. Finally, there is some evidence of positive nonlinear
blending of x\ and x2.
4.7 Summary
In this chapter, we have introdced a new class of combined designs for mixture
experiments with process variables. These designs axe specifically created to support a
second-order Taylor series approximation of the true response function. These designs
employ an intuitive balance which is very appealing and also support the fitting of
some lack of fit terms. On the whole, they are quite competitive to D-optimal designs
generated by common software.
In addition, the combined second-order model proposed in terms of both the
mixture components and the process variables represents an explicit compromise be
tween the mixture and standard response surface schools of thought. The proposed


116
Table 43: Original Parameter Estimates for the Example
Variable
DF
Parameter
Estimate
Standard
Error
T for HO:
Parameter=0
Prob > |T|
1
11.43
2.74
4.172
0.0529
X2
1
16.93
2.74
6.179
0.0252
x3
1
11.43
2.74
4.172
0.0529
X\X2
1
-6.45
6.45
-1.000
0.4228
x,x3
1
-11.45
6.45
-1.775
0.2179
x2x3
1
-20.45
6.45
-3.171
0.0867
ZiZi
1
-0.50
1.07
-0.468
0.6858
Z2Z2
1
-1.50
1.07
-1.404
0.2955
Z\Z2
1
2.93
0.79
3.681
0.0665
V1Z1
1
2.15
0.61
3.536
0.0715
X \ Z2
1
-0.15
0.61
-0.254
0.8230
X2zx
1
1.90
0.61
3.125
0.0889
x2z2
1
-0.90
0.61
-1.485
0.2759
X3zx
1
2.15
0.61
3.536
0.0715
x3z2
1
0.85
0.61
1.387
0.2998


127
This is done for each whole plot. These observations along with the design matrix
and the model axe inputted into Proc Mixed in SAS. Using the code provided in
Letsinger, Myers and Lentner (1996) and given in Appendix C, the estimated variance
components, aj and d2, are computed. These estimates are used to get Vc. Finally,
the determinant
(x,V71x)-1x'vc1vv;1x(x/vc-1x)-1
is calculated. This procedure is repeated 10,000 times and the average determinant
is used for comparison to the asymptotic determinant.
For the method using pure error estimates, simulation is also used. However,
the only response values that need to be simulated are at the replicate points. The
simulation is carried out the same way as described for the REML case. In other
words, first W = /(z) + AT(0, d) is simulated and then y = /(x; z) + W + N(0,1) is
obtained for each whole plot.
A total of r replicates at the center of the process variables are used. Since a
balanced design is desired, in each of these whole plots a total of m replicates of the
centroid are used. This leads to the rm 1 total degrees of freedom being partitioned
into r 1 df for the pure error term involving the process variables (whole plot factors)
and r{m 1) df for the pure error term involving the mixture components (subplot
factors). The estimated whole plot pure error term is
where Y im is the mean of the simulated values in the Th whole plot and Y is the
overall mean of the simulated values. From the replicated centroids at the center of


41
number of factors, k. Even when resources are available, we may not want to estimate
all of the 2k 1 factorial effects. As an example, with k > 3, interactions involving
3 or more factors are generally considered to be negligible or of little importance.
Thus, a single replicate of a 27 requires 128 experimental units and provides a 64-fold
replication of each main effect. Of the 127 effects that can be estimated, only 28 may
be of major interest (seven main effects and 21 two-factor interactions).
3.1 Fractional Factorials
Finney (1945) proposed reducing the size of the experiment by using only a frac
tion of the total number of possible treatment combinations. Such experiments are
called fractional factorials. He outlined methods of constructing fractions for 2k and
3k experiments. For screening purposes, Plackett and Burman (1946) gave designs
for the minimum possible number of experimental units, N = k + 1 where N is a
multiple of 4, and pointed out their utility in physical and industrial research. Since
then, these designs have found many applications, particularly in industrial research
and development. Their chief appeal is that they enable a large number of factors,
generally 5 or more, to be included in an experiment of practical size so that the
investigator can discover quickly which factors have an effect on the response. In this
chapter, the discussion will be limited to the case where every factor has only two
levels.
A 2k experiment that is reduced by a factor of 2~p will be called a 2k~p fractional
factorial experiment. These experiments have two major problems which can limit
their usefulness:


79
Table 34: Condition Numbers for the Various Cases
# of WP Factors
# of SP Factors
Type of Confounding
Condition Number, k
2
4
Same Fraction
4.05
2
4
Split-Plot Confounding
3.36
3
3
Same Fraction
2.00
3
3
Split-Plot Confounding
2.00
3
4
Same Fraction
4.05
3
4
Split-Plot Confounding
3.36
4
Neither
2.00
4
3
Same Fraction
5.83
4
3
Split-Plot Confounding
2.00
4
4
Same Fraction
5.83
4
4
Split-Plot Confounding
3.36
the 24 run designs.
The phrase the chains are broken used throughout this chapter does not mean
that all of the effects are no longer aliased. Sometimes the effects are aliased at
a lower degree than unity (completely aliased). Therefore, there is some degree of
collinearity between the effects. To measure the magnitude of this collinearity, the
condition number
k =
N
Largest Eigenvalue of (X'X)
Smallest Eigenvalue of (X'X)
is calculated for each case (see Table 34). Many textbooks declare collinearity to be
a problem if k > 30. It is seen from Table 34 that collinearity does not seem to be a
problem for the cases considered in this chapter. The variance inflation factors (VIF)


156
Table 65: Design Points for 4 WP Factors and 2 SP Factors Using Separate Fractions
A
B
C
D
P
Q
-1
1
-1
-1
1
-l
-1
1
-1
-1
-1
i
-1
1
-1
-1
1
i
-1
1
-1
-1
-1
-l
-1
-1
1
-1
1
-l
-1
-1
1
-1
-1
l
-1
-1
1
-1
1
i
-1
-1
1
-1
-1
-l
1
-1
-1
1
1
-l
1
-1
-1
1
-1
i
1
-1
-1
1
1
l
1
-1
-1
1
-1
-l
1
1
1
1
1
-l
1
1
1
1
-1
l
1
1
1
1
1
l
1
1
1
1
-1
-l
-1
-1
-1
1
1
-l
-1
-1
-1
1
-1
l
-1
-1
-1
1
1
i
-1
-1
-1
1
-1
-l
-1
1
1
1
1
-l
-1
1
1
1
-1
i
-1
1
1
1
1
i
-1
1
1
1
-1
-l


CHAPTER 3
INCOMPLETE SPLIT-PLOT EXPERIMENTS
The focus of attention in factorial experiments centers on the effects of numerous
factors and their interactions. An important class of factorial experiments is the 2k
factorials where each of the k factors is assigned two levels. These experiments are
very useful in exploratory investigations as well as optimization problems because
they allow a large number of factors and their interactions to be examined.
Since there are only two levels of each factor, they will be denoted as low and high
for ease of reference. A treatment combination pertains to a level of each and every
factor and will be designated by lower case letters using the following conventions:
If a factor is at its low level, the corresponding letter is omitted from the treat
ment designation. Conversely, if a factor is at its high level, the corresponding
letter is included.
When all factors are at their low levels, the treatment will be designated by the
symbol (1).
Under this notation, the treatments for a 22 factorial experiment in factors P and Q
are designated as (1), p, q, and pq. Factors and their effects will be designated by
capital letters.
Factorial experiments become large very rapidly so that often a single replicate
of the N 2k runs requires more resources than are available, even with a moderate
40


61
Therefore, the additional eight runs will be added in the form of two extra whole
plots. This leads to a 24 run design with 6 whole plots with 4 subplots per whole
plot. The whole plot treatments used in the two additional whole plots are found
by semifolding on a whole plot factor. However, which subplot treatments should be
used in the two additional whole plots is case specific. To illustrate how to apply
these methods, the seven remaining cases involving 2, 3 and 4 whole plot and subplot
factors will be presented. For all the cases, tables which show the designs in highs
and lows for each factor are given in Appendix A.
For these designs, the analysis could use one normal probability plot for the whole
plot effects and a separate plot for the effects involving the subplot factors and the
interactions between whole plot and subplot factors. One assumption of a normal
probability plot is that the effects are independent. This is not the case here. However,
it will be shown later in this chapter that the correlations are low (near zero) and
that a normal probability plot is therefore valid. In some cases there are degrees
of freedom left at the subplot level so that if desired they can be used to estimate
the an error variance. It should be reiterated that the goal of the experiment is to
estimate as well as test for significance the main effects for the whole plot factors, the
two-factor interactions between whole plot and subplot factors, the main effects of
the subplot factors, and if possible two-factor interactions among the subplot factors.
2 WP Factors (A, B) and 4 SP Factors (P, Q, R, S)
To obtain a 16 point design under this situation, only the subplot treatments need
to be fractionated or confounded. First, consider fractionating the subplot treatments.


121
If a balanced design is desired, then the same number of subplots needs to be
run in each whole plot. This means that at each of the combinations of the process
variables, the same number of mixture blends should be run. The center of the process
variables will be used as a replicate point in order to get estimates of the pure error
for the whole plots and the subplots.
Consider the case of three mixture components and two process variables. At the
22 = 4 factorial points of the process variables, either the vertices or the midedge
points in the mixture components are run according to Figures 2 and 3 of Chapter 4.
However, the centroid will also be included at each of the four points. This is done
so that fewer replicates of the center of the process variables will be needed to get a
good estimate of the pure error at the subplot level. A more thorough explanation
of this will be given in the next subsection. In each whole plot there will be four
subplots. At the center of the process variables, four replicates of the centroid will
be included. Also, the entire whole plot at the center of the process variables will be
replicated in order to get an estimate of the whole plot error variance. Figure 8 gives
the proposed design.
It should be reiterated that while the design looks similar to those proposed in
Chapter 4, the difference is in the way the experiment is carried out. In this chapter,
the levels of the process variables are fixed and then the corresponding four mixture
blends are run in some random order. Then, the levels of the process variables are
fixed again and the four blends are run and so on. This of course is different from a
completely randomized design and can be thought of as a split-plot experiment.


114
Table 41: Higher Order terms Supported by the Proposed Design With Four Mixture
Components and the Simplex-Centroid at the Center of the Process Variables
Binary Blend by Process Variable Interactions
4MC, 2PV
4MC, 3PV
Without Special
Cubic Terms
The 6 Binary Blends with
One of the Process Variables
Or 3 Binary Blends with
Both Process Variables
All 18 Terms
With Special
Cubic Terms
1 Special Cubic Plus the
3 Binary Blends of the MCs
Involved in the Special Cubic
With Either Process Variable
All 18 Terms
lack of fit, then some of the special cubic terms and the interactions between a binary
blend and a process variable can be supported by the proposed design without any
additional points.
4.6 Example
Consider an experiment involving three mixture components (aq, x%, x3) and two
process variables {z\, z2). The data for this example is constructed for illustrative
purposes and is given in Table 42. The analysis is carried out in the manner described
in this chapter. The parameter estimates and the results of performing f-tests on the
estimates are shown in Table 43.
The analysis should begin with the mixture component by process variable inter
actions. The "main" question needs to be answered which asks if the effect of the
process variable is the same for all blends of the mixture components. The appropriate


170
[27] Martin, F. G. (1992). Statistical Design and Analysis.
[28] Mee, R. W. and Bates, R. L. (1998). "Split-Lot Designs: Experiments for
Multistage Batch Processes." Technometrics 40. pp. 127-140.
'29] Miller, A. (1997). Strip-Plot Configurations of Fractional Factorials." Tech
nometrics 39. pp. 153-161.
[30] Mitchell, T. J. (1974). "An Algorithm for the Construction of D-Optimal
Experimental Designs. Technometrics 20, pp. 203-210.
[31] Montgomery, D. C. (1997). Design and Analysis of Experiments 4th edition.
John Wiley &: Sons, New York, NY.
[32] Myers, R. H. and Montgomery, D. C. (1995). Response Surface Methodol
ogy: Process and Product Optimization Using Designed Experiments. John Wiley
& Sons, New York. NY.
[33] ODonnell, E. M. and Vining, G. G. (1997). Mean Squared Error of
Prediction Approach to the Analysis of the Combined Array. Journal of Applied
Statistics 24, pp. 733-746.
[34] Phadke, M. S. (1989). Oualtiy Enqineerinq Usinq Robust Design. Prentice
J Hall, Englewood Clifts, NJ
[35] Plackett, R. L. and Burman, J. T. (1946). The Design of Optimum
Multifactorial Experiments." Biometrika 33, pp. 305-325.
[36] Robinson, J. (1967). Incomplete Split Plot Designs. Biometrics 23. pp. 793-
802.
[37] Robinson, J. (1970). "Blocking in Incomplete Split Plot Designs." Biometrika
57, pp. 347-350.
38] Russell, T. S. and Bradley, R. A. (1958). "One-Way Variances in the
Two-Way Classification. Biometrika 45, pp. 111-129.
[39] Santer, T. J. and Pan, G. (1997). Subset Selection in Two-Factor Experi
ments Using Randomization Restricted Designs. Journal of Statistical Planning
and Inference 62, pp. 339-363.
[40] SAS Institute (1989). SAS/QC Software: Reference Guide, Ver. 6. 1st ed. SAS
Institute, Cary, NC.
[41] Shoemaker, A. C. ; Tsui, K. ; and Wu, C. F. J. (1991). "Economical
Experimentation Methods for Robust Designs. Technometrics 33. pp. 415-427.
[42] Snee, R. D. (1985). Computer-Aided Design of ExperimentsSome Practical
Experience. Journal of Quality Technology 17, pp. 222-236.


91
while the second design with just the centroid consists of 17 points. Either could be
used to estimate the 15 terms in the model.
The designs in Figures 2 and 3 can be extended to experiments involving more
than 3 mixture components (MC) and/or more than 2 process variables (PV). The
extension is straightforward. Following the same general notion of balance described
earlier, one can generate the needed designs. In this paper, a total of five cases are
discussed: 3 MC, 2 PV; 3 MC, 3 PV; 4 MC, 2 PV; 4 MC, 3 PV; and 3 MC, 2 PV with
upper and lower bound constraints on the mixture component proportions. For four
mixture components, there are four vertices and six edge midpoints of the tetrahedron.
For three process variables, the layout is a cube with 23 = 8 factorial points, six axial
points, and a center point. Placing upper and lower bound constraints on the mixture
component proportions creates a more complicated mixture region than the simplex.
The constrained region is typically an irregular polygon. The example in this paper
(3 MC, 2 PV) uses the following constraints:
0.25 < X! < 0.40 0.25 < x2 < 0.40 0.25 < x3 < 0.40.
The resulting mixture region is a hexagon. Generally, the original components are
transformed to L-pseudocomponents, x\ (x L)/( 1 J2=i Li) i = 1,2,... ,q, to
make the construction of the design and the fitting of the model easier. For the exam
ple in this paper, the mixture components can be transformed to L-pseudocomponents
using
~ 0-25 _xi~ 025 -
Xi 1 -(0.25 + 0.25 + 0.25) 0.25 '
Candidate points for the two designs in this case consist of the six vertices and


139
Table 51: Relative Efficiencies for Comparing Methods of Estimating
V With 4 Mixture Components and 2 Process Variables
d
r = 2
r = 3
OLS
REML
PE
OLS
REML
PE
.11
1.06
1.06
1.06
1.06
1.06
1.06
.43
1.45
1.18
1.13
1.46
1.15
1.08
1.0
2.25
1.29
1.30
2.28
1.23
1.14
2.3
4.15
1.44
1.55
4.21
1.29
1.24
4.0
6.64
1.42
1.83
6.75
1.27
1.30
r = 4
r = 5
d
OLS
REML
PE
OLS
REML
PE
.11
1.06
1.06
1.06
1.06
1.05
1.06
.43
1.46
1.12
1.06
1.45
1.11
1.05
1.0
2.28
1.18
1.09
2.26
1.13
1.07
2.3
4.20
1.17
1.13
4.17
1.12
1.08
4.0
6.74
1.16
1.14
6.68
1.09
1.09
the relative efficiencies for the second-order case which are shown in Tables 54 58.
OLS is not presented since it performed so poorly in the first-order case and does
so in the second-order case as well. REML performs well for all cases, d values, and
number of replicates at the center of the process variables. The pure error method
does not perform well when there are only 2 replicates at the center of the process
variables. With 4 or 5 replicates, the pure error method compares very well with
REML.
5.3 Summary
Often times, cost or time constraints make mixture experiments with process
variables difficult to run in a completely randomized order. In this situation, fixing


no
rejected in the main question. If H0 for testing Zk is not rejected, then there is no
additive effect due to the process variable.
After a thorough investigation of all the mixture component by process variable
interactions, the rest of the model should be studied. Even if the mixture component
by process variable interactions exist, it is still beneficial to investigate the mixture
component part of the model as well as the curvelinear terms involving the process
variables.
To investigate the mixture component part of the model, generally the linear
blending terms are not tested. However, if the nonlinear blending terms cannot be
shown to be different from zero, then one can test whether the linear blending terms
are all equal (see Cornell (1990), Chapter 5). Simple i-tests can be performed on the
nonlinear blending coefficient estimates to ascertain which pairs of components blend
nonlinearly. In other words, detecting curvature in the shape of the mixture surface
through the nonlinear blending of the components is of primary interest.
When data are collected from a simplex-centroid design and curvature in the
surface is detected by finding the coefficient estimate of the binary crossproduct, this
generally means the binary blend produces a response value which differs from the
average of the response values collected at the two vertices. If the coefficient is positive
and the test is significant, this means the binary blend produces a higher response
value than is obtained from simply averaging the response values from the individual
components. Similarly, for a significant negative parameter estimate, the response to
the binary blend is lower than the average of the responses at the vertices. The tests


120
5.1 First-Order Model in the Process Variables
First, we shall consider the case where a first-order plus ionteractions model is
assumed for the process variables which are to serve as the whole plot factors. There
fore, the model for the process variables is the model given in Equation (6) of Chapter
4. When this model is combined with the model in Equation (5) of Chapter 4 under
a Taylor series approach, the resulting model is
q q n q n
7?(X,Z) = ^+E + E E OLklZkZl + EE 'Yik^'iZk
=1 i Now, a design is needed that will support the fit of this model.
Proposed Design
The designs proposed in Chapter 4 use a ccd in the process variables with a fraction
of the mixture blends run at each location in the ccd. The vertices of the simplex are
run at half of the the 2" factorial points in the process variables with the midedge
points being run at the other half. Also, the axial points in the process variables are
paired with the centroid of the simplex. Finally, either the simplex-centroid or simply
the centroid is run at the center of the process variables. Figures 2 and 3 in Chapter
4 show the proposed designs for three mixture components and two process variables.
Initially in this section, we are assuming only a first order model in the process
variables. Hence, it is unnecessary to include the axial points. It is important to keep
in mind that the nature of the experiment is that of a split-plot experiment. This
will influence how the design is chosen.


13
The concept of the split-plot design can be extended if further randomization
restrictions exist. For example, suppose there are two levels of randomization restric
tions within a block in which case we might have a split-split-plot design. For a more
detailed discussion of split-plot designs and their extensions see Yates (1937), Cox
(1958), Wooding (1973) and Montgomery (1997).
1.3 Dissertation Goals
The focus of this dissertation is to enhance our understanding of the design and
analysis of split-plot experiments. The experiments considered will be industrial in
nature. As much as possible, the dissertation will focus on or discuss the types
of experiments that would be run in industry in terms of size and resources. An
important goal is to come up with methods that are clear, practical, and easy to
implement. In other words, this dissertation will address issues of real concern to
applied statisticians working in industry and provide them some tools that can be
used with split-plot experiments. Below two industrial statisticians have been kind
enough to share real situations that help to show the relevance of the work in this
dissertation.
A Food Industry Example
Frozen heat-and-serve pastries, along with shelf-stable ready-to-eat pastries, rep
resent a large segment of the convenience foods that todays consumers crave. Op
timized proofing and baking operations are critical to the successful manufacture of
high quality baked goods such as these. However, as this market segment has grown,


3.5 Discussion of Minimum-Aberration in Split-Plot Designs 55
3.6 Adding Runs to Improve Estimation 57
3.7 An Example 72
3.8 Summary 78
4 A NEW MODEL AND CLASS OF DESIGNS FOR MIXTURE
EXPERIMENTS WITH PROCESS VARIABLES 81
4.1 Experimental Situation 84
4.2 The Combined Mixture Component-Process Variable Model 85
4.3 Design Approach 89
4.4 Analysis 104
4.5 Lack of Fit Ill
4.6 Example 114
4.7 Summary 117
5 MIXTURE EXPERIMENTS WITH PROCESS VARIABLES IN A
SPLIT-PLOT SETTING 119
5.1 First-Order Model for the Process Variables 120
5.2 Second-Order Model for the Process Variables 135
5.3 Summary 139
6 SUMMARY AND CONCLUSIONS 147
APPENDICES
A: TABLES FOR CHAPTER 3 DESIGNS 149
B: TABLES FOR CHAPTER 4 DESIGNS 161
C: SAS CODE FOR PROC MIXED 167
IV


157
Table 66: Design Points for 4 WP Factors and 3 SP Factors Using Separate Fractions
A
B
C
D
P
Q
R
-1
1
-1
-1
1
-l
-1
-1
1
-1
-1
-1
i
-1
-1
1
-1
-1
-1
-l
1
-1
1
-1
-1
1
i
1
-1
-1
1
-1
1
-l
-1
-1
-1
1
-1
-1
i
-1
-1
-1
1
-1
-1
-l
1
-1
-1
1
-1
1
i
1
1
-1
-1
1
1
-l
-1
1
-1
-1
1
-1
i
-1
1
-1
-1
1
-1
-l
1
1
-1
-1
1
1
i
1
1
1
1
1
1
-l
-1
1
1
1
1
-1
i
-1
1
1
1
1
-1
-l
1
1
1
1
1
1
i
1
-1
-1
-1
1
1
-l
-1
-1
-1
-1
1
-1
i
-1
-1
-1
-1
1
-1
-l
1
-1
-1
-1
1
1
i
1
-1
1
1
1
1
i
-1
-1
1
1
1
1
-l
1
-1
1
1
1
-1
i
1
-1
1
1
1
-1
-l
-1


74
array. For our purposes, we shall only consider the first three factors in the inner
array along with the three factors in the outer array. The outer array will make up
the whole plot factors while the inner array will have the subplot factors. The original
experiment was not run using restricted randomization, but it will be assumed that
it was in order to present the analysis.
The 24 run designs discussed earlier can be utilized. For this example, the design
given in Table 23 with separate fractions at the whole plot and subplot levels is used.
The only difference is that the Taguchi example uses the negative fraction of the whole
plot factors instead of the positive fraction. To correspond with the factor names in
the Taguchi example, let X, Y and Z be the whole plot factors and A. B and C be
the subplot factors. The design along with the responses is shown in Table 32.
The analysis involves fitting the 18-term model involving the main effects of the
whole plot factors, the main effects of the subplot factors, the two-factor interactions
among subplot factors, and twofactor interactions between whole plot and subplot
factors. This leaves five degrees of freedom for a subplot error term. Table 33 gives
the estimated effects and t-tests. The tests for the three whole plot factor main effects
are not correct and should be ignored. From Table 33, there appears to be an effect
due to the interaction of factors B and C. Since there are only 5 df for error, one
might choose to use a normal probability plot to investigate at the subplot level. The
design is not completely balanced or orthogonal which leads to some effects having
one standard error and others having a different standard error. Therefore, instead
of just plotting the effects, the effects are divided by their standard error and then
plotted. The plot is shown in Figure 1 and gives BC and ZC as significant effects.


94
midpoints of the six edges of the hexagon plus the centroid at each of the nine design
points of the process variables. The designs in the L-pseudocomponents are given in
Appendix B.
The five cases are not intended to be exhaustive, but rather are used because
they encompass a typical industrial experiment. Also, they can be used to evaluate
the performance of the designs and to discuss their analysis. It is assumed that the
designs can be extended to higher dimensions without complications although they
may be difficult to view geometrically. To help illustrate the extension of the designs,
the simplex-centroid-ccd design for the 3 MC, 3 PV case is shown in Figure 4. Each
ccd axial point contains the centroid of the simplex while the ccd center point contains
either the 7-point simplex-centroid or just the centroid. The coordinates of the two
proposed designs for all five cases are listed in Appendix A.
When the number of design points is limited to being less than the total number
generated by crossing the ccd in the process variables with the full design in the
mixture components then the typical user will rely on the computer to generate a
design based on some optimality criterion, such as D-optimality. Such generated
designs, while optimal statistically speaking, may not be very intuitively appealing.
There could be a design that is close to optimal, but has some other nice properties
such as symmetry and near orthogonality. In other words, the statistician should
not use the computer generated design blindly. Snee (1985) discusses some practical
aspects of choosing computer-aided designs. The main point he makes is that while
computer-aided designs should be used with caution, they can be helpful when the


53
Table 16: Design Layout for Split-Plot Confounding
With Defining Contrast / = ABC = CPQR = ABPQR
a
b
c
abc
(1)
(1)
P
V
pq
pq
Q
q
pr
pr
r
r
qr
qr
pqr
pqr
factor interactions can be estimated using this design. Also, on a consulting level,
some experimenters would feel more comfortable with this design since it uses all 8
subplot treatments.
The fourth method uses eight whole plots and split-plot confounding. Since there
are eight whole plots, the complete 23 factorial can be used for the whole plot fac
tors. However, we must now reduce the number of subplots to two per whole plot.
This implies that we must confound two members in the defining contrast and their
generalized interaction completes the defining contrast. Using split-plot confounding,
the defining contrast is
I = ABPQ = ACQR = BCPR,
with the layout given in Table 18 and the alias structure given in Table 19. This design
is good for estimating main effects but has some serious deficiencies with interactions.
One possible problem with designs that use eight whole plots is cost. If the whole
plot factors are costly to change, then using eight whole plots as opposed to four
might be impractical. Another problem with designs using eight whole plots is the
breakdown of the degrees of freedom. There are 7 df for the whole plot design and


25
It is of great interest to the researcher to learn how and which environmental
factors influence the design variables. This information is contained in the subplot
x whole plot interactions. However, Taguchis analysis is commonly conducted in
terms of a performance statistic, such as the signal to noise ratio (SNR). The SNR
is calculated for each point in the inner array using data obtained from the outer
array about that point. Therefore, Taguchi ignores any information contained in the
interactions of the design and environmental factors. This is generally considered to
be a serious drawback to the Taguchi analysis.
Phadke (1989) presented an example involving a polysilicon deposition process
which he analyzed using Taguchis SNRs. Polysilicon film is typically deposited
on top of the oxide layer of the wafers using a hot-wall, reduced pressure reactor.
The reactant gases are introduced into one end of a three-zone furnace tube and are
pumped into the other end. The wafers enter the low-pressure chemical vapor depo
sition furnace in two quartz boats, each with 25 wafers, and polysilicon is deposited
simultaneously on all 50 wafers. The desired output of this process is a wafer which
has a uniform layer of film of a specified thickness. Six design factors each at three
levels were identified: temperature, pressure, nitrogen flow, silane flow, setting time,
and cleaning method. Tube location and die location were considered noise factors.
Three responses, film thickness, particle counts, and deposition rate, were of interest.
The smaller the better SNR was used in the analysis for particles, the target is best
SNR was used for thickness, and a 20 logio transformation was used for deposition
rate. The data were analyzed using ANOVA techniques to determine the effect of


143
Table 55: Relative Efficiencies for Comparing Methods of Estimating V With
3 Mixture Components and 3 Process Variables With a Quadratic Model
d
r
2
r
3
REML
PE
REML
PE
.11
1.25
1.08
1.23
1.07
.43
1.10
1.10
1.09
1.07
1.0
1.06
1.20
1.04
1.11
2.3
1.04
1.39
1.03
1.17
4.0
1.04
1.58
1.03
1.22
r =
4
r =
5
d
REML
PE
REML
PE
.11
1.23
1.06
1.22
1.06
.43
1.09
1.05
1.08
1.04
1.0
1.03
1.07
1.04
1.05
2.3
1.02
1.10
1.02
1.07
4.0
1.02
1.12
1.02
1.07


38
1. Select a row design that consists of a 2A_ 2. Select a column design that consists of a 2k~q design in b blocks;
3. Consider the product of the designs in steps 1 and 2 and select a Latin-Square
fraction of this product.
The selection of the design in steps 1 and 2 can be made on the basis that the analyses
for the row stratum and the column stratum will essentially be the analyses of these
designs. The Latin-Square fraction is selected so that the confounding array effects
in the unit stratum have desirable properties.
Mee and Bates (1998) consider split-lot experiments involving the etching of silicon
wafers. These experiments are performed in steps where a different factor is applied
at each step. Thus, there are an equal number of steps and factors. Specifying a
split-lot design involves determining the following:
1. the number of process steps with experimentation;
2. the number of factors and their levels at each processing step with experimen
tation;
3. the subplot size at each processing step;
4. the number of wafers (experimental units) in the entire experiment;
5. a plan that details for each experimental wafer the process subplot at each step.
Mee and Bates emphasize symmetric designs, which are designs having the same
subplot size at each experimentation step.


8
Pi ~ N (o, of) 8ij ~ AT (o, of) eijk ~ A7' (o, a2) ,
and where Sij and eljk are independent.
Montgomery (1997) uses a restricted mixed model as the basis for his analysis of
the following form
Vijkh = fi + Ti+Pj + + 7fc + (Tj)ik + (Py)jk + (rPy)ijk + eijkh,
where
h = 1,2,..., r is the number of replicates,
(r/3)ij is the random block by whole plot treatment interaction,
is the whole plot treatment x subplot treatment interaction,
{Pi)jk is the random block by subplot treatment interaction,
{r/3y)ijk is the random block by whole plot treatment by subplot treatment
interaction.
Under this restricted mixed model, the random interactions involving a fixed factor
are assumed subject to the constraint that the sum of that interactions effects over
the levels of the fixed factor is zero. Table 4 gives the resulting expected mean squares,
which suggests that there are three distinct error terms. The block by whole plot by
subplot interaction is used to test the whole plot by subplot interaction; the block
by subplot interaction is used to test the main effect of the subplot treatment; and
the block by whole plot interaction is used to test the main effect of the whole plot.


67
Table 25: 24 Point Design for the Case of 3 WP Factors and 4 SP Factors Using
the Same Fraction [HP(HS)-denotes high P(S) and LP(LS)-denotes low P(5)]
a
b
c
abc
9
9
9
9
r
r
r
r
ps
ps
ps
ps
pqrs
pqrs
pqrs
pqrs
HP
Folc
LP
on
HS
LS
s
pq
P
qs
qrs
pr
pqr
rs
Next, consider split-plot confounding. The defining contrast is I = ABC =
BCPQR = ACQRS = ABPS = APQR BQRS = CPS which is resolution
III. Not much of anything is estimable free of two-factor interactions. Again, the
additional eight points are obtained by semifolding on factors P and S. The 24 point
design is shown in Table 26. Again, the chains are almost completely broken. Three
resulting chains C = PS, AP = PS and BS = PS are left. The aliasing here means
that the sum of C, AP, and BS equals PS. Therefore, the model can accomodate
the fitting of any three of the four factors. So for example, assuming PS is negligible
allows for C, AP and BS to be estimated. Also, any two of the remaining five
two-factor interactions among the subplot factors can be estimated.
4 WP Factors (A, B, C, D) and 2 SP Factors (P, Q)
In this case, only the whole plot treatments need to be fractionated. Note, with
four whole plot factors there are only 3 df for whole plot factor effects. Hence, whole


133
Table 47: Values for the Variance of /3 (4 Mixture
Components and 3 Process Variables)
d
Asymptotic
OLS
REML
Pure Error
r = 2
.11
1.1 x HT13
1.2 x 10"13
1.2 x 10 13(5.2 x 10"17)
1.1 x 10-13(4.9 x 10-17)
.43
2.8 x 10"11
4.1 x 10"11
3.2 x 10_11(4.7 x HT14)
3.3 x 10_11(4.4 x 10-13)
1.0
3.3 x 10~9
7.5 x 109
4.1 x 10"9 (1.3 x HT11)
4.6 x 10~9 (1.4 x 10-11)
2.3
6.6 x 10-7
2.7 x 10~6
8.2 x 10"7 (4.0 x 10"9)
1.2 x 10~6 (6.7 x 109)
4.0
2.6 x 10~5
1.7 x 10~4
3.1 x 105 (1.7 x IQ7)
5.5 x 10~5 (4.5 x IQ7)
r = 3
.11
9.2 x 10~14
9.8 x 10~14
9.7 x 10_14(4.1 x 10"17)
9.8 x 10 14(7.1 x 10-17)
.43
2.5 x HT11
3.6 x 10"11
2.7 x 10"n(3.7 x 1014)
2.7 x 10"n(2.5 x 10"14)
1.0
2.9 x 10~9
6.7 x 10-9
3.5 x 10~9 (9.8 x 10"12)
3.4 x 1098 (7.5 x 10-12)
2.3
5.9 x 10"7
2.5 x 10~6
6.9 x 10"7 (3.0 x 10-9)
7.3 x 10-7 (3.1 x 109)
4.0
2.4 x 10~5
1.6 x 10"4
2.6 x 105 (1.1 x 10-7)
3.1 x 10~5 (1.9 x 10-7)
r = 4
.11
8.0 x 10~14
8.5 x 1014
8.4 x 10_14(3.7 x 10-17)
8.8 x 10 14(8.2 x 10-17)
.43
2.2 x 10"n
3.2 x lo-11
2.4 x 10_11(2.9 x 1014)
2.3 x 10-11(1.5 x 10-14)
1.0
2.7 x 10-9
6.1 x 10~9
3.0 x 10~9 (7.6 x 10"12)
2.9 x 1098 (3.5 x 10-12)
2.3
5.4 x 107
2.3 x 10"6
5.9 x 10"7 (2.0 x 10-9)
5.8 x 107 (1.4 x HT9)
4.0
2.2 x 105
1.5 x 10~4
2.3 x 10~5 (7.6 x 10"8)
2.3 x IQ*5 (7.6 x 108)
r = 5
.11
7.i x nr14
7.5 x 10-14
7.4 x 10_14(3.3 x nr17)
8.2 x 10~14(8.6 x 10-lV)
.43
1.9 x 10-11
2.9 x nr11
2.i x io_11(2.6 x nr14)
2.1 x 10-11(1.3 x 10"14)
1.0
2.4 x 10-9
5.5 x 10~9
2.7 x 10-9 (5.6 x 10-12)
2.6 x 109 (1.8 x 10-12)
2.3
4.9 x 10-7
2.1 x 106
5.3 x 10-7 (1.3 x 10~9)
5.1 x 10-7 (5.9 x lo-10)
4.0
i.9 x nr5
1.3 x 10-4
2.1 x 105 (4.9 x 10"8)
2.1 x IQ'5 (3.1 x HT8)
Standard Error for the Simulation are in Parentheses


136
Table 49: Relative Efficiencies for Comparing Methods of Estimating
V With 3 Mixture Components and 2 Process Variables
d
r 2
r = 3
OLS
REML
PE
OLS
REML
PE
.11
1.03
1.06
1.07
1.03
1.06
1.08
.43
1.26
1.13
1.10
1.27
1.12
1.07
1.0
1.78
1.25
1.20
1.79
1.20
1.11
2.3
3.02
1.38
1.40
3.07
1.23
1.18
4.0
4.67
1.48
1.62
4.76
1.33
1.24
r = 4
r = 5
d
OLS
REML
PE
OLS
REML
PE
.11
1.03
1.06
1.08
1.03
1.05
1.08
.43
1.27
1.10
1.06
1.26
1.09
1.05
1.0
1.79
1.13
1.08
1.79
1.13
1.06
2.3
3.06
1.18
1.12
3.05
1.11
1.08
4.0
4.75
1.14
1.13
4.72
1.14
1.08
A design similar to the one presented in the previous section is needed which will
support the fit of this model.
Proposed Design
Since a second-order model in the process variables is considered, a ccd is now
appropriate. The addition of the axial points will allow for the fitting of this model.
Therefore, the design given in Figure 8 will be augmented with axial points. In these
axial points, m replicates of the centroid blend will be run. This will preserve a
balanced design and provide an additional 2r[m 1) df for estimating the subplot
error variance in the pure error approach. There are still r replicates of the center of
the process variables and m replicates of the centroid in each of these. The design


126
Using REML changes the estimate of (3 to 3 (X'VC *X) 1X/VC 1y where Vc =
of J + of I. The variance is
Var(3) = (X'V^XJ^X'V-1 (Var(y)) V^X^'V^X)"1
= (x/vc-1x)-1x'vc-1vv-1x(x/v-1x)-1
and the determinant is used for comparison. Therefore, better estimates of of and of
will make the determinant of the REML estimate of the variance of (3 closer to the
determinant of the asymptotic value. Clearly, in practice V is not available to the
experimenter. However, we do know V for our simulation. Hence, we can calculate
the correct variance-covariance structure.
To obtain Vc, a simulation is conducted under the assumption that of = 1. Then,
for a fixed known value of d, Vc can be computed. Since there is no prior knowledge of
which terms in the model will be important, all model coefficients should be assumed
equal. For convenience, they will be set equal to 1 (ie. (3 = 1). The simulation of the
values needed to compute Vc is a two-step procedure corresponding to the split-plot
nature of the experiment. The whole plot is simulated by
W = f(z) + N(0,d)
where /(z) consists of the whole plot terms in the model evaluated at the point z and
N(0, d) is the whole plot error term. Then, the observations are given by
y = /(x;z) + kE + JV(0,1)
where /(x; z) consists of the subplot terms and the whole plot x subplot interactions
in the model evaluated at the point (x; z) and 7V(0,1) is the subplot error term.


OF FLORIDA


Table 17: Alias Structure for Split-Plot Confounding
A
=
BC
B
=
AC
C
=
AB
P
=
Q
=
R

AP
=
AQ
=
AR
=
BP
=
BQ
=
BR
=
CP
=
QR
CQ
=
PR
CR
=
PQ
Table 18: Design Layout for Split-Plot Confounding
in 8 Whole plots With / = ABPQ = ACQR = BCPR
(1)
a
b
ab
c
ac
be
abc
AB+
AB-
AB~
AB+
AB+
AB~
AB~
AB+
AC+
AC~
AC+
AC~
AC-
AC+
AC-
AC+
PQ+
PQ-
PQ-
PQ+
PQ+
PQ-
PQ-
PQ+
QR+
QR-
QR+
QR-
QR-
QR+
QR-
QR+
pqr
pr
qr
pq
pq
qr
pr
pqr
(1)
Q
P
r
r
P
Q
(1)


14
so has the manufacturing capacity, which has necessitated the installation of new
proofers and ovens. Given the complexity of these operations, qualifying a new piece
of proofing or baking equipment poses a challenging experimental design problem:
how do you design an experiment to explore the operating profile for a new proofer
or oven?
As an example, consider a continuous oven, in which dough-based products move
through on a belt. The oven has two zones, which are controlled independently. In
each zone you can adjust the Temperature, the Relative Humidity (RH), the Air Flow
Speed (AF), and the Residence Time. In general, the conditions in each zone will be
different, as each zone is used to impart different characteristics to the product. All
of these variables will impact the quality of the finished product.
Experimenting with this type of oven requires a restricted randomization. You
can easily reset the air flow and residence time in each zone on the fly, but changes in
the temperature and relative humidity require a waiting period to allow the oven to
return to steady state. Thus, oven experiments are typically conducted as a split-plot
design with four whole plot treatments, namely
Zone 1 Temp, Zone 1 RH, Zone 2 Temp, and Zone 2 RH,
and four split plot treatments, namely
Zone 1 AF, Zone 1 Res Time, Zone 2 AF, and Zone 2 Res Time.
In addition, we typically want to evaluate the effect of the oven on at least two prod
ucts (say Products A and B) to see if they respond differently to the oven. Each


29
For both crossed and non-crossed BRDs, the two randomizations complicate the
error structure. The first randomization leads to the whole plot error variance, erf,
while the second randomization leads to the sub-plot variance, the covariance between any two observations on the same whole plot is constant over
all whole plots and that observations on two sub-plots from different whole plots are
uncorrelated. The response surface model is
y = X/3 + 6 + e,
where 6 -1- e ~ V(0, V) with V = erf J -I- erf I, where J is a block diagonal matrix of
lb, x l'j,., and where 6, is the number of observations within the zth whole plot. Now
using generalized least squares (GLS), the maximum likelihood estimate (MLE) of
the response surface model is
3 = (X'V-'X)1 X'V-'y (2)
with
Var (3) = (x'V-1x)_1 (3)
From Equation (2), it is seen that the model estimation depends on the matrix V
and thus both erf and erf.
Suppose that the response surface model is partitioned into the whole plot and
sub-plot terms as
y = Z7 + X*(3* + Z'AX*,
where A is a matrix of whole plot x sub-plot interaction parameters. The response
surface design should be large enough to test for general lack of fit as well as lack of


75
Table 32: 24 Point Design for the Example


CHAPTER 4
A NEW MODEL AND CLASS OF DESIGNS FOR
MIXTURE EXPERIMENTS WITH PROCESS VARIABLES
Experiments that involve the blending of two or more components to produce high
quality products are known as mixture experiments. The quality of the end product
depends on the relative proportions of the components in the mixture. For example,
suppose we wish to study the flavor of a fruit punch consisting of juices from apples,
pineapples, and oranges. The flavor of the punch depends on the relative proportion
of the juices in the blend.
Consider a mixture experiment consisting of q components. Let Xi, for i
1,2,,q, represent the fractional proportion contributed by component i. Then the
proportions must satisfy the following constraints
9
0 < Xi < 1, xi = 1,
i= 1
and the experimental region is a (q l)-dimensional simplex, Sq. For q = 3, S3 is an
equilateral triangle and for q = 4, S4 is a tetrahedron. Typically, the blends used in
a mixture experiment are the vertices or single-component blends, the midpoints of
the edges, centroids of faces, etc., and the centroid of the simplex.
In some mixture experiments, the quality of the product depends not only on the
proportions of the components in the blend, but also on the processing conditions.
Process variables are factors that do not form any portion of the mixture but whose
levels, when changed, could affect the blending properties of the components. Cornell
81


113
Table 40: Higher Order terms Supported by the Proposed Design With Three Mixture
Components and the Simplex-Centroid at the Center of the Process Variables
Binary Blend by Process Variable Interactions
3MC, 2PV
3MC, 3PV
Constrained 3MC, 2PV
Without Special
Cubic Terms
1 Binary Blend With
Both Process Variables
All 9 Terms
All 6 Terms
With Special
Cubic Terms
Any 2 of the 3 Binary
Blends With Either
Process Variable
OR
1 Binary Blend With
Both Process Variables
All 9 Terms
All 6 Terms
of freedom left over for lack of fit terms than the design with just the mixture centroid
at the center of the process variables. To simplify the many possible combinations
of higher order terms that could be fit, only the terms in (i) and (ii), namely, the
special cubic terms and the interactions between a binary blend and the linear effect
of a process variable, are considered. Tables 40 and 41 give the additional terms the
proposed design can accomodate for each of the five cases. It is seen from Tables 40
and 41 that the proposed design supports some of the higher order terms in most
cases and all of the higher order terms in the case of three mixture components and
three process variables.
If check points are going to be used to test for lack of fit, one alternative to the
method described in Cornell (1990) is to add points that would aid in the fitting of
higher order terms, such as face center points. If the proposed model suffers from


49
and three subplot factors, P, Q, and R where all factors have two levels. Suppose
only 16 runs are possible among the 64 total number of combinations. There are two
ways to allocate the whole plots and subplots for this experiment. We can use four
whole plots with each whole plot containing four subplots or we can use eight whole
plots with each whole plot containing two subplots. The goal of the experiment is to
estimate all six main effects and as many of the nine two-factor interactions between
the whole plot and subplot factors as is possible, although it is believed that some
two-factor interactions among the subplot factors might be significant. To conserve
space in the tables, the confounding structure or alias chains will be given only up to
order two. Therefore, if there is a blank space in the alias table, it means that the
effect is aliased with interactions of order higher than two.
First, suppose that the experimenter ignores the split-plot structure by consid
ering the factors as a 26 factorial in a completely randomized design. Actually, the
experimenter would use a 26-2 fractional factorial design to obtain the 16 runs. The
best defining contrast is
/ = ABCP = CPQR = ABQR,
which has Resolution IV. The layout is given in Table 12 and the alias chains are
given in Table 13. All main effects can be estimated, but two-factor interactions are
aliased with each other. Even if we assume that all two-factor interactions among A,
B, and C and all two-factor interactions among P, Q, and R are negligible, there is
still a problem since AQ is aliased with BR and AR is aliased with BQ. In other
words, some of the interactions we are interested in are aliased with each other.


x_1 = 1 z_2
Figure 5: 23 Point Design Generated by SAS for the 3-2 Case
CD
00


45
they are no longer separately estimable. Generally, these effects will be three-factor
interactions or higher so that all main effects and two-factor interactions can be
estimated.
When the block size of 2k is reduced by 2~p, each block will contain 2k~p experi
mental units and each complete replicate will contain 2P blocks. In this case, it will
be necessary to confound 2P 1 effects in each replicate. The experimenter chooses p
of these effects with the remaining 2P p 1 effects being the generalized interactions
of the original p effects. When more than one replicate of the 2k~p fractional factorial
is performed, two types of confounding are possible:
1. Complete the same set of effects is confounded in each replicate;
2. Partial different sets of effects are confounded in different replicates.
Complete confounding is used whenever all information on the confounded effects
can be sacrificed. This should only be used when all confounded effects are believed
to be negligible. Complete confounding creates no problems with the analysis. It is
only necessary to find the effect totals for all unconfounded effects.
There are situations where effects believed to be important must be confounded,
for example, when available resources force the use of small block sizes. In these cases,
partial confounding is used. Partial confounding means confounding different effects
in different replicates so as to allow estimation of all effects. These estimates use only
the data from the replicates in which the effect is unconfounded. Thus, there will be
greater precision on effects that are unconfounded than on effects that are partially


47
a criterion based on estimability. Throughout this chapter, wherever the phase best
design is used it will be under the above setting.
3.4 Combining Fractional Factorials and
Confounding in Split-Plot Experiments
Splitting the plots or experimental units is possible with any experimental design.
The design refers to the assignment of the whole plot and subplot treatments and is
selected in order to control the known sources of extraneous variation. Regardless of
the choice of design, the subplot treatments can be thought of as being arranged in
blocks where the whole plots are the blocks. In each whole plot, if all the subplot
treatments can be run, then the situation resembles that of a complete block design
as far as the subplot treatments are concerned. However, there are situations where
in each whole plot not all of the subplot treatments can be performed so that some
form of an incomplete block design must be used. If the subplot treatments result
from a 2k factorial structure, then the methods discussed in the previous sections of
this chapter can be applied to reduce the number of subplot treatments in a whole
plot.
Consider the situation where both the whole plot treatments and the subplot
treatments have a 2k factorial structure. Assume that the design for the whole plot
treatments is a CRD. Suppose, only a fraction of the whole plot treatments are of
interest and only a fraction of the subplot treatments can be run for each whole plot.
We will consider the situation involving noise factors and design factors. The noise
factors will be the whole plot factors. Therefore, the goal of the experiment is to
estimate the following:


7
Yijk [i + Ti + [3j + [r(3)ij +7 fe + (T'y)ik + ejfc (1)
* = 1,2,... ,t j = l,2,...,b k = l,2,...,s,
where
n is the overall mean,
Ti is the effect of whole plot treatment i,
(3j is the effect of block j,
(r/3)ij is the block x whole plot treatment interaction,
7*, is the effect of subplot treatment k,
ChOik is the whole plot treatment x subplot treatment interaction, and
ijk is the subplot error.
The (r/3)ij term will be the whole plot error term for the case of an RCB design under
the usual assumption of no block x whole plot treatment interaction. The analysis
of variance table associated with the model in Equation (1), assuming whole plot
treatments and subplot treatments are fixed and blocks are random, is given in Table
3. If the block by whole plot interaction is called the whole plot error, then Tables 2
and 3 suggest the same basic testing procedures. The following additional constraints
and assumptions are needed for hypothesis testing:
^ = , 7fc = 0 ,
i k


124
diagonal matrix of l¡,xi x l/lx6 where b is the number of observations in each whole plot
and I is the identity matrix of dimension b. The structure implies that observations
in different whole plots are independent while observations within a whole plot are
correlated.
To complicate matters even more, the same mixture blends are not run at each
combination of the process variables. Because of this deviation in treatment structure,
ordinary least squares is not equivalent to generalized least squares. Therefore, the
estimating equation is
3 = (X/V-1x)1X/V-1y
with
Var(3) = (X'V^X)-1 .
The estimation of the model coefficients and of the variances of these estimates de
pends on the matrix V and thus on estimation of the two error variances. Three methods, ordinary least squares, re
stricted maximum likelihood and a method based on pure error, will be considered
for estimating these error variances.
Ordinary least squares (OLS) assumes the observations are independent. Thus, it
ignores the dependent structure of the split-plot design. This is a naive approach in
light of the restricted randomization and one would expect that it will not perform
well. Restricted maximum likelihood (REML) is similar to maximum likelihood esti
mation in that it uses the likelihood of a transformation of y which is based on the
residuals (see Russell and Bradley (1958)). The proposed design uses r replicates of
the whole plot at the center of the process variables. Within each of these whole plots,


APPENDIX C
SAS CODE FOR PROC MIXED
Consider the model of the form
y = X/3 + 6 +
where 6 4- e ~ V(0, V) and V = of J + a^I. Then, the following code can be used to
obtain estimates of of and of.
PROC MIXED METHOD = REML;
CLASS WP;
MODEL Y = FIXED EFFECTS;
RANDOM WP;
where WP is a classification variable defining into which whole plot each observation
falls, the model statement defines the model matrix, X.
167


141
Table 53: Relative Efficiencies for Comparing Methods of Estimating
V With 3 Mixture Components and 2 Process Variables in a Constrained Region
d
r = 2
r = 3
OLS
REML
PE
OLS
REML
PE
.11
1.03
1.23
1.07
1.03
1.08
1.08
.43
1.26
1.15
1.10
1.27
1.09
1.07
1.0
1.78
1.22
1.20
1.79
1.13
1.11
2.3
3.02
1.30
1.40
3.07
1.17
1.18
4.0
4.67
1.39
1.62
4.76
1.22
1.24
r = 4
r 5
d
OLS
REML
PE
OLS
REML
PE
.11
1.03
1.03
1.08
1.03
1.02
1.08
.43
1.27
1.06
1.06
1.26
1.04
1.05
1.0
1.79
1.09
1.08
1.79
1.06
1.06
2.3
3.06
1.12
1.12
3.05
1.10
1.08
4.0
4.75
1.15
1.13
4.72
1.13
1.08
Both first-order and second-order models are considered for the process variables.
In the first-order case, the pure error method performs well when there are 3, 4, or
5 replicates of the center of the process variables. REML performs well in all cases
and especially when there are only 2 or 3 replicates. With the second-order model,
REML performs well across the board while the pure error method performs well only
with 4 or 5 replicates. If the experimenter can afford a few extra runs, then the pure
error approach is a simple method and has the added feature of being able to test the
model for lack of fit.


77
Effect/Standard Error
Figure 1: Normal Probability Plot for the Example


96
experimental region is irregular or when the number of available runs is limited or for
special models such as those that contain nonlinear parameters.
Computer generated designs are typically not unique. Therefore, if the computer
is used to generate two or three designs with the same number of points, the designs
will contain some points that are not common to all the designs. Because of this, one
design may be more appealing than the others for one reason or another. This leads
to the question: Wouldnt it be better to generate an acceptable or appealing design
and compare it to the computer generated design? This is what is done in this paper.
Another idea is to use the computer to generate a design based on some criterion and
then alter it slightly so that it is more appealing, and finally compare it back to the
original design. Ultimately, if the computer produces a design that is unacceptable,
an acceptable design will have to be constructed anyway. The computer can be used
to provide an initial design as a start toward generating a good design. The basic
idea is to use the computer generated designs wisely.
The designs presented in this paper are compared to the designs chosen by PROC
OPTEX (1989) in SAS. PROC OPTEX requires the user to provide a candidate point
list and the model to be fit. The candidate point list for the examples in this paper
consists of the simplex-centroid design at each point of the ccd. With the 7-point
simplex-centroid design in three components and a 9-point ccd in two process vari
ables, for example, the candidate point list for three mixture components and two
process variables consists of 7 x 9 = 63 points. The model to be fit is the model in
Equation (10) with 15 terms. PROC OPTEX uses a random seed and performs ten
searches from the random starting points. A few options are used. The first is to


108
are
Ho : 7n 721 = 73i
H^ : at least one not equal.
The testing procedure can be rewritten as the simultaneous test of
H0 : 7n 721 = 0 H0 : 7n 731=0
H/i : 7n 72i ^ 0 HA : 7n 731 ^ 0.
If the three terms here are entered last in the model, then in matrix notation the
above hypotheses become
where L/ =
0
2,p3)
model, and c =
0
0
H0 : L//3 = c
, (3 is a p x 1 vector of all the parameters in the
. The appropriate F-test is
-1 0
0 -1
_ Q/s Ho p
MSE sn~p~1
where s (=2 in our case) is the rank of L', MSE is the mean squared error from the
fit of the model, and
Q = (l'3 c)' [v (x'xr1 l] 1 (l'3 c)
with f3 being the least squares estimate of the parameter vector (3. The test can
easily be performed in SAS using PROC REG. After the model statement, the test
statement is used with a separate test statement for each process variable. For the
above example, the SAS code is
PROC REG;
MODEL Y = XiX2 x\Z\X2Z\X-iZ\ / noint;
TEST x\Z\ x2z\ = 0, X\Z\ x2Z\ = 0;
TEST xxz2 x2z2 = 0, xxz2 x3z2 = 0;
TEST
RUN;


155
Table 64: Design Points for 3 WP Factors and 4
SP Factors Using Split-Plot Confounding
A
B
C
P
Q
R
S
1
-1
-1
1
-l
-1
-1
1
-1
-1
1
i
1
-1
1
-1
-1
-1
i
-1
1
1
-1
-1
-1
-l
1
1
-1
1
-1
-1
-l
-1
1
-1
1
-1
1
i
-1
-1
-1
1
-1
1
-l
1
-1
-1
1
-1
-1
l
1
1
-1
-1
1
-1
i
1
-1
-1
-1
1
1
i
-1
1
-1
-1
1
1
-l
1
1
-1
-1
1
-1
-l
-1
-1
1
1
1
-1
i
-1
-1
1
1
1
-1
-l
1
-1
1
1
1
1
-l
-1
1
1
1
1
1
i
1
1
1
-1
-1
-1
-l
-1
-1
1
-1
-1
-1
i
1
-1
-1
1
-1
1
-l
-1
1
-1
1
-1
1
i
1
1
-1
-1
1
-1
l
1
1
-1
-1
1
-1
-l
-1
1
1
1
1
1
-l
-1
-1
1
1
1
1
l
1
-1


48
main effects for the whole plot factors;
main effects for the subplot factors;
two-factor interactions between the whole plot and subplot factors;
and if possible, two-factor interactions among the subplot factors.
Note that if there were sufficient resources to run all whole plot treatments and
subplot treatments, then all four goals would be automatically satisfied. However, in
most situations, this is not economically possible. Therefore, we shall try to estimate
as many effects as is possible within the restrictions on the resources available.
The idea of confounding effects in order to reduce the number of subplot treat
ments per whole plot treatment and achieve the second goal has been around for some
time. Kempthorne (1952) has a section devoted to confounding in split-plot experi
ments. Addelman (1964) also discusses ways of accomplishing this. Recently, the use
of split-plot experiments in industry has generated renewed interest in confounding.
Huang, Chen, and Voelkel (1998) and Bingham and Sitter (1999) discuss minimum-
aberration designs for factors with two-levels. This technique helps to improve the
estimation problem by raising the resolution concerning the subplot factors, but one
must be careful with the whole plot x subplot interactions. Bisgaard (1999) uses
inner and outer arrays, with factors at two-levels, as in robust parameter design and
provides the standard errors for various contrasts among the whole plot and subplot
factors.
We will use an example to compare the use of confounding in a split-plot exper
iment. Consider a split-plot experiment with three whole plot factors, A, B, and C,


165
Table 73: Design Points for 4-3 Case
xx
%3
X4
Z\
Z2
Z3
X\
x2
X3
x4
Z\
Z2
Z3
1
0
0
0
-1
-1
1
0
0
.5
.5
-1
-1
-1
1
0
0
0
-1
1
-1
0
0
.5
.5
-1
1
1
1
0
0
0
1
1
1
0
0
.5
.5
1
-1
1
1
0
0
0
1
-1
-1
0
0
.5
.5
1
1
-1
0
1
0
0
-1
-1
1
.25
.25
.25
.25
-1
0
0
0
1
0
0
-1
1
-1
.25
.25
.25
.25
1
0
0
0
1
0
0
-1
1
-1
.25
.25
.25
.25
0
-1
0
0
1
0
0
1
1
1
.25
.25
.25
.25
0
1
0
0
0
1
0
-1
-1
1
.25
.25
.25
.25
0
0
-1
0
0
1
0
-1
1
-1
.25
.25
.25
.25
0
0
1
0
0
1
0
1
1
1
.25
.25
.25
.25
0
0
0
0
0
1
0
1
-1
-1
Additional Points for
0
0
0
1
-1
-1
1
Simplex-Centroid
0
0
0
1
-1
1
-1
1
0
0
0
0
0
0
0
0
0
1
1
1
1
0
1
0
0
0
0
0
0
0
0
1
1
-1
-1
0
0
1
0
0
0
0
.5
.5
0
0
-1
-1
-1
0
0
0
1
0
0
0
.5
.5
0
0
-1
1
1
.5
.5
0
0
0
0
0
.5
.5
0
0
1
-1
1
.5
0
.5
0
0
0
0
.5
.5
0
0
1
1
-1
.5
0
0
.5
0
0
0
.5
0
.5
0
-1
-1
-1
0
.5
.5
0
0
0
0
.5
0
.5
0
-1
1
1
0
.5
0
.5
0
0
0
.5
0
.5
0
1
-1
1
0
0
.5
.5
0
0
0
.5
0
.5
0
1
1
-1
.5
0
0
.5
-1
-1
-1
.5
0
0
.5
-1
1
1
.5
0
0
.5
1
-1
1
.5
0
0
.5
1
1
-1
0
.5
.5
0
-1
-1
-1
0
.5
.5
0
-1
1
1
0
.5
.5
0
1
-1
1
0
.5
.5
0
1
1
-1
0
.5
0
.5
-1
-1
-1
0
.5
0
.5
-1
1
1
0
.5
0
.5
1
-1
1
0
.5
0
.5
1
1
-1


Ill
for these parameters involve the least squares estimates and are performed in SAS
using PROC REG.
For the process variables, in addition to testing their effects on the linear blending
properties of the mixture components, tests on the pure quadratic effects and the
two-factor interactions among the process variables are of interest. If the test for a
pure quadratic term is significant, then the response is curvilinear as one moves from
the low to the high level of that process variable. A significant interaction implies
that the relationship between the two process variables is not always the same. Before
eliminating any terms from the model resulting testing results being nonsignificant,
the model should be tested for lack of fit.
4.5 Lack of Fit
The model in Equation (10) is assumed to be the true model. However, it is
possible that this is not the case. There may be some important higher order terms
that were not included in the model. Therefore, before making any inferences about
significant effects, the model should be tested for lack of fit.
One way to test the model for lack of fit is to replicate one or more points in
the design. Then, the error or residual sum of squares in the ANOVA table can be
partitioned into sum of squares due to lack of fit and sum of squares for pure error.
The ratio of these two divided by their appropriate degrees of freedom forms an F-test
which can be used to determine if the model suffers from lack of fit. Another way is
to use check points which involves three steps:
1. Fit the proposed model using data collected at the design points.


76
Table 33: Effects Table for the Example
Term
Effect
Coeff
Std Error
t-value
P-value
Constant
21.266
0.4288
49.60
0.000
X
-0.937
-0.469
0.4951
-0.95
0.387f
Y
1.406
0.703
0.4288
1.64
0.1621
Z
-2.156
-1.078
0.4288
-2.51
0.054*
A
1.000
0.500
0.4951
1.01
0.359
B
-0.812
-0.406
0.4951
-0.82
0.449
C
0.937
0.469
0.4951
0.95
0.387
X*A
0.031
0.016
0.4288
0.04
0.972
X*B
-0.187
-0.094
0.4951
-0.19
0.857
X*C
-0.563
-0.281
0.4951
-0.57
0.595
Y*A
-0.094
-0.047
0.4288
-0.11
0.917
Y*B
0.312
0.156
0.4288
0.36
0.730
Y*C
0.875
0.438
0.4288
1.02
0.354
Z*A
0.344
0.172
0.4288
0.40
0.705
Z*B
0.500
0.250
0.4288
0.58
0.585
Z*C
-1.312
-0.656
0.4288
-1.53
0.186
A*B
1.375
0.687
0.4951
1.39
0.224
A*C
1.250
0.625
0.4951
1.26
0.262
B*C
-4.438
-2.219
0.4951
-4.48
0.007
^ Not Valid Tests


86
Equation (4) becomes
V = fio )+&X+ J+EA*^
\i=l / =1 =1 ft j^i J i 9 9 9 9
^ 1 ifio t ft Pa) y {0ii%i y {Xj + y ^ y ^ fiijx^xj
i= 1 i=1 = E#3*+ EE#a*rJ>
=1 i where fi¡ = fio+fii + /ft and /?* = fi fij for i, j = 1,2,..., q, i < j.
(5)
Suppose that an experiment is to be performed with q mixture components,
xi, x2,..., xq, and n process variables, Z\, , zn. In the process variables, let us
consider the model
n n
1lPV = OLq + E akZk + E E aklZkZl (6)
fc=l k Then there are two main types of combined models (see Cornell (1990)) that can be
used in this situation. The first type is a model which crosses the mixture model
terms in Equation (5) with each and every term of Equation (6). This produces the
combined model
9 9 9 n 9 n
*?(x,z) = E #* + E E P*i3xixi+E E rlikX%Z}i T ZEE liklXiZkZi
=1 i + EEE ^ijkxixjZk T El y El y 1 1ijklxixjZkZi (7)
i which includes parameters for three and four factor interactions. Depending on the
design, the model of Equation (7) provides a measure of the linear and nonlinear
blending properties of the mixture components averaged across the settings of the
process variables as well as the effects of the process variables on the linear and
nonlinear blending properties.


x 1 = 1 Z_2
Figure 6:17 Point Design Generated by SAS for the 3-2 Case
CO
CO


153
Table 62: Design Points for 3 WP Factors and 3
SP Factors Using Split-Plot Confounding
A
B
C
P
Q
R
1
-1
-1
1
i
-1
1
-1
-1
1
-l
1
1
-1
-1
-1
i
1
1
-1
-1
-1
-l
-1
-1
1
-1
1
i
-1
-1
1
-1
1
-l
1
-1
1
-1
-1
i
1
-1
1
-1
-1
-l
-1
-1
-1
1
1
-l
-1
-1
-1
1
-1
l
-1
-1
-1
1
-1
-l
1
-1
-1
1
1
l
1
1
1
1
1
-l
-1
1
1
1
-1
i
-1
1
1
1
-1
-l
1
1
1
1
1
i
1
1
-1
-1
-1
i
-1
1
-1
-1
-1
-l
1
-1
1
-1
1
i
1
-1
1
-1
1
-l
-1
-1
-1
1
1
l
-1
-1
-1
1
1
-l
1
1
1
1
-1
-l
-1
1
1
1
-1
i
1


ACKNOWLEDGMENTS
I would like to express my sincere gratitude to Dr. G. Geoffrey Vining for serving
as my dissertation advisor. Many thanks go out to him for allowing me the oppor
tunity to serve as an editorial assistant for the Journal of Quality Technology. His
generosity has given me experiences that most graduate students could only dream
of having. Also, not only as he been a mentor for me statistically, but he has also
been my friend.
I would also like to thank Dr. John Cornell for his extreme interest in the work
that I have done and for being a constant resource for me. In addition, I would like
to thank Drs. Jim Hobert, Richard Scheaffer, and Diane Schaub for serving on my
committee. Also, I would like to thank Dr. Frank Martin for sitting in on my defense
and for developing my interest in Design of Experiments through his course. I extend
a big thank you to Dr. Ronald Randles, chairman of the department of statistics, for
supporting me through my many years at the University of Florida.
I thank my parents for their endless love and support. They fully believed in me
when, at times, I wasnt so sure I would finish. Finally, I have to thank my wife for
her love and especially her patience. She has stood by me through everything and I
owe her more than I could ever repay.
ii
i


78
3.8 Summary
The main goal of this chapter is to understand some of the complications involved
with using the two types of confounding in split-plot experiments. If the design is
chosen using the MA criterion, then designs constructed by combining the fractional
factorial of the whole plot treatments with a fractional factorial of the subplot treat
ments are not considered. This is because their partial resolution is too low. However,
it has been shown that depending on what is to be estimated, it may not be wise
to eliminate such designs. Also, for 16 run designs the MA criterion tends to select
designs with eight whole plots and two subplots per whole plot. These designs do
not take into consideration the possible increase in the cost of experimentation or of
factors whose levels are hard to change.
We have presented a 24 run design which is a compromise between the 16 run
and 32 run designs. We begin with a 16 run design using four whole plots with four
subplots per whole plot. Such a strategy accommodates hard-to-change factors as
well as cost considerations involving the experiment. Then, eight additional runs are
added using semifolding of one or two factors. These runs preserve the balance of
high and low levels of each subplot factor as well as maintain the same number of
subplots for each whole plot. Except for the cases involving four whole plot factors,
the additional runs are at the subplot level. Thus, as long as adding the runs is
feasible, it should not be too costly. Also, the extra runs allow for additional effects to
be estimated and/or add degrees of freedom for estimating the subplot error variance.
Though designs using split-plot confounding and separate fractions differ in what
they can estimate in 16 runs, there is not much difference in estimability when using


32
2. Y ew + es.
3. If the level of aq of the current run is different from that of the previous run, a
new value of both ew and es is generated. Otherwise, generate a new value for
es only.
4. Go to step 2 until all 28 runs are completed.
The data were generated for completely randomized, completely restricted, and
partially restricted run orders. For a partially restricted run order, each level of
the hard-to-change factor was visited exactly twice and the runs at each level were
randomly divided into two equal groups. Each time a data set was generated, the
least squares estimates of the /?s were computed and the residual error was estimated.
The simulation procedure was repeated 1,000 times.
Lucas and Ju (1992) summarized their simulation results in a table with a listing
of the standard deviations of the regression coefficients for the three different ways
of running the experiment. The restricted randomization case has a much smaller
residual standard deviation and much smaller standard deviations for all the regres
sion coefficients except those associated with the hard-to-change factor, f3\ and /3n.
These results correspond with the general result that split-plot designs will produce
increased precision on the subplot factors while sacrificing precision on the whole plot
factors. The magnitudes of the coefficients of the estimated standard deviations for
the partially restricted case were greater than those with the completely randomized
case but less than the corresponding estimates for the completely restricted case.


66
Table 24: 24 Point Design for the Case of 3 WP Factors and 3 SP Factors
Using Split-Plot Confounding [HP-denotes high P and LP-denotes low P]
a b c abc
PQ
PQ
P
P
pr
pr
Q
Q
qr
qr
r
r
(1)
(1)
pqr
pqr
Folc
HP LP
on
LP HP
Q
pqr
PQ
(1)
r
P
pr
qr
3 WP Factors (A, B, C) and 4 SP Factors (P, Q, R, S)
To obtain a 16 point design in this situation, both the whole plot and subplot
treatments need to be fractionated or confounded. First, consider fractionating the
whole plot and subplot treatments separately. The defining contrast is I ABC
PQR = QRS = PS = ABCPQR = ABCQRS = ABCPS which is Resolution
II. In order to estimate the subplot factor main effects and possibly the two-factor
interactions among the subplot factors, the two chains PQR and QRS with resulting
chain PS need to be broken. The additional eight points are obtained by semifolding
on both factors P and S. The 24 points design is shown in Table 25. The chains
are almost completely broken. Two resulting chains .4P = PS and AS = PS are
left. The aliasing here means that the sum of AP and AS equals PS. Therefore, the
model can accomodate the fitting of any two of the three factors. So for example, if
PS is assumed negligible, then the effects of .45 and AP are estimable. There are 4
df left that can be used as an error term or used to estimate PQ, PR, QS and RS.


163
Table 71: Design Points for 3-3 Case
Xi
X3
Z\
Z2
Z3
1
0
0
-1
-1
1
1
0
0
-1
1
-1
1
0
0
1
1
1
1
0
0
1
-1
-1
0
1
0
-1
-1
1
0
1
0
-1
1
-1
0
1
0
1
1
1
0
1
0
1
-1
-1
0
0
1
-1
-1
1
0
0
1
-1
1
-1
0
0
1
1
1
1
0
0
1
1
-1
-1
.5
.5
0
-1
-1
-1
.5
.5
0
-1
1
1
.5
.5
0
1
-1
1
.5
.5
0
1
1
-1
.5
0
.5
-1
-1
-1
.5
0
.5
-1
1
1
.5
0
.5
1
-1
1
.5
0
.5
1
1
-1
0
.5
.5
-1
-1
-1
0
.5
.5
-1
1
1
0
.5
.5
1
-1
1
0
.5
.5
1
1
-1
.33
.33
.33
-1
0
0
.33
.33
.33
1
0
0
.33
.33
.33
0
-1
0
.33
.33
.33
0
1
0
.33
.33
.33
0
0
-1
.33
.33
.33
0
0
1
.33
.33
.33
0
0
0
Additional Points for
Simplex-Centroid
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
.5
.5
0
0
0
0
.5
0
.5
0
0
0
0
.5
.5
0
0
0



PAGE 1

7+( '(6,*1 $1' $1$/<6,6 2) 63/,73/27 (;3(5,0(176 ,1 ,1'8675< %\ 6&277 0 .2:$/6., $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 2) 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

$&.12:/('*0(176 ZRXOG OLNH WR H[SUHVV P\ VLQFHUH JUDWLWXGH WR 'U *HRIIUH\ 9LQLQJ IRU VHUYLQJ DV P\ GLVVHUWDWLRQ DGYLVRU 0DQ\ WKDQNV JR RXW WR KLP IRU DOORZLQJ PH WKH RSSRUn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fW VR VXUH ZRXOG ILQLVK )LQDOO\ KDYH WR WKDQN P\ ZLIH IRU KHU ORYH DQG HVSHFLDOO\ KHU SDWLHQFH 6KH KDV VWRRG E\ PH WKURXJK HYHU\WKLQJ DQG RZH KHU PRUH WKDQ FRXOG HYHU UHSD\ LL L

PAGE 3

7$%/( 2) &217(176 $&.12:/('*0(176 $%675$&7 YL &+$37(56 ,1752'8&7,21 5HVSRQVH 6XUIDFH 0HWKRGRORJ\ 6SOLW3ORW 'HVLJQV 'LVVHUWDWLRQ *RDOV 2YHUYLHZ /,7(5$785( 5(9,(: 6SOLW3ORW &RQIRXQGLQJ 6SOLW3ORWV LQ 5REXVW 3DUDPHWHU 'HVLJQV %L5DQGRPL]DWLRQ 'HVLJQV 6SOLW3ORWV LQ ,QGXVWULDO ([SHULPHQWV ,1&203/(7( 63/,73/27 (;3(5,0(176 )UDFWLRQDO )DFWRULDOV &RQIRXQGLQJ &RQIRXQGLQJ LQ )UDFWLRQDO )DFWRULDOV &RPELQLQJ )UDFWLRQDO )DFWRULDOV DQG &RQIRXQGLQJ LQ 6SOLW3ORW ([SHULPHQWV LLL

PAGE 4

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

PAGE 5

5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ Y

PAGE 6

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

PAGE 7

FRPSRQHQWV EXW DOVR RQ WKH SURFHVVLQJ FRQGLWLRQV $ FRPELQHG PRGHO LV SURSRVHG ZKLFK LV D FRPSURPLVH EHWZHHQ WKH DGGLWLYH DQG FRPSOHWHO\ FURVVHG FRPELQHG PL[WXUH E\ SURFHVV YDULDEOH PRGHOV $OVR D QHZ FODVV RI GHVLJQV WKDW ZLOO DFFRPRGDWH WKH ILWWLQJ RI WKH QHZ PRGHO LV FRQVLGHUHG 7KH GHVLJQ DQG DQDO\VLV RI WKH PL[WXUH H[SHULPHQWV ZLWK SURFHVV YDULDEOHV LV GLVn FXVVHG IRU ERWK D FRPSOHWHO\ UDQGRPL]HG VWUXFWXUH DQG D VSOLWSORW VWUXFWXUH :KHQ WKH VWUXFWXUH LV WKDW RI D VSOLWSORW H[SHULPHQW WKH DQDO\VLV LV PRUH FRPSOLFDWHG VLQFH RUGLQDU\ OHDVW VTXDUHV LV QR ORQJHU DSSURSULDWH :LWK WKH SURFHVV YDULDEOHV VHUYLQJ DV WKH ZKROH SORW IDFWRUV WKUHH PHWKRGV IRU HVWLPDWLRQ DUH FRPSDUHG XVLQJ D VLPXODWLRQ VWXG\ 7KHVH DUH RUGLQDU\ OHDVW VTXDUHV WR VHH KRZ LQDSSURSULDWH LW LVf UHVWULFWHG PD[LPXP OLNHOLKRRG DQG XVLQJ UHSOLFDWH SRLQWV WR JHW DQ HVWLPDWH RI SXUH HUURU 7KH ODVW PHWKRG DSSHDUV WR EH WKH EHVW LQ WHUPV RI WKH LQFUHDVH LQ WKH VL]H RI WKH FRQILn GHQFH HOOLSVRLG IRU WKH SDUDPHWHUV DQG KDV WKH DGGHG IHDWXUH RI QRW GHSHQGLQJ RQ WKH PRGHO 9OO

PAGE 8

&+$37(5 ,1752'8&7,21 $ FRPPRQ H[HUFLVH LQ WKH LQGXVWULDO ZRUOG LV WKDW RI GHVLJQLQJ H[SHULPHQWV H[SORUn LQJ FRPSOH[ UHJLRQV DQG RSWLPL]LQJ SURFHVVHV 7KH VHWWLQJ XVXDOO\ FRQVLVWV RI VHYHUDO LQSXW IDFWRUV WKDW SRWHQWLDOO\ LQIOXHQFH VRPH TXDOLW\ FKDUDFWHULVWLF RI WKH SURFHVV ZKLFK LV FDOOHG WKH UHVSRQVH %R[ DQG :LOVRQ f LQWURGXFHG VWDWLVWLFDO PHWKRGV WR DWWDLQ RSWLPDO VHWWLQJV RQ WKH GHVLJQ YDULDEOHV 7KHVH PHWKRGV DUH FRPPRQO\ NQRZQ DV UHVSRQVH VXUIDFH PHWKRGRORJ\ 560f ZKLFK FRQWLQXHV WR EH DQ LPSRUWDQW DQG DFWLYH DUHD RI UHVHDUFK IRU LQGXVWULDO VWDWLVWLFLDQV 0DQ\ WLPHV LQ LQGXVWULDO H[SHULPHQWV WKH IDFWRUV FRQVLVW RI WZR W\SHV VRPH ZLWK OHYHOV WKDW DUH HDV\ WR FKDQJH DQG RQH RU PRUH ZLWK OHYHOV WKDW DUH GLIILFXOW RU FRVWO\ WR FKDQJH 6XSSRVH IRU LOOXVWUDWLRQ WKDW WKHUH LV RQO\ RQH IDFWRU WKDW LV GLIILFXOW WR FKDQJH :KHQ WKLV LV WKH FDVH WKH H[SHULPHQWHU XVXDOO\ ZLOO IL[ WKH OHYHO RI WKLV IDFWRU LH UHVWULFW WKH UDQGRPL]DWLRQ VFKHPHf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f [ E\

PAGE 9

U" [fH ZKHUH WKH IRUP RI WKH IXQFWLRQ J LV XQNQRZQ DQG H LV D UDQGRP HUURU WHUP 7KH JRDO LV WR ILQG LQ WKH VPDOOHVW QXPEHU RI H[SHULPHQWV WKH VHWWLQJV DPRQJ WKH OHYHOV RI [ ZLWKLQ WKH UHJLRQ RI LQWHUHVW DW ZKLFK U" LV D PD[LPXP RU PLQLPXP %HFDXVH WKH IRUP RI J LV XQNQRZQ LW PXVW EH DSSUR[LPDWHG 560 XVHV 7D\ORU VHULHV H[SDQVLRQ WR DSSUR[LPDWH J[f RYHU VRPH UHJLRQ RI LQWHUHVW 7\SLFDOO\ ILUVW RU VHFRQG RUGHU PRGHOV DUH XVHG WR DSSUR[LPDWH J[f 7KH WUDGLWLRQDO 560 PRGHO ZRXOG EH 9L [f H ZKHUH f LML LV WKH rWK UHVSRQVH f [ LV WKH LWK VHWWLQJ RI WKH GHVLJQ IDFWRUV f [f LV WKH DSSURSULDWH SRO\QRPLDO H[SDQVLRQ RI [ f c LV D YHFWRU RI XQNQRZQ FRHIILFLHQWV DQG f WKH HV DUH DVVXPHG WR EH LQGHSHQGHQW DQG LGHQWLFDOO\ LLGf GLVWULEXWHG DV M9Df )RU D PRUH GHWDLOHG GLVFXVVLRQ RQ 560 VHH .OLXUL DQG &RUQHOO f %R[ DQG 'UDSHU f DQG 0\HUV DQG 0RQWJRPHU\ f

PAGE 10

6SOLW3ORW 'HVLJQV $ VSOLWSORW GHVLJQ RIWHQ UHIHUV WR D GHVLJQ ZLWK TXDOLWDWLYH IDFWRUV EXW FDQ HDVLO\ KDQGOH TXDQWLWDWLYH IDFWRUV $OVR D VSOLWSORW GHVLJQ XVXDOO\ KDV UHSOLFDWLRQ +RZHYHU LQ WKH OLWHUDWXUH LW KDV EHHQ FRPPRQ SUDFWLFH WR UHIHU WR DQ\ GHVLJQ WKDW XVHV RQH OHYHO RI UHVWULFWHG UDQGRPL]DWLRQ UHJDUGOHVV RI UHSOLFDWLRQ DV D fVSOLWSORWf GHVLJQ 7KHUHIRUH LQ WKLV GLVVHUWDWLRQ ZH ZLOO XVH WKH WHUP VSOLWSORW GHVLJQ WKURXJKRXW :KHQ SHUIRUPLQJ PXOWLIDFWRU H[SHULPHQWV WKHUH PD\ EH VLWXDWLRQV ZKHUH FRPn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f VKRZV WKDW WKHUH DUH DSSOLFDWLRQV IRU VSOLWSORW H[SHULPHQWV LQ LQGXVWULDO VHWn WLQJV $ SDSHU PDQXIDFWXUHU LV LQWHUHVWHG LQ VWXG\LQJ WKH WHQVLOH VWUHQJWK RI SDSHU EDVHG RQ WKUHH GLIIHUHQW SXOS SUHSDUDWLRQ PHWKRGV DQG IRXU FRRNLQJ WHPSHUDWXUHV IRU WKH

PAGE 11

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

PAGE 12

7DEOH 'DWD IRU 7HQVLOH 6WUHQJWK RI 3DSHU IURP 0RQWJRPHU\ ff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f DQG WKH UDQGRPL]HG FRPSOHWH EORFN 5&%f GHn VLJQ $V LQ WKH H[DPSOH IURP 0RQWJRPHU\ f VXSSRVH WKH VSOLWSORW H[SHULPHQW LV SHUIRUPHG XVLQJ D 5&% GHVLJQ /HW GHQRWH WKH REVHUYDWLRQ IRU VXESORW WUHDWn PHQW N UHFHLYLQJ ZKROH SORW WUHDWPHQW L LQ EORFN M .HPSWKRUQH f XVHV DV KLV PRGHO 9LMN L 7 OM 7 LM f\N 7 7n\nMLN 7 WLMN IRU L W  N V

PAGE 13

ZKHUH f W LV WKH QXPEHU RI OHYHOV IRU WKH ZKROH SORW WUHDWPHQW f E LV WKH QXPEHU RI EORFNV RU UHSOLFDWHV RI WKH EDVLF ZKROH SORW H[SHULPHQW f V LV WKH QXPEHU RI OHYHOV IRU WKH VXESORW WUHDWPHQW f U LV WKH RYHUDOO PHDQ f LV WKH HIIHFW RI WKH LWK ZKROH SORW WUHDWPHQW f M LV WKH HIIHFW RI WKH MWK EORFN f 6LM LV WKH ZKROH SORW HUURU WHUP f N LV WKH HIIHFW RI WKH NWK VXESORW WUHDWPHQW f WfLN LV WKH ZKROH SORW WUHDWPHQW E\ VXESORW WUHDWPHQW LQWHUDFWLRQ DQG f fLMN LV WKH VXESORW HUURU +H XVHV UDQGRPL]DWLRQ WKHRU\ WR GHULYH WKH H[SHFWHG PHDQ VTXDUHV VXPPDUL]HG LQ 7DEOH ,Q WKLV WDEOH HUI LV WKH H[SHULPHQWDO HUURU YDULDQFH IRU WKH ZKROH SORW WUHDWn PHQWV DQG D LV WKH H[SHULPHQWDO HUURU YDULDQFH IRU WKH VXESORW WUHDWPHQWV 0DQ\ DQDO\VWV DVVXPH WKDW WKH EORFNV DUH UDQGRP DQG XVH DQ XQUHVWULFWHG PL[HG PRGHO WR GHULYH WKH DSSURSULDWH PHDQ VTXDUHV 7KH PRVW FRPPRQ PRGHO IRU WKLV DSSURDFK LV

PAGE 14

L 7L >M >UfLM IH 7n\fLN HMIF f r W M OE N OV ZKHUH f Q LV WKH RYHUDOO PHDQ f 7L LV WKH HIIHFW RI ZKROH SORW WUHDWPHQW L f M LV WKH HIIHFW RI EORFN M f UfLM LV WKH EORFN [ ZKROH SORW WUHDWPHQW LQWHUDFWLRQ f r LV WKH HIIHFW RI VXESORW WUHDWPHQW N f &K2LN LV WKH ZKROH SORW WUHDWPHQW [ VXESORW WUHDWPHQW LQWHUDFWLRQ DQG f fLMN LV WKH VXESORW HUURU 7KH UfLM WHUP ZLOO EH WKH ZKROH SORW HUURU WHUP IRU WKH FDVH RI DQ 5&% GHVLJQ XQGHU WKH XVXDO DVVXPSWLRQ RI QR EORFN [ ZKROH SORW WUHDWPHQW LQWHUDFWLRQ 7KH DQDO\VLV RI YDULDQFH WDEOH DVVRFLDWHG ZLWK WKH PRGHO LQ (TXDWLRQ f DVVXPLQJ ZKROH SORW WUHDWPHQWV DQG VXESORW WUHDWPHQWV DUH IL[HG DQG EORFNV DUH UDQGRP LV JLYHQ LQ 7DEOH ,I WKH EORFN E\ ZKROH SORW LQWHUDFWLRQ LV FDOOHG WKH ZKROH SORW HUURU WKHQ 7DEOHV DQG VXJJHVW WKH VDPH EDVLF WHVWLQJ SURFHGXUHV 7KH IROORZLQJ DGGLWLRQDO FRQVWUDLQWV DQG DVVXPSWLRQV DUH QHHGHG IRU K\SRWKHVLV WHVWLQJ A r IF L N

PAGE 15

3L a 1 R RIf LM a $7 R RIf HLMN a $n R Df DQG ZKHUH 6LM DQG HOMN DUH LQGHSHQGHQW 0RQWJRPHU\ f XVHV D UHVWULFWHG PL[HG PRGHO DV WKH EDVLV IRU KLV DQDO\VLV RI WKH IROORZLQJ IRUP 9LMNK IL 7L3M IF 7MfLN 3\fMN U3\fLMN HLMNK ZKHUH f K U LV WKH QXPEHU RI UHSOLFDWHV f UfLM LV WKH UDQGRP EORFN E\ ZKROH SORW WUHDWPHQW LQWHUDFWLRQ f LV WKH ZKROH SORW WUHDWPHQW [ VXESORW WUHDWPHQW LQWHUDFWLRQ f ^3LfMN LV WKH UDQGRP EORFN E\ VXESORW WUHDWPHQW LQWHUDFWLRQ f ^U\fLMN LV WKH UDQGRP EORFN E\ ZKROH SORW WUHDWPHQW E\ VXESORW WUHDWPHQW LQWHUDFWLRQ 8QGHU WKLV UHVWULFWHG PL[HG PRGHO WKH UDQGRP LQWHUDFWLRQV LQYROYLQJ D IL[HG IDFWRU DUH DVVXPHG VXEMHFW WR WKH FRQVWUDLQW WKDW WKH VXP RI WKDW LQWHUDFWLRQfV HIIHFWV RYHU WKH OHYHOV RI WKH IL[HG IDFWRU LV ]HUR 7DEOH JLYHV WKH UHVXOWLQJ H[SHFWHG PHDQ VTXDUHV ZKLFK VXJJHVWV WKDW WKHUH DUH WKUHH GLVWLQFW HUURU WHUPV 7KH EORFN E\ ZKROH SORW E\ VXESORW LQWHUDFWLRQ LV XVHG WR WHVW WKH ZKROH SORW E\ VXESORW LQWHUDFWLRQ WKH EORFN E\ VXESORW LQWHUDFWLRQ LV XVHG WR WHVW WKH PDLQ HIIHFW RI WKH VXESORW WUHDWPHQW DQG WKH EORFN E\ ZKROH SORW LQWHUDFWLRQ LV XVHG WR WHVW WKH PDLQ HIIHFW RI WKH ZKROH SORW

PAGE 16

7DEOH ([SHFWHG 0HDQ 6TXDUHV 7DEOH 8QGHU 5DQGRPL]DWLRQ 7KHRU\ 6RXUFH GI ([SHFWHG 0HDQ 6TXDUH :KROH 3ORW 7UHDWPHQW W n L EV = r n Wf§ nL %ORFNV :KROH 3ORW (UURU E D 6&7J 6XESORW 7UHDWPHQW V f§ N :KROH 3ORW [ 6XESORW W OfV f U PA]LL(/,(/7fIL 6XESORW (UURU WEOfVOf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f f f UW YHUVXV DW OHDVW RQH QRW HTXDO LV WHVWHG XVLQJ WKH %ORFN [ :KROH 3ORW 7UHDWPHQW LQWHUDFWLRQ DV

PAGE 17

7DEOH ([SHFWHG 0HDQ 6TXDUHV 7DEOH 8QGHU WKH 0RVW &RPPRQ 8QUHVWULFWHG 0L[HG 0RGHO 6RXUFH GI ([SHFWHG 0HDQ 6TXDUH :KROH 3ORW 7UHDWPHQW W r   %ORFNV E D VD VWD %ORFN [ :KROH 3ORW 7UHDWPHQW WP f D VR 6XESORW 7UHDWPHQW V f§ r e‹ IF L :KROH 3ORW [ 6XESORW fm f U rBbBf < ^ULfN Y Y f Lf§? Nf§? (UURU WE fm f D 1RWH :KROH 3ORW DQG 6XESORW 7UHDWPHQWV DUH DVVXPHG IL[HG ZKLOH %ORFNV DUH DVVXPHG WR EH UDQGRP 7DEOH ([SHFWHG 0HDQ 6TXDUHV 7DEOH 8QGHU WKH 5HVWULFWHG 0L[HG 0RGHO 6RXUFH GI ([SHFWHG 0HDQ 6TXDUH :KROH 3ORW 7UHDWPHQW W f§ D VD e (f L 7L %ORFNV %ORFN [ :KROH R Wf§ rf§r D VD VWRS D VDO 6XESORW 7UHDWPHQW V f§ Rn WRe t (8 LO %ORFN [ 6XE fr f D WD :KROH 3ORW [ 6XESORW  fr f D D7 e L (8LQ26} %ORFN [ :KROH [ 6XE W f OfD f J

PAGE 18

WKH HUURU WHUP 7KH K\SRWKHVLV RI QR VXESORW WUHDWPHQW HIIHFW + L f f f 6f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f§ GHJUHHV RI IUHHGRP GIf IRU WKH ZKROH SORW WUHDWPHQWV FDQ EH SDUWLWLRQHG LQWR VLQJOH GI FRQWUDVWV =? ] DQG =?= /LNHZLVH WKH V f§ f§ GI IRU WKH VXESORW WUHDWPHQWV FDQ EH SDUWLWLRQHG LQWR D VLQJOH GI FRQWUDVWV DT [ DQG ;L; $OVR WKH W f§ OfV f§ f GI IRU WKH ZKROH SORW [ VXESORW WUHDWPHQW LQWHUDFWLRQ FDQ EH EURNHQ GRZQ LQWR VLQJOH GI HIIHFWV LQYROYLQJ ]` ] ;L DQG [ VHH 7DEOH f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

PAGE 19

7DEOH $QDO\VLV RI 9DULDQFH 7DEOH IRU D 6SOLW3ORW ([SHULPHQW 5XQ 8VLQJ D 5&% 'HVLJQ :LWK )DFWRULDO 6WUXFWXUH DQG WKH 0RVW &RPPRQ 8QUHVWULFWHG 0RGHO 6RXUFH GI :KROH 3ORW 7UHDWPHQW Wf§ 9 I OI %ORFNV E %ORFN [ :KROH 3ORW 7UHDWPHQW r;tf 6XESORW 7UHDWPHQW V ;L G [ WI [?[ G :KROH 3ORW [ 6XESORW WOfVOf ]L[L G =?; IW = Q ][ IW =?;L[ G =;; G =?=;? G =?=; G =?=;?; G (UURU L OfV f I 7KHVH WHUPV DUH WHVWHG XVLQJ WKH %ORFN [ :KROH 3ORW 7UHDWPHQW LQWHUDFWLRQ IW 7KHVH WHUPV DUH WHVWHG XVLQJ (UURU

PAGE 20

7KH FRQFHSW RI WKH VSOLWSORW GHVLJQ FDQ EH H[WHQGHG LI IXUWKHU UDQGRPL]DWLRQ UHVWULFWLRQV H[LVW )RU H[DPSOH VXSSRVH WKHUH DUH WZR OHYHOV RI UDQGRPL]DWLRQ UHVWULFn WLRQV ZLWKLQ D EORFN LQ ZKLFK FDVH ZH PLJKW KDYH D VSOLWVSOLWSORW GHVLJQ )RU D PRUH GHWDLOHG GLVFXVVLRQ RI VSOLWSORW GHVLJQV DQG WKHLU H[WHQVLRQV VHH
PAGE 21

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f WKH $LU )ORZ 6SHHG $)f DQG WKH 5HVLGHQFH 7LPH ,Q JHQHUDO WKH FRQGLWLRQV LQ HDFK ]RQH ZLOO EH GLIIHUHQW DV HDFK ]RQH LV XVHG WR LPSDUW GLIIHUHQW FKDUDFWHULVWLFV WR WKH SURGXFW $OO RI WKHVH YDULDEOHV ZLOO LPSDFW WKH TXDOLW\ RI WKH ILQLVKHG SURGXFW ([SHULPHQWLQJ ZLWK WKLV W\SH RI RYHQ UHTXLUHV D UHVWULFWHG UDQGRPL]DWLRQ
PAGE 22

SURGXFW WKDW LV EDNHG LV HYDOXDWHG LQ D YDULHW\ RI ZD\V LQFOXGLQJ VHQVRU\ FKDUDFWHULn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n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

PAGE 23

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

PAGE 24

&+$37(5 /,7(5$785( 5(9,(: 7KH VSOLWSORW HUURU VWUXFWXUH KDV EHHQ XQGHUXWLOL]HG LQ 560 0RVW 560 H[n SHULPHQWV DVVXPH D FRPSOHWHO\ UDQGRPL]HG HUURU VWUXFWXUH /HWVLQJHU 0\HUV DQG /HQWQHU SJ f SRLQW RXW f8QIRUWXQDWHO\ ZKLOH WKLV FRPSOHWHO\ UDQGRPL]HG DVVXPSWLRQ VLPSOLILHV DQDO\VLV DQG UHVHDUFK LQGHSHQGHQW UHVHWWLQJ RI YDULDEOH OHYHOV IRU HDFK GHVLJQ UXQ PD\ QRW EH IHDVLEOH GXH QRW RQO\ WR HTXLSPHQW DQG UHVRXUFH FRQn VWUDLQWV EXW DOVR EXGJHW UHVWULFWLRQVf 7KLV FKDSWHU IRFXVHV RQ WKH OLWHUDWXUH LQYROYLQJ UHVWULFWHG UDQGRPL]DWLRQ ZLWKLQ 560 6SOLW3ORW &RQIRXQGLQJ :KHQ WKH ZKROH SORW DQGRU VXESORW WUHDWPHQWV DUH RI D IDFWRULDO QDWXUH LW LV SRVVLEOH WR UHGXFH WKH QXPEHU RI ZKROH SORWV DQGRU VXESORWV QHHGHG WKURXJK IUDFn WLRQDWLQJ 7KLV LV LPSRUWDQW LQ LQGXVWULDO H[SHULPHQWV ZKHUH FRQVWUDLQWV OLPLW WKH VL]H RI WKH H[SHULPHQW %DUWOHWW f VXJJHVWHG WKH SRVVLELOLW\ RI FRQIRXQGLQJ KLJKHU RUGHU VXESORW LQWHUDFWLRQV WR UHGXFH WKH QXPEHU RI VXESORWV QHHGHG ZLWKLQ HDFK ZKROH SORW /DWHU VSOLWSORW FRQIRXQGLQJ ZDV VWXGLHG E\ $GGHOPDQ f +H SURYLGHG D WDEOH FRQWDLQLQJ IDFWRULDO DQG IUDFWLRQDOIDFWRULDO DUUDQJHPHQWV WKDW LQYROYH VSOLW SORW FRQIRXQGLQJ +RZHYHU KH GLG QRW FRQVLGHU FRQIRXQGLQJ ZLWKLQ WKH ZKROH SORWV /HWVLQJHU 0\HUV DQG /HQWQHU f GLVFXVV WKH SRVVLELOLW\ RI VSOLWSORW FRQIRXQGLQJ

PAGE 25

ZLWK WKH XVH RI WKHLU QRQFURVVHG ELUDQGRPL]DWLRQ GHVLJQV %R[ DQG -RQHV f LOOXVWUDWH VSOLWSORW FRQIRXQGLQJ XVLQJ D FDNH PL[ H[DPSOH ,Q VRPH H[SHULPHQWV WKHUH DUH FRQVWUDLQWV RQ WKH QXPEHU RI VXESORWV ZLWKLQ HDFK ZKROH SORW :KHQ WKH ZKROH SORWV DUH DUUDQJHG LQ D &5' 5RELQVRQ f GLVFXVVHG VLWXDWLRQV ZKHUH WKH QXPEHU RI VXESORWV SHU ZKROH SORW LV OHVV WKDQ WKH QXPEHU RI VXESORW WUHDWPHQWV 7KH ZKROH SORWV DUH WUHDWHG DV EORFNV DQG WKHQ D EDODQFHG LQFRPSOHWH EORFN %,%f GHVLJQ LV XVHG WR DOORFDWH WKH VXESORW WUHDWPHQWV WR WKH ZKROH SORWV ,I WKH ZKROH SORWV DUH DUUDQJHG LQ DQ 5&% GHVLJQ WKH VDPH SURFHGXUH FDQ EH DSSOLHG ,I WKH QXPEHU RI ZKROH SORWV SHU EORFN LV OHVV WKDQ WKH QXPEHU RI ZKROH SORW WUHDWPHQWV WKHQ DQ LQFRPSOHWH EORFN GHVLJQ FDQ EH XVHG WKHUH DV ZHOO 5RELQVRQ f JDYH GHWDLOV RQ WKH FDVH ZKHQ ERWK ZKROH SORW DQG VXESORW WUHDWPHQWV DUH DUUDQJHG LQ LQFRPSOHWH EORFN GHVLJQV (VVHQWLDOO\ WKH SURFHGXUH DPRXQWV WR DUUDQJLQJ WKH ZKROH SORW WUHDWPHQWV LQ EORFNV XVLQJ D %,% GHVLJQ DQG WKHQ FRQVLGHULQJ WKH ZKROH SORWV DV EORFNV DQG DUUDQJLQJ WKH VXESORW WUHDWPHQWV LQ DQRWKHU %,% GHVLJQ 5RELQVRQ f SURYLGHG IRUPXODV IRU WKH HVWLPDWHV RI WKH PDLQ HIIHFWV DQG LQWHUDFWLRQV IRU WKUHH FDVHV ZLWKLQ ZKROH SORW EHWZHHQ ZKROH SORW ZLWKLQ EORFN DQG EHWZHHQ EORFNV )RUPXODV DUH DOVR JLYHQ IRU WKH YDULDQFH RI WKH GLIIHUHQFHV RI WKHVH HVWLPDWHV IRU HDFK FDVH +XDQJ &KHQ DQG 9RHONHO f DOVR LQYHVWLJDWH IUDFWLRQDWLQJ WZROHYHO VSOLWSORW GHVLJQV DW ERWK WKH ZKROH SORWV DQG WKH VXESORWV 7KH\ FRQVLGHU AQLQABAIFOIFA VSOLWSORW GHVLJQV ZKLFK DUH DVVRFLDWHG ZLWK D VXEVHW RI WKH QaN IUDFWLRQDO IDFWRULDO GHVLJQV ZKHUH Q Q? UF DQG N N 7KH FULWHULRQ XVHG WR VHOHFW WKH RSWLPDO GHVLJQ LV WKDW RI PLQLPXPDEHUUDWLRQ ZKLFK LV WKH GHVLJQ WKDW KDV VPDOOHVW QXPEHU RI

PAGE 26

ZRUGV LQ WKH GHILQLQJ FRQWUDVW ZLWK WKH IHZHVW OHWWHUV 7ZR PHWKRGV DUH SUHVHQWHG IRU FRQVWUXFWLQJ PLQLPXPDEHUUDWLRQ VSOLWSORW GHVLJQV 7KH ILUVW PHWKRG GHFRPSRVHV WKH QaN GHVLJQ LQWR WKH QfBIFOIFf VSOLWSORW GHVLJQ 7KLV PHWKRG LV XVHG WR GHULYH H[WHQVLYH WKRXJK LQFRPSOHWH WDEOHV RI WKH GHVLJQV 7KH VHFRQG DQG PRUH FRPSOLFDWHG PHWKRG ZKLFK LQYROYHV OLQHDU LQWHJHU SURJUDPPLQJ LV XVHG ZKHQ WKH ILUVW PHWKRG IDLOV 0LQLPXPDEHUUDWLRQ WZROHYHO VSOLWSORW GHVLJQV DUH DOVR GLVFXVVHG LQ %LQJKDP DQG 6LWWHU f $ FRPELQHG VHDUFK DQG VHTXHQWLDO DOJRULWKP LV SUHVHQWHG IRU FRQn VWUXFWLQJ DOO QRQLVRPRUSKLF PLQLPXPDEHUUDWLRQ VSOLWSORW GHVLJQV ZKLFK LQFOXGH WKH WDEOHV RI +XDQJ &KHQ DQG 9RHONHO f %LQJKDP DQG 6LWWHU f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n WRUV WKDW DUH GLIILFXOW WR FRQWURO DQG FDQ FDXVH YDULDWLRQ LQ WKH XVH RU SHUIRUPDQFH RI SURGXFWV 7KH H[SHULPHQWDO GHVLJQ RU fFURVVHG DUUD\f FRQVLVWV RI FURVVLQJ HDFK

PAGE 27

H[SHULPHQWDO GHVLJQ VHWWLQJ RI WKH LQQHU DUUD\ ZLWK HDFK H[SHULPHQWDO GHVLJQ VHWWLQJ RI WKH RXWHU DUUD\ 8QOHVV WKH QXPEHU RI IDFWRUV LQ WKHVH DUUD\V LV VPDOO 7DJXFKLfV GHVLJQV EHFRPH ODUJH DQG H[SHQVLYH $Q DOWHUQDWLYH WR 7DJXFKLfV FURVVHG DUUD\ LV WKH fFRPELQHGf DUUD\ 7KH FRPELQHG DUUD\ XWLOL]HV D VLQJOH H[SHULPHQWDO GHVLJQ LQ ERWK WKH GHVLJQ DQG HQYLURQPHQWDO IDFn WRUV 7KHUHIRUH WKH UHVSRQVH LV PRGHOHG GLUHFWO\ DV D IXQFWLRQ RI WKH GHVLJQ IDFWRUV DQG WKH HQYLURQPHQWDO IDFWRUV XVLQJ D VLQJOH OLQHDU PRGHO 0RUH GHWDLOV RQ WKH FRPn ELQHG DUUD\ FDQ EH IRXQG LQ :HOFK .DQJ DQG 6DFNV f 6KRHPDNHU 7VXL DQG :X f DQG 2f'RQQHOO DQG 9LQLQJ f %LVJDDUG f GLVFXVVHV VSOLWSORW GHVLJQV LQ DVVRFLDWLRQ ZLWK LQQHU DQG RXWHU DUUD\ GHVLJQV +H IRFXVHV RQ VFUHHQLQJ H[SHULPHQWV WKDW XVH UHVWULFWHG UDQGRPL]DWLRQ 7KH SDSHU JLYHV D QLFH RYHUYLHZ RI GHILQLQJ UHODWLRQV DQG FRQIRXQGLQJ VWUXFWXUHV IRU WKH NaS [ TaU VSOLWSORW GHVLJQV ,Q DGGLWLRQ WR VSOLWSORW FRQIRXQGLQJ %LVJDDUG f SRLQWV RXW WKDW WKH VDPH IUDFWLRQ RI WKH VXESORW IDFWRUV FDQ EH UXQ LQ HDFK ZKROH SORW 7KH DSSURSULDWH VWDQGDUG HUURUV IRU WHVWLQJ HIIHFWV ZKHQ XVLQJ VSOLWSORW FRQIRXQGLQJ DUH DOVR JLYHQ %R[ DQG -RQHV f LQYHVWLJDWH WKH XVH RI VSOLWSORW GHVLJQV DV DQ DOWHUQDWLYH WR WKH FURVVHG DUUD\ 7KH\ FRQVLGHU WKUHH H[SHULPHQWDO DUUDQJHPHQWV ZKHUH WKH UREXVW SDUDPHWHU GHVLJQ LV VHW XS DV D VSOLWSORW GHVLJQ DUUDQJHPHQW Df f§ WKH ZKROH SORWV FRQWDLQ WKH HQYLURQPHQWDO IDFWRUV DQG WKH VXESORWV FRQWDLQ WKH GHVLJQ IDFWRUV

PAGE 28

DUUDQJHPHQW Ef f§ WKH ZKROH SORWV FRQWDLQ WKH GHVLJQ IDFWRUV DQG WKH VXESORWV FRQWDLQ WKH HQYLURQPHQWDO IDFWRUV DUUDQJHPHQW Ff f§ WKH VXESORW IDFWRUV DUH DVVLJQHG LQ fVWULSVf DFURVV WKH ZKROH SORW IDFWRUV FRPPRQO\ FDOOHG D VWULSEORFN H[SHULPHQWf 7KHVH WKUHH DUUDQJHPHQWV DUH LOOXVWUDWHG WKURXJK DQ H[DPSOH VHHNLQJ WKH EHVW UHFLSH IRU D FDNH PL[ 7KUHH GHVLJQ IDFWRUV KDYH EHHQ LGHQWLILHG DV DIIHFWLQJ WDVWH 7KH\ DUH IORXU VKRUWHQLQJ DQG HJJ SRZGHU DQG DUH VWXGLHG XVLQJ D IDFWRULDO GHVLJQ 7KH FRQVXPHU PD\ KDYH DQ RYHQ LQ ZKLFK WKH WHPSHUDWXUH LV ELDVHG XS RU GRZQ $OVR WKH FRQVXPHU PD\ RYHUFRRN RU XQGHUFRRN WKH FDNH 7KHUHIRUH WKH UHFLSH LV WR EH UREXVW WR WZR HQYLURQPHQWDO IDFWRUV RYHQ WHPSHUDWXUH DQG EDNLQJ WLPH ZKRVH OHYHOV DUH FRPELQHG XVLQJ D IDFWRULDO GHVLJQ $UUDQJHPHQW Df 8QGHU WKLV DUUDQJHPHQW WKH ZKROH SORWV FRQWDLQ WKH HQYLURQPHQWDO IDFWRUV DQG WKH VXESORWV FRQWDLQ WKH GHVLJQ IDFWRUV 6XSSRVH WKHUH DUH P OHYHOV RI WKH HQYLn URQPHQWDO IDFWRUV (L(c (M (P DSSOLHG WR WKH ZKROH SORWV Q OHYHOV RI WKH GHVLJQ IDFWRUV 7IL 'O ' 'Q DSSOLHG WR WKH VXESORWV DQG O UHSOLFDWHV 7? U UX U ZLWK WKH ZKROH SORWV LQ O UDQGRPL]HG EORFNV )RU WKH FDNH PL[ H[DPSOH P Q DQG $UUDQJHPHQW Df UHTXLUHV P [ Q [ O VXESORWV DQG P [ O ZKROH SORWV 7KXV IRU WKH FDNH PL[ H[DPSOH [ [ FDNH PL[ EDWFKHV DUH UHTXLUHG EXW RQO\ [ RSHUDWLRQV RI WKH RYHQ DUH QHFHVVDU\ %\ FRPSDULVRQ D FRPSOHWHO\ UDQGRPL]HG FURVVSURGXFW DUUD\ ZRXOG UHTXLUH FDNH PL[

PAGE 29

EDWFKHV DQG RSHUDWLRQV RI WKH RYHQ 7KXV WKH VSOLWSORW DUUDQJHPHQW KDV VDYHG WLPH E\ UHGXFLQJ WKH QXPEHU RI RSHUDWLRQV RI WKH RYHQ 7KH PRGHO IRU DUUDQJHPHQW Df LV 9LMN f§ + IF DM 9MN L 4fLM HLMN ZKHUH \LMN LV WKH UHVSRQVH RI WKH NWK UHSOLFDWH RI WKH ]WK OHYHO RI IDFWRU DQG WKH MWK OHYHO RI IDFWRU ( LV WKH RYHUDOO PHDQ r LV WKH UDQGRP HIIHFW RI WKH $WK UHSOLFDWH ZLWK r a 1 &7\f DM LV WKH IL[HG HIIHFW RI WKH MWK OHYHO RI IDFWRU ( LV WKH IL[HG HIIHFW RI WKH WK OHYHO RI IDFWRU DfA LV WKH LQWHUDFWLRQ HIIHFW RI WKH ]WK OHYHO RI DQG WKH MWK OHYHO RI ( UcMN a $A LV WKH ZKROH SORW HUURU HWMN a 1 DLV WKH VXESORW HUURU DQG fMN DQG DUH LQGHSHQGHQW $UUDQJHPHQW Ef :LWK WKLV DUUDQJHPHQW WKH ZKROH SORWV FRQWDLQ WKH GHVLJQ IDFWRUV ZKLOH WKH VXEn SORWV FRQWDLQ WKH HQYLURQPHQWDO IDFWRUV $UUDQJHPHQW Ef UHTXLUHV RQO\ [ FDNH PL[ EDWFKHV EXW UHTXLUHV [ [ RSHUDWLRQV RI WKH RYHQ $JDLQ D FRPSOHWHO\ UDQGRPL]HG FURVVSURGXFW DUUD\ ZRXOG XVH FDNH PL[ EDWFKHV DQG RSHUDWLRQV RI WKH RYHQ +HUH WKH VDYLQJV RI WKH VSOLWSORW GHVLJQ LV QRW DV JUHDW VLQFH RQO\ WKH QXPEHU RI FDNH PL[ EDWFKHV LV UHGXFHG 7KLV LV QRW DQ LGHDO VLWXDWLRQ IRU LQGXVn WULDO H[SHULPHQWV )LUVW RI DOO WKH GHVLJQ IDFWRUV DUH RI JUHDWHU LQWHUHVW 7KHUHIRUH DSSO\LQJ WKH GHVLJQ IDFWRUV WR WKH ZKROH SORWV UHVXOWV LQ D ORVV RI SUHFLVLRQ IRU WKH GHVLJQ IDFWRUV +HQFH LW LV SRVVLEOH WR KDYH ODUJH GLIIHUHQFHV EHWZHHQ WKH OHYHOV RI WKH GHVLJQ IDFWRUV WKDW DUH LQVLJQLILFDQW ZKHQ WHVWHG $OVR IURP DQ HFRQRPLF SRLQW

PAGE 30

RI YLHZ DUUDQJHPHQW Ef LV FRVWO\ ,W UHTXLUHV DQ LQHIILFLHQW XVH RI WKH HQYLURQPHQn WDO IDFWRUV ZKLFK LQ LQGXVWULDO H[SHULPHQWV DUH W\SLFDOO\ WKH GLIILFXOW RU H[SHQVLYH WR FKDQJH IDFWRUV 7KH PRGHO IRU DUUDQJHPHQW Ef LV 9LMN Ir , 9LN DM fLMN ZKHUH LV WKH UHVSRQVH RI WKH NWK UHSOLFDWH RI WKH LWK OHYHO RI IDFWRU DQG WKH MWK OHYHO RI IDFWRU ( \ LV WKH RYHUDOO PHDQ r LV WKH UDQGRP HIIHFW RI WKH NWK UHSOLFDWH ZLWK r a 1 DAf DM LV WKH IL[HG HIIHFW RI WKH MWK OHYHO RI IDFWRU ( 6L LV WKH IL[HG HIIHFW RI WKH ]WK OHYHO RI IDFWRU DfA LV WKH LQWHUDFWLRQ HIIHFW RI WKH LWLO OHYHO RI DQG WKH MWK OHYHO RI ( GLN a 9 WAf LV WKH ZKROH SORW HUURU HAN a L9 f LV WKH VXESORW HUURU DQG WN DQG HAN DUH LQGHSHQGHQW $UUDQJHPHQW Ff 1RZ FRQVLGHU WKH DUUDQJHPHQW ZKHUH WKH VXESORW WUHDWPHQWV DUH UDQGRPO\ DVn VLJQHG LQ VWULSV DFURVV HDFK EORFN RI ZKROH SORW WUHDWPHQWV VHH 7DEOH f )RU WKH FDNH PL[ H[DPSOH VXSSRVH HDFK RI WKH Q f§ EDWFKHV RI FDNH PL[ LV VXEGLYLGHG LQWR P VXEJURXSV 2QH VXEJURXS IURP HDFK EDWFK LV WKHQ VHOHFWHG DQG WKHVH HLJKW DUH EDNHG LQ WKH VDPH RYHQ DW WKH DSSURSULDWH WHPSHUDWXUH IRU WKH DSSURSULDWH WLPH 7KLV DUUDQJHPHQW UHTXLUHV RQO\ FDNH PL[ EDWFKHV DQG RQO\ RSHUDWLRQV RI WKH RYHQ 7KHUHIRUH WKH VWULSEORFN H[SHULPHQW LV HDVLHU WR UXQ WKDQ WKH FRPSOHWHO\ UDQGRPL]HG FURVVSURGXFW GHVLJQ DV ZHOO DV ERWK DUUDQJHPHQWV Df DQG Ef

PAGE 31

7DEOH 6WULS%ORFN $UUDQJHPHQW %R[ DQG -RQHV ff %ORFN %ORFN %ORFN m D m 2L 2O m EL L! EL 7KH PRGHO IRU WKH VWULSEORFN DUUDQJHPHQW LV 9LMN f§ 9 ,N 2WM fMN 2LN DAfLM HLMN ZKHUH \OMN LV WKH UHVSRQVH RI WKH NWK UHSOLFDWH RI WKH WK OHYHO RI IDFWRU DQG WKH MWK OHYHO RI IDFWRU ( cL LV WKH RYHUDOO PHDQ r LV WKH UDQGRP HIIHFW RI WKH NWK UHSOLFDWH ZLWK N a 1 Df FWM LV WKH IL[HG HIIHFW RI WKH MWK OHYHO RI IDFWRU ( O LV WKH IL[HG HIIHFW RI WKH WK OHYHO RI IDFWRU D6fA LV WKH LQWHUDFWLRQ HIIHFW RI WKH WK OHYHO RI DQG WKH MWK OHYHO RI ( ,Q DUUDQJHPHQW Ff UMMN a $UD_f LN a 1D'f HLMN a 1 Df DQG UcMNO LN DQG HLMN DUH LQGHSHQGHQW $129$ WDEOHV IRU DOO WKUHH DUUDQJHPHQWV DUH JLYHQ LQ %R[ DQG -RQHV f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

PAGE 32

,W LV RI JUHDW LQWHUHVW WR WKH UHVHDUFKHU WR OHDUQ KRZ DQG ZKLFK HQYLURQPHQWDO IDFWRUV LQIOXHQFH WKH GHVLJQ YDULDEOHV 7KLV LQIRUPDWLRQ LV FRQWDLQHG LQ WKH VXESORW [ ZKROH SORW LQWHUDFWLRQV +RZHYHU 7DJXFKLfV DQDO\VLV LV FRPPRQO\ FRQGXFWHG LQ WHUPV RI D SHUIRUPDQFH VWDWLVWLF VXFK DV WKH VLJQDO WR QRLVH UDWLR 615f 7KH 615 LV FDOFXODWHG IRU HDFK SRLQW LQ WKH LQQHU DUUD\ XVLQJ GDWD REWDLQHG IURP WKH RXWHU DUUD\ DERXW WKDW SRLQW 7KHUHIRUH 7DJXFKL LJQRUHV DQ\ LQIRUPDWLRQ FRQWDLQHG LQ WKH LQWHUDFWLRQV RI WKH GHVLJQ DQG HQYLURQPHQWDO IDFWRUV 7KLV LV JHQHUDOO\ FRQVLGHUHG WR EH D VHULRXV GUDZEDFN WR WKH 7DJXFKL DQDO\VLV 3KDGNH f SUHVHQWHG DQ H[DPSOH LQYROYLQJ D SRO\VLOLFRQ GHSRVLWLRQ SURFHVV ZKLFK KH DQDO\]HG XVLQJ 7DJXFKLfV 615fV 3RO\VLOLFRQ ILOP LV W\SLFDOO\ GHSRVLWHG RQ WRS RI WKH R[LGH OD\HU RI WKH ZDIHUV XVLQJ D KRWZDOO UHGXFHG SUHVVXUH UHDFWRU 7KH UHDFWDQW JDVHV DUH LQWURGXFHG LQWR RQH HQG RI D WKUHH]RQH IXUQDFH WXEH DQG DUH SXPSHG LQWR WKH RWKHU HQG 7KH ZDIHUV HQWHU WKH ORZSUHVVXUH FKHPLFDO YDSRU GHSRn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

PAGE 33

HDFK GHVLJQ IDFWRU RQ WKH UHVSRQVHV $ PRUH GHWDLOHG GLVFXVVLRQ RI WKH VHOHFWLRQ RI IDFWRUV GHVLJQ DQG DQDO\VLV RI WKH 615fV LV FRQWDLQHG LQ &KDSWHU RI 3KDGNH f 7KH DFWXDO VWUXFWXUH RI WKLV H[SHULPHQW ZDV D VSOLWVSOLWSORW GHVLJQ EHFDXVH WKHUH DUH WKUHH VL]HV RI H[SHULPHQWDO XQLWV ZLWK GLIIHUHQW VRXUFHV RI YDULDWLRQ 7KH GHVLJQ IDFWRUV DUH DSSOLHG WR WKH WXEH UXQWRUXQ YDULDELOLW\f WKH ORFDWLRQ LQ WKH WXEH DIIHFWV WKH ZDIHU ZDIHUWRZDIHU YDULDELOLW\f ZKHUHDV ORFDWLRQ LQ WKH ZDIHU DIIHFWV WKH GLH GLHWRGLH YDULDELOLW\f 7KHUHIRUH XVLQJ 7DJXFKLfV 615fV WR DQDO\]H WKLV H[SHULPHQW ZLOO UHVXOW LQ D FRPSOHWH ORVV RI LQIRUPDWLRQ LQ WKH GHVLJQ [ QRLVH IDFWRU LQWHUDFWLRQV &DQWHOO DQG 5DPLUH] f UHDQDO\]HG WKH GDWD DV LI LW ZHUH D VSOLWVSOLWSORW GHVLJQ 7KH\ SRROHG KLJKHU RUGHU LQWHUDFWLRQV WR JHW WKH QHFHVVDU\ HUURU WHUPV LQ RUGHU WR SHUIRUP K\SRWKHVLV WHVWV RQ WKH GHVLJQ IDFWRUV DQG WKH GHVLJQ [ QRLVH IDFWRU LQWHUn DFWLRQV ,QWHUDFWLRQ SORWV ZHUH XVHG WR GHWHUPLQH WKH OHYHO RI WKH GHVLJQ IDFWRU WKDW PLQLPL]HG WKH YDULDWLRQ DFURVV WKH OHYHOV RI WKH QRLVH IDFWRUV $OWKRXJK WKH ILQDO UHFRPPHQGDWLRQV RQ WKH GHVLJQ IDFWRU OHYHOV E\ &DQWHOO DQG 5DPLUH] f GLIIHUHG IURP 3KDGNH f RQ RQO\ RQH RI WKH VL[ GHVLJQ IDFWRUV WKH XVH RI WKH VSOLWVSOLWSORW GHVLJQ KDV DOORZHG WKH SURFHVV HQJLQHHU WR KDYH D EHWWHU XQGHUVWDQGLQJ RI WKH VRXUFHV RI YDULDWLRQ 7KLV DGGHG LQIRUPDWLRQ PD\ OHDG WR SURFHVV LPSURYHPHQW LQ WKH IXWXUH .HPSWKRUQH f DQG %R[ DQG -RQHV f SURYLGH GHWDLOV RQ WKH UHODWLYH HIILFLHQF\ RI WKHVH VSOLWSORW GHVLJQV FRPSDUHG WR WKH &5' DQG 5&% $ VXPPDU\ RI WKHLU FRQFOXVLRQV LV SURYLGHG KHUH &RQVLGHU WKH VSOLWSORW H[SHULPHQW DV D XQLIRUPLW\ WULDO ,I WKH XQLIRUPLW\ WULDO ZDV UXQ DV D &5' RU D 5&% H[SHULPHQW WKHQ IRU DUUDQJHPHQWV Df DQG Ef WKH VXESORW IDFWRU HIIHFWV DQG WKH VXESORW [ ZKROH SORW LQWHUDFWLRQV DUH HVWLPDWHG PRUH SUHFLVHO\ WKDQ WKH ZKROH SORW IDFWRU HIIHFWV &RPSDUHG

PAGE 34

ZLWK DUUDQJHPHQWV Df DQG Ef WKH VWULSEORFN GHVLJQ HVWLPDWHV WKH VXESORW [ ZKROH SORW LQWHUDFWLRQV PRUH SUHFLVHO\ EXW WKH VXESORW IDFWRU HIIHFWV ZLWK OHVV SUHFLVLRQ +RZHYHU WKH ZKROH SORW IDFWRU HIIHFWV DUH HVWLPDWHG ZLWK HTXDO SUHFLVLRQ %DVHG RQ WKHVH UHVXOWV DUUDQJHPHQW Df ZLWK WKH HQYLURQPHQWDO IDFWRUV DSSOLHG WR WKH ZKROH SORWV LV JHQHUDOO\ SUHIHUUHG RYHU DUUDQJHPHQW Ef %RWK WKH VWULSEORFN GHVLJQ DQG WKH VSOLWSORW GHVLJQ ZLWK WKH GHVLJQ IDFWRUV DSSOLHG WR WKH VXESORWV FDQ EH H[WUHPHO\ XVHIXO LQ UREXVW SDUDPHWHU GHVLJQ %L5DQGRPL]DWLRQ 'HVLJQV /HWVLQJHU 0\HUV DQG /HQWQHU f LQWURGXFHG ELUDQGRPL]DWLRQ GHVLJQV %5'fVf %5'fV UHIHU WR GHVLJQV ZLWK WZR UDQGRPL]DWLRQV VLPLODU WR WKDW RI D VSOLWSORW GHVLJQ 7KH ZKROH SORW YDULDEOHV DUH GHQRWHG E\ ] ]L] ‘ ‘ ‘ ]]f ZKLOH WKH VXESORW YDULn DEOHV DUH GHQRWHG E\ [ [L [ [[f +HQFH WKH ]WK GHVLJQ UXQ LV ] [f %5'fV DUH EURNHQ LQWR WZR FODVVHV FURVVHG DQG QRQFURVVHG &URVVHG %5'fV DUH FRQVWUXFWHG DV IROORZV UDQGRPL]H WKH D XQLTXH FRPELQDWLRQV RI ] WR WKH ZKROH SORW H[SHULPHQWDO XQLWV (8fVf WKHQ UDQGRPL]H WKH E OHYHOV RI [ WR WKH VPDOOHU (8fV ZLWKLQ HDFK ZKROH SORW VHH 7DEOH f 7KXV HYHU\ OHYHO RI [ LV fFURVVHGf ZLWK HYHU\ OHYHO RI ] 7KHVH GHVLJQV DUH WKH XVXDO VSOLWSORW GHVLJQV

PAGE 35

7DEOH &URVVHG %5' )URP /HWVLQJHU 0\HUV DQG /HQWQHU f =L ;O ;t ] ;O ;E ? ] D ;O ; 7DEOH 1RQFURVVHG %5' )URP /HWVLQJHU 0\HUV DQG /HQWQHU f =O ;X ;OO = ; ; ‘ =D ;DO ;DD 7KH QRQFURVVHG %5'fV GLIIHU IURP WKH FURVVHG %5'fV LQ WKDW QRW DOO OHYHOV RI [ DUH DVVRFLDWHG ZLWK ] 7KH ZKROH SORWV KDYH GLIIHUHQW OHYHOV RI WKH VXESORWV DQG QHHG QRW KDYH WKH VDPH QXPEHU RI OHYHOV 1RQFURVVHG %5'fV DUH FRQVWUXFWHG DV IROORZV UDQGRPL]H WKH D XQLTXH FRPELQDWLRQV RI ] WR WKH ZKROH SORW (8fV WKHQ UDQGRPL]H WKH EO OHYHOV RI [ WR WKH VPDOOHU (8fV ZLWKLQ HDFK ZKROH SORW VHH 7DEOH f 7KH GLVWLQFWLRQ EHWZHHQ WKHVH WZR FDQ EH WKRXJKW RI LQ WHUPV RI WKH VXESORW IDFWRUV 7KH FURVVHG %5' PLJKW EH UHSUHVHQWHG E\ D N IDFWRULDO LQ WKH VXESORW IDFWRUV ZKLOH WKH QRQFURVVHG %5' PLJKW XVH D NaS IUDFWLRQDOIDFWRULDO LQ WKH VXESORW IDFWRUV EXW QRW WKH VDPH NaS VHW RI WUHDWPHQWV

PAGE 36

)RU ERWK FURVVHG DQG QRQFURVVHG %5'fV WKH WZR UDQGRPL]DWLRQV FRPSOLFDWH WKH HUURU VWUXFWXUH 7KH ILUVW UDQGRPL]DWLRQ OHDGV WR WKH ZKROH SORW HUURU YDULDQFH HUI ZKLOH WKH VHFRQG UDQGRPL]DWLRQ OHDGV WR WKH VXESORW YDULDQFH UI ,W LV DVVXPHG WKDW WKH FRYDULDQFH EHWZHHQ DQ\ WZR REVHUYDWLRQV RQ WKH VDPH ZKROH SORW LV FRQVWDQW RYHU DOO ZKROH SORWV DQG WKDW REVHUYDWLRQV RQ WZR VXESORWV IURP GLIIHUHQW ZKROH SORWV DUH XQFRUUHODWHG 7KH UHVSRQVH VXUIDFH PRGHO LV \ ; H ZKHUH H a 9 9f ZLWK 9 HUI HUI ZKHUH LV D EORFN GLDJRQDO PDWUL[ RI OE [ OnM DQG ZKHUH LV WKH QXPEHU RI REVHUYDWLRQV ZLWKLQ WKH ]WK ZKROH SORW 1RZ XVLQJ JHQHUDOL]HG OHDVW VTXDUHV */6f WKH PD[LPXP OLNHOLKRRG HVWLPDWH 0/(f RI WKH UHVSRQVH VXUIDFH PRGHO LV ;n9n;ff ;n9n\ f ZLWK 9DU f [n9[fB f )URP (TXDWLRQ f LW LV VHHQ WKDW WKH PRGHO HVWLPDWLRQ GHSHQGV RQ WKH PDWUL[ 9 DQG WKXV ERWK HUI DQG HUI 6XSSRVH WKDW WKH UHVSRQVH VXUIDFH PRGHO LV SDUWLWLRQHG LQWR WKH ZKROH SORW DQG VXESORW WHUPV DV \ = ;rr =n$;r ZKHUH $ LV D PDWUL[ RI ZKROH SORW [ VXESORW LQWHUDFWLRQ SDUDPHWHUV 7KH UHVSRQVH VXUIDFH GHVLJQ VKRXOG EH ODUJH HQRXJK WR WHVW IRU JHQHUDO ODFN RI ILW DV ZHOO DV ODFN RI

PAGE 37

ILW IURP WKH ZKROH SORWV 7KHUHIRUH WKH QXPEHU RI ZKROH SORWV DYDLODEOH PXVW H[FHHG WKH QXPEHU RI SDUDPHWHUV LQ )RU WKH FURVVHG %5' WKHUH LV DQ HTXLYDOHQFH EHWZHHQ RUGLQDU\ OHDVW VTXDUHV 2/6f DQG */6 7KLV HTXLYDOHQFH PHDQV WKDW (TXDWLRQ f EHFRPHV ;n;f;n\ DQG WKH PRGHO HVWLPDWLRQ QR ORQJHU GHSHQGV RQ WKH HUURU YDULDQFH +RZHYHU IRU WHVWLQJ SXUSRVHV WKH HUURU YDULDQFH PXVW EH HVWLPDWHG /HWVLQJHU 0\HUV DQG /HQWQHU f VXJJHVW DXJPHQWLQJ WKH UHVSRQVH VXUIDFH PRGHO ZLWK LQVLJQLILFDQW ZKROH SORW WHUPV =rS WR VDWXUDWH WKH D f§ ZKROH SORW GHJUHHV RI IUHHGRP 7KH ZKROH SORW VDWXUDWHG PRGHO FDQ EH XVHG WR FDOFXODWH ODFN RI ILW VXPV RI VTXDUHV IRU ERWK WKH ZKROH SORWV DQG WKH VXESORWV 7KHQ DSSUR[LPDWH IWHVWV FDQ EH IRUPHG E\ VXEVWLWXWLQJ WKH HVWLPDWHG HUURU YDULDQFHV LQWR (TXDWLRQ f 1RQFURVVHG %5'fV SUHVHQW D PRUH FRPSOLFDWHG VLWXDWLRQ 7KH HTXLYDOHQF\ RI 2/6 DQG */6 LV RQO\ WUXH LQ WKH FDVH RI D ILUVWRUGHU PRGHO $OWKRXJK PRUH FRPSOH[ WKH DERYH PHWKRG FDQ EH DGDSWHG IRU WKH ILUVWRUGHU FDVH /HWVLQJHU 0\HUV DQG /HQWQHU f FRPSDUH WKUHH PHWKRGV IRU WKH VHFRQGRUGHU FDVH 7KH\ DUH 2/6 LWHUDWLYH UHZHLJKWHG OHDVW VTXDUHV ,5/6f DQG UHVWULFWHG PD[LPXP OLNHOLKRRG 5(0/f 7KRXJK ,5/6 DQG 5(0/ DSSHDU WR EH EHWWHU PHWKRGV WKH fEHVWf PHWKRG GHSHQGV RQ WKH GHVLJQ PRGHO DQG DQ\ SULRU LQIRUPDWLRQ %LUDQGRPL]DWLRQ LQWURGXFHV WKH QHHG IRU QHZ GHILQLWLRQV IRU GHVLJQ HIILFLHQF\ EHn FDXVH HIILFLHQW GHVLJQV LQ WKH OLWHUDWXUH DUH EDVHG RQ D FRPSOHWHO\ UDQGRPL]HG HUURU VWUXFWXUH )RU H[DPSOH IRU WKH %5' WKH 'RSWLPDOLW\ FULWHULRQ VHH HJ .LHIHU DQG

PAGE 38

:ROIRZLW] ff EHFRPHV PLQ 1 ;n9B;f RYHU DOO GHVLJQV /HWVLQJHU 0\HUV DQG /HQWQHU f SURYLGH FRPSDULVRQV RI VHYHUDO ILUVW DQG VHFRQGRUGHU GHVLJQV )RU WKH VHFRQGRUGHU GHVLJQV WKH SRSXODU FHQWUDO FRPSRVLWH GHVLJQ &&'f SURYHV WR EH D JRRG GHVLJQ 6SOLW3ORWV LQ ,QGXVWULDO ([SHULPHQWV /XFDV DQG -X f LQYHVWLJDWHG WKH XVH RI VSOLWSORW GHVLJQV LQ LQGXVWULDO H[SHULn PHQWV ZKHUH RQH IDFWRU ZDV GLIILFXOW WR FKDQJH DQG LWV OHYHOV VHUYHG DV WKH ZKROH SORW WUHDWPHQWV 7KH\ EHJDQ WKHLU VWXG\ ZLWK D VLPXODWLRQ H[HUFLVH XVLQJ D IRXU IDFWRU IDFHFHQWHUHG FXEH GHVLJQ ZLWK IRXU FHQWHU SRLQWV 7KH\ OHW ;? FRUUHVSRQG WR WKH KDUGWRFKDQJH IDFWRU ZKLOH [ [ DQG [ ZHUH HDV\ WR YDU\ 7KLV GHVLJQ DOORZHG IRU WKH ILWWLQJ RI WKH TXDGUDWLF PRGHO \ ID tL;L 6 ( ( 3LM[L[M H f   L  M WL +RZHYHU VLQFH WKH HUURU ZDV WKH RQO\ WHUP RI LQWHUHVW DOO WKH UHJUHVVLRQ FRHIILFLHQWV FDQ EH ]HUR 7KHUHIRUH GDWD ZDV JHQHUDWHG XVLQJ \ f§ f‘ f§ HV ZKHUH Hf a 1 FUf ZDV WKH HUURU WHUP DVVRFLDWHG ZLWK FKDQJLQJ WKH OHYHO RI ;? DQG HV a 1 DIf ZDV WKH HUURU DVVRFLDWHG ZLWK DQ\ QHZ H[SHULPHQWDO UXQ 7ZHQW\HLJKW UXQV ZHUH JHQHUDWHG XVLQJ WKH IROORZLQJ VWHSV *HQHUDWH Hf a L9 Dbf DQG HV a 1DAf

PAGE 39

< f§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fV ZHUH FRPSXWHG DQG WKH UHVLGXDO HUURU ZDV HVWLPDWHG 7KH VLPXODWLRQ SURFHGXUH ZDV UHSHDWHG WLPHV /XFDV DQG -X f VXPPDUL]HG WKHLU VLPXODWLRQ UHVXOWV LQ D WDEOH ZLWK D OLVWLQJ RI WKH VWDQGDUG GHYLDWLRQV RI WKH UHJUHVVLRQ FRHIILFLHQWV IRU WKH WKUHH GLIIHUHQW ZD\V RI UXQQLQJ WKH H[SHULPHQW 7KH UHVWULFWHG UDQGRPL]DWLRQ FDVH KDV D PXFK VPDOOHU UHVLGXDO VWDQGDUG GHYLDWLRQ DQG PXFK VPDOOHU VWDQGDUG GHYLDWLRQV IRU DOO WKH UHJUHVn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

PAGE 40

$ VLPLODU VLPXODWLRQ ZDV FRQGXFWHG IRU WZROHYHO IDFWRULDOV VHH /XFDV DQG -X ff 7KH\ FRQVLGHUHG D IDFWRULDO ZLWK [[ DV WKH KDUGWRFKDQJH IDFWRU 7KLV DOORZV WKH ILWWLQJ RI D UHJUHVVLRQ PRGHO WKDW LQFOXGHV WKH OLQHDU DQG LQWHUDFWLRQ WHUPV $JDLQ D VXPPDU\ WDEOH LV SURYLGHG E\ /XFDV DQG -X f VKRZLQJ V LPLODU UHVXOWV WR WKH RWKHU H[SHULPHQWDO VFHQDULRV 7KH FRPSOHWHO\ UHVWULFWHG H[SHULPHQW KDG VPDOOHU VWDQGDUG GHYLDWLRQV IRU DOO WKH UHJUHVVLRQ FRHIILFLHQWV H[FHSW c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f GLVFXVVHV WKH DQDO\VLV RI GDWD IURP PL[WXUH H[SHULPHQWV ZLWK SURn FHVV YDULDEOHV ZKHUH WKH PL[WXUH EOHQGV DUH HPEHGGHG LQ WKH SURFHVV YDULDEOH FRPn ELQDWLRQV DV LQ fD VSOLWSORW GHVLJQf 7KH PL[WXUH SURFHVV YDULDEOHV DUH IDFWRUV WKDW DUH QRW PL[WXUH LQJUHGLHQWV EXW ZKRVH OHYHOV FRXOG DIIHFW WKH EOHQGLQJ SURSHUWLHV RI WKH PL[WXUH FRPSRQHQWV 7R LOOXVWUDWH WKLV VLWXDWLRQ &RUQHOO XVHV DQ H[DPSOH IURP &RUQHOO DQG *RUPDQ f LQYROYLQJ ILVK SDWWLHV 7KH PL[WXUH H[SHULPHQW LQYROYHV PDNLQJ ILVK SDWWLHV IURP GLIIHUHQW EOHQGV RI WKUHH ILVK VSHFLHV PXOOHW VKHHSVKHDG DQG FURDNHUf 7KH SDWWLHV ZHUH VXEMHFWHG WR IDFWRU OHYHO FRPELQDWLRQV RI WKUHH SURFHVV YDULDEOHV FRRNLQJ WHPSHUDWXUH FRRNLQJ WLPH DQG GHHSIU\LQJ WLPHf (DFK SURFHVV YDULDEOH ZDV VWXGLHG DW WZR OHYHOV :KHQ SURFHVV YDULDEOHV DUH LQFOXGHG LQ D PL[WXUH

PAGE 41

H[SHULPHQW FRPSOHWH UDQGRPL]DWLRQ WHQGV WR EH LPSUDFWLFDO 7KLV OHDGV WR D UHVWULFn WLRQ RQ UDQGRPL]DWLRQ DQG OHQGV LWVHOI WR WKH VSOLWSORW GHVLJQ &RUQHOO f FRQVLGHUV IDFWRUOHYHO FRPELQDWLRQV RI WKH SURFHVV YDULDEOHV DV WKH ZKROH SORW WUHDWPHQWV DQG WKH PL[WXUH FRPSRQHQW EOHQGV DV WKH VXESORW WUHDWPHQWV EXW SRLQWV RXW WKDW WKHLU UROHV FDQ EH VZLWFKHG +HQFH D FRPELQDWLRQ RI WKH OHYHOV RI WKH SURFHVV YDULDEOHV LV VHOHFWHG DQG DOO EOHQGV DUH UXQ DW WKLV FRPELQDWLRQ $Qn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f GLVFXVV VXEVHW VHOHFWLRQ SURFHGXUHV IRU VFUHHQLQJ LQ WZR IDFWRU WUHDWPHQW GHVLJQV 7KH SDSHU GHDOV PDLQO\ ZLWK VSOLWSORW GHVLJQV UXQ LQ FRPn SOHWH EORFNV KRZHYHU WKH VWULSSORW GHVLJQ LV DOVR GLVFXVVHG 2QH IDFWRU VHUYHV DV WKH ZKROH SORW IDFWRU ZKLOH WKH RWKHU LV WKH VXESORW IDFWRU 7KH JRDO LV WR VHOHFW D VXEVHW RI WKH WUHDWPHQW FRPELQDWLRQV DVVRFLDWHG ZLWK WKH ODUJHVW PHDQ 6XEVHW VHOHFWLRQ SURFHGXUHV DUH JLYHQ IRU DGGLWLYH DQG QRQDGGLWLYH IDFWRU FDVHV ZKHUH QHLWKHU RI WKH

PAGE 42

7DEOH 9DULDQFH RI WKH 5HJUHVVLRQ &RHIILFLHQWV )RU D N :LWK 2QH +DUG7R&KDQJH )DFWRU IURP /XFDV DQG -X ff 9DUf $FW %FW +DUG 7R &KDQJH 9DULDEOH 2WKHU 7HUPV $ % $ % OS L S AU L g LS L g 3 rn GHVLJQ 3 IRU 3 Na f IRU WKH FRPSOHWHO\ UDQGRPL]HG WKH FRPSOHWHO\ UHVWULFWHG GHVLJQ f§ f>IFB f§ f@ IRU WKH SDUWLDOO\ UHVWULFWHG GHVLJQ SURFHGXUHV GHSHQG RQ WKH EORFN YDULDQFH 0LOOHU f FRQVLGHUV YDULRXV IUDFWLRQDOIDFWRULDO VWUXFWXUHV LQ VWULSSORW H[SHULn PHQWV 7KHVH VWULSSORW H[SHULPHQWV DUH LGHQWLFDO LQ QDWXUH WR WKH VWULSEORFN H[SHULn PHQWV DUUDQJHPHQW Ff GLVFXVVHG LQ %R[ DQG -RQHV f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n PHQWDO XQLWV LQ ZKLFK WKH OHYHOV RI RQH WUHDWPHQW IDFWRU RU VHW RI IDFWRUVf DUH DVVLJQHG

PAGE 43

WR WKH URZV DQG WKH OHYHOV RI D VHFRQG WUHDWPHQW IDFWRU RU VHW RI IDFWRUVf DUH DVVLJQHG WR WKH FROXPQV 7DEOH UHSUHVHQWV WKH ODXQGU\ H[SHULPHQW LQ ZKLFK HDFK VTXDUH UHSUHVHQWV D FORWK VDPSOH URZV UHSUHVHQW VHWV RI VDPSOHV WKDW ZHUH ZDVKHG WRJHWKHU DQG FROXPQV UHSUHVHQW VHWV RI VDPSOHV WKDW ZHUH GULHG WRJHWKHU 7KH $129$ WDEOH IRU WKH ODXQGU\ H[DPSOH ZKLFK LV GLYLGHG LQWR fVWUDWDf FRUUHVSRQGLQJ WR EORFNV URZV FROXPQV DQG XQLWV LV JLYHQ LQ 7DEOH :KHQ PDNLQJ LQIHUHQFHV DERXW WKH HIIHFWV LQ D SDUWLFXODU VWUDWXP WKH HVWLPDWH RI YDULDWLRQ PXVW EH EDVHG RQ WKH UHVLGXDO WHUP IRU WKDW VWUDWXP 0LOOHU f SURSRVHV D PHWKRG IRU FRQVWUXFWLQJ VWULSSORW FRQILJXUDWLRQV IRU IUDFWLRQDOIDFWRULDO GHVLJQV ZKLFK FRQVLVWV RI WKUHH VWHSV ,GHQWLI\ D VXLWDEOH GHVLJQ IRU DSSO\LQJ URZ WUHDWPHQWV WR URZV LJQRULQJ FROXPQV ,GHQWLI\ D VXLWDEOH GHVLJQ IRU DSSO\LQJ FROXPQ WUHDWPHQWV WR FROXPQV LJQRULQJ URZV 6HOHFW D VXLWDEOH IUDFWLRQ RI WKH SURGXFW RI WKH URZ DQG FROXPQ GHVLJQV 7KH PHWKRG LV DSSOLHG IRU WZROHYHO GHVLJQV DQG WKHQ H[WHQGHG WR UQOHYHO DQG PL[HG OHYHO GHVLJQV 7KH SURFHGXUH IRU WZROHYHO GHVLJQV LV SUHVHQWHG KHUH IRU GHWDLOV RQ WKH H[WHQGHG FDVHV VHH 0LOOHU f &RQVLGHU WKH VLWXDWLRQ LQ ZKLFK D SURSHU IUDFWLRQ RI D WZROHYHO IDFWRULDO GHVLJQ LV WR EH UXQ LQ D VWULSSORW DUUDQJHPHQW XVLQJ E : EORFNV (DFK EORFN KDV U 0 URZV DQG F f§ P FROXPQV /HW DQG N UHSUHVHQW WKH QXPEHU RI URZ DQG FROXPQ IDFWRUV UHVSHFWLYHO\ DQG GHILQH 4 Z 0f DQG T N Z Pf 7KHQ WKH SURFHGXUH LV DV IROORZnV

PAGE 44

7DEOH 6WULSSORW &RQILJXUDWLRQ RI WKH /DXQGU\ ([SHULPHQW IURP 0LOOHU ff 'U\HU 'U\HU :DVKHU :DVKHU %ORFN RFN 7DEOH $129$ 7DEOH IRU WKH /DXQGU\ ([DPSOH IURP 0LOOHU ff 6WUDWD 6RXUFH GI (06f %ORFN %ORFNV D FUA FUe FUA 5RZ ::DVKHU D A f  : D R? 5RZ 5HVLGXDO &ROXPQ ''U\HU R $DF f '? &ROXPQ 5HVLGXDO D 2T 8QLW : [ 8QLW 5HVLGXDO 7 f ( ( ,:'@b N D

PAGE 45

6HOHFW D URZ GHVLJQ WKDW FRQVLVWV RI D $B" GHVLJQ LQ E EORFNV 6HOHFW D FROXPQ GHVLJQ WKDW FRQVLVWV RI D NaT GHVLJQ LQ E EORFNV &RQVLGHU WKH SURGXFW RI WKH GHVLJQV LQ VWHSV DQG DQG VHOHFW D /DWLQ6TXDUH IUDFWLRQ RI WKLV SURGXFW 7KH VHOHFWLRQ RI WKH GHVLJQ LQ VWHSV DQG FDQ EH PDGH RQ WKH EDVLV WKDW WKH DQDO\VHV IRU WKH URZ VWUDWXP DQG WKH FROXPQ VWUDWXP ZLOO HVVHQWLDOO\ EH WKH DQDO\VHV RI WKHVH GHVLJQV 7KH /DWLQ6TXDUH IUDFWLRQ LV VHOHFWHG VR WKDW WKH FRQIRXQGLQJ DUUD\ HIIHFWV LQ WKH XQLW VWUDWXP KDYH GHVLUDEOH SURSHUWLHV 0HH DQG %DWHV f FRQVLGHU VSOLWORW H[SHULPHQWV LQYROYLQJ WKH HWFKLQJ RI VLOLFRQ ZDIHUV 7KHVH H[SHULPHQWV DUH SHUIRUPHG LQ VWHSV ZKHUH D GLIIHUHQW IDFWRU LV DSSOLHG DW HDFK VWHS 7KXV WKHUH DUH DQ HTXDO QXPEHU RI VWHSV DQG IDFWRUV 6SHFLI\LQJ D VSOLWORW GHVLJQ LQYROYHV GHWHUPLQLQJ WKH IROORZLQJ WKH QXPEHU RI SURFHVV VWHSV ZLWK H[SHULPHQWDWLRQ WKH QXPEHU RI IDFWRUV DQG WKHLU OHYHOV DW HDFK SURFHVVLQJ VWHS ZLWK H[SHULPHQn WDWLRQ WKH VXESORW VL]H DW HDFK SURFHVVLQJ VWHS WKH QXPEHU RI ZDIHUV H[SHULPHQWDO XQLWVf LQ WKH HQWLUH H[SHULPHQW D SODQ WKDW GHWDLOV IRU HDFK H[SHULPHQWDO ZDIHU WKH SURFHVV VXESORW DW HDFK VWHS 0HH DQG %DWHV HPSKDVL]H V\PPHWULF GHVLJQV ZKLFK DUH GHVLJQV KDYLQJ WKH VDPH VXESORW VL]H DW HDFK H[SHULPHQWDWLRQ VWHS

PAGE 46

7KH H[SHULPHQWDO SODQ LQ LWHP DERYH ZLOO EH GHWHUPLQHG DV IROORZV )LUVW WR GHILQH VXESORWV DW HDFK VWHS REWDLQ f§ FRQWUDVWV IRU HDFK H[SHULPHQWDO VWHS 7KHQ DVVLJQ IDFWRUV WR FRQWUDVWV ZLWKLQ WKH JURXS LQWHQGHG IRU WKHLU UHVSHFWLYH SURFHVVLQJ VWHS 7KLV LV GRQH LQ D ZD\ WKDW JLYHV WKH PRVW LQIRUPDWLRQ RQ WKH LQWHUDFWLRQ HIIHFWV RI LQWHUHVW 7KH DSSURDFK LV WR GHWHUPLQH D VHW RI LQGHSHQGHQW FRQWUDVWV WKDW FDQ EH F\FOHG WR SURGXFH DGGLWLRQDO VHWV 7KH LQLWLDO VHW RI LQGHSHQGHQW FRQWUDVWV PXVW EH FKRVHQ VR WKDW WKH JURXSV RI HIIHFWV UHPDLQ GLVMRLQW 7KLV SURFHVV DQG WKH UHVXOWn LQJ GHVLJQV DUH LOOXVWUDWHG IRU D YDULHW\ RI ZDIHU H[SHULPHQWV VHH 0HH DQG %DWHV ff 6SOLWORW GHVLJQV IRU WKUHHOHYHO IDFWRUV DUH DOVR GLVFXVVHG ,W VKRXOG EH QRWHG WKDW LI WKHUH DUH RQO\ WZR VWHSV WKH SURFHGXUHV E\ 0LOOHU f FDQ EH DSSOLHG ZLWK RQH RU ZLWK PDQ\ IDFWRUV DW HDFK VWHS

PAGE 47

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f ,I D IDFWRU LV DW LWV ORZ OHYHO WKH FRUUHVSRQGLQJ OHWWHU LV RPLWWHG IURP WKH WUHDWn PHQW GHVLJQDWLRQ &RQYHUVHO\ LI D IDFWRU LV DW LWV KLJK OHYHO WKH FRUUHVSRQGLQJ OHWWHU LV LQFOXGHG f :KHQ DOO IDFWRUV DUH DW WKHLU ORZ OHYHOV WKH WUHDWPHQW ZLOO EH GHVLJQDWHG E\ WKH V\PERO f 8QGHU WKLV QRWDWLRQ WKH WUHDWPHQWV IRU D IDFWRULDO H[SHULPHQW LQ IDFWRUV 3 DQG 4 DUH GHVLJQDWHG DV f S T DQG ST )DFWRUV DQG WKHLU HIIHFWV ZLOO EH GHVLJQDWHG E\ FDSLWDO OHWWHUV )DFWRULDO H[SHULPHQWV EHFRPH ODUJH YHU\ UDSLGO\ VR WKDW RIWHQ D VLQJOH UHSOLFDWH RI WKH 1 f§ N UXQV UHTXLUHV PRUH UHVRXUFHV WKDQ DUH DYDLODEOH HYHQ ZLWK D PRGHUDWH

PAGE 48

QXPEHU RI IDFWRUV N (YHQ ZKHQ UHVRXUFHV DUH DYDLODEOH ZH PD\ QRW ZDQW WR HVWLPDWH DOO RI WKH N f§ IDFWRULDO HIIHFWV $V DQ H[DPSOH ZLWK N LQWHUDFWLRQV LQYROYLQJ RU PRUH IDFWRUV DUH JHQHUDOO\ FRQVLGHUHG WR EH QHJOLJLEOH RU RI OLWWOH LPSRUWDQFH 7KXV D VLQJOH UHSOLFDWH RI D UHTXLUHV H[SHULPHQWDO XQLWV DQG SURYLGHV D IROG UHSOLFDWLRQ RI HDFK PDLQ HIIHFW 2I WKH HIIHFWV WKDW FDQ EH HVWLPDWHG RQO\ PD\ EH RI PDMRU LQWHUHVW VHYHQ PDLQ HIIHFWV DQG WZRIDFWRU LQWHUDFWLRQVf )UDFWLRQDO )DFWRULDOV )LQQH\ f SURSRVHG UHGXFLQJ WKH VL]H RI WKH H[SHULPHQW E\ XVLQJ RQO\ D IUDFn WLRQ RI WKH WRWDO QXPEHU RI SRVVLEOH WUHDWPHQW FRPELQDWLRQV 6XFK H[SHULPHQWV DUH FDOOHG IUDFWLRQDO IDFWRULDOV +H RXWOLQHG PHWKRGV RI FRQVWUXFWLQJ IUDFWLRQV IRU N DQG N H[SHULPHQWV )RU VFUHHQLQJ SXUSRVHV 3ODFNHWW DQG %XUPDQ f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aS ZLOO EH FDOOHG D NaS IUDFWLRQDO IDFWRULDO H[SHULPHQW 7KHVH H[SHULPHQWV KDYH WZR PDMRU SUREOHPV ZKLFK FDQ OLPLW WKHLU XVHIXOQHVV

PAGE 49

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n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f WKDW LV FRQIRXQGHG ZLWK WKH FRQVWDQW HIIHFW ,W FDQ EH UHSUHVHQWHG DV DQ HTXDWLRQ E\ VHWWLQJ WKH FRQIRXQGHG HIIHFW HTXDO WR 7KH DOLDV FKDLQ IRU DQ HIIHFW LV IRXQG E\ IRUPLQJ WKH JHQHUDOL]HG LQWHUDFWLRQ RI WKH HIIHFW ZLWK DOO WHUPV LQ WKH GHILQLQJ FRQWUDVW )RU H[DPSOH LI D IUDFWLRQ LQ IDFWRUV $ % DQG & LV UXQ ZLWK GHILQLQJ FRQWUDVW $%& WKHQ WKH DOLDV RI WKH PDLQ HIIHFW $ LV $,f $%& ZKLFK JLYHV $ %& VLQFH $ 7KHUHIRUH WKH

PAGE 50

DOLDV FKDLQV IRU WKH PDLQ HIIHFWV $ % DQG & DUH DV IROORZV $ %& % $& & $% )RU D NaS WKHUH DUH 3 f§ HIIHFWV LQ WKH GHILQLQJ FRQWUDVW 7KH H[SHULPHQWHU FDQ VHOHFW DQ\ S IDFWRULDO HIIHFWV WR EH WKH GHILQLQJ FRQWUDVW 7KH UHPDLQLQJ 3 f§ S f§ IDFWRULDO HIIHFWV DUH DXWRPDWLFDOO\ GHWHUPLQHG DV EHLQJ WKH JHQHUDOL]HG LQWHUDFWLRQV DPRQJ WKH S HIIHFWV %R[ DQG +XQWHU D Ef FODVVLILHG IUDFWLRQDO IDFWRULDO SODQV E\ WKHLU GHJUHH RI DOLDVLQJ RI HIIHFWV 7KLV PHDVXUH LV FDOOHG WKH UHVROXWLRQ RI WKH SODQ 7KH QXPEHU RI OHWWHUV LQ WKH VKRUWHVW PHPEHU RI D VHW RI GHILQLQJ FRQWUDVWV GHWHUPLQHV WKH GHVLJQfV UHVROXWLRQ 7KUHH LPSRUWDQW UHVROXWLRQV DUH 5HVROXWLRQ ,OI LQ ZKLFK PDLQ HIIHFWV DUH DOLDVHG ZLWK WZRIDFWRU LQWHUDFWLRQV 5HVROXWLRQ ,9 LQ ZKLFK PDLQ HIIHFWV DUH DOLDVHG ZLWK WKUHHIDFWRU LQWHUDFWLRQV DQG WZRIDFWRU LQWHUDFWLRQV DUH DOLDVHG ZLWK RWKHU WZRIDFWRU LQWHUDFWLRQV 5HVROXWLRQ 9 ZKHUH WZRIDFWRU LQWHUDFWLRQV DUH DOLDVHG ZLWK WKUHHIDFWRU LQn WHUDFWLRQV 2I FRXUVH LI DOO WKUHHIDFWRU DQG KLJKHU LQWHUDFWLRQV DUH QHJOLJLEOH D GHVLJQ ZLWK 5HVROXWLRQ 9 LV GHVLUHG EHFDXVH LW ZLOO DOORZ WKH HVWLPDWLRQ RI DOO PDLQ HIIHFWV DQG WZRIDFWRU LQWHUDFWLRQV VLQFH WKH\ DUH DOLDVHG ZLWK QHJOLJLEOH HIIHFWV

PAGE 51

&RQIRXQGLQJ 6XSSRVH WKDW D N IDFWRULDO H[SHULPHQW LV WR EH UXQ LQ EORFNV $V QRWHG HDUOLHU WKH PDLQ GLVDGYDQWDJH RI N IDFWRULDO H[SHULPHQWV LV WKHLU VL]H &RQVHTXHQWO\ HYHQ IRU D PRGHUDWH QXPEHU RI IDFWRUV LW PD\ QRW EH SRVVLEOH WR ILQG EORFNV ZLWK WKH UHTXLUHG QXPEHU RI KRPRJHQHRXV H[SHULPHQWDO XQLWV :KHQ WKLV RFFXUV LW LV QHFHVVDU\ WR XVH VPDOOHUVL]HG EORFNV RU LQFRPSOHWH EORFN GHVLJQV :LWK DQ LQFRPSOHWH EORFN GHVLJQ WKHUH PXVW EH VRPH ORVV RI LQIRUPDWLRQ $ EDODQFHG LQFRPSOHWH EORFN GHVLJQ LI LW H[LVWV GLVWULEXWHV WKLV ORVV HTXDOO\ WR DOO WUHDWn PHQWV +RZHYHU LQ IDFWRULDO H[SHULPHQWV LW LV WKH PDLQ HIIHFWV DQG LQWHUDFWLRQV WKDW DUH LPSRUWDQW )RU PRVW IDFWRULDO H[SHULPHQWV ZLWK PRUH WKDQ WKUHH IDFWRUV LW LV KLJKO\ XQOLNHO\ WKDW DOO HIIHFWV HVSHFLDOO\ WKH KLJKHURUGHU LQWHUDFWLRQV DUH LPSRUWDQW ,I VRPH HIIHFWV FDQ EH DVVXPHG QHJOLJLEOH SULRU WR SHUIRUPLQJ WKH H[SHULPHQW WKHQ D EHWWHU SURFHGXUH IRU FRQVWUXFWLQJ LQFRPSOHWH EORFNV RULJLQDOO\ VXJJHVWHG E\ )LVKHU f ZRXOG EH ILQGLQJ DUUDQJHPHQWV ZKLFK FRPSOHWHO\ RU SDUWLDOO\ VDFULILFH WKH LQn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

PAGE 52

WKH\ DUH QR ORQJHU VHSDUDWHO\ HVWLPDEOH *HQHUDOO\ WKHVH HIIHFWV ZLOO EH WKUHHIDFWRU LQWHUDFWLRQV RU KLJKHU VR WKDW DOO PDLQ HIIHFWV DQG WZRIDFWRU LQWHUDFWLRQV FDQ EH HVWLPDWHG :KHQ WKH EORFN VL]H RI N LV UHGXFHG E\ aS HDFK EORFN ZLOO FRQWDLQ NaS H[SHULn PHQWDO XQLWV DQG HDFK FRPSOHWH UHSOLFDWH ZLOO FRQWDLQ 3 EORFNV ,Q WKLV FDVH LW ZLOO EH QHFHVVDU\ WR FRQIRXQG 3 f§ HIIHFWV LQ HDFK UHSOLFDWH 7KH H[SHULPHQWHU FKRRVHV S RI WKHVH HIIHFWV ZLWK WKH UHPDLQLQJ 3 f§ S f§ HIIHFWV EHLQJ WKH JHQHUDOL]HG LQWHUDFWLRQV RI WKH RULJLQDO S HIIHFWV :KHQ PRUH WKDQ RQH UHSOLFDWH RI WKH NaS IUDFWLRQDO IDFWRULDO LV SHUIRUPHG WZR W\SHV RI FRQIRXQGLQJ DUH SRVVLEOH &RPSOHWH f§ WKH VDPH VHW RI HIIHFWV LV FRQIRXQGHG LQ HDFK UHSOLFDWH 3DUWLDO f§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

PAGE 53

FRQIRXQGHG :KLOH WKH DPRXQW RI LQIRUPDWLRQ LV UHGXFHG VWDWLVWLFDO VLJQLILFDQFH RI HDFK HIIHFW FDQ EH DVFHUWDLQHG &RQIRXQGLQJ LQ )UDFWLRQDO )DFWRULDOV $OWKRXJK RQO\ D IUDFWLRQ RI WKH WUHDWPHQWV DUH LQFOXGHG LQ D NaS H[SHULPHQW WKLV QXPEHU PD\ VWLOO EH WRR ODUJH IRU DYDLODEOH EORFNV $V LQ DQ\ IDFWRULDO H[SHULPHQW FRQIRXQGLQJ LV XVHG WR UHGXFH WKH EORFN VL]H &RQIRXQGLQJ DQ HIIHFW LQ D IUDFWLRQDO IDFWRULDO H[SHULPHQW DOVR FRQIRXQGV DOO RI LWV DOLDVHV &RQVLGHU D B IUDFWLRQDO IDFWRULDO H[SHULPHQW XVLQJ WKH fEHVWf GHILQLQJ FRQWUDVW IRU D KDOIUHSOLFDWH $%&'() 7KLV UHTXLUHV KRPRJHQHRXV H[SHULPHQWDO XQLWV ,I WKHVH DUH QRW DYDLODEOH WKHQ EORFNV RI VPDOOHU VL]H FDQ EH FUHDWHG E\ FRQn IRXQGLQJ DGGLWLRQDO HIIHFWV 6XSSRVH EORFNV RI VL]H H[SHULPHQWDO XQLWV DUH DYDLODEOH 7R FUHDWH WZR EORFNV RI VL]H IRU WKH WUHDWPHQWV LW LV QHFHVVDU\ WR FRQIRXQG RQH HIIHFW 6LQFH $%&'() ZDV XVHG WR GHILQH WKH KDOIIUDFWLRQ LW ZRXOG DSSHDU ORJLFDO WR VHOHFW D ILYHIDFWRU LQWHUDFWLRQ VD\ $%2'( +RZHYHU WKH DOLDV RI WKLV LQWHUDFn WLRQ RU JHQHUDOL]HG LQWHUDFWLRQ RI WKH HIIHFW ZLWK $%&'() LV ) DQG ZLOO DOVR EH FRQIRXQGHG ZLWK EORFNV $ EHWWHU FKRLFH LV WR FRQIRXQG DQ\ WKUHHIDFWRU LQWHUDFWLRQ VLQFH LWV DOLDV ZLOO DOVR EH D WKUHHIDFWRU LQWHUDFWLRQ $V D UHVXOW QR LQIRUPDWLRQ LV ORVW RQ SRWHQWLDOO\ LPSRUWDQW HIIHFWV 7KH ZRUG fEHVWf VKRXOG EH FODULILHG ,W LV UHIHUULQJ WR WKH GHVLJQ ZKLFK KDV WKH OHDVW DPRXQW RI DOLDVLQJ DPRQJ LPSRUWDQW HIIHFWV ZKLFK DUH XVXDOO\ WKRXJKW WR EH PDLQ HIIHFWV DQG WZRIDFWRU LQWHUDFWLRQV ,I LPSRUWDQW HIIHFWV DUH QRW DOLDVHG ZLWK HDFK RWKHU WKHQ EHVWf UHIHUV WR WKH GHVLJQ ZLWK KLJKHVW 5HVROXWLRQ 7KHUHIRUH fEHVWf LV

PAGE 54

D FULWHULRQ EDVHG RQ HVWLPDELOLW\ 7KURXJKRXW WKLV FKDSWHU ZKHUHYHU WKH SKDVH fEHVW GHVLJQf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

PAGE 55

f PDLQ HIIHFWV IRU WKH ZKROH SORW IDFWRUV f PDLQ HIIHFWV IRU WKH VXESORW IDFWRUV f WZRIDFWRU LQWHUDFWLRQV EHWZHHQ WKH ZKROH SORW DQG VXESORW IDFWRUV f DQG LI SRVVLEOH WZRIDFWRU LQWHUDFWLRQV DPRQJ WKH VXESORW IDFWRUV 1RWH WKDW LI WKHUH ZHUH VXIILFLHQW UHVRXUFHV WR UXQ DOO ZKROH SORW WUHDWPHQWV DQG VXESORW WUHDWPHQWV WKHQ DOO IRXU JRDOV ZRXOG EH DXWRPDWLFDOO\ VDWLVILHG +RZHYHU LQ PRVW VLWXDWLRQV WKLV LV QRW HFRQRPLFDOO\ SRVVLEOH 7KHUHIRUH ZH VKDOO WU\ WR HVWLPDWH DV PDQ\ HIIHFWV DV LV SRVVLEOH ZLWKLQ WKH UHVWULFWLRQV RQ WKH UHVRXUFHV DYDLODEOH 7KH LGHD RI FRQIRXQGLQJ HIIHFWV LQ RUGHU WR UHGXFH WKH QXPEHU RI VXESORW WUHDWn PHQWV SHU ZKROH SORW WUHDWPHQW DQG DFKLHYH WKH VHFRQG JRDO KDV EHHQ DURXQG IRU VRPH WLPH .HPSWKRUQH f KDV D VHFWLRQ GHYRWHG WR FRQIRXQGLQJ LQ VSOLWSORW H[SHULn PHQWV $GGHOPDQ f DOVR GLVFXVVHV ZD\V RI DFFRPSOLVKLQJ WKLV 5HFHQWO\ WKH XVH RI VSOLWSORW H[SHULPHQWV LQ LQGXVWU\ KDV JHQHUDWHG UHQHZHG LQWHUHVW LQ FRQIRXQGLQJ +XDQJ &KHQ DQG 9RHONHO f DQG %LQJKDP DQG 6LWWHU f GLVFXVV PLQLPXP DEHUUDWLRQ GHVLJQV IRU IDFWRUV ZLWK WZROHYHOV 7KLV WHFKQLTXH KHOSV WR LPSURYH WKH HVWLPDWLRQ SUREOHP E\ UDLVLQJ WKH UHVROXWLRQ FRQFHUQLQJ WKH VXESORW IDFWRUV EXW RQH PXVW EH FDUHIXO ZLWK WKH ZKROH SORW [ VXESORW LQWHUDFWLRQV %LVJDDUG f XVHV LQQHU DQG RXWHU DUUD\V ZLWK IDFWRUV DW WZROHYHOV DV LQ UREXVW SDUDPHWHU GHVLJQ DQG SURYLGHV WKH VWDQGDUG HUURUV IRU YDULRXV FRQWUDVWV DPRQJ WKH ZKROH SORW DQG VXESORW IDFWRUV :H ZLOO XVH DQ H[DPSOH WR FRPSDUH WKH XVH RI FRQIRXQGLQJ LQ D VSOLWSORW H[SHUn LPHQW &RQVLGHU D VSOLWSORW H[SHULPHQW ZLWK WKUHH ZKROH SORW IDFWRUV $ % DQG &

PAGE 56

DQG WKUHH VXESORW IDFWRUV 3 4 DQG 5 ZKHUH DOO IDFWRUV KDYH WZR OHYHOV 6XSSRVH RQO\ UXQV DUH SRVVLEOH DPRQJ WKH WRWDO QXPEHU RI FRPELQDWLRQV 7KHUH DUH WZR ZD\V WR DOORFDWH WKH ZKROH SORWV DQG VXESORWV IRU WKLV H[SHULPHQW :H FDQ XVH IRXU ZKROH SORWV ZLWK HDFK ZKROH SORW FRQWDLQLQJ IRXU VXESORWV RU ZH FDQ XVH HLJKW ZKROH SORWV ZLWK HDFK ZKROH SORW FRQWDLQLQJ WZR VXESORWV 7KH JRDO RI WKH H[SHULPHQW LV WR HVWLPDWH DOO VL[ PDLQ HIIHFWV DQG DV PDQ\ RI WKH QLQH WZRIDFWRU LQWHUDFWLRQV EHWZHHQ WKH ZKROH SORW DQG VXESORW IDFWRUV DV LV SRVVLEOH DOWKRXJK LW LV EHOLHYHG WKDW VRPH WZRIDFWRU LQWHUDFWLRQV DPRQJ WKH VXESORW IDFWRUV PLJKW EH VLJQLILFDQW 7R FRQVHUYH VSDFH LQ WKH WDEOHV WKH FRQIRXQGLQJ VWUXFWXUH RU DOLDV FKDLQV ZLOO EH JLYHQ RQO\ XS WR RUGHU WZR 7KHUHIRUH LI WKHUH LV D EODQN VSDFH LQ WKH DOLDV WDEOH LW PHDQV WKDW WKH HIIHFW LV DOLDVHG ZLWK LQWHUDFWLRQV RI RUGHU KLJKHU WKDQ WZR )LUVW VXSSRVH WKDW WKH H[SHULPHQWHU LJQRUHV WKH VSOLWSORW VWUXFWXUH E\ FRQVLGn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

PAGE 57

7DEOH 'HVLJQ /D\RXW IRU :LWK 'HILQLQJ &RQWUDVW $%&3 &345 $%45 DEFS DE FS DEFSTU DFU DFT FSTU DETU EFT EFU ESU EST DST TU DSU f 7DEOH $OLDV 6WUXFWXUH IRU $ % & 3 4 5 $% &3 45 $& %3 $3 %& $4 f§ %5 $5 %4 &4 35 &5 34

PAGE 58

7DEOH 'HVLJQ /D\RXW IRU WKH &RPELQHG [ :LWK 'HILQLQJ &RQWUDVW $%& f§ 345 $%&345 D E F DEF 3 3 3 3 4 T T T U U U U STU STU STU STU $ VHFRQG PHWKRG LQFRUSRUDWLQJ WKH VSOLWSORW QDWXUH DQG XVLQJ IRXU ZKROH SORWV LV WR FRQVLGHU UHGXFLQJ WKH ZKROH SORW IDFWRUV DQG VXESORW IDFWRUV VHSDUDWHO\ XVLQJ IUDFWLRQDO IDFWRULDOV $ B IUDFWLRQDO IDFWRULDO ZLWK GHILQLQJ FRQWUDVW $%& ZLOO EH XVHG IRU VHOHFWLQJ WKH ZKROH SORW WUHDWPHQWV DQG FRPELQHG ZLWK D B ZLWK GHILQLQJ FRQWUDVW 345 LQ VHOHFWLQJ WKH VXESORW WUHDWPHQWV VHH 7DEOH f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

PAGE 59

7DEOH $OLDV 6WUXFWXUH IRU [ $ %& % f§ $& & $% 3 45 4 35 5 34 $3 $4 $5 %3 %4 %5 &3 &4 &5 f§ VXESORW WUHDWPHQWV WR IRXU SHU ZKROH SORW VHH 7DEOH f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

PAGE 60

7DEOH 'HVLJQ /D\RXW IRU 6SOLW3ORW &RQIRXQGLQJ :LWK 'HILQLQJ &RQWUDVW $%& &345 $%345 D E F DEF f f 3 9 ST ST 4 T SU SU U U TU TU STU STU IDFWRU LQWHUDFWLRQV FDQ EH HVWLPDWHG XVLQJ WKLV GHVLJQ $OVR RQ D FRQVXOWLQJ OHYHO VRPH H[SHULPHQWHUV ZRXOG IHHO PRUH FRPIRUWDEOH ZLWK WKLV GHVLJQ VLQFH LW XVHV DOO VXESORW WUHDWPHQWV 7KH IRXUWK PHWKRG XVHV HLJKW ZKROH SORWV DQG VSOLWSORW FRQIRXQGLQJ 6LQFH WKHUH DUH HLJKW ZKROH SORWV WKH FRPSOHWH IDFWRULDO FDQ EH XVHG IRU WKH ZKROH SORW IDFn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

PAGE 61

7DEOH $OLDV 6WUXFWXUH IRU 6SOLW3ORW &RQIRXQGLQJ $ %& % $& & $% 3 4 5 f§ $3 $4 $5 %3 %4 %5 &3 45 &4 35 &5 34 7DEOH 'HVLJQ /D\RXW IRU 6SOLW3ORW &RQIRXQGLQJ LQ :KROH SORWV :LWK $%34 $&45 %&35 f D E DE F DF EH DEF $% $% $%a $% $% $%a $%a $% $& $&a $& $&a $& $& $& $& 34 34 34 34 34 34 34 34 45 45 45 45 45 45 45 45 STU SU TU ST ST TU SU STU f 4 3 U U 3 4 f

PAGE 62

7DEOH $OLDV 6WUXFWXUH IRU 6SOLW3ORW &RQIRXQGLQJ )RU :KROH 3ORWV :LWK $%34 $&45 %&35 $ % & $% 34 $& f§ 45 %& 35 3 4 f§ 5 $3 f§ %4 $4 f§ %3 &5 $5 f§ &4 %5 f§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

PAGE 63

WKH GHILQLQJ FRQWUDVW ZLWK WKH IHZHVW OHWWHUV 7KHUHIRUH LW LV ORRNLQJ DW WKH RYHUDOO UHVROXWLRQ RI WKH GHVLJQ DQG QRW WKH SDUWLDO UHVROXWLRQ +XDQJ &KHQ DQG 9RHONHO f DQG %LQJKDP DQG 6LWWHU f KDYH WDEOHG PLQLPXPDEHUUDWLRQ 0$f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f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

PAGE 64

WKHVH IDFWRUV LV H[SHQVLYH WKHQ DJDLQ FKDQJLQJ WKHP IRXU WLPHV VHHPV PRUH UHDVRQDEOH ,W LV QRW DQ HIILFLHQW DOORFDWLRQ RI WKH GHJUHHV RI IUHHGRP 8VLQJ HLJKW ZKROH SORWV ZLWK WZR VXESORWV SHU ZKROH SORW JLYHV GI IRU ZKROH SORW IDFWRUV DQG GI IRU VXESORW IDFWRUV DQG ZKROH SORW [ VXESORW IDFWRU LQWHUDFWLRQV 7KLV DOORFDWHV D GLVSURSRUWLRQDWH QXPEHU RI GHJUHHV RI IUHHGRP WR WKH ZKROH SORW IDFWRUV ,Q FRQWUDVW XVLQJ IRXU ZKROH SORWV ZLWK IRXU VXESORWV SHU ZKROH SORW JLYHV GI IRU ZKROH SORW IDFWRUV DQG GI IRU WHUPV LQYROYLQJ VXESORW IDFWRUV 8VLQJ WZR VXESORWV SHU ZKROH SORW LV VLPLODU WR XVLQJ EORFNV RI VL]H WZR LQ D EORFN GHVLJQ ZKLFK LV QRW JHQHUDOO\ UHFRPPHQGHG 0$ GHVLJQV DUH LQ JHQHUDO fJRRGf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ff 7KH EHVW

PAGE 65

GHVLJQ LV IRXQG XVLQJ WKH fPLQLPXPDEHUUDWLRQf 0$f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f DQG %LQJKDP DQG 6LWWHU f :KLFK HLJKW WUHDWPHQWV VKRXOG EH DGGHG LV WKH TXHVWLRQ WR EH DQVZHUHG QH[W EXW ILUVW ZH EULHIO\ GLVFXVV IROGRYHU GHVLJQV 7KH FRQFHSW RI D IROGRYHU GHVLJQ ZDV LQWURGXFHG LQ %R[ DQG +XQWHU Ef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f 7KH DGGLWLRQ RI WKH QHZ SRLQWV EUHDNV FHUWDLQ DOLDV FKDLQV DQG DOORZV HVWLPDWHV RI LQWHUDFWLRQV LQYROYLQJ WKH IDFWRU WKDW LV VHPLIROGHG WR EH FDOFXODWHG ZKLOH DGGLQJ RQO\ KDOI DV PDQ\ SRLQWV DV D FRPSOHWH IROGRYHU GHVLJQ ,Q WKH UHVW RI WKLV FKDSWHU ZH DSSO\ VHPLIROGLQJ WR VSOLWSORW H[SHULPHQWV

PAGE 66

,Q PRVW RI WKH FDVHV VWXGLHG KHUH WKH HLJKW DGGLWLRQDO SRLQWV DUH DGGHG WR WKH LQLWLDO UXQ GHVLJQ E\ VHPLIROGLQJ RQ HLWKHU RQH RU WZR VXESORW IDFWRUV ZKLFK UHVXOWV LQ D SRLQW GHVLJQ FRQVLVWLQJ RI IRXU ZKROH SORWV ZLWK VL[ VXESORWV SHU ZKROH SORW 7KHVH GHVLJQV ZLOO KDYH GI IRU WKH ZKROH SORW WUHDWPHQWV DQG GI IRU WKH VXESORW WUHDWPHQWV 7KH LQLWLDO SRLQW GHVLJQ LV EDODQFHG RYHU WKH VXESORW IDFWRUVf§HDFK IDFWRU KDV WKH VDPH QXPEHU RI KLJK DQG ORZ OHYHOV SUHVHQWf§ZKLFK DOORZV IRU WKH HIIHFWV WR HVWLPDWHG ZLWK HTXDO SUHFLVLRQ ,W LV GHVLUHG WR SUHVHUYH WKLV EDODQFH RI WKH VXESORW IDFWRUV LQ WKH SRLQW GHVLJQ DV ZHOO DV PDLQWDLQ WKH VDPH QXPEHU RI VXESORW WUHDWPHQWV SHU ZKROH SORW 7KHUHIRUH LQ KDOI RI WKH ZKROH SORWV WKH VHPLIROGLQJ LV RQ WKH KLJK OHYHO RI D VXESORW IDFWRU ZKLOH LQ WKH RWKHU KDOI WKH VHPLIROGLQJ LV RQ WKH ORZ OHYHO RI WKDW IDFWRU ,Q VRPH FDVHV LW LV QHFHVVDU\ WR IROG RQ D ZKROH SORW IDFWRU LQ RUGHU WR HVWLPDWH WKH PDLQ HIIHFWV RI WKH ZKROH SORW IDFWRUV ,Q WKHVH FDVHV WZR ZKROH SORWV DUH DGGHG VR WKDW WKH SRLQW GHVLJQ FRQVLVWV RI VL[ ZKROH SORWV ZLWK IRXU VXESORWV SHU ZKROH SORW 7KHVH GHVLJQV ZLOO KDYH GI IRU WKH ZKROH SORW WUHDWPHQWV DQG GI IRU WKH VXESORW WUHDWPHQWV $OO QLQH FDVHV LQYROYLQJ DQG ZKROH SORW DQG VXESORW IDFWRUV DUH FRQVLGHUHG +RZHYHU WZR FDVHV GR QRW QHHG WR EH LPSURYHG XSRQ 7ZR ZKROH SORW IDFWRUV DQG WZR VXESORW IDFWRUV WKH SRLQWV UHSUHVHQW WKH IXOO IDFWRULDO 6LQFH QR IUDFWLRQDWLQJ RU FRQIRXQGLQJ LV QHHGHG WKHUH LV QRWKLQJ WR LPSURYH XSRQ 7ZR ZKROH SORW IDFWRUV DQG WKUHH VXESORW IDFWRUV LQ WKLV FDVH WKH 0$ GHVLJQ SUHVHQWHG LQ +XDQJ &KHQ DQG 9RHONHO f LV WKH EHVW GHVLJQ SRVVLEOH DQG

PAGE 67

DOORZV HVWLPDWHV DOO RI WKH PDLQ HIIHFWV DQG DOO RI WKH WZRIDFWRU LQWHUDFWLRQV )RU DOO VLWXDWLRQV LQYROYLQJ OHVV WKDQ IRXU ZKROH SORW IDFWRUV D JHQHUDO PHWKRG FDQ EH XVHG WR FRQVWUXFW UXQ GHVLJQV )LUVW FRQVWUXFW D UXQ GHVLJQ WKDW XVHV IRXU ZKROH SORWV ZLWK IRXU VXESORWV LQ HDFK ZKROH SORW f ,I WKHUH DUH WZR ZKROH SORW IDFWRUV WKHQ XVH WKH FRPSOHWH IDFWRULDO LQ WKH ZKROH SORW IDFWRUV f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f ,Q WKH UHPDLQLQJ WZR ZKROH SORWV WKH VXESORW WUHDWPHQWV DUH IROGHG RQ D GLIIHUHQW VXESORW IDFWRU DJDLQ RQ WKH KLJK OHYHO LQ RQH ZKROH SORW DQG WKH ORZ OHYHO LQ WKH RWKHU ZKROH SORWf ,Q WKH VSHFLDO FDVH RI WKUHH VXESORW IDFWRUV WKH VHPLIROGLQJ LV GRQH RQ MXVW RQH IDFWRU VLQFH WKHUH LV RQO\ RQH DOLDV FKDLQ LQ WKH GHILQLQJ FRQWUDVW :KHQ WKHUH DUH IRXU RU PRUH ZKROH SORW IDFWRUV XVLQJ ZKROH SORWV UHVXOWV LQ LQVXIILFLHQW GHJUHHV RI IUHHGRP WR HVWLPDWH WKH PDLQ HIIHFWV RI WKH ZKROH SORW IDFWRUV

PAGE 68

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f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f DQG 63 )DFWRUV 3 4 5 6f 7R REWDLQ D SRLQW GHVLJQ XQGHU WKLV VLWXDWLRQ RQO\ WKH VXESORW WUHDWPHQWV QHHG WR EH IUDFWLRQDWHG RU FRQIRXQGHG )LUVW FRQVLGHU IUDFWLRQDWLQJ WKH VXESORW WUHDWPHQWV

PAGE 69

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f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

PAGE 70

7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ WKH 6DPH )UDFWLRQ >+3+6fGHQRWHV KLJK 36f DQG /3/6fGHQRWHV ORZ 36f` D E DE f 4 4 4 4 U U U U SV SV SV SV STUV STUV STUV STUV )ROF +3 /3 RQ +6 /6 V ST 3 TV TUV SU STU UV 7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ >+3+6fGHQRWHV KLJK 36f DQG /3/6fGHQRWHV ORZ 36f@ D E DE f 3 V 4 TU STU ST U STV TV SU SV SUV UV TUV STUV f )ROF +3 /3 RQ +6 /6 f SV 3 TUV TU STUV STU V

PAGE 71

7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ D )UDFWLRQ )DFWRULDO RI WKH :KROH 3ORW )DFWRUV D E F DEF S S S S T T T T ST ST ST ST f Lf Lf Lf :3 )DFWRUV $ % &f DQG 63 )DFWRUV 3 4f ,Q WKLV FDVH RQO\ WKH ZKROH SORW IDFWRUV QHHG WR EH IUDFWLRQDWHG 6LQFH QRWKLQJ QHHGV WR EH GRQH WR WKH VXESORW IDFWRUV WKHUH LV RQO\ RQH SRLQW GHVLJQ 7KH GHILQLQJ FRQWUDVW LV f§ $%& ZKLFK LV 5HVROXWLRQ ,,, 6LQFH WKH GHVLJQ HVWLPDWHV HYHU\WKLQJ VHW RXW LQ WKH JRDO RI WKH H[SHULPHQW QR SRLQWV QHHG WR EH DGGHG WR WKLV GHVLJQ +RZHYHU QRWH WKDW WKLV LV QRW WKH 0$ GHVLJQ ZKLFK LV UXQ XVLQJ HLJKW ZKROH SORWV ZLWK VXESORWV SHU ZKROH SORW 7KH SRLQW GHVLJQ LQ IRXU ZKROH SORWV ZLWK IRXU VXESORWV SHU ZKROH SORW LV VKRZQ LQ 7DEOH :3 )DFWRUV $ % &f DQG 63 )DFWRUV 3 4 5f 7R REWDLQ D SRLQW GHVLJQ LQ WKLV VLWXDWLRQ ERWK WKH ZKROH SORW DQG VXESORW WUHDWPHQWV QHHG WR EH IUDFWLRQDWHG RU FRQIRXQGHG )LUVW FRQVLGHU IUDFWLRQDWLQJ WKH ZKROH SORW DQG VXESORW WUHDWPHQWV VHSDUDWHO\ 7KH GHILQLQJ FRQWUDVW LV $%& 345 $%&345 ZKLFK LV UHVROXWLRQ ,,, 7KH WZRIDFWRU LQWHUDFWLRQV EHWZHHQ ZKROH SORW DQG VXESORW IDFWRUV DUH DOUHDG\ HVWLPDEOH 7KHUHIRUH WKHUH LV RQO\ RQH DOLDV FKDLQ WKDW QHHGV WR EH EURNHQ DQG WKDW LV DVVRFLDWHG ZLWK 345 7KH DGGLWLRQDO

PAGE 72

7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ WKH 6DPH )UDFWLRQ >+3GHQRWHV KLJK 3 DQG /3GHQRWHV ORZ 3@ D E F DEF 3 3 3 3 T T T T U U U U STU STU STU STU )ROF +3 /3 RQ /3 +3 f ST ST f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

PAGE 73

7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ >+3GHQRWHV KLJK 3 DQG /3GHQRWHV ORZ 3@ D E F DEF 34 34 3 3 SU SU 4 4 TU TU U U f f STU STU )ROF +3 /3 RQ /3 +3 4 STU 34 f U 3 SU TU :3 )DFWRUV $ % &f DQG 63 )DFWRUV 3 4 5 6f 7R REWDLQ D SRLQW GHVLJQ LQ WKLV VLWXDWLRQ ERWK WKH ZKROH SORW DQG VXESORW WUHDWPHQWV QHHG WR EH IUDFWLRQDWHG RU FRQIRXQGHG )LUVW FRQVLGHU IUDFWLRQDWLQJ WKH ZKROH SORW DQG VXESORW WUHDWPHQWV VHSDUDWHO\ 7KH GHILQLQJ FRQWUDVW LV f§ $%& f§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

PAGE 74

7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ WKH 6DPH )UDFWLRQ >+3+6fGHQRWHV KLJK 36f DQG /3/6fGHQRWHV ORZ 3f@ D E F DEF U U U U SV SV SV SV STUV STUV STUV STUV +3 )ROF /3 RQ +6 /6 V ST 3 TV TUV SU STU UV 1H[W FRQVLGHU VSOLWSORW FRQIRXQGLQJ 7KH GHILQLQJ FRQWUDVW LV $%& %&345 $&456 $%36 $345 f§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f DQG 63 )DFWRUV 3 4f ,Q WKLV FDVH RQO\ WKH ZKROH SORW WUHDWPHQWV QHHG WR EH IUDFWLRQDWHG 1RWH ZLWK IRXU ZKROH SORW IDFWRUV WKHUH DUH RQO\ GI IRU ZKROH SORW IDFWRU HIIHFWV +HQFH ZKROH

PAGE 75

7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ >+3+6fGHQRWHV KLJK 36f DQG /3/6fGHQRWHV ORZ 36f@ DEH DEH 3 V TU 4 STU ST STV U TV SU SUV SV UV TUV f STUV )ROF +3 /3 RQ /6 +6 f SV TUV 3 TU STUV V STU 7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ D )UDFWLRQDO )DFWRULDO RI WKH :KROH 3ORW )DFWRUV )ROG RQ $ E F DG DEHG G EHG 3 3 S 3 4 T 4 4 ST ST ST ST f f f f SORWV ZLOO QHHG WR EH DGGHG IRU DOO FDVHV LQYROYLQJ IRXU ZKROH SORW IDFWRUV 6LQFH QRWKLQJ QHHGV WR EH GRQH WR WKH VXESORW IDFWRUV WKHUH LV RQO\ RQH SRLQW GHVLJQ 7KH GHILQLQJ FRQWUDVW LV f§ $%& f§ %&' $' ZKLFK LV UHVROXWLRQ ,, 7KH DGGLWLRQDO ZKROH SORWV DUH REWDLQHG E\ VHPLIROGLQJ RQ IDFWRU $ 7KH SRLQW GHVLJQ LV VKRZQ LQ 7DEOH 7KH FKDLQV DUH EURNHQ DQG HYHU\WKLQJ LV HVWLPDEOH

PAGE 76

:3 )DFWRUV $ % & 'f DQG 63 )DFWRUV 3 4 5f 7R REWDLQ D SRLQW GHVLJQ LQ WKLV VLWXDWLRQ ERWK WKH ZKROH SORW DQG VXESORW WUHDWPHQWV QHHG WR EH IUDFWLRQDWHG RU FRQIRXQGHG )LUVW FRQVLGHU IUDFWLRQDWLQJ WKH ZKROH SORW DQG VXESORW WUHDWPHQWV VHSDUDWHO\ 7KH GHILQLQJ FRQWUDVW LV $%& %&' $' 345 $%&345 %&'345 $'345 ZKLFK LV UHVROXWLRQ ,, 7KH DGGLWLRQDO ZKROH SORW WUHDWPHQWV DUH REWDLQHG E\ VHPLIROGLQJ RQ IDFWRU $ 7KH SRVLWLYH IUDFWLRQ 345 LV UXQ LQ RQH ZKROH SORW ZKLOH WKH QHJDWLYH IUDFWLRQ f§345 LV UXQ LQ WKH RWKHU ZKROH SORW 7KH QHJDWLYH IUDFWLRQ FDQ EH WKRXJKW RI DV VHPLIROGLQJ RQ DQ\ VXESORW IDFWRU DQG SODFLQJ DOO RI WKH SRLQWV LQ RQH ZKROH SORW LQVWHDG RI WZR DV ZDV GRQH LQ DOO WKH FDVHV XS XQWLO QRZ 7KH SRLQW GHVLJQ LV VKRZQ LQ 7DEOH 7KH FKDLQV DUH EURNHQ DQG HYHU\WKLQJ LV HVWLPDEOH 1H[W FRQVLGHU VSOLWSORW FRQIRXQGLQJ 7KH GHILQLQJ FRQWUDVW LV $%& %&' $' &345 $%345 f§ %'345 f§ $&'345 ZKLFK LV UHVROXWLRQ ,, %HVLGHV EUHDNLQJ FKDLQV DPRQJ WKH ZKROH SORW IDFWRUV WKH FKDLQ &345 QHHGV WR EH EURNHQ 7KH DGGLWLRQDO ZKROH SORW WUHDWPHQWV DUH REWDLQHG E\ VHPLIROGLQJ RQ IDFWRU $ $JDLQ WKH SRVLWLYH IUDFWLRQ 345 LV UXQ LQ RQH ZKROH SORW ZLWK WKH QHJDWLYH IUDFWLRQ f§345 LV UXQ LQ WKH RWKHU ZKROH SORW $JDLQ WKLV FDQ EH WKRXJKW RI DV VHPLIROGLQJ HDFK IUDFWLRQ RQ DQ\ VXESORW IDFWRU DQG SODFLQJ DOO IRXU SRLQWV LQ WKH VDPH ZKROH SORW 7KH SRLQW GHVLJQ LV VKRZQ LQ 7DEOH 7KH FKDLQV DUH EURNHQ DQG HYHU\WKLQJ LV HVWLPDEOH

PAGE 77

7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ WKH 6DPH )UDFWLRQ E F DG DEHG S 3 3 3 T T T T U U U U STU STU STU STU )ROG RQ $ G EHG 3 ST T SU U TU STU f 7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ E F DG DEHG ST 3 ST 3 SU T SU T TU U TU U f STU f STU )ROG RQ $ G EHG 3 ST T SU U TU STU f

PAGE 78

:3 )DFWRUV $ % & 'f DQG 63 )DFWRUV 3 4 3 6f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f§ $' 345 $'456 $' 36 ZKLFK LV UHVROXWLRQ ,, %Hn VLGHV EUHDNLQJ FKDLQV DPRQJ WKH ZKROH SORW IDFWRUV WKH FKDLQV $'36 DQG 36 QHHG WR EH EURNHQ 7KH DGGLWLRQDO ZKROH SORW WUHDWPHQWV DUH REWDLQHG E\ VHPLIROGLQJ RQ IDFWRU $ 7KH VXESORWV DUH VHPLIROGHG RQ IDFWRU 3 LQ RQH ZKROH SORW DQG IDFWRU 6 LQ DQRWKHU ZKROH SORW 7KH SRLQW GHVLJQ LV VKRZQ LQ 7DEOH 1H[W FRQVLGHU VSOLWSORW FRQIRXQGLQJ 7KH GHILQLQJ FRQWUDVW LV f§ $%& %&' $' $&345 %'456 $%&'36 %345 $%'345 &'345 $&'456 &456 $%456 '36 $36 %&36 ZKLFK LV UHVROXWLRQ ,, 7KH DGGLWLRQDO ZKROH SORW WUHDWPHQWV DUH REWDLQHG E\ VHPLIROGLQJ RQ IDFWRU $ $JDLQ WKH VXESORW IDFWRUV DUH VHPLIROGHG RQ 3 DQG 6 &DUH PXVW EH WDNHQ

PAGE 79

ZKHQ FKRRVLQJ ZKLFK ZKROH SORWV WKH VXESORW IDFWRUV DUH VHPLIROGHG 2WKHUZLVH WKH VDPH WUHDWPHQW FRPELQDWLRQV ZLOO RFFXU LQ ERWK DGGLWLRQDO ZKROH SORWV 7KLV RFFXUV ZKHQ WKH VHPLIROGLQJ XVHV WKH ZKROH SORWV FRQWDLQLQJ WKH VXESORW WUHDWPHQWV GHILQHG E\ 345 456 DQG 345a 456a RU 345 456a DQG 345a 456 $Q\ RWKHU FRPELQDWLRQ LV ILQH ,Q WKLV VHFWLRQ 3 LV VHPLIROGHG LQ WKH ZKROH SORW FRQWDLQLQJ ZKROH WUHDWPHQW F 345a456f DQG 6 LV VHPLIROGHG LQ WKH ZKROH SORW FRQWDLQLQJ ZKROH SORW WUHDWPHQW DEHG 345 456f 7KH SRLQWV GHVLJQ LV VKRZQ LQ 7DEOH 0RVW RI WKH FKDLQV DUH EURNHQ EXW VRPH RI WKH WZRIDFWRU LQWHUDFWLRQV DPRQJ WKH VXESORW IDFWRUV DUH DOLDVHG ZLWK HDFK RWKHU $OVR IRXU RI WKH VL[WHHQ WZRIDFWRU LQWHUn DFWLRQV EHWZHHQ ZKROH SORW DQG VXESORW IDFWRUV PXVW EH DVVXPHG QHJOLJLEOH 7KHVH WHUPV DUH $6 &6 '6 DQG '3 7KLV LV IDLUO\ QLFH VLQFH WKUHH RI WKHVH WHUPV LQYROYH VXESORW IDFWRU 6 7KHUHIRUH LI LW LV EHOLHYHG WKDW RQH RI WKH VXESORW IDFWRUV LV XQOLNHO\ WR LQWHUDFW ZLWK WKH ZKROH SORW IDFWRUV WKHVH WHUPV RU HIIHFWV FRXOG EH DVVXPHG QHJOLn JLEOH 7KLV GRHV QRW VHHP XQUHDVRQDEOH 1RZ WKH VXESORW GI DUH SDUWLWLRQHG LQWR GI IRU WKH VXESORW IDFWRU PDLQ HIIHFWV GI IRU WKH ZKROH SORW E\ VXESORW LQWHUDFWLRQV DQG IRU WZRIDFWRU LQWHUDFWLRQV DPRQJ VXESORW IDFWRUV WKHVH WZR HIIHFWV FDQ EH DQ\ SDLU H[FHSW 34 DQG 46 RU 35 DQG 56f $Q ([DPSOH 7R LOOXVWUDWH KRZ DQ H[SHULPHQW FRXOG EH FDUULHG RXW DQG DQDO\]HG DQ H[DPSOH LV SUHVHQWHG 7KH H[DPSOH IURP 7DJXFKL f LQYROYHV WKH VWXG\ RI D ZRRO ZDVKLQJ DQG FDUGLQJ SURFHVV 7KH RULJLQDO H[SHULPHQW XVHG D [ LQQHU DQG RXWHU

PAGE 80

7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ WKH 6DPH )UDFWLRQ )ROG RQ $ E F DG DEHG G EHG ST TV SU UV V 3 TUV STU 4 T 4 T U U U U SV SV SV SV STUV STUV STUV STUV 7DEOH 3RLQW 'HVLJQ IRU WKH &DVH RI :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ E F DG DEHG 9 TUV f T STU V TU U TV ST STV SV UV SU SUV STUV )ROG RQ $ G EHG STUV TV SV UV T 3 U STU

PAGE 81

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f§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

PAGE 82

7DEOH 3RLQW 'HVLJQ IRU WKH ([DPSOH

PAGE 83

7DEOH (IIHFWV 7DEOH IRU WKH ([DPSOH 7HUP (IIHFW &RHII 6WG (UURU WYDOXH 3YDOXH &RQVWDQW ; I < = r $ % & ;r$ ;r% ;r&
PAGE 84

(IIHFW6WDQGDUG (UURU )LJXUH 1RUPDO 3UREDELOLW\ 3ORW IRU WKH ([DPSOH

PAGE 85

6XPPDU\ 7KH PDLQ JRDO RI WKLV FKDSWHU LV WR XQGHUVWDQG VRPH RI WKH FRPSOLFDWLRQV LQYROYHG ZLWK XVLQJ WKH WZR W\SHV RI FRQIRXQGLQJ LQ VSOLWSORW H[SHULPHQWV ,I WKH GHVLJQ LV FKRVHQ XVLQJ WKH 0$ FULWHULRQ WKHQ GHVLJQV FRQVWUXFWHG E\ FRPELQLQJ WKH IUDFWLRQDO IDFWRULDO RI WKH ZKROH SORW WUHDWPHQWV ZLWK D IUDFWLRQDO IDFWRULDO RI WKH VXESORW WUHDWn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

PAGE 86

7DEOH &RQGLWLRQ 1XPEHUV IRU WKH 9DULRXV &DVHV RI :3 )DFWRUV RI 63 )DFWRUV 7\SH RI &RQIRXQGLQJ &RQGLWLRQ 1XPEHU N 6DPH )UDFWLRQ 6SOLW3ORW &RQIRXQGLQJ 6DPH )UDFWLRQ 6SOLW3ORW &RQIRXQGLQJ 6DPH )UDFWLRQ 6SOLW3ORW &RQIRXQGLQJ 1HLWKHU 6DPH )UDFWLRQ 6SOLW3ORW &RQIRXQGLQJ 6DPH )UDFWLRQ 6SOLW3ORW &RQIRXQGLQJ WKH UXQ GHVLJQV 7KH SKUDVH fWKH FKDLQV DUH EURNHQf XVHG WKURXJKRXW WKLV FKDSWHU GRHV QRW PHDQ WKDW DOO RI WKH HIIHFWV DUH QR ORQJHU DOLDVHG 6RPHWLPHV WKH HIIHFWV DUH DOLDVHG DW D ORZHU GHJUHH WKDQ XQLW\ FRPSOHWHO\ DOLDVHGf 7KHUHIRUH WKHUH LV VRPH GHJUHH RI FROOLQHDULW\ EHWZHHQ WKH HIIHFWV 7R PHDVXUH WKH PDJQLWXGH RI WKLV FROOLQHDULW\ WKH FRQGLWLRQ QXPEHU N 1 /DUJHVW (LJHQYDOXH RI ;n;f 6PDOOHVW (LJHQYDOXH RI ;n;f LV FDOFXODWHG IRU HDFK FDVH VHH 7DEOH f 0DQ\ WH[WERRNV GHFODUH FROOLQHDULW\ WR EH D SUREOHP LI N ,W LV VHHQ IURP 7DEOH WKDW FROOLQHDULW\ GRHV QRW VHHP WR EH D SUREOHP IRU WKH FDVHV FRQVLGHUHG LQ WKLV FKDSWHU 7KH YDULDQFH LQIODWLRQ IDFWRUV 9,)f

PAGE 87

DUH DOVR FDOFXODWHG 7KH\ DUH QRW UHSRUWHG KHUH EXW QRQH RI WKH 9,)f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

PAGE 88

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f§ T UHSUHVHQW WKH IUDFWLRQDO SURSRUWLRQ FRQWULEXWHG E\ FRPSRQHQW L 7KHQ WKH SURSRUWLRQV PXVW VDWLVI\ WKH IROORZLQJ FRQVWUDLQWV ;L [L L DQG WKH H[SHULPHQWDO UHJLRQ LV D T f§ Of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

PAGE 89

f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f SUHVHQWHG FRPELQHG PL[WXUH FRPSRQHQWSURFHVV YDULn DEOH GHVLJQV IRU Q SURFHVV YDULDEOHV WKDW XVH RQO\ D IUDFWLRQ RI WKH WRWDO QXPEHU RI SRVVLEOH GHVLJQ SRLQWV 7KH\ FRQVLGHUHG SURFHVV YDULDEOHV HDFK DW WZR OHYHOV DQG VXJJHVW IUDFWLRQV RI WKH IDFWRULDO EH FRQVLGHUHG 7ZR SODQV LQYROYLQJ WKH IUDFn WLRQDO IDFWRULDO GHVLJQ LQ WKH SURFHVV YDULDEOHV ZHUH GLVFXVVHG 7KH ILUVW SODQ FDOOHG D PDWFKHG IUDFWLRQ SODFHV WKH VDPH IUDFWLRQDO UHSOLFDWH GHVLJQ DW HDFK PL[WXUH FRPSRVLWLRQ SRLQW 7KH RWKHU SODQ FDOOHG D PL[HG IUDFWLRQ XVHV GLIIHUHQW IUDFWLRQV DW WKH FRPSRVLWLRQ SRLQWV (DFK SODQ ZDV DSSOLHG WR WKH VLWXDWLRQ LQYROYLQJ WKUHH PL[n WXUH FRPSRQHQWV DQG WKUHH SURFHVV YDULDEOHV ZLWK WKH WRWDO QXPEHU RI GHVLJQ SRLQWV UDQJLQJ IURP IRU WKH FRPELQHG VLPSOH[FHQWURLG E\ IXOO IDFWRULDO WR RQO\ ZKLFK UHOLHG RQ UXQQLQJ WKH RQHTXDUWHU IUDFWLRQ ,W VKRXOG EH QRWHG WKDW LI LQWHUn DFWLRQV DPRQJ WKH SURFHVV YDULDEOHV DUH OLNHO\ WR EH SUHVHQW WKH XVH RI D IUDFWLRQDO IDFWRULDO ZLOO UHVXOW LQ ELDV EHLQJ SUHVHQW LQ WKH FRHIILFLHQW HVWLPDWHV &RUQHOO DQG

PAGE 90

*RUPDQ JLYH UHFRPPHQGDWLRQV UHJDUGLQJ WKH FKRLFH RI GHVLJQ ZKLFK GHSHQG RQ WKH IRUP RI WKH PRGHO WR EH ILWWHG DQG ZKHWKHU RU QRW WKHUH LV SULRU NQRZOHGJH RQ WKH PDJQLWXGH RI WKH H[SHULPHQWDO HUURU YDULDQFH &]LWURP f FRQVLGHUHG WKH EORFNLQJ RI PL[WXUH H[SHULPHQWV FRQVLVWLQJ RI WKUHH DQG IRXU PL[WXUH FRPSRQHQWV 6KH XVHG WZR RUWKRJRQDO EORFNV WR FRQVWUXFW 'RSWLPDO GHVLJQV 'UDSHU HW DO f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f DOWKRXJK RXU IUDFWLRQ ZLOO LQYROYH WKH PL[WXUH FRPSRQHQW EOHQGV DV ZHOO 7R HYDOXDWH WKH IUDFWLRQ ZH VKDOO PDNH XVH RI WKH 'FULWHULRQ FULWHULRQ &]LWURP ff 7KH QH[W VHFWLRQ SURYLGHV D W\SH RI H[SHULPHQWDO VLWXDWLRQ ZKLFK OHG WR WKLV UHVHDUFK ,Q WKH VHFWLRQ WKDW IROORZV D FRPELQHG PRGHO ZKLFK LV VOLJKWO\ GLIIHUHQW LQ IRUP IURP WKH FRPELQHG PL[WXUHSURFHVV YDULDEOH PRGHOV RUGLQDULO\ XVHG LV SUHVHQWHG 7KH PHWKRG IRU FRQVWUXFWLQJ WKH GHVLJQ DQG FRPSDULQJ LWfV 'FULWHULRQ LV GLVFXVVHG LQ WKH IRXUWK VHFWLRQ 7KH ILQDO VHFWLRQ RI WKLV SDSHU FRQWDLQV GHWDLOV RQ WKH DQDO\VLV RI WKH H[SHULPHQWV XVLQJ WKH SURSRVHG GHVLJQV DQG PRGHO IRUPV

PAGE 91

([SHULPHQWDO 6LWXDWLRQ +LVWRULFDOO\ LQ WKH PL[WXUH OLWHUDWXUH WKH LQWHUHVW LQ WKH EOHQGLQJ SURSHUWLHV RI WKH PL[WXUH FRPSRQHQWV KDV EHHQ KLJKHU WKDQ WKDW RI VWXG\LQJ WKH HIIHFWV RI WKH SURFHVV YDULDEOHV *HQHUDOO\ WKH SURFHVV YDULDEOHV KDYH EHHQ WUHDWHG DV fQRLVHf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n FHVV 'XULQJ WKLV SHULRG WKH SODQW SHUVRQQHO DUH WU\LQJ WR ILQG WKH SURSHU SURFHVVLQJ FRQGLWLRQV WR SURGXFH D XVHDEOH SURGXFW SURILWDEO\ $W VRPH SRLQW SODQW SHUVRQQHO QHHG WR UHHYDOXDWH WKH SRO\PHUfV IRUPXODWLRQ LQ OLJKW RI WKH DFWXDO UDZ PDWHULDOV DQG WKH SODQWfV IXOO VFDOH SURGXFWLRQ FDSDELOLWLHV 3ODQW SHUVRQQHO QHHG WR ILQG WKH fRSWLPDOf FRPELQDWLRQ RI WKH IRUPXODWLRQ DQG SURn FHVVLQJ FRQGLWLRQV 7UDGLWLRQDOO\ LQ UHVSRQVH VXUIDFH DSSOLFDWLRQV WKH PRGHO DVVXPHG IRU SURFHVV RSn WLPL]DWLRQ LV D VHFRQGRUGHU 7D\ORU VHULHV 6XFK DQ DVVXPSWLRQ LV EDVHG RQ EDFNJURXQG NQRZOHGJH LQ NQRZLQJ WKH WUXH VXUIDFH RYHU WKH H[SHULPHQWDO UHJLRQ FDQ EH DSSUR[n LPDWHG E\ ILWWLQJ D VHFRQGRUGHU PRGHO )XUWKHUPRUH LQ RXU SRO\PHU H[DPSOH DOO VHFRQGRUGHU WHUPV LQYROYLQJ PL[WXUH FRPSRQHQWV SURFHVV YDULDEOHV DQG WKH PL[WXUH

PAGE 92

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n DEOHV GHSHQGV RQ WKH EOHQGLQJ SURSHUWLHV RI WKH PL[WXUH FRPSRQHQWV WKH HIIHFWV RI WKH SURFHVV YDULDEOHV DQG DQ\ LQWHUDFWLRQV EHWZHHQ WKH PL[WXUH FRPSRQHQWV DQG SURn FHVV YDULDEOHV 7KHVH PRGHOV DUH W\SLFDOO\ VHFRQGRUGHU PRGHOV WKDW DOORZ IRU SXUH TXDGUDWLF DQG WZRIDFWRU LQWHUDFWLRQ WHUPV 7KH JHQHUDO VHFRQGRUGHU SRO\QRPLDO LQ T PL[WXUH FRPSRQHQWV LV Y $f r[L e 3L[L[ ‘ LM 1RZ XVLQJ WKH FRQVWUDLQWV A ;L DQG ;M ;L L ;L L LrL f

PAGE 93

(TXDWLRQ f EHFRPHV 9 ILR f‹t; -(‹$rA ?L O   IW MAL LM A LILR fW IW 3Df \ ^LLbL \ ^;M ‘‘ \ A \ A ILLM[A[M L L M (r ((DrU-!  LM ZKHUH ILc ILRILL IW DQG "rf IL ILM£ IRU L M T L M f 6XSSRVH WKDW DQ H[SHULPHQW LV WR EH SHUIRUPHG ZLWK T PL[WXUH FRPSRQHQWV [L [ [T DQG Q SURFHVV YDULDEOHV =? ‘ ‘ ‘ ]Q ,Q WKH SURFHVV YDULDEOHV OHW XV FRQVLGHU WKH PRGHO Q Q O39 2/T ( DN=N ( ( DNO=N=O f IF O NO 7KHQ WKHUH DUH WZR PDLQ W\SHV RI FRPELQHG PRGHOV VHH &RUQHOO ff WKDW FDQ EH XVHG LQ WKLV VLWXDWLRQ 7KH ILUVW W\SH LV D PRGHO ZKLFK FURVVHV WKH PL[WXUH PRGHO WHUPV LQ (TXDWLRQ f ZLWK HDFK DQG HYHU\ WHUP RI (TXDWLRQ f 7KLV SURGXFHV WKH FRPELQHG PRGHO Q Q r"[]f ( r} ( ( 3rL[L[L( ( UOLN;b=`L 7 =(( OLNO;L=N=L  LM  IF O L O NO ((( ALMN[L[M=N 7 (O \ (O \ LMNO[L[M=N=L f LM IH O LM NO ZKLFK LQFOXGHV SDUDPHWHUV IRU WKUHH DQG IRXU IDFWRU LQWHUDFWLRQV 'HSHQGLQJ RQ WKH GHVLJQ WKH PRGHO RI (TXDWLRQ f SURYLGHV D PHDVXUH RI WKH OLQHDU DQG QRQOLQHDU EOHQGLQJ SURSHUWLHV RI WKH PL[WXUH FRPSRQHQWV DYHUDJHG DFURVV WKH VHWWLQJV RI WKH SURFHVV YDULDEOHV DV ZHOO DV WKH HIIHFWV RI WKH SURFHVV YDULDEOHV RQ WKH OLQHDU DQG QRQOLQHDU EOHQGLQJ SURSHUWLHV

PAGE 94

7KH VHFRQG W\SH RI FRPELQHG PRGHO LV WKH DGGLWLYH PRGHO ZKLFK FRPELQHV WKH PRGHOV LQ (TXDWLRQV f DQG f ZLWKRXW FURVVLQJ DQ\ RI WKH [W DQG =M WHUPV 7KLV SURGXFHV WKH PRGHO YIF]f er ee 3LM;L;M L LM Q Q A tN]N AA2LNL=N=L f N NO (TXDWLRQf SURYLGHV D PHDVXUH RI WKH TXDGUDWLF EOHQGLQJ RI WKH PL[WXUH FRPSRQHQWV RQ WKH UHVSRQVH DV ZHOO DV XS WR WZRIDFWRU LQWHUDFWLRQV EHWZHHQ WKH SURFHVV YDULDEOHV RQ WKH UHVSRQVH 6LQFH WKH PRGHO GRHV QRW FRQWDLQ DQ\ FURVVSURGXFW WHUPV EHWZHHQ WKH PL[WXUH FRPSRQHQWV DQG WKH SURFHVV YDULDEOHV ZKHQ ILWWLQJ (TXDWLRQ f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n DEOHV LQFUHDVHV WKH PRGHO LQ (TXDWLRQ f ZLOO UHTXLUH D GHVLJQ ZLWK D ODUJH QXPEHU RI SRLQWV :KLOH WKH ILWWLQJ RI WKH PRGHO LQ (TXDWLRQ f SHUPLWV WKH XVH RI D VPDOOHU GHVLJQ WKDQ WKH ILWWLQJ RI WKH PRGHO LQ (TXDWLRQ f LW GRHV QRW DV SRLQWHG RXW HDUOLHU DGGUHVV WKH HVWLPDWLRQ RI WKH PL[WXUH FRPSRQHQWV E\ SURFHVV YDULDEOH LQWHUn DFWLRQV ,I FRVW FRQVWUDLQWV OLPLW WKH VL]H RI WKH H[SHULPHQW \HW LQWHUDFWLRQV EHWZHHQ

PAGE 95

PL[WXUH FRPSRQHQWV DQG SURFHVV YDULDEOHV DUH EHOLHYHG WR EH LPSRUWDQW VRPH VRUW RI FRPSURPLVH EHWZHHQ WKHVH WZR PRGHOV LV QHHGHG 0RVW RI WKH PRGHO IRUPV WKDW KDYH EHHQ SURSRVHG IRU UHVSRQVH VXUIDFH LQYHVWLn JDWLRQV DUH EDVHG RQ D 7D\ORU VHULHV DSSUR[LPDWLRQ ,Q NHHSLQJ ZLWK WKLV WUDGLWLRQ VXSSRVH WKDW WKH WUXH PRGHO IRU WKH Q SURFHVV YDULDEOHV LV D VHFRQGRUGHU PRGHO ,QVWHDG RI (TXDWLRQ f VXFK D PRGHO ZRXOG EH Q Q Q 9SY D DA=N DNN=N e DNL=N=L f f IF L IF L NL (TXDWLRQ f LV (TXDWLRQ f SOXV WKH Q SXUH TXDGUDWLF WHUPV $OVR D 7D\ORU VHULHV DSSUR[LPDWLRQ IRU D FRPELQHG VHFRQGRUGHU PRGHO ZRXOG LQFOXGH RQO\ XS WR WZR IDFWRU LQWHUDFWLRQV DQG ZRXOG QRW EH WKH PRGHO LQ (TXDWLRQ f &RPELQLQJ (TXDWLRQ f ZLWK (TXDWLRQ f RXU SURSRVHG FRPELQHG VHFRQGRUGHU PRGHO LV T T Q L O LM N O Q T Q (( tNO=N=O ((ZZ f NO  IF O ZKLFK LQFOXGHV WKH PL[WXUH PRGHO SOXV SXUH TXDGUDWLF DV ZHOO DV WZRIDFWRU LQWHUn DFWLRQ HIIHFWV DPRQJ WKH SURFHVV YDULDEOHV DQG WZRIDFWRU LQWHUDFWLRQV EHWZHHQ WKH OLQHDU EOHQGLQJ WHUPV LQ WKH PL[WXUH FRPSRQHQWV DQG WKH PDLQ HIIHFW WHUPV LQ WKH SURFHVV YDULDEOHV 7KH PLQLPXP QXPEHU RI GHVLJQ SRLQWV QHHGHG IRU WKH SURSRVHG PRGHO f LV OHVV WKDQ ZKDW LV QHHGHG IRU WKH FRPSOHWHO\ FURVVHG PRGHO f EXW LV PRUH WKDQ LV QHHGHG IRU WKH DGGLWLYH PRGHO f $OVR WKH SURSRVHG PRGHO FDQ EH XVHG HYHQ LI RQH GRHV QRW IHHO WKH QHHG IRU SXUH TXDGUDWLF WHUPV LQ WKH SURFHVV YDULDEOHV E\ VLPSO\ RPLWWLQJ WKRVH Q WHUPV

PAGE 96

7R VXSSRUW WKH ILWWLQJ RI (TXDWLRQ f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fT Q Of SRLQW GHVLJQ LV QHHGHG WR VXSSRUW WKH ILWWLQJ RI WKH PRGHO LQ (TXDWLRQ f $ SRSXODU UHVSRQVH VXUIDFH GHVLJQ IRU ILWWLQJ D VHFRQGRUGHU PRGHO RI WKH IRUP LQ (TXDWLRQ f LV WKH FHQWUDO FRPSRVLWH GHVLJQ FFGf ZKLFK FRQVLVWV RI D FRPSOHWH Q RU D 5HVROXWLRQ 9 IUDFWLRQ RI D f IDFWRULDO GHVLJQ Q D[LDO SRLQWV ZLWK OHYHOV sD IRU RQH IDFWRU DQG ]HUR IRU WKH UHVW DQG DW OHDVW RQH FHQWHU SRLQW ,I D LV VHOHFWHG WKH GHVLJQ UHJLRQ LV D K\SHUFXEH 7KH DSSURDFK WR UHGXFLQJ WKH QXPEHU RI REVHUYDWLRQV QHHGHG LQ D PL[WXUH H[SHUn LPHQW EHJLQV ZLWK D FFG LQ WKH SURFHVV YDULDEOHV $ VLPSOH[ LV WKHQ SODFHG DW HDFK SRLQW LQ WKH FFG ZLWK RQO\ D IUDFWLRQ RI WKH PL[WXUH EOHQGV DW HDFK SRLQW 7KH PL[WXUH EOHQGV DW HDFK GHVLJQ SRLQW DUH VHOHFWHG IURP WKH IXOO VLPSOH[FHQWURLG $ JHQHUDO QRWLRQ RI EDODQFH DPRQJ WKH PL[WXUH FRPSRQHQWV DFURVV WKH SURFHVV YDULDEOHV LV GHn VLUHG )LUVW RI DOO OHW XV LQVLVW RQ WKH VDPH QXPEHU RI PL[WXUH EOHQGV WR EH SUHVHQW

PAGE 97

DW ERWK WKH KLJK DQG ORZ OHYHOV RI HDFK SURFHVV YDULDEOH 6HFRQGO\ OHW XV LQVLVW RQ DOO RI WKH PL[WXUH EOHQGV EH SUHVHQW DW HDFK s IDFWRULDO OHYHO IRU HDFK SURFHVV YDULDEOH 7KHVH LGHDV VHHP YHU\ LQWXLWLYH DQG OHDG XV WR VHOHFW VRPH RI WKH PL[WXUH EOHQGV WR EH XVHG DW FHUWDLQ GHVLJQ SRLQWV DQG GLIIHUHQW PL[WXUH EOHQGV WR EH XVHG DW RWKHU GHVLJQ SRLQWV 7ZR GHVLJQV DUH FRQVLGHUHG IRU WKH ILWWLQJ RI WKH PRGHO LQ (TXDWLRQ f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n DEOHV 7KH PRGHO IRU WKLV H[DPSOH XVLQJ (TXDWLRQ f FRQWDLQV WHUPV 7KH WZR GHVLJQV DUH VKRZQ LQ )LJXUHV DQG )RU WKUHH PL[WXUH FRPSRQHQWV WKH GHVLJQ ZLWK WKH IXOO VLPSOH[FHQWURLG DW WKH FHQWHU RI WKH SURFHVV YDULDEOHV FRQVLVWV RI SRLQWV

PAGE 98

ZKLOH WKH VHFRQG GHVLJQ ZLWK MXVW WKH FHQWURLG FRQVLVWV RI SRLQWV (LWKHU FRXOG EH XVHG WR HVWLPDWH WKH WHUPV LQ WKH PRGHO 7KH GHVLJQV LQ )LJXUHV DQG FDQ EH H[WHQGHG WR H[SHULPHQWV LQYROYLQJ PRUH WKDQ PL[WXUH FRPSRQHQWV 0&f DQGRU PRUH WKDQ SURFHVV YDULDEOHV 39f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f XVHV WKH IROORZLQJ FRQVWUDLQWV ; [ [ 7KH UHVXOWLQJ PL[WXUH UHJLRQ LV D KH[DJRQ *HQHUDOO\ WKH RULJLQDO FRPSRQHQWV DUH WUDQVIRUPHG WR /SVHXGRFRPSRQHQWV [? f§ [ f§ /f f§ - L /Lf L T WR PDNH WKH FRQVWUXFWLRQ RI WKH GHVLJQ DQG WKH ILWWLQJ RI WKH PRGHO HDVLHU )RU WKH H[DPn SOH LQ WKLV SDSHU WKH PL[WXUH FRPSRQHQWV FDQ EH WUDQVIRUPHG WR /SVHXGRFRPSRQHQWV XVLQJ a B[La f ;L f f f f f n &DQGLGDWH SRLQWV IRU WKH WZR GHVLJQV LQ WKLV FDVH FRQVLVW RI WKH VL[ YHUWLFHV DQG

PAGE 99

[B ]B )LJXUH 3URSRVHG 'HVLJQ IRU WKH &DVH :LWK )XOO 6LPSOH[ FR WR

PAGE 100

[B ]B )LJXUH 3URSRVHG 'HVLJQ IRU WKH &DVH :LWK -XVW WKH &HQWURLG &2 &2

PAGE 101

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f GLVFXVVHV VRPH SUDFWLFDO DVSHFWV RI FKRRVLQJ FRPSXWHUDLGHG GHVLJQV 7KH PDLQ SRLQW KH PDNHV LV WKDW ZKLOH FRPSXWHUDLGHG GHVLJQV VKRXOG EH XVHG ZLWK FDXWLRQ WKH\ FDQ EH KHOSIXO ZKHQ WKH

PAGE 102

&HQWHU )LJXUH 3URSRVHG 'HVLJQ IRU WKH &DVH WR &Q

PAGE 103

H[SHULPHQWDO UHJLRQ LV LUUHJXODU RU ZKHQ WKH QXPEHU RI DYDLODEOH UXQV LV OLPLWHG RU IRU VSHFLDO PRGHOV VXFK DV WKRVH WKDW FRQWDLQ QRQOLQHDU SDUDPHWHUV &RPSXWHU JHQHUDWHG GHVLJQV DUH W\SLFDOO\ QRW XQLTXH 7KHUHIRUH LI WKH FRPSXWHU LV XVHG WR JHQHUDWH WZR RU WKUHH GHVLJQV ZLWK WKH VDPH QXPEHU RI SRLQWV WKH GHVLJQV ZLOO FRQWDLQ VRPH SRLQWV WKDW DUH QRW FRPPRQ WR DOO WKH GHVLJQV %HFDXVH RI WKLV RQH GHVLJQ PD\ EH PRUH DSSHDOLQJ WKDQ WKH RWKHUV IRU RQH UHDVRQ RU DQRWKHU 7KLV OHDGV WR WKH TXHVWLRQ f:RXOGQfW LW EH EHWWHU WR JHQHUDWH DQ DFFHSWDEOH RU DSSHDOLQJ GHVLJQ DQG FRPSDUH LW WR WKH FRPSXWHU JHQHUDWHG GHVLJQ"f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f LQ 6$6 352& 237(; UHTXLUHV WKH XVHU WR SURYLGH D FDQGLGDWH SRLQW OLVW DQG WKH PRGHO WR EH ILW 7KH FDQGLGDWH SRLQW OLVW IRU WKH H[DPSOHV LQ WKLV SDSHU FRQVLVWV RI WKH VLPSOH[FHQWURLG GHVLJQ DW HDFK SRLQW RI WKH FFG :LWK WKH SRLQW VLPSOH[FHQWURLG GHVLJQ LQ WKUHH FRPSRQHQWV DQG D SRLQW FFG LQ WZR SURFHVV YDULn DEOHV IRU H[DPSOH WKH FDQGLGDWH SRLQW OLVW IRU WKUHH PL[WXUH FRPSRQHQWV DQG WZR SURFHVV YDULDEOHV FRQVLVWV RI [ SRLQWV 7KH PRGHO WR EH ILW LV WKH PRGHO LQ (TXDWLRQ f ZLWK WHUPV 352& 237(; XVHV D UDQGRP VHHG DQG SHUIRUPV WHQ VHDUFKHV IURP WKH UDQGRP VWDUWLQJ SRLQWV $ IHZ RSWLRQV DUH XVHG 7KH ILUVW LV WR

PAGE 104

VSHFLI\ WKH QXPEHU RI SRLQWV LQ WKH ILQDO GHVLJQ 1 7KHQ WKH QXPEHU RI SRLQWV LQ WKH ILQDO GHVLJQ FKRVHQ E\ WKH FRPSXWHU LV IL[HG DW 1 RU WKH GHIDXOW ZKLFK LV WKH QXPEHU RI WHUPV LQ WKH PRGHO S f§ SOXV WHQ DGGLWLRQDO SRLQWV $ VHFRQG RSWLRQ LV WKH '(70$; RSWLRQ 7KLV RSWLRQ XVHV WKH GHWPD[ URXWLQH GHYHORSHG E\ 0LWFKHOO f WR REWDLQ D GHVLJQ ZLWK WKH PD[LPXP GHWHUPLQDQW RI ;n; ZKHUH ; LV D 1 [ S PDWUL[ FRQWDLQLQJ WKH GHVLJQ UXQV 352& RSWH[ UHWXUQV WKH FKRVHQ GHVLJQ DQG LWfV 'FULWHULRQ ZKLFK LV GHILQHG DV 'FULWHULRQ [ GHW ;n;f3 1 DQG LV XVHG WR FRPSDUH WKH GHVLJQV 7ZR GHVLJQV JHQHUDWHG E\ 352& 237(; DUH VKRZQ LQ )LJXUHV DQG 7KHVH GHVLJQV FRQVLVW RI DQG SRLQWV UHVSHFWLYHO\ DQG DUH WR EH FRPSDUHG WR RXU GHVLJQV LQ )LJXUHV DQG UHVSHFWLYHO\ 1RWH WKDW XQOHVV 352& RSWH[ LV UXQ ZLWK WKH VDPH RSWLRQV DQG VWDUWLQJ VHHG WKH GHVLJQV JHQHUDWHG FRXOG EH GLIIHUHQW 7DEOHV DQG VKRZ WKH 'FULWHULRQ IRU RXU SURSRVHG GHVLJQV DQG WKH GHVLJQV JHQHUDWHG E\ 352& RSWH[ IRU WKH FDVHV ZKHUH T f§ RU DQG Q f§ RU 7KH GHVLJQV IURP 352& 237(; DUH ZLWK 1 UHVWULFWHG WR WKH QXPEHU RI SRLQWV IURP WKH SURSRVHG GHVLJQ DV ZHOO DV WKH GHIDXOW 1 $OO ILYH FDVHV XQGHU FRQVLGHUDWLRQ DUH JLYHQ 7KH UHODWLYH HIILFLHQFLHV RI RXU WZR SURSRVHG GHVLJQV DUH GHILQHG DV 'FULWHULRQ IRU WKH SURSRVHG GHVLJQ 'FULWHULRQ IRU 352& RSWH[ GHVLJQV ZLWK VDPH QXPEHU RI SRLQWV f DQG DUH JLYHQ LQ 7DEOH )URP 7DEOH LW LV VHHQ WKDW LQ DOO FDVHV ERWK RI RXU GHVLJQV DUH DW OHDVW b DV HIILFLHQW DV WKH FRPSXWHU JHQHUDWHG GHVLJQV DQG H[FOXGLQJ

PAGE 105

[B ]B )LJXUH 3RLQW 'HVLJQ *HQHUDWHG E\ 6$6 IRU WKH &DVH &'

PAGE 106

[ =B )LJXUH 3RLQW 'HVLJQ *HQHUDWHG E\ 6$6 IRU WKH &DVH &2 &2

PAGE 107

7DEOH &RPSDULVRQ RI 2XU 'HVLJQ :LWK D )XOO 6LPSOH[ DW WKH &HQWHU WR WKH 'HVLJQV &KRVHQ E\ 352& 237(; 0L[WXUH &RPSRQHQWV 'FULWHULRQ 3URFHVV 352& 237(; 9DULDEOHV AUHVWULFWHGf 352& 237(; 1 f§ GHIDXOWf 2XU 'HVLJQ $7f f f f f f f f f f f f f 8SSHU DQG /RZHU &RQVWUDLQW f f f 7DEOH &RPSDULVRQ RI 2XU 'HVLJQ :LWK -XVW WKH &HQWURLG DW WKH &HQWHU WR WKH 'HVLJQV &KRVHQ E\ 352& RSWH[ 0L[WXUH &RPSRQHQWV 3URFHVV 9DULDEOHV 'FULWHULRQ 352& 237(; 9UHVWULFWHGf 352& 237(; 1 GHIDXOWf 2XU 'HVLJQ 1f f f f f f f f f f f f f 8SSHU DQG /RZHU &RQVWUDLQW f f f

PAGE 108

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f 6XSSRVH WKH PRGHO LQ (TXDWLRQ f LV XVHG IRU WKH SURFHVV YDULDEOHV LQVWHDG RI WKH PRGHO LQ (TXDWLRQ f LH WKH PRGHO IRU WKH SURFHVV YDULDEOHV LV EHOLHYHG WR EH ILUVWRUGHU SOXV LQWHUDFWLRQVf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

PAGE 109

'&ULWHULRQ '&ULWHULRQ &DVH R /2 2 T FR R FYL 'HVLJQ 6L]H &DVH 'HVLJQ 6L]H )LJXUH 3ORW RI '&ULWHULRQ 9HUVXV 'HVLJQ 6L]H & 'HVLJQ :LWK -XVW WKH &HQWURLG ) 'HVLJQ :LWK WKH )XOO 6LPSOH[&HQWURLG DW &HQWHU RI 3URFHVV 9DULDEOHVf

PAGE 110

&ULWHULRQ '&ULWHULRQ &DVH 'HVLJQ 6L]H &DVH )LJXUH &RQWLQXHG

PAGE 111

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f GLVFXVVHV WKH DQDO\VLV IRU WKH DGGLWLYH DQG FURVVHG PRGHOV WKDW XVH VWDQGDUG GHVLJQV ,Q WKLV SDSHU WKH DSSURSULDWH DQDO\VLV XSRQ ILWWLQJ WKH FRPSURn PLVHG PRGHO JLYHQ LQ (TXDWLRQ f XVLQJ HLWKHU RI WKH WZR SURSRVHG GHVLJQV LV QHHGHG ,W ZLOO EH DVVXPHG WKURXJKRXW WKLV VHFWLRQ WKDW WKH GHVLJQ LV UXQ DV D FRPSOHWHO\ UDQGRPL]HG GHVLJQ )RU WKH SURSRVHG GHVLJQV WKHUH LV D WRWDO RI 1 f§ GHJUHHV RI IUHHGRP (VWLPDWLQJ WKH WHUPV LQ WKH PRGHO UHTXLUHV S f§ GI 8VLQJ HIIHFW VSDUVLW\ WKH UHPDLQLQJ 1 f§ S GI FDQ EH SRROHG WR IRUP DQ HUURU VRXUFH ZKLFK FDQ EH XVHG WR WHVW WKH VLJQLILFDQFH RI WKH WHUPV LQ WKH PRGHO 7KH $129$ WDEOH LV VKRZQ LQ 7DEOH

PAGE 112

7DEOH $129$ 7DEOH IRU WKH 3URSRVHG 'HVLJQV 8VLQJ WKH &RPSURPLVHG 0RGHO 6RXUFH ') PL[WXUH FRPSRQHQWV OLQHDU T TXDGUDWLF TT Of SURFHVV YDULDEOHV IDFWRU LQW Q^Q f§ Of TXDGUDWLF Q 0& [ 39 LQW IDFWRU T ‘ Q 7RWDO (IIHFWV L B JfOfQQOfTLQ S (UURU 1S 7RWDO 1 f§ O

PAGE 113

7DEOH (UURU 'HJUHHV RI )UHHGRP IRU WKH 7ZR 3URSRVHG 'HVLJQV 8QGHU WKH &DVHV (UURU ') 0L[WXUH 3URFHVV )XOO -XVW &RPSRQHQWV 9DULDEOHV 6LPSOH[ &HQWURLG I r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

PAGE 114

RQ WKH OLQHDU EOHQGLQJ SURSHUWLHV RI WKH PL[WXUH FRPSRQHQWV 2QFH WKHVH KDYH EHHQ LQYHVWLJDWHG WKH QRQOLQHDU EOHQGLQJ SURSHUWLHV RI WKH PL[WXUH FRPSRQHQWV DQG WKH TXDGUDWLF HIIHFWV RI WKH SURFHVV YDULDEOHV FDQ EH WHVWHG DQG WKLV PD\ UHVXOW LQ FKDQJLQJ WKH WHUPV LQ WKH RULJLQDO PRGHO 7KH LQYHVWLJDWLRQ RI WKH LQWHUDFWLRQ WHUPV LQYROYHV D PDLQ TXHVWLRQ DQG WKH WZR VXEVHTXHQW TXHVWLRQV f 0DLQ ,V WKH HIIHFW RI WKH SURFHVV YDULDEOH WKH VDPH IRU DOO EOHQGV RI WKH PL[WXUH FRPSRQHQWV" ,I VR LV WKLV HIIHFW VLJQLILFDQWO\ GLIIHUHQW IURP ,I QRW ZKHUH DQG KRZ LV WKH HIIHFW GLIIHUHQW" IRU HDFK RI WKH Q SURFHVV YDULDEOHV )RU D JLYHQ SURFHVV YDULDEOH WKHUH DUH T LQWHUDFWLRQ WHUPV RI WKH IRUP O?N;?=N ON;]N f f ‘ TN;T]N IRU N Q 7KH TXHVWLRQV DERYH SDUWLWLRQ WKH T GI LQYROYLQJ WKH T LQWHUDFWLRQ WHUPV LQWR T f§ GI IRU WHVWLQJ WKH fPDLQf TXHVWLRQ DQG GI IRU WHVWLQJ VXETXHVWLRQ QXPEHU 7KH fPDLQf TXHVWLRQ DERYH DVNV LI WKH HIIHFW RI =N LV WKH VDPH RQ HDFK RI WKH PL[WXUH FRPSRQHQWV 7KHUHIRUH WKH K\SRWKHVHV DUH +R L N f§ N ‘ ‘ TN DW OHDVW RQH QRW HTXDO DQG WKH WHVWV DUH FDUULHG RXW IRU HDFK SURFHVV YDULDEOH 7R LOOXVWUDWH KRZ WKH WHVWV RQ WKH LQWHUDFWLRQ FRHIILFLHQWV DUH FDUULHG RXW FRQVLGHU WKH FDVH RI WKUHH PL[WXUH FRPSRQHQWV 7KHQ IRU SURFHVV YDULDEOH =? WKH K\SRWKHVHV

PAGE 115

DUH +R Q f§ L +A DW OHDVW RQH QRW HTXDO 7KH WHVWLQJ SURFHGXUH FDQ EH UHZULWWHQ DV WKH VLPXOWDQHRXV WHVW RI + Q + Q f +L Q L A +$ Q A ,I WKH WKUHH WHUPV KHUH DUH HQWHUHG ODVW LQ WKH PRGHO WKHQ LQ PDWUL[ QRWDWLRQ WKH DERYH K\SRWKHVHV EHFRPH ZKHUH / Sf§f PRGHO DQG F + / F LV D S [ YHFWRU RI DOO WKH SDUDPHWHUV LQ WKH 7KH DSSURSULDWH )WHVW LV B 4V +R S 06( VfQaSa ZKHUH V LQ RXU FDVHf LV WKH UDQN RI /n 06( LV WKH PHDQ VTXDUHG HUURU IURP WKH ILW RI WKH PRGHO DQG 4 On Ffn >Y [n[U O@ On Ff ZLWK I EHLQJ WKH OHDVW VTXDUHV HVWLPDWH RI WKH SDUDPHWHU YHFWRU 7KH WHVW FDQ HDVLO\ EH SHUIRUPHG LQ 6$6 XVLQJ 352& 5(* $IWHU WKH PRGHO VWDWHPHQW WKH WHVW VWDWHPHQW LV XVHG ZLWK D VHSDUDWH WHVW VWDWHPHQW IRU HDFK SURFHVV YDULDEOH )RU WKH DERYH H[DPSOH WKH 6$6 FRGH LV 352& 5(* 02'(/ < ;L; ‘ f ‘ [?=?;=?;L=? QRLQW 7(67 [?=? f§ []? ;?=? f§ [=? 7(67 [[] [] [[] [] 7(67 581

PAGE 116

ZRXOG SURGXFH WKH FRUUHFW VLPXOWDQHRXV WHVW RI WKH K\SRWKHVHV ,I + LV UHMHFWHG WKHQ WKHUH LV DQ LQWHUDFWLRQ SUHVHQW EHWZHHQ WKH PL[WXUH FRPSRQHQWV DQG WKH SURFHVV YDULDEOH +HQFH WKH HIIHFW RI WKH SURFHVV YDULDEOH LV QRW WKH VDPH IRU DOO PL[WXUH FRPSRQHQWV $ IXUWKHU LQYHVWLJDWLRQ RI WKH IRUP RI WKH LQWHUDFWLRQ QHHGV WR EH FDUULHG RXW 7KLV LV HDVLO\ DFFRPSOLVKHG ZLWK SORWV ,I + LV QRW UHMHFWHG WKHQ IRU D IL[HG SURFHVV YDULDEOH N WKH rfV DUH HTXDO ,Q RWKHU ZRUGV WKH SURFHVV YDULDEOH DIIHFWV WKH PL[WXUH FRPSRQHQWV HTXDOO\ ZKHQ WKH PL[WXUH UHJLRQ LV D VLPSOH[ $ IROORZXS WHVW LV QHHGHG VLQFH WKH HTXDO PDJQLWXGH FRXOG EH ]HUR $V LW WXUQV RXW WKLV WHVW ZLOO LQYHVWLJDWH ZKHWKHU RU QRW WKHUH LV DQ DGGLWLYH HIIHFW GXH WR WKH SURFHVV YDULDEOH 6LQFH + r N f§ f f f IF LV QRW UHMHFWHG WKH WHUPV LQ WKH PRGHO LQ (TXDWLRQ f LQYROYLQJ WKH LQWHUDFWLRQ RI WKH PL[WXUH FRPSRQHQWV DQG SURFHVV YDULDEOH N FDQ EH UHZULWWHQ DV O?N;?=N aN[]N + TN;T=N rA ]N N[]N K rrrrr ON=N[[ [ f f f [Tf ON]N ‘ 7KHUHIRUH WKH IROORZXS WHVW LV D WHVW RI WKH VLJQLILFDQFH RI WKH DGGLWLYH HIIHFW RI SURFHVV YDULDEOH N 1RWH WKLV ZRXOG KDYH EHHQ WKH WHVW IRU WKH DGGLWLYH PRGHO $OO WKDW QHHGV WR EH GRQH IRU WKH DQDO\VLV LV WR UHILW WKH PRGHO ZLWK ]N UHSODFLQJ WKH LQWHUDFWLRQV RI WKH PL[WXUH FRPSRQHQWV ZLWK ]N 7KLV ZLOO SURYLGH WKH GI WWHVW WKDW FRPELQHG ZLWK WKH T f§ GI IURP DERYH SDUWLWLRQV WKH T GI LQYROYLQJ WKH LQWHUDFWLRQV RI WKH PL[WXUH FRPSRQHQWV DQG WKH SURFHVV YDULDEOHV 7KLV LV WKH WHVW IRU WKH DGGLWLYH HIIHFW RI WKH SURFHVV YDULDEOH ]N DQG LV GRQH IRU HDFK SURFHVV YDULDEOH WKDW + LV QRW

PAGE 117

QR UHMHFWHG LQ WKH fPDLQf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f &KDSWHU f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

PAGE 118

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f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

PAGE 119

&ROOHFW DGGLWLRQDO REVHUYDWLRQV DW SRLQWV RWKHU WKDQ WKH GHVLJQ SRLQWV 7KHVH DUH FDOOHG FKHFN SRLQWV &RPSDUH WKH REVHUYHG GDWD DW WKH FKHFN SRLQWV WR WKH SUHGLFWHG UHVSRQVH IURP WKH ILWWHG PRGHO DW WKH FKHFN SRLQWV 7KH EDVLF LGHD LV WKDW LI WKH SUHGLFWLRQV DUH FORVH WR WKH REVHUYHG YDOXHV WKHQ WKH PRGHO GRHV QRW VXIIHU IURP ODFN RI ILW VHH &RUQHOO f IRU PRUH GHWDLOVf ,I ODFN RI ILW LV GHWHFWHG WKHQ WKH SURSRVHG PRGHO QHHGV WR EH XSJUDGHG E\ WKH DGGLWLRQ RI KLJKHURUGHU WHUPV 6LJQLILFDQW WHUPV LQ WKH ILWWHG PRGHO PLJKW VXJJHVW SRVVLEOH FDQGLGDWHV IRU ODFN RI ILW )RU WKH SURSRVHG PRGHO VRPH OLNHO\ FDQGLGDWHV DUH L WKH VSHFLDO FXELF WHUPV LQYROYLQJ WKH PL[WXUH FRPSRQHQWV LH ;?;; f LL WKH LQWHUDFWLRQV EHWZHHQ ELQDU\ EOHQGV DQG WKH OLQHDU HIIHFWV RI D SURFHVV YDULDEOH LH ;L;]L f LLL WKH LQWHUDFWLRQV EHWZHHQ WKH OLQHDU EOHQGLQJ WHUPV DQG WKH WZRIDFWRU LQWHUDFWLRQV DPRQJ WKH SURFHVV YDULDEOHV LH ;?=?= f 2QFH WKH QHFHVVDU\ WHUPV DUH DGGHG WR WKH PRGHO WKH QHZ PRGHO VKRXOG DOVR EH FKHFNHG IRU ODFN RI ILW 2QH QLFH SURSHUW\ RI RXU SURSRVHG GHVLJQV LV WKDW ERWK GHVLJQV FDQ VXSSRUW WKH ILWWLQJ RI KLJKHU RUGHU WHUPV ZLWKRXW DGGLQJ PRUH GHVLJQ SRLQWV 7KH GHVLJQ ZLWK WKH VLPSOH[FHQWURLG DW WKH FHQWHU RI WKH SURFHVV YDULDEOHV LV XVHG VLQFH LW KDV PRUH GHJUHHV

PAGE 120

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f DQG LLf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f LV WR DGG SRLQWV WKDW ZRXOG DLG LQ WKH ILWWLQJ RI KLJKHU RUGHU WHUPV VXFK DV IDFH FHQWHU SRLQWV ,I WKH SURSRVHG PRGHO VXIIHUV IURP

PAGE 121

7DEOH +LJKHU 2UGHU WHUPV 6XSSRUWHG E\ WKH 3URSRVHG 'HVLJQ :LWK )RXU 0L[WXUH &RPSRQHQWV DQG WKH 6LPSOH[&HQWURLG DW WKH &HQWHU RI WKH 3URFHVV 9DULDEOHV %LQDU\ %OHQG E\ 3URFHVV 9DULDEOH ,QWHUDFWLRQV 0& 39 0& 39 :LWKRXW 6SHFLDO &XELF 7HUPV 7KH %LQDU\ %OHQGV ZLWK 2QH RI WKH 3URFHVV 9DULDEOHV 2U %LQDU\ %OHQGV ZLWK %RWK 3URFHVV 9DULDEOHV $OO 7HUPV :LWK 6SHFLDO &XELF 7HUPV 6SHFLDO &XELF 3OXV WKH %LQDU\ %OHQGV RI WKH 0&fV ,QYROYHG LQ WKH 6SHFLDO &XELF :LWK (LWKHU 3URFHVV 9DULDEOH $OO 7HUPV ODFN RI ILW WKHQ VRPH RI WKH VSHFLDO FXELF WHUPV DQG WKH LQWHUDFWLRQV EHWZHHQ D ELQDU\ EOHQG DQG D SURFHVV YDULDEOH FDQ EH VXSSRUWHG E\ WKH SURSRVHG GHVLJQ ZLWKRXW DQ\ DGGLWLRQDO SRLQWV ([DPSOH &RQVLGHU DQ H[SHULPHQW LQYROYLQJ WKUHH PL[WXUH FRPSRQHQWV DT [b [f DQG WZR SURFHVV YDULDEOHV ^]? ]f 7KH GDWD IRU WKLV H[DPSOH LV FRQVWUXFWHG IRU LOOXVWUDWLYH SXUSRVHV DQG LV JLYHQ LQ 7DEOH 7KH DQDO\VLV LV FDUULHG RXW LQ WKH PDQQHU GHVFULEHG LQ WKLV FKDSWHU 7KH SDUDPHWHU HVWLPDWHV DQG WKH UHVXOWV RI SHUIRUPLQJ IWHVWV RQ WKH HVWLPDWHV DUH VKRZQ LQ 7DEOH 7KH DQDO\VLV VKRXOG EHJLQ ZLWK WKH PL[WXUH FRPSRQHQW E\ SURFHVV YDULDEOH LQWHUn DFWLRQV 7KH PDLQ TXHVWLRQ QHHGV WR EH DQVZHUHG ZKLFK DVNV LI WKH HIIHFW RI WKH SURFHVV YDULDEOH LV WKH VDPH IRU DOO EOHQGV RI WKH PL[WXUH FRPSRQHQWV 7KH DSSURSULDWH

PAGE 122

7DEOH 'DWD IRU WKH ([DPSOH ;? ; [ =? ] 5HVSRQVH

PAGE 123

7DEOH 2ULJLQDO 3DUDPHWHU (VWLPDWHV IRU WKH ([DPSOH 9DULDEOH ') 3DUDPHWHU (VWLPDWH 6WDQGDUG (UURU 7 IRU +2 3DUDPHWHU 3URE _7_ ; [ ;?; [[ [[ =L=L == =?= 9= ; ? = ;][ [] ;][ []

PAGE 124

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n WZHHQ WKH PL[WXUH DQG VWDQGDUG UHVSRQVH VXUIDFH VFKRROV RI WKRXJKW 7KH SURSRVHG

PAGE 125

PRGHO DOORZV WKH H[SHULPHQWHU WR VHH GLUHFWO\ WKH LQWHUDFWLRQ EHWZHHQ VSHFLILF PL[WXUH FRPSRQHQWV DQG VSHFLILF SURFHVV YDULDEOHV 7KHVH LQWHUDFWLRQV FDQ SURYLGH YDOXDEOH LQVLJKWV LQWR WKH HQWLUH SURFHVV EHLQJ VWXGLHG

PAGE 126

&+$37(5 0,;785( (;3(5,0(176 :,7+ 352&(66 9$5,$%/(6 ,1 $ 63/,73/27 6(77,1* :LWK PDQ\ LQGXVWULDO H[SHULPHQWV LW LV RIWHQ GLIILFXOW RU FRVWO\ WR UXQ WKH H[SHUn LPHQW LQ D FRPSOHWHO\ UDQGRP RUGHU 2IWHQ WLPHV D OHYHO RI RQH RU PRUH IDFWRUV LV IL[HG DQG WKHQ DOO RU D IUDFWLRQ RI DOO RI WKH FRPELQDWLRQV RI WKH RWKHU IDFWRUV DUH UXQ 7KLV SURFHVV LV UHSHDWHG XQWLO WKH GHVLUHG QXPEHU RI UXQV KDYH EHHQ XVHG 7KH UHVXOW LV D GHVLJQ WKDW XVHV UHVWULFWHG UDQGRPL]DWLRQ DQG UHVHPEOHV D VSOLWSORW GHVLJQ 0L[WXUH H[SHULPHQWV ZLWK SURFHVV YDULDEOHV DUH RIWHQ UXQ XQGHU WKH DERYH VHWWLQJ 7\SLFDOO\ WKH SURFHVV YDULDEOHV VHUYH DV WKH ZKROH SORW IDFWRUV DQG WKH PL[WXUH FRPn SRQHQWV PDNH XS WKH VXESORW IDFWRUV +RZHYHU LQ VRPH H[SHULPHQWV WKHLU UROHV FRXOG EH UHYHUVHG )RU D GLVFXVVLRQ RI PL[WXUH H[SHULPHQWV UXQ XVLQJ VSOLWSORW GHVLJQV VHH &RUQHOO f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

PAGE 127

)LUVW2UGHU 0RGHO LQ WKH 3URFHVV 9DULDEOHV )LUVW ZH VKDOO FRQVLGHU WKH FDVH ZKHUH D ILUVWRUGHU SOXV LRQWHUDFWLRQV PRGHO LV DVVXPHG IRU WKH SURFHVV YDULDEOHV ZKLFK DUH WR VHUYH DV WKH ZKROH SORW IDFWRUV 7KHUHn IRUH WKH PRGHO IRU WKH SURFHVV YDULDEOHV LV WKH PRGHO JLYHQ LQ (TXDWLRQ f RI &KDSWHU :KHQ WKLV PRGHO LV FRPELQHG ZLWK WKH PRGHO LQ (TXDWLRQ f RI &KDSWHU XQGHU D 7D\ORU VHULHV DSSURDFK WKH UHVXOWLQJ PRGHO LV T T Q T Q ";=f ‹A(‹ ( ( 2/NO=N=O (( n
PAGE 128

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

PAGE 129

[ =B )LJXUH 3URSRVHG 'HVLJQ IRU 6SOLW3ORW 6WUXFWXUH :LWK D )LUVW2UGHU 0RGHO WR WR

PAGE 130

7KH SURSRVHG GHVLJQ IRU WKUHH PL[WXUH FRPSRQHQWV DQG WZR SURFHVV YDULDEOHV FDQ EH H[WHQGHG WR KLJKHU GLPHQVLRQV )RU WKUHH SURFHVV YDULDEOHV WKH ZKROH SORW WUHDWPHQWV DUH QRZ RQ WKH YHUWLFHV RI D FXEH 7KHUH DUH HLJKW IDFWRULDO SRLQWV DQG WKHQ VRPH UHSOLFDWHV RI WKH FHQWHU 7KH FDVH RI IRXU PL[WXUH FRPSRQHQWV LV QRW TXLWH DV VWUDLJKWIRUZDUG VLQFH WKHUH DUH IRXU YHUWLFHV EXW VL[ PLGHGJH SRLQWV ,Q WKLV FDVH WZR UHSOLFDWHV RI WKH FHQWURLG ZLOO EH UXQ LQ WKH ZKROH SORWV WKDW FRQWDLQ WKH YHUWLFHV $OVR WKH FHQWHU RI WKH SURFHVV YDULDEOHV ZLOO KDYH VL[ UHSOLFDWHV RI WKH FHQWURLG 7KH GHVLJQ ZLOO WKHQ EH EDODQFHG ZLWK VL[ VXESORWV LQ HDFK ZKROH SORW (VWLPDWLRQ 0RGHO HVWLPDWLRQ XQGHU WKH VSOLWSORW VWUXFWXUH RI WKH H[SHULPHQW LV PRUH FRPSOH[ WKDQ ZKHQ WKH WKH H[SHULPHQW LV FRPSOHWHO\ UDQGRPL]HG 7KH LQLWLDO UDQGRPL]DWLRQ RI WKH SURFHVV YDULDEOHV FRUUHVSRQGV WR WKH ILUVW UDQGRPL]DWLRQ 7KLV JHQHUDWHV WKH ZKROH SORW HUURU YDULDQFH HUI 7KH UDQGRPL]DWLRQ RI WKH PL[WXUH EOHQGV FRQVWLWXWHV D VHFRQG UDQGRPL]DWLRQ ZKLFK JHQHUDWHV WKH VXESORW HUURU YDULDQFH HUI +HQFH D PRGHO IRU WKH H[SHULPHQW LV \ ; H ZKHUH  H a 1 9f 7KH PDWUL[ 9 HUI HUI UHSUHVHQWV WKH YDULDQFHFRYDULDQFH VWUXFWXUH RI WKH VSOLWSORW H[SHULPHQW LV D EORFN

PAGE 131

GLDJRQDO PDWUL[ RI Oc[L [ OO[ ZKHUH E LV WKH QXPEHU RI REVHUYDWLRQV LQ HDFK ZKROH SORW DQG LV WKH LGHQWLW\ PDWUL[ RI GLPHQVLRQ E 7KH VWUXFWXUH LPSOLHV WKDW REVHUYDWLRQV LQ GLIIHUHQW ZKROH SORWV DUH LQGHSHQGHQW ZKLOH REVHUYDWLRQV ZLWKLQ D ZKROH SORW DUH FRUUHODWHG 7R FRPSOLFDWH PDWWHUV HYHQ PRUH WKH VDPH PL[WXUH EOHQGV DUH QRW UXQ DW HDFK FRPELQDWLRQ RI WKH SURFHVV YDULDEOHV %HFDXVH RI WKLV GHYLDWLRQ LQ WUHDWPHQW VWUXFWXUH RUGLQDU\ OHDVW VTXDUHV LV QRW HTXLYDOHQW WR JHQHUDOL]HG OHDVW VTXDUHV 7KHUHIRUH WKH HVWLPDWLQJ HTXDWLRQ LV ;9[ff;9\ ZLWK 9DUf ;n9A;f 7KH HVWLPDWLRQ RI WKH PRGHO FRHIILFLHQWV DQG RI WKH YDULDQFHV RI WKHVH HVWLPDWHV GHn SHQGV RQ WKH PDWUL[ 9 DQG WKXV RQ UI DQG RI *HQHUDOO\ WKH DQDO\VLV UHTXLUHV WKH HVWLPDWLRQ RI WKH WZR HUURU YDULDQFHV 7KUHH PHWKRGV RUGLQDU\ OHDVW VTXDUHV UHn VWULFWHG PD[LPXP OLNHOLKRRG DQG D PHWKRG EDVHG RQ SXUH HUURU ZLOO EH FRQVLGHUHG IRU HVWLPDWLQJ WKHVH HUURU YDULDQFHV 2UGLQDU\ OHDVW VTXDUHV 2/6f DVVXPHV WKH REVHUYDWLRQV DUH LQGHSHQGHQW 7KXV LW LJQRUHV WKH GHSHQGHQW VWUXFWXUH RI WKH VSOLWSORW GHVLJQ 7KLV LV D QDLYH DSSURDFK LQ OLJKW RI WKH UHVWULFWHG UDQGRPL]DWLRQ DQG RQH ZRXOG H[SHFW WKDW LW ZLOO QRW SHUIRUP ZHOO 5HVWULFWHG PD[LPXP OLNHOLKRRG 5(0/f LV VLPLODU WR PD[LPXP OLNHOLKRRG HVWLn PDWLRQ LQ WKDW LW XVHV WKH OLNHOLKRRG RI D WUDQVIRUPDWLRQ RI \ ZKLFK LV EDVHG RQ WKH UHVLGXDOV VHH 5XVVHOO DQG %UDGOH\ ff 7KH SURSRVHG GHVLJQ XVHV U UHSOLFDWHV RI WKH ZKROH SORW DW WKH FHQWHU RI WKH SURFHVV YDULDEOHV :LWKLQ HDFK RI WKHVH ZKROH SORWV

PAGE 132

WKHUH DUH P UHSOLFDWHV RI WKH FHQWURLG 7KH ILQDO PHWKRG XVHV WKHVH UHSOLFDWH SRLQWV WR IRUP SXUH HUURU WHUPV ZKLFK FDQ EH XVHG WR HVWLPDWH WKH WZR HUURU YDULDQFHV 6LPXODWLRQ 6WXG\ $Q LQYHVWLJDWLRQ LV FRQGXFWHG WR HYDOXDWH WKH SHUIRUPDQFHV RI WKH WKUHH PHWKRGV $ VLPXODWLRQ VWXG\ LV FDUULHG RXW WR REWDLQ DSSUR[LPDWLRQV IRU WKH HOHPHQWV RI WKH PDWUL[ 9DUf 7KHQ WKH GHW 9DUf@ LV FRPSXWHG DQG FRPSDUHG WR WKH DV\PSWRWLF YDOXHV ZKLFK XVH NQRZQ 9 %\ ORRNLQJ DW WKH GHW 9DUfZH DUH FRPSDULQJ WKH VL]H RI WKH MRLQW FRQILGHQFH HOOLSVRLG DURXQG WKH SDUDPHWHU HVWLPDWHV 7KH ILYH FDVHV IURP &KDSWHU DUH FRQVLGHUHG ,W LV FRQYHQLHQW WR GHILQH WKH UHODWLRQVKLS $V G LQFUHDVHV LQ YDOXH JUHDWHU WKDQ XQLW\ WKH ZKROH SORW HUURU YDULDQFH EHFRPHV PXFK ODUJHU WKDQ WKH VXESORW HUURU YDULDQFH DQG WKXV WKH FRUUHODWLRQV DPRQJ WKH REVHUYDWLRQV EHFRPH VWURQJHU :LWKRXW ORVV RI JHQHUDOLW\ ZH ZLOO DVVXPH WKDW R? f§ DQG WKXV G HUI UHSUHVHQWV WKH ZKROH SORW HUURU YDULDQFH 7KH DV\PSWRWLF YDOXH IRU WKH 9DUf LV ;n9B;fB ZKHUH 9 G, 7KH VXPPDU\ YDOXH _;n9B;fB_ ZLOO EH XVHG IRU FRPSDULQJ WKH WKUHH PHWKRGV )RU 2/6 WKH HVWLPDWH RI LV ;n;MA;n\ 7KH HVWLPDWHG YDULDQFH RI LV 9DUf ;n;-A;n 9DU\ff ;;n;f ;;f;9;;;f ZKHUH 9 G, DV DERYH 7KHUHIRUH WKH TXDQWLW\ _;n;fB;n9;;n;f_ FDQ EH FDOFXODWHG DQG FRPSDUHG WR WKH DV\PSWRWLF YDOXH

PAGE 133

8VLQJ 5(0/ FKDQJHV WKH HVWLPDWH RI WR f§ ;n9& r;f ;9& \ ZKHUH 9F RI RI 7KH YDULDQFH LV 9DUf ;n9A;-A;n9 9DU\ff 9A;An9A;f [YF[f[nYFYY[[Y[f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f 7KH VLPXODWLRQ RI WKH YDOXHV QHHGHG WR FRPSXWH 9F LV D WZRVWHS SURFHGXUH FRUUHVSRQGLQJ WR WKH VSOLWSORW QDWXUH RI WKH H[SHULPHQW 7KH ZKROH SORW LV VLPXODWHG E\ : I]f 1Gf ZKHUH ]f FRQVLVWV RI WKH ZKROH SORW WHUPV LQ WKH PRGHO HYDOXDWHG DW WKH SRLQW ] DQG 1 Gf LV WKH ZKROH SORW HUURU WHUP 7KHQ WKH REVHUYDWLRQV DUH JLYHQ E\ \ []f N( -9f ZKHUH [ ]f FRQVLVWV RI WKH VXESORW WHUPV DQG WKH ZKROH SORW [ VXESORW LQWHUDFWLRQV LQ WKH PRGHO HYDOXDWHG DW WKH SRLQW [ ]f DQG 9f LV WKH VXESORW HUURU WHUP

PAGE 134

7KLV LV GRQH IRU HDFK ZKROH SORW 7KHVH REVHUYDWLRQV DORQJ ZLWK WKH GHVLJQ PDWUL[ DQG WKH PRGHO D[H LQSXWWHG LQWR 3URF 0L[HG LQ 6$6 8VLQJ WKH FRGH SURYLGHG LQ /HWVLQJHU 0\HUV DQG /HQWQHU f DQG JLYHQ LQ $SSHQGL[ & WKH HVWLPDWHG YDULDQFH FRPSRQHQWV DM DQG G DUH FRPSXWHG 7KHVH HVWLPDWHV DUH XVHG WR JHW 9F )LQDOO\ WKH GHWHUPLQDQW [9[f[nYFfYY[[YF[f LV FDOFXODWHG 7KLV SURFHGXUH LV UHSHDWHG WLPHV DQG WKH DYHUDJH GHWHUPLQDQW LV XVHG IRU FRPSDULVRQ WR WKH DV\PSWRWLF GHWHUPLQDQW )RU WKH PHWKRG XVLQJ SXUH HUURU HVWLPDWHV VLPXODWLRQ LV DOVR XVHG +RZHYHU WKH RQO\ UHVSRQVH YDOXHV WKDW QHHG WR EH VLPXODWHG DUH DW WKH UHSOLFDWH SRLQWV 7KH VLPXODWLRQ LV FDUULHG RXW WKH VDPH ZD\ DV GHVFULEHG IRU WKH 5(0/ FDVH ,Q RWKHU ZRUGV ILUVW : ]f $7 Gf LV VLPXODWHG DQG WKHQ \ [ ]f : 1f LV REWDLQHG IRU HDFK ZKROH SORW $ WRWDO RI U UHSOLFDWHV DW WKH FHQWHU RI WKH SURFHVV YDULDEOHV DUH XVHG 6LQFH D EDODQFHG GHVLJQ LV GHVLUHG LQ HDFK RI WKHVH ZKROH SORWV D WRWDO RI P UHSOLFDWHV RI WKH FHQWURLG DUH XVHG 7KLV OHDGV WR WKH UPf§ WRWDO GHJUHHV RI IUHHGRP EHLQJ SDUWLWLRQHG LQWR U f§ GI IRU WKH SXUH HUURU WHUP LQYROYLQJ WKH SURFHVV YDULDEOHV ZKROH SORW IDFWRUVf DQG U^P f§ f GI IRU WKH SXUH HUURU WHUP LQYROYLQJ WKH PL[WXUH FRPSRQHQWV VXESORW IDFWRUVf 7KH HVWLPDWHG ZKROH SORW SXUH HUURU WHUP LV ZKHUH < LP LV WKH PHDQ RI WKH VLPXODWHG YDOXHV LQ WKH 7K ZKROH SORW DQG < LV WKH RYHUDOO PHDQ RI WKH VLPXODWHG YDOXHV )URP WKH UHSOLFDWHG FHQWURLGV DW WKH FHQWHU RI

PAGE 135

WKH SURFHVV YDULDEOHV WKH HVWLPDWHG VXESORW SXUH HUURU WHUP LV f§ ? LQ f§ ?  H H \m \!f T B M A VSa U :KHQ WKHUH DUH WKUHH PL[WXUH FRPSRQHQWV WKH DERYH WHUP LV WKH HVWLPDWHG VXESORW HUURU WHUP +RZHYHU ZKHQ WKHUH DUH IRXU PL[WXUH FRPSRQHQWV WZR WRWDO UHSOLFDWHV RI WKH FHQWURLG DUH DGGHG WR WKH ZKROH SORWV WKDW FRQWDLQ WKH YHUWLFHV $V ZDV SRLQWHG RXW HDUOLHU WKLV LV GRQH WR KDYH D EDODQFHG GHVLJQ 6R LQ HDFK RI WKHVH ZKROH SORWV RQH H[WUD GHJUHH RI IUHHGRP FDQ EH XVHG WR HVWLPDWH WKH VXESORW HUURU WHUP 7KH HVWLPDWH RI WKH VXESORW HUURU WHUP EHFRPHV D ZHLJKWHG HUURU WHUP DQG LV JLYHQ E\ H B a f§ fA f f f Of6AS 9KL 6JS4 VS a f f f f f f U^P f R B 6VS9 6VS9 aO A f6nVS9AO IIDS2 IFB UP f§ f ZKHUH 9 DQG 6OS9N f§ 9Lf§
PAGE 136

&RPSDULVRQ RI 0HWKRGV 7KH PHWKRGV DUH FRPSDUHG IRU YDULRXV YDOXHV RI G 6LQFH LW LV DVVXPHG WKDW D? WKHQ G f§ D UHSUHVHQWV WKH ZKROH SORW HUURU YDULDQFH DV ZHOO DV WKH UDWLR RI WKH WZR HUURU WHUPV )ROORZLQJ /HWVLQJHU 0\HUV DQG /HQWQHU f ILYH YDOXHV DUH XVHG IRU G DQG 7KLV LQFOXGHV YDOXHV ZKHUH WKH ZKROH SORW HUURU YDULDQFH LV VPDOOHU WKDQ f HTXDO WR f DQG JUHDWHU WKDQ f WKH VXESORW HUURU YDULDQFH 7KH GHVLJQV FRQVLVWHG RI U DQG WRWDO UHSOLFDWHV RI WKH FHQWHU RI WKH SURFHVV YDULDEOHV ZKROH SORW IDFWRUVf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n;ff;n9;;n;f!_ 5H/ (II .;n9A;f ZKHUH 9 LV WKH WUXH YDULDQFHFRYDULDQFH PDWUL[ )RU 5(0/ DQG WKH SXUH HUURU PHWKRG UHODWLYH HIILFLHQF\ LV GHILQHG DV DYJ _;9a;fa;n9a99a;;9U;f_ ‘ ‘ N[nY[f ZKHUH WKH DYHUDJH LV RYHU WKH VLPXODWHG GHWHUPLQDQWV 7DEOHV VKRZ WKH UHVXOWV IRU WKH ILYH YDOXHV RI G DQG U DQG

PAGE 137

7DEOH 9DOXHV IRU WKH 9DULDQFH RI 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHVf G $V\PSWRWLF 2/6 5(0/ 3XUH (UURU U f 22O22f f f f f f f f f U 22O22f 22O22f f f f f f f f f U 22O22f 22O22f f f f f f f f f U 22222f 22O22f f f f f f f f f 6WDQGDUG (UURU IRU WKH 6LPXODWLRQ DUH LQ 3DUHQWKHVHV

PAGE 138

7DEOH 9DOXHV IRU WKH 9DULDQFH RI 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHVf G $V\PSWRWLF 2/6 5(0/ 3XUH (UURU U [ [ a [ B [ af [ B [ f [ [ OR [ BOR [ af [ B [ f [ a [ [ f [ f [ f [ f [ a [ [ [ af [ f [ ff [ a [ a [ f [ af [ [ f U [ +7 [ [ [ +7f [ B [ ff [ [ [ B [ ff [ BOR [ f [ f [ [ a [ af [ [ f [ [ a [ [ ff [ [ f [ a [ a [ [ f [ [ f U [ [ [ [ +7f [ B [ f [ [ a [ BOR [ O2f [ B [ O2ff [ f [ O2 [ [ O2f [ f [ aQf [ [ O2 [ f [ O2f [ O2f [ O2f [ f [ [ a [ f [ O2 [ ff U [ [ f [ [ ff [ B [ O2f [ O2 [ O2 [ BOR [ O2ff [ B [ f [ a [ +7 [ [ f [ f [ O2ff [ f [ f [ f [ f [ f [ O2f [ f [ O2f [ O2f [ O2f [ +7 [ O2ff 6WDQGDUG (UURU IRU WKH 6LPXODWLRQ DUH LQ 3DUHQWKHVHV

PAGE 139

7DEOH 9DOXHV IRU WKH 9DULDQFH RI 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHVf G $V\PSWRWLF 2/6 5(0/ 3XUH (UURU U f f f f f f f f f f U f f f f f f f f f f U f f f f f f f f f f U f f f f f f f f f f 6WDQGDUG (UURU IRU WKH 6LPX DWLRQ DUH LQ 3DUHQWKHVHV

PAGE 140

7DEOH 9DOXHV IRU WKH 9DULDQFH RI 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHVf G $V\PSWRWLF 2/6 5(0/ 3XUH (UURU U [ +7 [ [ [ f [ [ f [ [ [ B [ +7f [ B [ f [ a [ f [ [ +7f [ a [ f [ [ a [ [ f [ a [ ff [ a [ a [ f [ ,4ff [ a [ ,4ff U [ a [ a [ B [ f [ [ f [ +7 [ [ Q [ ff [ Q [ f [ a [ [ a [ f [ [ f [ [ a [ [ f [ [ ff [ a [ [ f [ f [ a [ f U [ a [ f [ B [ f [ [ f [ Q [ OR [ B [ ff [ [ f [ [ a [ a [ f [ [ f [ f [ [ [ f [ f [ +7f [ f [ a [ a [ f [ ,4r [ ff U L [ QU [ [ B [ QUf [ a [ O9f [ [ QU L [ LRB [ QUf [ [ f [ [ a [ [ f [ f [ f [ [ f [ [ af [ [ ORf L [ QU [ [ f [ f [ ,4n [ +7f 6WDQGDUG (UURU IRU WKH 6LPXODWLRQ DUH LQ 3DUHQWKHVHV

PAGE 141

7DEOH 9DOXHV IRU WKH 9DULDQFH RI 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV LQ D &RQVWUDLQHG 5HJLRQf G $V\PSWRWLF 2/6 5(0/ 3XUH (UURU U f f f f f f f f f f U f§ f f f f f f f f f f U f f f f f f f f f f U f f f f f f f f f f 6WDQGDUG (UURU IRU WKH 6LPXODWLRQ DUH LQ 3DUHQWKHVHV

PAGE 142

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f ( .[L < ( 3LStL = ( RWNL=N=L DA]O << L O LM NO N O L N

PAGE 143

7DEOH 5HODWLYH (IILFLHQFLHV IRU &RPSDULQJ 0HWKRGV RI (VWLPDWLQJ 9 :LWK 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV G U f§ U 2/6 5(0/ 3( 2/6 5(0/ 3( U U G 2/6 5(0/ 3( 2/6 5(0/ 3( $ GHVLJQ VLPLODU WR WKH RQH SUHVHQWHG LQ WKH SUHYLRXV VHFWLRQ LV QHHGHG ZKLFK ZLOO VXSSRUW WKH ILW RI WKLV PRGHO 3URSRVHG 'HVLJQ 6LQFH D VHFRQGRUGHU PRGHO LQ WKH SURFHVV YDULDEOHV LV FRQVLGHUHG D FFG LV QRZ DSSURSULDWH 7KH DGGLWLRQ RI WKH D[LDO SRLQWV ZLOO DOORZ IRU WKH ILWWLQJ RI WKLV PRGHO 7KHUHIRUH WKH GHVLJQ JLYHQ LQ )LJXUH ZLOO EH DXJPHQWHG ZLWK D[LDO SRLQWV ,Q WKHVH D[LDO SRLQWV P UHSOLFDWHV RI WKH FHQWURLG EOHQG ZLOO EH UXQ 7KLV ZLOO SUHVHUYH D EDODQFHG GHVLJQ DQG SURYLGH DQ DGGLWLRQDO U>P f§ f GI IRU HVWLPDWLQJ WKH VXESORW HUURU YDULDQFH LQ WKH SXUH HUURU DSSURDFK 7KHUH DUH VWLOO U UHSOLFDWHV RI WKH FHQWHU RI WKH SURFHVV YDULDEOHV DQG P UHSOLFDWHV RI WKH FHQWURLG LQ HDFK RI WKHVH 7KH GHVLJQ

PAGE 144

7DEOH 5HODWLYH (IILFLHQFLHV IRU &RPSDULQJ 0HWKRGV RI (VWLPDWLQJ 9 :LWK 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV G U U f§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

PAGE 145

[ =B )LJXUH 3URSRVHG 'HVLJQ IRU 6SOLW3ORW 6WUXFWXUH :LWK D 6HFRQG2UGHU 0RGHO &2 RR

PAGE 146

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

PAGE 147

7DEOH 5HODWLYH (IILFLHQFLHV IRU &RPSDULQJ 0HWKRGV RI (VWLPDWLQJ 9 :LWK 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV G U U 2/6 5(0/ 3( 2/6 5(0/ 3( U f§ U f§ G 2/6 5(0/ 3( 2/6 5(0/ 3( D OHYHO RI VRPH IDFWRUV DQG WKHQ UXQQLQJ FRPELQDWLRQV RI WKH RWKHU IDFWRUV OHDGV WR DQ H[SHULPHQW ZLWK D VSOLWSORW VWUXFWXUH 'HVLJQV DUH SURSRVHG ZKLFK FRQVLGHU WKH SURFHVV YDULDEOHV DV WKH ZKROH SORW IDFWRUV DQG WKH PL[WXUH FRPSRQHQWV DV WKH VXESORW IDFWRUV 7KHVH GHVLJQV DUH DQ H[WHQVLRQ WR WKH GHVLJQV JLYHQ LQ &KDSWHU 7KH VSOLWSORW VWUXFWXUH RI WKH H[SHULPHQW FRPSOLFDWHV WKH HVWLPDWLRQ RI WKH YDULn DQFH FRPSRQHQWV EHFDXVH 2/6 LV QR ORQJHU YDOLG 7ZR DOWHUQDWLYH PHWKRGV DUH SUHn VHQWHG 5(0/ DQG D SXUH HUURU DSSURDFK $ VLPXODWLRQ LV FRQGXFWHG WR JHW HVWLPDWHV RI WKH YDULDQFH FRPSRQHQWV 7KH WZR PHWKRGV DORQJ ZLWK 2/6 DUH FRPSDUHG XVLQJ WKH GHWHUPLQDQW RI WKH YDULDQFH RI DQG IRUPLQJ D UHODWLYH HIILFLHQF\ LQ WHUPV RI WKH DV\PSWRWLF YDOXH 7KH UHODWLYH HIILFLHQFLHV JLYH WKH LQIODWLRQ IDFWRU RI WKH VL]H RI WKH FRQILGHQFH HOOLSVRLG DURXQG UHODWLYH WR WKH WUXH VL]H

PAGE 148

7DEOH 5HODWLYH (IILFLHQFLHV IRU &RPSDULQJ 0HWKRGV RI (VWLPDWLQJ 9 :LWK 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV LQ D &RQVWUDLQHG 5HJLRQ G U U 2/6 5(0/ 3( 2/6 5(0/ 3( U U f§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

PAGE 149

7DEOH 5HODWLYH (IILFLHQFLHV IRU &RPSDULQJ 0HWKRGV RI (VWLPDWLQJ 9 :LWK 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV :LWK D 4XDGUDWLF 0RGHO G U U 5(0/ 3( 5(0/ 3( U U G 5(0/ 3( 5(0/ 3(

PAGE 150

7DEOH 5HODWLYH (IILFLHQFLHV IRU &RPSDULQJ 0HWKRGV RI (VWLPDWLQJ 9 :LWK 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV :LWK D 4XDGUDWLF 0RGHO G U f§ U f§ 5(0/ 3( 5(0/ 3( U U G 5(0/ 3( 5(0/ 3(

PAGE 151

7DEOH 5HODWLYH (IILFLHQFLHV IRU &RPSDULQJ 0HWKRGV RI (VWLPDWLQJ 9 :LWK 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV :LWK D 4XDGUDWLF 0RGHO G U U 5(0/ 3( 5(0/ 3( U U f§ G 5(0/ 3( 5(0/ 3(

PAGE 152

7DEOH 5HODWLYH (IILFLHQFLHV IRU &RPSDULQJ 0HWKRGV RI (VWLPDWLQJ 9 :LWK 0L[WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV :LWK D 4XDGUDWLF 0RGHO G U f§ U 5(0/ 3( 5(0/ 3( U U G 5(0/ 3( 5(0/ 3(

PAGE 153

7DEOH 5HODWLYH (IILFLHQFLHV IRU &RPSDULQJ 0HWKRGV RI (VWLPDWLQJ 9 :LWK 0L[n WXUH &RPSRQHQWV DQG 3URFHVV 9DULDEOHV LQ D &RQVWUDLQHG 5HJLRQ :LWK D 4XDGUDWLF 0RGHO G U f§ U 5(0/ 3( 5(0/ 3( U U G 5(0/ 3( 5(0/ 3(

PAGE 154

&+$37(5 6800$5< $1' &21&/86,216 0DQ\ H[SHULPHQWV WKDW DUH SHUIRUPHG LQ LQGXVWU\ KDYH FRVW DQGRU WLPH FRQn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

PAGE 155

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n SURDFK LV QHHGHG :H LQYHVWLJDWH KRZ RUGLQDU\ OHDVW VTXDUHV KROGV XS WR UHVWULFWHG PD[LPXP OLNHOLKRRG XQGHU YDULRXV UDWLRV RI WKH ZKROH SORW HUURU YDULDQFH WR WKH VXEn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

PAGE 156

$33(1',; $ 7$%/(6 )25 &+$37(5 '(6,*16 7KH WDEOHV LQ WKLV $SSHQGL[ JLYH WKH GHVLJQ SRLQWV IRU WKH VHYHQ FDVHV GLVFXVVHG LQ &KDSWHU 7KH FRGLQJ FRQYHQWLRQ LV DV IROORZV f f§ LV WKH ORZ OHYHO RI D IDFWRU f LV WKH KLJK OHYHO RI D IDFWRU 7KH SRLQWV IRU UXQ GHVLJQV DUH OLVWHG DW WKH EHJLQQLQJ RI WKH WDEOH IROORZHG E\ WKH DGGLWLRQDO SRLQWV

PAGE 157

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6HSDUDWH )UDFWLRQV $ % 3 4 5 6 O O O O O O O O O O O O O O O O O O O O O O O O

PAGE 158

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ $ % 3 4 5 6 O O O O O L O O L O O O O L O O O O O L O O L O

PAGE 159

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6HSDUDWH )UDFWLRQV $ % & 3 4 5 O L O O O L O O O O O O O L O L O O L O L O O O

PAGE 160

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ $ % & 3 4 5 L O L O L O L O O O O O O L O L L O L O O O O L

PAGE 161

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6HSDUDWH )UDFWLRQV $ % & 3 4 5 6 O O O O O O O O O O O O O O O O O O O O O O O O

PAGE 162

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ $ % & 3 4 5 6 O L L O O L O O L L O O L O O L O L O L O O O O

PAGE 163

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6HSDUDWH )UDFWLRQV $ % & 3 4 O L L O O O L O O L O O O O O O O O L O O L L O

PAGE 164

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6HSDUDWH )UDFWLRQV $ % & 3 4 5 O L O L O L O L O L O L O L O L O L O L L O L O

PAGE 165

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ $ % & 3 4 5 O O O O O L O L L O L O O L O L O L O L L O O O

PAGE 166

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6DPH )UDFWLRQ $ % & 3 4 5 6 L O O O O O O L L O O L L O O L O O O L L O O L

PAGE 167

7DEOH 'HVLJQ 3RLQWV IRU :3 )DFWRUV DQG 63 )DFWRUV 8VLQJ 6SOLW3ORW &RQIRXQGLQJ $ % & 3 4 5 6 O L L O L O O O O L O O O O O L L O O O L O O L

PAGE 168

$33(1',; % 7$%/(6 )25 &+$37(5 '(6,*16 7KH WDEOHV LQ WKLV $SSHQGL[ JLYH WKH GHVLJQ SRLQWV IRU WKH WZR SURSRVHG GHVLJQV DQG DOO FDVHV GLVFXVVHG LQ &KDSWHU 7KH FRGLQJ FRQYHQWLRQ LV DV IROORZV f WKH PL[WXUH FRPSRQHQWV DUH RQ D VFDOH RI WR DQG WKH GHFLPDO YDOXHV UHIHU WR SURSRUWLRQV RU IUDFWLRQV RI WKH FRPSRQHQWV f f§ LV WKH ORZ OHYHO RI D SURFHVV YDULDEOH f LV WKH PLGGOH OHYHO RI D SURFHVV YDULDEOH f LV WKH KLJK OHYHO RI D SURFHVV YDULDEOH 7KH DGGLWLRQDO GHVLJQ SRLQWV QHFHVVDU\ IRU WKH VLPSOH[FHQWURLG DW WKH FHQWHU RI WKH SURFHVV YDULDEOH DUH JLYHQ EHORZ WKH GHVLJQ XVLQJ MXVW WKH FHQWURLG DW WKH FHQWHU

PAGE 169

7DEOH 'HVLJQ 3RLQWV IRU &DVH ;? [ =O ] $GGLWLRQDO 3RLQWV IRU 6LPSOH[&HQWURLG

PAGE 170

7DEOH 'HVLJQ 3RLQWV IRU &DVH ;L ; =? = = $GGLWLRQDO 3RLQWV IRU 6LPSOH[&HQWURLG

PAGE 171

7DEOH 'HVLJQ 3RLQWV IRU &DVH ;? = ; =O = $GGLWLRQDO 3RLQWV IRU 6LPSOH[&HQWURLG

PAGE 172

7DEOH 'HVLJQ 3RLQWV IRU &DVH [[ b ; =? = = ;? [ ; [ =? = = $GGLWLRQDO 3RLQWV IRU 6LPSOH[&HQWURLG

PAGE 173

7DEOH 'HVLJQ 3RLQWV IRU &DVH ZLWK 8SSHU DQG /RZHU &RQVWUDLQWV $ $ $ =O = $ $ $ =O = $GGLWLRQDO 3RLQWV IRU 6 LPSOH[ &HQWURLG

PAGE 174

$33(1',; & 6$6 &2'( )25 352& 0,;(' &RQVLGHU WKH PRGHO RI WKH IRUP \ ; f ZKHUH H a 9 9f DQG 9 RI DA, 7KHQ WKH IROORZLQJ FRGH FDQ EH XVHG WR REWDLQ HVWLPDWHV RI RI DQG RI 352& 0,;(' 0(7+2' 5(0/ &/$66 :3 02'(/ < ),;(' ())(&76 5$1'20 :3 ZKHUH :3 LV D FODVVLILFDWLRQ YDULDEOH GHILQLQJ LQWR ZKLFK ZKROH SORW HDFK REVHUYDWLRQ IDOOV WKH PRGHO VWDWHPHQW GHILQHV WKH PRGHO PDWUL[ ;

PAGE 175

5()(5(1&(6 >@ $GGHOPDQ 6 f f6RPH 7ZR/HYHO )DFWRULDO 3ODQV :LWK 6SOLW3ORW &RQn IRXQGLQJf 7HFKQRPHWULFV SS >@ $QGHUVRQ 5 / f 6WDWLVWLFDO 7KHRU\ LQ 5HVHDUFK 0F*UDZ+LOO 1HZ @ %LQJKDP DQG 6LWWHU 5 6 f f0LQLPXP$EHUUDWLRQ 7ZR/HYHO )UDFn WLRQDO )DFWRULDO 6SOLW3ORW 'HVLJQVf 7HFKQRPHWULFV SS >@ %LVJDDUG 6 f f7KH 'HVLJQ DQG $QDO\VLV RI NaS [ TaU 6SOLW 3ORW ([SHULPHQWVf DFFHSWHG E\ -RXUQDO RI 4XDOLW\ 7HFKQRORJ\ @ %R[ ( 3 DQG 'UDSHU 1 5 f (PSLULFDO 0RGHO %XLOGLQJ DQG 5HVSRQVH 6XUIDFHV -RKQ :LOH\  6RQV 1HZ @ %R[ ( 3 DQG +XQWHU 6 Df 7KH NaS )UDFWLRQDO )DFWRULDO 'HVLJQV ,f 7HFKQRPHWULFV SS >@ %R[ ( 3 DQG +XQWHU 6 Ef 7KH NaS )UDFWLRQDO )DFWRULDO 'HVLJQV ,,f 7HFKQRPHWULFV SS >@ %R[ ( 3 DQG -RQHV 6 f f6SOLW3ORW 'HVLJQV IRU 5REXVW 3URGXFW ([SHULPHQWDWLRQf -RXUQDO RI $SSOLHG 6WDWLVWLFV SS >@ %R[ ( 3 DQG :LOVRQ % f f2Q WKH ([SHULPHQWDO $WWDLQPHQW RI 2SWLPXP &RQGLWLRQVf -RXUQDO RI WKH 5R\DO 6WDWLVWLFDO 6RFLHW\ 6HU % SS >@ &DQWHOO % DQG 5DPLUH] f f5REXVW 'HVLJQ RI D 3RO\VLOLFRQ 'HSRVLWLRQ 3URFHVV 8VLQJ D 6SOLW3ORW $QDO\VLVf 4XDOLW\ 5HOLDELOLW\ (QJLQHHULQJ ,QWHUQDWLRQDO SS @ &RUQHOO $ f ([SHULPHQWV :LWK 0L[WXUHV 'HVLJQV 0RGHOV DQG WKH $QDO\VLV RI 0L[WXUH 'DWD QG HG -RKQ :LOH\ t 6RQV 1HZ
PAGE 176

>@&RUQHOO $ f $QDO\]LQJ 'DWD IURP 0L[WXUH ([SHULPHQWV &RQWDLQn LQJ 3URFHVV 9DULDEOHV $ 6SOLW3ORW $SSURDFK -RXUQDO RI 4XDOLW\ 7HFKQRORJ\ SS >@ &RUQHOO $ DQG *RUPDQ : f )UDFWLRQDO 'HVLJQ 3ODQV IRU 3URFHVV 9DULDEOHV LQ 0L[WXUH ([SHULPHQWVf -RXUQDO RI 4XDOLW\ 7HFKQRORJ\ SS >@ &R[ 5 f 3ODQQLQJ ([SHULPHQWV -RKQ :LOH\ t 6RQV 1HZ @ &]LWURP 9 f f0L[WXUH ([SHULPHQWV ZLWK 3URFHVV 9DULDEOHV '2SWLPDO 2UWKRJRQDO ([SHULPHQWDO 'HVLJQVf &RPPXQLFDWLRQV LQ 6WDWLVWLFVf§7KHRU\ DQG 0HWKRGV SS >@ &]LWURP 9 f ([SHULPHQWDO 'HVLJQV IRU )RXU 0L[WXUH &RPSRQHQWV ZLWK 3URFHVV 9DULDEOHVf &RPPXQLFDWLRQV LQ 6WDWLVWLFVf§7KHRU\ DQG 0HWKRGV SS >@ 'UDSHU 1 5 3UHVFRWW 3 /HZLV 6 0 'HDQ $ 0 -RKQ 3 : 0 DQG 7XFN 0 f f0L[WXUH 'HVLJQV IRU )RXU &RPSRQHQWV LQ 2UWKRJRQDO %ORFNVf 7HFKQRPHWULFV SS >@ )LQQH\ f 7KH )UDFWLRQDO 5HSOLFDWLRQ RI )DFWRULDO $UUDQJHPHQWVf $QQDOV RI (XJHQLFV SS >@ )LVKHU 5 $ f 7KH $UUDQJHPHQW RI )LHOG ([SHULPHQWVf -RXUQDO RI WKH 0LQLVWU\ RI $JULFXOWXUH SS >@ +XDQJ 3 &KHQ DQG 9RHONHO 2 f f0LQLPXP$EHUUDWLRQ 7ZR/HYHO 6SOLW3ORW 'HVLJQVf 7HFKQRPHWULFV SS >@ .HPSWKRUQH 2 f 7KH 'HVLJQ DQG $QDO\VLV RI ([SHULPHQWV -RKQ :LOH\ t 6RQV 1HZ @ .KXUL $ DQG &RUQHOO $ f 5HVSRQVH 6XUIDFHV QG HGLWLRQ 0DUFHO 'HNNHU 1HZ @ .LHIHU DQG :ROIRZLW] f f2SWLPXP 'HVLJQV LQ 5HJUHVVLRQ 3UREOHPVf $QQDOV RI 0DWKHPDWLFDO 6WDWLVWLFV SS >@ /HWVLQJHU 0\HUV 5 + DQG /HQWQHU 0 f f5HVSRQVH 6XUIDFH 0HWKRGV IRU %L5DQGRPL]DWLRQ 6WUXFWXUHVf -RXUQDO RI 4XDOLW\ 7HFKQROn RJ\ SS >@ /XFDV 0 DQG -X + / f f6SOLW 3ORWWLQJ DQG 5DQGRPL]DWLRQ LQ ,Qn GXVWULDO ([SHULPHQWVf $64& 4XDOLW\ &RQJUHVV 7UDQVDFWLRQV $PHULFDQ 6RFLHW\ IRU 4XDOLW\ &RQWURO 1DVKYLOOH 71 SS

PAGE 177

>@ 0DUWLQ ) f 6WDWLVWLFDO 'HVLJQ DQG $QDO\VLV >@ 0HH 5 : DQG %DWHV 5 / f 6SOLW/RW 'HVLJQV ([SHULPHQWV IRU 0XOWLVWDJH %DWFK 3URFHVVHV 7HFKQRPHWULFV SS n@ 0LOOHU $ f f6WULS3ORW &RQILJXUDWLRQV RI )UDFWLRQDO )DFWRULDOV 7HFKn QRPHWULFV SS >@ 0LWFKHOO 7 f $Q $OJRULWKP IRU WKH &RQVWUXFWLRQ RI '2SWLPDO ([SHULPHQWDO 'HVLJQVf 7HFKQRPHWULFV SS >@ 0RQWJRPHU\ & f 'HVLJQ DQG $QDO\VLV RI ([SHULPHQWV WK HGLWLRQ -RKQ :LOH\ t 6RQV 1HZ @ 0\HUV 5 + DQG 0RQWJRPHU\ & f 5HVSRQVH 6XUIDFH 0HWKRGROn RJ\ 3URFHVV DQG 3URGXFW 2SWLPL]DWLRQ 8VLQJ 'HVLJQHG ([SHULPHQWV -RKQ :LOH\ t 6RQV 1HZ @ 2f'RQQHOO ( 0 DQG 9LQLQJ * f f0HDQ 6TXDUHG (UURU RI 3UHGLFWLRQ $SSURDFK WR WKH $QDO\VLV RI WKH &RPELQHG $UUD\f -RXUQDO RI $SSOLHG 6WDWLVWLFV SS >@ 3KDGNH 0 6 f 2XDOWL\ (QTLQHHULQT 8VLQT 5REXVW 'HVLJQ 3UHQWLFH f +DOO (QJOHZRRG &OLIWV 1>@ 3ODFNHWW 5 / DQG %XUPDQ 7 f f7KH 'HVLJQ RI 2SWLPXP 0XOWLIDFWRULDO ([SHULPHQWV %LRPHWULND SS >@ 5RELQVRQ f f,QFRPSOHWH 6SOLW 3ORW 'HVLJQVf %LRPHWULFV SS >@ 5RELQVRQ f %ORFNLQJ LQ ,QFRPSOHWH 6SOLW 3ORW 'HVLJQV %LRPHWULND SS @ 5XVVHOO 7 6 DQG %UDGOH\ 5 $ f 2QH:D\ 9DULDQFHV LQ WKH 7ZR:D\ &ODVVLILFDWLRQf %LRPHWULND SS >@ 6DQWHU 7 DQG 3DQ f f6XEVHW 6HOHFWLRQ LQ 7ZR)DFWRU ([SHULn PHQWV 8VLQJ 5DQGRPL]DWLRQ 5HVWULFWHG 'HVLJQVf -RXUQDO RI 6WDWLVWLFDO 3ODQQLQJ DQG ,QIHUHQFH SS >@ 6$6 ,QVWLWXWH f 6$64& 6RIWZDUH 5HIHUHQFH *XLGH 9HU VW HG 6$6 ,QVWLWXWH &DU\ 1& >@ 6KRHPDNHU $ & 7VXL DQG :X & ) f (FRQRPLFDO ([SHULPHQWDWLRQ 0HWKRGV IRU 5REXVW 'HVLJQVf 7HFKQRPHWULFV SS >@ 6QHH 5 f f&RPSXWHU$LGHG 'HVLJQ RI ([SHULPHQWVf§6RPH 3UDFWLFDO ([SHULHQFHf -RXUQDO RI 4XDOLW\ 7HFKQRORJ\ SS

PAGE 178

@ 7DJXFKL f 6\VWHPV RI ([SHULPHQWDO 'HVLJQ 9RO DQG .UDXV ,QWHUQDWLRQDO 3XEOLFDWLRQV :KLWH 3ODLQV 1< >@ :HOFK : .DQJ 7 DQG 6DFNV f &RPSXWHU ([SHULPHQWV IRU 4XDOLW\ &RQWURO E\ 3DUDPHWHU 'HVLJQf -RXUQDO RI 4XDOLW\ 7HFKQRORJ\ SS >@ :RRGLQJ : 0 f 7KH 6SOLW3ORW 'HVLJQf -RXUQDO RI 4XDOLW\ 7HFKn QRORJ\ SS >@
PAGE 179

%,2*5$3+,&$/ 6.(7&+ 6FRWW .RZDOVNL ZDV ERUQ RQ )HEUXDU\ LQ .LWFKHQHU 2QWDULR &DQDGD +H UHFHLYHG KLV EDFKHORUfV GHJUHH LQ VWDWLVWLFV DW WKH 8QLYHUVLW\ RI )ORULGD LQ DQG KLV PDVWHUfV GHJUHH LQ VWDLVWLFV DOVR IURP WKH 8QLYHUVLW\ RI )ORULGD 8QGHU WKH VXSHUYLVLRQ RI 3URIHVVRU *HRIIUH\ 9LQLQJ KH UHFHLYHG D 'RFWRU RI 3KLORVRSK\ GHJUHH LQ VWDWLVWLFV LQ 'HFHPEHU +H KDV EHHQ PDUULHG WR .LP VLQFH 7KH\ KDYH D EHDXWLIXO GDXJKWHU 5HJDQ 0LFKHOOH

PAGE 180

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ *HRIIUH\ 9LQLQJ &KDLUPDQ $VVRFLDWH 3URIHVVRU RI 6WDWLVWLFV FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ ANJ}$ -RKQ $ &RUQHOO 3URIHVVRU RI 6WDWLVWLFV FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 5LFKDUG / 6FKHDIIHU 3URIHVVRU RI 6WDWLVWLFV

PAGE 181

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 0WX G VƒƒRAf§ 'LDQH $ 6FKDXE $VVLVWDQW 3URIHVVRU RI ,QGXVWULDO DQG 6\VWHPV (QJLQHHULQJ 7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH 'HSDUWPHQW RI 6WDWLVWLFV LQ WKH &ROOHJH RI /LEHUDO $UWV DQG 6FLHQFHV DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 'HFHPEHU 'HDQ *UDGXDWH 6FKRRO

PAGE 182

2) )/25,'$


REFERENCES 168
BIOGRAPHICAL SKETCH 172
v


72
when choosing which whole plots the subplot factors are semifolded. Otherwise, the
same treatment combinations will occur in both additional whole plots. This occurs
when the semifolding uses the whole plots containing the subplot treatments defined
by PQR+, QRS+ and PQR~, QRS~ or PQR+, QRS~ and PQR~ ,QRS+. Any other
combination is fine. In this section, P is semifolded in the whole plot containing
whole treatment c (PQR~,QRS+) and S is semifolded in the whole plot containing
whole plot treatment abed (PQR+, QRS+). The 24 points design is shown in Table
31.
Most of the chains are broken but some of the two-factor interactions among the
subplot factors are aliased with each other. Also, four of the sixteen two-factor inter
actions between whole plot and subplot factors must be assumed negligible. These
terms are AS, CS, DS, and DP. This is fairly nice since three of these terms involve
subplot factor S. Therefore, if it is believed that one of the subplot factors is unlikely
to interact with the whole plot factors, these terms or effects could be assumed negli
gible. This does not seem unreasonable. Now the 18 subplot df are partitioned into 4
df for the subplot factor main effects, 12 df for the whole plot by subplot interactions,
and 2 for two-factor interactions among subplot factors (these two effects can be any
pair except PQ and QS or PR and RS).
3.7 An Example
To illustrate how an experiment could be carried out and analyzed, an example is
presented. The example, from Taguchi (1987), involves the study of a wool washing
and carding process. The original experiment used a 213-9 x 23-1 inner and outer


152
Table 61: Design Points for 3 WP Factors and 3 SP Factors Using Separate Fractions
A
B
C
P
Q
R
1
-1
-1
1
-l
-1
1
-1
-1
-1
i
-1
1
-1
-1
-1
-l
1
1
-1
-1
1
l
1
-1
1
-1
1
-l
-1
-1
1
-1
-1
i
-1
-1
1
-1
-1
-l
1
-1
1
-1
1
l
1
-1
-1
1
1
-l
-1
-1
-1
1
-1
l
-1
-1
-1
1
-1
-l
1
-1
-1
1
1
l
1
1
1
1
1
-l
-1
1
1
1
-1
i
-1
1
1
1
-1
-l
1
1
1
1
1
i
1
1
-1
-1
-1
-l
-1
1
-1
-1
-1
l
1
-1
1
-1
1
i
-1
-1
1
-1
1
-l
1
-1
-1
1
1
i
-1
-1
-1
1
1
-l
1
1
1
1
-1
-l
-1
1
1
1
-1
l
1


71
4 WP Factors (A, B, C, .D) and 4 SP Factors (P, Q, P, S)
As the number of both whole plot factors and subplot factors increases, it becomes
impossible to break all of the relationships and estimate all of the important effects.
Therefore, some effects will need to be assumed negligible. Also, in the case of four
whole plot factors and four subplot factors, there is insufficient degrees of freedom to
estimate the four subplot factor main effects and the sixteen two-factor whole plot
by subplot interactions. Thus, some effects cannot be estimated anyway. Assuming
these effects to be negligible enables the estimation of the remaining effects.
To obtain a 16 point design in this situation, both the whole plot and subplot
treatments need to be fractionated. First, consider fractionating the wrhole plot and
subplot treatments separately. The defining contrast is I = ABC = BCD = AD =
PQR = QRS = PS = ABCPQR = ABCQRS = ABCPS = BCDPQR =
BCDQRS = BCDPS AD PQR = ADQRS = AD PS which is resolution II. Be
sides breaking chains among the whole plot factors, the chains, ADPS and PS need
to be broken. The additional whole plot treatments are obtained by semifolding on
factor A. The subplots are semifolded on factor P in one whole plot and factor S in
another whole plot. The 24 point design is shown in Table 30.
Next, consider split-plot confounding. The defining contrast is I ABC =
BCD = AD = ACPQR = BDQRS = ABCDPS = BPQR = ABDPQR =
CDPQR = ACDQRS = CQRS = ABQRS = DPS = APS = BCPS which is
resolution II. The additional whole plot treatments are obtained by semifolding on
factor A. Again, the subplot factors are semifolded on P and S. Care must be taken


18
with the use of their noncrossed bi-randomization designs. Box and Jones (1992)
illustrate split-plot confounding using a cake mix example.
In some experiments, there are constraints on the number of subplots within each
whole plot. When the whole plots are arranged in a CRD, Robinson (1967) discussed
situations where the number of subplots per whole plot is less than the number
of subplot treatments. The whole plots are treated as blocks and then a balanced
incomplete block (BIB) design is used to allocate the subplot treatments to the whole
plots. If the whole plots are arranged in an RCB design, the same procedure can be
applied. If the number of whole plots per block is less than the number of whole plot
treatments, then an incomplete block design can be used there as well. Robinson
(1970) gave details on the case when both whole plot and subplot treatments are
arranged in incomplete block designs. Essentially, the procedure amounts to arranging
the whole plot treatments in blocks using a BIB design and then considering the whole
plots as blocks and arranging the subplot treatments in another BIB design. Robinson
(1970) provided formulas for the estimates of the main effects and interactions for
three cases: within whole plot, between whole plot within block, and between blocks.
Formulas are also given for the variance of the differences of these estimates for each
case.
Huang, Chen and Voelkel (1998) also investigate fractionating two-level split-plot
designs at both the whole plots and the subplots. They consider 2^ni+n2^_^fcl+fc2^
split-plot designs which are associated with a subset of the 2n~k fractional factorial
designs where n = n\ 4- rc2 and k = + k2. The criterion used to select the optimal
design is that of minimum-aberration which is the design that has smallest number of


11
the error term. The hypothesis of no subplot treatment effect, H0 : 7i = 72 = = 7S)
versus at least one not equal, is tested using Error which is also used to test the
significance of the whole plot x subplot treatment interaction.
Suppose the whole plot and subplot treatments are a factorial structure. In this
case, after the hypothesis tests above are performed, a more detailed investigation of
the individual factors and their interactions can be carried out. For example, consider
the situation discussed above with t = 4 whole plot treatments consisting of a 22
factorial in Z\ and z2 and s = 4 subplot treatments also consisting of a 22 factorial
in X\ and x2. The i 1 = 3 degrees of freedom (df) for the whole plot treatments
can be partitioned into single df contrasts Z\, z2, and Z\Z2- Likewise, the s 1 3
df for the subplot treatments can be partitioned into a single df contrasts aq, x2,
and XiX2. Also, the (t, l)(s 1) = 9 df for the whole plot x subplot treatment
interaction can be broken down into 9 single df effects involving z}, z2, Xi, and x2
(see Table 5). Orthogonal contrasts should be calculated and tested for each factor
and the interactions using the appropriate error term from the original analysis. This
can be accomplished in SAS by using PROC GLM and the CONTRAST statement along
with the option E = error term after the model statement for the whole factors and
interactions. For the subplot factors, interactions among subplot factors, and whole
plot x subplot factor interactions, the analysis can be run a second time. In this
second analysis, the treatments in the model statement can be entered as factors and
interactions, similar to a regression model. The correct tests for the subplot factors
and whole plot x subplot factor interactions are given by SAS.


50
Table 12: Design Layout for 26 2 With Defining
Contrast / = ABCP = CPQR = ABQR
abcp
ab
cp
abcpqr
acr
acq
cpqr
abqr
bcq
bcr
bpr
bpq
apq
qr
apr
(1)
Table 13: Alias Structure for 26 2
A
=
B
=
C
=
P
=
Q
=
R
=
AB
=
CP + QR
AC
=
BP
AP
=
BC
AQ

BR
AR
=
BQ
CQ
=
PR
CR
=
PQ


90
at both the high and low levels of each process variable. Secondly, let us insist on all
of the mixture blends be present at each 1 factorial level for each process variable.
These ideas seem very intuitive and lead us to select some of the mixture blends to be
used at certain design points and different mixture blends to be used at other design
points.
Two designs are considered for the fitting of the model in Equation (10). With
both designs, the vertices of the simplex are run at one-half of the 2" factorial points
in the process variables with the midedge points of the simplex being run at the other
half. This is done in a such a way, that if the design is collapsed across the levels
of each process variable then one gets a simplex with vertices and midpoints at both
the low and high level of the remaining process variables. Hence, the information in
the mixture blends is spread evenly among the process variables. This is intuitively
appealing since if a process variable is deemed negligible then there is still complete
information on the mixture blends for the other process variables. Next, the axial
points in the process variables are paired with just the centroid of the simplex. This
allows for the centroid to also be present if the design is collapsed. The two designs
differ only in the number of points placed at the center of the process variables. With
one design the entire simplex-centroid is performed at the center while with the other
only the centroid mixture blend is performed at the center of the process variables.
Consider an example involving three mixture components and two process vari
ables. The model for this example, using Equation (10), contains 15 terms. The two
designs are shown in Figures 2 and 3. For three mixture components, the design with
the full simplex-centroid at the center of the process variables consists of 23 points


123
The proposed design for three mixture components and two process variables
can be extended to higher dimensions. For three process variables the whole plot
treatments are now on the vertices of a cube. There are eight factorial points and
then some replicates of the center. The case of four mixture components is not quite
as straightforward since there are four vertices but six midedge points. In this case,
two replicates of the centroid will be run in the whole plots that contain the vertices.
Also, the center of the process variables will have six replicates of the centroid. The
design will then be balanced with six subplots in each whole plot.
Estimation
Model estimation under the split-plot structure of the experiment is more complex
than when the the experiment is completely randomized. The initial randomization
of the process variables corresponds to the first randomization. This generates the
whole plot error variance, erf. The randomization of the mixture blends constitutes a
second randomization which generates the subplot error variance, erf. Hence, a model
for the experiment is
y = X/3 + 6 + e
where
+ e ~ N(0, V).
The matrix
V = erf J + erf I
represents the variance-covariance structure of the split-plot experiment. J is a block


166
Table 74: Design Points for 3-2 Case with Upper and Lower Constraints
A
A
A
Zl
Z2
A
A
A
Zl
Z2
0
.4
.6
1
-1
Additional Points for
0
.4
.6
-1
1
S implex- Centroid
0
.6
.4
-1
1
0
.4
.6
0
0
0
.6
.4
1
-1
0
.6
.4
0
0
.4
.6
0
1
-1
.4
.6
0
0
0
.4
.6
0
-1
1
.4
0
.6
0
0
.4
0
.6
1
-1
.6
.4
0
0
0
.4
0
.6
-1
1
.6
0
.4
0
0
.6
.4
0
-1
1
.5
.5
0
0
0
.6
.4
0
1
-1
.5
0
.5
0
0
.6
0
.4
-1
1
0
.5
.5
0
0
.6
0
.4
1
-1
.2
.2
.6
0
0
0
.5
.5
-1
-1
.2
.6
.2
0
0
0
.5
.5
1
1
.6
.2
.2
0
0
.6
.2
.2
-1
-1
.6
.2
.2
1
1
.5
0
.5
-1
-1
.5
0
.5
1
1
.2
.6
.2
-1
-1
.2
.6
.2
1
1
.5
.5
0
-1
-1
.5
.5
0
1
1
.2
.2
.6
-1
-1
.2
.2
.6
1
1
.33
.33
.33
-1
0
.33
.33
.33
1
0
.33
.33
.33
0
-1
.33
.33
.33
0
1
.33
.33
.33
0
0


2
r? = 0(x)+e,
where the form of the function g is unknown and e is a random error term. The goal
is to find, in the smallest number of experiments, the settings among the levels of
x within the region of interest at which r? is a maximum or minimum. Because the
form of g is unknown, it must be approximated. RSM uses Taylor series expansion to
approximate g(x) over some region of interest. Typically, first or second order models
are used to approximate g(x). The traditional RSM model would be
Vi = /(x)73 + e,
where
iji is the *th response,
x is the ith setting of the design factors,
/(x) is the appropriate polynomial expansion of x,
¡3 is a vector of unknown coefficients, and
the e/s are assumed to be independent and identically (i.i.d.) distributed as
jV(0,a2).
For a more detailed discussion on RSM see Kliuri and Cornell (1996), Box and Draper
(1987), and Myers and Montgomery (1995).


36
to the rows and the levels of a second treatment factor (or set of factors) are assigned
to the columns. Table 10 represents the laundry experiment in which each square
represents a cloth sample, rows represent sets of samples that were washed together,
and columns represent sets of samples that were dried together. The ANOVA table
for the laundry example, which is divided into strata corresponding to blocks, rows,
columns, and units, is given in Table 11. When making inferences about the effects
in a particular stratum, the estimate of variation must be based on the residual term
for that stratum.
Miller (1997) proposes a method for constructing strip-plot configurations for
fractional-factorial designs which consists of three steps:
1. Identify a suitable design for applying row treatments to rows ignoring columns;
2. Identify a suitable design for applying column treatments to columns ignoring
rows;
3. Select a suitable fraction of the product of the row and column designs.
The method is applied for two-level designs and then extended to rn-level and mixed-
level designs. The procedure for two-level designs is presented here; for details on the
extended cases, see Miller (1997).
Consider the situation in which a proper fraction of a two-level factorial design is
to be run in a strip-plot arrangement using b = 2W blocks. Each block has r = 2M
rows and c 2m columns. Let K and k represent the number of row and column
factors, respectively, and define Q = K (w + M) and q = k (w + m). Then, the
procedure is as follow's:


100
Table 35: Comparison of Our Design With a Full Simplex
at the Center to the Designs Chosen by PROC OPTEX
Mixture
Components
D-criterion
Process PROC OPTEX
Variables (^-restricted)
PROC OPTEX
(N default)
Our Design
(AT)
3
2 11.21 (23)
11.18 (25)
8.24 (23)
3
3 14.62 (37)
14.21 (31)
12.33 (37)
4
2 5.91 (35)
5.79 (31)
4.66 (35)
4
3 8.32 (57)
7.94 (38)
7.01 (57)
Upper and Lower Constraint
3
2 5.12 (41)
5.01 (25)
4.25 (41)
Table 36: Comparison of Our Design With Just the Centroid
at the Center to the Designs Chosen by PROC optex
Mixture
Components
Process
Variables
D-criterion
PROC OPTEX
(V-restricted)
PROC OPTEX
(N = default)
Our Design
(N)
3
2
10.26 (17)
11.18 (25)
7.70 (17)
3
3
14.21 (31)
14.21 (31)
13.11 (31)
4
2
5.39 (25)
5.79 (31)
4.55 (25)
4
3
8.14 (47)
7.94 (38)
7.47 (47)
Upper and Lower Constraint
3
2
5.07 (29)
5.01 (25)
4.22 (29)


x 1 = 1 Z_2
Figure 8: Proposed Design for Split-Plot Structure With a First-Order Model
to
to


85
by process variable interactions are of equal importance. In fact, the specific mixture
component by process variable interaction terms may provide a significant amount of
insight into which operating conditions are optimal. For instance, the engineer truly
needs to know if a specific mixture component makes the reaction especially sensitive
to the reaction temperature.
This type of experimental situation leads us to propose a new model for extracting
information from a mixture experiment with process variables. The time and cost
constraints faced by plant personnel leads us to propose a new class of designs based
upon this model.
4.2 The Combined Mixture Component-Process Variable Model
In mixture experiments involving process variables, the form of the combined
model consisting of terms in the mixture proportions as well as in the process vari
ables depends on the blending properties of the mixture components, the effects of
the process variables, and any interactions between the mixture components and pro
cess variables. These models are typically second-order models that allow for pure
quadratic and two-factor interaction terms.
The general second-order polynomial in q mixture components is
7 7 7
v = A) + 0*xi + £ Pixix3
2= 1 2=1 i Now using the constraints
^2 Xi = 1 and
Xj, = Xi
i= 1
Xi
i= 1
i*i
(4)


CHAPTER 5
MIXTURE EXPERIMENTS WITH PROCESS
VARIABLES IN A SPLIT-PLOT SETTING
With many industrial experiments, it is often difficult or costly to run the exper
iment in a completely random order. Often times, a level of one or more factors is
fixed and then all or a fraction of all of the combinations of the other factors are run.
This process is repeated until the desired number of runs have been used. The result
is a design that uses restricted randomization and resembles a split-plot design.
Mixture experiments with process variables are often run under the above setting.
Typically, the process variables serve as the whole plot factors and the mixture com
ponents make up the subplot factors. However, in some experiments their roles could
be reversed. For a discussion of mixture experiments run using split-plot designs, see
Cornell (1988).
Consider the proposed model and class of designs for a mixture experiment with
process variables presented earlier. In this chapter, we extend the ideas presented
in Chapter 4 to the situation where complete randomization is not possible. This
may be due to physical constraints of the experiment or to cost constraints. We will
assume that the experiment is conducted by fixing the levels of the process variables
and then running some of the mixture blends. Therefore, the process variables are
the whole plot factors while the mixture components are the subplot factors. The
goal of this chapter is to consider different methods of estimation for the effects under
the restricted randomization.
119


3
1.2 Split-Plot Designs
A split-plot design often refers to a design with qualitative factors but can easily
handle quantitative factors. Also, a split-plot design usually has replication. However,
in the literature it has been common practice to refer to any design that uses one
level of restricted randomization regardless of replication as a split-plot design.
Therefore, in this dissertation, we will use the term split-plot design throughout.
When performing multifactor experiments, there may be situations where com
plete randomization might not be feasible. A common situation is when the nature
of the experiment or factor levels preclude the use of small experimental units. Often
a second factor can be studied by dividing the experimental units into sub-units. In
these situations, the split-plot experiment can be utilized. The experimental unit
is referred to as the whole plot while the sub-units are referred to as the subplots.
For every split-plot experiment there are two randomizations. Whole plot treatments
are randomly assigned to whole plots based on the whole plot design. Within each
whole plot, subplot treatments are randomly assigned to subplots with a separate
randomization for each whole plot. This leads to two error terms, one for the whole
plot treatments and one for subplot treatments as well as the interaction between
whole plot treatments and subplot treatments. Split-plot experiments have been used
extensively in agricultural settings. Even so, the following example from Montgomery
(1997) shows that there are applications for split-plot experiments in industrial set
tings.
A paper manufacturer is interested in studying the tensile strength of paper based
on three different pulp preparation methods and four cooking temperatures for the


89
To support the fitting of Equation (10) we shall require a design that will support
nonlinear blending of the mixture components as well as the fitting of the full second
order model in the process variables. In the next section, we discuss a design approach
that will accomodate these terms.
4.3 Design Approach
In mixture experiments as in most response surface investigations, the design and
the form of the model to be fitted go hand in hand. For example, if a second order
model is suspected, it is necessary to select a design that will support the fit of this
model. The design chosen must have at least as many points as there are parameters
in the model. Therefore, a (q + n)(q + n + l)/2 point design is needed to support the
fitting of the model in Equation (10).
A popular response surface design for fitting a second-order model of the form in
Equation (9) is the central composite design (ccd) which consists of a complete 2n (or
a Resolution V fraction of a 2") factorial design, 2n axial points with levels a for
one factor and zero for the rest, and at least one center point. If a = 1 is selected,
the design region is a hypercube.
The approach to reducing the number of observations needed in a mixture exper
iment begins with a ccd in the process variables. A simplex is then placed at each
point in the ccd with only a fraction of the mixture blends at each point. The mixture
blends at each design point are selected from the full simplex-centroid. A general
notion of balance among the mixture components across the process variables is de
sired. First of all, let us insist on the same number of mixture blends to be present


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements of the Degree of Doctor of Philosophy
THE DESIGN AND ANALYSIS OF SPLIT-PLOT
EXPERIMENTS IN INDUSTRY
By
Scott M. Kowalski
December, 1999
Chairman: G. Geoffrey Vining
Major Department: Statistics
Split-plot experiments where the whole plot treatments and the subplot treatments
are made up of combinations of two-level factors are considered. Due to cost and/or
time constraints, the size of the experiment needs to be kept small. Using fractional
factorials and confounding, a method for constructing sixteen run designs is presented.
Along with this, semifolding is used to add eight more runs. The resulting twenty-four
run design has better estimating properties and gives some degrees of freedom which
can be used for estimating the subplot error variance.
Experiments that involve the blending of several components to produce high
quality products are known as mixture experiments. In some mixture experiments,
the quality of the product depends not only on the relative proportions of the mixture
vi


34
experiment, complete randomization tends to be impractical. This leads to a restric
tion on randomization and lends itself to the split-plot design.
Cornell (1988) considers factor-level combinations of the process variables as the
whole plot treatments and the mixture component blends as the subplot treatments,
but points out that their roles can be switched. Hence, a combination of the levels
of the process variables is selected and all blends are run at this combination. An
other combination of the process variable levels is chosen and all blends are run at
this combination. This procedure is continued until all combinations of the process
variables are performed. Following a replication of the complete design, the split-
plot nature of the experiment leads to two error terms which are used to assess the
significance of the effects of the whole plot treatments, the subplot treatments, and
their interaction. Several regression-type models are considered for estimating the
effects of the process variables, the blending properties of the mixture components,
and interactions between the two. The paper explains how to estimate the regression
coefficients as well as how to obtain variances and perform hypotheses tests. Both
balanced and unbalanced cases are considered. The hypothesis testing procedures are
illustrated with two completely worked-out numerical examples.
Santer and Pan (1997) discuss subset selection procedures for screening in two-
factor treatment designs. The paper deals mainly with split-plot designs run in com
plete blocks; however, the strip-plot design is also discussed. One factor serves as the
whole plot factor while the other is the subplot factor. The goal is to select a subset
of the treatment combinations associated with the largest mean. Subset selection
procedures are given for additive and nonadditive factor cases, where neither of the


137
Table 50: Relative Efficiencies for Comparing Methods of Estimating
V With 3 Mixture Components and 3 Process Variables
d
r = 2
r 3
OLS
REML
PE
OLS
REML
PE
.11
1.03
1.21
1.06
1.03
1.18
1.10
.43
1.26
1.22
1.12
1.26
1.10
1.09
1.0
1.79
1.12
1.28
1.77
1.11
1.13
2.3
3.05
1.22
1.60
3.01
1.13
1.21
4.0
4.72
1.24
1.91
4.66
1.16
1.28
r = 4
r = 5
d
OLS
REML
PE
OLS
REML
PE
.11
1.03
1.16
1.10
1.03
1.15
1.09
.43
1.26
1.09
1.07
1.25
1.09
1.06
1.0
1.76
1.09
1.09
1.75
1.08
1.07
2.3
2.98
1.12
1.13
2.95
1.08
1.09
4.0
4.59
1.09
1.16
4.54
1.07
1.10
is shown in Figure 9 for 3 mixture components and 2 process variables. Again, the
designs can be extended to higher dimensions with m changing appropriately.
The estimation procedure and simulation study are carried out the same way as
discussed in the first-order case. The only addition is that the m replicates of the
centroid blend in the axial points contribute to the estimate of the subplot pure error
variance.
Comparison of Methods
The same values of d are used and it is still assumed that of = 1. Also, the designs
consisted of r = 2, 3, 4 and 5 total replicates of the center of the process variables.
The formulas for the ratios of the determinants given for the first-order case still give


82
(1990) discusses an experiment involving fish patties. The texture of the fish patties
depends not only on the proportions of three fish species that are blended but also
on three process variables which are cooking temperature, cooking time and deep fat
frying time.
A concern with mixture experiments involving process variables is that the size
of the experiment increases rapidly as the number of process variables, n increases.
In the fruit punch or fish patty examples above, it may not be necessary to limit the
size of the experiment. However, in most industrial experiments, cost and time do
impose restrictions on the number of runs permitted. Therefore, a design strategy
that uses fewer observations is preferred over a design that does not.
Cornell and Gorman (1984) presented combined mixture component-process vari
able designs for n > 3 process variables that use only a fraction of the total number
of possible design points. They considered process variables each at two levels and
suggest fractions of the 2" factorial be considered. Two plans involving the frac
tional factorial design in the process variables were discussed. The first plan, called
a matched fraction, places the same 23-1 fractional replicate design at each mixture
composition point. The other plan, called a mixed fraction, uses different fractions at
the composition points. Each plan was applied to the situation involving three mix
ture components and three process variables with the total number of design points
ranging from 56 for the combined simplex-centroid by full 23 factorial, to only 16,
which relied on running the one-quarter fraction. It should be noted that if inter
actions among the process variables are likely to be present, the use of a fractional
factorial will result in bias being present in the coefficient estimates. Cornell and


87
The second type of combined model is the additive model which combines the
models in Equations (5) and (6) without crossing any of the xt and Zj terms. This
produces the model
vfcz) = £#*< + ££ PijXiXj
i=1 i n n
+ ^2 &kzk + ^2^2OikiZkZi. (8)
k=1 k Equation(8) provides a measure of the quadratic blending of the mixture components
on the response as well as up to two-factor interactions between the process variables
on the response. Since the model does not contain any crossproduct terms between
the mixture components and the process variables, when fitting Equation (8) the
user assumes the blending of the mixture components is the same at all factor-level
combinations of the process variables. This assumption is probably unrealistic in most
situations. Also, in some experiments like the one described in the previous section,
the mixture component by process variable interactions may be the most important
terms in the model.
A major concern with mixture experiments involving process variables is their
size. Many industrial situations require the use of small experiments due to time
and/or cost constraints. As the number of mixture components and/or process vari
ables increases, the model in Equation (7) will require a design with a large number
of points. While the fitting of the model in Equation (8) permits the use of a smaller
design than the fitting of the model in Equation (7), it does not, as pointed out
earlier, address the estimation of the mixture components by process variable inter
actions. If cost constraints limit the size of the experiment yet interactions between


42
1. Every linear contrast of the treatments estimates more than one effect; hence,
each effect is aliased with one or more other effects. This can lead to the
misinterpretation of an effect which is not likely to happen with a complete
factorial experiment.
2. There is no independent estimate of experimental error.
Despite these limitations, fractional factorial experiments are used in exploratory
research and in situations that permit follow-up experiments to be performed. They
have been especially useful in industrial research and development where experimen
tal errors tend to be small, the number of factors being investigated is large, and
experimentation is sequential. As a tool for exploratory research, fractional factorials
provide a means to efficiently evaluate a large number of factors using a relatively
small number of experimental units. This allows important factors to be detected
and unimportant factors to be screened or discarded rather than committing a large
amount of experimental resources on all of the factors.
Effects that are estimated by the same linear combination of treatments are called
aliases. Which effects are aliased depends on the factorial effects used to select the
treatments. The defining contrast is the effect(s) that is confounded with the constant
effect, /. It can be represented as an equation by setting the confounded effect equal
to /. The alias chain for an effect is found by forming the generalized interaction of
the effect with all terms in the defining contrast. For example, if a 23-1 fraction in
factors A, B, and C is run with defining contrast I = ABC, then the alias of the
main effect A is A(I) = A2BC which gives A = BC since A2 = /. Therefore, the


125
there are m replicates of the centroid. The final method uses these replicate points
to form pure error terms which can be used to estimate the two error variances.
Simulation Study
An investigation is conducted to evaluate the performances of the three methods.
A simulation study is carried out to obtain approximations for the elements of the
matrix, Var(/3). Then the det Var(/3)] is computed and compared to the asymptotic
values which use known V. By looking at the det Var(/3)J, we are comparing the
size of the joint confidence ellipsoid around the parameter estimates. The five cases
from Chapter 4 are considered.
It is convenient to define the relationship
As d increases in value greater than unity, the whole plot error variance becomes
much larger than the subplot error variance and thus the correlations among the
observations become stronger. Without loss of generality, we will assume that o\ 1
and thus d = erf represents the whole plot error variance. The asymptotic value for
the Var(3) is (X'V_1X)_1 where V = dJ + I. The summary value |(X'V_1X)_1|
will be used for comparing the three methods.
For OLS, the estimate of (3 is (3 = (X'Xj^X'y- The estimated variance of (3 is
Var(/3) = (X'XJ^X' (Var(y)) X(X'X)"1
= (X/X)-1X/VX(X/X)-1
where V = dJ + I as above. Therefore, the quantity |(X'X)_1X'VX(X'X)-1| can be
calculated and compared to the asymptotic value.