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1 Forida State University Libraries Eectronic Theses, Treatises and Dissertations The Graduate Schoo Construction of Efficient Fractiona Factoria Mixed-Leve Designs Yong Guo Foow this and additiona works at the FSU Digita Library. For more information, pease contact

2 THE FLORIDA STATE UNIVERSITY COLLEGE OF ENGINEERING CONSTRUCTION OF EFFICIENT FRACTIONAL FACTORIAL MIXED-LEVEL DESIGNS By YONG GUO A Thesis submitted to the Department of Industria Engineering in partia fufiment of the requirements for the degree of Master of Science Degree Awarded Fa Semester,

3 The members of the Committee approve the thesis of Yong Guo defended on November,. James R. Simpson Professor Directing Thesis Samue A. Awoniyi Committee Member Joseph J. Pignatieo, Jr. Committee Member Approved: Ben Wang, Chair, Department of Industria Engineering Ching-Jen Chen, Dean, FAMU-FSU Coege of Engineering The Office of Graduate Studies has verified and approved the above named committee members. ii

4 AKNOWLEDGEMENTS I wish to thank my mother for giving birth to me and supporting a of my goas. Dr. James Simpson, not ony for his ski in research, but aso because he woud not accept anything ess than my best when conducting research. Dr. Samue Awoniyi taught me critica thinking skis that heped me sove industria engineering probems. Dr. Joseph Pigniateo showed me the beauty of mathematics and how I can appy it to different situations. I am aso gratefu for the support I have received from the Quaity Lab especiay Lisa Hughes, Francisco Ortiz, Marcus Perry, and Rupert Giroux. My friend Todd Kramer heped me edit this thesis and is becoming a great Chinese schoar. Trisha Shaw from the Engish Department at Forida State University for editing severa chapters of my thesis. Lasty, to a of my friends in the Industria Engineering Department who guided me aong this ourney. iii

5 TABLE OF CONTENTS List of Tabes...v List of Figures...vi ABSTRACT...viii INTRODUCTION... - Statement of the Probem... - Research Obective and Methodoogy... - Thesis Research Scope... LITERATURE REVIEW Genera Concepts of Design of Experiments Construction of Orthogona Designs Based on Difference Matrices Construction of Orthogona Designs Based upon J Optimaity Genetic Agorithms Construction of Supersaturated Designs by Genetic Agorithms... 5 METHODOLOGY... - Baance Coefficient for Factoria Designs... - Standardized J -Optimaity Construction of Mixed-eve Designs by Genetic Agorithms... 4 RESULTS AND ANALYSIS Investigation of Baance Coefficient and J -optimaity for Designs with Different Numbers of Runs Generation of the Baanced Orthogona Mixed-eve Designs and Performance of the Program Generation of the Efficient Mixed-eve Designs... 6 CONCLUSIONS AND FUTURE RESEARCH Concusions Future Research Suggestions APPENDIX.MATLAB SOURCE CODES...7 REFERENCES...78 BIOGRAPHICAL SKETCH...8 iv

6 List of Tabes Tabe. Fu Factoria Design Tabe. Treatments Combinations for Factors with Leves Each... 8 Tabe. Pus and Minus Signs for the Fractiona Designs... Tabe.4 An Orthogona Pair... Tabe.5 A Unbaanced Design... Tabe.6 Possibe Designs Generated by the D m,m; and L m... 8 Tabe.7 OA (, 4 )... 9 Tabe.8 Coincidence Matrix of OA (, 4 )... 9 Tabe. Computation of Baance Coefficient for the Design Shown in Figure... 6 Tabe. Fu Size 4 Factoria Design... 4 Tabe. Continued... 4 Tabe. Chromosome Representation Using Binary Encoding... 4 Tabe.4 Chromosome Representation Using Rea-vaue Encoding... 4 Tabe 4. Statistics for EA (, 5 7 ) with Sampe= Tabe 4. Ĥ and Ĵ for Different Number of Runs Tabe 4. Statistics of Ĥ and Ĵ Tabe 4.4 Computation of the Idea Standardized Baance Coefficient... 5 Tabe 4.5 Ĵ with Different Number of Runs (Vaidation with More Generations) Tabe 4.6 Statistics of Generated EA (5, 5 7 )... 6 Tabe 4.7 Statistics of Generated EA (, 5 7 ) Tabe 4.8 Statistics of Generated EA (, 5 7 ) v

7 List of Figures Figure. The idea of the efficient mixed-eve design construction method Figure. The components of a process... 6 Figure. A fowchart of the working principe of a genetic agorithm... Figure. Optima soution output... 9 Figure. Graphica representation of a -eve, -run coumn... Figure. Feasibe soutions of + + =... Figure. An unbaanced design... 6 Figure.4 A - design Figure 4. Pot of Ĥ and Ĵ vs. number of runs Figure 4. Adusted pot of Ĥ and Ĵ vs. number of runs Figure 4. Enarged adusted pot of Ĥ vs. number of runs Figure 4.4 The optima standardized baance coefficient vs. number of runs Figure 4.5 Design matrices with two runs and four runs Figure 4.6 Increasing Ĵ trend with, generations Figure 4.7 OA(4, ) generated by GA Figure 4.8 Pot of the best vaue vs generations for OA(4, ) Figure 4.9 OA(8, 4) generated by GA Figure 4. Pot of the best vaue vs generations for OA(8, 4) Figure 4. OA(, 4 ) generated by GA... 6 Figure 4. Pot of the best vaue vs. generations for OA(, 4 ) Figure 4.4 A pre-convergence case OA(, 4 ) vi

8 Figure 4.5 EA (5, 5 7 ) generated by GA Figure 4.6 EA (, 5 7 ) generated by GA Figure 4.7 EA (, 5 7 ) generated by GA vii

9 ABSTRACT Mixed-eve factoria designs are experimenta designs whose factors have different numbers of eves. These designs are very usefu in experiments invoving both quaitative and quantitative factors. One design approach is to run a possibe combinations of the factor eves. However, as the number of factors or factor eves increases, the number of experiments increases dramaticay. As a resut, research has focused on deveoping orthogona or near-orthogona fractiona factoria designs. The property of design baance, that the same number of runs is performed for each factor eve, has been maintained in currenty proposed designs. In some cases, maintaining baance requires too many experimenta runs. The obective of this thesis is to deveop fractiona mixed-eve factoria designs with economica run size that have desirabe properties associated with near-baance and near-orthogonaity. Two criteria are deveoped to assess the degree of near-baance for comparing and constructing designs. A modified J-optimaity criterion is used for comparing design near-orthogonaity. These criteria are combined to assess different design aternatives. A genetic agorithm is then used to buid designs with the most desirabe combination of near-baance and nearorthogonaity. viii

10 CHAPTER INTRODUCTION - Statement of the Probem Traditiona two-eve factoria designs are widey used in industria research and deveopment. However, in some situations, factors with more than two eves are more desirabe, especiay when those factors are quaitative factors. In fact, quaitative factors are the main motivation of research on mixed-eve designs. As a resut, designs with mixed-eve factors have been used more often in designed experiments in modern industries, especiay when ony imited resources are aowed. Two of the more desirabe properties of factoria designs are baance and orthogonaity. Baance requires that each eve of a factor is run the same number of times in an experiment. Orthogona designs are coumn pairwise ineary independent, usefu for assessing factor significance. Construction of mixed-eve designs with efficient run sizes and desirabe baance and orthogonaity properties is the primary focus of research in this area. Severa agorithms have been deveoped by different authors to generate baanced orthogona mixed-eve designs. Additionay, near-orthogona designs have been generated as the aternatives to stricty orthogona designs, when orthogona designs are either difficut or impossibe to produce. Wang & Wu (99) proposed an approach for constructing orthogona designs based upon difference matrices. Their construction method essentiay appies the generaized Kronecker sum and uses the technique of adding coumns. DeCock & Stufken () proposed an agorithm for constructing orthogona mixed-eve designs through searching some existing two-eve orthogona designs. Xu () proposed an agorithm to construct orthogona and near-orthogona designs based on the concept of J -optimaity. The J criterion is equivaent to the other orthogonaity criteria, such as the

11 (M, S) criterion (Ecceston and Hedayat, 974), the A -optimaity (Xu, ), and the ave(s ) criterion (Xu, ). However, the J criterion is more appropriate for situations where factor eves are not denoted by contrasts, because J -optimaity does not require computing the vaue of X T X, where X is the contrast matrix of the design. Xu () aso shows that a design is J -optima and orthogona if J equas its ower bound. Unfortunatey, as the number of factors or factor eves increases, the number of runs increases dramaticay. Maintaining the baance property requires too many runs in some situations. For exampe, consider a design with four factors, one with eves, one with eves, one with 4 eves and the ast with 5 eves. To generate a baanced design, at east 6 runs are needed. Suppose an engineer ony has resources for tests and the test obective is screening. The intent of this thesis is to deveop a mechanism for creating mixed-eve designs with desirabe properties capabe of meeting resource requirements. In this thesis, efficient near-baanced, near-orthogona mixed-eve designs wi be constructed by using the genetic agorithm. - Research Obective and Methodoogy This thesis focuses on the construction of efficient mixed-eve factoria designs with desirabe baance and orthogonaity properties. In order to evauate these baance and orthogonaity properties, some criteria must be proposed and used. These criteria can then be combined into an obective function to be optimized, and a genetic agorithm approach wi be deveoped to buid these desirabe, efficient designs. Currenty, no existing criterion is avaiabe to evauate the baance property of the design matrix. Therefore, a new optimaity criterion, caed the baance coefficient, wi be defined and formuated. Some investigations on the baance coefficient wi be performed and anayzed. The J -optimaity is used to measure the orthogonaity property of the design. Essentiay, J -optimaity is consistent with the orthogonaity property. Note that the ower bound for the J -optimaity, defined by Xu (), can ony be used in the baanced design situations. This ower bound is not appicabe to unbaanced design situations. As such, this research proposes a modified J criterion that can be used to measure the degree of orthogonaity of unbaanced design matrices.

12 In terms of the baance coefficient and J -optimaity, this thesis wi empoy genetic agorithms to generate efficient mixed-eve designs. The basic purpose of the genetic agorithm is to choose a subset of rows from a matrix of a possibe combinations. For exampe, suppose a design invoves three factors: factor A with two eves, factor B with three eves and factor C with four eves. The fu factoria design, with 4 runs, is shown in Figure.a. Desiring to reduce the experimenta runs, eight runs are chosen and combined into a reduced sized design, which is shown in Figure.b. The function of the genetic agorithm is to seect the eight runs that optimize the obective function and combine these runs into a suitabe mixed-eve design. - Thesis Research Scope This thesis wi focus on experimenta design construction issues ony. Using the concept of the east common mutipe and degrees of freedom necessary for ow order modes, the number of runs for efficient designs wi be suggested. This thesis wi propose a method to construct efficient mixed-eve factoria designs by appying genetic agorithms. A MATLAB code wi be deveoped to achieve this goa. A new baance criterion using aternative definitions and associated formuations wi be deveoped. A J -optimaity criterion for unbaanced designs wi be deveoped and combined with the baance coefficient to serve as the obective function for the genetic agorithm. This thesis wi aso investigate the reationship between these two criteria and the design sizes. The anaysis of the mixed-eve designs wi not be discussed in this thesis.

13 Run Factor A Factor B Factor C Run Factor A Factor B Factor C a) A fu factoria design b) An efficient design Figure. The idea of the efficient mixed-eve design construction method. 4

14 CHAPTER LITERATURE REVIEW Mixed-eve designs have drawn a considerabe amount of attention in terms of construction. This chapter is intended to provide an overview of experimenta design concepts and a through discussion of mixed-eve research performed to date. The basic concepts of the design of experiments wi be introduced in Section - so as to provide the background knowedge of this research. This thesis focuses on the construction of mixed-eve designs, and two existing orthogona mixed-eve design construction methods wi be anayzed and summarized in Sections - and -. One method is based upon difference matrices and another method is based upon J -optimaity. In Section -4, the principe of the genetic agorithms wi be briefy introduced. In order to iustrate how genetic agorithms are used in the design of experiments, a construction method of supersaturated designs by genetic agorithms wi be introduced in Section Genera Concepts of Design of Experiments An experiment is a test or series of tests conducted under controed conditions made to demonstrate a known truth, examine the vaidity of a hypothesis, or determine the efficacy of something previousy untried. In an experiment, one or more input process variabes are changed deiberatey in order to observe the effect the changes have on one or more response variabes. Experiments are performed a number of times in order to evauate the output response variabes under the different input process variabe conditions. The design of experiments is an efficient method for panning experiments so that the data obtained can be anayzed to yied vaid and obective concusions. The method for conducting designed experiments begins with determining the obectives of an experiment and seecting the process factors for the study. A designed experiment 5

15 requires estabishing a detaied experimenta pan in advance of conducting the experiment, which resuts in a streamined approach in the data coection stage. Appropriatey choosing experimenta designs maximizes the amount of information that can be obtained for a given amount of experimenta effort. -- Process Modes for Designed Experiment Experimenta designs are used to investigate industria systems or processes. A typica process mode is given in Figure.. Purposefu changes are made to the controabe input factors of a process so as to observe and identify the reasons for changes that may be observed in the output responses. The noise factors are considered as random factors that cannot be controed. Experimenta data are used to derive a statistica empirica mode inking the outputs and inputs. These empirica modes generay contain first and second-order terms. For more information regarding the statistica empirica mode, see Montgomery,. Noise Factors Process Controabe Output Responses Input Factors Figure. The components of a process. -- Factoria Designs Many experiments invove the study of the effects of two or more factors. Fu factoria designs are test matrices that contain a possibe combinations of the eves of 6

16 the factors. For exampe, if factor A has a eves and factor B has b eves, then the twofactor fu factoria design contains ab combinations. Tabe. shows another exampe, a fu factoria design with three factors: one with two eves, one with three and one with five. The shorthand notation for this design is ( 5 ), which dispays the factor eves as the base numbers and the number of factors with that many eves as the exponent. Tabe. Fu Factoria Design - 5 Run Factor A Factor B Factor C

17 One purpose of factoria designs is to study the effects of these factors on the response. The main effect of a factor is defined to be the change in response produced by a change in the eve of the factor. The term main effect is used because it refers to the primary factors of interest in the experiment. A main effect refects the individua effect of each factor. One-factor-at-a-time testing is an extensivey used experimentation strategy. This method consists of seecting a starting point setting for each factor, then successivey varying the settings of each factor over its range, with the other factors hed constant (Montgomery, ). Compared with one-factor-at-a-time experiments, factoria designs are more efficient. Factoria designs aow the effects of a factor to be estimated at severa eves of the other factors because the difference in response between the eves of one factor may not be the same at a eves of the other factor. Therefore, factoria designs are necessary when interactions may be present. A specific case of genera factoria designs is the k factoria design. That is, these designs have k factors, each at ony two eves. These eves may be either quantitative or quaitative. Normay + is used to represent the high eve and is used to represent the ow eve in the -eve factoria designs. A compete repicate of such a design requires k observations and is caed a k factoria design. Tabe. gives an exampe for k= in three repicates. Tabe. Treatments Combinations for Factors with Leves Each Factor Treatment Repicate A B Combination I II III A ow, B ow A high, B ow 6 + A ow, B high A high, B high 9 8

18 The interaction effect AB is defined as the average change in response between the effect of A at high eve of B and the effect of A at the ow eve of B. The methods used for generating k factoria designs are straightforward. Each coumn represents a factor. The eves for the first coumn foow The eves for the second coumn foow the pattern of For the n th coumn, the pattern wi be + + and the number for minus signs and pus signs is n for each. -- Fractiona Factoria Designs If the experimenter can reasonaby assume that certain high-order interactions are negigibe, information on the main effects and ow-order interactions may be obtained by running ony a fraction of the compete factoria experiment. The concept of fractions used in factoria designs is creative. The use of fractiona factoria designs is especiay popuar in screening experiments. Screening experiments are experiments in which many factors are considered, and the obective is to identify those factors with arge effects. Screening experiments are usuay performed in the earier stages of a proect when it is ikey that many of the factors initiay considered have itte or no effect on the response. The factors that are identified as significant are then investigated more thoroughy in subsequent experiments. Since the concept of the fractiona factoria designs is that ony a fraction of the compete factoria experiment is used, it raises the questions of how to choose appropriate fractions and how to perform experiment augmentation. Experiment augmentation denotes expansion of factoria designs into arger size designs. The approach used to generate fractions is based on design generators. Design generators are used to generate a particuar fraction. Consider a situation in which three factors, each at two eves are of interest, but the experimenters cannot afford to run a =8 treatment combinations. They can, however, afford four runs. This restriction suggests the use of a one-haf fraction of a design. Because the design contains - =4 treatment combinations, a one-haf fraction of the design is often caed a - design. 9

19 Tabe. Pus and Minus Signs for the Fractiona Designs Factoria Effect I A B C AB AC BC ABC From Tabe. it can be seen that the - design in the first four runs is formed by seecting ony those treatment combinations with a + sign in the ABC coumn. In this situation, ABC is caed the generator of this particuar fraction. Furthermore, the identity coumn I is aso aways pus, so I=ABC is caed the defining reation for the design incuding ony the first four runs. Generay, the defining reation for a fractiona factoria wi aways be the set of a coumns that are equa to the identity coumn I. ABC in this exampe is a word in the defining reation. More highy fractionated designs have more than one word in the defining reation. It is important to consider the resoution of fractiona factoria design when designing a fraction. In genera, the resoution of a -eve fractiona factoria design is equa to the smaest word (in terms of number of etters) in the defining reation. Usuay, fractiona factoria designs with highest resoution are preferred. The higher the resoution, the ess restrictive the assumptions that are required regarding which interactions are negigibe to obtain a unique interpretation of the data. Resoution III designs are designs in which no main effects are aiased with any other main effect, but main effects are aiased with -factor interactions and -factor interactions may be aiased with each other. Resoution IV designs are designs in which no main effect is aiased with any other main effect or with any -factor interaction, but -factor interactions are aiased with each other. Resoution V designs are designs in which no main effect or -factor

20 interaction is aiased with any other main effect or -factor interaction, but -factor interactions are aiased with -factor interactions (Montgomery, ). Construction of n-k fractiona factoria designs can be accompished reativey easiy using a basic design and a ist of design generations. For additiona discussion of construction of fractiona factoria designs, see Montgomery (). --4 Mixed-Leve Designs Mixed-eve designs are factoria designs whose factors have varying eves. Montgomery () mentions that the k factoria designs are the cornerstone of industria experimentation. However, in some situations it is necessary to incude factors that have more than two eves. It may be undesirabe to reduce the factor eves to two, particuary with quaitative factors, because choosing ony two eves can require not considering potentiay infuentia factor settings. Usuay, mixed-eve designs are used when both quantitative and quaitative factors are invoved in the experiment. The exampe given in Tabe. shows a simpe mixed-eve design. Two of the more desirabe properties of factoria designs are baance and orthogonaity. In Sections --4- and --4-, these two properties wi be briefy introduced. Some notation for mixedeve designs wi then be discussed in Section Orthogonaity property. An orthogona design of strength is an experimenta design in which a pairwise coumns are orthogona. This thesis wi not consider the designs whose orthogona strength are arger than, because ony orthogona designs with strength are currenty used in experimenta designs. Two coumns are orthogona if each of their eve combinations appears equay often. Tabe.4 shows an orthogona pair. Tabe.4 An Orthogona Pair

21 Normay the degree of orthogonaity is measured by orthogona optimaity criteria. Some optimaity criteria are (M, S) criterion (Ecceston and Hedayat, 974), A - optimaity (Xu, ), ave(s ) criterion (Xu, ) and J -optimaity (Xu, ) Baance property. In addition to orthogonaity, baance is another property commony considered in experimenta designs. By definition, a baanced design is a design in which a the coumns are baanced. A coumn is baanced if each factor eve contains the same number of runs. An exampe is shown in Tabe.5. Tabe.5 A Unbaanced Design A B C 4 4 In this exampe, factor C has three eves and these three eves do not appear with the same frequency. Therefore, this design is not a baanced design. Regarding the baance property of design matrices, there is no criterion currenty avaiabe to measure this property. So it is necessary to deveop a proper criterion to evauate and compare the design matrices in terms of the baance property. The baance criterion is discussed in Chapter Notation for mixed-eve designs. In order to measure the baance and orthogonaity properties, factor eves have to be coded. There are severa forms of notation with regard to factor eves. At east two notationa forms are

22 commony used. In -eve factoria designs, the two eves are denoted by using pus and minus signs. However, in or higher-eve designs, the,, representation is used. In this case, each number is ust a abe, so it does not matter which number represents that eve. In mixed-eve design situations, factor eves are coded by natura numbers. But for the -eve design particuary, the pus-minus sign notation other than shoud be used in order to avoid confusion in the sign interpretation for the interaction effect estimate. Regarding orthogona mixed-eve designs, there are different notations commony used. One commony used notation for orthogona design is k k k OA( n, L ), which denotes an orthogona design of size n and strength having k i r r k k k coumns with eves. In this thesis the notation EA( n, L ) is used to denote a near-baanced, near-orthogona, efficient mixed-eve design. r r Construction of mixed-eve designs. The concept of orthogona designs dates back to Rao (947). Construction methods incude combinatoria, geometrica, agebraic, coding theoretic, and agorithmic approaches. State-of-art construction of orthogona designs has been described by Hedayat, Soane, and Stufken (999). Nguyen (996) proposed an interchange agorithm for constructing supersaturated designs. His program can be used to construct two-eve orthogona designs and the argest one constructed and pubished is OA(, 9 ). DeCock and Stufken () proposed an agorithm for constructing mixed-eve orthogona designs through searching some existing two-eve orthogona designs. Their purpose is to construct mixed-eve orthogona designs with as many two-eve coumns as possibe, and their agorithm succeeded in constructing severa new arge mixed-eve orthogona designs. However, their agorithm fais to produce some orthogona designs. Nguyen (996) proposed an agorithm for constructing near-orthogona designs by adding two-coumns to existing orthogona designs. Wang and Wu (99) systematicay studied near-orthogona designs and proposed some genera combinatoria construction methods, such as the construction method based on the difference matrix.

23 Ma et a. () proposed two agorithms for constructing near-orthogona designs by minimizing some combinatoria criteria via the threshoding accepting technique. Xu () proposed an agorithm for constructing orthogona and near-orthogona designs based upon the J -optimaity. His agorithm is easier to achieve by program. The method works with coumns of more than two eves and appropriate weights are aso considered for different coumns. - Construction of Orthogona Designs Based on Difference Matrices Wang and Wu (99) proposed a method to construct orthogona mixed-eve designs based upon the existing ideas or their extensions, and synthesized a these ideas in a singe approach. The construction method empoys two mathematica concepts: difference matrices and Kronecker sums. -- Genera Ideas Let G be an additive group of p eements denoted by {,,, p-}, where the sum of any two eements x and y is x+y (mod p) for prime p, and is defined as in a Gaois fied for prime power p. A λp r matrix with eements from G denoted by D λp,r;p is caed a difference matrix if, among the differences moduus p of the corresponding eements of any of its two coumns, each eement of G occurs exacty λ times. For p=, D n,n; is a Hadamard matrix of order n. For two matrices A=[a i ] of order n r and B of order m s, both with entries from G, their Kronecker sum is defined to be a [ B i ] i n, r A B =, where B k =(B+kJ) mod p is obtained from adding k (mod p) to the eements of B, and J is an m s matrix of s. 4

24 The generaized Kronecker sum of A and B is defined as A B = [ A B, L, A n B ]. n If A is an orthogona design with the partition, then s A = [ OA( N, p ), L, OA( N, p s n n )] and where D M k i ; pi B = D M, L, D ], [, k; p M, K n ; pn, is a difference matrix, and N and M are both mutipes of the p i s. Then their generaized Kronecker sum, s sn A B = [ OA( N, p ) DM, k ; p, L, OA( N, pn ) DM, k ; ks r is an orthogona design (, s n OA NM p L p ). k r n pn ] -- Genera Construction Procedures Based on the principe of this construction method, the genera procedures to generate mixed-eve designs can be summarized as foows. First construct the orthogona design A B. Then et r L = * OA( N, q Lq N r r be a matrix consisting of N copies of an orthogona design, with OA( N, q Lq m ) as its rows, and N is the N vector of zeros. By adding the coumns of L to r m m ) m A B, the k resuting matrix [ A B, L] is an orthogona design (, s krsn r rm OA NM p L p q Lq ). The proof of the orthogonaity between A B is given in Wang and Wu (99). The variety number types of orthogona designs generated by this method are imited. Consider a difference matrix D 6,6;, which is An orthogona design can be constructed, namey. r m 5

25 6 = ) (, OA. By the construction procedures ust mentioned, it can be shown that + + = (mod ) (mod ) ) (, 6,6; 6,6; 6,6; 6,6; D D D D OA. This resuting design is an ) (8, 6 OA, which is dispayed beow, F E D C B A Run. The constructed design wi be used to buid orthogona mixed-eve designs. Let L 6 denote a six-run orthogona design, and L 6 is chosen to be L 6 =[ ] T 5 4, then [ ] 6 6,6;, ), ( L D OA is an orthogona design ) (8,6 6 OA, which is

26 F E D C B A G Run -- Generate Orthogona Mixed-eve Designs by D m,m; In order to see the imitation of this mixed-eve design construction method, a series of orthogona mixed-eve designs are being constructed by using D m,m;. In this simpe exampe, ) (, OA =(,) T and D m,m; are used to construct orthogona mixedeve designs. Then by the construction procedures, ), ( ) (, O ;, ;, ;, m m m m m m m m OA D D D A = + =. If m L * L = is seected, the generated designs are expressed as ], ) (, [ ;, m m D m L OA. By choosing different m, the possibe designs generated by the D m,m; and L m are shown in Tabe.6.

27 Tabe.6 Possibe Designs Generated by the D m,m; and L m m Possibe L m Fina designs,5, 7, OA(, ); OA(5, 5 ); OA(7, 7 ); OA(, () ) OA(6, ); OA(, 5 5 ); OA(4, 7 7 ); OA(, () ) 4 OA(4, 4 ) or OA(4, ) OA(8, 4 4 ) or OA(8, 6 ) 6 OA(6, 6 ) or OA(6, ) OA(, 6 6 ) or OA(, 7 ) 8 OA(8, 8 ) or OA(8, ) or OA(8, 4 ) OA(6, 8 8 )or OA(6, )or OA(6, 4 ) 9 OA(9, 9 ) or OA(9, ) OA(8, 9 9 )or OA(8, 9 ) OA(, () ) or OA(, 5 ) OA(, () ) or OA(, 5 ) OA(, () ) or OA(, 6 ) or OA(, ) or OA(, 4 ) OA(4, () ) or OA(4, 6 ) or OA(4, 4 ) or OA(4, 4 ) There are severa designs generated by this method. A these designs are orthogona designs and can be used in different situations. However, this construction method reveas its imitation. That is, the number of possibe designs that can be generated by this method is imited by the avaiabiity of difference matrices. - Construction of Orthogona Designs Based upon J Optimaity J -optimaity is used to measure the degree of orthogonaity of mixed-eve designs. Based upon this criterion, Xu () proposed a method to generate baanced orthogona designs. His method can efficienty construct various designs with good baance and orthogonaity properties. -- Definition of J -Optimaity Before discussing this mixed-eve design construction method, the definition of J -optimaity is first introduced. Consider a n m matrix D = [ x ik ], with weights w k >, and s k eves for each factor. The weights indicate the reative importance of each factor and if w k = are chosen, δ ( ) is the number of coincidences between the i th and th rows. For i, n, i, d 8

28 where δ ( x, y) m δ, ( d) = w δ ( x, x ), i = if x = y and otherwise. k = Then the J -optimaity of this design matrix d is defined as k ik k [ ( d ] J ( = δ. d) i, ) i< n Tabe.7 gives an exampe of an orthogona design, OA (, 4 ), with J vaue. The corresponding coincidence matrix δ ( ) of this matrix is given in Tabe.8. i, d Tabe.7 OA (, 4 ) Tabe.8 Coincidence Matrix of OA (, 4 ). 9

29 This coincidence matrix is symmetric. The eements on the diagona wi aways equa k. In fact, J is the sum of squares of the eements above the diagona. It is easy to see that J = for this exampe. -- Lower bound of J -optimaity for baanced designs There exists a ower bound for the J -optimaity. When the J -optimaity of a baanced design matrix reaches this ower bound, the design matrix is orthogona. This concusion was proved by Xu (). The ower bound, L(n), is given as foows, m m = + ( )( ) m L ( n) nsk wk sk nsk wk n wk. k = k = k = Reca the orthogona design OA (, 4 ) discussed in Section --. It is easy to verify the ower bound is aso. Therefore, this design is orthogona, because the J vaue is equa to the ower bound. -- Construction Agorithm Based on J -optimaity Xu () provides an agorithm to construct orthogona designs based on J - optimaity. The basic idea of the agorithm is to add coumns sequentiay to an existing design. The genera construction agorithm is stated as foows. Considering the change in the J vaue if a coumn is added to d or if two eves are switched in a coumn. A coumn c=(c, c,, c n ) T is added to d and et d + be the new n ( m +) design. Then for < i, < n δ i, ( d ) = δi, ( d) + wkδ i, ( c), + where δ ( c) = δ ( c, c ), i, i and J ( d ) = J ( d) + w ( ns ) δ ( d) δ ( c) + nw. + k i, i, k k i< n

30 Then a δ ( ) are unchanged except that δ c) = δ ( ) and δ c) = δ ( ) are i, c a, (, a c b, (, b c switched for a, b. Hence, J ( d ) is reduced by w k ( a, b), where + [ δ a, ( d) δ b, ( d) ] [ δ a, ( c) δ b ( c) ] ( a, b) =, a, b n. The foowing cited procedure to construct orthogona mixed-eve designs is summarized by Xu in.. For k=,,n, compute the ower bound L(k) according to ().. Specify an initia design d with two coumns: (,,,,,,, s -,, s -) and (,, s -,,, s -,,,, s -). Compute δ ( ) and J ( ). If J ( d ) = (), et n = and T = T ; otherwise, et n = and T = T.. For k =,..., n, do the foowing: i, d d L (a) Randomy generate a baanced s k -eve coumn c. Compute J ( d ) by definition. If J ( d ) = L( ) k +, go to (d). (b) For a pairs of rows a and b with distinct eves, compute ( a, b) as in (5). Choose a pair of rows with argest ( a, b) and exchange the symbos in rows a and b of coumn c. Reduce J ( d ) by ( a, b). + + w k If J ( d ) = L( ) k +, go to (d); otherwise, repeat (b) unti no further improvement is made. (c) Repeat (a) and (b) T times and choose a coumn c that produces the smaest J ( d ). + (d) Add coumn c as the k th coumn of d, et If J d) = J ( d ) and update δ ( ) by (). If J ( d ) = L( ) k +, et n =k; otherwise, et T=T. ( + i, d -4 Genetic Agorithms The foundation of genetic agorithms was deveoped by John Hoand in the 96s. By the midde of 98s, genetic agorithms had provided soutions to compex probems in optimization, machine earning, programming, and ob scheduing. Instead of using gradient information, a genetic agorithm can be used to perform a direct search.

31 Genetic agorithms are attractive not ony because they are reativey easy to impement, but aso mathematicay they do not require differentiabe obective functions. As Gen and Cheng () mentions, genetic agorithms are evoutionary search strategies based on simpified rues, bioogica popuation genetics and theories of evoution. A genetic agorithm maintains a popuation of candidate soutions for a probem, and then seects those candidates most fit to sove the probem. After the seection process, the most fit candidate soutions are combined and atered by reproduction operators to produce new soutions for the next generation. The process continues, with each generation evoving more fit soutions unti an acceptabe soution has evoved. Figure. shows a fowchart of the working principe of a genetic agorithm. For any genetic agorithm, the construction of the genetic representation of a chromosome is required. In genera, a genetic agorithm has five basic components:. A genetic representation of soutions to the probem. A way to create an initia popuation of soutions. An evauation function rating soutions in terms of their fitness 4. Genetic operators that ater the genetic composition of chidren during reproduction, and 5. Vaues for the parameters of genetic agorithms. -4- Encode Chromosomes Impementation of genetic agorithms are based upon the chromosomes. A chromosome represents a potentia soution to the probem of interest and traditionay is represented by a string of genes that are either binary encoded or rea number encoded. In a rea number representation, genes are encoded with rea number whie in a binary representation; genes are encoded or. Besides binary encoding and rea number encoding, integer or itera permutation encoding and genera data structure encoding are aso used for genetic agorithms.

32 Begin Initiaize Popuation Generation = Evauation Assign Fitness Satisfy Stop Criterion? No Increment of Generations Yes Stop Reproduction Crossover Mutation New Popuation Figure. A fowchart of the working principe of a genetic agorithm.

33 -4- Genetic Operations Typicay there are two types of search behaviors: random search and oca search. Random search expores the entire soution space and is capabe of achieving an escape from a oca optimum. Loca search expoits the best soution and is capabe of cimbing upward toward a oca optimum. The two types of search capabiities form the mutuay compementary components of optimization. An idea search shoud possess both types simutaneousy. It is amost impossibe to design such a search method using traditiona techniques. Genetic agorithms are a cass of genera-purpose search methods combining eements of directed and stochastic searches, which can make a good baance between exporation and expoitation of the search space. In conventiona genetic agorithms, the crossover operator is used as the principa operator and the performance of a genetic system is heaviy dependent on it. The mutation operator that produces spontaneous random changes in various chromosomes is used as a background operator. In essence, genetic operators perform a random search and cannot guarantee to yied improved offspring because of its heuristic sense. Commony used genetic operations are crossover and mutation Crossover operation and crossover rate. The crossover operation is one of the most commony used forms of recombination. A typica crossover invoves two chromosomes, caed parents. These two distinct chromosomes are seected randomy from the popuation and broken at the same ocation. The new chromosomes, caed offspring, are obtained by combining opposite portions of the parent chromosomes. The parent chromosomes can aso be broken in more than one ocation. The crossover rate refers to the percent of the parent popuation X that wi undergo a crossover operation. A singe point crossover technique is empoyed throughout our experimentation since the chromosome string ength is reativey short. Whie a high crossover rate typicay causes good points to be discarded, a ow crossover rate paces too much emphasis on parents and may stagnate the search Mutation operation and mutation rate. Mutation is used to ater the genetic materia of a very sma number of individuas in a random fashion, enhancing the diversity of the popuation and expanding the voume of the search space. 4

34 After a crossover is performed, mutation takes pace. This operation is intended to prevent a soutions in a popuation from converging to a oca optimum of a soved probem. Mutation randomy changes the new offspring chromosome. For binary encoding we can switch a few randomy chosen genes from to or from to. Mutation refers to the atering of one or more genes in a chromosome. Mutation types are characterized by how many genes are atered and the degree to which they are changed. As Ortiz () mentions, two types appropriate for rea-vaued coding are uniform and Gaussian mutation. Uniform mutation changes ony one gene according to a uniform distribution. The Gaussian type changes a the genes by a sma amount according to normay distributed disturbance. A chromosome atered by Gaussian mutation resuts in a point ocated in the neighborhood of its parent. Gaussian mutation aters the vector X by adding a randomy generated vector m = m, m,..., m ) consisting of mid, normay distributed ( k * perturbations. The new design space point becomes X = X + m. Bashir () first appies genetic agorithms to construct supersaturated designs. In his dissertation, supersaturated designs are constructed using different approaches to create the initia popuation. The principe of his method wi be introduced in the foowing section. -5 Construction of Supersaturated Designs by Genetic Agorithms A supersaturated design is a fractiona factoria design in which the number of factors m exceeds the number of runs n. A design is caed supersaturated when m n. Because of the specia features of the supersaturated designs, there exist a kinds of construction methods. However, the designs may not be the best in terms of the criteria used for generation. Bashir () proposed a construction method for improved supersaturated designs by the criterion of E ( s ). 5

35 The E ( s ) optimaity criterion has been the primary criterion used for comparing different supersaturated designs. A smaer vaue is preferred and a vaue of zero represents orthogonaity. The E ( s ) criterion was used in his research for comparing different supersaturated designs, and it was considered as the fitness function in a genetic agorithm approach. Regarding the approach, supersaturated designs are constructed based on the E ( s ) criterion using different approaches to create the initia popuation for genetic agorithm. The basic idea is to find a reasonabe initia popuation of candidate coumns which wi be used by the genetic agorithm to seect the best coumns using the E ( s ) criterion. The operating matrices incude the Hadamard matrix, a genera baanced matrix containing a baanced combinations of the minus and pus signs, and a combined matrix formed from a Hadamard and baanced incompete bock designs. The genetic agorithm was executed by first constructing an initia soution which consists of certain candidate coumns; each candidate coumn is considered a gene. This chromosome ength is equa to the number of factors in the design. Each gene in the chromosome wi represent a coumn in the supersaturated design. In order to construct a chromosome, binary coding is used. The vaue indicates that the coumn is not seected, and the vaue indicates that the coumn is seected. Then the fitness function E ( s ) is cacuated for that soution. Using the crossover and the mutation operations, the agorithms then perform reproduction of a new soution. The fitness function for the new design is cacuated and compared to the previous one. This operation is repeated unti no further improvement in the fitness function is achieved. -5- The E ( S ) Criterion Bashir () constructs supersaturated designs based upon the E(s ) criterion, which is used to measure orthogonaity of design matrices. Let s i denote the dot product of coumns i and in the design matrix X, then the criterion is given by 6

36 E( s = si ). m The E ( s ) criterion gives an intuitive measure of the degree of non-orthogonaity, the smaer the better. Tang and Wu (997) derived a ower bound for E ( s ), meaning optima with respect to E ( s ). The ower bound of E ( s ), which is a function of n and m, is defined as n ( m n + ) E ( s ) ( n) ( m ) ( n ) m, where m is the number of coumns of the design and n is the number of runs. -5- An Exampe The foowing exampe is given in order to understand his method. Consider the N Hadamard matrix H beow. From this matrix, a supersaturated design with 6 = runs for a maximum of N = factors coud be produced. A process contains eight factors is required to be studied and investigated by using a supersaturated design approach. The foowing matrix shows the Hadamard matrix H, consisting of rows and coumns. The H matrix is orthogona and a its entries are minus one and pus one. It is noted that the first coumn and the first row of this matrix is ones. By arbitrariy choosing any coumn as the branching coumn of matrix H, the tota of runs can be spit into two groups. Suppose the ast coumn is seected as the branching coumn, so group I has the sign of one in the branching coumn of H and group II has the minus one in the branching coumn. 7

37 Deeting the first coumn and the branching coumns from GI, resuts in a matrix with 6 rows and coumns. This matrix is a supersaturated design that examines N N = factors in 6 = runs. In the above matrix, there are 6 rows and candidate coumns to construct a supersaturated design for 8 factors. Any combination consists of 8 factors out of the 8

38 candidate coumns coud be a candidate soution. There wi be 45 candidate soutions in this case. As an exampe, one soution coud be formed by seecting the first eight coumns of the matrix. The vaue of s i for the design matrix, for exampe the vaue of s 4 is equa to and is cacuated as foows: s = 4 = The vaue of E ( s ) for every design is cacuated and the owest vaue is picked to be the best design among the others. Figure. shows the optima soution based on E ( s ) for the X matrix. Figure. Optima soution output. This supersaturated design construction method provides optima designs most of the time. The genetic agorithm chooses the best coumns from the candidate coumns to 9

39 buid an optima supersaturated design based on E ( s ). However, in this thesis, genetic agorithms wi be used to choose rows from fu design matrices based on different criteria.

40 CHAPTER METHODOLOGY This thesis concentrates on the construction of efficient mixed-eve designs with desirabe baance and orthogonaity properties. A new optimaity criterion, caed the baance coefficient, is proposed in Section -. This baance coefficient is defined using two aternate formuations. These definitions provide for a numerica evauation of design matrices in terms of their baance property. In Section -, the J -optimaity criterion is modified to the standardized J -optimaity, which is independent of design sizes. Based on these baance and orthogonaity criteria, in Section -, a genetic agorithm is deveoped to construct efficient mixed-eve designs. - Baance Coefficient for Factoria Designs Baance property is an important property for factoria designs. For a baanced design, there is consistency in the variance of the difference between two treatment combinations. As such, it is necessary to deveop a new criterion caed the baance coefficient, to assess the degree of baance for design matrices. Before introducing the definition of the baance coefficient, some reated notation is given as foows. Let n represent the number of rows and m represent the number of coumns for design matrices. For each coumn, w is the corresponding weight, which shows the reative importance of this coumn compared to the others. The coumn contains eves and i are the frequency of the i th eve in that coumn. The standardized number of eves for a specific eve is defined as i f i =. n

41 For a coumn, the number of each eve is shown by a pattern of [ ]. For exampe, for a -eve, -run coumn, [ 4 ] means this coumn contains three st eves, four nd eves, and three rd eves. The baance situation of a design can be represented by these i, which denote the numbers of i th eve in factor. Two aternate formuations of baance coefficient wi be given in Sections -- and --. Before the statement of these definitions, the graphica representation of [ ] wi be first introduced in Section Graphica Representation The pattern of [ ] indicates the baance situation of a coumn. When the i equa each other, that coumn is baanced. Because the summation of i is the number of runs, which is a constant, these i can be represented by a coordinate pane. For exampe, a tweve-run coumn has three eves. These three eves are represented by the notation (,, ). Then the number of eves can be expressed by an equation + +. = This equation has two degrees of freedom, and represents a coordinate pane in threedimensiona space, which is shown in Figure.. [ ] [ ] [ ] [ ] Figure. Graphica representation of a -eve, -run coumn.

42 A the feasibe soutions of + + are discretey scattered on this pane, = which is shown in Figure.. The point in the center, [4 4 4], represents the most baanced situation. In genera, the coumn with k eves can be represented by which denotes a hyperpane. k i = n, i= [ ] [ ] [ ] [ ] [9 ] [9 ] [8 4] [8 ] [8 ] [7 5] [7 4] [7 ] [6 6] [6 5] [6 4] [6 ] [5 7] [5 6] [5 5] [5 4] [4 8] [4 7] [4 6] [4 5] [4 4 4] [ ] [ ] Figure. Feasibe soutions of + + =.

43 Based on the nature of i, a baance coefficient needs to be defined. The obective of this new optimaity criterion is to provide numerica evauation for design matrices in terms of baance property. The definition is given in two forms. -- Formuation of Baance Coefficient - Form I In form I, the motivation behind the definition of the baance coefficient is a simpe optimization probem. The baance coefficient of design matrices wi be derived from this optimization probem. The probem can be formuated as, Max G = K k = X k Subect to K k = where C is a constant. It can be shown that when X X k = C C = X for a i and, K i = K C max( G) =. K To appy this concusion to experimenta designs, coefficient for coumn, and F = i, i=, F are defined as the baance where i is the number of the i th eve. When i are equa to each other, and then the n i =, F reach their maximum vaues, which are n max( F ) =. 4

44 The baance coefficient for design matrix d, F (d), is defined as the combination of the baanced coefficient of each coumn, F, m m F( d) = w = F i w, = = i= where w are the weights for the corresponding coumns. This baance coefficient depends on the number of runs. To avoid this situation, a standardized baance coefficient is defined by using a standardized number of eves. The baance coefficient is standardized when the number of eves is standardized. The notation f i is used instead of i. In this case, the definitions of the standardized baance coefficient for a specific coumn and for a design matrix are given as foows: i i f = ˆ = i =, i= F n When w i ˆ m m i= ( d) = w Fˆ =. = = F f i =, n n max( Fˆ ) = =. n Consider the unbaanced design matrix in Figure. with seven factors: A, B, C, D, E, F and G. Factor G has 4 eves and a other factors have eves. This design contains experimenta runs, so n=, and m=7. The numbers of eves and the baance coefficient for each factor are shown in Tabe.. 5