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Design and Analysis of Response Surface Designs with Restricted Randomization Wayne R Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2006 Design and Analysis of Response Surface Designs with Restricted Randomization Wayne R. Wesley Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ENGINEERING DESIGN AND ANALYSIS OF RESPONSE SURFACE DESIGNS WITH RESTRICTED RANDOMIZATION By Wayne R. Wesley A Dissertation submitted to the Department of Industrial Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2006 Copyright © 2006 Wayne R. Wesley All Rights Reserved The members of the Committee approve the dissertation of Wayne R. Wesley defended on July 6, 2006. ______________________________ James R. Simpson Professor Directing Dissertation ______________________________ Anuj Srivastava Outside Committee Member ______________________________ Peter A. Parker Outside Committee Member ______________________________ Joseph J. Pignatiello, Jr. Committee Member Approved: _________________________________ Chuck Zhang, Chair, Department of Industrial and Manufacturing Engineering _________________________________ Ching-Jen Chen, Dean, College of Engineering The Office of Graduate Studies has verified and approved the above named committee members. ii I dedicate this dissertation to my son and daughter Jowayne and Jowaynah Wesley. iii ACKNOWLEDGEMENTS I must first give thanks to the almighty God for good health, strength and wisdom to successfully complete my dissertation. It is with a deep sense of gratitude that I express my sincere appreciation to Dr. James Simpson and Dr. Peter Parker for their invaluable advice and contributions in directing the dissertation. Special thanks to Dr. Joseph Pignatiello and Dr. Anuj Srivastava for their meaningful comments and suggestions that helped to improve the quality of the dissertation. Additionally, I would like to give special recognition to my wife Joy Wesley for her unwavering support and commitment throughout my endeavors at The Florida State University. Special thanks to my mother Mary White for her tremendous sacrifice in giving me the opportunity to develop and pursue my career goals. To my fellow students and friends, Michelle, Lisa, Francisco, Rupert and Young-Guo thanks for your support and encouragement. Finally thanks to all those who supported me with their prayers, financially and otherwise. iv TABLE OF CONTENTS LIST OF TABLES...........................................................................................................viii LIST OF FIGURES ........................................................................................................... xi ABSTRACT...................................................................................................................... xii CHAPTER 1 ....................................................................................................................... 1 1.0 INTRODUCTION .................................................................................................. 1 1.1 Background................................................................................................. 1 1.2 The Split Plot Model................................................................................... 4 1.3 Design Optimality....................................................................................... 6 1.4 Problem Statement...................................................................................... 9 1.5 Research Objectives.................................................................................. 10 CHAPTER 2 ..................................................................................................................... 11 2.0 REVIEW OF LITERATURE ............................................................................... 11 2.1 Introduction............................................................................................... 11 2.2 First-Order Split Plot Designs................................................................... 12 2.3 Second-Order Split Plot Designs .............................................................. 14 2.4 Design Optimality Criteria........................................................................ 16 2.5 Summary................................................................................................... 18 CHAPTER 3 ..................................................................................................................... 19 3.0 Impact of Restricted Randomization on the Structure of the Information Matrix 19 3.1 Abstract..................................................................................................... 19 3.2 Introduction............................................................................................... 20 3.3 The Split Plot Model................................................................................. 20 3.4 Structure of Second-Order SPD................................................................ 24 3.5 Critical Functions of the Variance Ratio and Whole Plot Sizes ............... 28 3.6 Information Matrix for CCD and BBD Structures ................................... 29 3.7 Impact of Design Structure on the Application of the Critical Functions 33 3.8 Outline of Analytical Characterization Scheme ....................................... 38 −1 3.9 Characterization of (XR' −1 X) for Split Plot CCD.................................. 40 −1 3.10 Characterization of (XR' −1 X) for Split Plot BBD.................................. 45 3.11 Generalized Variance of Parameter Estimates.......................................... 46 3.11.1 Determinant for split plot CCD................................................... 47 v 3.11.2 Determinant for split plot BBD................................................... 47 3.12 Application................................................................................................ 48 3.13 Conclusion ................................................................................................ 54 CHAPTER 4 ..................................................................................................................... 56 4.0 Prediction Variance Properties and G-Criterion Location for Second-Order Split Plot Designs .......................................................................................................... 56 4.1 Abstract..................................................................................................... 56 4.2 Introduction............................................................................................... 57 4.3 The Split Plot Model................................................................................. 58 4.4 Prediction Optimality Criteria................................................................... 60 4.5 Second-Order Split Plot Designs .............................................................. 61 4.6 Critical functions of the variance ratio and whole plot sizes.................... 62 4.7 Information Matrix for the Split Plot CCD............................................... 63 4.8 Information Matrix for the Split Plot BBD............................................... 67 4.9 Minimum and Maximum Prediction Variance ......................................... 70 4.10 Vminρ and Vmaxρ for Spherical Regions .................................................. 72 4.11 Vmin and Vmax for Whole Plot Design Space for Spherical Regions ρz ρz ................................................................................................................... 73 4.12 Vmin and Vmax for Subplot Design Space for Spherical Regions .... 74 ρx ρx 4.13 Vminρ and Vmaxρ for Combined Design Space for Spherical Regions.... 74 4.14 Vminρ and Vmaxρ for Cuboidal Regions................................................... 75 4.15 Vmin and Vmax for Whole Plot Design Space for Cuboidal Regions 76 ρz ρz 4.16 Vmin and Vmax for Subplot Design Space for Cuboidal Regions.... 76 ρx ρx 4.17 Vminρ and Vmaxρ for the Combined Design Space for Cuboidal Regions77 4.18 Delta G-Criterion Measures for Whole Plot and Subplot Design Spaces 79 4.19 Prediction Variance Assessment of Second-Order SPD........................... 79 4.19.1 CCD with one whole plot factor and two subplot factors........... 79 4.19.2 BBD with two whole plot factors and two subplot factors......... 85 4.19.3 FCC with two whole plot factors and three subplots factors...... 88 4.20 Conclusion ................................................................................................ 91 CHAPT ER 5 ..................................................................................................................... 93 5.0 Integrated Prediction Variance for Response Surface Designs ............................ 93 5.1 Abstract..................................................................................................... 93 5.2 Introduction............................................................................................... 94 5.3 The Split Plot Model................................................................................. 95 5.4 IV – Optimality Criterion.......................................................................... 97 −1 5.5 Analytical Characterization of (XR' −1 X) ............................................ 100 5.5.1 Critical functions of the variance ratio and whole plot sizes...... 100 5.5.2 Information
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