RESPONSE SURFACE METHODOLOGY

Size: px

Start display at page:

Download "RESPONSE SURFACE METHODOLOGY"

Ronald Carpenter
6 years ago
Views:

3 RESPONSE SURFACE METHODOLOGY

4 WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg Editors Emeriti: Vic Barnett, J. Stuart Hunter, Jozef L. Teugels A complete list of the titles in this series appears at the end of this volume.

5 RESPONSE SURFACE METHODOLOGY Process and Product Optimization Using Designed Experiments Third Edition RAYMOND H. MYERS Virginia Polytechnic University, Department of Statistics, Blacksburg, VA DOUGLAS C. MONTGOMERY Arizona State University, Department of Industrial Engineering, Tempe, AZ CHRISTINE M. ANDERSON-COOK Los Alamos National Laboratory, Los Alamos, NM

6 Copyright # 2009 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) , outside the United States at (317) or fax (317) Wiley also publishes its books in variety of electronic formats. Some content that appears in print may not be available in electronic format. For more information about Wiley products, visit our web site at Library of Congress Cataloging-in-Publication Data: Myers, Raymond H. Response surface methodology : process and product optimization using designed experiments rd ed. / Raymond H. Myers, Douglas C. Montgomery, Christine M. Anderson-Cook. p. cm. - - (Wiley series in probability and statistics) Includes bibliographical references and index. ISBN (cloth) 1. Experimental design. 2. Response surfaces (Statistics). I. Montgomery, Douglas C. II. Anderson-Cook, Christine M. III. Title. QA279.M dc Printed in the United States of America

7 CONTENTS Preface xi 1 Introduction Response Surface Methodology, Approximating Response Functions, The Sequential Nature of RSM, Objectives and Typical Applications of RSM, RSM and the Philosophy of Quality Improvement, Product Design and Formulation (Mixture Problems), Robust Design and Process Robustness Studies, Useful References on RSM, 11 2 Building Empirical Models Linear Regression Models, Estimation of the Parameters in Linear Regression Models, Properties of the Least Squares Estimators and Estimation of s 2, Hypothesis Testing in Multiple Regression, Test for Significance of Regression, Tests on Individual Regression Coefficients and Groups of Coefficients, Confidence Intervals in Multiple Regression, Confidence Intervals on the Individual Regression Coefficients b, A Joint Confidence Region on the Regression Coefficients b, Confidence Interval on the Mean Response, Prediction of New Response Observations, Model Adequacy Checking, 36 v

8 vi CONTENTS Residual Analysis, Scaling Residuals, Influence Diagnostics, Testing for Lack of Fit, Fitting a Second-Order Model, Qualitative Regressor Variables, Transformation of the Response Variable, 58 Exercises, 63 3 Two-Level Factorial Designs Introduction, The 2 2 Design, The 2 3 Design, The General 2 k Design, A Single Replicate of the 2 k Design, The Addition of Center Points to the 2 k Design, Blocking in the 2 k Factorial Design, Blocking in the Replicated Design, Confounding in the 2 k Design, Split-Plot Designs, 121 Exercises, Two-Level Fractional Factorial Designs Introduction, The One-Half Fraction of the 2 k Design, The One-Quarter Fraction of the 2 k Design, The General 2 k2p Fractional Factorial Design, Resolution III Designs, Resolution IV and V Designs, Fractional Factorial Split-Plot Designs, Summary, 172 Exercises, Process Improvement with Steepest Ascent Determining the Path of Steepest Ascent, Development of the Procedure, Practical Application of the Method of Steepest Ascent, Consideration of Interaction and Curvature, What About a Second Phase?, What Happens Following Steepest Ascent?, Effect of Scale (Choosing Range of Factors), Confidence Region for Direction of Steepest Ascent, Steepest Ascent Subject to a Linear Constraint, Steepest Ascent in a Split-Plot Experiment, 202 Exercises, 210

9 CONTENTS vii 6 The Analysis of Second-Order Response Surfaces Second-Order Response Surface, Second-Order Approximating Function, The Nature of the Second-Order Function and Second-Order Surface, Illustration of Second-Order Response Surfaces, A Formal Analytical Approach to the Second-Order Model, Location of the Stationary Point, Nature of the Stationary Point (Canonical Analysis), Ridge Systems, Role of Contour Plots, Ridge Analysis of the Response Surface, What is the Value of Ridge Analysis?, Mathematical Development of Ridge Analysis, Sampling Properties of Response Surface Results, Standard Error of Predicted Response, Confidence Region on the Location of the Stationary Point, Use and Computation of the Confidence Region on the Location of the Stationary Point, Confidence Intervals on Eigenvalues in Canonical Analysis, Multiple Response Optimization, Further Comments Concerning Response Surface Analysis, 264 Exercises, Experimental Designs for Fitting Response Surfaces I Desirable Properties of Response Surface Designs, Operability Region, Region of Interest, and Model Inadequacy, Model Inadequacy and Model Bias, Design of Experiments for First-Order Models, The First-Order Orthogonal Design, Orthogonal Designs for Models Containing Interaction, Other First-Order Orthogonal Designs The Simplex Design, Another Variance Property Prediction Variance, Designs for Fitting Second-Order Models, The Class of Central Composite Designs, Design Moments and Property of Rotatability, Rotatability and the CCD, More on Prediction Variance Scaled, Unscaled, and Estimated, The Cuboidal Region and the Face-Centered Cube, When is the Design Region Spherical?, Summary Statements Regarding CCD, The Box Behnken Design, Other Spherical RSM Designs; Equiradial Designs, Orthogonal Blocking in Second-Order Designs, 325 Exercises, 336

10 viii CONTENTS 8 Experimental Designs for Fitting Response Surfaces II Designs that Require a Relatively Small Run Size, The Hoke Designs, Koshal Design, Hybrid Designs, The Small Composite Design, Some Saturated or Near-Saturated Cuboidal Designs, General Criteria for Constructing, Evaluating, and Comparing Experimental Designs, Practical Design Optimality, Use of Design Efficiencies for Comparison of Standard Second-Order Designs, Graphical Procedure for Evaluating the Prediction Capability of an RSM Design, Computer-Generated Designs in RSM, Important Relationship between Prediction Variance and Design Augmentation for D-Optimality, Illustrations Involving Computer-Generated Design, Some Final Comments Concerning Design Optimality and Computer-Generated Design, 405 Exercises, Advanced Topics in Response Surface Methodology Effects of Model Bias on the Fitted Model and Design, A Design Criterion Involving Bias and Variance, The Case of a First-Order Fitted Model and Cuboidal Region, Minimum Bias Designs for a Spherical Region of Interest, Simultaneous Consideration of Bias and Variance, How Important is Bias?, Errors in Control of Design Levels, Experiments with Computer Models, Minimum Bias Estimation of Response Surface Models, Neural Networks, RSM for Non-Normal Responses Generalized Linear Models, Model Framework: The Link Function, The Canonical Link Function, Estimation of Model Coefficients, Properties of Model Coefficients, Model Deviance, Overdispersion, Examples, Diagnostic Plots and Other Aspects of the GLM, Split-Plot Designs for Second-Order Models, 466 Exercises, 476

11 CONTENTS ix 10 Robust Parameter Design and Process Robustness Studies Introduction, What is Parameter Design?, Examples of Noise Variables, An Example of Robust Product Design, The Taguchi Approach, Crossed Array Designs and Signal-to-Noise Ratios, Analysis Methods, Further Comments, The Response Surface Approach, The Role of the Control Noise Interaction, A Model Containing Both Control and Noise Variables, Generalization of Mean and Variance Modeling, Analysis Procedures Associated with the Two Response Surfaces, Estimation of the Process Variance, Direct Variance Modeling, Use of Generalized Linear Models, Experimental Designs for RPD and Process Robustness Studies, Combined Array Designs, Second-Order Designs, Other Aspects of Design, Dispersion Effects in Highly Fractionated Designs, The Use of Residuals, Further Diagnostic Information from Residuals, Further Comments Concerning Variance Modeling, 544 Exercises, Experiments with Mixtures Introduction, Simplex Designs and Canonical Mixture Polynomials, Simplex Lattice Designs, The Simplex-Centroid Design and Its Associated Polynomial, Augmentation of Simplex Designs with Axial Runs, Response Trace Plots, Reparameterizing Canonical Mixture Models to Contain a Constant Term (b 0 ), 577 Exercises, Other Mixture Design and Analysis Techniques Constraints on the Component Proportions, Lower-Bound Constraints on the Component Proportions, Upper-Bound Constraints on the Component Proportions, 599

12 x CONTENTS Active Upper- and Lower-Bound Constraints, Multicomponent Constraints, Mixture Experiments Using Ratios of Components, Process Variables in Mixture Experiments, Mixture-Process Model and Design Basics, Split-Plot Designs for Mixture-Process Experiments, Robust Parameter Designs for Mixture-Process Experiments, Screening Mixture Components, 641 Exercises, 643 Appendix 1 Moment Matrix of a Rotatable Design 655 Appendix 2 Rotatability of a Second-Order Equiradial Design 661 References 665 Index 677

13 PREFACE This book deals with the exploration and optimization of response surfaces. This is a problem faced by experimenters in many technical fields, where, in general, the response variable of interest is y and there is a set of predictor variables x 1, x 2,..., x k. For example, y might be the viscosity of a polymer and x 1, x 2, and x 3 might be the reaction time, the reactor temperature, and the catalyst feed rate in the process. In some systems the nature of the relationship between y and the x s might be known exactly, based on the underlying engineering, chemical, or physical principles. Then we could write a model of the form y ¼ g(x 1, x 2,..., x k ) þ 1, where 1 represents the error in the system. This type of relationship is often called a mechanistic model. We consider the more common situation where the underlying mechanism is not fully understood, and the experimenter must approximate the unknown function g with an appropriate empirical model y ¼ f(x 1, x 2,..., x k ) þ 1. Usually the function f is a first-order or second-order polynomial. This empirical model is called a response surface model. Identifying and fitting an appropriate response surface model from experimental data requires some knowledge of statistical experimental design fundamentals, regression modeling techniques, and elementary optimization methods. This book integrates all three of these topics into what has been popularly called response surface methodology (RSM). We assume that the reader has some previous exposure to statistical methods and matrix algebra. Formal coursework in basic principles of experimental design and regression analysis would be helpful, but are not essential, because the important elements of these topics are presented early in the text. We have used this book in a graduate-level course on RSM for statisticians, engineers, and chemical/physical scientists. We have also used it in industrial short courses and seminars for individuals with a wide variety of technical backgrounds. This third edition is a substantial revision of the book. We have rewritten many sections to incorporate new material, ideas, and examples, and to more fully explain some topics that were only briefly mentioned in previous editions. We have also woven the computer more xi

14 xii PREFACE tightly into the presentation, relying on JMP 7 and Design-Expert Version 7 for much of the computing, but also continuing to employ SAS for a few applications. Chapters 1 through 4 contain the preliminary material essential to studying RSM. Chapter 1 is an introduction to the general field of RSM, describing typical applications such as (a) finding the levels of process variables that optimize a response of interest or (b) discovering what levels of these process variables will result in a product satisfying certain requirements or specifications on responses such as yield, molecular weight, purity, or viscosity. Chapter 2 is a summary of regression methods useful in response surface work, focusing on the basic ideas of least squares model fitting, diagnostic checking, and inference for the linear regression model. Chapters 3 and 4 describe two-level factorial and fractional factorial designs. These designs are essential for factor screening or identifying the correct set of process variables to use in the RSM study. They are also basic building blocks for many of the response surface designs discussed later in the text. Chapter 5 presents the method of steepest ascent, a simple but powerful optimization procedure used at the early stages of RSM to move the process from a region of relatively poor performance to one of greater potential. Chapter 6 introduces the analysis and optimization of a second-order response surface model. Both graphical and numerical techniques are presented. This chapter also includes techniques for the simultaneous optimization of several responses, a common problem in the application of RSM. Chapters 7 and 8 present detailed information on the choice of experimental designs for fitting response surface models. Chapter 7 is devoted to standard designs, including the central composite and Box Behnken designs, and the important topic of blocking a response surface design. Chapter 8 covers small response surface designs, design optimality criteria, the use of computer-generated designs in RSM, and methods for evaluation of the prediction properties of response surface models constructed from various designs. We focus on variance dispersion graphs and fraction of design space plots, which are very important ways to summarize prediction properties. Chapter 9 contains more advanced RSM topics, including the use of mean square error as a design criterion, the effect of errors in controllable variables, RSM experiments for computer models, neural networks and RSM, split-plot type designs in a response surface setting, and the use of generalized linear models in the analysis of response surface experiments. Chapter 10 describes how the problem of robust parameter design originally proposed by Taguchi can be efficiently solved in the RSM framework. We show how RSM not only makes the original problem posed by Taguchi easier to solve, but also provides much more information to the analyst about process or system performance. This chapter also contains much information on robust parameter design and process robustness studies. Chapters 11 and 12 present techniques for designing and analyzing experiments that involve mixtures. A mixture experiment is a special type of response surface experiment in which the design factors are the components or ingredients of a mixture, and the response depends on the proportions of the ingredients that are present. Extensive sets of end-of-chapter problems are provided, along with a reference section. The previous two editions of the text were written to emphasize methods that are useful in industry and that we have found useful in our own consulting experience. We have continued that applied focus in this new edition, though much new material has been added. We develop enough of the underlying theory to allow the reader to gain an understanding of the assumptions and conditions necessary to successfully apply RSM. We are grateful to many individuals that have contributed meaningfully to this book. In particular, Dr. Bradley Jones, Mr. Pat Whitcomb, Dr. Geoff Vining, Dr. Soren Bisgaard, Dr. Connie Borror, Dr. Scott Kowalski, Dr. Dennis Lin, Dr. George Runger,

15 PREFACE xiii and Dr. Enrique Del Castillo made many useful suggestions. Dr. Matt Carlyle and Dr. Enrique Del Castillo also provided some figures that were most helpful. We also thank the many classes of graduate students that have studied from the book and the instructors that have used the book. They have made many helpful comments and suggestions to improve the clarity of the presentation. We have tried to incorporate many of their suggestions. We also thank John Wiley & Sons for permission to use and adapt copyrighted material. Blacksburg, Virginia Tempe, Arizona Los Alamos, New Mexico March, 2008 RAYMOND H. MYERS DOUGLAS C. MONTGOMERY CHRISTINE M. ANDERSON-COOK

Response Surface Methodology

Response Surface Methodology Process and Product Optimization Using Designed Experiments Second Edition RAYMOND H. MYERS Virginia Polytechnic Institute and State University DOUGLAS C. MONTGOMERY Arizona