Response Surface Methodology:

Transcription

1 Response Surface Methodology: Process and Product Optimization Using Designed Experiments RAYMOND H. MYERS Virginia Polytechnic Institute and State University DOUGLAS C. MONTGOMERY Arizona State University A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Brisbane Toronto Singapore

2 Contents Preface 1. Introduction 1.1 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 Product and Process Design, Building Empirical Models 2.1 Linear Regression Models, Estimation of the Parameters in Linear Regression Models, Properties of the Least Squares Estimators and Estimation of o- 2, Hypothesis Testing in Multiple Regression, Test for Significance of Regression, Tests on Individual Regression Coefficients and Groups of Coefficients, Conndence Intervals in Multiple Regression, Confidence Intervals on the Individual Regression Coefficients ß, A Joint Confidence Region on the Regression Coefficients ß, Confidence Interval on the Mean Response, Prediction of New Response Observation, 39

3 vi CONTENTS 2.7 Model Adequacy Checking, Residual Analysis, Scaling Residuais, Influence Diagnostics, Testing for Lack of Fit, Fitting a Second-Order Model, Qualitative Regressor Variables, 58 Exercises, 67 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, 116 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 k ~ p Fractional Factorial Design, Resolution III Designs, Resolution IV and V Designs, Summary, 173 Exercises, Process Improvement with Steepest Ascent The Computation of the Path of Steepest Ascent, Consideration of Interaction and Curvature, What About a Second Phase?, What Transpires Following Steepest Ascent?, Effect of Scale (Choosing Range of Factors), Confidence Region for Direction of Steepest Ascent, Steepest Ascent Subject to a Linear Constraint, 198 Exercises, 203

4 CONTENTS vii 6. The Analysis of 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 Surface, A Formal Analytical Approach to the Second-Order Analysis, 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 Result, 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, Response Surface Analysis with Multiple Responses, The Use of Many Responses: The Desirability Function, Other Multiple Response Technique, Choice of Metrie for the Response, Other Reasons for Data Transformation, 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, Design of Experiments for First-Order Models, The First-Order Orthogonal Design, Orthogonal Designs for Models Containing Interaction, 286

5 viii CONTENTS Other First-Order Orthogonal Designs The Simplex Design, Another Variance Type Property-Prediction Variance, Designs for Fitting Second-Order Models, The Class of Central Composite Designs, Property of Rotatability, Rotatability and the CCD, The Cuboidal Region and the Face Center 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, 328 Exercises, Experimental Designs for Fitting Response Surfaces II Designs That Require a Relatively Small Run Size, The Small Composite Design, Koshai Design, Hybrid Designs, 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 Prediction Capability of an RSM Design, Computer-Generated Design in RSM, Important Relationship Between Prediction Variance and Design Augmentation for D-Optimality, Illustration Involving Computer-Generated Design, Some Final Comments Concerning Design Optimality and Computer-Generated Design, 393 Exercises, 394

6 CONTENTS ix 9. Miscellaneous Response Surface Topics Impact 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?, RSM in the Presence of Qualitative Variables, Models First Order in the Quantitative Design Variable (Two-Level Design), First-Order Models with More Than Two Levels of the Qualitative Factors, Models with Interaction Among Qualitative and Quantitative Variables, Design Considerations: First-Order Models with and without Interaction, Design Considerations: Second-Order Models in the Quantitative Variables Use of Computer Generated Design, Further Comments, Errors in Control of Design Levels, Restrictions on Randomization in RSM, The Dilemma of "Difficult to Change" Factors, Split Plot Structures, RSM Estimation and Testing Under a Split Plot Structure, 453 Exercises, Response Surface Methods and Taguchi's Robust Parameter Design Introduction, What Is Parameter Design?, Examples of Noise Variables, Examples of a Robust Product, Crossed Array Designs and Signal-to-Noise Ratios, The Taguchi Approach to Analysis, 471

7 X CONTENTS 10.5 Further Comments on Design and Analysis in Parameter Design, Experimental Design, The Taguchi Analysis, Response Surface Alternatives for Parameter Design Problems, The Role of the Control X Noise Interaction, Use of the Model Containing Both Control and Noise Variables, A Generalization of the Mean and Variance Modeling, Analysis Procedures Associated with the Two Response Surfaces, Appropriate Estimation of the Process Variance with Application, The Log Linear Variance Model, Designs for Robust Parameter Design; Crossed and Combined Arrays, The Combined Array, Second-Order Designs for Robust Parameter Design, Summary Remarks, Dispersion Effects in Highly Fractionated Designs, The Use of Residuais, Further Diagnostic Information from Residuais, Further Comments Concerning Variance Modeling, 521 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 (ß 0 ), 560 Exercises, 562

8 CONTENTS XI 12. 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, Active Upper- and Lower-Bound Constraints, Multicomponent Constraints, Mixture Experiments Using Ratios of Components, Process Variables in Mixture Experiments, Screening Mixture Components, 611 Exercises, Continuous Process Improvement with Evolutionary Operation 13.1 Introduction, An Example of EVOP, EVOP Using Computer Software, Simplex EVOP, Some Practical Advice About Using EVOP, 636 Exercises, Appendices Appendix 1. Variable Selection and Model-Building in Regression 640 Appendix 2. Multicollineariry and Biased Estimation in Regression 656 Appendix 3. Robust Regression 667 Appendix 4. Some Mathematical Insights into Ridge Analysis 674 Appendix 5. Moment Matrix of a Rotatable Design 675 Appendix 6. Rotatability of a Second-Order Equiradial Design 680 Appendix 7. Relationship Between D-Optimality and the Volume of a Joint Confidence Ellipsoid on ß Appendix 8. Relationship Between Maximum Prediction Variance 683 in a Region and the Number of Parameters 685

9 Xll CONTENTS Appendix 9. The Development of Equation (8.21) 686 Appendix 10. Determination of Data Augmentation Result (Choice of x r+1 for the Sequential Development ofa D-Optimal Design) 687 References 689 Index 697