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
Contents Preface 1. Introduction 1.1 Response Surface Methodology, 1 1.1.1 Approximating Response Functions, 3 1.1.2 The Sequential Nature of RSM, 10 1.1.3 Objectives and Typical Applications of RSM, 12 1.1.4 RSM and the Philosophy of Quality Improvement, 13 1.2 Product Design and Formulation (Mixture Problems), 14 1.3 Robust Product and Process Design, 14 2. Building Empirical Models 2.1 Linear Regression Models, 16 2.2 Estimation of the Parameters in Linear Regression Models, 17 2.3 Properties of the Least Squares Estimators and Estimation of o- 2, 24 2.4 Hypothesis Testing in Multiple Regression, 27 2.4.1 Test for Significance of Regression, 28 2.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients, 31 2.5 Conndence Intervals in Multiple Regression, 36 2.5.1 Confidence Intervals on the Individual Regression Coefficients ß, 37 2.5.2 A Joint Confidence Region on the Regression Coefficients ß, 37 2.5.3 Confidence Interval on the Mean Response, 38 2.6 Prediction of New Response Observation, 39
vi CONTENTS 2.7 Model Adequacy Checking, 41 2.7.1 Residual Analysis, 42 2.7.2 Scaling Residuais, 43 2.7.3 Influence Diagnostics, 48 2.7.4 Testing for Lack of Fit, 50 2.8 Fitting a Second-Order Model, 55 2.9 Qualitative Regressor Variables, 58 Exercises, 67 3 Two-Level Factorial Designs 79 3.1 Introduction, 79 3.2 The 2 2 Design, 79 3.3 The 2 3 Design, 91 3.4 The General 2 k Design, 103 3.5 A Single Replicate of the 2 k Design, 104 3.6 The Addition of Center Points to the 2 k Design, 111 3.7 Blocking in the 2 k Factorial Design, 114 3.7.1 Blocking in the Replicated Design, 115 3.7.2 Confounding in the 2 k Design, 116 Exercises, 124 4 Two-Level Fractional Factorial Designs 134 4.1 Introduction, 134 4.2 The One-Half Fraction of the 2 k Design, 135 4.3 The One-Quarter Fraction of the 2 k Design, 150 4.4 The General 2 k ~ p Fractional Factorial Design, 157 4.5 Resolution III Designs, 162 4.6 Resolution IV and V Designs, 172 4.7 Summary, 173 Exercises, 174 5 Process Improvement with Steepest Ascent 183 5.1 The Computation of the Path of Steepest Ascent, 184 5.2 Consideration of Interaction and Curvature, 188 5.2.1 What About a Second Phase?, 190 5.2.2 What Transpires Following Steepest Ascent?, 190 5.3 Effect of Scale (Choosing Range of Factors), 192 5.4 Confidence Region for Direction of Steepest Ascent, 194 5.5 Steepest Ascent Subject to a Linear Constraint, 198 Exercises, 203
CONTENTS vii 6. The Analysis of Response Surfaces 208 6.1 Second-Order Response Surface, 208 6.2 Second-Order Approximating Function, 208 6.2.1 The Nature of the Second-Order Function and Second-Order Surface, 209 6.2.2 Illustration of Second-Order Response Surface, 211 6.3 A Formal Analytical Approach to the Second-Order Analysis, 217 6.3.1 Nature of\the Stationary Point (Canonical Analysis), 218 6.3.2 Ridge Systems, 220 6.3.3 Role of Contour Plots, 222 6.4 Ridge Analysis of the Response Surface, 226 6.4.1 What Is the Value of Ridge Analysis?, 228 6.4.2 Mathematical Development of Ridge Analysis, 230 6.5 Sampling Properties of Response Surface Result, 235 6.5.1 Standard Error of Predicted Response, 235 6.5.2 Confidence Region on the Location of the Stationary Point, 237 6.5.3 Use and Computation of the Confidence Region on the Location of the Stationary Point, 239 6.6 Response Surface Analysis with Multiple Responses, 244 6.6.1 The Use of Many Responses: The Desirability Function, 247 6.6.2 Other Multiple Response Technique, 258 6.7 Choice of Metrie for the Response, 260 6.7.1 Other Reasons for Data Transformation, 261 6.8 Further Comments Concerning Response Surface Analysis, 264 Exercises, 265 7. Experimental Designs for Fitting Response Surfaces I 279 7.1 Desirable Properties of Response Surface Designs, 279 7.2 Operability Region, Region of Interest, and Model Inadequacy, 280 7.3 Design of Experiments for First-Order Models, 283 7.3.1 The First-Order Orthogonal Design, 284 7.3.2 Orthogonal Designs for Models Containing Interaction, 286
viii CONTENTS 7.3.3 Other First-Order Orthogonal Designs The Simplex Design, 289 7.3.4 Another Variance Type Property-Prediction Variance, 294 7.4 Designs for Fitting Second-Order Models, 297 7.4.1 The Class of Central Composite Designs, 297 7.4.2 Property of Rotatability, 306 7.4.3 Rotatability and the CCD, 309 7.4.4 The Cuboidal Region and the Face Center Cube, 312 7.4.5 When Is the Design Region Spherical?, 314 7.4.6 Summary Statements Regarding CCD, 317 7.4.7 The Box-Behnken Design, 318 7.4.8 Other Spherical RSM Designs; Equiradial Designs, 324 7.4.9 Orthogonal Blocking in Second-Order Designs, 328 Exercises, 341 8. Experimental Designs for Fitting Response Surfaces II 351 8.1 Designs That Require a Relatively Small Run Size, 351 8.1.1 The Small Composite Design, 351 8.1.2 Koshai Design, 357 8.1.3 Hybrid Designs, 359 8.1.4 Some Saturated or Near-Saturated Cuboidal Designs, 363 8.2 General Criteria for Constructing, Evaluating, and Comparing Experimental Designs, 363 8.2.1 Practical Design Optimality, 364 8.2.2 Use of Design Efficiencies for Comparison of Standard Second-Order Designs, 370 8.2.3 Graphical Procedure for Evaluating Prediction Capability of an RSM Design, 373 8.3 Computer-Generated Design in RSM, 383 8.3.1 Important Relationship Between Prediction Variance and Design Augmentation for D-Optimality, 384 8.3.2 Illustration Involving Computer-Generated Design, 387 8.4 Some Final Comments Concerning Design Optimality and Computer-Generated Design, 393 Exercises, 394
CONTENTS ix 9. Miscellaneous Response Surface Topics 402 9.1 Impact of Model Bias on the Fitted Model and Design, 402 9.2 A Design Criterion Involving Bias and Variance, 406 9.2.1 The Case of a First-Order Fitted Model and Cuboidal Region, 409 9.2.2 Minimum Bias Designs for a Spherical Region of Interest, 416 9.2.3 Simultaneous Consideration of Bias and Variance, 418 9.2.4 How Important Is Bias?, 419 9.3 RSM in the Presence of Qualitative Variables, 421 9.3.1 Models First Order in the Quantitative Design Variable (Two-Level Design), 422 9.3.2 First-Order Models with More Than Two Levels of the Qualitative Factors, 424 9.3.3 Models with Interaction Among Qualitative and Quantitative Variables, 425 9.3.4 Design Considerations: First-Order Models with and without Interaction, 426 9.3.5 Design Considerations: Second-Order Models in the Quantitative Variables 9.3.6 Use of Computer Generated Design, 434 9.4 Further Comments, 442 9.5 Errors in Control of Design Levels, 444 9.6 Restrictions on Randomization in RSM, 447 9.6.1 The Dilemma of "Difficult to Change" Factors, 448 9.6.2 Split Plot Structures, 449 9.6.3 RSM Estimation and Testing Under a Split Plot Structure, 453 Exercises, 455 10 Response Surface Methods and Taguchi's Robust Parameter Design 462 10.1 Introduction, 462 10.2 What Is Parameter Design?, 462 10.2.1 Examples of Noise Variables, 463 10.2.2 Examples of a Robust Product, 464 10.3 Crossed Array Designs and Signal-to-Noise Ratios, 465 10.4 The Taguchi Approach to Analysis, 471
X CONTENTS 10.5 Further Comments on Design and Analysis in Parameter Design, 478 10.5.1 Experimental Design, 479 10.5.2 The Taguchi Analysis, 480 10.6 Response Surface Alternatives for Parameter Design Problems, 480 10.6.1 The Role of the Control X Noise Interaction, 481 10.6.2 Use of the Model Containing Both Control and Noise Variables, 486 10.6.3 A Generalization of the Mean and Variance Modeling, 492 10.6.4 Analysis Procedures Associated with the Two Response Surfaces, 495 10.6.5 Appropriate Estimation of the Process Variance with Application, 503 10.6.6 The Log Linear Variance Model, 508 10.7 Designs for Robust Parameter Design; Crossed and Combined Arrays, 511 10.7.1 The Combined Array, 511 10.7.2 Second-Order Designs for Robust Parameter Design, 514 10.7.3 Summary Remarks, 515 10.8 Dispersion Effects in Highly Fractionated Designs, 515 10.8.1 The Use of Residuais, 516 10.8.2 Further Diagnostic Information from Residuais, 518 10.8.3 Further Comments Concerning Variance Modeling, 521 Exercises, 529 11. Experiments with Mixtures 535 11.1 Introduction, 535 11.2 Simplex Designs and Canonical Mixture Polynomials, 539 11.2.1 Simplex Lattice Designs, 539 11.2.2 The Simplex-Centroid Design and Its Associated Polynomial, 549 11.2.3 Augmentation of Simplex Designs with Axial Runs, 551 11.3 Response Trace Plots, 558 11.4 Reparameterizing Canonical Mixture Models to Contain a Constant Term (ß 0 ), 560 Exercises, 562
CONTENTS XI 12. Other Mixture Design and Analysis Techniques 570 12.1 Constraints on the Component Proportions, 571 12.1.1 Lower-Bound Constraints on the Component Proportions, 571 12.1.2 Upper-Bound Constraints on the Component Proportions, 584 12.1.3 Active Upper- and Lower-Bound Constraints, 587 12.1.4 Multicomponent Constraints, 603 12.2 Mixture Experiments Using Ratios of Components, 604 12.3 Process Variables in Mixture Experiments, 609 12.4 Screening Mixture Components, 611 Exercises, 615 13. Continuous Process Improvement with Evolutionary Operation 13.1 Introduction, 624 13.2 An Example of EVOP, 625 13.3 EVOP Using Computer Software, 630 13.4 Simplex EVOP, 634 13.5 Some Practical Advice About Using EVOP, 636 Exercises, 637 624 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
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