Response Surface Methodology

Preface xiii 1 Introduction 1 1.1 Response Surface Methodology, 1 1.1.1 Approximating Response Functions, 2 1.1.2 The Sequential Nature of RSM, 7 1.1.3 Objectives and Typical Applications of RSM, 9 1.1.4 RSM and the Philosophy of Quality Improvement, 11 1.2 Product Design and Formulation (Mixture Problems), 11 1.3 Robust Design and Process Robustness Studies, 12 1.4 Useful References on RSM, 12 2 Building Empirical Models 13 2.1 Linear Regression Models, 13 2.2 Estimation of the Parameters in Linear Regression Models, 14 2.3 Properties of the Least Squares Estimators and Estimation of 𝜎2, 22 2.4 Hypothesis Testing in Multiple Regression, 24 2.4.1 Test for Significance of Regression, 24 2.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients, 27 2.5 Confidence Intervals in Multiple Regression, 31 2.5.1 Confidence Intervals on the Individual Regression Coefficients β, 32 2.5.2 A Joint Confidence Region on the Regression Coefficients β, 32 2.5.3 Confidence Interval on the Mean Response, 33 2.6 Prediction of New Response Observations, 35 2.7 Model Adequacy Checking, 36 2.7.1 Residual Analysis, 36 2.7.2 Scaling Residuals, 38 2.7.3 Influence Diagnostics, 42 2.7.4 Testing for Lack of Fit, 43 2.8 Fitting a Second-Order Model, 47 2.9 Qualitative Regressor Variables, 55 2.10 Transformation of the Response Variable, 61 Exercises, 66 3 Two-Level Factorial Designs 81 3.1 Introduction, 81 3.2 The 22 Design, 82 3.3 The 23 Design, 94 3.4 The General 2k Design, 103 3.5 A Single Replicate of the 2k Design, 108 3.6 2k Designs are Optimal Designs, 125 3.7 The Addition of Center Points to the 2k Design, 130 3.8 Blocking in the 2k Factorial Design, 135 3.8.1 Blocking in the Replicated Design, 135 3.8.2 Confounding in the 2k Design, 137 3.9 Split-Plot Designs, 141 Exercises, 146 4 Two-Level Fractional Factorial Designs 161 4.1 Introduction, 161 4.2 The One-Half Fraction of the 2k Design, 162 4.3 The One-Quarter Fraction of the 2k Design, 174 4.4 The General 2k−p Fractional Factorial Design, 184 4.5 Resolution III Designs, 188 4.6 Resolution IV and V Designs, 197 4.7 Alias Structures in Fractional Factorial and Other Designs, 198 4.8 Nonregular Fractional Factorial Designs, 200 4.8.1 Nonregular Fractional Factorial Designs for 6, 7, and 8 Factors in 16 Runs, 203 4.8.2 Nonregular Fractional Factorial Designs for 9 Through 14 Factors in 16 Runs, 209 4.8.3 Analysis of Nonregular Fractional Factorial Designs, 213 4.9 Fractional Factorial Split-Plot Designs, 216 4.10 Summary, 219 Exercises, 220 5 Process Improvement with Steepest Ascent 233 5.1 Determining the Path of Steepest Ascent, 234 5.1.1 Development of the Procedure, 234 5.1.2 Practical Application of the Method of Steepest Ascent, 237 5.2 Consideration of Interaction and Curvature, 241 5.2.1 What About a Second Phase?, 244 5.2.2 What Happens Following Steepest Ascent?, 244 5.3 Effect of Scale (Choosing Range of Factors), 245 5.4 Confidence Region for Direction of Steepest Ascent, 247 5.5 Steepest Ascent Subject to a Linear Constraint, 250 5.6 Steepest Ascent in a Split-Plot Experiment, 254 Exercises, 262 6 The Analysis of Second-Order Response Surfaces 273 6.1 Second-Order Response Surface, 273 6.2 Second-Order Approximating Function, 274 6.2.1 The Nature of the Second-Order Function and Second-Order Surface, 274 6.2.2 Illustration of Second-Order Response Surfaces, 276 6.3 A Formal Analytical Approach to the Second-Order Model, 277 6.3.1 Location of the Stationary Point, 278 6.3.2 Nature of the Stationary Point (Canonical Analysis), 278 6.3.3 Ridge Systems, 282 6.3.4 Role of Contour Plots, 286 6.4 Ridge Analysis of the Response Surface, 289 6.4.1 Benefits of Ridge Analysis, 290 6.4.2 Mathematical Development of Ridge Analysis, 291 6.5 Sampling Properties of Response Surface Results, 296 6.5.1 Standard Error of Predicted Response, 296 6.5.2 Confidence Region on the Location of the Stationary Point, 299 6.5.3 Use and Computation of the Confidence Region on the Location of the Stationary Point, 300 6.5.4 Confidence Intervals on Eigenvalues in Canonical Analysis, 304 6.6 Further Comments Concerning Response Surface Analysis, 307 Exercises, 307 7 Multiple Response Optimization 325 7.1 Balancing Multiple Objectives, 325 7.2 Strategies for Multiple Response Optimization, 338 7.2.1 Overlaying Contour Plots, 339 7.2.2 Constrained Optimization, 340 7.2.3 Desirability Functions, 341 7.2.4 Pareto Front Optimization, 343 7.2.5 Other Options for Optimization, 349 7.3 A Sequential Process for Optimization—DMRCS, 350 7.4 Incorporating Uncertainty of Response Predictions into Optimization, 352 Exercises, 357 8 Design of Experiments for Fitting Response Surfaces—I 369 8.1 Desirable Properties of Response Surface Designs, 369 8.2 Operability Region, Region of Interest, and Metrics for Desirable Properties, 371 8.2.1 Metrics for Desirable Properties, 372 8.2.2 Model Inadequacy and Model Bias, 373 8.3 Design of Experiments for First-Order Models and First-Order Models with Interactions, 375 8.3.1 The First-Order Orthogonal Design, 376 8.3.2 Orthogonal Designs for Models Containing Interaction, 378 8.3.3 Other First-Order Orthogonal Designs—The Simplex Design, 381 8.3.4 Definitive Screening Designs, 385 8.3.5 Another Variance Property—Prediction Variance, 389 8.4 Designs for Fitting Second-Order Models, 393 8.4.1 The Class of Central Composite Designs, 393 8.4.2 Design Moments and Property of Rotatability, 399 8.4.3 Rotatability and the CCD, 403 8.4.4 More on Prediction Variance—Scaled, Unscaled, and Estimated, 406 8.4.5 The Face-Centered Cube in Cuboidal Regions, 408 8.4.6 Choosing between Spherical and Cuboidal Regions, 411 8.4.7 The Box–Behnken Design, 413 8.4.8 Definitive Screening Designs for Fitting Second-Order Models, 417 8.4.9 Orthogonal Blocking in Second-Order Designs, 422 Exercises, 434 9 Experimental Designs for Fitting Response Surfaces—II 451 9.1 Designs that Require a Relatively Small Run Size, 452 9.1.1 The Hoke Designs, 452 9.1.2 Koshal Design, 454 9.1.3 Hybrid Designs, 455 9.1.4 The Small Composite Design, 458 9.1.5 Some Saturated or Near-Saturated Cuboidal Designs, 462 9.1.6 Equiradial Designs, 463 9.2 General Criteria for Constructing, Evaluating, and Comparing Designed Experiments, 465 9.2.1 Practical Design Optimality, 467 9.2.2 Use of Design Efficiencies for Comparison of Standard Second-Order Designs, 474 9.2.3 Graphical Procedure for Evaluating the Prediction Capability of an RSM Design, 477 9.3 Computer-Generated Designs in RSM, 488 9.3.1 Important Relationship Between Prediction Variance and Design Augmentation for D-Optimality, 491 9.3.2 Algorithms for Computer-Generated Designs, 494 9.3.3 Comparison of D-, G-, and I-Optimal Designs, 497 9.3.4 Illustrations Involving Computer-Generated Design, 499 9.3.5 Computer-Generated Designs Involving Qualitative Variables, 508 9.4 Multiple Objective Computer-Generated Designs for RSM, 517 9.4.1 Pareto Front Optimization for Selecting a Design, 518 9.4.2 Pareto Aggregating Point Exchange Algorithm, 519 9.4.3 Using DMRCS for Design Optimization, 520 9.5 Some Final Comments Concerning Design Optimality and Computer-Generated Design, 525 Exercises, 527 10 Advanced Topics in Response Surface Methodology 543 10.1 Effects of Model Bias on the Fitted Model and Design, 543 10.2 A Design Criterion Involving Bias and Variance, 547 10.2.1 The Case of a First-Order Fitted Model and Cuboidal Region, 550 10.2.2 Minimum Bias Designs for a Spherical Region of Interest, 556 10.2.3 Simultaneous Consideration of Bias and Variance, 558 10.2.4 How Important Is Bias?, 558 10.3 Errors in Control of Design Levels, 560 10.4 Experiments with Computer Models, 563 10.4.1 Design for Computer Experiments, 567 10.4.2 Analysis for Computer Experiments, 570 10.4.3 Combining Information from Physical and Computer Experiments, 574 10.5 Minimum Bias Estimation of Response Surface Models, 575 10.6 Neural Networks, 579 10.7 Split-Plot Designs for Second-Order Models, 581 10.8 RSM for Non-Normal Responses—Generalized Linear Models, 591 10.8.1 Model Framework: The Link Function, 592 10.8.2 The Canonical Link Function, 593 10.8.3 Estimation of Model Coefficients, 593 10.8.4 Properties of Model Coefficients, 595 10.8.5 Model Deviance, 595 10.8.6 Overdispersion, 597 10.8.7 Examples, 598 10.8.8 Diagnostic Plots and Other Aspects of the GLM, 605 Exercises, 609 11 Robust Parameter Design and Process Robustness Studies 619 11.1 Introduction, 619 11.2 What is Parameter Design?, 619 11.2.1 Examples of Noise Variables, 620 11.2.2 An Example of Robust Product Design, 621 11.3 The Taguchi Approach, 622 11.3.1 Crossed Array Designs and Signal-to-Noise Ratios, 622 11.3.2 Analysis Methods, 625 11.3.3 Further Comments, 630 11.4 The Response Surface Approach, 631 11.4.1 The Role of the Control × Noise Interaction, 631 11.4.2 A Model Containing Both Control and Noise Variables, 635 11.4.3 Generalization of Mean and Variance Modeling, 638 11.4.4 Analysis Procedures Associated with the Two Response Surfaces, 642 11.4.5 Estimation of the Process Variance, 651 11.4.6 Direct Variance Modeling, 655 11.4.7 Use of Generalized Linear Models, 657 11.5 Experimental Designs For RPD and Process Robustness Studies, 661 11.5.1 Combined Array Designs, 661 11.5.2 Second-Order Designs, 663 11.5.3 Other Aspects of Design, 665 11.6 Dispersion Effects in Highly Fractionated Designs, 672 11.6.1 The Use of Residuals, 673 11.6.2 Further Diagnostic Information from Residuals, 674 11.6.3 Further Comments Concerning Variance Modeling, 680 Exercises, 684 12 Experiments with Mixtures 693 12.1 Introduction, 693 12.2 Simplex Designs and Canonical Mixture Polynomials, 696 12.2.1 Simplex Lattice Designs, 696 12.2.2 The Simplex-Centroid Design and Its Associated Polynomial, 704 12.2.3 Augmentation of Simplex Designs with Axial Runs, 707 12.3 Response Trace Plots, 716 12.4 Reparameterizing Canonical Mixture Models to Contain A Constant Term (𝛽0), 716 Exercises, 720 13 Other Mixture Design and Analysis Techniques 731 13.1 Constraints on the Component Proportions, 731 13.1.1 Lower-Bound Constraints on the Component Proportions, 732 13.1.2 Upper-Bound Constraints on the Component Proportions, 743 13.1.3 Active Upper- and Lower-Bound Constraints, 747 13.1.4 Multicomponent Constraints, 758 13.2 Mixture Experiments Using Ratios of Components, 759 13.3 Process Variables in Mixture Experiments, 763 13.3.1 Mixture-Process Model and Design Basics, 763 13.3.2 Split-Plot Designs for Mixture-Process Experiments, 767 13.3.3 Robust Parameter Designs for Mixture-Process Experiments, 778 13.4 Screening Mixture Components, 783 Exercises, 785 Appendix 1 Moment Matrix of a Rotatable Design 797 Appendix 2 Rotatability of a Second-Order Equiradial Design 803 References 807 Index 821