Response Surface Methodology: Process and Product Optimization Using Designed Experiments

Preface.

1. Introduction. 

1.1 Response Surface Methodology.

1.1.1 Approximating Response Functions.

1.1.2 The Sequential Nature of RSM.

1.1.3 Objectives and Typical Applications of RSM.

1.1.4 RSM and the Philosophy of Quality Improvement.

1.2 Product Design and Formulation (Mixture Problems).

1.3 Robust Design and Process Robustness Studies.

1.4 Useful References on RSM.

2. Building Empirical Models. 

2.1 Linear Regression Models.

2.2 Estimation of the Parameters in Linear Regression Models.

2.3 Properties of the Least Squares Estimators and Estimation of.

2.4 Hypothesis Testing in Multiple Regression.

2.4.1 Test for Significance of Regression.

2.4.2 Tests on Individual Regression Coefficients and Groups of Coefficients.

2.5 Confidence Intervals in Multiple Regression.

2.5.1 Confidence Intervals on the Individual Regression Coefficients, 38.

2.5.2 A Joint Confidence Region on the Regression Coefficients.

2.5.3 Confidence Interval on the Mean Response.

2.6 Prediction of New Response Observations.

2.7 Model Adequacy Checking.

2.7.1 Residual Analysis.

2.7.2 Scaling Residuals.

2.7.3 Influence Diagnostics.

2.7.4 Testing for Lack of Fit.

2.8 Fitting a Second-Order Model.

2.9 Qualitative Regressor Variables.

2.10 Transformation of the Response Variable.

Exercises.

3. Two-Level Factorial Designs. 

3.1 Introduction.

3.2 The Design.

3.3 The Design, 1.

3.4 The General Design.

3.5 A Single Replicate of the Design.

3.6 The Addition of Center Points to the Design.

3.7 Blocking in the Factorial Design.

3.7.1 Blocking in the Replicated Design.

3.7.2 Confounding in the Design.

3.8 Split-Plot Designs,.

Exercises.

4. Two-Level Fractional Factorial Designs. 

4.1 Introduction.

4.2 The One-Half Fraction of the Design.

4.3 The One-Quarter Fraction of the Design.

4.4 The General Fractional Factorial Design.

4.5 Resolution III Designs.

4.6 Resolution IV and V Designs.

4.7 Fractional Factorial Split-Plot Designs.

4.8 Summary.

Exercises.

5. Process Improvement with Steepest Ascent. 

5.1 Determining the Path of Steepest Ascent.

5.1.1 Development of the Procedure.

5.1.2 Practical Application of the Method of Steepest Ascent.

5.2 Consideration of Interaction and Curvature,.

5.2.1 What About a Second Phase?.

5.2.2 What Happens Following Steepest Ascent?.

5.3 Effect of Scale (Choosing Range of Factors).

5.4 Confidence Region for Direction of Steepest Ascent.

5.5 Steepest Ascent Subject to a Linear Constraint.

5.6 Steepest Ascent in a Split-Plot Experiment.

Exercises.

6. The Analysis of Second-Order Response Surfaces. 

6.1 Second-Order Response Surface.

6.2 Second-Order Approximating Function.

6.2.1 The Nature of the Second-Order Function and Second-Order Surface.

6.2.2 Illustration of Second-Order Response Surfaces.

6.3 A Formal Analytical Approach to the Second-Order Model.

6.3.1 Location of the Stationary Point.

6.3.2 Nature of the Stationary Point (Canonical Analysis).

6.3.3 Ridge Systems.

6.3.4 Role of Contour Plots.

6.4 Ridge Analysis of the Response Surface.

6.4.1 What Is the Value of Ridge Analysis?.

6.4.2 Mathematical Development of Ridge Analysis.

6.5 Sampling Properties of Response Surface Results.

6.5.1 Standard Error of Predicted Response.

6.5.2 Confidence Region on the Location of the Stationary Point.

6.5.3 Use and Computation of the Confidence Region on the Location of the Stationary Point.

6.5.4 Confidence Intervals on Eigenvalues in Canonical Analysis.

6.6 Multiple Response Optimization.

6.7 Further Comments Concerning Response Surface Analysis.

Exercises.

7. Experimental Designs for Fitting Response Surfaces--I. 

7.1 Desirable Properties of Response Surface Designs.

7.2 Operability Region, Region of Interest, and Model Inadequacy.

7.3 Design of Experiments for First-Order Models.

7.3.1 The First-Order Orthogonal Design.

7.3.2 Orthogonal Designs for Models Containing Interaction.

7.3.3 Other First-Order Orthogonal Designs - The Simplex Design.

7.3.4 Another Variance Property - Prediction Variance.

7.4 Designs for Fitting Second-Order Models.

7.4.1 The Class of Central Composite Designs.

7.4.2 Design Moments and Property of Rotatability.

7.4.3 Rotatability and the CCD.

7.4.4 More on Prediction Variance - Scaled, Unscaled and Estimated.

7.4.5 The Cuboidal Region and the Face-Centered Cube.

7.4.6 When Is the Design Region Spherical?.

7.4.7 Summary Statements Regarding CCD.

7.4.8 The Box-Behnken Design.

7.4.9 Other Spherical RSM Designs; Equiradial Designs.

7.4.10 Orthogonal Blocking in Second-Order Designs.

Exercises.

8. Experimental Designs for Fitting Response Surfaces--II. 

8.1 Designs That Require a Relatively Small Run Size.

8.1.1 The Hoke Design.

8.1.2 Koshal Design.

8.1.3 Hybrid Designs.

8.1.4 The Small Composite Design.

8.1.5 Some Saturated or Near-Saturated Cuboidal Designs.

8.2 General Criteria for Constructing, Evaluating, and Comparing Experimental Designs.

8.2.1 Practical Design Optimality.

8.2.2 Use of Design Efficiencies for Comparison of Standard Second-Order Designs.

8.2.3 Graphical Procedure for Evaluating the Prediction Capability of an RSM Design.

8.3 Computer-Generated Designs in RSM.

8.3.1 Important Relationship Between Prediction Variance and Design Augmentation for D-Optimality.

8.3.2 Illustrations Involving Computer-Generated Design.

8.4 Some Final Comments Concerning Design Optimality and Computer-Generated Design.

Exercises.

9. Advanced Topics in Response Surface Methodology. 

9.1 Effects of Model Bias on the Fitted Model and Design.

9.2 A Design Criterion Involving Bias and Variance.

9.2.1 The Case of a First-Order Fitted Model and Cuboidal Region.

9.2.2 Minimum Bias Designs for a Spherical Region of Interest.

9.2.3 Simultaneous Consideration of Bias and Variance.

9.2.4 How Important Is Bias?.

9.3 Errors in Control of Design Levels.

9.4 Experiments with Computer Models.

9.5 Minimum Bias Estimation of Response Surface Models.

9.6 Neural Networks,.

9.7 RSM for Nonnormal Responses - Generalized Linear Models.

9.7.1 Model Framework: The Link Function.

9.7.2 The Canonical Link Function.

9.7.3 Estimation of Model Coefficients.

9.7.4 Properties of Model Coefficients.

9.7.5 Model Deviance.

9.7.6 Overdispersion.

9.7.7 Examples.

9.7.8 Diagnostic Plots and Other Aspects of the GLM.

9.8 Split-Plot Designs for Second-Order Models.

Exercises.

10. Robust Parameter Design and Process Robustness Studies. 

10.1 Introduction.

10.2 What Is Parameter Design?.

10.2.1 Examples of Noise Variables.

10.2.2 An Example of Robust Product Design.

10.3 The Taguchi Approach.

10.3.1 Crossed Array Designs and Signal-to-Noise Ratios.

10.3.2 Analysis Methods.

10.3.3 Further Comments.

10.4 The Response Surface Approach.

10.4.1 The Role of the Control x Noise Interaction,.

10.4.2 A Model Containing Both Control and Noise Variables.

10.4.3 Generalization of Mean and Variance Modeling,.

10.4.4 Analysis Procedures Associated with the Two Response Surfaces.

10.4.5 Estimation of the Process Variance.

10.4.6 Direct Variance Modeling.

10.4.7 Use of Generalized Linear Models.

10.5 Experimental Designs for RPD and Process Robustness Studies.

10.5.1 Combined Array Designs.

10.5.2 Second-Order Designs.

10.5.3 Other Aspects of Design.

10.6 Dispersion Effects in Highly Fractionated Designs.

10.6.1 The Use of Residuals.

10.6.2 Further Diagnostic Information from Residuals.

10.6.3 Further Comments Concerning Variance Modeling.

Exercises.

11. Experiments with Mixtures. 

11.1 Introduction.

11.2 Simplex Designs and Canonical Mixture Polynomials,.

11.2.1 Simplex Lattice Designs.

11.2.2 The Simplex-Centroid Design and Its Associated Polynomial.

11.2.3 Augmentation of Simplex Designs with Axial Runs.

11.3 Response Trace Plots.

11.4 Reparameterizing Canonical Mixture Models to Contain a Constant Term,.

Exercises.

12. Other Mixture Design and Analysis Techniques. 

12.1 Constraints on the Component Proportions.

12.1.1 Lower-Bound Constraints on the Component Proportions.

12.1.2 Upper-Bound Constraints on the Component Proportions.

12.1.3 Active Upper- and Lower-Bound Constraints.

12.1.4 Multicomponent Constraints.

12.2 Mixture Experiments Using Ratios of Components.

12.3 Process Variables in Mixture Experiments.

12.3.1 Mixture-Process Model and Design Basics.

12.3.2 Split-Plot Designs for Mixture-Process Experiments,.

12.3.3 Robust Parameter Designs for Mixture-Process Experiments,.

12.4 Screening Mixture Components.

Exercises.

References.

Appendix 1. Moment Matrix of a Rotatable Design.

Appendix 2. Rotatability of a Second-Order Equiradial Design.

Index.