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Design and Analysis of Engineering Experiments. Ali Ahmad, PhD. Response Surface Methodology. Text reference, Chapter 11 Primary focus of previous chapters is factor screening Two-level factorials, fractional factorials are widely used Objective of RSM is optimization
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Design and Analysis of Engineering Experiments Ali Ahmad, PhD Based on Design & Analysis of Experiments 7E 2009 Montgomery
Response Surface Methodology Design & Analysis of Experiments 7E 2009 Montgomery
Text reference, Chapter 11 • Primary focus of previous chapters is factorscreening • Two-level factorials, fractional factorials are widely used • Objective of RSM is optimization • RSM dates from the 1950s; early applications in chemical industry • Modern applications of RSM span many industrial and business settings Design & Analysis of Experiments 7E 2009 Montgomery
Response Surface Methodology • Collection of mathematical and statistical techniques useful for the modeling and analysis of problems in which a response of interest is influenced by several variables • Objective is to optimize the response Design & Analysis of Experiments 7E 2009 Montgomery
Steps in RSM Find a suitable approximation for y = f(x) using LS {maybe a low – order polynomial} Move towards the region of the optimum When curvature is found find a new approximation for y = f(x) {generally a higher order polynomial} and perform the “Response Surface Analysis” Design & Analysis of Experiments 7E 2009 Montgomery
Response Surface Models • Screening • Steepest ascent • Optimization Design & Analysis of Experiments 7E 2009 Montgomery
RSM is a Sequential Procedure • Factor screening • Finding the region of the optimum • Modeling & Optimization of the response Design & Analysis of Experiments 7E 2009 Montgomery
The Method of Steepest Ascent • Text, Section 11.2 • A procedure for moving sequentially from an initial “guess” towards to region of the optimum • Based on the fitted first-order model • Steepest ascent is a gradient procedure Design & Analysis of Experiments 7E 2009 Montgomery
Example 11.1: An Example of Steepest Ascent Design & Analysis of Experiments 7E 2009 Montgomery
Points on the path of steepest ascent are proportional to the magnitudes of the model regression coefficients • The direction depends on the sign of the regression coefficient • Step-by-step procedure: Design & Analysis of Experiments 7E 2009 Montgomery
Second-Order Models in RSM • These models are used widely in practice • The Taylor series analogy • Fitting the model is easy, some nice designs are available • Optimization is easy • There is a lot of empirical evidence that they work very well Design & Analysis of Experiments 7E 2009 Montgomery
Characterization of the Response Surface • Find out where our stationary point is • Find what type of surface we have • Graphical Analysis • Canonical Analysis • Determine the sensitivity of the response variable to the optimum value • Canonical Analysis Design & Analysis of Experiments 7E 2009 Montgomery
Finding the Stationary Point • After fitting a second order model take the partial derivatives with respect to the xi’s and set to zero • δy / δx1 = . . . = δy / δxk = 0 • Stationary point represents… • Maximum Point • Minimum Point • Saddle Point Design & Analysis of Experiments 7E 2009 Montgomery
Stationary Point Design & Analysis of Experiments 7E 2009 Montgomery
Canonical Analysis • Used for sensitivity analysis and stationary point identification • Based on the analysis of a transformed model called: canonical form of the model • Canonical Model form: • y = ys + λ1w12 + λ2w22 + . . . + λkwk2 Design & Analysis of Experiments 7E 2009 Montgomery
Eigenvalues • The nature of the response can be determined by the signs and magnitudes of the eigenvalues • {e} all positive: a minimum is found • {e} all negative: a maximum is found • {e} mixed: a saddle point is found • Eigenvalues can be used to determine the sensitivity of the response with respect to the design factors • The response surface is steepest in the direction (canonical) corresponding to the largest absolute eigenvalue Design & Analysis of Experiments 7E 2009 Montgomery
Ridge Systems Design & Analysis of Experiments 7E 2009 Montgomery
Overlay Contour Plots Design & Analysis of Experiments 7E 2009 Montgomery
Mathematical Programming Formulation Design & Analysis of Experiments 7E 2009 Montgomery
Desirability Function Method Design & Analysis of Experiments 7E 2009 Montgomery
Addition of center points is usually a good idea Design & Analysis of Experiments 7E 2009 Montgomery
The Rotatable CCD Design & Analysis of Experiments 7E 2009 Montgomery
The Box-Behnken Design Design & Analysis of Experiments 7E 2009 Montgomery
A Design on A Cube – The Face-Centered CCD Design & Analysis of Experiments 7E 2009 Montgomery
