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Design of Engineering Experiments Part 8 – Overview of Response Surface Methods

Design of Engineering Experiments Part 8 – Overview of Response Surface Methods. Text reference, Chapter 11, Sections 11-1 through 11-4 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 of Engineering Experiments Part 8 – Overview of Response Surface Methods

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  1. Design of Engineering Experiments Part 8 – Overview of Response Surface Methods • Text reference, Chapter 11, Sections 11-1 through 11-4 • 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 DOX 6E Montgomery

  2. RSM is a Sequential Procedure • Factor screening • Finding the region of the optimum • Modeling & Optimization of the response DOX 6E Montgomery

  3. Response Surface Models • Screening • Steepest ascent • Optimization DOX 6E Montgomery

  4. The Method of Steepest Ascent Text, page 407 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 DOX 6E Montgomery

  5. An Example of Steepest AscentExample 11-1, pg. 409 DOX 6E Montgomery

  6. An Example of Steepest AscentExample 11-1, pg. 409 • An approximate stepsize and path can be determined graphically • Formal methods can also be used (pp. 407-412) • Types of experiments along the path: • Single runs • Replicated runs DOX 6E Montgomery

  7. Results from the Example (pg. 434) The step size is 5 minutes of reaction time and 2 degrees F What happens at the conclusion of steepest ascent? DOX 6E Montgomery

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  9. 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 DOX 6E Montgomery

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  13. Analysis of the Second-Order Response Surface Model (pg. 413) This is a central composite design DOX 6E Montgomery

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  15. Example 11-2 DOX 6E Montgomery

  16. Example 11-2 DOX 6E Montgomery

  17. Contour Plots for Example 11-2 The contour plot is given in the natural variables The optimum is at about 87 minutes and 176.5 degrees Formal optimization methods can also be used (particularly when k > 2) DOX 6E Montgomery

  18. Multiple Responses • Example 11-2 illustrated three response variables (yield, viscosity and molecular weight) • Multiple responses are common in practice • Typically, we want to simultaneously optimize all responses, or find a set of conditions where certain product properties are achieved • A simple approach is to model all responses and overlay the contour plots • See Section 11-3.4, pp. 423 -427. DOX 6E Montgomery

  19. Designs for Fitting Response Surface Models • Section 11-4, page 427 • For the first-order model, two-level factorials (and fractional factorials) augmented with centerpoints are appropriate choices • The central composite design is the most widely used design for fitting the second-order model • Selection of a second-order design is an interesting problem • There are numerous excellent second-order designs available DOX 6E Montgomery

  20. Other Aspects of RSM • Robust parameter design and process robustness studies (Chapter 12) • Find levels of controllable variables that optimize mean response and minimize variability in the response transmitted from “noise” variables • Original approaches due to Taguchi (1980s) • Modern approach based on RSM • Experiments with mixtures • Special type of RSM problem • Design factors are components (ingredients) of a mixture • Response depends only on the proportions • Many applications in product formulation DOX 6E Montgomery

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