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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 1
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 2 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 RSM dates from the 1950s; early applications in chemical industry Modern applications of RSM span many industrial and business settings
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 3 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
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 4 Steps in RSM 1.Find a suitable approximation for y = f(x) using LS {maybe a low – order polynomial} 2.Move towards the region of the optimum 3.When curvature is found find a new approximation for y = f(x) {generally a higher order polynomial} and perform the “Response Surface Analysis”
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 5 Response Surface Models Screening Steepest ascent Optimization
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 6 RSM is a Sequential Procedure Factor screening Finding the region of the optimum Modeling & Optimization of the response
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 7 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
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 8 Example 11.1: An Example of Steepest Ascent
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 16 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:
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 17 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
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 21 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
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 22 Finding the Stationary Point After fitting a second order model take the partial derivatives with respect to the x i ’s and set to zero –δy / δx 1 =... = δy / δx k = 0 Stationary point represents… –Maximum Point –Minimum Point –Saddle Point
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 23 Stationary Point
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 24 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 = y s + λ 1 w 1 2 + λ 2 w 2 2 +... + λ k w k 2
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 26 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
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 33 Ridge Systems
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 38 Overlay Contour Plots
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 39 Mathematical Programming Formulation
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 40 Desirability Function Method
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 45 Addition of center points is usually a good idea
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 47 The Rotatable CCD
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 49 The Box-Behnken Design
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 50 A Design on A Cube – The Face-Centered CCD
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 51 Note that the design isn’t rotatable but the prediction variance is very good in the center of the region of experimentation
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 52 Other Designs Equiradial designs (k = 2 only) The small composite design (SCD) –Not a great choice because of poor prediction variance properties Hybrid designs –Excellent prediction variance properties –Unusual factor levels Computer-generated designs
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 56 Blocking in a Second-Order Design
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 58 Computer-Generated (Optimal) Designs These designs are good choices whenever –The experimental region is irregular –The model isn’t a standard one –There are unusual sample size or blocking requirements These designs are constructed using a computer algorithm and a specified “optimality criterion” Many “standard” designs are either optimal or very nearly optimal
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 62 Which Criterion Should I Use? For fitting a first-order model, D is a good choice –Focus on estimating parameters –Useful in screening For fitting a second-order model, I is a good choice –Focus on response prediction –Appropriate for optimization
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 63 Algorithms Point exchange –Requires a candidate set of points –The design is chosen from the candidate set –Random start, several (many) restarts to ensure that a highly efficient design is found Coordinate exchange –No candidate set required –Search over each coordinate one-at-a-time –Many random starts used
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 64 The Adhesive Pull-Off Force Experiment – a “Standard” Design
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 65 A D-Optimal Design
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 66 Relative efficiency of the standard inscribed design The standard design would have to be replicated approximately twice to estimate the parameters as precisely as the optimal design
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 78 Designs for Computer Experiments Optimal designs are appropriate if a polynomial model is used Space-filling designs are also widely used for non-polynomial models –Latin hypercube designs –Sphere-packing designs –Uniform designs –Maximum entropy designs
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 81 The Gaussian Process Model Spatial correlation model Interpolates the data One parameter for each factor – more parsimonious that polynomials Often a good choice for a deterministic computer model
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 90 Mixture Models
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 94 Constraints
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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 103 Evolutionary Operation (EVOP) An experimental deign based technique for continuous monitoring and improvement of a process Small changes are continuously introduced in the important variables of a process and the effects evaluated The 2-level factorial is recommended There are usually only 2 or 3 factors considered EVOP has not been widely used in practice The text has a complete example
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