Text reference, Chapter 11 Primary focus of previous chapters is factor screening
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Transcript of Text reference, Chapter 11 Primary focus of previous chapters is factor screening
Design & Analysis of Experiments 8E 2012 Montgomery
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• 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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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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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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Response Surface Models
0 1 1 2 2 12 1 2y x x x x
0 1 1 2 2y x x
2 20 1 1 2 2 12 1 2 11 1 22 2y x x x x x x
• Screening
• Steepest ascent
• Optimization
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RSM is a Sequential Procedure
• Factor screening• Finding the
region of the optimum
• Modeling & Optimization of the response
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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
0 1 1 2 2ˆ ˆ ˆy x x
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Example 11.1: An Example of Steepest Ascent
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• 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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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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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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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
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Stationary Point
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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 + λ1w1
2 + λ2w22 + . . . + λkwk
2
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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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Ridge Systems
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Overlay Contour Plots
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Mathematical Programming Formulation
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Desirability Function Method
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1/1 2( ... ) m
mD d d d
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Addition of center points is usually a good idea
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The Rotatable CCD 1/ 4F
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The Box-Behnken Design
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A Design on A Cube – The Face-Centered CCD
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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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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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Blocking in a Second-Order Design
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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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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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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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The Adhesive Pull-Off Force Experiment – a “Standard” Design
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A D-Optimal Design
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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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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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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
2Correlation Matrix R(θ)
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Mixture Models
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Constraints
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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