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Chapter 11 Design & Analysis of Experiments 7E 2009 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; earlyRSM dates from the 1950s; early
applications in chemical industry• Modern applications of RSM span many• Modern applications of RSM span many
industrial and business settings
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Response Surface MethodologyResponse Surface Methodology
• Collection of mathematical andCollection of mathematical and statistical techniques useful for the modeling and analysis of problems inmodeling and analysis of problems in which a response of interest is influenced by several variablesby several variables
• Objective is to optimize the response
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Steps in RSMSteps 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 “R S f A l i ”“Response Surface Analysis”
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Response Surface ModelsResponse Surface Models
• Screening
0 1 1 2 2 12 1 2y x x x x • Steepest ascent
0 1 1 2 2y x x • Optimization
2 20 1 1 2 2 12 1 2 11 1 22 2y x x x x x x
Optimization
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RSM is a Sequential ProcedureRSM is a Sequential Procedure
• Factor screening• Finding the g
region of the optimum
• Modeling & Optimization of ththe response
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The Method of Steepest AscentThe Method of Steepest Ascent• Text, Section 11.2,• A procedure for moving
sequentially from an initial “ ” t d t i“guess” towards to region of the optimum
• Based on the fitted first-Based on the fitted firstorder model
0 1 1 2 2ˆ ˆ ˆŷ x x
• Steepest ascent is a gradient procedure
0 1 1 2 2y x x
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gradient procedure
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Example 11.1: An Example of Steepest Ascenta p e a p e o Steepest sce t
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• Points on the path of steepest ascent are proportional to p p p pthe magnitudes of the model regression coefficients
• The direction depends on the sign of the regression ffi i tcoefficient
• Step-by-step procedure:
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Second-Order Models in RSMSecond Order Models in RSM
• These models are used widely in practicey p
• 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 SurfaceCharacterization of the Response Surface
• Find out where our stationary point is • Find what type of surface we have
– Graphical Analysis p y– Canonical Analysis
• Determine the sensitivity of theDetermine the sensitivity of the response variable to the optimum value
Canonical Analysis– Canonical Analysis
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Finding the Stationary PointFinding 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 PointStationary Point
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Canonical AnalysisCanonical Analysis
• Used for sensitivity analysis andUsed for sensitivity analysis and stationary point identification
• Based on the analysis of a transformed• Based on the analysis of a transformed model called: canonical form of the modelmodel
• Canonical Model form: y = ys + λ1w12 + λ2w22 + . . . + λkwk2
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EigenvaluesEigenvalues• The nature of the response can be determined by the
i d it d f th i lsigns and magnitudes of the eigenvalues – {e} all positive: a minimum is found– {e} all negative: a maximum is found { } g– {e} mixed: a saddle point is found
• Eigenvalues can be used to determine the sensitivity of the response with respect to the design factorsof the response with respect to the design factors
• The response surface is steepest in the direction (canonical) corresponding to the largest absolute ( ) p g geigenvalue
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Ridge SystemsRidge Systems
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Overlay Contour PlotsOverlay Contour Plots
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Mathematical Programming FormulationMathematical Programming Formulation
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Desirability Function MethodDesirability Function Method
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1/1 2( ... )
mmD d d d
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Addition of center points is usually a good idea
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The Rotatable CCD 1/ 4F1/ 4F
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The Box-Behnken DesignThe Box Behnken Design
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A Design on A Cube – The Face-Centered CCDes g o Cube e ace Ce te ed CC
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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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good in the center of the region of experimentation