Chapter 11 Design & Analysis of Experiments 1 7E 2009 …noordin/s/ch11 rev [Compatibility... ·...

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

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

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