text reference, chapter 11 primary focus of previous chapters is factor screening

Design & Analysis of Experiments 8E 2012 Montgomery

1Chapter 11


2Chapter 11

• 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


3Chapter 11

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


4Chapter 11

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”


5Chapter 11

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


6Chapter 11

RSM is a Sequential Procedure

• Factor screening• Finding the

region of the optimum

• Modeling & Optimization of the response


7Chapter 11

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


8Chapter 11

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:


17Chapter 11

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


22Chapter 11

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


23Chapter 11

Stationary Point


24Chapter 11

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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26Chapter 11

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


39Chapter 11

Mathematical Programming Formulation


40Chapter 11

Desirability Function Method


41Chapter 11

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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47Chapter 11

The Rotatable CCD 1/ 4F


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The Box-Behnken Design


50Chapter 11

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


52Chapter 11

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


63Chapter 11

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


64Chapter 11

The Adhesive Pull-Off Force Experiment – a “Standard” Design


65Chapter 11

A D-Optimal Design


66Chapter 11

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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81Chapter 11

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

text reference, chapter 11 primary focus of previous chapters is factor screening

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