Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 1.

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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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


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


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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

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Chapter 11Design & Analysis of Experiments 8E 2012 Montgomery 1.

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