Design and Analysis of
Engineering Experiments
Ali Ahmad, PhD
Chapter 11
Based on Design & Analysis of
Experiments 7E 2009 Montgomery
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Response
Surface
Methodology
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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
• Screening
y  0  1 x1   2 x2  12 x1 x2  
• Steepest ascent
• Optimization
y  0  1 x1   2 x2  
y   0  1 x1   2 x2  12 x1 x2  11 x12   22 x22  
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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 firstorder model
ŷ  ˆ0  ˆ1 x1  ˆ2 x2
• Steepest ascent is a
gradient procedure
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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 + λ1w12 + λ2w22 + . . . + λkwk2
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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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D  (d1d 2 ...d m )1/ m
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Addition of center points is usually a good idea
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The Rotatable CCD
F
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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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The Adhesive Pull-Off Force Experiment – a
“Standard” Design
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A D-Optimal Design
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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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