1. Design and Analysis of Engineering Experiments
Ali Ahmad, PhD
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Based on Design & Analysis of Experiments 7E 2009 Montgomery

2. Response Surface Methodology
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Design & Analysis of Experiments 7E 2009 Montgomery

3. Design & Analysis of Experiments 7E 2009 Montgomery
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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4. 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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5. Design & Analysis of Experiments 7E 2009 Montgomery
Steps in RSM
Find a suitable approximation for y = f(x) using LS {maybe a low – order polynomial}
Move towards the region of the optimum
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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6. Response Surface Models
Screening
Steepest ascent
Optimization
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7. RSM is a Sequential Procedure
Factor screening
Finding the region of the optimum
Modeling & Optimization of the response
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8. 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
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9. 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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18. 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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22. 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
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23. 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 + λ2w 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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39. Mathematical Programming Formulation
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40. Desirability Function Method
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Addition of center points is usually a good idea
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The Rotatable CCD
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49. The Box-Behnken Design
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50. 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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56. Blocking in a Second-Order Design
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58. 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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62. 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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63. The Adhesive Pull-Off Force Experiment – a “Standard” Design
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A D-Optimal Design
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70. 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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Design & Analysis of Experiments 7E 2009 Montgomery