Transcript Design of Engineering Experiments Part 7 – The 2k

Design of Engineering Experiments
Part 8 – Overview of
Response Surface Methods
• Text reference, Chapter 11, Sections 11-1 through 114
• 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
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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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Response Surface Models
• Screening
y  0  1x1  2 x2  12 x1x2  
• Steepest ascent
y  0  1x1  2 x2  
• Optimization
y  0  1x1  2 x2  12 x1x2   x   x  
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The Method of Steepest Ascent
Text, page 407
A procedure for moving
sequentially from an initial
“guess” towards to region of
the optimum
Based on the fitted first-order
model
yˆ  ˆ0  ˆ1x1  ˆ2 x2
Steepest ascent is a gradient
procedure
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An Example of Steepest Ascent
Example 11-1, pg. 409
yˆ  40.44  0.775x1  0.325x2
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An Example of Steepest Ascent
Example 11-1, pg. 409
PERT Plot
yield
2.000
An approximate step
size and path can be
determined
graphically
ion T ime
p
oints
B: Temp
1.250
40.5
0.500
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Formal methods can
also be used (pp. 407412)
41.5
5
40
-0.250
Types of experiments
along the path:
39.5
-1.000
-1.000
-0.250
0.500
1.250
A: Reaction Tim e
2.000
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•Single runs
•Replicated runs
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Results from the Example (pg. 434)
The step size is 5 minutes of reaction time and 2 degrees F
What happens at the conclusion of steepest ascent?
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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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Analysis of the Second-Order Response Surface Model (pg. 413)
This is a central composite design
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Example 11-2
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Example 11-2
yˆ  79.94  0.99x1  0.52x2  0.25x1x2 1.38x12  1.00x22
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Contour Plots for Example 11-2
The contour plot is given in the natural variables
The optimum is at about 87 minutes and 176.5 degrees
Formal optimization methods can also be used (particularly when k > 2)
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Multiple Responses
• Example 11-2 illustrated three response variables
(yield, viscosity and molecular weight)
• Multiple responses are common in practice
• Typically, we want to simultaneously optimize all
responses, or find a set of conditions where certain
product properties are achieved
• A simple approach is to model all responses and
overlay the contour plots
• See Section 11-3.4, pp. 423 -427.
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Designs for Fitting Response
Surface Models
• Section 11-4, page 427
• For the first-order model, two-level factorials (and
fractional factorials) augmented with center
points are appropriate choices
• The central composite design is the most widely
used design for fitting the second-order model
• Selection of a second-order design is an
interesting problem
• There are numerous excellent second-order
designs available
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Other Aspects of RSM
• Robust parameter design and process robustness studies
(Chapter 12)
– Find levels of controllable variables that optimize mean response and
minimize variability in the response transmitted from “noise”
variables
– Original approaches due to Taguchi (1980s)
– Modern approach based on RSM
• Experiments with mixtures
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–
–
–
Special type of RSM problem
Design factors are components (ingredients) of a mixture
Response depends only on the proportions
Many applications in product formulation
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