Design of Engineering Experiments Part 8 Overview of Response Surface Methods

1
Design of Engineering Experiments Part 8
Overview of Response Surface Methods

• Text reference, Chapter 11, Sections 11-1 through
  11-4
• 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

2
RSM is a Sequential Procedure

• Factor screening
• Finding the region of the optimum
• Modeling Optimization of the response

3
Response Surface Models

• Screening
• Steepest ascent
• Optimization

4
The Method of Steepest Ascent
Text, page 430 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
5
An Example of Steepest AscentExample 11-1, pg.
431
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An Example of Steepest AscentExample 11-1, pg.
431

• An approximate step size and path can be
  determined graphically
• Formal methods can also be used (pp. 434-436)
• Types of experiments along the path
• Single runs
• Replicated runs

7
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?
8
Analysis of the Second-Order Response Surface
Model (pg. 436)
This is a central composite design
9
The Second-Order Response Surface Model

• 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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Example 11-2
Sequential Model Sum of Squares Sum
of Mean F Source Squares DF Square Value Prob gt
F Mean 80062.16 1 80062.16 Linear 10.04 2 5.02 2
.69 0.1166 2FI 0.25 1 0.25 0.12 0.7350 Quadratic
17.95 2 8.98 126.88 lt 0.0001 Suggested Cubic 2.0
42E-003 2 1.021E-003 0.010 0.9897 Aliased Residua
l 0.49 5 0.099 Total 80090.90 13 6160.84
Model Summary Statistics Std. Adjusted Predi
cted Source Dev. R-Squared R-Squared R-Squared PR
ESS Linear 1.37 0.3494 0.2193 -0.0435 29.99 2FI
1.43 0.3581 0.1441 -0.2730 36.59 Quadratic 0.27 0
.9828 0.9705 0.9184 2.35 Suggested Cubic 0.31 0.9
828 0.9588 0.3622 18.33 Aliased
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Example 11-2
ANOVA for Response Surface Quadratic
Model Analysis of variance table Partial sum of
squares Sum of Mean F Source Squares DF Squa
re Value Prob gt F Model 28.25 5 5.65 79.85 lt
0.0001 A 7.92 1 7.92 111.93 lt 0.0001 B 2.12 1 2.
12 30.01 0.0009 A2 13.18 1 13.18 186.22 lt
0.0001 B2 6.97 1 6.97 98.56 lt 0.0001 AB 0.25 1 0
.25 3.53 0.1022 Residual 0.50 7 0.071 Lack of
Fit 0.28 3 0.094 1.78 0.2897 Pure
Error 0.21 4 0.053 Cor Total 28.74 12
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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 gt 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. 448 and page 451

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Designs for Fitting Response Surface Models

• Section 11-4, page 455
• 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 Response Surface Methodology

• Robust parameter design and process robustness
  studies
• 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
• Special type of RSM problem
• Design factors are components (ingredients) of a
  mixture
• Response depends only on the proportions
• Many applications in product formulation