Chapter 11 Based on Design & Analysis of Experiments 7E 2009 Montgomery 1 Design and Analysis of Engineering Experiments Ali Ahmad, PhD
Feb 22, 2016
Chapter 11 Based on Design & Analysis of Experiments 7E 2009 Montgomery
1
Design and Analysis of Engineering Experiments
Ali Ahmad, PhD
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
2
Response Surface
Methodology
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
3
• 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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
5
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”
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
6
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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
7
RSM is a Sequential Procedure
• Factor screening• Finding the
region of the optimum
• Modeling & Optimization of the response
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
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
0 1 1 2 2ˆ ˆ ˆy x x
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
9
Example 11.1: An Example of Steepest Ascent
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
10
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
11
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
12
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
13
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
14
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
15
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
16
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
17
• 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:
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
19
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
20
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
21
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
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– Canonical Analysis
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
24
Stationary Point
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
25
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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
26
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
27
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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
28
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
29
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
30
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
31
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
32
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
33
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
34
Ridge Systems
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
35
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
36
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
37
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
38
Overlay Contour Plots
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
39
Mathematical Programming Formulation
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
40
Desirability Function Method
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
41
1/1 2( ... ) m
mD d d d
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
42
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
43
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
44
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
45
Addition of center points is usually a good idea
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
46
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
47
The Rotatable CCD 1/ 4F
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
48
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
49
The Box-Behnken Design
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
50
A Design on A Cube – The Face-Centered CCD
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
51
Note that the design isn’t rotatable but the prediction variance is very good in the center of the region of experimentation
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
52
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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
53
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
54
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
55
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
56
Blocking in a Second-Order Design
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
57
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
59
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
60
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
61
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
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
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
63
The Adhesive Pull-Off Force Experiment – a “Standard” Design
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
64
A D-Optimal Design
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
65
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
66
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
67
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
68
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
69
Chapter 11 Design & Analysis of Experiments 7E 2009 Montgomery
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