![Page 1: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/1.jpg)
Design of Engineering Experiments The 2k-p Fractional Factorial Design
• Text reference, Chapter 8Text reference, Chapter 8• Motivation for fractional factorials is obvious; as the
number of factors becomes large enough to be “interesting” the size of the designs grows very quicklyinteresting , the size of the designs grows very quickly
• Emphasis is on factor screening; efficiently identify the factors with large effects
• There may be many variables (often because we don’t know much about the system)
• Almost always run as unreplicated factorials, but oftenAlmost always run as unreplicated factorials, but often with center points
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
1
![Page 2: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/2.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
2
![Page 3: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/3.jpg)
Why do Fractional FactorialWhy do Fractional Factorial Designs Work?
• The sparsity of effects principle– There may be lots of factors, but few are important
i d i d b i ff l d– System is dominated by main effects, low-order interactions
• The projection propertyThe projection property– Every fractional factorial contains full factorials in
fewer factorsS i i i• Sequential experimentation– Can add runs to a fractional factorial to resolve
difficulties (or ambiguities) in interpretation
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
3
d cu t es (o a b gu t es) te p etat o
![Page 4: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/4.jpg)
The One-Half Fraction of the 2k
• Section 8.2, page 290• Notation: because the design has 2k/2 runs, it’s referred to as a 2k-1
C id ll i l th 23 1• Consider a really simple case, the 23-1
• Note that I =ABC
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
4
![Page 5: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/5.jpg)
The One-Half Fraction of the 23
For the principal fraction, notice that the contrast for estimating the main effect A is exactly the same as the contrast used for estimating the BCy ginteraction.
This phenomena is called aliasing and it occurs in all fractional designs
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
5
Aliases can be found directly from the columns in the table of + and - signs
![Page 6: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/6.jpg)
Aliasing in the One-Half Fraction of the 23g
A = BC, B = AC, C = AB (or me = 2fi)
Aliases can be found from the defining relation I = ABCby multiplication:
AI = A(ABC) = A2BC = BC
BI =B(ABC) = AC
CI = C(ABC) = AB
Textbook notation for aliased effects:Textbook notation for aliased effects:
[ ] , [ ] , [ ]A A BC B B AC C C AB
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
6
![Page 7: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/7.jpg)
The Alternate Fraction of the 23-1
• I = -ABC is the defining relation• Implies slightly different aliases: A = -BC,
B= -AC, and C = -AB• Both designs belong to the same family, defined
by I ABC
• Suppose that after running the principal fraction, the alternate fraction was also run
I ABC
the alternate fraction was also run• The two groups of runs can be combined to form a
full factorial – an example of sequential
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
7
experimentation
![Page 8: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/8.jpg)
Design Resolution
• Resolution III Designs:– me = 2fi
l 3 1– example • Resolution IV Designs:
2fi = 2fi
3 12III
– 2fi = 2fi– example
• Resolution V Designs:
4 12IV
g– 2fi = 3fi– example 5 12V
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
8
![Page 9: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/9.jpg)
Construction of a One half FractionConstruction of a One-half FractionThe basic design; the design generator
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
9
![Page 10: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/10.jpg)
Projection of Fractional Factorialsj
Every fractional factorial contains full factorials incontains full factorials in fewer factors
The “flashlight” analogy
A one-half fraction will project into a full factorial in any k – 1 of the original factors
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
10
![Page 11: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/11.jpg)
Example 8.1
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
11
![Page 12: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/12.jpg)
Ex. 6.2 The Resin Plant Experiment
DOX 6E Montgomery 12
![Page 13: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/13.jpg)
Example 8.1pInterpretation of results often relies on making some assumptionsOckham’s razorOckham s razorConfirmation experiments can be importantAdding the alternate fraction – see page 301g p g
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
13
![Page 14: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/14.jpg)
Fig_08_04 Projection of the 2IV 4-1 design into a 23 design in A, C and D for Ex 8.1
![Page 15: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/15.jpg)
DOX 6E Montgomery 15
![Page 16: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/16.jpg)
DOX 6E Montgomery 16
Figure 8.5 A 25–1 Design for Example 8.2
![Page 17: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/17.jpg)
DOX 6E Montgomery 17
![Page 18: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/18.jpg)
DOX 6E Montgomery 18
![Page 19: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/19.jpg)
The AC and AD interactions can be verified by inspection of the cube plot
DOX 6E Montgomery 19
verified by inspection of the cube plot
![Page 20: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/20.jpg)
Possible Strategies forStrategies for
Follow-Up Experimentation
Following a Fractional
F t i l D iFactorial Design
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
20
![Page 21: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/21.jpg)
Effect Alias Effect Alias estimate structure estimate structure
for principal for principal for alternate for principalfraction fraction fraction fraction
[i] [i] [i]' 1/2([i]+[i]') 1/2([i]-[i]')A 19 A+BCD 24.25 A-BCD 21.63 A -2.63 BCDB 1 5 B ACD 4 75 B ACD 3 13 B 1 63 ACDB 1.5 B+ACD 4.75 B-ACD 3.13 B -1.63 ACDC 14 C+ABD 5.75 C-ABD 9.88 C 4.13 ABDD 16.5 D+ABC 12.75 D-ABC 14.63 D 1.88 ABC
AB -1 AB+CD 1.25 AB-CD 0.13 AB -1.13 CDAC -18.5 AC+BD -17.75 AC-BD -18.13 AC -0.38 BD
DOX 6E Montgomery 21
AC 18.5 AC BD 17.75 AC BD 18.13 AC 0.38 BDAD 19 AD+BC 14.25 AD-BC 16.63 AD 2.38 BC
![Page 22: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/22.jpg)
Confirmation experiment for this example:Confirmation experiment for this example: see page 302
U h d l di h bi i f iUse the model to predict the response at a test combination of interest in the design space – not one of the points in the current design.
Run this test combination – then compare predicted and observed.
For Example 8.1, consider the point +, +, -, +. The predicted response is
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
22
Actual response is 104.
![Page 23: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/23.jpg)
The One-Quarter Fraction of the 26-2
Complete defining relation: I ABCE BCDF ADEFComplete defining relation: I = ABCE = BCDF = ADEF
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
23
![Page 24: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/24.jpg)
The One-Quarter Fraction of the 2k
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
24
![Page 25: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/25.jpg)
The One Quarter Fraction of the 26-2The One-Quarter Fraction of the 26 2
• Uses of the alternate fractions, E ABC F BCD
• Projection of the design into subsets of the original six variables
• Any subset of the original six variables that is not• Any subset of the original six variables that is not a word in the complete defining relation will result in a full factorial designg– Consider ABCD (full factorial)– Consider ABCE (replicated half fraction)
C id ABCF (f ll f t i l)Chapter 8 Design and Analysis of Experiments
7E 2009 Montgomery25
– Consider ABCF (full factorial)
![Page 26: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/26.jpg)
A One-Quarter Fraction of the 26-2:A One-Quarter Fraction of the 2 :Example 8.4, Page 305
• Injection molding process with six factors• Design matrix, page 305g , p g• Calculation of effects, normal probability
plot of effectsp• Two factors (A, B) and the AB interaction
are importantp• Residual analysis indicates there are some
dispersion effects (see page 307) Chapter 8 Design and Analysis of Experiments
7E 2009 Montgomery26
p ( p g )
![Page 27: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/27.jpg)
The General 2k-p FractionalThe General 2 p Fractional Factorial Design
S i 8 4 309• Section 8.4, page 309• 2k-1 = one-half fraction, 2k-2 = one-quarter fraction,
2k-3 = one-eighth fraction 2k-p = 1/ 2p fraction2 = one-eighth fraction, …, 2 p = 1/ 2p fraction• Add p columns to the basic design; select p
independent generatorsp g• Important to select generators so as to maximize
resolution, see Table 8.14• Projection – a design of resolution R contains full
factorials in any R – 1 of the factorsBl ki
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
27
• Blocking
![Page 28: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/28.jpg)
The General 2k-p Design: Resolution may t b S ffi i tnot be Sufficient
• Minimum abberation designsg
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
28
![Page 29: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/29.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
29
![Page 30: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/30.jpg)
table_08_15
![Page 31: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/31.jpg)
fig_08_18
![Page 32: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/32.jpg)
![Page 33: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/33.jpg)
![Page 34: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/34.jpg)
fig_08_19
![Page 35: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/35.jpg)
![Page 36: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/36.jpg)
fig_08_20
![Page 37: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/37.jpg)
fig_08_21
![Page 38: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/38.jpg)
fig_08_22
![Page 39: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/39.jpg)
Resolution III Designs: Section 8 5Resolution III Designs: Section 8.5, page 320
• Designs with main effects aliased with two-factor interactions
• Used for screening (5 – 7 variables in 8 runs 9 - 15 variables in 16 runs forruns, 9 15 variables in 16 runs, for example)
• A saturated design has k = N 1 variables• A saturated design has k = N – 1 variables• See Table 8.19, page 320 for a 7 42III
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
39
![Page 40: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/40.jpg)
Resolution III Designs
Saturated 2III7-4 design used for
studying 7 factors in 8 runsstudying 7 factors in 8 runs.
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
40
![Page 41: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/41.jpg)
Saturated 2III7-4 design can be used to obtain resolutionIII g
III for studying fewer than 7 factors in 8 runs. Simplydrop any one on the column in the previous design
![Page 42: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/42.jpg)
Resolution III Designs• Sequential assembly of fractions to separate aliased effects
(page 322)S it hi th i i one l id ti t f• Switching the signs in one column provides estimates of that factor and all of its two-factor interactions
• Switching the signs in all columns dealiases all main effects from their two-factor interaction alias chains –called a full fold-over
• Defining relation for a fold-over (page 325)g (p g )• Be careful – these rules only work for Resolution III
designs• There are other rules for Resolution IV designs and other• There are other rules for Resolution IV designs, and other
methods for adding runs to fractions to dealias effects of interest
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
42
• Example 8.7, eye focus time, page 323
![Page 43: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/43.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
43
![Page 44: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/44.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
44
![Page 45: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/45.jpg)
R b th t th f ll f ld t h iRemember that the full fold-over technique illustrated in this example (running a “mirror image” design with all signs reversed) only works in adesign with all signs reversed) only works in a Resolution III design.
Defining relation for a fold over design see pageDefining relation for a fold-over design – see page 325.
Bl ki b i id i iBlocking can be an important consideration in a fold-over design – see page 325.
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
45
![Page 46: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/46.jpg)
Plackett-Burman Designs
• These are a different class of resolution III design• The number of runs, N, need only be a multiple of , , y p
four• N = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …• The designs where N = 12, 20, 24, etc. are called
nongeometric PB designs• See text, page 326 for comments on construction
of Plackett-Burman designs
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
46
![Page 47: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/47.jpg)
Plackett-Burman Designs
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
47
![Page 48: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/48.jpg)
This is a non-regular design
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
48
![Page 49: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/49.jpg)
Plackett-Burman Designs
Projection of the
g
Projection of the 12-run design into 3 and 4 factors
All PB designs have projectivity 3 (contrast with other(contrast with other resolution III fractions)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
49
![Page 50: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/50.jpg)
Plackett-Burman Designs
Th li t t i l i th PB d i• The alias structure is complex in the PB designs• For example, with N = 12 and k = 11, every main
effect is aliased with every 2FI not involving itselfeffect is aliased with every 2FI not involving itself• Every 2FI alias chain has 45 terms
P ti l li i t ti ll tl li t• Partial aliasing can potentially greatly complicate interpretation if there are several large interactions
• Use very very carefully but there are some• Use very, very carefully – but there are some excellent opportunities
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
50
![Page 51: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/51.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
51
![Page 52: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/52.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
52
![Page 53: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/53.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
53
![Page 54: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/54.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
54
![Page 55: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/55.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
55
![Page 56: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/56.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
56
![Page 57: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/57.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
57
![Page 58: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/58.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
58
![Page 59: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/59.jpg)
R l i IV d V D i (P 322)Resolution IV and V Designs (Page 322)
A resolution IV design must have at least 2k runs.
“optimal” designs may often prove useful.
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
59
![Page 60: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/60.jpg)
Sequential Experimentation with ResolutionSequential Experimentation with Resolution IV Designs – Page 339
We can’t use the full fold-over procedure given previously for Resolution III designs – it will result in replicating the runs in the
i i l d ioriginal design.
Switching the signs in a single column allows all of the two-factor interactions involving that column to be separated.
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
60
![Page 61: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/61.jpg)
The spin coater experiment – page 340
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
61
![Page 62: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/62.jpg)
[AB] = AB + CE
We need to dealias these interactionsinteractions
The fold-over design switches the signs in column A
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
62
![Page 63: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/63.jpg)
The aliases from the complete design following the fold-over (32 runs) are as follows:
Finding the aliases is somewhat beyond the scope of this course (Chapter 10 provided details) but it can be determined using Design-Expert.
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
63
de e ed us g es g pe .
![Page 64: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/64.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
64
![Page 65: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/65.jpg)
A full fold-over of a Resolution IV design is usually not necessary, and it’s potentially very inefficient.
In the spin coater example, there were seven degrees of freedom available to estimate two-factor interaction alias chains.
Af ddi h f ld (16 ) h l 12 d fAfter adding the fold-over (16 more runs), there are only 12 degrees of freedom available for estimating two-factor interactions (16 new runs yields only five more degrees of freedom).
A partial fold-over (semifold) may be a better choice of follow-up design. To construct a partial fold-over:
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
65
![Page 66: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/66.jpg)
N tNot an orthogonal design – but that’s notthat s not such a big deal
l dCorrelated parameter estimates
Larger standard errors of regression model coefficients
ff t
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
66
or effects
![Page 67: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/67.jpg)
There are still 12 degrees of freedom available to estimate
two-factor interactions
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
67
![Page 68: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/68.jpg)
R l i V D i P 331Resolution V Designs – Page 331
We used a Resolution V design (a 25-2) in Example 8.2
Generally, these are large designs (at least 32 runs) for six or more factors
Irregular designs can be found using optimal design construction methods
JMP h ll biliJMP has excellent capability
Examples for k = 6 and 8 factors are illustrated in the book
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
68
![Page 69: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/69.jpg)
Supersaturated Designs
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
69
![Page 70: Design of Engineering Experiments The 2k-p Fractional Factorial Designnoordin/s/aqe ch08 rev.pdf · Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery 1 Chapter 8 Design](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b76e76a7f8b9ad2498ba4f9/html5/thumbnails/70.jpg)
Chapter 8 Design and Analysis of Experiments 7E 2009 Montgomery
70