6. fractional factorial designs (ch.8. two-level...
TRANSCRIPT
Hae-Jin Choi School of Mechanical Engineering,
Chung-Ang University
6. Fractional Factorial Designs
(Ch.8. Two-Level Fractional Factorial Designs)
1 DOE and Optimization
Introduction to The 2k-p Fractional Factorial Design
Motivation for fractional factorials is obvious; as the number of factors becomes large enough to be “interesting”, 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
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Why do Fractional Factorial Designs Work? The sparsity of effects principle There may be lots of factors, but few are important System is dominated by main effects, low-order interactions
The projection property Every fractional factorial contains full factorials in fewer factors
Sequential experimentation Can add runs to a fractional factorial to resolve difficulties (or
ambiguities) in interpretation
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The One-Half Fraction of the 2k
Notation: because the design has 2k/2 runs, it’s referred to as a 2k-1
Consider a really simple case, the 23-1 Note that I =ABC
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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 BC interaction.
This phenomena is called aliasing and it occurs in all fractional designs
Aliases can be found directly from the columns in the table of + and - signs
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Projection of Fractional Factorials
Every fractional factorial contains 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
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Aliasing in the One-Half Fraction of the 23
A = BC, B = AC, C = AB (or me = 2fi)
Aliases can be found from the defining relation I = ABC by multiplication
ABC is called the generator.
AI = A(ABC) = A2BC = BC
BI =B(ABC) = AC
CI = C(ABC) = AB
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Aliasing in the One-Half Fraction of the 23
Main effect
[ ] , [ ] , [ ]A A BC B B AC C C AB→ + → + → +
121212
A a b c abc
B a b c abc
C a b c abc
Two factor interaction effect
121212
BC a b c abc
AC a b c abc
AB a b c abc
Alias structure of effects
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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= ±
[ ]' , [ ]' , [ ]'A A BC B B AC C C AB→ − → − → −
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Design Resolution Resolution III Designs: me = 2fi (i.e., main effect = 2 factor interaction) example
Resolution IV Designs: 2fi = 2fi example
Resolution V Designs: 2fi = 3fi example
3 12III−
4 12IV−
5 12V−
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Construction of a One-half Fraction
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Resin Plant Experiment – the 24-1 Design A chemical product is produced in a pressure vessel. A factorial
experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product .
The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate
A 24-1 fractional factorial was used to investigate the effects of four factors on the filtration rate of a resin
Generator I = ABCD
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Resin Plant Experiment – the 24-1 Design
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Aliasing the 2IV4-1 Factorial Design
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Resolution IV design with the generator I=ABCD
Main effect is aliased with three factor interaction A=A2BCD=BCD; B=AB2CD=ACD; C=ABC2D=ABD; D=ABCD2=ABC;
Two factor interaction is aliased with other two factor interaction AB=CD; AC=BD; AD=BC;
Resin Plant Experiment – the 24-1 Design
Interpretation of results often relies on making some assumptions Ockham’s razor Confirmation experiments can be important Adding the alternate fraction – see page 301
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Resin Plant Experiment – MINITAB Results
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Resin Plant Experiment – MINITAB Results
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0 1 1 2 2 3 3 4 4 5 1 2 6 1 3 7 1 4y x x x x x x x x x x
Zero degree of freedom for residuals
0 1 1 3 3 4 4 6 1 3 7 1 4y x x x x x x x
2 degree of freedom for residuals
Resin Plant Experiment – MINITAB Results
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0 1 1 3 3 4 4 6 1 3 7 1 4ˆ ˆ ˆ ˆ ˆ ˆy x x x x x x x
1 3 4 1 3 1 419.00 14.00 16.50 18.50 19.00ˆ 70.75
2 2 2 2 2y x x x x x x x
Resin Plant Experiment – MINITAB Results
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1 3 4 1 3 1 419.00 14.00 16.50 18.50 19.00ˆ 70.75
2 2 2 2 2y x x x x x x x
For example the residual at ˆ
19.00 14.00 16.50 18.50 19.00100 70.75 (1) ( 1) (1) (1)( 1) (1)(1)2 2 2 2 2
100 100.25 0.25
y y
1 2 3 41, 1, 1, 1x x x x
Resin Plant Experiment – MINITAB Results
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Manufacturing Process for a Circuit
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Five factors in a manufacturing process for an integrated circuit were investigated in a 25-1 design with the objective of improving the process yield.
Select ABCDE as the generator (Resolution V design) I=ABCDE ; E=ABCD ; Every main effect is aliased with a four-factor interaction. E.g., [A] -> A+BCDE Every two factor interaction is aliased with a three-factor interaction. E.g., [AB]-> AB+CDE
Manufacturing Process – MINITAB Results
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Manufacturing Process – MINITAB Results
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A, B, C, and AB are significant
Manufacturing Process – MINITAB Results
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Selecting only A, B, C, and AB
This implies 23 Design with 2 replicates at each experimental point
Manufacturing Process – MINITAB Results
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ANOVA
Residual analysis
Manufacturing Process – MINITAB Results
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Interaction Plot of AB
Cube Plot
The Sequential Experimentation
Suppose that after running the principal fraction, the alternate fraction was also run
The two groups of runs can be combined to form a full factorial – an example of sequential experimentation
De-aliased estimates of the effects can be obtained by adding and subtracting 1 1([ ] [ ]') ( )
2 2A A A BC A BC A
1 1([ ] [ ]') ( )2 2
A A A BC A BC BC
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The Sequential Experimentation
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If it is necessary to resolve ambiguities, we can run the alternate fraction and complete 2k design.
Run 1 Run 2
Resin Plant Experiment – Alternate Fraction
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Recall the resin plant experiment Generator I=-ABCD
[ ] 19 (from main fraction)1[ ]' ( 43 71 48 104 68 86 70 65)4
24.25 (from alternative fraction)
Main Effect of original design1 [ ] [ ]' 21.632
A A BCD
A
A BCD
A A A
The One-Quarter Fraction of the 2k
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The One-Quarter Fraction of the 26-2
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The General 2k-p Fractional Factorial Design
2k-1 = one-half fraction, 2k-2 = one-quarter fraction, 2k-3 = one-eighth fraction, …, 2k-p = 1/ 2p fraction
Add p columns to the basic design; select p independent generators
Important to select generators so as to maximize resolution, see the table in the next slide
Projection – a design of resolution R contains full factorials in any R – 1 of the factors
Effects of factors are
( / 2)
= number of observations
ii
ContrastEffectN
whereN
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The General 2k-p Design Resolution may not be sufficient Minimum abberation designs
Our choice
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