6. Fractional Factorial Designs (Ch.8. Two-Level...

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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


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


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


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


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


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


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


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→ − → − → −


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−


Construction of a One-half Fraction


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


Resin Plant Experiment – the 24-1 Design


Aliasing the 2IV4-1 Factorial Design

DOE and Optimization 14

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


Resin Plant Experiment – MINITAB Results




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



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



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

Manufacturing Process for a Circuit


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




A, B, C, and AB are significant



Selecting only A, B, C, and AB

This implies 23 Design with 2 replicates at each experimental point



ANOVA

Residual analysis



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


The Sequential Experimentation


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


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


The One-Quarter Fraction of the 26-2


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


The General 2k-p Design Resolution may not be sufficient Minimum abberation designs

Our choice


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