design of engineering experiments the 2k-p fractional factorial designnoordin/s/aqe ch08 rev.pdf ·...

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

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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


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d cu t es (o a b gu t es) te p etat o

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


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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 BCy ginteraction.

This phenomena is called aliasing and it occurs in all fractional designs


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Aliases can be found directly from the columns in the table of + and - signs

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


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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

• 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


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experimentation

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


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Construction of a One half FractionConstruction of a One-half FractionThe basic design; the design generator


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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


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Example 8.1


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Ex. 6.2 The Resin Plant Experiment

DOX 6E Montgomery 12

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


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Fig_08_04 Projection of the 2IV 4-1 design into a 23 design in A, C and D for Ex 8.1


Figure 8.5 A 25–1 Design for Example 8.2

The AC and AD interactions can be verified by inspection of the cube plot


verified by inspection of the cube plot

Possible Strategies forStrategies for

Follow-Up Experimentation

Following a Fractional

F t i l D iFactorial Design


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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


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

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


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Actual response is 104.

The One-Quarter Fraction of the 26-2

Complete defining relation: I ABCE BCDF ADEFComplete defining relation: I = ABCE = BCDF = ADEF


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The One-Quarter Fraction of the 2k


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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)

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 )

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


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• Blocking

The General 2k-p Design: Resolution may t b S ffi i tnot be Sufficient

• Minimum abberation designsg


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table_08_15

fig_08_18

fig_08_19

fig_08_20

fig_08_21

fig_08_22

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


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Resolution III Designs

Saturated 2III7-4 design used for

studying 7 factors in 8 runsstudying 7 factors in 8 runs.


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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

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


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• Example 8.7, eye focus time, page 323


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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.


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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


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This is a non-regular design


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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)


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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


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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.


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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.


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The spin coater experiment – page 340


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[AB] = AB + CE

We need to dealias these interactionsinteractions

The fold-over design switches the signs in column A


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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.


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de e ed us g es g pe .


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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:


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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


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or effects

There are still 12 degrees of freedom available to estimate

two-factor interactions


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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


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Supersaturated Designs


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