Javier Garcia - Verdugo Sanchez - Six Sigma Training - W3 Fractional Factorial Designs

Fractional FactorialFractional FactorialDesignsDesigns

Week 3

Knorr-Bremse Group

About this Module

Full factorial designs at 2 levels readily becomeFull factorial designs at 2 levels readily become

unaffordable. In these cases we have the possibility to

d t f ti l f t i l d i Th d t i thconduct fractional factorial designs. The advantage is the

reduced number of runs. On the other hand we have to

pay these financial savings with possible restrictions

regarding the interpretation of the results.g g p

To save experimental runs one can use blocked designs, p g ,

fractional factorial designs and designs as described by

Plackett-BurnamPlackett-Burnam.

Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 2/33

Content

• Fractional factorial designs: basics

• Evaluation of fractional factorials

• Nomenclature

• Examples of fractional factorials

• Example for a Plackett-Burnam design


The Strategy of Experimentation

Collect information Fractional FactorialCollect information

Validate factors

A l b h i f

Mirror Plackett-Burnam

2k F t i lAnalyze behavior of important factors

E t bli h d l

2k FactorialCenter PointsBlockingEstablish a model

Determine optimized dj t t

BlockingFull Factorial

Box-Behnkenadjustments

RSM

Taguchi

EVOP

Taguchi


Knowledge and complexity define the type of experiment

The Strategy of Experimentation

Fractional factorial designs

Sort out uncritical factors Fold overSort out uncritical factors, Fold over

Plackett Burman Designs

2k factorial designs2 factorial designs

Center points

BlocksBlocks

Evaluate co variables

Full factorial designsg

RSM

Box Behnken

Evop

Taguchi

Mixed design


Knowledge and complexity define the type of experiment

The Application of Fractional Factorial Designs

• The number of factors determine the number of runs:

2 f t 4– 2 factors = 4 runs

– 4 factors = 16 runs

– 6 factors = 64 runs… and so on

• Fractional factorial designs are often used for screening.Fractional factorial designs are often used for screening. The purpose is to get a quick overview of significant factors with a small number of runs.

• Screening experiments usually are used in the analysis phase of a project. This type of DOE is the alternative to a M lti V t dMulti-Vary study.

• We follow the rule of thumb that high order interactions are ld t ib ti t th d l Th f ti f th f llseldom contributing to the model. Thus a fraction of the full

factorial design is sufficient to evaluate the main effects.


Fractional Factorial Designs

The advantages of fractional experiments:

Hi h d i t ti t ft i ifi t• High order interactions are not very often significant:

– Main effects and low order interactions usually are sufficient to describe the interrelations of a processto describe the interrelations of a process.

– Fractional designs can be converted in full factorial designs as soon as main effects can be omitted (if not significant)as soon as main effects can be omitted (if not significant)

• Sequential experimentation:• Sequential experimentation:

– Fractional designs can be extended to full factorial designs.

– Full factorial experiments can be conducted step by step. (Folded fractional experiments).

Th i d i f ti d t i f th l i f– The received information determines our further planning of investigation and experimentation.


Fractional Factorial DesignsExample for a fractional factorial experiment:

4 factors are assessed in 8 experimental runs. We start with a plan for 3 factors. Next we use the column of the highest order interaction AxBxC for the introduction of factor D. As one result of this procedure we can no more assign any effect in this column clearly. The effect of factor D and the effect of the 3-y yway interaction are called confounded.

StdO d A B C A*B*C DStdOrder A B C A*B*C D1 -1 -1 -1 -1 -12 1 -1 -1 1 13 1 1 1 1 13 -1 1 -1 1 14 1 1 -1 -1 -15 -1 -1 1 1 16 1 1 1 1 16 1 -1 1 -1 -17 -1 1 1 -1 -18 1 1 1 1 1

Limited interpretation vs. fewer experiments



Minitab displays the design in the worksheet:

StdO d A B C DStdOrder A B C D1 -1 -1 -1 -12 1 -1 -1 13 -1 1 -1 13 -1 1 -1 14 1 1 -1 -15 -1 -1 1 16 1 -1 1 -1

i l i l i7 -1 1 1 -18 1 1 1 1

Fractional Factorial DesignFactors: 4Runs: 8 Fraction: 1/2/Resolution: IVDesign Generators: D = ABC Alias StructureI + ABCD

In the session window Minitab informs us about the degree of confounding: I + ABCD

A + BCDB + ACDC + ABD

us about the degree of confounding:

For this example:All main effects are confounded with 3-

D + ABCAB + CDAC + BDAD + BC

All main effects are confounded with 3way interactions2-way interactions are confounded with each other


AD + BCeach other.


The degree of confounding defines the resolution of a design:

• Resolution - III - Designs:– Main effects are confounded with 2-way interactions

• Resolution - IV - Designs (example from page 9):Resolution IV Designs (example from page 9):

– Main effects are confounded with 3-way interactions

2-way interactions are confounded with 2-way– 2-way interactions are confounded with 2-way interactions

––

• Resolution - V - Designs:M i ff t f d d ith 4 i t ti– Main effects are confounded with 4-way interactions

– 2-way interaction are confounded with 3-way interactions


interactions

Fractional Factorial DesignsAvailable designs in MinitabStat

>DOE

>Factorial>Factorial

>Create Factorial Designs…

>Display Available Designs…p y g


The Minitab overview shows the resolution

The Nomenclature

pk2 − • k indicates the numbers of factors under investigation

k

R2 • 2k-p indicates the number of runs

• R indicates the resolution

Another way of describing fractional factorial designs is by stating the number of runs compared to a full factorial design (1/2; 1/4; 1/8; 1/16; 1/32 etc.).

Lets create some fractional factorial designs in Minitab


Fractional Factorial Designs1. 4 Factors – Resolution IV – 8 runsStat

>DOE

>Factorial

14

IV2 −

>Factorial

>Create Factorial Design…

>Number of factors > 4IV

>Designs

Fractional Factorial DesignFactors: 4Runs: 8 Fraction: 1/2

StdOrder A B C D1 -1 -1 -1 -1

Fraction: 1/2Resolution: IVDesign Generators: D = ABC Alias Structure 1 -1 -1 -1 -1

2 1 -1 -1 13 -1 1 -1 14 1 1 -1 -1

I + ABCDA + BCDB + ACDC + ABD

5 -1 -1 1 16 1 -1 1 -17 -1 1 1 -1

C + ABDD + ABCAB + CDAC + BD


8 1 1 1 1AD + BC

Fractional Factorial Designs2. 6 Factors – Resolution III – 8 runsStat

>DOE

>Factorial

36

III2 −

>Factorial

>Create Factorial Design…

>Number of factors > 6III

>Designs

StdOrder A B C D E F1 -1 -1 -1 1 1 1I + ABD + ACE + BCF + DEF + ABEF + ACDF + BCDE 1 1 1 1 1 1 12 1 -1 -1 -1 -1 13 -1 1 -1 -1 1 -14 1 1 -1 1 -1 -1

A + BD + CE + BEF + CDF + ABCF + ADEF + ABCDEB + AD + CF + AEF + CDE + ABCE + BDEF + ABCDFC + AE + BF + ADF + BDE + ABCD + CDEF + ABCEFD + AB + EF + ACF + BCE + ACDE + BCDF + ABDEF 5 -1 -1 1 1 -1 -1

6 1 -1 1 -1 1 -17 -1 1 1 -1 -1 18 1 1 1 1 1 1

D + AB + EF + ACF + BCE + ACDE + BCDF + ABDEFE + AC + DF + ABF + BCD + ABDE + BCEF + ACDEFF + BC + DE + ABE + ACD + ABDF + ACEF + BCDEFAF + BE + CD + ABC + ADE + BDF + CEF + ABCDEF


8 1 1 1 1 1 1

Folded DOE with 5 Factors

Start with 8 runs

Evaluation and fold over according of the founded results

8 further runs with focus on A

StdOrder A B C D E1 1 1 1 1 1

252 − 152 −

1 -1 -1 -1 1 12 1 -1 -1 -1 -13 -1 1 -1 -1 14 1 1 1 1 1

I + ABD + ACE + BCDEI + BCDE

III IV

4 1 1 -1 1 -15 -1 -1 1 1 -16 1 -1 1 -1 17 -1 1 1 -1 -1

A + BD + CE + ABCDEB + AD + CDE + ABCEC + AE + BDE + ABCD

A + ABCDEB + CDEC + BDED + BCE

8 1 1 1 1 1

9 1 1 1 -1 -1

C + AE + BDE + ABCDD + AB + BCE + ACDEE + AC + BCD + ABDEBC + DE + ABE + ACD

E + BCDAB + ACDEAC + ABDEAD + ABCE10 -1 1 1 1 1

11 1 -1 1 1 -112 -1 -1 1 -1 113 1 1 1 1 1

BE + CD + ABC + ADEAD + ABCEAE + ABCDBC + DEBD + CE

13 1 1 -1 -1 114 -1 1 -1 1 -115 1 -1 -1 1 116 -1 -1 -1 -1 -1

BE + CDABC + ADEABD + ACEABE + ACD


16 -1 -1 -1 -1 -1 ABE + ACD

Folding of the Experiment in MinitabStat

>DOE

>F t i l>Factorial

>Create Factorial Design

>Number of factors 5

>Design > 8 Runs > OK

>Options > Fold on all factors


The Plackett-Burnam Design

Using this design we can assess 11 factors within 12 runs

StdOrder A B C D E F G H J K L1 + - + - - - + + + - +2 + + - + - - - + + + -3 - + + - + - - - + + +4 + - + + - + - - - + +4 + + + + + +5 + + - + + - + - - - +6 + + + - + + - + - - -7 + + + + + +7 - + + + - + + - + - -8 - - + + + - + + - + -9 - - - + + + - + + - +10 + + + + + +10 + - - - + + + - + + -11 - + - - - + + + - + +12 - - - - - - - - - - -

This again saves runs compared to the fractional experiment.


Example of an ApplicationProblem with thermostat valves:

Large spread during leakage testing of a batch

Factor low highNi content low high

M t l b t h 1 2Metal batch 1 2Machine 1 2Washing short long

The output is the StDev of leakage n = 50

g g

Batch Machine Washing Ni StDev-1 -1 -1 -1 2,491 1 1 1 3 651 -1 -1 1 3,65-1 1 -1 1 2,001 1 -1 -1 2,441 1 1 1 2 36

File: Leakage mtw-1 -1 1 1 2,36

1 -1 1 -1 2,41-1 1 1 -1 1,201 1 1 1 1 10

Leakage.mtw


1 1 1 1 1,10

Another Example

• Goal: Creation and evaluation of a fractional factorial experiment with Minitab

• Output-Variable: Yield in %

• Inputs:

– Feed rate (l/min) 10; 15Feed rate (l/min) 10; 15

– Catalyst type 1; 2

– Agitator speed (U/min) 100; 120

– Temperature (C) 140; 180Temperature (C) 140; 180

– Concentration Cat (%) 3; 6

• Your budget only allows 16 experimental runs


Your budget only allows 16 experimental runs

Another Example

File: Fractional factorial.mtw

Feed rate Catalyst Agitator Temp Concentration Yield10 1 100 140 6 5615 1 100 140 3 5310 2 100 140 3 6310 2 100 140 3 6315 2 100 140 6 6510 1 120 140 3 5315 1 120 140 6 5515 1 120 140 6 5510 2 120 140 6 6715 2 120 140 3 6110 1 100 180 3 6915 1 100 180 6 4510 2 100 180 6 7815 2 100 180 3 9310 1 120 180 6 4910 1 120 180 6 4915 1 120 180 3 6010 2 120 180 3 9515 2 120 180 6 82

Exercise: Evaluation and interpretation, f d ti


go-forward suggestions

Summary

• Fractional factorial designs: basics

• Evaluation of fractional factorials

• Nomenclature

• Examples of fractional factorials

• Example for a Plackett-Burnam design


Appendix:Evaluation of the

lexamples


Entries in Minitab for the EvaluationStat

>DOE

>Factorial

Example:

File: Fractional factorial.mtw>Factorial

>Define Custom Factorial Design…


Entries in Minitab for the EvaluationStat

>DOE

>F t i l 1


>Factorial

>Analyze Factorial Design…

1

2

3


The Graphical Evaluation in Minitab

B

4,82

A Feed rateFactor Name

Pareto Chart of the Effects(response is Yield, Alpha = 0,05)


ACE

EDEBDD

rm

A Feed rateB C ataly st ty peC A gitator speedD Temp.E C oncentration

CDACADAE

BEBCAB

Ter

Pareto Chart of the Standardized EffectsC

CD

20151050Effect

Lenth's PSE = 1,875B

2,23

B C ataly st ty peD Temp

Factor Name

Pareto Chart of the Standardized Effects(response is Yield, Alpha = 0,05)

D

Bm

D Temp.E C oncentration

Step 1:Start with the overview and the reduce to the best

DE

BD

Term

reduce to the best model

E

1614121086420Standardized Effect


Standardized Effect

The Evaluation in Minitab, the Reduced Model

Factorial Fit: Yield versus Catalyst type; Temp.; Concentration

Estimated Effects and Coefficients for Yield (coded units)


Term Effect Coef SE Coef T PConstant 65,250 0,6626 98,47 0,000Catalyst type 20,500 10,250 0,6626 15,47 0,000T 12 250 6 125 0 6626 9 24 0 000Temp. 12,250 6,125 0,6626 9,24 0,000Concentration -6,250 -3,125 0,6626 -4,72 0,001Catalyst type*Temp. 10,750 5,375 0,6626 8,11 0,000Temp.*Concentration -9,500 -4,750 0,6626 -7,17 0,000

S = 2,65047 PRESS = 179,84R-Sq = 97,89% R-Sq(pred) = 94,60% R-Sq(adj) = 96,84%

Analysis of Variance for Yield (coded units)

Source DF Seq SS Adj SS Adj MS F Pq j jMain Effects 3 2437,50 2437,50 812,500 115,66 0,0002-Way Interactions 2 823,25 823,25 411,625 58,59 0,000Residual Error 10 70,25 70,25 7,025

Lack of Fit 2 7,25 7,25 3,625 0,46 0,647Pure Error 8 63,00 63,00 7,875

Total 15 3331,00

This model assigns the variation properly


This model assigns the variation properly

The Residual Diagnostics in MinitabStat

>DOE

>Factorial


>Factorial

>Analyze Factorial Designs…

>Graphs… “Four in one”99

Normal Probability Plot Versus Fits

Residual Plots for Yield

S

or

99

90

50

Per

cent

N 16AD 0,326P-Value 0,486

4

2

0

Res

idua

l

Stat

>Regression

>Regression…

5,02,50,0-2,5-5,0

10

1

Residual9080706050

-2

-4

Fitted Value

Histogram Versus OrderRegression…


3

2quen

cy

4

2

0esid

ual

Histogram Versus Order

You have to store the fits and residuals before

420-2-4

1

0

Residual

Fre

16151413121110987654321

0

-2

-4

Observation Order

Re

The behavior of the residuals supports the model


The behavior of the residuals supports the model

The Evaluation in Minitab, InterpretationStat

>DOE Stat


>Factorial

>Factorial Plots…

>ANOVA

>Interaction Plots…

or

90t

Catalyst

Interaction Plot for YieldData Means

80140

Temp.

Interaction Plot for YieldData Means

80

70ea

n

12

type

75

70

ea

n

140180

70

60

Me

65

60

Me

18014050

Temp.63

Concentration

The temperature of 180°C in combination with catalyst 2 and a low concentration obtains the best result


concentration obtains the best result

The Evaluation in Minitab, InterpretationStat

>Quality Tools

>Multi Vari Chart


100

180140

3 6t pe

Catalyst

Multi-Vari Chart for Yield by Catalyst type - Concentration>Multi-Vari Chart…

100

90

80

12

type

80

70

60

Yie

ld

60

50

40180140

40

Temp.

Panel variable: Concentration

The temperature of 180°C in combination with catalyst 2 and a low concentration obtains the best result


concentration obtains the best result

The Graphical Evaluation in Minitab

B

2,188Factor NameA BatchB Machine

Pareto Chart of the Effects(response is StDev, Alpha = ,05) File:

Leakage.mtw

Term A

AC

C NiC WashingD

D

AD

AB

2 52 01 51 00 50 0Effect

2,52,01,51,00,50,0

Lenth's PSE = 0,58125

2,776

Pareto Chart of the Standardized Effects(response is StDev, Alpha = ,05)

Machine

Term

B t h

WashingStep 1:Start with the overview and the reduce to the best

Standardized Effect

Batch

43210

reduce to the best model


The Evaluation in Minitab, the Reduced ModelFactorial Fit: StDev versus Machine; Washing

Estimated Effects and Coefficients for StDev (coded units)

File: Leakage.mtw

( )

Term Effect Coef SE Coef T PConstant 2,2062 0,1458 15,14 0,000Machine -1,0425 -0,5212 0,1458 -3,58 0,016Washing -0,8775 -0,4388 0,1458 -3,01 0,030

S = 0,412301 PRESS = 2,17590R-Sq = 81,38% R-Sq(pred) = 52,32% R-Sq(adj) = 73,93%

Analysis of Variance for StDev (coded units)

Source DF Seq SS Adj SS Adj MS F PM i Eff t 2 3 71363 3 71363 1 85681 10 92 0 015Main Effects 2 3,71363 3,71363 1,85681 10,92 0,015Residual Error 5 0,84996 0,84996 0,16999

Lack of Fit 1 0,07411 0,07411 0,07411 0,38 0,570Pure Error 4 0,77585 0,77585 0,19396

Total 7 4 56359Total 7 4,56359

Unusual Observations for StDev

Obs StdOrder StDev Fit SE Fit Residual St Resid1 1 2,49000 3,16625 0,25248 -0,67625 -2,07R

R denotes an observation with a large standardized residual.


g

The Residual Diagnostics in MinitabStat

>DOE

>F t i l

File: Leakage.mtw

>Factorial

>Analyze Factorial Designs…

>Graphs… “Four in one” Residual Plots for StDevp

Stat

or 99

90

50cent

N 8AD 0,270P-Value 0,570

0,50

0,25

0,00dua

l

Normal Probability Plot Versus Fits

Stat

>Regression

>Regression…1,00,50,0-0,5-1,0

50

10

1

Residual

Per

c

3,02,52,01,51,0

-0,25

-0,50

Fitted Value

Res

i

g


3

enc

y

0,50

0,25

0 00ual

Histogram Versus Order

You have to store the fits and residuals before 0,500,250,00-0,25-0,50-0,75

2

1

0

R id l

Fre

que

87654321

0,00

-0,25

-0,50

Ob ti O d

Res

idu

Residual Observation Order


The Evaluation in Minitab, Interpretation

Stat

>DOE

File: Leakage.mtw

Stat

>Quality Tools

>Factorial

>Factorial Plots

>Multi-Vari Chart…>Main Effects

2,8Machine Washing

Main Effects Plot for StDevData Means 4,0

3,5

-11

Machine

Multi-Vari Chart for StDev by Machine - Washing

2,6

2,4

2 2Me

an

3,5

3,0

2,5

StD

ev

2,2

2,0

1,8

M

2,0

1,5

1,0

1-11,6

1-1 1-1

1,0

Washing


Engineering

Javier Garcia - Verdugo Sanchez - Six Sigma Training - W3 Fractional Factorial Designs