Javier Garcia - Verdugo Sanchez - Six Sigma Training - W3 Fractional Factorial Designs
-
Upload
j-garcia-verdugo -
Category
Engineering
-
view
191 -
download
7
Transcript of 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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 3/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 4/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 5/33
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.
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 6/33
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.
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 7/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 8/33
Fractional Factorial Designs
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 9/33
AD + BCeach other.
Fractional Factorial Designs
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 10/33
interactions
Fractional Factorial DesignsAvailable designs in MinitabStat
>DOE
>Factorial>Factorial
>Create Factorial Designs…
>Display Available Designs…p y g
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 11/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 12/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 13/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 14/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 15/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 16/33
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.
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 17/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 18/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 19/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 20/33
go-forward suggestions
Summary
• Fractional factorial designs: basics
• Evaluation of fractional factorials
• Nomenclature
• Examples of fractional factorials
• Example for a Plackett-Burnam design
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 21/33
Appendix:Evaluation of the
lexamples
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 22/33
Entries in Minitab for the EvaluationStat
>DOE
>Factorial
Example:
File: Fractional factorial.mtw>Factorial
>Define Custom Factorial Design…
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 23/33
Entries in Minitab for the EvaluationStat
>DOE
>F t i l 1
File: Fractional factorial.mtw
>Factorial
>Analyze Factorial Design…
1
2
3
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 24/33
The Graphical Evaluation in Minitab
B
4,82
A Feed rateFactor Name
Pareto Chart of the Effects(response is Yield, Alpha = 0,05)
File: Fractional factorial.mtw
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 25/33
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)
File: Fractional factorial.mtw
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 26/33
This model assigns the variation properly
The Residual Diagnostics in MinitabStat
>DOE
>Factorial
File: Fractional factorial.mtw
>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…
>Graphs… “Four in one”4
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 27/33
The behavior of the residuals supports the model
The Evaluation in Minitab, InterpretationStat
>DOE Stat
File: Fractional factorial.mtw
>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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 28/33
concentration obtains the best result
The Evaluation in Minitab, InterpretationStat
>Quality Tools
>Multi Vari Chart
File: Fractional factorial.mtw
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 29/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 30/33
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.
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 31/33
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
>Graphs… “Four in one”4
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 32/33
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
Knorr-Bremse Group 03 BB W3 fractional factorials designs 08, D. Szemkus/H. Winkler Page 33/33