3274090
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D e s i g n
f o r
DFSS-1Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Executing Robust Design
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D e s i g n
f o r
DFSS-3Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Why We Need to Reduce Variation
C o s t
Low Variation;
Minimum Cost
LSL
LSL
USL
USL
Nom
Nom
C o s t
High Variation;High Cost
LSL
LSL USL
USLNom
Nom
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D e s i g n
f o r
DFSS-'Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Purpose of this Module
To introduce a variation improvement investigationstrategy – Can noise factors be manipulated?
To provide the !"!TA# steps to design$ e%ecute$
and analy&e a variability response e%periment To provide the !"!TA# steps to optimi&e a design
for both mean and variation effects
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D e s i g n
f o r
DFSS-(Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Objecties of this Module
At the end of this module$ participants will be able to :
!dentify possible variation effects from residual plots
Create a variability response from replicates
!dentify possible mean and variance ad'ustmentfactors from noise-factor interaction plots
(se the !"!TA# Response )ptimi&er to achieve a
process on target with minimum variation
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D e s i g n
f o r
DFSS-)Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
!trategies to Detect VariationE"ects
*assive Approach – "oise factors are *+$ included$ manipulated or controlled in
the e%perimental design
– *ossible variation effects are identified through analysis of
the variability of replicates from an e%perimental design
Active Approach – "oise factors AR included in the e%perimental design in
order to force variability to occur
– Analysis is similar to the passive approach
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D e s i g n
f o r
DFSS-/Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
$ Passie Exa(ple
A and # are control factors. 4ithin each treatment combination$noise factors are allowed to naturally fluctuate. 4ithin treatmentvariation is largely driven by this bac5ground noise.
A
"
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D e s i g n
f o r
DFSS-10Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Exa(ple output fro( a PassieDesign
The graphs at rightillustrate the type of outputwhich might be obtainedfrom a Robust *arameteresign 1%periment. #othare ain 1ffects plots with
the top row showing themain effects of factors Aand # on the mean andthe bottom row showingthe main effects of factors
A and # on the variation.
"ote that in this e%amplethe mean and variationcan be ad'ustedindependently of eachother6
e an<
# A
> ar i a t i on<
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D e s i g n
f o r
DFSS-12Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Exa(ple) $ Passie NoiseExperi(ent
A design engineer has evaluated the output performance of acircuit design and performed an initial capability analysis of this
design to determine if there is a problem with the mean and2or
the variability.
+tat Duality Tools Capability Analysis "ormal
< E <F!nitialG 0ower +pec E 9HG (pper +pec E 7;
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D e s i g n
f o r
DFSS-13Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Design *apability $nalysis
!s there a problem?
61.560.058.557.055.554.052.5
LSL USL
Process Data
Sample N 100
StDev(Withi! 1.47217
StDev("verall! 1.5414
LSL 58 #ar$et %
USL 62
Sample &ea 56.25'
Potetial (Withi! apa)ilit*
"verall apa)ilit*
Pp 0.4'
PPL +0.'8
PPU 1.24
Pp, +0.'8
pm
p
%
0.45
PL +0.40
PU 1.'0
p, +0.40
")serve- Perormace
PP& / LSL 860000.00
PP& USL 0.00
PP& #otal 860000.00
p. Withi Perormace
PP& / LSL 882'24.02
PP& USL 47.'5
PP& #otal 882'71.'7
p. "verall Perormace
PP& / LSL 871474.48
PP& USL 36.''
PP& #otal 871570.81
Withi
"verall
Process *apability of +,-nitial
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D e s i g n
f o r
DFSS-1'Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
./ 0ull 0actorial Experi(ent
A ;
8
full factorial has been designed to determine if four factorshave an effect on the mean and2or variability of voltage drop<FB. There are five replicates for a total of H= runs$ with nocenter points or bloc5s. – Resistor R AB
– !nductor 0 #B
– Capacitor C; 1B – Capacitor C;I 3B
4or5sheet: Passive Design/
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D e s i g n
f o r
DFSS-1(Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Passie $nalysis Road(ap 1 Part ,2for Mean Only3
Analy&e the response of interest – 3actorial *lots ain 1ffects$ !nteractionB
– +tatistical Results A")>A table and p-valuesB
– Residual *lots by factor
Reduce model using statistical results
(se the residuals plot to evaluate potential e%istence
of variation effects – !f residuals plot indicates a possible variation effect$ go to
*assive Analysis Roadmap - *art ;
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D e s i g n
f o r
DFSS-1)Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
-nteraction Plot
#ased on the interaction plot$ a few of the interactions may besignificant. Chec5 the statistical output for verification.
+tat )1 3actorial 3actorial *lots !nteraction *lot
$
E
0
4
51 1'.512.5 636'
60.0
57.5
55.0
60.0
57.5
55.0
60.0
57.5
55.0
10
20
1
5
12.5
1'.5
-nteraction Plot 2data (eans3 for +,
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D e s i g n
f o r
DFSS-1Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Main E"ects Plot
The main effects plot indicates that factors # and 1 have thelargest effects. 3actor A also has a moderate positive effect.
3actor 3 does not seem to be important. 0ets loo5 at the results.
+tat )1 3actorial 3actorial *lots ain 1ffects *lot
M e a n o f + ,
2010
58
57
56
55
5451
1'.512.5
58
57
56
55
54636'
Main Effects Plot 2data (eans3 for +,
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D e s i g n
f o r
DFSS-1Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
0actorial $nalysis
A preliminary loo5 at the statistical output of the e%perimentindicates factor 3 may not be significant. id we ma5e amista5e by including it in the e%perimental design?
J "ote that the -way and 8-way interactions are still in the model but notpresented in the output above
+tat )1 3actorial Analy&e 3actorial esign
Estimated Effects and Coefficients for Y1 (coded units)
Term Effect Coef SE Coef T P
Constant 55.991 0.2041 274.27 0.000A 2.! 1.1"1 0.2041 5.79 0.000
# $.192 $1.59! 0.2041 $7."2 0.000
E .12 1.!5! 0.2041 ".11 0.000
% 0.4! 0.21 0.2041 1.1 0.2!2
A&# 0.17" 0.0"9 0.2041 0.4 0.!!5
A&E 0.002 0.001 0.2041 0.01 0.995
A&% $0.1" $0.0!9 0.2041 $0.4 0.77
#&E $0.9 $0.4!! 0.2041 $2.2" 0.02!#&% $0.!!2 $0.1 0.2041 $1.!2 0.110
E&% $0.007 $0.004 0.2041 $0.02 0.9"5
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D e s i g n
f o r
DFSS-1/Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Reduce Model to !ignicant#er(s
)ur final model indicates that factors A$ #$ and 1 aresignificant$ along with interactions #1$ #3$ A#3$ and
#13$ using a p-value cut-off of =.;
Estimated Effects and Coefficients for Y1 (coded units)
Term Effect Coef SE Coef T P
Constant 55.991 0.194! 2"7.74 0.000
A 2.! 1.1"1 0.194! !.07 0.000
# $.192 $1.59! 0.194! $".20 0.000
E .12 1.!5! 0.194! ".51 0.000
% 0.4! 0.21 0.194! 1.19 0.29
#&E $0.92 $0.4!! 0.194! $2.40 0.019
#&% $0.!!2 $0.1 0.194! $1.70 0.09A&#&% $0.51 $0.25! 0.194! $1.2 0.192
#&E&% 0."" 0.419 0.194! 2.15 0.05
S ' 1.7404! $S ' 7.11* $S(ad+) ' 70.0"*
+tat )1 3actorial Analy&e 3actorial esign
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D e s i g n
f o r
DFSS-20Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
#he Role of Residual Plots in RD
!n Robust *arameter esign$ the residual plots can show thepossibility for a variation effect
Remember from A")>A and Regression$ we stated one of the
assumptions on the residuals was constant variance and we
chec5ed this via plots
+tat )1 3actorial Analy&e 3actorial esign
Choose Kraphs Residuals vs >ariables A # 1 3
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D e s i g n
f o r
DFSS-21Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
What Next'
After reducing the model$ the Residuals versus 3actor 3/ plotstill indicates that 3 contributes to a variation effect. This finding
should encourage us to move further in the analysis of this data
to create a variability response and analy&e the data. Thus we
move on to *art ; of the roadmap.
0
! t a n d a r d i 5 e d R e s i d u a l
6368676665646'
'
2
1
0
+1
+2
+'
Residuals Versus 0(respose is 1!
+tat )1 3actorial
Analy&e 3actorial esign
Choose Kraphs
Residuals vs >ariables 3
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D e s i g n
f o r
DFSS-22Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Create a >ariability Response Analy&e >ariability
– 3actorial *lots ain 1ffects$ !nteractionB
– +tatistical Results A")>A table and p-valuesB
– Reduce model using statistical results
Compare main effects plots for mean and variabilityto determine which are ean Ad'ustment 3actors andwhich are >ariance Ad'ustment 3actors or bothB
(se the ultiple Response )ptimi&er to find optimal
settings of the factors – ean on target
– inimum variability
*erform a capability study 2 analysis on the resultingfactor settings
Passie $nalysis Road(ap 1 Part .2if Variation e"ect present3
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D e s i g n
f o r
DFSS-23Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
*reate a Variability Response
4e are now going to use the replications to ma5e anew response in order to model the variability. )ncewe have modeled the variability$ we can use the!"!TA# Response )ptimi&er to find the settings ofthe control factors that will put < on target with
minimum variation.
!"!TA# ma5es this easy with a pre-processing ofthe responses in preparation for a variability analysis
<ou will see that !"!TA# will use the standarddeviation as the measure of variability$ rather than thevariance – the results are e,uivalent
6 N t l 7 2!t d d D
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D e s i g n
f o r
DFSS-2'Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
6ses Natural 7og 2!tandard Deof +3
All of the statistical techni,ues that we are using to analy&e this)1 assume that the data is symmetric because we aretesting for mean differencesB
(nfortunately$ when we use a calculated standard deviation as aresponse$ we do not meet this assumption because thesampling distribution of variances is e%pected to be s5ewed$
hence the distribution of standard deviations would also bes5ewed
Raw +t ev 0n +t evB
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D e s i g n
f o r
DFSS-2(Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
*reate a Variability Response
+tat )1 3actorial *re-*rocess Responses for Analy&e >ariability
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D e s i g n
f o r
DFSS-2Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
$naly5e the Variability
Analysis of the variability will be essentially identical to theanalysis for the mean
4ill select the Terms/ to estimate in the model
4ill use the *areto of 1ffects Kraph/ in order to facilitate the
first model reduction
+tat )1 3actorial
Analy&e >ariability
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D e s i g n
f o r
DFSS-2Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
$naly5e the Variability
$RS RAS
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D e s i g n
f o r
DFSS-2/Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Pareto *hart of the E"ects
#ecause we dont have any degrees of freedom for error$ wemust loo5 at the *areto of effects to decide which term to drop
into the error and begin to reduce the model
# e r (
Effect
D
D
D
D
D
D
D
D
1.41.21.00.80.60.40.20.0
0.1'0actor
Name
D
Pareto *hart of the Effects(espose is at9ral lo$ o StDev1: lpha ; 0.20!
Leth<s PS ; 0.0878'1'
rop A#
interaction first
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D e s i g n
f o r
DFSS-31Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
-nteraction Plot for !tDe+,
The interaction plot indicates a moderately strong interaction betweenfactors A L 1 and A L 3
+tat )1 3actorial 3actorial *lots !nteraction *lot
$
E
0
4
51 1'.512.5 636'
'
2
1
'
2
1
'
2
1
10
20
1
5
12.51'.5
-nteraction Plot 2data (eans3 for !tDe+,
4here shoul
5!ctors A, ", !n F 6e set in
orer to
minimi7e the
v!ri!6ility in
volt!ge rop,
819
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D e s i g n
f o r
DFSS-32Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Main E"ects Plot for !tDe+,
3actor 3 has the largest effect on the variability. !ncreasing 3 should
reduce variability. #ut what did the interaction plot show? 3actor A is the ne%t strongest. +et A E F=. 4hat did the interaction plot
show? 3actors # and 1 are wea5 but what did the interaction plot show?
M e a n
o f ! t D e + ,
2010
2.5
2.0
1.5
1.0
0.5
51
1'.512.5
2.5
2.0
1.5
1.0
0.5
636'
Main Effects Plot 2data (eans3 for !tDe+,
+tat )1 3actorial 3actorial *lots ain 1ffects *lot
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D e s i g n
f o r
DFSS-33Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Deter(ine Mean 8 Variation E"ects
The graphs at right allowus to directly compareeach factors singulareffect on both the meanand variation
#ased on these graphs$ – 3actors # and 1 are
ean Ad'ustment3actors since theyaffect the mean withlittle or no effect on thevariability
– 3actor 3 is a >ariance Ad'ustment 3actor
since it affects thevariability with little orno effect on the mean
– 3actor A appears toaffect both mean andvariability
M e a n o f + ,
2010
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56
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54
51
M e a n o f + ,
1'.512.5
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M e a n o f !
t D e + ,
2010
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M e a n o f !
t D e + ,
1'.512.5
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636'
Main Effects Plot 2data (eans3 for +, Main Effects Plot 2data (eans3 for +,
Main Effects Plot 2data (eans3 for !tDe+, Main Effects Plot 2data (eans3 for !tDe+,
A55 ects "oth A55ects e!n A55ects e!n A55ects :!ri!tion
9uality *hec:) !tatus of +our
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D e s i g n
f o r
DFSS-3'Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
9uality *hec:) !tatus of +ourModels
(se the +how esign/ iconto chec5 on the status of the
analysis. <ou should ma5e
sure that the correct model
has been fit for each
response that you intend to
specify in the response
optimi&er.
As shown in this window$
models have been fit for both
the <F and +tev<F
responses
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D e s i g n
f o r
DFSS-3(Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Multiple Response Opti(i5er
+tat )1 3actorial Response )ptimi&er
+et the 4eight for <F to F= to ensure hitting 7=$ tight lower L upper range
Read first-guess/ target L upper values for +tev<F from interaction plots – "ote that +tev<F is in regular units here$ ")T logged units6
3or +tev<F$ set weight low to protect against a bad first guess
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D e s i g n
f o r
DFSS-3)Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Multiple Response Opti(i5er
4e use the multiple response optimi&er to provide a stac5edmain effects plot. This plot allows us to interactively manipulate
the values of each factor in the model and see the effect on both
the mean and the variation.
<ou can use the red sliders to tune each of the factors
Mi
0o=.I7IH
)ptimal
Cur
d E =.HNHI7
inimum+tev<F
d E =.IIHH8
Targ: 7=.=
y E =.9N;9
y E 7=.===F
7.=
7I.=
F;.9=
F.9=
F.=
9.=
F=.=
;=.=# 1 3 A
OF.8HNHP OF.=P OF.9=P O7I.=P
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D e s i g n
f o r
DFSS-3Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
6se the E;uations to *onr( +,
0ets use the model coefficients to predict and seethat it matches ma5e sure to use un-codedB6
3rom the optimi&ed solution$ A E F.8HNH$ # E F$ 1 E
F.9$ 3 E 7I
Y1 = $22.0994 0.214!70& A 45.2!19&B 4.71125&E 0.24270"&F $ .2!279&B&E $ 0.!07!7!&B&F 0.00010"9""& A &B&F 0.04271"&B&E&F
Y1 = $22.0994 0.214!70&13.4878 45.2!19&1
4.71125&13.5 0.24270"&69 $ .2!279&1&13.5 $0.!07!7!&1&69 0.00010"9""&13.4878&1&69 0.04271"&1&13.5&69
Y1 = 60.00013 as seen in te otimi6er indo
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D e s i g n
f o r
DFSS-3Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
6se the E;uations to *onr( !tDe+,
Again$ ma5e sure to use the un-coded coefficients6
Again$ A E F.8HNH$ # E F$ 1 E F.9$ 3 E 7I
lnStDevY1 = 15.4015 $ 0.00177"& A 0.727714&B $0.927015&E $ 0.05!!905&F $ 0.152!2!& A &B 0.05!52& A &E $ 0.0097"99"& A &F 0.024219"&B&F
0.009"241& A &B&E $ 0.002"29!!&B&E&F 0.000017& A &B&E&F
lnStDevY1 = 15.4015 $ 0.00177"&13.4878 0.727714&1 $ 0.927015&13.5 $ 0.05!!905&69 $0.152!2!&13.4878&1 0.05!52&13.4878&13.5 $
0.0097"99"&13.4878&69 0.024219"&1&69 0.009"241&13.4878&1&13.5 $ 0.002"29!!&1&13.5&69 0.000017&13.4878&1&13.5&69
lnStDevY1 = $0.5577
StDevY1 = e-0.5577 = 0.57253 as seen in te otimi6er
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D e s i g n
f o r
DFSS-3/Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
0inal Design *apability $nalysis
id we achieve our ob'ectives?
61.6061.0560.5053.3553.4058.8558.'0
LSL USL
Process Data
Sample N 100StDev(Withi! 0.120084
StDev("verall! 0.115355
LSL 58
#ar$et %
USL 62
Sample &ea 60.'18'
Potetial (Withi! apa)ilit*
"verall apa)ilit*
Pp 5.75
PPL 6.66
PPU 4.8'
Pp, 4.8'
pm
p
%
5.55
PL 6.44PU 4.67
p, 4.67
")serve- Perormace
PP& / LSL 0.00
PP& USL 0.00
PP& #otal 0.00
p. Withi Perormace
PP& / LSL 0.00
PP& USL 0.00
PP& #otal 0.00
p. "verall Perormace
PP& / LSL 0.00
PP& USL 0.00
PP& #otal 0.00
Withi
"verall
Process *apability of +,0inal
wor5sheet passive capability /
+tat Duality Tools Capability
Analysis "ormal
< E <F3inalG 0ower +pec E 9HG(pper +pec E 7;
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D e s i g n
f o r
DFSS-'0Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Re(e(ber the #&o !trategies'
4e 'ust reviewed the *assive Approach – "oise factors are *+$ included$ manipulated or controlled in
the e%perimental design
– 4e analy&ed the variability of replicates from an
e%perimental design
"ow we will loo5 at the Active Approach – "oise factors AR included in the e%perimental design in
order to force variability to occur
–4e will see that the analysis is similar to the passiveapproach
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D e s i g n
f o r
DFSS-'1Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
#he $ctie $pproach
A factorial e%periment is performed using Control A" "oisefactors in the same e%periment. Analysis can be performed by
characteri&ing ControlJ"oise interactions only or by moving
forward to analy&e the variability by dropping the noise factors
into the error term.
*ros – +imple e%tension of standard e%perimental techni,ues
– Kuarantees noise in the "oise factors
– *rovides for fle%ibility in analysis methods
– Can allow for reduced replication
Cons – Re,uires ability to manipulate and control "oise factors
– )ptimal designs for minimi&ation of unneeded effects noise by
noise interactionsB can be difficult to create
Exa(ple) $n $ctie Noise
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D e s i g n
f o r
DFSS-'2Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
control factors
F noise factor
;8 full factorial design
Two Approaches to Analysis – (se only interpretation of interaction plots to choose settings
of the control factors to minimi&e effect of noise
– odel the variability by dropping the noise factors into the
error and analy&e li5e the passive approach
Exa(ple) $n $ctie NoiseExperi(ent
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D e s i g n
f o r
DFSS-'3Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
$ctie $nalysis Road(ap < Plots Only
Create and e%ecute with noise included as a factor Analy&e the response of interest
– 3actorial *lots ain 1ffects$ !nteractionB
– +tatistical Results A")>A table and p-valuesB
– Reduce model using statistical results
Review !nteractions *lot – !nterpret the interaction plots to loo5 for evidence of variation effects
Review ain 1ffects *lot if applicableB
(se the ultiple Response )ptimi&er to find the optimal settings
of the factors such that the mean is on target
– 4ill force in settings obtained from the interaction plots *erform a capability study 2 analysis on the resulting factor
settings
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D e s i g n
f o r
DFSS-''Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Exa(ple) $n $ctie Noise Experi(ent
An engineer is interested in improving the stability androbustness of a filtration product
Review the capability of the current performance to determine
the opportunity to apply robust design techni,ues
+tat Duality Tools Capability Analysis "ormal
< E <8!nitialG 0ower +pec E 7=G (pper +pec E H=
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D e s i g n
f o r
DFSS-'(Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Design *apability $nalysis
4hat conclusions can you draw from this graph?
807672686460
LSL USL
Process Data
Sample N 100
StDev(Withi! 4.006'8
StDev("verall! '.37663
LSL 60
#ar$et %
USL 80
Sample &ea 70.'21
Potetial (Withi! apa)ilit*
"verall apa)ilit*
Pp 0.84
PPL 0.87
PPU 0.81
Pp, 0.81pm
p
%
0.8'
PL 0.86
PU 0.81
p, 0.81
")serve- Perormace
PP& / LSL 0.00
PP& USL 10000.00
PP& #otal 10000.00
p. Withi Perormace
PP& / LSL 4335.46
PP& USL 7848.20
PP& #otal 1284'.66
p. "verall Perormace
PP& / LSL 4724.40
PP& USL 7467.87
PP& #otal 12132.28
Withi
"verall
Process *apability of +/-nitial
Exa(ple) $n $ctie Noise
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D e s i g n
f o r
DFSS-')Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Exa(ple) $n $ctie NoiseExperi(ent
The device contains several control factors fromwhich three were identified as )1 candidates – *ressure AB
– Concentration #B
– +tir Rate CB
Ambient Temperature was identified as being
significant$ but not economically controllable – Temperature would not change appreciably during the time
in which it would ta5e to e%ecute a three factor e%periment – ecided to include it as a factor in the design to force it to
change
– Call this 3actor K
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D e s i g n
f o r
DFSS-'Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
#he Experi(ental Design A ;8-F fractional design Res !>B was re'ected because
the ;-way interactions are of great interest in thise%periment
A ;8 full factorial design was used
#ecause of time constraints$ only F replicate was
performed The variables are listed below:
– A E *ressure
– # E Concentration
–
C E +tir Rate – K E Temperature
The data is in wor5sheet Active Design/
)pen wor5sheet Active Design/ within Robust
Design.mpj /
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D e s i g n
f o r
DFSS-'Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
0actorial $nalysis
This )1 is an unreplicated$ 8-factor full factorial – 4e need to create the *areto of 1ffects/ chart
+tat )1 3actorial Analy&e 3actorial esign
# e r (
Effect
D
DD
D
D
DD
D
20151050
0.08actor
=
Name
D
Pareto *hart of the Effects(respose is 4: lpha ; .20!
Leth<s PS ; 0.05625
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D e s i g n
f o r
DFSS-'/Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
0it the Reduced Model
+tat )1 3actorial Analy&e 3actorial esign
#ased on the p-values$ the AJK and #JK and CJK interactionsare important. +ince we have identified some importantcontrolJnoise interactions$ the ne%t step is to e%amine theinteraction plots.
Estimated Effects and Coefficients for Y4 (coded units)
Term Effect Coef SE Coef T P
Constant 70.0"1 0.00!250 1121.00 0.000
A 5.9"7 2.994 0.00!250 479.00 0.000
# 9.7"7 4."94 0.00!250 7".00 0.000
C 14.!12 7.0! 0.00!250 11!9.00 0.000
8 21."7 10.!94 0.00!250 1711.00 0.000
A&# $0.0" $0.019 0.00!250 $.00 0.05"
A&8 $0.0"7 $0.044 0.00!250 $7.00 0.00!
#&C $0.0!2 $0.01 0.00!250 $5.00 0.015#&8 $1".0" $9.019 0.00!250 $144.00 0.000
C&8 1!.!" ".19 0.00!250 11.00 0.000
A&#&C 0.0!2 0.01 0.00!250 5.00 0.015
A&#&8 0.17 0.0!9 0.00!250 11.00 0.002
A&#&C&8 0.07 0.019 0.00!250 .00 0.05"
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D e s i g n
f o r
DFSS-(0Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
-nteraction Plot
The interaction plot show the valuable interactions available tothe designer. 0ets ta5e a closer loo5 at the strongest ones.
+tat )1 3actorial 3actorial *lots !nteraction *lot
$
*
=
4
51 200150 1+1
100
75
50
100
75
50
100
75
50
10
20
1
5
150
200
-nteraction Plot 2data (eans3 for +/
n
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D e s i g n
f o r
DFSS-(1Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
-nteraction Plot < $ *loser 7oo:
The two interaction plots at right
indicate that both # and C can be
e%ploited to desensiti&e <8 to the
noise variable K
– !f # is set at its high level$ the
slope of the K effect line is
minimi&ed – !f C is set at its low level$ the
slope of the K effect line is also
minimi&ed
To minimi&e output variation due to
noise in ambient temperature KB$
the above two settings should becontrolled in the design
=
M e a n
1+1
100
80
60
40
1
5
=
M e a n
1+1
100
80
60
40
150
200
-nteraction Plot 2data (eans3 for +/
n
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D e s i g n
f o r
DFSS-(2Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Main E"ects Plot
+ince factor A is not involved in a significant interaction and itsmain effect is significant$ we should ta5e a loo5 at its main effectplot to see if there is some potential value in controlling A
This plot indicates that factor A has about Q2- units of controlover the nominal value of <8. Thus$ if manipulating one of theother factors ta5es the mean value off target$ this factor could beused to e%ert some control over the mean value of <.
+tat )1 3actorial 3actorial *lots ain 1ffects *lot
$
M e a n o f + /
2010
7'
72
71
70
63
68
67
Main Effects Plot 2data (eans3 for +/
n
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D e s i g n
f o r
DFSS-(3Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Ne& !ettings ) *apability $nalysis
# was set to 9$ C was set to F9=$ A was left at nominal F9B !t appears that the changes to # L C were successful in reducing
variation in <8 but the mean is now off target
(se factor A to ad'ust bac5 to target6
78757263666'60
LSL USL
Process Data
Sample N 100
StDev(Withi! 2.06587
StDev("verall! 2.15042
LSL 60
#ar$et %
USL 80
Sample &ea 67.838
Potetial (Withi! apa)ilit*
"verall apa)ilit*
Pp 1.55
PPL 1.22
PPU 1.88
Pp, 1.22
pm
p
%
1.61
PL 1.27
PU 1.35
p, 1.27
")serve- Perormace
PP& / LSL 0.00
PP& USL 0.00
PP& #otal 0.00
p. Withi Perormace
PP& / LSL 65.30
PP& USL 0.00
PP& #otal 65.30
p. "verall Perormace
PP& / LSL 113.37
PP& USL 0.01
PP& #otal 113.38
Withi
"verall
Process *apability of +/Valid
n %o& (uch should &e shift
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D e s i g n
f o r
DFSS-('Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
%o& (uch should &e shiftfactor $'
+et up the responseoptimi&er to target N=. Thelower and upper limits arenot important since we willmanually manipulate this.
+et factor #E9 and factorCEF9=. The optimi&er
indicates a nominal<8E7N.N$ very close to thatobserved in the validationstudy.
3inally$ slowly slide the barfor factor A to the right
while observing thepredicted value of <8.This indicates that anominal setting of AEFH.Hshould achieve <8EN=.
Mi
0o=.III8
"ew
Cur
d E =.III8
Targ: N=.=<8
y E 7I.IIHN
-F.=
F.=
F9=.=
;==.=
F.=
9.=
F=.=
;=.=# C K A
OFH.H=P O9.=P OF9=.=P O-=.==N7P
4or5sheet active design”
+tat )1 3actorial Response )ptimi&er
Mi
0o=.=====
"ew
Cur
d E =.=====
Targ: N=.=
y E 7N.N9=7
-F.=
F.=
F9=.=
;==.=
F.=
9.=
F=.=
;=.=
# C K A
OF9.=P O9.=P OF9=.=P O-=.==N7P
n Ne& !etting for $ ) *apability
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D e s i g n
f o r
DFSS-((Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
Ne& !etting for $ ) *apability$nalysis
4hen we shift factor A to this new nominal value$ we succeed atshifting the response to put it on target without degrading the
variation
78757263666'60
LSL USL
Process Data
Sample N 100
StDev(Withi! 2.0310'
StDev("verall! 2.2058'
LSL 60
#ar$et %
USL 80
Sample &ea 70.2302
Potetial (Withi! apa)ilit*
"verall apa)ilit*
Pp 1.51
PPL 1.56
PPU 1.47
Pp, 1.47
pm
p
%
1.53
PL 1.64
PU 1.55
p, 1.55
")serve- Perormace
PP& / LSL 0.00
PP& USL 0.00
PP& #otal 0.00
p. Withi Perormace
PP& / LSL 0.4'
PP& USL 1.71
PP& #otal 2.14
p. "verall Perormace
PP& / LSL 1.54
PP& USL 5.'7
PP& #otal 6.31
Withi
"verall
Process *apability of +/Reduced
n
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D e s i g n
f o r
DFSS-()Copyright 2003 Cummins, Inc. All Rights Reserve.
Copyright 2000-2002 Sigm! "re!#through $echnologies, Inc. %se &ith permission.
!u((ary
>ariation improvement strategies can ta5e two forms: – *assive Approach
– Active Approach
The choice of which strategy to use depends on the ability to
control or manipulate noise factors at least for the duration of
the e%perimentB +tandard full and fractional designs can be used
>ariation effects must be calculated using replications
A log transform of the variability response is automatically used
to minimi&e the effects of asymmetry in the variance distribution
The response optimi&er can be used to simultaneously optimi&eboth mean and variability responses
A validation study must be made at the end of a Robust
*arameter esign study
n
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D e s i g n
f o r Objecties Reisited
At the end of this module$ participants should be able to :
!dentify possible variation effects from residual plots
Create a variability response from replicates
!dentify possible mean and variance ad'ustmentfactors from noise-factor interaction plots
(se the !"!TA# Response )ptimi&er to achieve a
process on target with minimum variation
Complete validation capability studies
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