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I Library of Cowanss CataJoging-in-Pablicatioo Data Roy.Ranjit_
Aprimer 011the TagucbimethodIRanjit Roy. p.eat. --
(Competitivemanufacturingseries) BibliograpIIJ:p. Includesindex.
ISBN 0-442-237294 1.Tagucbi methods(Quality
control)2.Taguchi.Gen'icbi.1924-I. Title.H.Series. TSI56.R69.1990
658.S'62--dcZO89-\4736 ':IP l TomywifeKrishnaand mydaughters
Purbaand Paala. f I .I ! .- -----"- ."--.--.-_._----- -/ t / I t f
I Peopleusetheir eyes.Theynevu see a bird.theysee a "'r
sparrow.Theynever see a tree,they see abirch.Theysee concepts.
-'"-t' jJ?;;-Joyce Cary p:,UP- VNR COMPEl II IVE MANUfACTURING
SERIES-.................... PRACTICAL EXPERIMENT DESIGN "W'ilfiamJ.
Diamond VALUE ANALYSIS INDESIGN by 1heoc:IoIe C. fowler A PRIMER
0,. THE TAGUCHI UEIHOD by Ranjit Roy MANAGING NEW-PRODUCT
DEVELOPMENT by Geoft'Vincent ART AND SCIENCE Of INVBI1DIG by GIbert
lQvenson RELlABlUlY ENGlNEERlNG IN SYSTEMS DESIGN-AND OPERAnoN by
Ba1bir S. DbiIon ., REUABIU1Y AND MAlNTAINABIUIY MAMGEUENT bv
Batbir s. DhiUon and Haas Reiche APPUED RElJAB1U1Y by Paul A Tobias
ancI bawd C.Trindad ................ INDUSTRIAL ROBOT HANDBOOIC:
CASE HISTORIES Of EFfECTIVE ROBOT USE tN 10 lNDtISTRIES by Richan:I
KMIler ROBOTIC TECHNOLOGY:PRINCIPLES AND PMCl1CE by Waner
G.HoIzbock MAOBNE VIStON by Ne1lo Zuech and Iic:hDtd It Mikr DESIGN
Of AUfOMATiC MACHINERY", ~W.l.entz.Jr. TRANSDUCERS .FOR
AUTOMA11ONby lIicDMl HordctsId MIO'tOPROCESSOR IN JNJ.>lJS1tW.,
IIidJMJ Hordold DISTRIBUTED CONTROL svstaI5 by Ilic:hHJP.LuJcas
BU1J( MATERIAtS HANDUNG HAtmBOOJ( by.Jacob fruddbaum MICROCOMPUTER
SOfTWARE f O ~MECHANICALby Howani fde ~.. WORI22II22 4-1--2>
2222II S--2---1>32I2I2 6-2---1 >32212I 7-2--2> 4II22I
8-2----2> 4I2II2 Table 5-22. Modified LaArray with One 4Level
Column EXlIRINENTSI COlUMNIII.WCQWMN4 , 61 IIIIII 2I2222 32J122
42221I S31212 632I2I 14I221 8421I2 Table 5-23.Modi&ed Lawith
Factors Assigned(One 4Level Column) FACTORS-+ABCDE
EXPERIMB\'TS/COWMNNEW COLUMN4 , 67 11 -; II11 212222 32I122 4222I1
S31212 632I2I 741221 842I12 WorkingMechanics of the TaguchiDesign
of Experiments81 2 ~ - - - - - - - - ~ ~ - - - - - - - - ~ 4 6
Figure 5-11.Preparation of an8 levelcolumn. can beselected on this
basis.A fourthline connecting the apexandthe base
representstheinteraction (AxBxC) asshown in Figure5-11.
Step2.Select thethree columnsto be usedtoformanSlevel column.
Selectthethreecolumnswherethethreefactors,A,B,andCare assigned.In
general,select each apex of thetriangle of thelinear graph
forthesettorepresent thecolumns.Theseare columnsI ~2,and 4for
theset.Theremaining four columns are eliminated,since thethree
col-umnsincludethe fourinteractive(AxB,AxC, BxC and AxB xC).
Step3.Combinethree2 levelcolumns intoan8level column. Compare
numbers in each row of thethree columns and combine them
usingtherulesshownin Table5-24.Notetheruleisnot theprevious
oneforthe4levelarraythoughitfollowsthesame pattern. Table S.l4.
Rules forPreparation of an 8 LevelColumn for an,LI6 Array COLUMNS
NEW FIRSTSECONDTHDIDCOLUMN 1I11 1122 12I3 I224 2115 2I26 2217 '2228
I' I . 12A Primer onthe TaguchiMethod Table 5-25.Converting Lu. to
Include an 8LevelColumn ColumnsCombinedtoFormNewColumns L I I ~
.NE\\' COLUMN EXPEJUMENTSI 1~ OOLUMNI248910II121314 II ..zt
II1II1I1I 211II2222222 . 3I122I1I .I 222 4II22222211I SI231I122112
6I231221122I 7I242It22221 8I242221III2 921SI1212I21 10215I21212I2
II , ~ , ~ [; I62I212212 121622I2I121 13 ':1 2271I22tI22
14227121122I1 152282t22t21I 1622822112I22 IS I 2 2 I 2 I I 2 2 I I
2 I 2 2 I For the set of columns under consideration,the
first,second, and third are columns1,2, and 3, respectively
(Fig.5-11). The modified LI6 array
withitsupgradedcolumnisshowninTable5-25.Notethatthelinear
graph(Fig.5-11)representssevencolumnsconsistingofthreemain
effectsandfourinteractions.Thuscombiningthecolumnrepresenting
thethreemain effectsincludesthefourinteractions. 5-7DUMMY TREATMENT
(COLUMN DEGRADING) Justas2levelcolumnsof
OAcanbecombinedtohigherlevels,soa higher
levelcolumncanbedecomposedintolowerlevelcolumns. The methodusedis
knownasDummy Treatment.I
ConsideranexperimentinvolvingfourfactorsA,B,C,andD.of
whichAhasonly2levels.andalltheothershave3levelseach.DOF
WorkingMecbmics of the TagucbiDesign of Experiments83 Table
5-26.Design with Degraded Column of L" FACTORS-+8 EXPERlMENI'SI
COLUMNI 11 21 3 . 1 42 52 62 73 83 -93 (') indicates
newmodifiedlevel I''"(level3) CA 21 11 22 3l' I2 2l' 31 Il' 21 32 D
4 1 1 3 ;. 3 t 2 2 3 I is 7.An 4array has four 3level columns with
8 DOF. It could be used if onecolumncan be reducedtothe2level
forfactor Aandthethree remaining columns occupied by factors B. C,
and D. In dummy treatment, thethirdlevelof A=A3isformallytreated as
A3,asif A3exists.But in reality A3isset to beeither AI or A2
Thedesignwiththemodifiedcolumn(3)of 4isshowninTable
5-26.FactorAcanbeassignedtoanycolumn.Notethatcolumn3 was,
selectedsuchthatthemodifiedlevel3=I'occursoncein each groupof
threetrialruns.Thisdistributionenhancestheexperiment. Example 5..
10 In a casting process used to manufacture engine blocks for a
passenger car,ninefactorsand their levelswereidentified(Table
5-27).Theop-timum process parameters forthe casting operation are
to be- detennined by experiment.Of the nine factors,twoare of 3
levels each and another of 4levels.The remaining sixfactorsare all
of 2levels each.The OOF isatleast13if nointeractionsare considered.
The Design Sincemostfactorsare2level,a2levelOAmaybesuitable.Each3
levelfactorcanbeaccommodatedby3columns(modified)andthe4 level
factor can also be described by 3 columns for a subtotal of 9
columns. I 84A Prima' onthe TagucbiMethod Table 5-27.Factors of the
Casting Process Experiment-Example S-10 VAlUABLESl.EVELILEVEL
2LEVEL 3LEVEL 4 A:Sand compactionPlant XPlantYPlant Z B:Gating
typePlantXPlantYPlant Z C:MetalbeadLowHigh D:Sand supplierSupplier
1Supplier 2 E:Coating typeTypeIType2Type3Type 4 F:Sand
permeability300 penn400 penn G:Metal temperature1430F1460F
H:Quenchtype450F725F I:Gas levelAbsentHighamount
Theremainingsix2level factors requireone column each.Thus
amin-imum15columnsisneeded.Ll6 satisfiesthis requirement.Nine
col-umnsaretobe converted tothree4levelcolumns,then2columnswill be
reducedby dummy treatment to 3 level columns for this experiment.
Nonnally a3level column willhave2 DOF.But whenit is prepared
byreducinga4levelcolumn,it mustbecountedas3DOFsinceit a
dummylevel.Thus, thetotal DOF for the experiment is: 6Variables at2
levels each6DOf IVariableat4levelseach3 DOf 2 Variablesat3 levels
each6DOF (Dummy Treated) ------------------TotalOOF=15 Ll6 has15OOF
and therefore is suitable for the design.The three sets of
interacting columnsused for column upgrading are1 23,48 12, and 7
914.The columnpreparation andassignment followsthesesteps.
I. usecolumnsIand2to preparea4level 00iWtiii: 'thentreat it to a
3 level column. Place it as column 1.Assign factor A(sand
compaction)to this column.
2.Discardcolumn12andusecolumns4and8tocreatea4level column
first,then dummy treat it to a 3 level column.Call the new
column,column 4.Assign factor B (gating type)to thiscolumn.
3.Discardcolumn14andusecolumns1and9tocreatea4level columnforfactor
E(coatingtype).Callit column7. WorkingMechanics of the
TagucbiDesignof Experiments15 Table 5-28-1.Casting Process
Optimization Design-Example 5-10 -,DesignVariablesand Their Levels
COWMN NUMBERFACTORNAMElVEL ILEVEL 2LEVELlLE\U-I ISand
compactionPlantXPlant YPlant Z 2(UsedwithCol .I)MIU 3(Usedwith
ColI)MIU 4Gating typePlantXPlantYPlant Z 5MetalheadLowHigh
6SandsupplierSupplierISupplier 2 7Coating typeTypetType2Type3Type4
8(UsedwithCol 4)M/U 9(UsedwithCol7)MlU 10Sandperm200Perm300 Perm
11Metaltemperature1430 F1460 F 12(Usedwith Col 4)MIU
13Quenchtype450 F725F 14(Usedwith Col 7)MIU 15GaslevelNoneHigb
Note:ModifiedCo/s_I:!34812and7914.Nointeractionobjective:Detennineprocessparameter
focbest casting Thehigher thebetterISiq or odIer) 4.Assign the
remaining seven 2 levelfactorsto the rest of the2 level columns
asshown inTable 5-28-1. The detail array modified to produce two,3
level and one 4 level column is shown in Table 5-28-2. Table 5-28-3
shows the modifications to create three 4level columns andthe dummy
treatment two columns to 3 level. .Notethat in new colun;m1, the
four dummy levelsl' occur together.In
thiscase,toavoidanyundesirablebiasduetolevell,theexperiment
shouldbe carried out byselectingtrial conditions in arandomorder.
Description of Experimental Conditions Once the factorsare
assigned,the16 trialruns are described bytherows of the OA
(modified).With experience, the run conditions are easily read from
the array.But for the inexperienced, and for large arrays.
translating thearraynotationsintoactualdescriptionsof
thefactorlevelsmaybe
subjecttoerror.Computersoftwareexiststoreduce/eliminatechances
86APrimer onthe Tagucbi Method Table 5-28-2.Casting Process
OptimizationDesign-Example 5-10 EXPER1MENf1
COLUMN234S678910II12Il14IS Expt1100II11001I0I0I
Expt2100211200220202 Expt3100322300I10202 Expt4100122400220I01
Expt5200I12300220I02 Expt62002I2400II0201 I; Expt720032I100220201
Expt8200121200I10102 "I i!Expt9300121400120202
Expt103002213002I0I01 ExptII3003I220012010I Expt123001I2I00210202
Expt 13100I22200210201 Expt14':;II,;'.;!00222I00120I02 I Expt15
00311400210I02 Expt16100111300I20201 . I I of such errors.A
printout of thetrialconditions forsampletrialrunsis showninTable
5-28-4. f Main Effect Plots for 3 Level and 4Level Factors
Theanalysisof experimentaldatafollowsthesamestepsasbefore.
Theresultsof a singletestrunateachof the16conditionsareshown
inTable5-28-5.Themaineffectsof thefactorsarepresentedinTable -;
5-28-6;theeffectsforthe3and4levelfactorsaredisplayedinFigure
5-12.Theoptimumcombinationiseasilydeterminedbyplottingmain
effectsof allfactorsorfromthedataof Table5-28-6of
themainef-fects,byselectingthehighervalues(sincethequalitycharacteristicis
"thebiggerthe Notethatforsandcompactionthemiddlelevel
yproducesthehighestvalue.Suchnonlinearbehaviorof thefactorwas
frompreviousexperience,hencethreelevelswereselected
fortheexperiment. WorkingMechanics of theTagucbiDesign of
Experiments87-Table 5-28-3.Casting Process Optimization Design
(Column Upgrading Procedure).' 1 2 and3 to fonna-3 levelcol."New1"
23 1---1----1 1---1---1 1----1---1 1--1----1 1----2----2 1----2---2
1---2----2 1---2----2 2----1---2 2--1--2 2---1---2 2----1--2
2----2---1 2----2----1 2----2----1 2----2---1 NEWt I I I I 2 2 2 2
3 3 3 3 4=l' 4= 1/ 4= 1/ 4=l' 48and12 to fQrma 3 level
col.New4812NEW 4 1---1-1 :t 2----2---1' 2---1-2 2--2-1 1--1-1
1---2-2 2----lr--2 2----2--1 1----1--1 1----2----2 2----1---2
2--2---1 1----1---1 I---J--l 2----1--2 2----2---1 1 'J:-II/If 3
4=l' I 2 3 4=l' 1 2 3 4= I' 1 2 3 4=I' Note: indicales
dummytreatedlevels_ 5-8COMBINATIONDESIGN 7 9and14 to form a 4level
col."New T 7914 1---1--1 1---2-2 2--1--2 2--2--1 2-1-2 2----2---1
1--1----1 1---2--2 2---2----1 2---1---2 1----2---2 1----1-1 1--2-2
1---1--1 2----2--1 2---1-2 I 2 3 4 3 4 1 2 4 3 2 t 2 I 4 3 ;..
Consideranexperimentinvolvingthree3levelfactorsandtwo 2level
factors.An experiment design could consider L16 OA with 3 columns
for each of the 3 level factors and 2 additional columns for the 2
level factors. Such a designwin utilize11of the available15columns
and require16
trialrunsfortheexperiment.Alternativelyconsiderthe4OA.- Three
columnssatisfythe3levelfactors.If thefourthcolumn can beusedto
accommodatetwo2levelfactors,then4withonly9trialruns.could beused.
:;. I'>Indeed,itispossibletocombinetwo2levelfactorsintoasingle3
i..levelfactor,withsomelossofconfidenceintheresultsandlossof /
tostudyinteractions.Theprocedureisgivenbelow. -
Defineanewfactor(XY)tobeformedoutof thecombinationof X andY
andassignit tocolumn 4.Fromthefourpossible combinations X I { , I I
; " 88A I'Iia:Ia' on the Tagucbi Method Table 5-28-4.Description of
IndividualTrial Conditions 11UAL NUMBERI s.d compactionMlC
Gatingtype Metalhead Smd supplier Orating type s-rperm
Metaltemperature Qlatcbtype GIs level 11UAL NUMBER2 Sad
compactionMlC Gating type Metalhead Sat-supplier OJating type Sad
perm -MdaI temppture Qracbtype Gralevel = PlantX =PlantX =Low
=SupplierI =Type1 =300perm = 1430 F = 450F = Absent/none = Plant X
=PlantY =Low =Supplier1 =Type2 =400 perm =1460 F =725F =High ..
LevelI .. Levell .. Levell .. LevelI .. LevelI .. LevelI .. LevelI
.. Levell .. LevelI .. LevelI .. Level2 .. Levell .. Levell ..
Level 2 .. Level2 .. Level 2 .. Level2 .. Level2 Table
5-28-5.Casting Process OptimizationData TIUM.RIRlR314RsRt. AVO.
I67.0067.00 166.0066.00 356.0056.00 467.0067.00 5- 78.0078.00
690.0090.00 768.0068.00 878.0078.00 989.0089.00 1078.0078.00
II69.0069.00 1276.0076.00 1378.0078.00 1466.0066.00 1577.0077.00
1687.0087.00 t I " 1 i \ I I I I ( ;.. WorkingMechanics of
theTaguchiDesign of Experiments89 Table 5-28-6.Casting Process
Optimization Design Main Effects \COL NUMBERFACTORNAMELEVELILEVEL
2(Lz- LI>LEVE;L 3LEVEL 4 1Sand
compaction70.5078.508.0078.0000.00 4Gating
type77.5075.00-2.5067.5000.00 5Metalhead76.2572.50-3.7500.0000.00
6Sand supplier76.2572.50-3.7500.0000.00 7Coating
type69.2572.753.5074.7580.75 10Sandperm75.2573.50-1.7500.0000.00
ItMetaltemperature75.0073.75-1.2500.0000.00 13Quench
type72.5076.253.7500.0000.00 15Gas level75.5073.25-2.2500.0000.00
andY (XIY"X2Y.,XIY2andX1YZ) ..select anythreeand labelthemas
statedbelow. callXIY.as(XY.r)1.i.e.,level1 of newfactor(XY) X2Yl
as(XYhi.e.,level2 of newfactor(Xy) XtY2 as(XYhi.e .level3 of
newfactor(XY). Note that one combination, X2Y2,is not included.
With factorXY assigned, 40A isshownin
Table5-29.Fromthearray.thetrialrunconditions
definedfortrialnumberI(row1)areAlBICI (XY) Iwhere(XY)tis
'XIY),whichwasdefinedabove. 82 80 78.50 17.5 76 ~74 0 i72 a: 70
70.50 6869.25 66 67.5 64 L,L2L3L4 CoatingSand CompactionGating
Figure5-12.Plotsof maineffects-Example 5-10. 90A Primer on the
Tagucbi Method Table 5-29.L9withFive Factors fACfoasABC(X)
NUMBERICOWMSI234 Y- t.t III1I \ 2I222 1.-1 3I3331 'Z. '4 2I23 1-'Z
52231 1- 1 623I2 'l.1 73I3~ 1.- 1 832t3 , ~ 9332t , 1 The total
data are analyzed with the two fadors X and Y treated as one
(XY).The analysis yields themain effect of (XY).The individual
effect of theconstituents, X andY isthen obtained as follows: Main
effect of X= (XY)t- (XYhand Main effect of Y =(XY)I- (XY)3 Notethat
the firstequation hasY constant as Y.and the second equation has X
constant as XI' After detennining themaineffects,theoptimum
condition,including the levels of the two factors used in
combination design, can be identified.
However,theinteractioneffectsbetween factorsX andYcannotbeob-tained
from the data by this method.Should interaction be important,the
experiment designmustbebased on alarger arraysuch
asL16-5-9DESIGNING EXPERIMENTS TO INVESTIGATE NOISE FACTORS
Throughoutthistextthetennsfactors.variablesandparametersare
synonymouslyusedtorefertofactorswhichinfluencetheoutcomeof
theproduct or processunderinvestigation.Taguchifurthercategorized
the factors as controllable factors and noise factors.The factors
identified for the baking process experiment, namely sugar, butter,
eggs, milk,and flour,wereeasilycontrolledfactors.Other
factorswhicharelesscon-trollable,suchas oventemperature
distribution,humidity,oventemper-t I i 1 , It i .-- - ..... - ---
-- ......_.-WorkingMechanics of theTagucbiDesignefExperiments91
aturecycleband etc.,mayalsoinfluencetheoptimumproduct.
Sincethegoalisaro'ffustoptimumwhichisinfluencedminimallyby
theseless controllable variables, thestudy of theimpact of noise
factors ontheoptimumparametersisdesirable. Taguchi fullyrecognizes
the potentialinfluence of uncontrollable fac-tors.Noattemptismadeto
removethemfromtheexperiment.Before describinghowthe
factorsaretreated,acWitionaldefini-tionsareneeded.
ControllableFactors-Factorswhoselevelscanbespeeifiedandcon-trolledduringtheexperimentandinthefinaldesignof
theproductor process.
NoiseFactors-Thesearefactorswhichhaveinfluence ontheproduct
orprocessresults,butgenerallyarenotmaintainedatspecificlevels
duringtheproduction processor applicationperiod. Inner Array-The OA
of the controllable factors.All experiment designs discussedtothis
point fallinthis category. OuterArray-The OAof
recognizednoisefactors.TIletermouteror inner refers to the usage
rather than the arrayitself, as wiD be clear soon. Experimenl-The
experimentreferstothewholeexperimentalprocess.
TrialCondition-Thecombinationof factorslJevelsat whicha trialrun .
isconducted. Conditionsof Experiment-Uniquecombinationsof
factorlevelsde-scribedbytheinner array (orthogonalarray).
ReRetitions or Runs-These define thenumber of obsemWons under the
same conditions of an experiment. Theexperimentrequiresaminimumof
onerunper conditionof the experiment.But
onerundoesnotrepresenttherange of possiblevari-ability in the
results. Repetition of runs enhances the available information
inthedata.Taguchisuggestsguidelinesforrepetitions.
5-10BENEFITINGFROM
Forsomeexperiments,trialrunsareeasilyandinexpensivelyrepeated. For
others,repetitions of testsare expensiveaswellas timeconsuming.
Wheneverpossibletrialsshouldberepeated.particularlyif strongnoise
92A Primer on the TagucbiMethod
factorsarepresent.Repetitionoffersseveraladvantages.First,thead-ditional
trial data confirms the original data points. Second, if noise
factors varyduringtheday.then.repeatingtrialsthroughthedaymayreveal
their influence. Third. additional data can be analyzed for
variance around atarget value. When the cost of
repetitivetrialsislow,repetition ishighly desirable. Whenthecostis
high,orinterferencewiththeoperationishigh,then the number of
repetitions should be determined by means of an expected payoff for
the added cost. Thepayoff can be the deveJopment of amore robust
pmduction procedureor process, or bytheintroduction of apro-duction
processthatgreatlyreducesproductvariance. Repetition permits
determination of a. variance index called the Signal. toNoise
Ratio(SIN).Thegreaterthisvalue,thesmallertheproduct
variancearoundtbetargetvalue.Thesignaltonoiseratioconcepthas been
used in the fieldsof acoustics, electrical and mechanical
vibrations, andother engineeringdisciplinesformanyyears.Itsbroader
definition and application will be coveredinChapter 6.The basic
definition of the SIN ratio is introduced here. 5-11DmNITION OF
SINRATIO SIN=-10 LoglO(MSD) Where
MSD=Meansquareddeviationfromthetargetvalueof
thequalitycharacteristic.
Consistentwithitsapplicationinengineeringandscience,thevalue of SIN
isintended to belarge,hencethevalue of MSD should be small.
Thusthemeansquareddeviation(MSD)isdefineddifferentlyfor each of the
three quality characteristics considered, smaller, nominal or
larger. For smaDer is better: MSD= (y!+ n + yj+.. .)/n For
nominalisthebest MSD= YI- m)2+(Y2- mf+.. .)/n For bigger isbetter
MSD=(llyi+l I y ~+l ! y ~+.. .)/n Working Mechanics of the
TagucbiDesign of Experiments93 Where Yl,Y2,etc.=Resultsof
experiments,observationsor qualitycharacteristics suchaslength,
weight,surface finish.etc. m=Target value of results(above) n=
Number of repetitions (Yi) an experiment with three
repetitions,using an L4orthogonal arrayasin Table 5-30.
Inthetable,trialnumberIisrepeatedthreetimeswithresults5,6, and7.The
average of these three is 6. The averageis used for thestudy of
themain effects in a manner similar to that described for
nonrepeated trials.Slight differences in the analysis of variance
for the repetitive case is covered in Chapter 6. For experiments
with repetitions, analysis should alwaysusethe SIN ratioscomputed
asfollows. Assume that bigger is better isthe quality
characteristic sought by the experimental data of Table5-30.Then,
MSD=(l/yi+ +lin +.. .)/n Now,forrow1, yi=5x5=25 rl =7x7=49 = 6x6=
36 . -n= 3 Therefore,MSD=(1/25+1136+1149)/3 =
(.04+.02777+.020408)/3 - .0881-85/3 orMSD=.029395,a smallvalue.
Table 5-30.L4with Results and Averages TRIAL NUMBER
COLUMN23R,R:R3AVERAGE III15676 2I223454 32I27898 42224565 94A
Primer onthe TagucbiMethod Table 5-31. L ..withResults. and SIN
Ratios TRIAL SIN COLntN3RlR2R)RATIO 1I1I56715.316 212234511.47
32I278917.92 422245613.62 The SINratio is calculated as: SIN=- 10
1.og1o (MSD) - -10 1.oglo (.029395) - 15.31 SIN valuesfor all
rowsareshown in Table 5-31. In.
theSINratioistreatedasasingledatapointateachof thetest run
conditions.Normalprocedure for studies of themain effects
willfollow.The only differencewillbe in theselection of theoptimum
levels.In SIN analysis,thevalue of MSD or greatest valueof SIN
rep-resentsamoredesirablecondition. 5-12REPETITIONSUNDERCONTROLLED
NOISE CONDITIONS Repetitionsshowthevariationof theproduct or
process.Thevariation occurs principally as a result of the
uncontrollable factors (noise factors). By expanding thedesign of
theexperimenttoincludenoisefactorsina controlled manner, optimum
conditions insensitive to the influence of the
noisefactorscanbefound.TheseareTagucbi'srobustconditionsthat
control production close to the target value despite noise in the
production process.Toincorporatenoisefactorsintothedesign of
theexperiment
thefactorsandtheirlevelsareidentifiedinamannersimilartothose used
for other product and process factors (control factors).For
example, if humidity is considered noise, the low and high levels
may be considered a factor for the design. After determining the
noise factors and their levels
forthetest,OAsareusedtodesigntheconditionsof thenoisefactors
WortingMechanics of the Taguchi Design of Experiments9S
whichdictatethenumber of repetitionsforthetrialruns.The OAused
fordesigningthenoise experimentiscalled anouter array.
Assumethatthreenoisefactorsareidentifiedforthecakebaking
experiment(Tables5-10and5-14)whichutilizedan LsOA.Thenoise
factorsaretobeinvestigatedat2levelseach.Therearefourpossible
combinations of these factors.To obtain complete data, each trial
run of Lsmustberepeated ,for each of
thefournoisecombinations.Thenoise arrayselectedisanL4OA.Thisouter
array,withfour combinations of noiseof thethreenoisefactors,tests
eachof the8 trialconditionsfour
times.Theexperimentdesignwithinnerandouter arrayisshownby Figure
5-13.Note that for the outer array,column 3 represents both the
thirdnoise factor and the interaction of thefirstand secondnoise
factor. Notealsothearrangementof eacharray,withthenoise(outer)array
perpendicular to the inner array.The complete design is shown by
Figure 5-14. For most simple applications,the outer array describes
the noise c0n-ditionsfortherepetitions.Thisformalarrangement of
thenoisefactors and the subsequent analysis influences the
combination of the contronable M .,.. Outer Array NN -~N B i .-
N...N u...-- -NN fA ~ VI c: Inner Array j.- NMq-8
CoatroIFactorsResults ~ Number t2345671234 Experiment Number 1
1111111 2 1112222 3 1221122 4 1222211 52121212 6 2122121 72211221 8
2212112 Figure5-13.Inner andouter orthogonal arrays. , . I' :
96APrimer ontheTagucbiMethod zm c:)( a . ~ ~ I :3 ..
(D"""Cl)U1","WW-~ ~ 3E" 13 Factor LevellLevel2 Description
NIUNN----" -Egg2 EggslEg NlW--NN- ....NMilk2 CupslCups --WWNW--W .
NI-W-w ....w.... "'" Butter1 Stick1.5Stieks -w ....WW-Nl- U1 Flour1
Extra Scoop2Extra Scoops -IUW--WW-Cl) N- ... W ....WW- .....Sugar1
Spoon2 Spoons . Type ofBaking OvenTimeHumidity 1.Gas1.+Smin.1.BO%
2.Electric2.- 5 min.2.60% Columns123 :D 1111 :D "" 2122 :0 ...3212
::0 ~ 4221 Figure5-14.Cakebaking experimentwithnoisefactors.
factorsfortheoptimumcondition.Theuseof SINratioinanalysis,is
strongly recommended. 5-13DESIGNANDANALYSISSUMMARY Applicationof
theTaguchitechniqueisaccomplishedintwophases: (I)designof
theexperiment,whichincludesdeterminingcontrollable
andnoisefactorsandthelevelstobeinvestigated,whichdetermines
thenumberof repetitions,and(2)analysisof theresultstodetermine
thebestpossiblefactorcombinationfromindividualfactprinfluences
andinteractions.Thetwoactivities,experimentdesignandanalysisof
Working Mechanics of theTagucbi Designof Experiments97 t Experiment
I Design I I1 Simple DesignUsing Designs with Mixedlevels Standard
Arraysand Interactions I AssignFactors to ColumnsModify Columns as
AppropriateAssignFactors Requiring Level Modification
AssignInteracting Factors Assign All Other Factors I I Consider
NoiseFactor Determine NoiseCondition Using Outer Arrav Determine
Number of Repetitions I Run Experiments InRandom Order When
Possible Figure5-15.Experiment designflow diagram. NominalIsBest
Smaller Is Better BiggerIsBetter Figure5-16.Analysisflowdiagram. t.
98A Primer on the Taguchi Method
testdataarepresented(flowcharts)inFigures5-15and5-16.Thesteps
involvedarebrieflydescribedhere. Design
Dependingonthefactorsandlevelsidentified,followoneof thetwo paths
(Fig.5-16).If allthefactorsareof thesame say2, one of thestandard
OAs can probablybe used.Inthis case,thefactorscan be assignedto
thecolumnswithoutmuchconsiderationaboutwherethey should be
placed.On the other hand,if the factors requiremanylevels, or one
or moreinteractionsaretobeinvestigated,thencarefullyselect
certainspecificcolumnsforfactorassignmentsorlevelchanges.No matter
howsimple thedesign,the applicable noise conditions should be
identified,andasecondarray(outerarray)selectedtoincludenoise The
number of repetitions will be dictated by the number of noise
factors.In the absence of a formallayout such as Figure 5-13 the
number of repetitionswillbeinfluencedbytimeandcost considerations.
Analysis Analysis of resultsfollowseitherpaths(Fig.5-16)of
repetitionsor no
repetition.Generally,forasingleobservationforeachtrialcondition,
thestandardanalysisapproachisfollowed.Whentherearerepetitions of
thetrialruns,whetherbyouterarraydesignednoisecondition,or under
randomnoisecondition,SINanalysisshouldbeperformed.The
finalanalysisfortheoptimumconditionisbasedononeof thethree
characteristics greatest,smallest,or nominalvalue of qUality.
EXERCISES 51.Identify each element (8.2.7, etc.) of
thenotationforanorthogonal array Ls(27).
5-2.DesignanexperimenttostudyfourfactorsA.B.C.andDandthreeinteractions
AxC.CxD and AxD.Select theorthogonal arrayandidentifythecolumns for
thethreeinteractions. 5.;3.Anexperimentwith three2
levelfactorsyieldedthe foBowingresults.Determine the average effect
of factor C at levels C Iand C2 TRIALSi ABCRESULTS 111130 212225
321234 422127 -">->------------------------. WorkingMechanics
of the TaguchiDesign of Experiments99 Table 5-31.DesignVariables
and Their Levels \COLUMN NUMBERVARIABLE NAMESLE\U-ILEVEL2
1Speed2100RPM250 RPM 2OilviscosityAtlowTPAt hight
3Interaction1x2NlANlA 4ClearanceLowHigh ;.
5PinstraightnessPerfectBend 6(Unused)MIU 7(Unused)MIU 5-4.Describe
the procedure you will followto design an experiment to study one 3
level factorandfour2 levelfactors. 5-5.In an experiment involving
piston bearings, an La OA was used in a manner sbown in
Table5-32.Determine the description of thetrial number 7.
5-6.Theaverageeffectsof
thefactorsinvolvedinProblem5areassbowninTable 5-33. If the quality
characteristic is "the bigger the better. to determine (a) the
optimum condition of the design.(b)thegrand averageof
performance,and (c)theperfor-manceat the optimum condition.(Ans.
(b) 35.01(c)37.03) Table 5-33.Average Factor Eft'eds COLUMN
NUMBERVAlUABLE NAMESLEVEL-ILEVEL-2 1Speed34.3935.63
2Oilviscosity35.5034.52 3Interaction.lx233.6036.42>
4Clearance35.6234.40 5Pinstraightness35.3134.70 -6 Analysisof
Variance(ANOV A) 6-1. THE ROLE OF ANOVA Tagucbi replaces the full
factorial experiment with a lean, less expensive,
faster,partialfactorialexperiment.Taguchi's designfor
thepartialfac-torialisbased on specially developed OAs.Since
thepartial experiment isonlyasampleof
thefullexperiment,theanalysisof
thepartialex-perimentmustincludeananalysisof theconfidencethatcan
beplaced inthe results.Fortunately,thereisa standard statistical
technique called AnalysisofVpce (ANOV
A)whichisroutinelyusedtoprovidea measure of conaence. The technique
does not directly analyze the data, butrather
determinesthevariability(variance)of thedata.Confidence
ismeasuredfro.. thevariance. Analysisprovidesthevarianceof
controllableandnoisefactors.By understandingthesourceandmagnitudeof
variance,robustoperating conditions can be predicted. This is a
second benefit of the methodology.
6-2ANOVATERMS,NOTATIONSANDDEVELOPMENT Intheanalysis of
variancemanyquantitiessuchasdegrees of freedom .sumsof
squares,meansquares,etc.,arecomputedandorganizedina standardtabular
fonnat.These quantitiesand their interrelationshipsare defined
belowand theirmathematicaldevelopmentispresented. DEfINmONS: C .F.
= Correction factor e =Error (experimental) F =Varianceratio 1=~ ~
s o f ~ o m Ie =Degreesof freedomof error h=Totaldegreesof freedom
tOO n=Number of trials r=Number of repetitions P=Percent
contribution T= Total(of results) S=Sumof squares S'= Puresumof
squares V=Meansquares(variance) \ J \ I J I "- -_.-
-------------------------" Analysisof Variance (A."IIOVA)(O( Total
Number of Trials In an experiment designed to detennine the effect
of factor A on response Y.factor Ais to be tested at L
levels.Assume n 1repetitions of each trial that includes A
1.Similarly at level A2 the trialis to be repeated 112times. The
total number of trials is the sum of the number of trials at each
level. i.e. , Degrees of Freedom (DOF) OOF is animportant and
usefulconcept that is difficult to define.Itisa measureof
theamountof infonnationthatcanbeuniquelydetermined fromagivensetof
data.OOFfordata concerning a factor equalsone lessthanthenumber of
levels.For afactorAwithfourlevels, A Idata can becomparedwith
A2,A3and ~dataandnotwithitself.lbus a4 levelfactorhas3
OOF.SimilarlyanL4OA with threecolumnsrepre-senting2
levelfactors,has 3 OOF. The concept of OOF can be extended to the
experiment. An experiment withn trialand rrepetitions of each
trialhasnxrtrial runs.Thetotal OOF becomes: fT=nXr-1
Similarly,theDOF for asum of squarestenn is equal to the number of
tenns usedtocompute thesum of squaresandtheDOF of theerror tenn
Ieisgivenby: ft!=IT- fA- fs- Ie Sum of Squares Thesumof
squaresisameasureof thedeviationof theexperimental data
fromthemeanvalue of the data.Summingeachsquared deviation
emphasizesthetotaldeviation.Thus n ST='L (Yj- Y)2 i= 1 Where Y
istheaveragevalueof Yio '-' t I unA Primer ontheTaguchiMethod
Similarly thesumof squares of deviations ST,from a target
valueYo,is givenby; n ST=L(Y,- y)2+n(Y- Yo)2(6-1-1)* ;= I
Variauc:emeasures' thedistributionof thedata aboutthemeanof the
data.Since thedataisrepresentative of onlya part of aUpossibledata,
DOF radler than the"number of observationsisusedin thecalculation.
Sum of Squares Variance-Degreesof Freedom orV =Sri!
Whendieaveragesumof squaresis calculatedabout themean,itis c a l l
~generalvariance.Thegeneralvariance a2isdefinedas. 1n a2= - ~(Yi -
y)2 .ni-I (6-1-2) *ST= ~(Y;- for ;=1 .. = L (Y;- Y + Y - YO)2 i=1
II =~[(Y;- y)2+ 2(f;- Y)(Y- Yo)+ (Y- YO)2] .,,"II = 2 (Yj- Yr+
L2(Y;- n(Y - Yo)+~(Y- Yo)2 i-I;=1 .... since~(Vi- 1) = ~Yi- L Y =
nY- nY= 0 ;=1i=J and~(j - Yor=n(Y- fo)2 pi The above
equationbecomes, .\ ST= L (Y.- n2 + n(Y- for i=1 'I 1 --- - - ~ - -
- - - - - - - - - . Analysisof Variance(ANOVA)103 Let m represent
thedeviationof themeanY fromthetargetvalueYo, I.e. , m=(Y-
Yo)(6-1-3) Substituting Eqs.(6-1-2)and(6-1-3)intoEq.(6-1-1),
(6-1-4) Thus the totalsum of squares of deviations (ST)from the
target valueYo is the sum of the variance about the mean., and the
square of the deviation ofthemem
fromthetargetvaluemultipliedbythetotalnumberof
observationsmadeintheexperiment. STof
Eq.(6-1-4)alsorepresentstheexpectedstatisticalvalueof
ST-Inthisbook,rigorousproofsareomitted unlessnecessarytoclarifyan
ideaorconcept.Further,thesymbolSTisusedforboththeexpected valueand
thecomputedvaluefor agivensample. Thetotalsum of squares
ST(Eq.6-1-4)givesan estimate of thesum of thevariationsof
theindividual observationsabout the mean Y of the
experimentaldataandthevariationof themeanaboutthetargetvalue
Yo.This information is valuable for controlling manufacturing
processes, as the corrective actions to reduce the variations
around the mean )7,i.e.,
toreduce0'2,areusuallynotidenticaltothoseactionswhichmovethe
meantowardthetargetvalue.Whenthetotalsumof squaresST,is
separatedintoitsconstituents,thevariationcanbeunderstoodandan
appropriatestrategytobringtheprocessundercontrolcanbeeasily
developed.Furthermore, the information thus acquired can be
effectively utilizedinStatisticalProcessControl (SPC). Mean Sum (of
Deviations) Squared n LetT=I(Yi - Yo)thesum of
alldeviationsfromthetargetvalue. i= 1 Then,themeansum of squaresof
thedeviationis: S.... =T2/n=[ ~(Y;- YOI]2/n(6-2) 1 ! 104APrimer
onthe TaguchiMethod Eq.(6-2)canthusbewrittenas: It
isimportanttonotethateventhoughfromanover-simplisticderi-vation of
thevalue of Sm= nm2,itsstatistical estimate or theexpected
value,includes one part of the generalvariance.Therefore,
representing thestatistically expectedvalueby E(Sm): (6-3)
Thetenn(ST- Sm>isusually,referredtoastheerror sumof squares
andcan be obtained from' Eqs.(6-14) and (6-3). Therefore,
RewritingSr= Se+ Sm.Thus tbe total effect of variance
STcanbedei!bmposedintothemeandeviationSmandthedeviationSe
aboutthemelt. Thusindividualeffects can be analyzed.Let Y1 -Yo= 3Y4
- Yo= 4 Y2- Yo= 5Ys - Yo= 6 Y3 - Yo= 7Y6 - Yo= 8 1 *5",==- (Y1 -
Yo)+ ... + (Ya- Yo)}2 n Whichcan also be expressed as: or or 1
5",=- [(Y.+ Y2 +... + Y..- nYu)]] n 1-5",==- [(nY- nYo)]2 n n2_
5",= - [(Y- YO)]2 n ;.. whereYoisatargetvalue,then and = 199
S",=(3+ 5+7+ 4+ 6+ 8)2/6 .332/6 = 181.5 Y= (3+ 5+7+ 4+ 6+ 8)/6 =5.5
Se= [(3- Y)2+(5- y)2+... ] Analysisof Variance(ANOVA,lOS =[(3-
5.5)2+ (5- 5.5f+ .... (8- 5.5)2) =17.5 Notethat Se= ST- S'"=199-
181.5=17.5 Also,sincethestandarddeviationof
thedata3,5,7,4,6,and8,is equalto1.8708. Se= (n- l)',., ~ l . =(6-
1)x(1.8708)2 =17.5 Degrees of Freedom Sums The DOF Ie,/T, and/mof
the sum of squares Se, ST.andSm are as follows: /T'= n= number of
datapoints. fm=I(alwaysforthemean) Ie=IT- 1m=(n- I) 106APrimer
ODiii: TaguchiMethod Aspointed rot earlier,theDOF ITisequalton
becausetherearen independentvaluesof (Yj- YO)2.For
investigatingtheeffect of factors
atdifferentletels,theOOFareusuallyonelessthanthenumberof
observations. Tosummarize: Sr=ncr+nm1 Sm= cr+ nm2 Se=Sr- Sm=(n-
1)0-2 Alsoasstated earlier,varianceV,is V=SII Therefore: VT
=SriITcr+ m2 Vm=S,,/I.cr+ nm2 Ve= (Sr - Sm)/Ie= cr (totalvariance)
(meanvariance) (error variance) (6-4) (6-5) (6-6) The example that
follows should clarify the application of the concepts
developedabcwe.The data for this examplearefictitiousbut sufficefor
thepurposeof illustrating theprinciples. 6-3ONE WAYANOV A One F8tor
ODeLevel Experiment
WhenonedimensionalexperimentaldataareanalyzedusingANOV A.
theprocedure is termedaonewayanalysisof variance.The following
problemisaDexampleof onewayANOY A.LaterANOY Awillbe
extendedtomulti-dimensionalproblems. Example 6.1 To obtain tke most
desirable iron castings for an engine block. a design engineer
wants to maintain the material hardness at 200 BHN. To measure
thequalityof dIecastings beingsuppliedbythefoundrythehardnessof
Aaalysis of Variance(ANOVA)107 Table 6-1.Hardness of Cylinder Block
Castings-Example 6-1 SAMPlEHARDNESS 1 2 3 4 5 240 190 210 230 220 6
7 8 9 10 ISO 195 205 215 215
10castingschosenatrandomfromalotismeasured,anddisplayedin Table
6-1. The analysis: IT=Total number of results- 1 = 10' - 1= 9
Yo=Desired value=200 themeanvalueis: Y= (240+190+210+230+
220+180+195+205 +215+ 215)/10 =210 then ST=(240- 200)2+(190- 200)2+
(210- 200)2 + (230- 200)2+ (220- 200)2+ (180- 200)2 +(195-
200)2+(205- 200)2+ (215- 200)2 + (215- 200)2 =4000 andSm=n(Y-
YO)2=10(210- 200f=1000 Se=ST- Sm=4000- 1000=3000 I I, 'j , I
1.APrimer on theTaguchiMethod Table 6-2.Analysis of Variance
(ANOVA)Table-Example 6-1 \:\RIA."ICEVARIANCEPUREPERCENT
SOUItCEOFSUM OF(MEA.'"SQUARE)RATIOSUM OF SQUARESCONTRIBUTION
VARIATIONfSQUARESvF 5' P Mean(m)I10001000.00 Error (e)93000333.33
Total104000 Andthevarianceiscalculatedasfollows: 10l.t" IH). D VT=
SrlfT= 4000/.9" = ~ Vm=1000/1=1000 V ~= (ST- Sm)//e=(4000- 1(00)/9=
333.33
TheseresultsaresummarizedinTable6-2.Table6-3representagen-eralizedformataf
theANOV Atable. Thedatacannotbeanalyzedfurther,but analysisof
thevarianceof thedata can provide additionalinformation about
thedata. LetFbetheratioof totalvariancetotheerror variance.Fcoupled
withthedegreesof freedomforVTandVeprovidesameasureforthe
confidenceintheresults.
Tocompletetheanalysis,theerrorvarianceVeisremovedfromSm andaddedtoS
~ .Thenewvaluesare renamed as S:"=puresum of squares S;=pure error.
Table 6-3.ANOV A Table for Randomized One Factor Designs-~Example
6-1 VARIANCEVARIANCEPUREPERCENT SOURCE OFSUMOF(MEANSQUARES)RATIOSUM
OF SQUARESCONTRIBUTION VAlUATIONfSQUARESVFs' p t:oan(m)1mSmSm1lm
r(e) h S ~S i / ~ Total IT_ST Analysisof Variance(ANOVA)109
Thisreformulationallowscalculationof thepercentcontribution,p.
forthemean,Pm,or foranyindividualfactor(P A.PB,etc.) Table
6-3presents the complete format for analysis F.Sand P. These
parameters aredescribedbelowingreater detail. VarianceRatio The
variance ratio, commonly called the F statistic, is the ratio of
variance duetotheeffect of'a factorandvariance duetotheerror
term.(The F statistic is named after Sir Ronald A.Fisher.) This
ratio is used to measure thesignificance of the factor under
investigation with respect to the vari-anceof
allthefactorsincludedintheerror term.The Fvalueobtained
intheanalysisiscomparedwithavaluefromstandardF -tablesfora
givenstatisticallevelof significance.The
tablesforvarioussignificance levelsand different degrees of
freedomareavailableinmost handbooks of
statistics.TableC-lthroughC-5inAppendixA providesabrief list of
Ffactorsforseverallevelsof significance.
TousethetablesentertheOOFof thenumeratortodeterminethe column and
theOOF of the denominator fortherow.The intersectionis the F
value.For example, the value of F.l (5,30) from the table is
2.0492, where5and30aretheDOFofthenumeratorandt,hedenominator,
respectively.When the computed F value is less than the value
determined fromthe F tables at the selected level of
significance,the factor does not - :contribute to the sum of the
squares within the confidence level. Computer software,such
asReference11. simplifiesandspeeds thedetennination of thelevelof
,significance of thecomputed F values. The Fvaluesarecalculated by:
Fe= V/Ve =1(6-7) andforafactor Aitisgivenby: (6-8) Pure Sum of
Squares InEquations (6-4).(6-5). and (6-6) for each of the sum of
squares,there isa generalvariance(J'2tennexpressedas(DOF)x(J':.
Whenthistermissubtractedfromthesumof squaresexpression.the -j I ,
.. ., ; 110A I'bmer ontheTaguchiMethod remainder is called the pure
sum of squares.Since Smhas only one degree of freedom,it therefore
contains only one a2i.e., VeeThus the pure sum of square forSmis:
The portionof errorvariancesubtractedfromthesumof squaresfor
Smisaddedtotheerror term.Therefore, (6-9) If
factODA,B,andC,havingOOF lA,Is,and Ieareincludedinan
experimmt,their puresum of squaresaredeterminedby: S ~=SA- IAXVe
S8=Sa- laXVe Sc=Sc- IeXVe S ~=Se+ (fA+ Is+ Ie>XVe Perce.
Contribution (6-10) (6-11) Thepm:ent
contributionforanyfactorisobtainedbydividingthepure sum of
squaresforthatfactorby S, andmUltiplyingtheresultby100. Thepen:ent
contribution is denoted by P and can becalculated using the
followiDgequations. Pm= - S:"XlOO/ST Pit=S ~Xl00/ST Pa =S8XlOO/ST
Pc= ScXl00/ST Pe = S;XtOO/ST (6-12) TheAHOV
ATable6-2cannowbecompletedasfollows:UsingEqs. (6-7)and (6-8)gives:
Fm= VmlVe=1000/333.33=3.00 F1=V ~ ,V ~=333.33,333.33=1.00 '/ ____
.-....- - 1-Analysisof Variance(A-';OVA)III Table 6-4.Analysis of
Variaace (ANOV A)Table-Example 6-1 VARIANCEVARIANCEPUREPERCENT
SOURCE OFSUM OF(MEAN SQUARES} RATIOSUMOFSQUARESCONTIUBtmON
VAlUATIONfSQUARESVFS'p Mean (m)110001000.003.00666.6716.67 Error
(e)93000333.331.003333.3383.33 Total104000100.00 The pure sum of
squares obtained using EqS.(6-9) and (6-10) is shown below: S:"=Sm-
Ve=1000- 333.33=666.67 S;=Se+Ve=3000+ 333.33=3333.33 Andthe percent
contribution iscalculatedusingEqs.(6-11)and(6-12):
Pm=S:"XlOO/ST=666.67/4000=16.67 Pe = S;XlOO/ST= 3333.33/4000=83.33
ThecompletedANOV A tables are showninTable 6-4. A generalized ANOV
A table for one factor randomized design is shown in Table 6-5.
Returningto Table 6-3, the computedvalue for Fm.3.00. is less than
thevaluefromTableC-lforF.I(1,9)i.e.,3.3603.Hencewith90%
confidence(10%risk)thecastingsappeartobesimilar.Theapparent
dataspreadcontJ:ibutesonly tothesamplevariability(sumof Table
6-5.ANOVA Table for Randomized One Factor Design-Example 6-1 SOURCE
OFVARIANCEVARIANCEPUREPERCENT VARIATIONSUM OFIMEAN
SQl'ARESlRATIOSt:M OFSQUARESfSQUARESvFS'PIOO Mean(m) 1mSIftVm=
S,.,IlmVSm-S' ".1ST Error(e) Sf! = Sm+ S',JST Total ITST I , IIIA
Primer onthe TaguchiMethod
squares)whereastheremaining83.33%variationiscausedbyother factors.
64ONE FACTOR TWO LEVEL EXPERIMENTS (ONE\\'AY A.'lOYA) Example ,.2
In Example6-1anexperiment withonefactoratonelevelwas con-sidered,
the factor being the hardness of the cylinder blocks being supplied
by one source. Now consider the case with two different vendors
suppling thecastings.Thesetwosourcesareassumedtouse similar casting
pro-cesses.Therefore,anew experiment is describedwith
onefactor,hard-ness of from two sourcesA. andA10The question to be
resolved is whether the castings being supplied by the two vendors
are statistically of the same qUality.If not,which one is
preferable. The target hardness, 200 BHN,isunchanged. Ten
fromeachof thetwocastingssourcesweredrawnat
randomandureirhardnesswasmeasured.Thetestyieldedtheresults shownin
Table 6-6. Theanalysisof thistestproceedsasfortheexperimentof
Example 6.1.Notethat the error sum of squares term, Se,as given in
Eq.(6.11), containsthevariation of themean andthat of the factor A.
Therefore to separatethe effect of vendors,i.e., factor A.the sum
of squares term SA mustbeisolatedfromSt!.Thesumof
squaresforthefactorAcanbe calculatedby: (6-13) n -: Table
6-6.Measured HardDess-Example 6-2 HARDNESS OF CASTINGSFROMHARDNESS
OF CASTINGS FROM VENOORAIVENDOIlA2 140180190191202198
195210205205203192 230215220208199195 115201 Analysisof
VIria.nce(ANOVA)113 Where, L= n u ~ b e rof levels ni,nk= number of
testsamplesatlevels Aiand Ab respectively T=sum totalof
alldeviationsfromthetarget value n= totalnumber of observations=nl+
n2+ ... + nj ~ . The Pin in Eq.(6-13) is a termsimilar to Smand is
called the correction factor,C.F. The expressionforthetotalsum of
squares cannow bewritten as (-14)* TheDOF equationwillbe: The
Analysis: Yo=200 (unchanged fromExample6.1) ST=(240- 200f+(180-
200)2+... +(215- 200)2 + (197- 200)2+... + (195- 2(0)2 + (201-
2(0)2- C.F. =4206- 500=3706 AsT2/n= C.F.= [(YJ - Yo)+ (Y2 - Yo)+...
FIn =[(40- 20+. . .- 15+15) + (- 3+2 ... - 5+1)]2/20 =500
*Taguchiconsidersde\'iationfromthetargetmoresignificantthanthat
aboutthemean.Thecost of quality is measured as afunction of the
deviations fromthe target.Therefore. Taguchi eliminates
thevariationaboutthemeanfromEq.(6-14)by redefining51asfollows: "
ST= :2(Yi- YO)2- C.F.= Se+ SA 1=1 114A Primer onthe TaguchiMethod
Using Eq.(6-13),thevalue of SA,thesquare sum for the effect of
factor A(venl+ (Y2 - fo)2+ ... +(YN - Y o ) ~ / N(6-18) The bigger
is better qualitycharacteristic: MSD= (tIff+I I Y
~+...+IIY"')IN(6-19) The Mean 8cJIaled Deviation (MSD) is a
statistical quantity that reflects
thedeviationft9mthetargetvalue.TheexpressionsfortheMSDare
differentfordiJerentqualitycharacteristics.Forthenominalisbest
characteristic, ttte standard definition of MSD isused.For the
other two characteristics,thedefinitionisslightly modified.For
smaller isbetter, theunstated target value is zero.For larger
isbetter,the inverse of each largevalue becomes a small value and
again,the unstated target iszero. Thusfor all three MSD
expressions.thesmallestmagnitude of MSDis being sought.In turn this
yields the greatest discrimination between
con-trolledanduncontrolledfactors.ThisisTaguchi'ssolutiontorobust
product or process Qesign. Alternate forms of definitions of the
SIN ratios exist (Ref.7, pp.172-173), particularly for the nominal
is the best characteristic. The definition in termsof MSD is
preferred asit is consistent with Taguchi's objective of
reducingvariationaroundthetarget.Conversionto SINratiocanbe.
viewedas ascaletransformationforconvenienceof better
datamanip-ulation. 6.. 8-2Advantage of SIN Ratio over Average To
analyze the results of experiments involvingmultiple runs,use of
the SINratiooverstandardanalysis(useaverageofresults)ispreferred.
Analysis using the SIN ratio will offer
thefollowingtwomainadvantages: Analysisof Variante(ANOVA)147 I.It
provides a guidance to a selection of the optimum level based on
least variation around the target and also on the average value
closest to thetarget., 2.It offers objective comparison of two sets
of experimental data with respect to variation around the target
and the deviation of the average fromthetarget value. To examine
how the SIN ratio is used inconsider the following twosetsof
observations whichhaveatargetvalueof 75. Observation A: 555860
6365Mean=60.2 Deviationof meanfromtarget=(75- 60.2)=14.8
Observation B:50 60 76 90100Average=75.00 Deviationof
meanfromtarget=(75- 75)= 0.0 Thesetwosetsof
observationsmayhavecomefromthetwodistri-butions shown in Figure
6--3.Observe that the set B has an average value Target Value
Figure1-3.Comparisonof(we --, i 148APrimer onthe TagucbiMethod
whichequalsthetargetvalue.buthasawidespreadaroundit.Onthe
otherhandforthesetA.thespreadarounditsaverageissmaller,but
theaverageitself isquitefarfromthetarget.Whichone of thetwois
better?Basedonaverage\"alue,theproductshownbyobservationB. appears
to be better.Based on consistency, product Ais better.How can one
credit Aforless variation?Howdoes one compare the distances of
theaveragesfromthetarget?Surely,comparingtheaveragesisone
method.Use of the SIN ratio offersan objective wayto look at thetwo
characteristicstogether. 6-8-3Computation of the SIN Ratio Consider
thefirstof thetwosetsof observationsshownabove.That is set A:
555860 63 65. CaseI. Thenominalisthebest. UsingEq.6-18,and with
thetargetvalue of 75, ~ MSD= s5iil.;,..75)2+ (58- 75)2+(60-
75)2+(63- 75)2 +(65- 75)2)15 =(400+289+ 225+144+100)15 =lt58/5 =
231.6 therefore, SIN=-10 LoglO(MSD) SIN=-10xLogIC)(231.6) - -23.65
Case2.The smaller is thebetter. UsingEq.6-17, MSD= (552 +582
+6()2+632 +652)/5 = (3025+3364+3600+3969+4425)15 - 18183/5 : " .
~16 ":.6t) Analysisof Variance(ANOVAJ149 and SIN=- 10Log (3636.6) -
-35.607 Case3.The bigger thebetter. -UsingEq.6-19, MSD=(1/552
+11582 +116()2+11632 +11652)/5
=(113025+113364+1/3600+113969+114425)15 = (3.305+2.972+ 2.777+
2.519+ 2.366)xIQ-415 =(13.939)xIQ-415 - .0002728 therefore, SIN"
=-10 Log(MSD)=-10 Log(.0002787)=35.548 The three SIN ratios
computed for the data sets A and Bunder the three different
qualitycharacteristicsareshownin Table 6-24.The threecol-umns N,
Sand B,under the heading "SIN ratios" are for nominal., smaller,
andthebigger thebetter characteristics,respectively.
Nowselectthebestdatasetonthebasisof minimumvariation.By
definitionlower deviation isindicated by a higher value of the SIN
ratio (regardlessofthecharacteristicsof quality).If
thenominalthebetter characteristic applies, then using column N,the
SIN ratio for A is- 23.65 andis- 25.32for B.Since- 23.65isgreater
than- 25.32.,set Ahas lessvariationthan st B,althoughset B hasan
average value equal to the desiredtargetvalue. Table6-24.SIN
Ratiosfor Three QualityCharacteristics SiS RATIOS
OBSERVATIOSSAVERAGE .\-5B A:55586063656 O _ ~- :3.65-35.6\35.5..1
3'1' ")C -.::: :"Ie .00 -5.)- ..." ,....- 3- .-6 36.')5J. 150A
Primer on the TaguchiMethod Similarly,set
Aisselectedforthesmallerisbetter characteristicand B isselected
forthebigger isbetter characteristic. 6-8-4EfI'ect of the SINRatio
on theAnalysis Use of the SIN ratio of the results, instead of the
average values, introduces some minor changesin theanalysis.
Degrees of freedom of theentire experiment isreduced. OOF with SIN
ratio=number of trial conditions- 1 (i.e., Number of repetitionsis
reduced to '1) Recall thattheOOF inthecase of thestandard
analysisis OOF= (Number of trialxNumber of repetition)- I The SIN
ratio calculationisbased on datafromallobservationsof a Irial
condition. The set of SIN ratios can thenbe considered as trial
resultswithoutrepetitions.HencetheDOF,incaseof SINisthe number.
trials- 1.Therestof theanalysisfollowsthestandard procedure. SIN
DlUStbe converted back to meaningful tenns. When the SIN ratio
isused,theresultsof theanalysis,suchasestimatedperfonnance
fromthemain effectsor confidenceintervalareexpressedintenns of
SIN.To expressthe analysisin termsof theexperimentalresult, ther
must be converted back to the original units of measurement. To
seethespecificdifferencesintheanalysisusing theSINratio,let us
compare the two analyses of the same observations for the
Cam-Lifter NoiseStudyshowninTable6-25-1(standardanalysis)andinTable
6-25-2(SINratioanalysis).Inthisstudythethreefactors(springrate, cam
profile,and weight of thepush rod) each at two
levels,wereinves-tigated.The L4OA definea the {pllr trial
conditions:Atof thefour trialthree (in someof 0to 60)were
recorded.Theresultswerethenanalyzedbothwaysasshowninthese two
tables." Asubtable"results"of
thestandardanalysis(Table6-25-1)presents theaverage of
thethreerepetitions for eachtrialrunat the extremeright hand
column.The averagesareused incalculatingthemain effects.The
valuesshowninthesubtabtetitled"MainEffects"havethesaineunits
astheoriginalobservations,Similarly, "/a!UeItAr.alysisof
Variance(ANOVA)151 Table 6-25-1.Cam-Lifter Noise Study Standard
Analy. COLL'MNFACTORSlEVEL Jl.E\U:!LEVEL 3LEVEL 4 ISpring
rateCurrentProposed 2Cam profileType1Type 2 3Wt.of push
rodUghterHe.33. 36, 92, 146,149,162 Mean squares.50
Mixedlevels,40,58 Moldingprocess.45.75 Monthlysavings,165
Multipleruns,142 Noisecondition.94 176 experiment.95 Nonparallel,3
Objective evaluation.19 Off-line strategy. 8 quality improvemeDt.8
Oneway ANOVA.106,112 Operating condition,16 Optimization,I, 29
Optimum combinatioa. SSt86 condition,10,16. 29.33. 44. 48. 49, 94,
96 performance, SS treatment,I Ortbogonalarrays (OA).14,93, 187.
202, 206 Outer array.28,91, 9S Out-of-to1erance,12 Overall
ev..uation aitaia (OEC),19,178. 179 Pace maker,1 Parallel. 3
Parameter design,10. 24 Partialfactorial experiment,40, 41 Percent
charactmstics,13 Percent contributioa,SO.51.110,liS,123 Percent
influence.SS Pooled effects, 54 Pooling,124,134 Poundcake,26
Priceandperformance,I Problem solving.18 Product parameter, 8
Product warranty,18 Profitability,I Puresum of squares,
SO,51.109.111.123. 133 QUALITEK-3 software.125 Quality
characteristic.19,20.156.iS8 Quality strategy.23 Random order.46.
47 Relativeinfluence.30 contribution.50 significance.72 Rqetition.
41. 91 leplication. 47,121 Ieproducibility,41 Response customer. 2
mean.2 Robustcondition,9,17,91 SecondWorldWar,7,14
Sessionadvisor,174 Sipal to noise ratio (SIN),28,31, 33,36, 92,
96.145,148,ISO,153,1S4 Simulation studies.18 Sack food.2
Societalloss,II.14 Sony,12 Specification limit,11 Standard
analysis,126 approacb,16 arrays,30,58 deviation,33,lOS,163,164
Statisticalprocess control(SPC),32,103 Subjective evaluation,19
Sugar,2.3 Sums of squares,SO,liS Supplier tolerance,161,168
Systemdesign,10 Tagucbi pbilosopby. "8 Target value.9.IS8
properties.II Team approach,17,18 Teamleader.174 Technique,I
Totaldqrees of freedom,SO Tolerance design,10 levels.37
limits,167.168 Traditional practices, 7 Transmission control cable,
204 TreatmeDt.2 Trialcondition,14. 89. 98 Trialrun.89 Triangular
table,60, 61. 71.72 Two wayA."'lOVA.117 Uncontrollable factors.27
Up-front thinking.31 Variability.100 Variance,102 Index147
ratio.SO.53,109.liS,120.133 data.163.164 Variation.9.20,lIS II ~ "
I ~ j ojI