Copyript C'910 by VanNostrandReinhold Library ofCODpiI Catalog CardNumber 89-14736 ISBN 0442-2l1.l9-4 AUrights ~No part of dUswork covered by the copyright benra IB&Jbe JeptOduced or usedin anyformor by any ~ .electronic.or meehanical.including photocopying.lIICORIing.taping.or information storage and mrieval ~written permission of the publisher. Printed in the U'ai(ed States of America V. Nosttmd Icinhold lIS Fifth Av.. NC\1tYark. NYloom VanNosuanct ReinholdInternationalCompanyLimited II NewFetter lane LondonEC4P .tEE. England VanNostrand Reinhold 480 La Trobe Sleet Melbourne.YtCtoria 3000,Australia Nelson Canada 1120 Birclunotml Road Scarborough. 0aIari0 MIK 504. Canada 1615141312 It 10 98765432 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