TAGUCHI METHOD for DYNAMIC PROBLEMS SIGNAL - TO ...

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012TAGUCHI METHODforDYNAMIC PROBLEMSDr. P. R. ApteIIT BombaySIGNAL - TO - NOISE RATIO28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 2Dyn S/N Ratio - 1

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012NOISEXP – DIAGRAMOUTPUTYZCONTROL FACTORSP – Diagram for STATIC Problems28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 3QUADRATIC LOSS FUNCTION Q = K' [ (µ - µ o ) 2 + σ 2 ] Quadratic Loss FunctionIdeal measure for quality of products as it is shippedby the supplier to the customer Minimizing Quadratic Loss Functionleads to quality improvement Signal-to-Noise ratios areLog form of Quadratic Loss Function28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 4Dyn S/N Ratio - 2

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012QUALITY LOSS AFTER ADJUSTMENTFirst Variance is reduced (“σ 2 term” in 'Q')Then Mean is brought on Target without disturbingthe Variance (σ 2 ) by a Scaling FactorHence the correct Quality Loss measure is Q aQ a = K σ 2 for (µ=µo) and NOTQ = K [(µ - µ o ) 2 + σ 2 ]Hence always use Q aand not Q for any estimation of quality28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 5RELATIONSHIP BETWEEN S/N RATIO AND Q a Q = K' [ (µ - µ o ) 2 + σ 2 ] If mean is µ and is to be ‘adjusted’ to µ o ,Adjust Thickness to ‘ µ o / µ’ times the earlier values Q a = K' [ (µ . {µ o / µ} - µ o ) 2 + (σ . {µ o / µ} ) 2 ] Q a = K' [ (µ o - µ o ) 2 + (σ . {µ o / µ} ) 2 ] Q a = K' [ (σ . {µ o / µ} ) 2 ]28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 6Dyn S/N Ratio - 3

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012RELATIONSHIP BETWEEN S/N RATIO AND Q a Q a = K' [ (σ . {µ o / µ} ) 2 ] Q a = K‘. {µ o } 2 [ (σ / µ) 2 ] Q a = K (σ / µ) 2 where K, new constant = K‘. {µ o } 2 Q a minimizing is equivalent to maximizing {1/Q a } S/N Ratio, η = Log Form of {1/Q a }Q a’ = η = 10 Log 10 [ µ 2 / σ 2 ]28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 7TAGUCHI'S TWO-STEP PROCEDUREWhat you need to do?1. Maximize η = 10 Log 10 [ µ 2 / σ 2 ]2. Adjust Mean on Target without disturbing the Varianceby a Scaling Factor (eg. deposition time)Any target can be adjusted without having to reoptimize Unconstrained optimization Development of sub-systems to take place in parallel R & D time reduced28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 8Dyn S/N Ratio - 4

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012ROBUST DESIGN METHODOLOGY2 - STEP OPTIMIZATIONSTEP 1 :REDUCE VARIATION IRRESPECTIVEOF TARGET VALUESTEP 2 :ADJUST PERFORMANCE ON TARGETLEAVING VARIATION UNDISTURBEDFREQUENCY OR PERFORMANCE6050403020100STEP 2 STEP 1STARTm µPARAMETER SETTINGS28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N Ratio (dynamic) 9IDENTIFICATION OF SCALING FACTOR DOES EVERY CONTROL FACTOR HAVE EFFECT ON 'η' AND 'µ' ?FACTORS THAT HAVE SIGNIFICANT EFFECT ON 'η' ---> MAXIMIZE 'η'FACTORS THAT HAVE SIGNIFICANT EFFECT ON 'µ' BUT NO EFFECT ON 'η‘---> SCALING FACTORFACTORS THAT NEITHER EFFECT 'η' NOR 'µ'---> NEUTRAL FACTORS( OPERATIONAL EASE, COST ETC.)28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 10Dyn S/N Ratio - 5

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012S/N RATIOS FOR STATIC PROBLEMS TYPES OF STATIC PROBLEMS– SMALLER - THE - BETTER TYPE– NOMINAL - THE - BEST TYPE– LARGER - THE - BETTER TYPE MORE TYPES– SIGNED - TARGET TYPE– FRACTION DEFECTIVE TYPE– ORDERED CATEGORIES TYPE– CURVE OR VECTOR RESPONSE TYPE28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 11KEY ACTIVITY OF ROBUST DESIGNIS TO :– DETERMINE CORRECT S/N RATIOSO AS TO ACHIEVE ADDITIVITY– DETERMINE PROPER ADJUSTMENTFACTORS TO BRING MEAN ONTARGET WITHOUT DISTURBINGVARIANCE28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 12Dyn S/N Ratio - 6

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012SMALLER - THE - BETTER TYPE PROBLEMQUALITY CHARACTERISTICS( CONTINUOUS AND NON-NEGATIVE )' 0 ' MOST DESIREDNO SCALING OR ADJUSTMENT FACTOR AS TARGET ISALWAYS ZEROVARIANCE TENDS TO ZERO AS MEAN TENDS TO ZEROMINIMIZE QUALITY LOSS WITHOUT ADJUSTMENTOR MAXIMIZE= – 10 Log101nO TO OO222( Y + Y + ... + Y )1 2 n28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 13NOMINAL - THE - BEST TYPE PROBLEMQUALITY CHARACTERISTICS( CONTINUOUS AND NON-NEGATIVE )MOST DESIRED OR TARGET VALUE IS NON-ZERO ANDFINITEPOSSIBLE TO FIND A SCALING OR ADJUSTMENTFACTORWHEN MEAN IS ZERO, STANDARD DEVIATION IS ALSOZEROMAXIMIZEO TO OOη = 10 Log 10 [ µ 2 / σ 2 ]28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 14Dyn S/N Ratio - 7

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012NOMINAL - THE - BEST TYPE PROBLEMMAXIMIZEη = 10 Log 10 [ µ 2 / σ 2 ]USE TWO - STEP PROCEDURE1. Minimize Variance { σ 2 }2. Adjust the mean ‘µ’ on to target ‘µ o ’28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 15LARGER - THE - BETTER TYPE PROBLEMQUALITY CHARACTERISTICS IS(CONTINUOUS AND NON-NEGATIVE)NO SCALING OR ADJUSTMENT FACTORCAN BE TRANSFORMED INTOSMALLER - THE - BETTER TYPECONSIDER THE RECIPROCAL OF QUALITYCHARACTERISTICSMAXIMIZE= – 10 Log101nO TO OO2 221 2 n( 1/Y + 1/Y + ... + 1/Y )28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 16Dyn S/N Ratio - 8

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012SIGNED - TARGET TYPE PROBLEMQUALITY CHARACTERISTICS CAN TAKEPOSITIVE AND NEGATIVE VALUESOFTEN, TARGET VALUE = ZEROIF NOT, IT CAN BE MADEZERO BY CHANGE OF REFERENCEWHEN MEAN IS ZERO,STANDARD DEVIATION IS NOT ZEROA SCALING OR ADJUSTMENT FACTOR EXISTSMAXIMIZEη = - 10 Log 10 [ σ 2 ]28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 17FRACTION DEFECTIVE TYPE PROBLEMQUALITY CHARACTERISTICS IS A FRACTION P ( 0 TO 1 )TARGET VALUE = ZERONO SCALING OR ADJUSTMENT FACTORMAXIMIZEη = – 10 Log 10 [ P / ( 1 – P ) ]OR MINIMIZENO. OF BAD PIECES TO PRODUCE ONE GOOD PIECE28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 18Dyn S/N Ratio - 9

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012ORDERED CATEGORICAL PROBLEMQUALITY CHARACTERISTICS TAKE ORDEREDCATEGORICAL VALUES( C1 = WORSE C2 = NO CHANGEC3 = GOOD C4 = EXCELLENT )EXTREME CATEGORY C4 IS MOST DESIRED (TARGET VALUE )FORM CUMULATIVE CATEGORIES.TREAT EACH CATEGORY ASA FRACTION DEFECTIVE PROBLEM28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 19CURVE OR VECTOR TYPE PROBLEMQUALITY CHARACTERISTICS IS ACURVE OR A VECTORBROKEN INTO SEVERAL SCALAR PROBLEMSOF THE TYPE ALREADY DESCRIBED28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 20Dyn S/N Ratio - 10

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012NOISEXSIGNALMP – DIAGRAMOUTPUTYZCONTROL FACTORSP – Diagram for DYNAMIC Problems28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 21S/N RATIOS FOR DYNAMIC PROBLEMSTYPES OF DYNAMIC PROBLEMS– CINTINUOUS - CONTINUOUS TYPE ( C - C )– CONTINUOUS - DIGITAL TYPE ( C - D )– DIGITAL - CONTINUOUS TYPE ( D - C )– DIGITAL - DIGITAL TYPE ( D - D )28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 22Dyn S/N Ratio - 11

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012CONTINUOUS - CONTINUOUS TYPE PROBLEMBOTH SIGNAL FACTOR AND QUALITYCHARACTERISTICS TAKE POSITIVE OR NEGATIVEVALUESWHEN SIGNAL M = 0 ,QUALITY CHARACTERISTIC = 0IDEAL FUNCTION y = MSCALING FACTOR EXISTS TO ADJUST SLOPE(PROPORTIONALITY CONSTANT) BETWEEN ' y ' AND ' M'28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 23S/N RATIO FOR A C - C TYPE PROBLEM QUALITY CHARACTERISTIC ' y ’ , FOLLOWS THE SIGNAL FACTOR ' M ' y = β M (usually β is taken as 1) y = MSIGNAL NOISE YM 1 X 1 , X 2 , . . . , X n Y 11 , Y 12 , . . , Y 1nM 2 X 1 , X 2 , . . . , X n Y 21 , Y 22 , . . , Y 2n..M m X 1 , X 2 , . . . , X nY m1 , Y m2 , . . , Y mn28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 24Dyn S/N Ratio - 12

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012Dynamic Characteristics, Y = M10.80.60.40.2Desired LineSlope 1Best-Fit LineSlope β OPTDesiredsample1Sample2Sample3Best-fit Line0M1 M2 M328Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 25S/N RATIO FOR A C - C TYPE PROBLEM Quality Loss for each Y ij = K ( Y ij - M i )AVERAGEQ Z =Km nmnI = 1 j = 1( Y ij - M i ) 2 QUALITY LOSS WITHOUT ADJUSTMENT HAS TWO COMPONENTS– LOSS DUE TO SLOPE NOT BEING EQUAL TO 1– LOSS DUE TO DEVIATION FROM LINARITY (VARIANCE)28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 26Dyn S/N Ratio - 13

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012REGRESSION OF ' Yij ' UPON Mi Let straight line fitting the ' Yij ‘ have a slope ‘β’ Quality Loss with respect to the straight lineQ Z’ =Km nm nI = 1 j = 1( Y ij - βM i ) 228Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 27 BY METHOD OF LEAST SQUARES (Differentiate wrt β and equate to zero)mn(Y ij M i )β OPT (slope of best fit line) =I = 1mj = 1nM i2I = 1j = 128Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 28Dyn S/N Ratio - 14

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012– BEST ADJUSTMENT Multiply all ' Yij ' by (1/β OPT )SUCH THAT Quality loss after adjustmentKQ a’ =m nm nI = 1 j = 1Y( - βM i ) 2ijβ OPT[ ADJUSTMENT FACTOR IS (1/β OPT ) ]28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 29Quality after adjustment, ‘Q a ’KQ a’ =m nm nI = 1 j = 12( Y ij - βM i )[ ]β OPT2Q a’ =1β2OPTKm nmnI = 1 j = 12( Y ij - βM i )Q a’ =σ2 eβ2OPTThis is σ e228Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 30Dyn S/N Ratio - 15

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012 Quality after adjustment, ‘Q a ’Q a’ =σ e2β OPT2 Minimize, ‘Q a ’ ( or maximize 1/ ‘Q a ’ )η = 10 Log 10β OPT2σ e2( REDUCES NON-LINEARITY ALONG WITH REDUCTIONIN SENSITIVITY TO NOISE FACTORS )28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 31TWO STEP PROCEDUREto Optimize Dynamic Problems STEP 1 MAXIMIZEη = 10 Log 10β OPT2σ e2 STEP 2 ADJUST SLOPE BY SUITABLE SCALING FACTOR28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 32Dyn S/N Ratio - 16

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012CONTINUOUS - DIGITAL TYPE ( C - D )TEMPERATURE CONTROLLER– INPUT TEMPERATURE SETTING - CONTINUOUS– OUTPUT OF HEATING UNIT - ' ON ' OR ' OFF '– DIVIDE INTO TWO SEPARATE PROBLEMSONE FOR ' ON ' FUNCTIONOTHER FOR ' OFF ' FUNCTION– EACH ONE CONTINUOUS - CONTINUOUS TYPE ORNOMINAL - THE - BEST TYPE PROBLEM28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 33DIGITAL - CONTINUOUS TYPE ( D - C )DIGITAL TO ANALOG CONVERTER– CONVERSION TO ' 0 ' AND ' 1 '– DIVIDE INTO TWO SEPARATE PROBLEMSONE FOR ' 0 ' FUNCTIONOTHER FOR ' 1 ' FUNCTION– EACH ONE CONTINUOUS - CONTINUOUS TYPE ORNOMINAL - THE - BEST TYPE PROBLEM28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 34Dyn S/N Ratio - 17

P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP 28Feb-1Mar 2012Thank You28Feb-1Mar 2012 P.R. Apte : 3-Day Taguchi Method Workshop at UNIMAP S/N ratio (Dynamic) - 35Dyn S/N Ratio - 18