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Taguchi and Response Surfaces

# Taguchi and Response Surfaces - Taguchi Methods and ‘...

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Unformatted text preview: Taguchi Methods and ‘ Response Surface ' Alternatives Once in control, goal is to produce uniformly near a target. I ° Mean on target ° Reduce variation Three possible situations: - Smaller is better 0 Larger is better ° On target is best On target is best. Taguchi considered deviation from target to be loss to society. Taguohi models this as: 120’) =K(Y—T)2 Y 2 Quality Control ' T 2 Target value On average, loss is proportional to: ‘ E(Y —- T)2 = Varﬂ’) + (Bier?)2 where B iaS : difference of mean and Target 5 For smaller is better, Taguehi has less as: L(Y) = Y2 _ i.e. on average E(Y 2). So Target is 0. For larger is better, Taguchi considers loss as: MY) =1/y2 0f E(1/y2). First step in Taguchi ' Method is parameter % I design. Parameters are inputs known or suspected to affect the quality characteristic. Goal: Design in quality, don’t sample to catch poor output. 8 Parameters are divided into two groups: ° Control variables- parameters which can be controlled in the process. ° Noise variables— parameters which vary and are usually not controlled (sometimes hard or expensive to control). Levels are chosen for each of the control variables. Levels are chosen 1 and temporarily ﬁxed for each of the - noise variables. Control variables are placed into inner array. Example: [\JNHMNNHh—d :Nl—tNr—tNr—twv—‘l Eight design points, each row one design point 11 Noise variables are ' placed in outer array Example: ' 2 l 2 2 Generally the levels span What is experienced in production conditions. 12 The structure of each array is typically ° Factorial design ° Fractional factorial design (sometimes referred to as - “orthogonal array”.) The arrays are then crossed and observa— tions collected. 13 Goal of analysis: Minimize _ E(Y -— N by choice of best control variable settings. Taguohi suggests look at “signal to noise” ratio. % Bigger is better. Then adjust mean. 14 Criticisms: -° Crossing inner and outer arrays can lead to unwieldy designs. ° Use of signal to noise ratios very ad-hoc and hard to generalize (various versions). '15 Alternatives proposed include ReSponse Surface Methodology ' (RSM). * Goals: - Model the response Y as a function of control and noise variables. - Use information on distribution of noise variables to model mean and variance. l6 Simple example: I Y = Lamina thickness X I : Viscosity (control variable) _ X 2 :— Ambient humidity (noise variable) . Fitted model: Y = 60+B‘1X1+B\12X1X2 Select X to minimize loss. 17' E(f)=6O+BIXI ‘ +6 X -E(X2) 12 1 Var()7) 2(612X1r -Var(X2) 2 +6 (Conditional mean and variance, conditional 0n ~ parameter estimates). 18 More complex example: XPX2 —C0ntr01 X3 —N0ise 17:50 +61X1+BA2X2 + B23X2 'X3 E®Zﬁo+61X1+62X2+323X2+623X2 '(EX3) Vam=(ﬁ23X2)-Vaz(X3)+o Can set X2 set X1 to control mean. to minimizevariance 19' ...
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