Chapter 12 Robust Parameter Design - UCLA Statisticshqxu/stat201A/ch12page.pdf · • Methods for...

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Chapter 12 Robust Parameter Design •  Goal is to make products and processes robust

or less sensitive to variability transmitted by factors that cannot be easily controlled

•  Methods for Robust Parameter Design (RPD) was developed by Taguchi starting in the 1950s and introduced to western industry in the 1980s

•  Taguchi methods generated much controversy •  Subsequent research produced an improved

approach based on RSM

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Controllable and Uncontrollable Factors •  Noise (or uncontrollable) variables transmit

variability into the response •  Noise variables cannot be controlled in the end

application, but can be controlled for purposes of an experiment (assumption)

•  Objective is to determine the levels of the controllable variables that minimize the variability transmitted from the noise variables

•  This approach is not always applicable – if the noise factors dominate, other methods must be considered

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12.2 Crossed Array Designs •  The leaf spring experiment

–  to investigate the effects of 5 factors on the free height of leaf springs used in an automotive application.

–  4 controllable factors: A = furnace temperature, B = heating time, C = transfer time, D = hold down time

–  1 noise factor: E=quench oil temperature –  Inner array: A 24-1 design I=ABCD –  Outer array: a 21 design –  Each run in an inner array is performed for all

combinations in the outer array.

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A cross array design provides important information about the interactions between controllable factors and noise factors.

A Cross Array Design

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Control by Noise Interaction

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Inner array: A 34-2 design for controllable factors A, B, C, D

Outer array: A 23 design for noise factors E, F, G Main disadvantage: require too many runs.

Another Crossed Array Design

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12.3 Analysis of Crossed Array Design •  Taguchi proposed to model the signal-to-noise

ratios, which are problematic. •  A better approach is to model the mean and

variance of the response directly •  For each run in the inner array, compute sample

mean and sample variance (over all combinations of the outer array).

•  Build two separate models –  Location model for mean responses –  Dispersion model for natural log of the variances

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Half-normal Plots for Location and Dispersion Models

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Results •  A, B, D are important for location (mean

response)

•  Only B is important for dispersion

•  To minimize variance, choose B=+ •  Then adjust A and D to bring mean

response to a desired target, say 7.75inch.

€

ˆ y = 7.63+ 0.12A − 0.081B + 0.044D

€

ln(ˆ s 2) = −3.74 −1.09B

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12.4 Combined Array Design and the Response Model Approach

•  A crossed array design may require a large number of runs.

•  A combined array (or single array) design contains both controllable and noise factors. –  Often requires much less runs

•  A disadvantage of the mean and variance modeling approach is that it does not take direct advantage of the interactions between controllable and noise variables.

•  The response model approach incorporates both controllable and noise variables and their interactions.

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Illustration •  Consider a response model

•  controllable variable x1 and x2 are fixed once chosen

•  noise variable z is random

•  A model for the mean response is

•  A model for the variance is €

E(z) = 0, V (z) =σ z2

€

Ez(y) = β0 +β1x1 +β2x2 + β12x1x2

€

Vz(y) =σ z2(γ1 +δ1x1 +δ2x2)

2 +σ 2

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y = β0 +β1x1 +β2x2 + β12x1x2 + γ1z + δ1x1z +δ2x2z + ε

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A Combined Array Design

Assume A=temperature is difficult to control in the full-scale process and treated as a noise factor

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A Combined Array Design (Continue) •  This is a combined array design

–  A=z1 is a noise variable –  B=x1, C=x2, D=x3 are controllable variables

•  The response model is

•  The mean model is

•  The variance model is

•  How to minimize the variance?

€

ˆ y (x,z) = 70.06 +10.81z1 + 4.94x2 + 7.31x3 − 9.06x2z1 +8.31x3z1

€

Ez[ˆ y (x,z)] = 70.06 + 4.94x2 + 7.31x3

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Vz[ˆ y (x,z)] =σ z2(10.81− 9.06x2 +8.31x3)2 +σ 2

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Control-by-noise Interaction plots

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12.5 Choice of Designs •  The selection of the experimental design is a

very important aspect of an RPD problem. •  The combined array approach will result in

smaller designs than the crossed array approach.

•  Research problem: How to define/construct (minimum aberration) combined arrays?

•  The estimation of the interactions between controllable and noise factors is the most important issue.

•  Resolution V designs are recommended if possible.