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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
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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
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Ez(y) = β0 +β1x1 +β2x2 + β12x1x2
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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?
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ˆ y (x,z) = 70.06 +10.81z1 + 4.94x2 + 7.31x3 − 9.06x2z1 +8.31x3z1
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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.