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Understanding Robustness in Experiments: Factors and Strategies for Variation Reduction - , Study notes of Systems Engineering

Georgia Institute of Technology - Main CampusSystems Engineering
Prof. Chien-Fu Wu

The concept of robustness in experiments, focusing on control factors and noise factors. It discusses strategies for reducing variation in noise factors and exploiting control-by-noise interactions through robust parameter design (rpd). Examples from layer growth and leaf spring experiments, as well as methods for analyzing data and making recommendations for control factor settings.

Typology: Study notes

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Uploaded on 08/05/2009

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Unit 8: Robust Parameter Design
Source : Chapter 10 (sections 10.1 - 10.6, part of sections 10.7 - 10.8 and 10.10).
•Revisiting two previous experiments.
•Strategies for reducing variation.
•Types of noise factors.
•Variation reduction through robust parameter design.
•Cross array, location-dispersion modeling, response modeling.
•Single arrays vs cross arrays.
•Signal-to-noise ratios and limitations.
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Download Understanding Robustness in Experiments: Factors and Strategies for Variation Reduction - and more Study notes Systems Engineering in PDF only on Docsity!

Unit 8: Robust Parameter Design

Source : Chapter 10 (sections 10.1 - 10.6, part of sections 10.7 - 10.8 and 10.10).

Revisiting two previous experiments.

Strategies for reducing variation.

Types of noise factors.

Variation reduction through robust parameter design.

Cross array, location-dispersion modeling, response modeling.

Single arrays vs cross arrays.

Signal-to-noise ratios and limitations.

Robust Parameter Design

Statistical/engineering method for product/process improvement (G.Taguchi).

Two types of factors in a system (product/process):

control factors: once chosen, values remain fixed.

noise factors: hard-to-control during normal process or usage.

Robust Parameter design

RPD

or

PD

): choose control factor settings to

make response less sensitive (i.e.more robust) to noise variation; exploitingcontrol-by-noise interactions.

Layer Growth Experiment: Factors and Levels

Table 1: Factors and Levels, Layer Growth Experiment

Level

Control Factor

A

susceptor-rotation method

continuous

oscillating

B

code of wafers

668G

678D

C

deposition temperature(

C)

D

deposition time

short

long

E

arsenic flow rate(%)

F

hydrochloric acid etch temperature(

C)

G

hydrochloric acid flow rate(%)

H

nozzle position

Level

Noise Factor

L

location

bottom

top

M

facet

Layer Growth Experiment: Thickness Data

Table 2: Cross Array and Thickness Data, Layer Growth Experiment

Noise Factor

Control Factor

L-Bottom

L-Top

A

B

C

D

E

F

G

H

M

M

M

M

M

M

M

M

Strategies for Variation Reduction

Sampling inspection

: passive, sometimes last resort.

Control charting and process monitoring

: can remove special causes. If

the process is stable, it can be

followed

by using a

designed experiment

Blocking, covariate adjustment

: passive measures but useful in reducing

variability, not for removing root causes.

Reducing variation in noise factors

: effective as it may reduce variation in

the response but can be expensive. Better approach is to change controlfactor settings (

cheaper

and

easier

to do) by exploiting control-by-noise

interactions, i.e., use robust parameter design!

Types of Noise Factors

1. Variation in process parameters.2. Variation in product parameters.3. Environmental variation.4. Load Factors.5. Upstream variation.6. Downstream or user conditions.7. Unit-to-unit and spatial variation.8. Variation over time.9. Degradation.

Traditional design uses 7 and 8

Exploitation of Nonlinearity

Nonlinearity between

y

and

x

can be exploited for robustness if

x

, nominal values

of

x

, are control factors and deviations of

x

around

x

are viewed as noise factors

(called

internal noise

). Expand

y

f

x

around

x

y

f

x

i

f

x

i

x

i

x

i

x

i

This leads to

i

f

x

i

x

i

2 i

where

var

y

2 i

var

x

i

, each component

x

i

has mean

x

i

and variance

2 i

From (1), it can be seen that

can be reduced by choosing

x

i

with a smaller slope

f

x

i

x

i

. This is demonstrated in Figure 1. Moving the nominal value

a

to

b

can

reduce

var

y

because the slope at

b

is more flat. This is a

parameter design

step.

On the other hand, reducing the variation of

x

around

a

can also reduce

var

y

. This

is a

tolerance design

step.

Exploitation of Nonlinearity to Reduce Variation

f(x)

x

(design parameter)

(response) y

a

b

a

b

Figure 1: Exploiting the Nonlinearity of

f

x

to Reduce Variation

Two-step Procedures for Parameter Design

Optimization

Two-Step Procedure for Nominal-the-Best Problem

i

select the levels o f the dispersion f actors to minimize dispersion

ii

select the level o f the ad justment f actor to bring the location on target

Two-Step Procedure for Larger-the-Better and Smaller-the-Better Problems

i

select the levels o f the location f actors to maximize

or minimize

the location

ii

select the levels o f the dispersion f actors that are not location

f actors to minimize dispersion

Note that the two steps in (3) are in reverse order from those in (2).Reason: It is usually harder to increase or decrease the response

y

in the latter

problem, so this step should be the first to perform.

Analysis of Layer Growth Experiment

From the ¯

y

i

and ln

s

2 i

columns of Table 5, compute the factorial effects for

location and dispersion respectively. (These numbers are not given in thebook.) From the half-normal plots of these effects (Figure 2),

D

is

significant for location and

H

A

for dispersion.

y

x

D

z

x

A

x

H

Two-step procedure:(i) choose

A

at the “

” level (continuous rotation) and

H

at the “

” level

(nozzle position 6).(ii) By solving

y

x

D

choose

x

D

Layer Growth Experiment: Plots

  • • • • • •

half-normal quantiles

absolute effects

0.8 0.6 0.4 0.2 0.

G

C

H

D

location

  • •
  • •

half-normal quantiles

absolute effects

2.0 1.5 1.0 0.5 0.

AE

D

A

H

dispersion

Figure 2:

Half-Normal Plots of Location and Dispersion Effects, Layer Growth Experiment

Analysis of Leaf Spring Experiment

Based on the half-normal plots in Figure 3,

B

C

and

E

are significant for

location,

C

is significant for dispersion:

y

x

B

x

C

x

E

z

x

C

Two-step procedure:(i) choose

C

at

(ii) With

x

C

y

x

B

x

E

To achieve ˆ

y

x

B

and

x

E

must be chosen beyond

1, i.e.,

x

B

x

E

78. This is too drastic, and not validated by current data. An

alternative is to select

x

B

x

E

x

C

1 (not to follow the two-step

procedure), then ˆ

y

=7.89 is closer to 8. (Note that ˆ

y

71 with

B

C

−

E

Reason for the breakdown of the 2-step procedure: its second step cannotachieve the target 8.0.

Leaf Spring Experiment: Plots

half-normal quantiles

absolute effects

0.15 0.

BC

BD

D

CD

E

C

B

location

half-normal quantiles

absolute effects

2.0 1.5 1.0 0.

B

BC

E

BD

D

CD

C

dispersion

Figure 3:

Half-Normal Plots of Location and Dispersion Effects, Leaf Spring Experiment

Response Modeling and Control-by-Noise

Interaction Plots

Response Model: model

y

i j

directly in terms of control, noise effects and

control-by-noise interactions.

half normal plot of various effects.

regression model fitting, obtaining ˆ

y

Make control-by-noise interaction plots for significant effects in ˆ

y

, choose

robust

control settings at which y has a flatter relationship with noise.

Compute

Var

y

with respect to variation in the noise factors. Call

Var

y

the

transmitted variance model

. Use it to identify control factor settings

with small transmitted variance.