Print Seminar Report

ASeminar Report

On

EVALUATION OF COMPOUND NOISE USING ROBUST DESIGN

Third YearMechanical Engineering

SUBMITTED BY:Aditya Sinha

Under the Guidance ofProf. R. R. Ghadge

DEPARTMENT OF MECHANICAL ENGINEERINGMaharashtra Institute of Technology, Pune.

1

MAEER’S

MAHARASHTRA INSTITUTE OF TECHNOLOGY

PUNE-38

DEPARTMENT OF MECHANICAL ENGINEERING

CERTIFICATE

This is to certify that ADITYA SINHA has successfully carried out the technical paper presentation on “EVALUATION OF COMPOUND NOISE USING ROBUST DESIGN” during the academic year 2010-2011, is approved.

Examination no:

Prof.R.R.Ghadge (Guide) Prof.P.B.Joshi

Professor, Mech Engg.Dept. Head of Dept.Mech.Engg

2

ACKNOWLEDGEMENT

It gives me an immense pleasure to submit this technical paper entitled

“EVALUATION OF COMPOUND NOISE USING ROBUST DESIGN” . I have

tried my level best to represent this topic into compact and to the point framework.

I wish to express my sincere thanks with profound gratitude to my guide Prof.

R.R.Ghadge for his valuable guidance and constant encouragement without which it

would have been impossible for me to present and complete this seminar successfully.

I would like to extend my sincere and true thanks to my H.O.D. Prof. P. B.

Joshi and all the staff members for impairing me the best of their knowledge and

guidance.

Last but not the least,I thank all my friends and family for their assistance and

help.

Aditya Sinha

TE (Mech.)

Roll No. 8020

3

INDEX

Sr. No. Topic Page No.1 Abstract 52 Introduction 63 Case Studies 84 Conditions for Compound Noise to be Completely

Effective

10

5 Effectiveness of Compound Noise In Real

Scenarios

13

6 Conclusions 177 General Procedure to implement Robust Design 198 References 20

4

AbstractThe following report evaluates compound noise as a robust design method.

Application of compound noise as a robust design method leads to a reduction in

experimental effort. The compound noise strategy was applied to two types of

situation: the first type has been described with active effects up to two-factor

interactions and the second type has been described with effects up to three-factor

interactions. These two situations are illustrated with help of case studies. The report

provides theoretical justification for the effectiveness of the compound noise strategy

as formulated by Taguchi and Phadke. For example, we found that the compound

noise strategy is very effective for systems which exhibit effect sparsity. Overall, the

paper studies the effectiveness of an alternative formulation, outlines scenarios where

compound noise as a robust design method can be effectively used and gives

alternative strategies for the systems on which compound noise cannot be effective.

5

Introduction

The negative effects of noise factors on quality are reduced by exploiting control- by-

noise interactions. A method for efficiently improving robustness was developed and

employed widely in Japanese industry by Taguchi1 in the 1950s. The methods became

popular in the United States and Europe in the 1980s, due to Japanese competition

especially in the automobile industry. In the past few decades, there have been

theoretical advances and new robust design methods have been developed.

Recently T.J Robinson2 presented paper, in which he reviewed how the research

efforts in robust design have led to improvement in performance measures, new

experimental designs, and also alternatives to Taguchi’s methods based on response

surface methodology. Most of the methods have been trying to reduce the

experimental run sizes & keep on attaining good results. The focus of this paper will

be compound noise, which is also intended to reduce run sizes.

In Compound noise technique we vary multiple noise factors simultaneously as

if they were one factor. Usually the outer array of noise factors is replaced by the

compound noise factor that is varied between two levels. According to Taguchi1 and

Phadke compound noise factors should be formed based on the directionality of the

noise factor effects. In a very simple case they can be opposite extremes.Du X

suggested forming compound noises based on the conditions of the two ‘most

probable points of inverse reliability’. This method allows us to account for the skew

in distribution of system performance. X.S Hou studied the conditions that will make

compound noise give robust settings for systems. Hou said ‘extreme settings should

exist for compound noise to work’. But we will find below that compound noise can

6

be effective even when extreme settings do not exist. The conditions mentioned in

‘compound noise factor theory’ are the sufficient conditions to for the theory to give

robust settings. In later sections we will extend the analysis to determine conditions

under which compound noise will predict a robust setting. Hou’s formulation was

limited to systems that had active effects up to two-factor interactions. We will extend

the formulation to systems that can have active effects up to three-factor interactions.

Compound noise can be considered as an extension of supersaturated

designs (SSD). SSDs were assumed to offer a potentially useful way to investigate

many factors with few experiments. Compound noise is an unbalanced SSDs.

D.R.Holcomb and W.M.Carlyle7 discussed the construction and evaluation of SSDs.

T.T.Allen and M.Bernshteyn discussed the advantages of unbalanced SSDs in terms

of performance and affordability. Y.V Heydon argued that rather than using it to find

main effects, SSDs can be used to estimate variance of response, which can be used as

a measure of robustness. SSDs ‘do not allow estimation of the effects of the

individual factors because of mixing or confusion between the main effects’.

However, estimation of the separate factor effects is not necessarily required in

improving robustness. Using compound noise as a robust design method we try to

estimate the robustness of the system at a given control factor setting. The setting that

improves this estimate is taken as the predicted robust setting.

The aim of this study is to explore the effectiveness of compound noise as a

robust design method. Engineering systems show certain regularities, one of which is

the hierarchical ordering principle. Systems can be classified on the basis of the

hierarchical ordering principle. The classes are as follows:

• Strong Hierarchy Systems: Systems that have main effects and are two-factor

interactions active only. Some small three-factor interactions might be present

in such systems, but they are not active.

• Weak Hierarchy Systems: Systems that also have active three-factor

interactions.

7

We apply compound noise to three strong hierarchy systems then three weak

hierarchy systems and analyze robustness gain. There are three main questions we

want to address are:

• Why is compound noise effective in achieving a robust setting in certain

cases, but ineffective in other cases?

• How can we measure the effectiveness of a compound noise strategy?

• Do we need to know the directionality of noise factors to use compound

noise?

We analyze a compound noise factor strategy for the six case studies and explore

reasons for their effectiveness or ineffectiveness. We then present the conditions for

compound noise to work ( these conditions are an extension of Hou ).

• First, we present conditions for strong and weak hierarchy systems.

• Second, we present the effectiveness of the compound noise strategy in real

scenarios using fractional factorial arrays for control and noise factors.

•Finally, we present a procedure to determine an outer array for robust design

experiments, which can be used by practitioners.

Case Studies

We analyzed six case studies. Three of them exhibited strong hierarchy:

•PNM: Passive Neuron Model ( Tawfik and Durand)

•Op Amp: Operational Amplifier (Phadke3)

• Journal Bearing: half-Sommerfeld solution (Hamrock).

Three case studies exhibited weak hierarchy:

•Continuous-Stirred Tank Reactor (CSTR; Kalagnanam and Diwekar)

• Temperature Control Circuit (Phadke3)

•The Slider Crank (Gao).

For each case study we ran full factorial control and noise factor crossed array

experiments. These runs were used to determine the effect coefficients for the main

8

effects, two-factor interactions and three-factor interactions. Lenth’s method was

employed to determine the active effects. The full factorial results were also used to

determine the most robust setting for all six systems.

In order to formulate compound noise for robust design experiments, we found

the directionality of noise factors on the system response for all six systems. For each

system we ran a full factorial design in control factors crossed with two-level

compound noise. Pure experimental error was introduced into the system’s response.

The error was of the order of the active main effects. From Lenth’s method, we can

estimate the average magnitude of active main effects. Pure experimental error

introduced into the system’s response was normally distributed with zero mean and

standard deviation as the average magnitude of active main effects. The compound

noise strategy performed extremely well for the PNM, Op Amp, journal bearing and

slider crank. Table I the shows results from the robust design experiments.

Table:I

System Hierarchy

Measure of

sparsity of effects

Existence of

extremesettings

Number of replications

Matching of all

control factors

(%)

Matching of control factories with their

robust setting (%)

Improvement Ratio(%)

Op-Amp Strong 0.105 Existing 100 96 99.2 99.2

PNM Strong 0.0132 Non-Existent 100 96 99 99.5

Journal Bearing Strong 0.28 Existing 100 97.8 98.8 98.54

CSTR Weak 0.647 Non-Existent 100 10 55.17 58.4

TCC Weak 0.725 Non-Existent 100 5 29.25 30.65

Slider Crank Weak 0.119 Existing 100 95 98 98.5

Results from full factorial control factor array and two-level compound noise

9

The third column of Table I shows the measure of sparsity of effects, which is the

ratio of active effects in a system to the total number of effects that can be present in

the system. The sparsity of effects means that among several experimental effects

examined in any experiment, a small fraction will usually prove to be significant . The

sixth column of Table I shows how well the predicted robust setting from compound

noise experiments matches with the actual robust setting as found by carrying out full

factorial control and noise array experiments. The seventh column of Table I shows

the percentage of control factors whose robust setting matched for full factorial and

compound noise experiments.

At the predicted robust setting of control factors from compound noise experiments,

we found the corresponding variance of the system’s response from full factorial

experiments. Full factorial experiments yield the optimal variance of the system’s

response and the average variance of the response at all control factor settings. With

this knowledge, variance can be improved from the average value to the optimal

value, which is the maximum possible improvement. For compound noise

experiments the variance of the response can be improved from the average value to

the variance at the predicted robust setting. The ratio of achieved improvement to

maximum possible improvement is called the improvement ratio. The average of this

measure for all replicates is shown in the eighth column of Table I.

From Table I we can see that compound noise strategy works reasonably well

for the Op Amp, PNM, journal bearing and slider crank, but not nearly as well for the

CSTR and the temperature control circuit.

Conditions for Compound Noise to be Completely Effective

Hou5 gave conditions under which a compound noise strategy will predict the robust

setting for a system. One of the conditions for compound noise to work is the

existence of extreme settings. The extreme settings of compound noise are those that

maximize and minimize the system’s response to capture noise variations. However,

we found that in some systems (e.g., PNM) the compound noise strategy would work

10

even if extreme settings do not exist. So the conditions outlined in Theorem 4 of Hou

are sufficient but not necessary conditions for compound noise to work.

For strong hierarchy systems the response y can be expressed as

390 J. SINGH ET AL.

improvement to maximum possible improvement is called the improvement ratio. The average of this measurefor all replicates is shown in the eighth column of Table I.

From Table I we can see that compound noise strategy works reasonably well for the Op Amp, PNM, journalbearing and slider crank, but not nearly as well for the CSTR and the temperature control circuit.

CONDITIONS FOR COMPOUND NOISE TO BE COMPLETELY EFFECTIVE

Hou5 gave conditions under which a compound noise strategy will predict the robust setting for a system.One of the conditions for compound noise to work is the existence of extreme settings. The extreme settingsof compound noise are those that maximize and minimize the system’s response to capture noise variations.However, we found that in some systems (e.g., PNM; Tawfik and Durand13) the compound noise strategy wouldwork even if extreme settings do not exist. So the conditions outlined in Theorem 4 of Hou5 are sufficient butnot necessary conditions for compound noise to work.


y = f (x1, x2, . . . , xl) +m!

j=1

!j zj +m!

j=1

l!

i=1

"ijxizj (1)

where f (x1, x2, . . . , xl) is a general function in the control factors xi, zj (j = 1, . . . , m) denotes the noisefactors affecting the system, !j denotes the effect intensity of noise factor j and "ij denotes the effect intensitiesof control-by-noise interactions present in the system. Control factors can have two settings, either !1 or 1.(Control factor variability can be represented as separate noise factors.) Noise factors are in the range !1 to 1and are independent, with zero mean and a variance of 1. Noise factors are symmetric about 0. The variance ofy with respect to zj terms is given by

Varz(y) = C + 2" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#(2)

where C is a constant. The robust setting of the system is such that the variance of the response is minimizedwith respect to noise factors. It can be represented as

X " arg$

minxi

" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#%(3)

where X is the setting of the xi terms that minimizes the variance.If we follow Taguchi’s formulation of compound noise, based on the directionality of noise factors, then

responses at two levels of compound noise will be given by

y+ = f (x1, x2, . . . , xl) +m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (4)

y! = f (x1, x2, . . . , xl) !m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (5)

The estimated variance with respect to compound noise levels is given by

&Var(y) = (y+ ! y)2 + (y! ! y)2 = 12 (y+ ! y!)2 (6)

&Var(y) = C1 + 4$ l!

i=1

" m!

j=1

|!j |m!

j=1

sign(!j )"ij

#xi +

l!

k=1i<k

" m!

j=1

sign(!j )"ij

m!

j=1

sign(!j )"kj

#xixk

%(7)

Copyright c# 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398DOI: 10.1002/qre

(Eq:1)

where f(x1,x2,...,xl) is a general function in the control factors xi,zj (j =1,...,m) denotes

the noise factors affecting the system, βj denotes the effect intensity of noise factor j

and γij denotes the effect intensities of control-by-noise interactions present in the

system. Control factors can have two settings, either −1 or 1. (Control factor

variability can be represented as separate noise factors.) Noise factors are in the range

−1 to 1 and are independent, with zero mean and a variance of 1. Noise factors are

symmetric about 0. The variance of y with respect to zj terms is given by

390 J. SINGH ET AL.






y = f (x1, x2, . . . , xl) +m!

j=1

!j zj +m!

j=1

l!

i=1

"ijxizj (1)


Varz(y) = C + 2" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#(2)


X " arg$

minxi

" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#%(3)



y+ = f (x1, x2, . . . , xl) +m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (4)

y! = f (x1, x2, . . . , xl) !m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (5)


&Var(y) = (y+ ! y)2 + (y! ! y)2 = 12 (y+ ! y!)2 (6)

&Var(y) = C1 + 4$ l!

i=1

" m!

j=1

|!j |m!

j=1

sign(!j )"ij

#xi +

l!

k=1i<k

" m!

j=1

sign(!j )"ij

m!

j=1

sign(!j )"kj

#xixk

%(7)


(Eq:2)

where C is a constant. The robust setting of the system is such that the variance of the

response is minimized with respect to noise factors. It can be represented as

(Eq:3)

where X is the setting of the xi terms that minimizes the variance. If we follow

Taguchi’s formulation of compound noise, based on the directionality of noise factors,

then responses at two levels of compound noise will be given by

390 J. SINGH ET AL.






y = f (x1, x2, . . . , xl) +m!

j=1

!j zj +m!

j=1

l!

i=1

"ijxizj (1)


Varz(y) = C + 2" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#(2)


X " arg$

minxi

" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#%(3)



y+ = f (x1, x2, . . . , xl) +m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (4)

y! = f (x1, x2, . . . , xl) !m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (5)


&Var(y) = (y+ ! y)2 + (y! ! y)2 = 12 (y+ ! y!)2 (6)

&Var(y) = C1 + 4$ l!

i=1

" m!

j=1

|!j |m!

j=1

sign(!j )"ij

#xi +

l!

k=1i<k

" m!

j=1

sign(!j )"ij

m!

j=1

sign(!j )"kj

#xixk

%(7)


11

(Eq: 4)

(Eq: 5)


390 J. SINGH ET AL.






y = f (x1, x2, . . . , xl) +m!

j=1

!j zj +m!

j=1

l!

i=1

"ijxizj (1)


Varz(y) = C + 2" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#(2)


X " arg$

minxi

" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#%(3)



y+ = f (x1, x2, . . . , xl) +m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (4)

y! = f (x1, x2, . . . , xl) !m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (5)


&Var(y) = (y+ ! y)2 + (y! ! y)2 = 12 (y+ ! y!)2 (6)

&Var(y) = C1 + 4$ l!

i=1

" m!

j=1

|!j |m!

j=1

sign(!j )"ij

#xi +

l!

k=1i<k

" m!

j=1

sign(!j )"ij

m!

j=1

sign(!j )"kj

#xixk

%(7)


(Eq: 6)

390 J. SINGH ET AL.






y = f (x1, x2, . . . , xl) +m!

j=1

!j zj +m!

j=1

l!

i=1

"ijxizj (1)


Varz(y) = C + 2" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#(2)


X " arg$

minxi

" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#%(3)



y+ = f (x1, x2, . . . , xl) +m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (4)

y! = f (x1, x2, . . . , xl) !m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (5)


&Var(y) = (y+ ! y)2 + (y! ! y)2 = 12 (y+ ! y!)2 (6)

&Var(y) = C1 + 4$ l!

i=1

" m!

j=1

|!j |m!

j=1

sign(!j )"ij

#xi +

l!

k=1i<k

" m!

j=1

sign(!j )"ij

m!

j=1

sign(!j )"kj

#xixk

%(7)


(Eq: 7)

where C1 is a constant. The estimated robust setting of the system is such that the

estimated variance of the response is minimized with respect to the compound noise

levels. It can be represented as

(Eq:8)

where XCompound is the setting of the xi terms that minimizes the estimated variance.

For compound noise to be completely effective the control factor setting that

minimizes the estimated variance should be the same as the control factor setting that

minimizes the actual variance of response, i.e.

(Eq:9)

390 J. SINGH ET AL.






y = f (x1, x2, . . . , xl) +m!

j=1

!j zj +m!

j=1

l!

i=1

"ijxizj (1)


Varz(y) = C + 2" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#(2)


X " arg$

minxi

" l!

i=1

" m!

j=1

!j"ij

#xi +

l!

k=1i<k

" m!

j=1

"ij"kj

#xixk

#%(3)



y+ = f (x1, x2, . . . , xl) +m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (4)

y! = f (x1, x2, . . . , xl) !m!

j=1

"!j +

l!

i=1

"ijxi

#sign(!j ) (5)


&Var(y) = (y+ ! y)2 + (y! ! y)2 = 12 (y+ ! y!)2 (6)

&Var(y) = C1 + 4$ l!

i=1

" m!

j=1

|!j |m!

j=1

sign(!j )"ij

#xi +

l!

k=1i<k

" m!

j=1

sign(!j )"ij

m!

j=1

sign(!j )"kj

#xixk

%(7)


COMPOUND NOISE: EVALUATION AS A ROBUST DESIGN METHOD 391

where C1 is a constant. The estimated robust setting of the system is such that the estimated variance of theresponse is minimized with respect to the compound noise levels. It can be represented as

XCompound ! arg!

minxi

" l#

i=1

" m#

j=1

|!j |m#

j=1

sign(!j )"ij

$xi +

l#

k=1i<k

" m#

j=1

sign(!j )"ij

m#

j=1

sign(!j )"kj

$xixk

$%(8)

where XCompound is the setting of the xi terms that minimizes the estimated variance. For compound noise to becompletely effective the control factor setting that minimizes the estimated variance should be the same as thecontrol factor setting that minimizes the actual variance of response, i.e.

X = XCompound (9)

Similarly, we can derive conditions for compound noise to be completely effective for weak hierarchysystems. The derivation of the conditions is given in Appendix A.

From the full factorial experiments performed on six case studies, we found effect intensities for noisefactors and control-by-noise interactions for strong hierarchy systems. For weak hierarchy systems we alsofound intensities of control-by-control-by-noise and control-by-noise-by-noise interactions. On plugging effectintensities for strong hierarchy systems into Equations (3) and (8), we found that all three systems satisfiedEquation (9). The effect intensities for weak hierarchy systems were plugged into Equations (A3) and (A7) andonly the slider crank satisfied Equation (9). The PNM is one of the systems that does not have extreme settings(Hou5), but compound noise still works for such a system.

Compound noise was effective on the systems that exhibited effect sparsity (Box and Meyer18 and Wu andHamada11,19). For all strong hierarchy systems there were only few active control-by-noise interactions in eachof the systems to be exploited during robust design experiments. The slider crank has only one significant noisefactor to which it needs to be desensitized. The temperature control circuit and CSTR did not exhibit effectsparsity. These systems have many significant noise factors and two-factor and three-factor interactions.

EFFECTIVENESS OF COMPOUND NOISE IN REAL SCENARIOS

In real scenarios it is often not possible to run full factorial inner arrays such as those reported in Table I.Therefore, we also studied the use of fractional factorial arrays by running robust design experiments which wereresolution III in both the control and noise factor array. We predicted robust settings from such experiments andcompared the results to those from full factorial experiments in Table II. We also ran robust design experimentswhich were resolution III in the control factor array combined with two-level compound noise. Results forthese experiments are summarized in Table III. Resolution III experiments formed the basis of comparisonfor the results we obtained from a resolution III control factor array crossed with two-level compound noise.Pure experimental error was introduced into the system’s response for all six case studies. The error introducedwas of the order of active main effects. The control factors were randomly assigned to a resolution III innerarray.

We did not study the PNM and journal bearing systems because these systems had only two control factorsand thus had no distinct resolution III control factor array.

Table II indicates that the robust setting for the Op Amp and slider crank can be achieved with high probabilitywhen a resolution III noise factor array is used. For the CSTR and temperature control circuit, the high meanimprovement ratio means that we will not attain the robust setting but will arrive near the optima. The highimprovement ratio suggests that we will achieve most of the improvement possible in the variance of thesystem’s response.

Table III indicates that even when using a resolution III control factor array, a compound noise strategywill predict the robust setting for the Op Amp and the slider crank with high probability. The compound noisestrategy does not work too well with a resolution III control factor array for the CSTR. The average improvementratio possible is only 40%, as opposed to nearly 97% if the resolution III noise factor array had been used.

Copyright c" 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398DOI: 10.1002/qre

COMPOUND NOISE: EVALUATION AS A ROBUST DESIGN METHOD 391

where C1 is a constant. The estimated robust setting of the system is such that the estimated variance of theresponse is minimized with respect to the compound noise levels. It can be represented as

XCompound ! arg!

minxi

" l#

i=1

" m#

j=1

|!j |m#

j=1

sign(!j )"ij

$xi +

l#

k=1i<k

" m#

j=1

sign(!j )"ij

m#

j=1

sign(!j )"kj

$xixk

$%(8)

where XCompound is the setting of the xi terms that minimizes the estimated variance. For compound noise to becompletely effective the control factor setting that minimizes the estimated variance should be the same as thecontrol factor setting that minimizes the actual variance of response, i.e.

X = XCompound (9)

Similarly, we can derive conditions for compound noise to be completely effective for weak hierarchysystems. The derivation of the conditions is given in Appendix A.

From the full factorial experiments performed on six case studies, we found effect intensities for noisefactors and control-by-noise interactions for strong hierarchy systems. For weak hierarchy systems we alsofound intensities of control-by-control-by-noise and control-by-noise-by-noise interactions. On plugging effectintensities for strong hierarchy systems into Equations (3) and (8), we found that all three systems satisfiedEquation (9). The effect intensities for weak hierarchy systems were plugged into Equations (A3) and (A7) andonly the slider crank satisfied Equation (9). The PNM is one of the systems that does not have extreme settings(Hou5), but compound noise still works for such a system.

Compound noise was effective on the systems that exhibited effect sparsity (Box and Meyer18 and Wu andHamada11,19). For all strong hierarchy systems there were only few active control-by-noise interactions in eachof the systems to be exploited during robust design experiments. The slider crank has only one significant noisefactor to which it needs to be desensitized. The temperature control circuit and CSTR did not exhibit effectsparsity. These systems have many significant noise factors and two-factor and three-factor interactions.

EFFECTIVENESS OF COMPOUND NOISE IN REAL SCENARIOS

In real scenarios it is often not possible to run full factorial inner arrays such as those reported in Table I.Therefore, we also studied the use of fractional factorial arrays by running robust design experiments which wereresolution III in both the control and noise factor array. We predicted robust settings from such experiments andcompared the results to those from full factorial experiments in Table II. We also ran robust design experimentswhich were resolution III in the control factor array combined with two-level compound noise. Results forthese experiments are summarized in Table III. Resolution III experiments formed the basis of comparisonfor the results we obtained from a resolution III control factor array crossed with two-level compound noise.Pure experimental error was introduced into the system’s response for all six case studies. The error introducedwas of the order of active main effects. The control factors were randomly assigned to a resolution III innerarray.

We did not study the PNM and journal bearing systems because these systems had only two control factorsand thus had no distinct resolution III control factor array.

Table II indicates that the robust setting for the Op Amp and slider crank can be achieved with high probabilitywhen a resolution III noise factor array is used. For the CSTR and temperature control circuit, the high meanimprovement ratio means that we will not attain the robust setting but will arrive near the optima. The highimprovement ratio suggests that we will achieve most of the improvement possible in the variance of thesystem’s response.

Table III indicates that even when using a resolution III control factor array, a compound noise strategywill predict the robust setting for the Op Amp and the slider crank with high probability. The compound noisestrategy does not work too well with a resolution III control factor array for the CSTR. The average improvementratio possible is only 40%, as opposed to nearly 97% if the resolution III noise factor array had been used.

Copyright c" 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398DOI: 10.1002/qre

12

Similarly, we can derive conditions for compound noise to be completely

effective for weak hierarchy systems. The derivation of the conditions is given in

Appendix A.

From the full factorial experiments performed on six case studies, we found

effect intensities for noise factors and control-by-noise interactions for strong

hierarchy systems. For weak hierarchy systems we also found intensities of control-

by-control-by-noise and control-by-noise-by-noise interactions. On substituting effect

intensities for strong hierarchy systems into Equations (3) and (8), we found that all

three systems satisfied Equation (9). The effect intensities for weak hierarchy systems

were plugged into Equations (A3) and (A7) and only the slider crank satisfied

Equation (9). The PNM is one of the systems that does not have extreme settings, but

compound noise still works for such a system.

Compound noise was effective on the systems that exhibited effect sparsity. For

all strong hierarchy systems there were only few active control-by-noise interactions

in each of the systems to be exploited during robust design experiments. The slider

crank has only one significant noise factor to which it needs to be desensitized. The

temperature control circuit and CSTR did not exhibit effect sparsity. These systems

have many significant noise factors and two-factor and three-factor interactions.

EFFECTIVENESS OF COMPOUND NOISE IN REAL SCENARIOS In real scenarios it is often not possible to run full factorial inner arrays such as

those reported in Table I. Therefore, we also studied the use of fractional factorial

arrays by running robust design experiments which were resolution III in both the

control and noise factor array. We also ran robust design experiments which were

resolution III in the control factor array combined with two-level compound noise.

Results for these experiments are summarized in Table III. Resolution III experiments

formed the basis of comparison for the results we obtained from a resolution III

control factor array crossed with two-level compound noise. Pure experimental error

was introduced into the system’s response for all six case studies. The error

13

introduced was of the order of active main effects. The control factors were randomly

assigned to a resolution III inner array. We did not study the PNM and journal bearing

systems because these systems had only two control factors and thus had no distinct

resolution III control factor array.

Table II:

System Hierarchy

Measure of

sparsity of effects

Number of replications

Matching of control factories with their

robust setting (%)

Matching of all

control factors (%)

Improvement Ratio(%)

Op-Amp Strong 0.105 100 99.2 96 99.2

CSTR Weak 0.647 100 55.17 10 58.4

TCC Weak 0.725 100 29.25 5 30.65

Slider Crank Weak 0.119 100 98 95 98.5

Results from Resolution III Control & Noise Factor Array

Table II indicates that the robust setting for the Op Amp and slider crank can be

achieved with high probability when a resolution III noise factor array is used. For the

CSTR and temperature control circuit, the high mean improvement ratio means that

we will not attain the robust setting but will arrive near the optima. The high

improvement ratio suggests that we will achieve most of the improvement possible in

the variance of the system’s response.

Table III:

System Hierarchy

Measure of

sparsity of effects

Number of

replications

Matching of all control factories with their

robust setting (%)

Matching of

control factors

with their robust setting

(%)

Improvement

Ratio(%)

Improvement ratio (from

resolution III noise array)

(%)

Op-Amp Strong 0.105 100 86 97.2 99.37 99.91

CSTR Weak 0.647 100 None 41.17 39.8 96.5

14

System Hierarchy

Measure of

sparsity of effects

Number of

replications

Matching of all control factories with their

robust setting (%)

Matching of

control factors


(%)

Improvement

Ratio(%)

Improvement ratio (from

resolution III noise array)

(%)

TCC Weak 0.725 100 None 43.5 25.2 87

Slider Crank Weak 0.119 100 89 94 96.5 97.6

Results from Resolution III Control & Noise Factor Array & Two-Level

Compounding Noise

Table III indicates that even when using a resolution III control factor array, a

compound noise strategy will predict the robust setting for the Op Amp and the slider

crank with high probability. The compound noise strategy does not work too well with

a resolution III control factor array for the CSTR. The average improvement ratio

possible is only 40%, as opposed to nearly 97% if the resolution III noise factor array

had been used.

Table IV:

System Hierarchy

Measure of

sparsity of effects

Number of

replications

Matching of all

control factories with their

robust setting

(%)

Matching of

control factors


(%)

Improvement

Ratio(%)

Improvementration (from Taguchiʼs compound

Noise)(%)

Op-Amp Strong 0.105 1000 30.8 73 80.2 99.2

PNM Strong 0.0132 24=16 50 68.75 79.9 99.5

Journal Bearing Strong 0.28 23=8 100 100 100 98.84

CSTR Weak 0.647 26=64 54.69 70.3 67.6 58.4

TCC Weak 0.725 25=32 68.75 76.56 74.75 30.65

Slider Crank Weak 0.119 25=32 50 70 55.2 98.5

Average results from Full Factorial Control Factor Array & Two-Level Random

Compound Noise

15

The compound noise strategy also does not perform well with a resolution III control

factor array for the temperature control circuit. The improvement ratio drops from

87% for a resolution III noise array to 25% for two-level compound noise.

In the formulation of compound noise as suggested by Taguchi1 and Phadke3,

we need to know the directionality of noise factors on the system’s response. We need

to run some fractional factorial experiments on the system to gather information about

the directionality of noise factors. We tried a different formulation of compound

noise. Noise factor settings can be picked randomly to form a two-level random

compound noise. We ran such a compound noise strategy with a full factorial control

factor array for each of the six case studies. Table IV presents the average results from

these experiments.

The Op Amp has 21 noise factors, so there can be 221 combinations of random

compound noise. Instead we ran 1000 such combinations of random compound noise

for the Op Amp. The PNM has four noise factors. Hence, there are 24 combinations

of random compound noise for the PNM. The journal bearing has three, the CSTR has

six and the temperature control circuit and slider crank have five noise factors each.

From Table IV we see that such a formulation of compound noise is very effective on

average. Except for the slider crank system, random compound noise can achieve

improvement ratios greater than 68% on average for all systems. For the slider crank

system, half of the replications gave a robust setting of the system and half of them

led to poor settings. Hence, had we used random compound noise with care for the

slider crank system we would have attained a robust setting.

16

CONCLUSIONS

The use of compound noise in robust design experiments leads to a reduction in

experimental effort. In this paper we tried to gage the effectiveness of compound

noise as a robust design method. We built upon the conditions given by Hou for the

effectiveness of compound noise. Also we extended the conditions to include the

possibility of three-factor interactions.

Compound noise as a robust design strategy is very effective on the systems that

exhibit effect sparsity. We ran compound noise on six case studies. The compound

noise strategy predicted robust settings for the systems that exhibited effect sparsity.

The given formulation of compound noise was adopted from Taguchi1 and Phadke3.

For such a formulation we need to know the directionality of noise factors on the

system’s response. To know the directionality of noise factors we need to run

fractional factorial experiments on the system. We tried to measure the effectiveness

of random compound noise for which we do not need to know the directionality of

noise factors. We found that such a formulation of compound noise can be very

effective in obtaining high robustness gains. The reason why such a formulation of

compound noise is very effective is that active effects in the system are sparse. Even

if we randomly compound noise there is a very low probability that two opposite

acting interactions get compounded with each other. The results outlined in Tables I–

IV did not change significantly on removing experimental error from observations. In

this case we will obtain the same conclusions regarding the effectiveness of all robust

design methods on all six systems that we studied.

These results may be used in the overall approach to deploying compound noise

as a robust design strategy. The flowchart in Figure 1 presents our suggestion for

implementing these findings in practice. First of all, practicing engineers must define

the scenario including the system to be improved, what objectives are being sought

and what design variables can be altered.

At this point, it may be possible to consider what assumptions can be made

regarding effect sparsity. It should be noted here that we do not argue that engineers

17

need to make a factual determination of effect sparsity. The experience on the system

should be used to make decision.

If engineers decide that effects are dense, then they should follow the procedure

on the right-hand side of Figure 1. In this case, the results in this paper suggest that

instead of using compound noise, a fractional factorial noise factor array should be

used as the outer array.

If engineers decide that effects are sparse, then they should follow the procedure

on the left-hand side of Figure 1. At this point, engineers should consider the

directionality of noise factors. If the directionality of noise factors is known, then they

should use Taguchi and Phadke’s formulation of compound noise. Otherwise they can

formulate random compound noise as the outer array. From Table IV, we can say that

random compound noise is effective for the systems for which effect sparsity holds.

However, experience, judgment and knowledge of engineering and science are

critical in formulating such a compound noise. It is hoped that the procedure proposed

here in Figure 1 will be of value to practitioners seeking to implement robust design

efficiently and using minimum resources.

18

General Procedure to implement Robust Design Figure 1:394 J. SINGH ET AL.

Define the robust design scenario

Assumptions about effect sparsity

Is the directionality of noise factors known?

Use compound noise (as defined by Taguchi, Phadke) as outer array

Use random compound noise

as outer array

Use fractional/full factorial array as

outer array (instead of compound

noise)

Carry out robust design using the chosen outer array

Sparse Dense

Yes No

Figure 1. Suggested procedure for compound noise in robust design

Acknowledgement

The support of Ford–MIT Alliance is gratefully acknowledged.

REFERENCES

1. Taguchi G. System of Experimental Design: Engineering Methods to Optimize Quality and Minimize Costs.UNIPUB/Kraus International: White Plains, NY, 1987.

2. Robinson TJ. Borror CM. Meyer RH. Robust parameter design: A review. Quality and Reliability EngineeringInternational 2004; 20:81–101.

3. Phadke MS. Quality Engineering Using Robust Design. Prentice-Hall: Englewood Cliffs, NJ, 1989.4. Du X, Sudjianto A, Chen W. An integrated framework for optimization under uncertainty using inverse reliability

strategy. ASME Journal of Mechanical Design 2004; 126(4):562–570.5. Hou XS. On the use of compound noise factor in parameter design experiments. Applied Stochastic Models in Business

and Industry 2002; 18:225–243.6. Satterthwaite FE. Random balanced experimentation (with discussion). Technometrics 1959; 1(2):111–137.7. Holcomb DR, Carlyle WM. Some notes on the construction and evaluation of supersaturated designs. Quality and

Reliability Engineering International 2002; 18:299–304.

Copyright c! 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2007; 23:387–398DOI: 10.1002/qre

19

ReferencesPapers:

1.“Achieving Robust Design Through Quality Engineering”;Shuji Takeshita, Tetsuo

Hosokawa;2007.

2.“Compound Noise”; Jagmeet Singh1, Daniel D. Frey, Nathan Soderborg and Rajesh

Jugulum;2008.

Books:

1.“System of Experimental Design: Engineering Methods to Optimize Quality and

Minimize Costs.” ;G.Taguchi;UNIPUB/Kraus International

2.“Robust parameter design: A review.”; Robinson TJ. Borror CM. Meyer RH.; Quality and Reliability Engineering International

3.”Quality Engineering Using Robust Design”; Phadke MS; Prentice-Hall

20

Documents

Print Seminar Report