7.Robust.design

8/16/2019 7.Robust.design

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7. Robust Design / Taguchi Method

(Ch.12. Robust Design)

Hae-Jin ChoiSchool of Mechanical Engineering,

Chung-Ang University



Robust Design : Taguchi Method

1. Introduction to Robust Design

2. Taguchi Method: MINITAB

Practice

SCHOOL OF MECHANICAL ENG.

CHUNG-ANG UNIVERSITY1

.

UCLLCL Target

o s

s

BAL(y)

DOE and Optimization



Understanding Quality

—

What is quality?— How does one achieve quality?

— Is quality subjective?



OE an d Optimization

— “The purpose of quality engineering is to conduct the research

necessary to develop robust technologies and methods that increase

the competitiveness of new products by reducing their cost and

improving their quality.”

– G.Taguchi



Quality and Robustness—Hit a Target or a Range?

Target Shooter

A




Sony Televisions

(Taguchi and C lausing)

B



The Loss Function

n“The quality of a product is the (minimum) loss imparted by the

product to the society from the time the product is shipped.”

– G . Taguchi

Target Shooter

Example:

A

BAL(y)

BAL(y)



4


L(Y) = K (Y T)

2

B

UCLLCL Target

L o s s

UCLLCL Target

L o s s



Taguchi’s Approach to Quality Engineering

· System Design

– Development of a newproduct or process

“Quality must be engineered !”



5


· Parameter Design

– Investigation to minimizeor reduce performancevariation

· Tolerance Design

– Setting and enforcingboundaries of variation



System Design

— Applying engineering and science

knowledge to configure a product or

process concept

— Defines initial settings of design



6


— Requires innovation to increase qualitySystem Design



Parameter Design

Types of factors:

— Control … factors that can be controlledto make a product robust

— Noise … factors that are difficult,

System

R

p

Control

Factors

Noise

Factors



impossi e, or too expensive to contro

— How to assess the effect of factors on theresponse? EXPERIMENTS

Parameter Design




Tolerance Design

— Minimize the sum of manufacturingand lifetime costs by determiningappropriate tolerances:—Narrowing tolerances increases

manufacturin costs



8


—Widening tolerances increases variationand lifetime costs

— Reducing variation in this stagerequires better materials,components, and machinery. Thisimplies increased costs.

Tolerance Design



Robust Design: Introduction

—One of the important tasks in engineering design is toaccount for variations that occur during manufacturingand operation.

— Robust design is designing a system to be insensitive tothe variations without eliminating or reducing them in

Genichi Taguchi



9


e sys em— In the early 1980’s, Dr. GenechiTaguchi introduced his

approach to quality engineering using experimentaldesign for— Designing products or processes that are robust to

environmental conditions

— Designing products or processes that are robust to componentvariation.



Type I Robust Design

Y

Control Factor

Response

x = a

x = b

Deviation

from noise

when

x = a

x

z

Y

y = f(x,z)

Type I Robust esign

: Insensitive design to variability in Noise

Factors



10


Z

oise

Factor

Deviation

from noise

when

x = a

Deviation

from noise

when

x = b

2∆Z

Minimization of the variation

is response y

Response Y is computed

as a function of control

factors, x, and mean noise

factors, z:

n is the no. noise variables



General approach for Robust Design1. Minimize sensitivity of performance wrt variations without removing

sources of variation

2. Approach the mean of performance to the target

y

Objective orDeviationFunction

y

Objective orDeviationFunction



11


x

robustµ

OptimizingSolution

RobustSolution

DesignVariable

optµ

x

robustµ robustµ

OptimizingSolution

RobustSolution

DesignVariable

optµ



— When working with random variable

s, it is necessary to propagate error (

variability) through systemic equatio

ns (or models).

— How to find the variance of perform

ance wrt variance of noise factors?

Propagation of Error (POE)Y

Response

( , ) f x z?

?



( , ) f x z y x

Control factor

z Noise factor

NID(μz , σ2z)

?

esponse

Z

oise

Factor




— For normal distributions, following equations are a method for estima

ting propagated errors.

— When a response is assumed as

— The mean model for the response is

Propagation of Error (POE)

1 1( ,..., , ,..., )n m y f x x z z e= +



— The variance model for the response is

— Where may be the response surface model of y

11ˆ( ) ( ,..., , ,..., )

mn z z E y f x x m m =

2

2

1

ˆ( )

i

n

z

i i

f V y MSE

z m

s

=

æ ö¶= +ç ÷ç ÷¶è ø

å

11ˆ( ,..., , ,..., )

nn z z f x x m m




Ex mple of the Resin Pl nt Experiment

— A chemical product is produced in a pressure vessel. A factorial

experiment is carried out in the pilot plant to study the factors thought to

influence the filtration rate of this product .

— The factors are A = temperature, B = pressure, C = mole ratio, D=

stirring rate



empera ure s e no se ac or, w c var es w , .




Ex mple: The Resin Pl nt Experiment






The Regression Model

x1 is reworded to z1, since temperature is assumed as the noise factor here

1 1 2 3 2 1 3 1

21.625 9.875 14.625 18.125 16.625ˆ( , ) 70.06

2 2 2 2 2 y z z x x x z x z

æ ö æ ö æ ö æ ö æ ö÷ ÷ ÷ ÷ ÷ç ç ç ç ç= + + + - +÷ ÷ ÷ ÷ ÷ç ç ç ç ç÷ ÷ ÷ ÷ ÷ç ç ç ç çè ø è ø è ø è ø è øx

1 2 3 1 2 1 3

21.625 9.875 14.625 18.125 16.625

ˆ( ) 70.06 2 2 2 2 2 y x x x x x x x

æ ö æ ö æ ö æ ö æ ö

÷ ÷ ÷ ÷ ÷ç ç ç ç ç= + + + - +÷ ÷ ÷ ÷ ÷ç ç ç ç ç÷ ÷ ÷ ÷ ÷ç ç ç ç çè ø è ø è ø è ø è øx



1 12 3

9.875 14.625ˆ[ ] ( , ) 70.06 since 0

2 2 z z E y y x xm m

æ ö æ ö÷ ÷ç ç= = + + =÷ ÷ç ç÷ ÷ç çè ø è øx

The variance model is

1 1

2 2

2 21

1 2 31

2

2 3

ˆ( , ) 21.625 18.125 16.625

ˆ( ) [ ( , )] 2 2 2

(10.81 9.06 8.31 ) 19.51

where 19.51 obtained from regression

z z

y z

V y V y z MSE x x MSE z

x x

MSE

s s

é ù é ùæ ö æ ö æ ö¶ ÷ ÷ ÷ç ç çê ú ê ú= = + = - + +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çê ú ê úè ø è ø è ø¶ ë ûë û

= - + +

=

x

x







The mean model of the response The variance model of the response





Robust optimum point



The mean model of the response The variance model of the response




Taguchi’s Philosophy for Robust Design

— Seek to hit a target rather than a range! Minimize variation around atarget value. Do not focus on tolerance ranges.

— No amount of inspection can improve a product … it is cheaper toensure quality at earlier stages.

—

ualit must be en ineerin into roducts and rocesses vs. bein



19

imposed on products and processes)!

— These issues are addressed in a stage of product development calledparameter design rather than in later, detail stages of design.




Taguchi Method: a Method for Type I Robust Design

— Orthogonal Array

— Signal-to-Noise (S/N) Ratio

— Nominal the best:

SNT = 10 log (y2

)



20

— Larger the better:

— Smaller the better:

— Maximizing S/N Ratio

SN L = -10 log ( 1n

1

yi2

Σi=1

n

)

SN S = -10 log ( 1n

yi2Σ

i=1

n

)




Example: Developing a New DrinkTask: Determine the appropriate amount of punch mix,

orange drink, and cherry drink mix to maximize taste scoresand make the new drink design robust with respect totemperature.

·Control factors:

– Amount of tro ical unch drink mix - or +



21


– Amount of orange drink mix (- or +) – Amount of cherry drink mix (- or +)

·Noise factors:

– Room or chilled temperature

•Response: Taste ScoreNew Drink

Noise factors

C

o

a

o

s

R

p



Taguchi’s “Inner” & “Outer” Array Approach

Taguchi advocates the use of “inner” and “outer” arrays when performing robustdesign.

Res onses

Outer Array

Inner Arra



22


· “Inner” array contains settings of the control factors to be run

· “Outer” array contains settings of the noise factors to be run for each set ofcontrol factors

The combination of these arrays is called the product array , also known as thecomplete parameter design layout.



Taguchi’s Approach to Experiment Design:

Orthogon l Arr ys

Orthogonal Arrays (OAs): “A sophisticated ‘switching system’ intowhich many different design variables and levels of change can beplugged.”

Most commonly used OAs: L4, L9, L12, L16, L18, L27



23




Taguchi-Style Product Array for Our New Drink

R

#

A

m

o

P

A

m

o

O

a

A

m

o

C

y

R

m

T

m

p

NOISEONTROLS

I

c



24


R

#

A

m

o

P

A

m

o

O

a

A

m

o

C

y

R

m

T

m

p

1

2

3

4

I

c

1 1 2

1 2 2

2 1 1

2 2 1

Results for

8 Experiments

‘1’ = 3 tsp per 1/2gallon

‘2’= 5 tsp per 1/2

gallon



Taste Scoring Scale

10

9

8

7

6

5

4

3

2

1

the ultimate taste sensation … great aftertaste … would drinkuntil sick … and then drink more!

extremely enjoyable … would buy occasionally for a treat

tastes ood but not extraordinar … somethin -- taste



25

10

9

8

7

6

5

4

3

2

1

aftertaste, smell, texture -- negatively affects overallexperience

“doesn’t repulse you” but you really don’t like it

repulses you … actively dislike the taste




Taguchi’s Approach for Interpreting Experimental

Results: S/N Ratios

· Taguchi suggests analyzing variation using an appropriately chosen signal-to-

noise (S/N) ratio

· Nominal the best:

SNT = 10 log (y2

S2)

n



· arger e e er:

· Smaller the better:

· It is interesting to note that S/N ratios originated in the communications

industry … they are more sensitive to the mean than to variance … other

variations are available

SN L = -10 og n y i2

i=1

SN S = -10 log ( 1n

y i2Σ

i=1

n

)




Taguchi’s Approach for Interpreting Experimental

Results: y and SN plots

· Employ a graphical approach for analyzing the data and “picking the winner”

· Plot “marginal means” of factors (i.e., average responses) and S/N ratiosagainst factor levels

y

y

SN

SN



27


· Identify factors that heavily influence mean (control factors) and factors that

heavily influence S/N ratio or variability (signals).· Adjust control factors to maximize S/N ratio or reduce the effect of noise.

Adjust signals to bring the mean on target.

Factor A

1 2 3

Factor B

1 2 3

y

Factor A

1 2 3

Factor B

1 2 3

SN



Taguchi Approach: Main Effects Plots

1 - 1 1 - 1 1 - 1

5.00

4.75

4.50

M e a n



28


CherryOrangePunch

4.25

4.00

Choose Factor Levels to

Maximize/Minimize Mean



Taguchi Approach: S/N Ratio Plots

1 - 1 1 - 1 1 - 1

13.4

12.8

12.2

/ N R a t i o



29


CherryOrangePunch

.

11.0

SN n y Lii

n

= -=å10

1 1

21log( )

Choose Factor Levels to

Maximize S/N



Alternative Approach: The Statistical Approach

Factor A

Factor B

F

o

C

(1 )

a

ab

ac

ab c

c

c

(

(

+

a

b

abc

c

(1)

ab

ac

bc

Fractional Factorial

Designs (2

3 1

)



30


Low ( )

High (+)

Factorial Design 2

3

Fractional Factorial

Designs (2

3 1

)

Run Punch Orange Cherry

1 - - -

2 + - -

…8 + + +

Run Punch Orange Cherry

1 - - -

2 + - +

3 - + +

4 + + -

Why Factorial Designs?

More Efficient Statistically (i.e., more info about

effects and interactions with fewer data runs)



Statistical Approach for Interpreting Experimental Results:

Main Effects, Interactions, and Variance

· Calculate the ‘main’ effects of factors, interactions between factors, andvariance

Main Effects

1

highn

ii

high

y

n=

å

1

lown

j j

low

y

n=

å



31


nhigh # data points with factor athigh level (+1)

nlow # data points with factor atlow level (-1)

y i response with factor at highlevel (+1)

y j response with factor at lowlevel (-1)Factor Level

Variance2

1

( )

1

high

highn

ii

high

y y

n=

-

-

å

2

1

( )

1

low

lown

j j

low

y y

n

=

-

-

å



Statistical Approach : Plots of Main Effects,

Interactions, and Variance

Factor 2Levels

Main Effects Interactions Variance

·Plot main effects, interactions, variance



32


Factor Level Factor 1 Level

Factor Level

· Identify factors that heavily influence mean and variance. Pay attention tointeractions.

· Adjust control factors to achieve a preferred tradeoff between on-targetresponse and minimum variance of the response.



Statistical Approach: Main Effects Plot for Means

1 - 1 1 - 1 1 - 1

5.00

4.75

4.50 e a n



33


CherryOrangePunch

4.25

4.00

1st: Look at Main Effects.At what level should each factor be set?



Statistical Approach: Interaction Plot for Means 1 - 1 1 - 1

Orange

Punch

1

-1



34


Cherry

1

-1

2nd: Look at Interaction Plots.

At what level should each factor be set?Has your answer changed?



Statistical Approach: Variance Plot

1 - 1 1 - 1 1 - 1

1.17

1.09

Variance

y y

n

i

i

n

=

-

-=

å ( )2

1

11

)(

tan

2

-

-

=

å

n

y y

iondardDeviat S

n

i

i



35


CherryOrangePunch

1.01

0.93

0.85

S t d D e v

3rd: Look at Variance or Std Deviation.At what levels should factors be set to minimize variation?



Confirmation Experiments

· Taguchi recommends conducting one or more runs at the chosen setting toverify that the predicted performance is in fact realized. This is especiallyimportant if your fractional factorial experimental design did not include thatcombination as a run!

· “Optimal” robust settings for our new drink:



36


· Amount of fruit punch =______

· Amount of orange =______

· Amount of cherry =______



Pros of the Taguchi Approach

· Minimizes “loss to society” through robust design of productsand processes

· Provides a “scientific way to get at robustness”

· “One shot” approach to robust design that does not require an



37


extensive statistica ac groun

· Useful to determine parameter settings that optimize functionalcharacteristics and minimize sensitivity to “noise”

· Has been utilized successfully in many fields; broadly applicable



Cons of the Taguchi Approach

· Requires engineering know-how about specific problems to beeffective

· Engineers generally know the least about problems at the startof a design process



38


· Parameter esign ac ieves a ro ust esign, ut oes not“characterize” the system

· Must be able to exploit interactions between control and noisefactors

· The difference between a control and noise factor is definitional



Cons of the Taguchi Approach … contd.

· Useful information is discarded when collapsing data to compute the S/Nratio … must analyze mean and variance separately

· S/N is ineffective at identifying variation … emphasis in S/N formulas is onthe mean

· “Assumes only main effects are important, ignores interactions between



39


factors” … must analyze main effects and interactions separately

· “Inner” and “outer” array approach may lead to a large number of experiments

· Confounding in Taguchi’s orthogonal outer arrays is high (i.e., main effectsmay be inseparable from interactions) … must use fractional factorial

experiments instead



— Robust design: (1) minimize variance of response and (2) approach totarget of response

— Response Surface Methodology with POE— Find Mean and Variance Models from the obtained response surface

— Plot the surface and find the location of the control factors

— -

Summary: Robust Design Methods



— Taguchi Method— Plan orthogonal Array and conduct experiments

— Find the control factors to maximize S/N ratio and approach the mean to target

performance

— There are pros and cons

— Statistical Method with main and interaction effect of Mean and Variances


S



1

2

3

6

Orthogonal Array

Level Selection

4

Confirmation

Experiment

Practice: Parameter Design Steps

Control factor

Noise factor



Task

Clarification

Problem

Definition

Factor

Selection

Optimum

Design

7Confirmation

5Experiment

S/N ratio

Optimum setting

Experimental

Design

Randomization

Repetition


Design objective

Problem description

Analysis objective



qq 단계단계 11:: Task Clarification

테마 : 자동차 윈도 소음 최소화실험목적 : 자동차가 고급화 되면서 윈도 개폐 시 소음을 최소화 하고자

한다.

::

파라미터 설계 예제 (Smaller-the-better)



브레인 스토밍과 특성요인도로 문제를 분석하였다.

자동차 윈도

소음원인

재질 윈도 오일

전류 홀더

제너레이터 성능

주위온도

틈새 및 각도


( )



qq 단계단계 33:: Factor Selection

특성요인도 분석으로부터 control factor 2수준 4개(A,B,C,D) 취하고, Interaction effect는 AxB만 고려하고, noise factor는 2수준 1개만 고려하여 수준별 2회 반복 실험했다.

인자구분 기호 내용 수준 1

(현재기준)

수준 2

(변경기준)

파라미터 설계 예제 (Smaller-the-better)



43

Control

Factor

A 재질 소 대

B 전류 저 중

C 윈도오일 저 중

D 홀더 저 중

AxB

Noise

Factor N 환경 보통 좋음


미니탭으로 분석하기 -Taguchi



qq 단계단계 4:4: Experimental design

먼저, 분석하고자 하는 인자에 대한 실험을 디자인한다.

Stat> DOE> Taguchi> Create Taguchi Design…




44

2수준 4인자를 선택하고, 먼저 Design… Tab을 선택한다.

실험횟수를 지정한다. 여기서는 L8(27) 이다.





Factors…Tab을 클릭하여 분석하고자 하는 factor를 선택한다.




45

Interaction effect를 분석하기 위해Interaction… Tab을

클릭하여 AB를 선택한다.





Worksheet에 디자인된 내용이 표시된다.

Noise 변수 N1, N2에서 2회 반복 데이터는 직접 입력한다.

실험결과 자료를

Outer arrayInner array

미니탭으로 분석하기 Taguchi



46

Interaction effect AxB는 Worksheet에는 표시되지 않지만, 내부적으로 정보를

가지고 있다. 처음 디자인 때 Interaction effect AxB를 지정했기 때문이다.





Stat> DOE> Taguchi> Analyze Taguchi Design…

Outer array에 있는 인자

N1y1, N1y2, N2y1, N2y2 를

qq 단계단계 5:5: Optimum design




47

선택한다.

분석 창에는 몇 가지 추가적인 Tab이 있다.





Tab은 main effect 및 Interaction effect 그래프를 그리도록 해 준다.




48

Interaction effect을 Plot하기 위해

Terms…를 클릭한다.

Interaction effect AB는 이미 선택되어 있다.이것은 디자인 때 고려된 인자이기 때문이다.








49

이것은 Session 창에 표시되는

간이분석표의 종류를 선택하게 한다.

SN비, 평균, 표준편차 표를

선택할 수 있다.

특성치의 종류를 선택하게 한다.

Larger-is-better, Nominal-is-best, (SN

비, 민감도 Sn), Smaller-is-better를 선택한다. 여기서는 Smaller-is-better 선택.





Response Table for Signal to Noise Ratios

Smaller is better

Level A B C D1 -19.3970 -20.1882 -18.7404 -20.92082 -20.2778 -19.4866 -20.9344 -18.7540Delta 0.8808 0.7015 2.1939 2.1667

Session 창에 S/N ratio 및 평균 관련 표가 나타난다.

Interaction effect은

어떻게 알 수

있을까?




50

an

Response Table for Means

Level A B C D1 9.3125 10.5625 8.6875 11.3752 10.6875 9.4375 11.3125 8.625Delta 1.3750 1.1250 2.6250 2.750Rank 3 4 2 1

S/N ratio의 수준간의 차이가 큰 C, D 인자가 deviation 관련 중요인자로 선택된다.





A B

Main Effects Plot (data means) for SN ratios

i l i ll i

21

Interaction Plot (data means) for SN ratios

i l i ll i

Graph 창에 S/N ratio의 main effect와 Interaction effect가 나타난다.

우리는 이 그래프로부터 S/N ratio를 크게 하는 조건을 찾을 수 있다.최적조건 : A1, B1, C1, D2 을 확인할 수 있다.




51

M e a n o f S N r a t i o s

21

-19.0

-19.5

-20.0

-20.5

-21.0

21

21

-19.0

-19.5

-20.0

-20.5

-21.0

21

C D

Signal-to-noise: Smaller is better

A

-18.0

-19.2

-20.4

-21.6

-22.8

21

-18.0

-19.2

-20.4

-21.6

-22.8

B

A

12

B

1

2

Signal-to-noise: Smaller is better





분석한 결과는 Session 창과 Graph 창에서 설계변수의 최적조건을 판단할 수

있다.

Session 창의 Rank와 Graph 창의 main effect 및 Interaction effect 그래프로「자동차 윈도의 소음을 최소화」하기 위한 설계변수의 최적조건




즉, S/N ratio를 최대화하는 조건은 A1 B1 C1 D2 임을 확인할 수 있다.

최적조건에서 특성치의 예측값을 구하기 위해

Predict Taguchi Results…를 실행한다.





Stat> DOE> Taguchi> Predict Taguchi Results…

qq 단계단계 6:6: Optimum design results

미니탭 분석하기 aguc



53

평균, S/N ratio, 표준편차, 표준편차의자연로그 등을 예측한다.





Tab을 눌러 최적조건을 지정해 준다.

최적조건에서의 S/N ratio는 Excel에서

구한 결과와 일치한다.

최적조건 : A1, B1, C1, D2

미니탭 분석하기 g



54

Predicted values

S/N Ratio Mean StDev Log(StDev)-15.4204 4.6875 0.773268 0.241744

Factor levels for predictionsA B C D

1 1 1 2

최적조건에서의 Y의 평균 : 4.69





※ 최종분석결과 : 현재와 최적조건에서의 S/N ratio와 추정값

분석에서 얻은 최적조건에서 특성치를 예측한다.

Stat> DOE> Taguchi> Predict Taguchi Results…

미니탭 분석하기 g



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현재:

A( 1 )B( 1 )C( 1 )D( 1 )최적: A( )B( )C( )D( )

S/N ratio -17.5871 -15.4204

Average 7.4375 4.6875





개선된 SN비 = (-15.4204 ) – ( -17.5871 ) = ( 2.1667 )손실금액의 감소 = 10(0.2167 ) = ( 1.65 ) 배 개선

만약 현재의 품목 단위당 손실 금액이 100원이라면, 100/1.65 = 60.6이므로,

100-60.6 = 39.4가 되어 단위당 손실 금액이 39.4원 정도 개선된다는 의미임.

qq 단계단계 7:7: Confirmation

g



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Control Factor: 최적조건

Noise Factor

확인(재현) 실험최적조건에서의

추정치

차이 30% 이내

기타 조건은 작업성, 편의성, 경제성 등을 고려하여

최적조건을 표준화 시킨다.





Taguchi 분석으로 결과를 얻을 수 있으며,

ANOVA 및 effect에 대한 Graph를 얻을 수 있다. 이로부터 최적조건의 설정 및최적조건에서의 예측치를 정확히 계산할 수 있으며,

또한 Taguchi의 Predict Taguchi Results…로 최적조건에서의 예측치를 얻을 수있다.




재현성 확인실험 결과 예측치와 차이가 30% 이내이면 재현성이 있다고 판단하며, 재현성이 없는 경우는

①① Interaction effectInteraction effect 존재존재 ②② noise factornoise factor의의 영향영향 ③③ 순수순수 실험오차실험오차

이 경우 적절한 조처가 필요하다.


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