Hae-Jin Choi School of Mechanical Engineering

Chung-Ang University

8 Robust Design Taguchi Method

(Ch12 Robust Design)

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -1-

Robust Design Taguchi Method

1 Introduction to Robust Design

2 Taguchi Method MINITAB Practice

3 Helicopter Design Game

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -1-

Robust Design Taguchi Method

1 Introduction to Robust Design

2 Taguchi Method MINITAB Practice

3 Helicopter Design Game

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -2-

Understanding Quality

DOE and Optimization

What is quality

How does one achieve quality

Is quality subjective

ldquoThe 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 qualityrdquo

ndash G Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -3-

Quality and RobustnessmdashHit a Target or a Range

DOE and Optimization

Sony Televisions (Taguchi and Clausing)

Target Shooter

A

B

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -4-

4

The Loss Function

DOE and Optimization

ldquoThe quality of a product is the (minimum) loss imparted by the product to the society from the time the product is shippedrdquo

ndash G Taguchi

L(Y) = K (Y - T)2

Target Shooter Example

A

B

UCLLCL Target

Loss

BAL(y)

UCLLCL Target

Loss

BAL(y)

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -5-

5

Taguchirsquos Approach to Quality Engineering

DOE and Optimization

bull System Design

ndash Development of a new product or process

bull Parameter Design

ndash Investigation to minimize or reduce performance variation

bull Tolerance Design

ndash Setting and enforcing boundaries of variation

ldquoQuality must be engineered rdquo

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -6-

6

System Design

DOE and Optimization

Applying engineering and science knowledge to configure a product or process concept

Defines initial settings of design characteristics

Requires innovation to increase quality System Design

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -7-

Parameter Design

Types of factors

Control hellip factors that can be controlled to make a product robust

Noise hellip factors that are difficult impossible or too expensive to control

How to assess the effect of factors on the response EXPERIMENTS

Parameter Design

System

Responses

Control Factors

Noise Factors

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -8-

8

Tolerance Design

DOE and Optimization

Minimize the sum of manufacturing and lifetime costs by determining appropriate tolerances Narrowing tolerances increases

manufacturing costs Widening tolerances increases variation

and lifetime costs Reducing variation in this stage

requires better materials components and machinery This implies increased costs

Tolerance Design

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -9-

9

Robust Design Introduction

DOE and Optimization

One of the important tasks in engineering design is to account for variations that occur during manufacturing and operation

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

In the early 1980rsquos Dr Genechi Taguchi introduced his approach to quality engineering using experimental design for Designing products or processes that are robust to

environmental conditions Designing products or processes that are robust to component

variation

Genichi Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -10-

10

Type I Robust Design

DOE and Optimization

Y

Z

Control Factor

Noise Factor

Response x = a

x = b

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

x

z

Y y = f(xz)

Type I Robust Design Insensitive design to variability in Noise Factors

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -11-

11

General approach for Robust Design

DOE and Optimization

1 Minimize sensitivity of performance wrt variations without removing sources of variation

2 Approach the mean of performance to the target

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -12-

When working with random variables it is necessary to propagate error (variability) through systemic equations (or models)

How to find the variance of performance wrt variance of noise factors

Propagation of Error (POE)

( )f x z y x Control factor

z Noise factor

NID(μz σ2 z)

Response

Y

Z Noise Factor

Response

( )f x z

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -13-

For normal distributions following equations are a method for estimating propagated errors

When a response is assumed as

The mean model for the response is

The variance model for the response is

Where may be the response surface model of y

Propagation of Error (POE)

1 1( )n my f x x z z e= +

11ˆ( ) ( )

mn z zE y f x x micro micro=

2

2

1

ˆ( )

i

n

zi i

fV y MSEz

micro

σ=

part= + part sum

11ˆ ( )

nn z zf x x micro micro

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -14-

Example of the Resin Plant 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

A = temperature is the noise factor which varies with NID(0 12)

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -15-

Example The Resin Plant Experiment

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -16-

The Regression Model

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

1 1 2 3 2 1 3 121625 9875 14625 18125 16625ˆ( ) 7006

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

x

The mean model of the response is

1 12 39875 14625ˆ[ ] ( ) 7006 since 0

2 2z zE y y x x

x

The variance model is

1 1

2 22 21

1 2 31

22 3

ˆ( ) 21625 18125 16625ˆ( ) [ ( )]2 2 2

(1081 906 831 ) 1951where 1951 obtained from regression

z zy zV y V y z MSE x x MSE

z

x xMSE

xx

1 2 3 1 2 1 321625 9875 14625 18125 16625ˆ( ) 7006

2 2 2 2 2y x x x x x x x

x

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -17-

Example The Resin Plant Experiment

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

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -18-

Example The Resin Plant Experiment

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

Robust optimum point

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -19-

19

Taguchirsquos Philosophy for Robust Design

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

No amount of inspection can improve a product hellip it is cheaper to ensure quality at earlier stages

Quality must be engineering into products and processes (vs being imposed on products and processes)

These issues are addressed in a stage of product development called parameter design rather than in later detail stages of design

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -20-

20

Taguchi Method a Method for Type I Robust Design

Orthogonal Array

Signal-to-Noise (SN) Ratio Nominal the best

Larger the better

Smaller the better

Maximizing SN Ratio

SNT = 10 log (y2

S2)

SN L = -10 log ( 1 n

1 y i 2 Σ

i=1

n )

SN S = -10 log ( 1 n y i

2 Σ i=1

n

)

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -21-

21

Example Developing a New Drink

DOE and Optimization

Task Determine the appropriate amount of punch mix orange drink and cherry drink mix to maximize taste scores and make the new drink design robust with respect to temperature

bullControl factors

ndashAmount of tropical punch drink mix (- or +) ndashAmount of orange drink mix (- or +) ndashAmount of cherry drink mix (- or +)

bullNoise factors

ndashRoom or chilled temperature

bullResponse Taste Score New Drink

Noise factors

Control fa

cto

rs

Response

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -22-

22

Taguchirsquos ldquoInnerrdquo amp ldquoOuterrdquo Array Approach

DOE and Optimization

Taguchi advocates the use of ldquoinnerrdquo and ldquoouterrdquo arrays when performing robust design

bull ldquoInnerrdquo array contains settings of the control factors to be run

bull ldquoOuterrdquo array contains settings of the noise factors to be run for each set of control factors

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

Responses

Outer Array

Inner Array

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23

Taguchirsquos Approach to Experiment Design Orthogonal Arrays

DOE and Optimization

Orthogonal Arrays (OAs) ldquoA sophisticated lsquoswitching systemrsquo into which many different design variables and levels of change can be pluggedrdquo

Most commonly used OAs L4 L9 L12 L16 L18 L27

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -24-

24

Taguchi-Style Product Array for Our New Drink

DOE and Optimization

Run

Am

ount

Punch

Am

ount

Ora

nge

Am

ount

Cherry

Room

T

em

p

NOISE CONTROLS

1

2

3

4

Iced

1 1 2

1 2 2

2 1 1

2 2 1

Results for 8 Experiments

lsquo1rsquo = 3 tsp per 12 gallon

lsquo2rsquo= 5 tsp per 12 gallon

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -25-

25

Taste Scoring Scale 10

9

8

7

6

5

4

3

2

1

the ultimate taste sensation hellip great aftertaste hellip would drink until sick hellip and then drink more extremely enjoyable hellip would buy occasionally for a treat tastes good but not extraordinary hellip something -- taste aftertaste smell texture -- negatively affects overall experience

ldquodoesnrsquot repulse yourdquo but you really donrsquot like it repulses you hellip actively dislike the taste

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -26-

Taguchirsquos Approach for Interpreting Experimental Results SN Ratios bull Taguchi suggests analyzing variation using an appropriately chosen signal-to-

noise (SN) ratio

bull Nominal the best

bull Larger the better

bull Smaller the better

bull It is interesting to note that SN ratios originated in the communications industry hellip they are more sensitive to the mean than to variance hellip other variations are available

SNT = 10 log (y2

S2)

SN L = -10 log ( 1 n

1 y i 2 Σ

i=1

n )

SN S = -10 log ( 1 n y i 2 Σ

i=1

n )

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -27-

27

Taguchirsquos Approach for Interpreting Experimental Results y and SN plots

DOE and Optimization

bull Employ a graphical approach for analyzing the data and ldquopicking the winnerrdquo

bull Plot ldquomarginal meansrdquo of factors (ie average responses) and SN ratios against factor levels

bull Identify factors that heavily influence mean (control factors) and factors that heavily influence SN ratio or variability (signals)

bull Adjust control factors to maximize SN ratio or reduce the effect of noise Adjust signals to bring the mean on target

Factor A

1 2 3

y

Factor B

1 2 3

y

Factor A

1 2 3

SN

Factor B

1 2 3

SN

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -28-

28

Taguchi Approach Main Effects Plots

DOE and Optimization

CherryOrangePunch

1-1 1-1 1-1

500

475

450

425

400

Mea

n

Choose Factor Levels to MaximizeMinimize Mean

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -29-

29

Taguchi Approach SN Ratio Plots

DOE and Optimization

CherryOrangePunch

1-1 1-1 1-1134

128

122

116

110

SN

Ratio

SNn yL

ii

n

= minus=sum10 1 1

21

log( ) Choose Factor Levels to Maximize SN

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -30-

30

Alternative Approach The Statistical Approach

DOE and Optimization

Factor A Factor B

Facto

r C

(1) a

ab b

ac

abc bc

c

Low (-) High (+)

(-)

(+)

Factorial Design 23

a

b

abc

c

(1)

ab

ac

bc

Fractional Factorial Designs (23-1)

Run Punch Orange Cherry

1 - - -

2 + - -

hellip

8 + + +

Run Punch Orange Cherry

1 - - -

2 + - +

3 - + +

4 + + -

Why Factorial Designs More Efficient Statistically (ie more info about effects and interactions with fewer data runs)

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -31-

31

Statistical Approach for Interpreting Experimental Results Main Effects Interactions and Variance

DOE and Optimization

bull Calculate the lsquomainrsquo effects of factors interactions between factors and variance

Factor Level

Main Effects

1

highn

ii

high

yn=sum

1

lown

jj

low

y

n=sum

nhigh data points with factor at high level (+1)

nlow data points with factor at low level (-1)

yi response with factor at high level (+1)

yj response with factor at low level (-1) Factor Level

Variance 2

1( )

1high

highn

ii

high

y y

n=

minus

minussum

21( )

1

low

lown

jj

low

y y

n=

minus

minus

sum

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -32-

32

Statistical Approach Plots of Main Effects Interactions and Variance

DOE and Optimization

Factor Level Factor 1 Level

Factor 2 Levels

Main Effects Interactions

Factor Level

Variance

bull Plot main effects interactions variance

bull Identify factors that heavily influence mean and variance Pay attention to interactions

bull Adjust control factors to achieve a preferred tradeoff between on-target response and minimum variance of the response

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33

Statistical Approach Main Effects Plot for Means

DOE and Optimization

CherryOrangePunch

1-1 1-1 1-1

500

475

450

425

400

Mea

n

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

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -34-

34

Statistical Approach Interaction Plot for Means

DOE and Optimization

1-1 1-1

Cherry

Orange

Punch

1

-1

1

-1

2nd Look at Interaction Plots At what level should each factor be set Has your answer changed

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -35-

35

Statistical Approach Variance Plot

DOE and Optimization

CherryOrangePunch

1-1 1-1 1-1

117

109

101

093

085

StdD

ev

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

Variancey y

n

ii

n

=minus

minus=sum ( )2

1

11

)(tan

2

minus

minus=sum

n

yyiondardDeviatS

n

ii

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -36-

36

Confirmation Experiments

DOE and Optimization

bull Taguchi recommends conducting one or more runs at the chosen setting to verify that the predicted performance is in fact realized This is especially important if your fractional factorial experimental design did not include that combination as a run

bull ldquoOptimalrdquo robust settings for our new drink

bull Amount of fruit punch =______

bull Amount of orange =______

bull Amount of cherry =______

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -37-

37

Pros of the Taguchi Approach

DOE and Optimization

bull Minimizes ldquoloss to societyrdquo through robust design of products and processes

bull Provides a ldquoscientific way to get at robustnessrdquo

bull ldquoOne shotrdquo approach to robust design that does not require an extensive statistical background

bull Useful to determine parameter settings that optimize functional characteristics and minimize sensitivity to ldquonoiserdquo

bull Has been utilized successfully in many fields broadly applicable

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -38-

38

Cons of the Taguchi Approach

DOE and Optimization

bull Requires engineering know-how about specific problems to be effective

bull Engineers generally know the least about problems at the start of a design process

bull Parameter design achieves a robust design but does not ldquocharacterizerdquo the system

bull Must be able to exploit interactions between control and noise factors

bull The difference between a control and noise factor is definitional

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -39-

39

Cons of the Taguchi Approach hellip contd

DOE and Optimization

bull Useful information is discarded when collapsing data to compute the SN ratio hellip must analyze mean and variance separately

bull SN is ineffective at identifying variation hellip emphasis in SN formulas is on the mean

bull ldquoAssumes only main effects are important ignores interactions between factorsrdquo hellip must analyze main effects and interactions separately

bull ldquoInnerrdquo and ldquoouterrdquo array approach may lead to a large number of experiments

bull Confounding in Taguchirsquos orthogonal outer arrays is high (ie main effects may be inseparable from interactions) hellip must use fractional factorial experiments instead

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -40-

Robust design (1) minimize variance of response and (2) approach to target 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 Systematic and useful method with multi-objective optimization techniques

Taguchi Method Plan orthogonal Array and conduct experiments Find the control factors to maximize SN ratio and approach the mean to target

performance There are pros and cons

Statistical Method with main and interaction effect of Mean and Variances

DOE and Optimization

Summary Robust Design Methods

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -41-

1

Task

Clarification

2

Problem

Definition

3

Factor

Selection

6

Optimum

Design

7

Confirmation

5

Experiment

Orthogonal Array

Level Selection

SN ratio Optimum setting

4

Experimental

Design

Confirmation

Experiment

Randomization

Repetition

Practice Parameter Design Steps

DOE and Optimization

Control factor Noise factor

Design objective

Problem description Analysis objective

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -42-

단계 1 Task Clarification

테마 자동차 윈도 소음 최소화

실험목적 자동차가 고급화 되면서 윈도 개폐 시 소음을 최소화 하고자 한다

단계 2 Problem Definition

브레인 스토밍과 특성요인도로 문제를 분석하였다

자동차 윈도

소음원인

재질 윈도 오일

전류 홀더

제너레이터 성능

주위온도

틈새 및 각도

파라미터 설계 예제 (Smaller-the-better)

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -43-

43

단계 3 Factor Selection

특성요인도 분석으로부터 control factor 2수준 4개(ABCD) 취하고 Interaction effect는 AxB만 고려하고 noise factor는 2수준 1개만 고려하여 수준별 2회 반복 실험했다

인자구분 기호 내용 수준 1

(현재기준) 수준 2

(변경기준)

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

DOE and Optimization

파라미터 설계 예제 (Smaller-the-better)

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -44-

44

단계 4 Experimental design

먼저 분석하고자 하는 인자에 대한 실험을 디자인한다

Statgt DOEgt Taguchigt Create Taguchi Designhellip

2수준 4인자를 선택하고 먼저 Designhellip Tab을 선택한다

실험횟수를 지정한다 여기서는 L8(27) 이다

미니탭으로 분석하기 -Taguchi

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -45-

45

FactorshellipTab을 클릭하여 분석하고자 하는 factor를 선택한다

Interaction effect를 분석하기 위해

Interactionhellip Tab을

클릭하여 AB를 선택한다

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -46-

46

Worksheet에 디자인된 내용이 표시된다

Noise 변수 N1 N2에서 2회 반복 데이터는 직접 입력한다

Interaction effect AxB는 Worksheet에는 표시되지 않지만 내부적으로 정보를

가지고 있다 처음 디자인 때 Interaction effect AxB를 지정했기 때문이다

실험결과 자료를

입력한다

Outer array Inner array

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -47-

47

Statgt DOEgt Taguchigt Analyze Taguchi Designhellip

Outer array에 있는 인자

N1y1 N1y2 N2y1 N2y2 를

선택한다

분석 창에는 몇 가지 추가적인 Tab이 있다

단계 5 Optimum design

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -48-

48

Tab은 main effect 및 Interaction effect 그래프를 그리도록 해 준다

Interaction effect을 Plot하기 위해

Termshellip를 클릭한다

Interaction effect AB는 이미 선택되어 있다

이것은 디자인 때 고려된 인자이기 때문이다

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -49-

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 선택

한다 여기서는 Smaller-is-better 선택

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -50-

50

Response Table for Signal to Noise Ratios Smaller is better Level A B C D 1 -193970 -201882 -187404 -209208 2 -202778 -194866 -209344 -187540 Delta 08808 07015 21939 21667 Rank 3 4 1 2 Response Table for Means Level A B C D 1 93125 105625 86875 11375 2 106875 94375 113125 8625 Delta 13750 11250 26250 2750 Rank 3 4 2 1

Session 창에 SN ratio 및 평균 관련 표가 나타난다

SN ratio의 수준간의 차이가 큰 C D 인자가 deviation 관련 중요인자로 선택된다

Interaction effect은 어떻게 알 수 있을까

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -51-

51

Mea

n of

SN

rati

os

21

-190

-195

-200

-205

-21021

21

-190

-195

-200

-205

-21021

A B

C D

Main Effects Plot (data means) for SN ratios

Signal-to-noise Smaller is better

A

21

-180

-192

-204

-216

-228

21

-180

-192

-204

-216

-228

B

A12

B12

Interaction Plot (data means) for SN ratios

Signal-to-noise Smaller is better

Graph 창에 SN ratio의 main effect와 Interaction effect가 나타난다

우리는 이 그래프로부터 SN ratio를 크게 하는 조건을 찾을 수 있다

최적조건 A1 B1 C1 D2 을 확인할 수 있다

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -52-

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 SN ratio를 최대화하는 조건은 A1 B1 C1 D2 임을 확인할 수 있다

최적조건에서 특성치의 예측값을 구하기 위해

Predict Taguchi Resultshellip를 실행한다

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -53-

53

Statgt DOEgt Taguchigt Predict Taguchi Resultshellip

평균 SN ratio 표준편차 표준편차의 자연로그 등을 예측한다

단계 6 Optimum design results

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -54-

54

Predicted values SN Ratio Mean StDev Log(StDev) -154204 46875 0773268 0241744 Factor levels for predictions A B C D 1 1 1 2

Tab을 눌러 최적조건을 지정해 준다

최적조건에서의 SN ratio는 Excel에서 구한 결과와 일치한다

최적조건 A1 B1 C1 D2

최적조건에서의 Y의 평균 469

DOE and Optimization

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55

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

SN ratio -175871 -154204

Average 74375 46875

최종분석결과 현재와 최적조건에서의 SN ratio와 추정값

분석에서 얻은 최적조건에서 특성치를 예측한다

Statgt DOEgt Taguchigt Predict Taguchi Resultshellip

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -56-

56

Control Factor 최적조건

개선된 SN비 = (-154204 ) ndash ( -175871 ) = ( 21667 )

손실금액의 감소 = 10(02167 ) = ( 165 ) 배 개선

만약 현재의 품목 단위당 손실 금액이 100원이라면 100165 = 606이므로

100-606 = 394가 되어 단위당 손실 금액이 394원 정도 개선된다는 의미임

Noise Factor

확인(재현) 실험 최적조건에서의

추정치

차이 30 이내

기타 조건은 작업성 편의성 경제성 등을 고려하여 최적조건을 표준화 시킨다

단계 7 Confirmation

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미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -57-

Taguchi 분석으로 결과를 얻을 수 있으며 ANOVA 및 effect에 대한 Graph를 얻을 수 있다 이로부터 최적조건의 설정 및 최적조건에서의 예측치를 정확히 계산할 수 있으며 또한 Taguchi의 Predict Taguchi Resultshellip로 최적조건에서의 예측치를 얻을 수 있다 재현성 확인실험 결과 예측치와 차이가 30 이내이면 재현성이 있다고 판단하며 재현성이 없는 경우는 ① Interaction effect 존재 ② noise factor의 영향 大 ③ 순수 실험오차 이 경우 적절한 조처가 필요하다

DOE and Optimization

미니탭으로 분석하기 -Taguchi

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -58-

① 설계변수들로 이루어진 실험(직교배열) 구성

② 성능특성치의 반복 관측치 측정

③ SN비와 와 감도(평균) 를 계산

④ SN비에 대한 분산분석으로 SN비에 영향이 큰 설계변수 찾음

⑤ 설계변수의 최적 수준 결정(SN비 최대로 하는 것)

⑥ 감도 반응표에서 감도에 영향이 큰 설계변수 찾고 최적수준 결정

그 밖의 변수 비용 편의성을 고려하여 적절한 수준으로 선택

예제2 파라미터 설계방법 (Nominal-the-best 경우)

DOE and Optimization

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망목특성의 파라미터 설계 예제

단계 1 Task Clarification

테마 금도금 조건의 최적화로 금 소비량 절감

실험목적 도금두께의 산포를 최소한으로 억제하는 control factor의

최적수준과 도금두께의 평균값을 목표값(8)에 맞추기

위한 수준을 찾는다

단계 2 Problem Definition

금도금 두께의

산포

온도 전류밀도

산도(pH) 금도금 두께

양극치수

장전량

니켈농도

주위온도

DOE and Optimization

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인자구분 기호 내용 수준 1

(현재기준)

수준 2

(변경기준)

control factor

A 금도금 두께 10 15

B 산도 5 pH 7 pH

C 전류밀도 8 10

D 온도 50 ordm 80 ordm

AxB

noise factor N 환경(청결) 보통 좋음

단계 3 Factor Selection

control factor로 금도금 두께 산도 전류밀도 온도를 2수준으로 잡고

두께와 산도와는 Interaction effect을 고려하고 noise factor는 환경의 청결도

2수순으로 수준별 2회 반복 실험했다

망목특성의 파라미터 설계 예제

DOE and Optimization

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망목특성의 파라미터 설계 예제

단계 4 Experimental Design and Experiment

L8 x M1 직교배열표에 인자를 할당하여 실험한 결과이다

R1 R2 R1 R2

1 0 0 0 0 0 0 0 5 8 9 7 725 1256

2 0 0 0 1 1 1 1 9 9 8 4 750 997

3 0 1 1 0 0 1 1 9 15 10 12 1150 1276

4 0 1 1 1 1 0 0 12 8 10 14 1100 1259

5 1 0 1 0 1 0 1 9 13 5 12 975 867

6 1 0 1 1 0 1 0 20 23 19 9 1775 931

7 1 1 0 0 1 1 0 6 5 7 7 625 1630

8 1 1 0 1 0 0 1 9 4 12 11 900 806

기본표시 a b ab c ac bc abc

군 1

실험번호

내 측 배 열 외측배열

A B AB C D e e SN비N1 N2

2 3

N(잡음인자)

Bar(y)

DOE and Optimization

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미니탭으로 분석하기

Nominal-the-best 특성의 미니탭 절차

Experimental design Taguchi로 orthogonal array 생성 Statgt DOEgt Taguchigt Create Taguchi Design

Optimum condition Statgt DOEgt Taguchigt Analyze Taguchi Design Prediction on optimum condition Statgt DOEgt Taguchigt Predict Taguchi Results

DOE and Optimization

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미니탭으로 분석하기-Taguchi

Statgt DOEgt Taguchigt Create Taguchi Designhellip

Taguchi로 디자인 생성 분석

Statgt DOEgt Taguchigt Analyze Taguchi Designhellip

Tab에서 SN비와 평균을 저장한다

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -64-

Mea

n of

Mea

ns

21

114

108

102

96

90

21

21

114

108

102

96

90

21

A B

C D

Main Effects Plot (data means) for Means

Mea

n of

SN

rati

os

21

12

11

10

21

21

12

11

10

21

A B

C D

Main Effects Plot (data means) for SN ratios

Signal-to-noise Nominal is best (10Log(Ybar2s2))

Taguchi에 의한 분석 결과

미니탭으로 분석하기-Taguchi

(1) SN비 (2) 평균(Mean)

DOE and Optimization

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(1) 1단계최적화 산포(SN비)

- 영향 큰 인자

- A( ) B( ) C( ) D( )

(2) 2단계최적화 감도(Slope)

- 영향 큰 인자

- A( ) B( ) C( ) D( )

there4 최종적인 최적조건은

- A( ) B( ) C( ) D( )

미니탭으로 분석하기-Taguchi

DOE and Optimization

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미니탭으로 분석하기-Taguchi

Taguchi로 분석한 결과는

Session 창과 Graph 창으로 설계변수의 최적조건을 판단한다

Session 창의 Rank와 Graph 창의 주효과 및 Interaction effect 그래프로

「금도금두께의 최적조건」을 찾기 위한 설계변수의 최적조건은

lsquoA( ) B( ) C( ) D( )rsquo 임을 확인할 수 있다

최적조건에서 특성치의 예측값을 구하기 위해

Predict Taguchi Resultshellip를 실행한다

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -67-

분석에서 얻은 최적조건에서 특성치를 예측한다

Statgt DOEgt Taguchigt Predict Taguchi Resultshellip

미니탭으로 분석하기-Taguchi

현재 A( )B( )C( )D( ) 최적 A( )B( )C( )D( )

SN비

평균

최종분석 결과 현재와 최적조건에서의 SN비와 추정값

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -68-

Factor 기호 인자 정의 1 수준 2 수준 3수준

A 압출 장치 A1형 A2형 -

B 속도 저속 중속 고속

C 온도 저온 보통 고온

D 절연재 D1 D2 D3

E CV압력 저압 중압 고압

F CV속도 저속 중속 고속

G 장력 저 중 고

H COATING I1 I2 I3

N1 작업 샘플 S1 S2

N2 샘플 위치 P1 P2

제어인자

잡음인자

1 Problem definition

자동차 점화 케이블의 코어 인장력의 규격을 40+-15 파운드에 맞도록 제품을 개발하고 있다 주요 인자 및 수준은 다음과 같다 A인자 만 2수준이고 나머지는 모두 3수준이다

Nominal-the-best 사례 (2)

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A B C D E F G H P1 P2 P1 P2 STD MEAN SN

1 1 1 1 1 1 1 1 1 30 40 38 49 78 393 14

2 1 1 2 2 2 2 2 2 10 15 25 25 75 188 796

3 1 1 3 3 3 3 3 3 49 53 53 55 252 525 264

4 1 2 1 1 2 2 3 3 62 58 52 68 673 60 19

5 1 2 2 2 3 3 1 1 30 50 49 62 132 478 112

6 1 2 3 3 1 1 2 2 10 25 29 36 11 25 714

7 1 3 1 2 1 3 2 3 58 42 41 50 793 478 156

8 1 3 2 3 2 1 3 1 28 29 32 31 183 30 243

9 1 3 3 1 3 2 1 2 110 74 94 115 185 983 145

10 2 1 1 3 3 2 2 1 76 88 66 104 164 835 142

11 2 1 2 1 1 3 3 2 52 37 54 59 947 505 145

12 2 1 3 2 2 1 1 3 55 79 62 98 192 735 117

13 2 2 1 2 3 1 3 2 5 35 16 42 17 245 316

14 2 2 2 3 1 2 1 3 52 96 79 91 197 795 121

15 2 2 3 1 2 3 2 1 50 70 56 65 896 603 166

16 2 3 1 3 2 3 1 2 15 20 18 21 265 185 169

17 2 3 2 1 3 1 2 3 51 62 59 71 826 608 173

18 2 3 3 2 1 2 3 1 77 83 66 74 707 75 205

N1No N2

외측 배열내측 배열(L18(2 13 7))

2 Experimental design

DOE and Optimization

Nominal-the-best 사례 (2)

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -70-

3 데이터 분석 ndash SN비

Level A B C D E F G H

1 155650 147893 138058 159940 139909 129406 133964 167864

2 141049 115237 145705 116735 160636 147115 131227 107016

3 181918 161285 168373 144503 168527 179858 170168

Delta 14602 66681 23227 51638 20727 39122 48631 63152

Rank 8 1 6 3 7 5 4 2

분산 분석 결과 유의한 인자로는 B D F G H로 볼 수 있고 SN를 최대로 해주는 조건은 B3D3F3G3H3 임을 알 수 있다

Analysis of Variance for SNRA2 using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

B 2 133411 133411 66705 1000 0009

D 2 92086 92086 46043 690 0022

F 2 46052 46052 23026 345 0091

G 2 89575 89575 44788 671 0024

H 2 153918 153918 76959 1154 0006

Error 7 46695 46695 6671

Total 17 561738

Pooling 후의 모습

Mea

n of

SN

rati

os 21

18

15

12

321 321

321

18

15

12

321 321

321

18

15

12

321

A B C

D E F

G H

Main Effects Plot (data means) for SN ratios

Signal-to-noise Nominal is best (10Log(Ybar2s2))

DOE and Optimization

Nominal-the-best 사례 (2)

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -71-

3 데이터 분석 - 평균

Level A B C D E F G H

1 465833 530000 455833 615000 528333 421667 594583 559583

2 584444 495000 478750 478750 435000 691667 493333 392500

3 550417 640833 481667 612083 462083 487500 623333

Delta 118611 55417 185000 136250 177083 270000 107083 230833

Rank 6 8 3 5 4 1 7 2

Analysis of Variance for MEAN2 using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

A 1 6331 6331 6331 208 0187

C 2 12204 12204 6102 200 0197

E 2 9417 9417 4708 155 0270

F 2 25448 25448 12724 418 0057

H 2 17053 17053 8526 280 0120

Error 8 24356 24356 3045

Total 17 94809

평균에 대한 분산분석 결과 평균에 대하여 P-Value 020 에서 유의한 인자는 ACFH이고 40에 가까운 수준은 A1C1F1H2임을 알 수 있다

Mea

n of

MEA

N1

21

70

60

50

40

321

321

70

60

50

40

321

A C

F H

Main Effects Plot (data means) for MEAN1

DOE and Optimization

Nominal-the-best 사례 (2)

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -72-

3 데이터 분석 - 요약

SN비 및 평균치를 분석한 결과 다음과 같은 결론을 내릴 수 있다

A B C D E F G H

산포제어인자(SN) B3 D3 F3 G3 H3

평균 조정인자 A1 C1 F1 H2

기타 인자 E1

F 와 H 는 산포와 평균값 모두에 영향을 주므로 산포 control factor로 분류 한다 기타 E 는 작업성이나 경제성을 고려하여 수준을 결정 한다 최적 조건은 A1B3C1D3E1F3G3H3 이다

DOE and Optimization

Nominal-the-best 사례 (2)

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -73-

Nominal-the-best 결론

최적조건을 탐색하는 과정은 2단계로 진행한다 2단계 최적화

1단계 산포를 감소시키는 인자 선택최적수준 결정 rarr

2단계 평균을 목표치에 맞추는 인자최적수준 결정 rarr

만약 산포와 평균에 모두 유의한 인자는 산포에 유의한 인자로

분류한다

최적조건에서 SN비의 추정 및 평균의 추정치를 구하여

SN비가 얼마나 개선되었으며 평균이 목표치에 얼마나 접근 하는지

파악한다

DOE and Optimization

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문제 정특성의 파라미터 설계 문제

1 정적 계량 망소특성으로 최적조건을 구하시오 2 정적 계량 망대특성으로 최적조건을 구하시오 3 정적 계량 망목특성으로 산포가 최소인 최적조건을 구하시오 4 정적 계량 망목특성으로 목표치 m=5인 최적조건을 구하시오

control factor 5개(A B C D F) noise factor 2수준(N0 N1)으로 2회 반복실험 데이터가 있다

DOE and Optimization

모형 Helicopter를 이용한 실습으로

다구찌 기법의 전체 과정을 이해한다

다구찌 기법 실습

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헬리콥터

noise factor 낙하각도

control factor Response y 체공시간

몸체모양 몸체넓이 몸체길이

날개모양 날개길이 날개넓이

Clip 수 조인트 폭 등

헬리콥터 시스템 例

낙하각도 바로(N1) 옆으로(N2)

DOE and Optimization

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헬리콥터

noise factor 낙하각도

control factor Response y 체공시간

몸체모양 몸체넓이 몸체길이

날개모양 날개길이 날개넓이

Clip 수 조인트 폭 등

헬리콥터 시스템 例

낙하각도 바로(N1) 옆으로(N2)

DOE and Optimization

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헬리콥터 실습

관련된 인자 수준 정의

인자구분 인자 1수준 2수준

control factor

noise factor

DOE and Optimization

날개

몸통

조인트

날개

몸통

조인트

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -79-

Helicopter 조립방법

① A 와 B 부분을 접는다

② C 부분을 접는다

③ 프로펠러 날개(D)를 편다

④ 일정한 높이(15m)에서 떨어뜨려 체공시간을 측정한다

C 부분에 클립을 창작할 수 있다

A B

C

D D

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -79-

Helicopter 조립방법

① A 와 B 부분을 접는다

② C 부분을 접는다

③ 프로펠러 날개(D)를 편다

④ 일정한 높이(15m)에서 떨어뜨려 체공시간을 측정한다

C 부분에 클립을 창작할 수 있다

A B

C

D D

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -80-

헬리콥터 실습

실험을 배치하고 실험데이터 수집

R1 R2 R1 R2 R1 R2 R1 R2

1 1 1 1 1 1 1 1

2 1 1 1 2 2 2 2

3 1 2 2 1 1 2 2

4 1 2 2 2 2 1 1

5 2 1 2 1 2 1 2

6 2 1 2 2 1 2 1

7 2 2 1 1 2 2 1

8 2 2 1 2 1 1 2

기본표시 a b ab c ac bc abc

군 1

7M1 M2

2 3

내측배열

1 2 3 4 5 6

외측배열

N0 N1 N0 감도SN비N1

Statgt DOEgt Taguchigt Create Taguchi Designhellip

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -81-

낙하높이 (15m)

모형 헬리콥터의 낙하시간은 손을 떠난 순간부터 바닥에 닿는 순간까지의 시간으로 한다

헬리콥터 낙하방법

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -82-

Statgt DOEgt Taguchigt Analyze Taguchi Designhellip

최적조건을 찾음

최적조건 도출

DOE and Optimization

SCHOOL OF MECHANICAL ENG CHUNG-ANG UNIVERSITY -83-

Statgt DOEgt Taguchigt Predict Taguchi Resultshellip

현재 최적

SN비

평균

최종분석결과 현재와 최적조건에서의 SN비와 추정값

최적조건으로부터 특성치 예측

특성치 예측

DOE and Optimization