Taguchi MethodsPrinciples and Practices of Quality DesignA Short Course

Huei-Huang Lee

Department of Engineering Science

National Cheng Kung University

hhlee@mail.ncku.edu.tw

More Resources: http://myweb.ncku.edu.tw/~hhlee/Myweb_at_NCKU/Taguchi4.html

Case Study: Polysilicon Deposition Process (1-32)

Case Study: Design of a Brake Assembly (33-44)

Summary (45)

Empirical Model & Interactions (46-52)

Orthogonal Arrays (53-60)

More Ideal Functions & SN Ratios (61)

Introduction to ANOVA (62)

Video: https://youtu.be/x2pbrcUzl9Q

1

Case Study: Polysilicon Deposition ProcessProblem Description

Polysilicon

Polysilicon SiO2

SiO2

Si Substrate

[2] In this case study, the focus is the deposition of a layer of

polysilicon of 3600 Å.

[1] Manufacturing IC chips involves hundreds

of steps.

[3] The polysilicon layer is critical for the quality

of the IC chips.

Video: https://youtu.be/vnK0TVVlTL8

2

Case Study: Polysilicon Deposition ProcessProblem DescriptionChemical Vapor Deposition (CVD)

[2] Wafers.

[1] Quartz tube.[3] Heater T0.

[4] Pressure P0.

[5] Silane (SiH4) and nitrogen (N2) introduced at one end...

[6] and pumped out at the other end.

[7] The silane decomposes into silicon and deposits on the wafers,

forming a layer of polysilicon.

Video: https://youtu.be/XyuWvnglbXg

3

Case Study: Polysilicon Deposition ProcessProblem DescriptionSurface Defects

[1] Spec: surface defects < 10 defect/cm2.

[2] Current status: average 600 defect/cm2; max 5,000 defect/cm2.

Video: https://youtu.be/Tbiyr2eQsxI

4

Case Study: Polysilicon Deposition ProcessProblem DescriptionThickness Uniformity

Variations within wafers and among wafers

Spec: 3600±8% (3600±288Å)

Current status: SD = 2.8% (101 Å), i.e., Cp = 0.95, or defect rate = 0.43%

Polysilicon

Polysilicon SiO2

SiO2

Si Substrate

Video: https://youtu.be/fR4sCRWh2zc

5

Case Study: Polysilicon Deposition ProcessQuality Characteristics & Ideal Functions

Quality Characteristics

Notation Ideal Function

Surface defect yd (defect/cm2) Smaller the better (SB)

yd = 0

Thickness yt (Å) Nominal the best (NB)

yt = 3600 Å

Deposition rate yr (Å/min) Larger the better (LB)

yr = ∞

Static Characteristics: The idea value is a constantDynamic Characteristics: The idea value is a function of "signal factors"

Video: https://youtu.be/vIWIzz51tbY

6

Case Study: Polysilicon Deposition ProcessControl Factors & Levels

T0

P0

N0 S0

t0

Factor Description Unit Level 1 Level 2 Level 3

A Temperature ℃ T0 - 25 T0 T0 + 25

B Pressure mtorr P0 - 200 P0 P0 + 200

C Nitrogen flow cc/min N0 N0 - 150 N0 - 75

D Silane flow cc/min S0 -100 S0 - 50 S0

E Settling time min t0 t0 + 8 t0 + 16

F Cleaning method 　 None Inside Outside

Note: Shaded valuees are originaal process setttings.

A,B,C,D,E,F yd ,yt ,yr Process

Video: https://youtu.be/y5J-Kq_r6As

7

Case Study: Polysilicon Deposition ProcessNoise Factors

A,B,C,D,E,F yd ,yt ,yr Process

Noise Factors

Robust process: Minimum variations of quality characteristics

Taguchi Methods: Set control factors such that

the process is robust to the noises, and

the average of the quality characteristics is close to the target.

Noise factors: Non-controllable factors causing variations.

In this case, the most significant noise factor is position.

The quality characteristics are measured at top, center, and bottom of the

3rd, 23rd, 48th wafers; totally 9 positions are measured for each trial.

Video: https://youtu.be/zyQs8wseryM

8

　 　 A B C D E FExp. 1 2 3 4 5 6 7 8

1 1 1 1 1 1 1 1 12 1 1 2 2 2 2 2 23 1 1 3 3 3 3 3 34 1 2 1 1 2 2 3 35 1 2 2 2 3 3 1 16 1 2 3 3 1 1 2 27 1 3 1 2 1 3 2 38 1 3 2 3 2 1 3 19 1 3 3 1 3 2 1 2

10 2 1 1 3 3 2 2 111 2 1 2 1 1 3 3 212 2 1 3 2 2 1 1 313 2 2 1 2 3 1 3 214 2 2 2 3 1 2 1 315 2 2 3 1 2 3 2 116 2 3 1 3 2 3 1 217 2 3 2 1 3 1 2 318 2 3 3 2 1 2 3 1

Case Study: Polysilicon Deposition ProcessExperimental Arrays

Each column for a control factor.

Allows changes of levels

systematically and efficiently.

18 trials (instead of 36).

Orthogonal arrays (OA): every two

columns are mutual orthogonal.

Facilitates construction of

empirical models.

Allow detection of interactions

Video: https://youtu.be/9Hn0ZFHxCEg

Video: https://youtu.be/kRF4qM9Lu2k

9

　 A. B. C. Nitrogen D. Silane E. Settling F. Cleaning　 Temperature Pressure flow flow time method

Exp. 2 3 4 5 6 81 T0 - 25 P0 - 200 N0 S0 - 100 t0 None2 T0 - 25 P0 N0 - 150 S0 - 50 t0 + 8 Inside3 T0 - 25 P0 + 200 N0 - 75 S0 t0 + 16 Outside4 T0 P0 - 200 N0 S0 - 50 t0 + 8 Outside5 T0 P0 N0 - 150 S0 t0 + 16 None6 T0 P0 + 200 N0 - 75 S0 - 100 t0 Inside7 T0 + 25 P0 - 200 N0 - 150 S0 - 100 t0 + 16 Outside8 T0 + 25 P0 N0 - 75 S0 - 50 t0 None9 T0 + 25 P0 + 200 N0 S0 t0 + 8 Inside

10 T0 - 25 P0 - 200 N0 - 75 S0 t0 + 8 None11 T0 - 25 P0 N0 S0 - 100 t0 + 16 Inside12 T0 - 25 P0 + 200 N0 - 150 S0 - 50 t0 Outside13 T0 P0 - 200 N0 - 150 S0 t0 Inside14 T0 P0 N0 - 75 S0 - 100 t0 + 8 Outside15 T0 P0 + 200 N0 S0 - 50 t0 + 16 None16 T0 + 25 P0 - 200 N0 - 75 S0 - 50 t0 + 16 Inside17 T0 + 25 P0 N0 S0 t0 Outside18 T0 + 25 P0 + 200 N0 - 150 S0 - 100 t0 + 8 None

Case Study: Polysilicon Deposition ProcessExperimental Arrays

Video: https://youtu.be/YU1_fLwj1vc

10

　 yd (unit: defectt/cm2) 　 　 　

　 Wafer 1 Wafer 2 Wafer 3 　 　 　

Exp. Top Center Bottom Top Center Bottom Top Center Bottom yd Sd ηd

1 1 0 1 2 0 0 1 1 0 1 1 0.51 2 1 2 8 180 5 0 126 3 1 36 64 -37.30 3 3 35 106 360 38 135 315 50 180 136 120 -45.17 4 6 15 6 17 20 16 15 40 18 17 9 -25.76 5 1720 1980 2000 487 810 400 2020 360 13 1088 781 -62.54 6 135 360 1620 2430 207 2 2500 270 35 840 983 -62.23 7 360 810 1215 1620 117 30 1800 720 315 776 609 -59.88 8 270 2730 5000 360 1 2 9999 225 1 2065 3237 -71.69 9 5000 1000 1000 3000 1000 1000 3000 2800 2000 2200 1303 -68.15

10 3 0 0 3 0 0 1 0 1 1 1 -3.47 11 1 0 1 5 0 0 1 0 1 1 1 -5.08 12 3 1620 90 216 5 4 270 8 3 247 495 -54.85 13 1 25 270 810 16 1 225 3 0 150 253 -49.38 14 3 21 162 90 6 1 63 15 39 44 50 -36.54 15 450 1200 1800 2530 2080 2080 1890 180 25 1359 876 -64.18 16 5 6 40 54 0 8 14 1 1 14 18 -27.31 17 1200 3500 3500 1000 3 1 9999 600 8 2201 3049 -71.51 18 8000 2500 3500 5000 1000 1000 5000 2000 2000 3333 2173 -72.00

Case Study: Polysilicon Deposition ProcessExperimental Data and SN Ratios (Surface Defects)

ηd = −10log yd2 +Sd

2( )

Video: https://youtu.be/wMaf7w4tNqc

11

　 yyt (unit: ÅÅ) 　 　 　 沉積 沉積 　

　 Wafer 1 Wafer 22 Wafer 33 　 　 　 時間 速率 　

Exp. Top Center Bottom Top Center Bottom Top Center Bottom y t St ηt(min) y r ηr

1 2029 1975 1961 1975 1934 1907 1952 1941 1949 1958 34 35.22 135 14.5 23.23 2 5375 5191 5242 5201 5254 5309 5323 5307 5091 5255 86 35.75 144 36.6 31.27 3 5989 5894 5874 6152 5910 5886 6077 5943 5962 5965 94 36.02 144 41.4 32.34 4 2118 2109 2099 2140 2125 2108 2149 2130 2111 2121 16 42.24 59 36.1 31.15 5 4102 4152 4174 4556 4504 4560 5031 5040 5032 4572 388 21.43 63 73.0 37.27 6 3022 2932 2913 2833 2837 2828 2934 2875 2841 2891 65 32.91 58 49.5 33.89 7 3030 3042 3028 3486 3333 3389 3709 3671 3687 3375 287 21.39 44 76.6 37.68 8 4707 4472 4336 4407 4156 4094 5073 4898 4599 4527 326 22.84 43 105.4 40.46 9 3859 3822 3850 3871 3922 3904 4110 4067 4110 3946 116 30.60 34 115.0 41.21

10 3227 3205 3242 3468 3450 3420 3599 3591 3535 3415 155 26.85 138 24.8 27.89 11 2521 2499 2499 2576 2537 2512 2551 2552 2570 2535 29 38.80 127 20.0 26.02 12 5921 5766 5844 5780 5695 5814 5691 5777 5743 5781 72 38.06 148 39.0 31.82 13 2792 2752 2716 2684 2635 2606 2765 2786 2773 2723 68 32.07 51 53.1 34.50 14 2863 2835 2859 2829 2864 2839 2891 2844 2841 2852 19 43.35 62 45.7 33.20 15 3218 3149 3124 3261 3205 3223 3241 3189 3197 3201 43 37.44 58 54.8 34.78 16 3020 3008 3016 3072 3151 3139 3235 3162 3140 3105 79 31.86 40 76.8 37.71 17 4277 4150 3992 3888 3681 3572 4593 4298 4219 4074 323 22.01 39 105.3 40.45 18 3125 3119 3127 3567 3563 3520 4120 4088 4138 3596 431 18.42 39 91.4 39.22

Case Study: Polysilicon Deposition ProcessExperimental Data and SN Ratios (Thickness)

ηt = −10log

St2

yt2

ηr = −10log

1yr

2

Video: https://youtu.be/Q0tUAHni8cI

12

Measure of Quality: SN RatiosQuality Loss Function

Qua

lity

loss

L(y

)

m Quality characteristics y

When a quality characteristics is on target, the quality loss is at its minimum.

When the quality characteristics deviates from the target, the quality loss

increases quadratically.

L(y) = k(y − m)2

Video: https://youtu.be/M-oUJxm-H8s

13

Q =

k(yi −m)2

i=1

n

∑n

= k(yi −m)2

i=1

n

∑n

= k MSD

where

MSD =

(yi − m)2

i=1

n

∑n

= (y − m)2 +S2

S =

(yi − y )2

i=1

n

∑n

Measure of Quality: SN RatiosAverage Quality Loss

MSD =1n

(yi − m)2

i=1

n

∑

=1n

(yi2 − 2myi + m2)

i=1

n

∑

=1n

yi2

i=1

n

∑ −1n

2myii=1

n

∑ +1n

m2

i=1

n

∑

=1n

yi2

i=1

n

∑ − 2my + m2

=1n

yi2

i=1

n

∑ − 2y 2 + y 2 + y 2 − 2my + m2

=1n

yi2

i=1

n

∑ −1n

2yiyi=1

n

∑ +1n

y 2

i=1

n

∑ + y 2 − 2my + m2( )=

1n

yi2 − 2yiy + y 2

i=1

n

∑ + y 2 − 2my + m2( )

=1n

yi − y( )2

i=1

n

∑ + y − m( )2

= S2 + y − m( )2

Video: https://youtu.be/CgyWs2LT_Gg

14

Q = k MSD�� �� could be used as a measure of quality; however, Taguchi proposed

a modified version,

SN = −10log MSD

The quality loss coefficient k is dropped, since it is a constant for a specific

product.

Logarithm transformation is to achieve better additivity.

Multiplication of -10 is to be consistent with the traditional definition of SN ratios.

Measure of Quality: SN RatiosDefinition

Video: https://youtu.be/odqh0K9tHZE

15

Measure of Quality: SN RatiosMean Squared Deviation (MSD)

For nominal-the-best cases,

MSD =

(yi − m)2

i=1

n

∑n

= (y − m)2 +S2

For smaller-the-better cases, m = 0,

MSDSB =

yi2

i=1

n

∑n

= y 2 +S2

For larger-the-better cases, we may inverse the quality characteristics and

then treat them as smaller-the-better cases,

MSDLB =

(1 yi )2

i=1

n

∑n

Video: https://youtu.be/Sy1oZgQCBV8

16

Measure of Quality: SN RatiosSB/LB Cases

For nominal-the-best cases,

SN = −10log MSD = −10log(yi − m)2

i=1

n

∑n

= −10log (y − m)2 +S2

For smaller-the-better cases,

SNSB = −10log MSDSB = −10logyi

2

i=1

n

∑n

= −10log y 2 +S2

For larger-the-better cases,

SNLB = −10log MSDLB = −10log(1 yi )

2

i=1

n

∑n

Video: https://youtu.be/mcvDPFIA4xQ

17

SN = −10log(yi − m)2

i=1

n

∑n

= −10log (y − m)2 +S2

Often, there exist "adjustment" factors so that the "bias" can be completely

eliminated (i.e., y = m ). In such cases,

SNNB2 = −10log(yi − y )2

i=1

n

∑n

= −10log S2

The deviation S usually enlarges as the average y increases. In order the

comparison be "fair", we divide the deviation by the average,

SNNB3 = −10log

S2

y 2

Measure of Quality: SN RatiosNominal-the-best Cases

Video: https://youtu.be/K94Qn1znlXk

18

Empirical Model

25

30

35

40

45

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3

20

25

30

35

40

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3

-70

-60

-50

-40

-30

-20

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3

ηd (A,B,C,D,E,F)

ηt (A,B,C,D,E,F)

ηr (A,B,C,D,E,F)

Video: https://youtu.be/pg51dhVkAvE

19

　 A B C D E F 　 　 　

Exp. 2 3 4 5 6 8 ηd ηt ηr

1 1 1 1 1 1 1 0.51 35.22 23.23 2 1 2 2 2 2 2 -37.30 35.75 31.27 3 1 3 3 3 3 3 -45.17 36.02 32.34 4 2 1 1 2 2 3 -25.76 42.24 31.15 5 2 2 2 3 3 1 -62.54 21.43 37.27 6 2 3 3 1 1 2 -62.23 32.91 33.89 7 3 1 2 1 3 3 -59.88 21.39 37.68 8 3 2 3 2 1 1 -71.69 22.84 40.46 9 3 3 1 3 2 2 -68.15 30.60 41.21

10 1 1 3 3 2 1 -3.47 26.85 27.89 11 1 2 1 1 3 2 -5.08 38.80 26.02 12 1 3 2 2 1 3 -54.85 38.06 31.82 13 2 1 2 3 1 2 -49.38 32.07 34.50 14 2 2 3 1 2 3 -36.54 43.35 33.20 15 2 3 1 2 3 1 -64.18 37.44 34.78 16 3 1 3 2 3 2 -27.31 31.86 37.71 17 3 2 1 3 1 3 -71.51 22.01 40.45 18 3 3 2 1 2 1 -72.00 18.42 39.22

Average = -45.36 31.52 34.12

Case Study: Polysilicon Deposition ProcessResponse Analysis

(ηd )A1 = (0.51− 37.30 − 45.17 − 3.47−5.08 − 54.85) / 6 = −24.23

(ηd )A2 = (−25.76 − 62.54 − 62.23 − 49.38−36.54 − 64.18) / 6 = −50.10

(ηd )A3 = (−59.88 − 71.69 − 68.15 − 27.31−71.51− 72.00) / 6 = −61.75

(ηd )B1 = (0.51− 25.76 − 59.88 − 3.47−49.38 − 27.31) / 6 = −27.55

(ηd )B2 = (−37.30 − 62.54 − 71.69 − 5.08−36.54 − 71.51) / 6 = −47.44

(ηd )B3 = (−45.17 − 62.23 − 68.15 − 54.85−64.18 − 72.00) / 6 = −61.10

Video: https://youtu.be/x23sEtrKWts

20

　 A B C D E F 　 　 　

Exp. 2 3 4 5 6 8 ηd ηt ηr

1 1 1 1 1 1 1 0.51 35.22 23.23 2 1 2 2 2 2 2 -37.30 35.75 31.27 3 1 3 3 3 3 3 -45.17 36.02 32.34

10 1 1 3 3 2 1 -3.47 26.85 27.89 11 1 2 1 1 3 2 -5.08 38.80 26.02 12 1 3 2 2 1 3 -54.85 38.06 31.82 4 2 1 1 2 2 3 -25.76 42.24 31.15 5 2 2 2 3 3 1 -62.54 21.43 37.27 6 2 3 3 1 1 2 -62.23 32.91 33.89 13 2 1 2 3 1 2 -49.38 32.07 34.50 14 2 2 3 1 2 3 -36.54 43.35 33.20 15 2 3 1 2 3 1 -64.18 37.44 34.78 7 3 1 2 1 3 3 -59.88 21.39 37.68 8 3 2 3 2 1 1 -71.69 22.84 40.46 9 3 3 1 3 2 2 -68.15 30.60 41.21 16 3 1 3 2 3 2 -27.31 31.86 37.71 17 3 2 1 3 1 3 -71.51 22.01 40.45 18 3 3 2 1 2 1 -72.00 18.42 39.22

Average = -45.36 31.52 34.12

Case Study: Polysilicon Deposition ProcessResponse Analysis (Factor A)

(ηd )A1 = (0.51− 37.30 − 45.17 − 3.47−5.08 − 54.85) / 6 = −24.23

(ηd )A2 = (−25.76 − 62.54 − 62.23 − 49.38−36.54 − 64.18) / 6 = −50.10

(ηd )A3 = (−59.88 − 71.69 − 68.15 − 27.31−71.51− 72.00) / 6 = −61.75

Video: https://youtu.be/ZOB-3tIKV_o

21

　 A B C D E F 　 　 　

Exp. 2 3 4 5 6 8 ηd ηt ηr

1 1 1 1 1 1 1 0.51 35.22 23.23 4 2 1 1 2 2 3 -25.76 42.24 31.15 7 3 1 2 1 3 3 -59.88 21.39 37.68

10 1 1 3 3 2 1 -3.47 26.85 27.89 13 2 1 2 3 1 2 -49.38 32.07 34.50 16 3 1 3 2 3 2 -27.31 31.86 37.71 2 1 2 2 2 2 2 -37.30 35.75 31.27 5 2 2 2 3 3 1 -62.54 21.43 37.27 8 3 2 3 2 1 1 -71.69 22.84 40.46 11 1 2 1 1 3 2 -5.08 38.80 26.02 14 2 2 3 1 2 3 -36.54 43.35 33.20 17 3 2 1 3 1 3 -71.51 22.01 40.45 3 1 3 3 3 3 3 -45.17 36.02 32.34 6 2 3 3 1 1 2 -62.23 32.91 33.89 9 3 3 1 3 2 2 -68.15 30.60 41.21 12 1 3 2 2 1 3 -54.85 38.06 31.82 15 2 3 1 2 3 1 -64.18 37.44 34.78 18 3 3 2 1 2 1 -72.00 18.42 39.22

Average = -45.36 31.52 34.12

Case Study: Polysilicon Deposition ProcessResponse Analysis (Factor B)

(ηd )B1 = (0.51− 25.76 − 59.88 − 3.47−49.38 − 27.31) / 6 = −27.55

(ηd )B2 = (−37.30 − 62.54 − 71.69 − 5.08−36.54 − 71.51) / 6 = −47.44

(ηd )B3 = (−45.17 − 62.23 − 68.15 − 54.85−64.18 − 72.00) / 6 = −61.10

Video: https://youtu.be/EOosMoGbJ4Y

22

-70

-60

-50

-40

-30

-20

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3

　 LevelFactor 1 2 3

A. Temperature -24.23 -50.10 -61.75 B. Pressure -27.55 -47.44 -61.10 C. Nitrogen -39.03 -55.99 -41.07 D. Silane -39.20 -46.85 -50.04 E. Settling time -51.52 -40.54 -44.03 F. Cleaning method -45.56 -41.58 -48.95

Case Study: Polysilicon Deposition ProcessResponse Table/Graph (Surface Defects ηd )

Video: https://youtu.be/CMiinNiH8i0

23

20

25

30

35

40

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3

　 LevelFactor 1 2 3

A. Temperature 35.12 34.91 24.52 B. Pressure 31.61 30.70 32.24 C. Nitrogen 34.39 27.86 32.31 D. Silane 31.69 34.70 28.16 E. Settling time 30.52 32.87 31.16 F. Cleaning method 27.04 33.67 33.85

Case Study: Polysilicon Deposition ProcessResponse Table/Graph (Thickness Uniformity ηt )

Video: https://youtu.be/euW9lxo3q_E

24

　 LevelFactor 1 2 3

A. Temperature 28.76 34.13 39.46 B. Pressure 32.03 34.78 35.54 C. Nitrogen 32.81 35.29 34.25 D. Silane 32.21 34.53 35.61 E. Settling time 34.06 33.99 34.30 F. Cleaning method 33.81 34.10 34.44

Case Study: Polysilicon Deposition ProcessResponse Table/Graph (Deposition Rate ηr )

25

30

35

40

45

A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3

Video: https://youtu.be/1p6f_e7ELI0

25

Case Study: Polysilicon Deposition ProcessDiscussion (A. Temperature)

Temperature is the most significant factor.

When the temperature decreases 25℃,

the surface defects improves 26 dB,

the thickness uniformity doesn't change,

and the deposition rate slows down by

5.4 dB.

Video: https://youtu.be/tBSiuGXk8zc

26

Case Study: Polysilicon Deposition ProcessDiscussion (B. Pressure)

Pressure is the second most significant

factor.

When the pressure decreases 200 mtorr,

the surface defects improves 20 dB,

the thickness uniformity doesn't change,

and the deposition rate slows down by

2.8 dB.

Video: https://youtu.be/As-Vfk47rM0

27

Case Study: Polysilicon Deposition ProcessDiscussion (C. Nitrogen Flow)

Nitrogen has medium effects on all three

quality characteristics.

Current setting is the best of the three

levels.

In the future, larger nitrogen flow may be

worth a trial.

Video: https://youtu.be/KdDFqhVcC7M

28

Case Study: Polysilicon Deposition ProcessDiscussion (D. Siliane Flow)

Silane has medium effects on all three

quality characteristics.

Reducing the silane flow by 50 cc/min

would improve both surface defects and

thickness uniformity, however sacrifies

some productivity.

Video: https://youtu.be/at6OOEOCZjs

29

Case Study: Polysilicon Deposition ProcessDiscussion (E. Settling Time)

Increasing the settling time by 8 min would

improve both surface defects and

thickness uniformity, without sacrificing

deposition rate.

Additional 8 min is acceptable.

Increasing the settling time by 16 min

would deteriorate both surface defects and

thickness uniformity.

Video: https://youtu.be/0XJbFmjfS5g

30

Case Study: Polysilicon Deposition ProcessDiscussion (F. Cleaning Method)

Cleaning has little effects on deposition

rate and surface defects but has

significant effects on thickness uniformity.

Cleaning inside or outside has little effects

on thickness uniformity.

Cleaning inside is more convenient than

outside.

Video: https://youtu.be/dZ027czgcXs

31

Factor Description Unit Level 1 Level 2 Level 3

A Temperature ℃ T0 - 25 T0 T0 + 25

B Pressure mtorr P0 - 200 P0 P0 + 200

C Nitrogen flow cc/min N0 N0 - 150 N0 - 75

D Silane flow cc/min S0 -100 S0 - 50 S0

E Settling time min t0 t0 + 8 t0 + 16

F Cleaning method 　 None Inside Outside

Case Study: Polysilicon Deposition ProcessProcess Optimization

Optimum condition: A1 B2 C1 D3 E2 F2

Original condition: A2 B2 C1 D3 E1 F1

Video: https://youtu.be/ABdW75jxtJk

32

0.000

0.005

0.010

0.015

0.020

3312 3408 3504 3600 3696 3792 3888

Pro

babi

lity

Den

sity

Deposition Thickness

Case Study: Polysilicon Deposition ProcessConfirmation Experiments

　 　 Original Optimum Improvement　 　 condition condition dB

Surface rms 600 defect/cm2 7 defect/cm2 　

defects ηd-55.6 -16.9 38.7

Deposition Std. Dev. 2.8% 1.3% 　

thickness ηt31.1 37.7 6.6

Deposition Rate 60 Å/min 35 Å/min 　

rate ηr35.6 30.9 -4.7

Optimum condition: A1 B2 C1 D3 E2 F2

Original condition: A2 B2 C1 D3 E1 F1Original

Optimum

Video: https://youtu.be/hbHKxLpp2h0

33

0 5

10 15 20 25 30 35 40 45 50

0.000 0.008 0.016 0.024 0.032 0.040 0.048 0.056 0.064 0.072

Bra

ke to

rque

(y)

Brake fluid pressure (M)

Case Study: Design of a Brake AssemblyProblem Description

[2] Caliper

[3] Pads

[4] Rotor

[1] Brake fluid

pressure

Brake AssemblyBrake pressure (M)

Braking torque (y)Heat, Sound

Video: https://youtu.be/HEYwYV9OlQs

34

Case Study: Design of a Brake AssemblyQuality Characteristics & Ideal Function

Quality characteristics: Braking torque y (kgf-mm).

Ideal function: Zero-point proportional

y = βM

In addition, the efficiency or sensitivity β should be as large as possible.

When the ideal values change according to a signal factor M, it is called a

dynamic characteristics.

Signal factors are not controlled by the engineers; they are controlled by the

users of the system; they are input to the system.

Video: https://youtu.be/5wDfXf2M60k

35

Factor Description Level 1 Level 2 Level 3

A Pad material Type-1 Type-2

B Pad shape Shape-1 Shape-2 Shape-3

C Pad curve profile Type-1 Type-2 Type-3

D Pad additive Low Medium High

E Rotor material Gray Cast Steel

F Pad taper Low Medium High

G Tapering thickness Low Medium HighH Rotor structure Type-1 Type-2 Type-3

Note: Shaded vaalues are original design.

Case Study: Design of a Brake AssemblySignal Factor & Control Factors

M = Brake pressure (kgf/mm2)

M1 = 0.008, M2 = 0.016, M3 = 0.032, M4 = 0.064

Video: https://youtu.be/5zSn5zxLtN8

36

Case Study: Design of a Brake AssemblyNoise Factors

The most significant noise factors

are pad temperature, pad wetness,

and pad wear.

Compound noise factor

N1 = 360°F, wet, 80% wear

N2 = 60°F, dry,10% wear

Another noise: measuring time

Q1 = Max brake torque

Q2 = Min brake torque

0 5

10 15 20 25 30 35 40 45 50

0.000 0.008 0.016 0.024 0.032 0.040 0.048 0.056 0.064 0.072

Bra

ke to

rque

(y)

Brake fluid pressure (M)

Video: https://youtu.be/oIkioejnYDw

37

　 　 　 　 　 　 　 　 　 MM = 00.0008 MM = 00.016 MM = 00.0332 MM = 00.064

　 　 　 　 　 　 　 　 　 NN1 NN2 NN1 NN2 NN1 NN2 NN1 NN2Exp. A B C D E F G H Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2

1 Type-1 Shape-1 Type-1 Low Gray Low Low Type-1 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

2 Type-1 Shape-1 Type-2 Medium Cast Medium Medium Type-2 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

3 Type-1 Shape-1 Type-3 High Steel High High Type-3 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

4 Type-1 Shape-2 Type-1 Low Cast Medium High Type-3 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

5 Type-1 Shape-2 Type-2 Medium Steel High Low Type-1 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

6 Type-1 Shape-2 Type-3 High Gray Low Medium Type-2 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

7 Type-1 Shape-3 Type-1 Medium Gray High Medium Type-3 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

8 Type-1 Shape-3 Type-2 High Cast Low High Type-1 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

9 Type-1 Shape-3 Type-3 Low Steel Medium Low Type-2 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

10 Type-2 Shape-1 Type-1 High Steel Medium Medium Type-1 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

11 Type-2 Shape-1 Type-2 Low Gray High High Type-2 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

12 Type-2 Shape-1 Type-3 Medium Cast Low Low Type-3 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

13 Type-2 Shape-2 Type-1 Medium Steel Low High Type-2 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

14 Type-2 Shape-2 Type-2 High Gray Medium Low Type-3 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

15 Type-2 Shape-2 Type-3 Low Cast High Medium Type-1 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

16 Type-2 Shape-3 Type-1 High Cast High Low Type-2 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

17 Type-2 Shape-3 Type-2 Low Steel Low Medium Type-3 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

18 Type-2 Shape-3 Type-3 Medium Gray Medium High Type-1 　 　 　 　 　 　 　 　 　 　 　 　 　 　 　

Case Study: Design of a Brake AssemblyExperimental Array

Video: https://youtu.be/iznRfk3TdnA

38

　 M = 0 0.008 M = 0 0.016 M = 0 0.032 M = 0 0.064 　 　 　

　 NN1 NN2 NN1 NN2 NN1 NN2 NN1 NN2 　 　 　

Exp. Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 β SZP SN

1 4.8 0.9 5.8 0.8 8.5 6.5 11.5 6.8 20.4 13.2 25.0 16.2 36.9 32.7 43.5 34.5 573 3.6 44.0 2 4.5 2.5 5.7 3.2 12.5 9.6 13.0 10.0 23.5 20.3 25.1 21.4 42.0 36.0 43.2 36.1 634 2.7 47.4 3 5.9 5.2 6.8 5.9 10.6 9.3 11.4 10.2 23.5 22.0 24.3 22.5 42.9 40.3 43.8 40.6 668 1.5 53.2 4 4.5 2.1 5.7 3.0 12.1 8.9 14.3 10.5 22.1 16.9 24.2 20.0 41.0 34.0 42.4 37.6 618 2.8 46.9 5 6.5 2.1 7.8 3.2 12.3 6.9 13.2 8.6 23.3 17.2 24.3 18.3 44.3 36.9 48.9 37.2 652 3.5 45.3 6 5.0 4.2 5.8 4.3 11.5 9.4 12.3 9.9 20.8 16.8 21.0 18.5 43.0 40.2 43.1 41.0 644 1.5 52.4 7 5.2 4.0 5.6 4.5 11.8 9.1 12.3 10.1 21.2 17.5 20.0 18.3 40.3 36.2 42.2 38.2 614 1.7 51.4 8 2.4 0.0 4.3 2.8 6.7 4.0 7.2 3.6 16.3 11.1 18.3 12.3 30.1 27.8 34.3 30.6 466 2.6 45.0 9 6.3 4.8 7.8 6.1 12.1 9.3 13.5 11.9 24.4 19.6 26.3 22.3 48.5 40.3 50.2 44.0 718 2.6 48.9

10 2.1 0.0 2.9 0.0 4.9 0.0 7.4 4.2 18.3 9.5 17.7 10.8 32.0 26.3 35.3 28.1 455 3.8 41.6 11 4.9 1.2 7.6 1.8 11.3 6.5 15.3 6.8 23.4 15.0 25.1 17.2 40.1 33.2 50.5 35.5 622 4.7 42.4 12 5.1 4.4 6.4 4.4 10.1 7.8 11.2 8.5 21.7 18.7 22.1 20.1 43.1 41.2 44.4 41.5 657 1.4 53.3 13 2.1 0.0 5.4 0.6 6.7 1.2 7.3 2.3 13.4 9.4 16.4 11.1 38.9 27.9 43.3 31.1 505 5.0 40.0 14 5.9 5.0 6.8 5.2 13.3 12.0 14.2 13.3 24.9 23.1 26.3 25.4 47.9 46.3 49.7 47.2 756 1.3 55.3 15 3.2 0.0 3.9 1.8 8.7 3.2 9.6 5.1 13.2 7.9 19.5 11.1 38.2 32.1 42.5 33.0 528 4.5 41.5 16 4.1 2.7 5.9 4.4 12.3 8.7 13.7 9.2 24.3 18.9 25.5 20.2 44.3 39.0 47.7 42.4 679 2.6 48.4 17 2.3 0.8 3.2 2.1 10.2 8.0 12.5 8.8 21.6 16.5 23.6 20.4 38.8 32.4 41.1 36.6 591 2.9 46.3 18 1.2 0.0 5.1 1.2 7.8 2.3 13.0 5.0 20.3 11.1 21.2 12.4 40.1 31.6 45.1 32.0 557 4.8 41.2

Average = 608 46.9

Case Study: Design of a Brake AssemblyRaw Data & SN Ratios

β =Mi yi

i=1

n

∑

Mi2

i=1

n

∑ SZP =

yi − βMi( )2

i=1

n

∑n −1

SN = −10log

SZP2

β 2

Video: https://youtu.be/57vH_yKbTQU

39

Consider n points (Mi ,yi ), i = 1, 2,... n, in the

M-y space.

What is the line y = βM, which passes

through the origin and "best-fit" (in the sense

of least squared errors) the n points?

The sum of squared errors is

SS = yi − βMi( )2

i=1

n

∑The least SS must satisfy dSS dβ = 0,

β =Miyi

i=1

n

∑

Mi2

i=1

n

∑

Measure of Quality: SN RatiosZero-Point Proportional Cases

M

y

(Mi ,yi )

y = βM

y i − βMi

We may define an MSD for the ZP

case:

MSDZP =

yi − βMi( )2

i=1

n

∑n −1

And the SN ratios can be defined:

SNZP = −10log MSDZP

Video: https://youtu.be/5cQ0K2U-BH8

40

Measure of Quality: SN RatiosZero-Point Proportional Cases

SNZP = −10log MSDZP = −10logyi − βMi( )2

i=1

n

∑n −1

Recall that SNNB2 = −10log S2 . If we define a “standard deviation,”

SZP =

yi − βMi( )2

i=1

n

∑n −1

Then

SNZP = −10log SZP

2

The deviation SZP usually enlarges as the slope β increases. In order the

comparison be "fair", we divide the deviation by the slope β ,

SNZP 2 = −10log

SZP2

β 2

Video: https://youtu.be/TObCOoS_Mio

41

　 A B C D E F G HLevel 1 48.3 47.0 45.4 45.0 47.8 46.8 49.2 43.1 Level 2 45.6 46.9 47.0 46.4 47.1 46.9 46.8 46.6 Level 3 　 46.9 48.4 49.3 45.9 47.0 44.8 51.1 Range 2.7 0.1 3.0 4.3 1.9 0.2 4.4 8.0 Rank 5 8 4 3 6 7 2 1

Case Study: Design of a Brake AssemblyResponse Analysis (SN Ratios)

42

44

46

48

50

52

A1 A2 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3 G1 G2 G3 H1 H2 H3

Video: https://youtu.be/APIUw-Jzwn4

42

　 A B C D E F G HLevel 1 621 602 574 608 628 573 673 538 Level 2 594 617 620 603 597 623 578 634 Level 3 　 604 629 611 598 627 573 651 Range 26 15 55 8 31 54 100 112 Rank 6 7 3 8 5 4 2 1

Case Study: Design of a Brake AssemblyResponse Analysis (Sensitivity β )

520

540

560

580

600

620

640

660

680

A1 A2 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3 G1 G2 G3 H1 H2 H3

Video: https://youtu.be/UMkZeFIi7JM

43

42

44

46

48

50

52

A1 A2 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3 G1 G2 G3 H1 H2 H3 520

540

560

580

600

620

640

660

680

A1 A2 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3 G1 G2 G3 H1 H2 H3

SN Ratios Sensitivity β

Case Study: Design of a Brake AssemblyDesign Optimization

Affect AffectType SN? β ? Control factors Usage

1 Yes Yes/No A, C, D, G, H Maximize SN

2 No Yes E, F Maximize Sensitivity β

3 No No B Minimize Cost

A1 B ? C3 D3 E ? F ? G1 H3

A1 B ? C3 D3 E1 F3 G1 H3

A1 B1 C3 D3 E1 F3 G1 H3

Video: https://youtu.be/I-Jgwm5k8Bo

44

　 　 　 　 　 　 　 　 　 M = 00.008 M = 00.016 M = 00.032 M = 00.064 　 　 　

　 　 　 　 　 　 　 　 　 NN1 NN2 NN1 NN2 NN1 NN2 NN1 NN2 　 　 　

Exp. A B C D E F G H Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 Q1 Q2 β SZP SN

Original 1 2 2 2 2 2 2 2 4.8 1.2 5.7 4.4 11.1 8.6 13.0 11.8 23.1 18.1 25.1 21.4 42.0 36.0 43.2 37.6 635 2.7 47.6

New 1 1 3 3 1 3 1 3 5.3 4.6 5.8 5.4 12.2 10.1 13.2 11.9 24.6 23.1 25.0 24.3 49.3 47.1 50.1 48.2 758 1.0 57.4

Gaain = 123 9.8

Case Study: Design of a Brake AssemblyConfirmation Experiments

0

10

20

30

40

50

0.000 0.008 0.016 0.024 0.032 0.040 0.048 0.056 0.064 0.072

Bra

ke to

rque

(y)

Brake fluid pressure (M)

New design

N1Q1

N1Q2

N2Q1

N2Q2

Linear fit

0

10

20

30

40

50

0.000 0.008 0.016 0.024 0.032 0.040 0.048 0.056 0.064 0.072

Bra

ke to

rque

(y)

Brake fluid pressure (M)

Original design

N1Q1

N1Q2

N2Q1

N2Q2

Linear fit

Video: https://youtu.be/alHcg359uxc

45

Summary

So far, we've introduced

Quality characteristics

Ideal functions & SN ratios

Signal factors & Levels

Control factors & Levels

Noise factors & Levels

Orthogonal array: L18

Response analysis

Process Optimization

Confirmation experiments

What we haven't covered are

Empirical models

Interactions

Other orthogonal arrays

More ideal functions and SN ratios

Analysis of variance (ANOVA)

Video: https://youtu.be/etUFHwlksqU

46

Empirical Models

The empirical model used in Taguchi Methods is called an additive model:

η(A,B,C,...) = ηC + a(A) + b(B) + c(C) + ...

It is not practical to determine the unknown functions (a, b, c, ...) from the experiment

data.

Instead, it is more practical to predict the response under arbitrary combination of

control factors' levels, e.g., η(A2,B1,C3,...) .

In this way, all we need to know are the function values at control factors' levels, i.e.,

a(A1), a(A2), a(A3), b(B1), b(B2), b(B3), c(C1), c(C2), c(C3), ...

Video: https://youtu.be/1ZFqsJRo-M0

47

Taguchi Methods use an additive model:

η(A,B,C,...) = ηC + a(A) + b(B) + c(C) + ...

Behaviors of engineering systems may deviate from this model; e.g., the tip

deflection of a cantilever beam

y =

PL3

3EI=

4PL3

EWH3 = f (P,L,E,W ,H)

If we apply the logarithmic transformation, then

log y = log4 + logP + 3logL − logE − logW − 3logHη = ηC + f1(P) + f2(L) − f3(E) − f4(W ) − f5(H)

It fits nicely into the additive model!

In a complex system, the logarithmic transformation may not work so

perfectly; however, it usually improve the additivity.

E,W ,H,L

PLogarithmic TransformationSeparation of Variables

Video: https://youtu.be/aiCYcHVo5Z0

48

Grand average ηη = 46.9

　 A B C D E F G HLevel 1 48.3 47.0 45.4 45.0 47.8 46.8 49.2 43.1 Level 2 45.6 46.9 47.0 46.4 47.1 46.9 46.8 46.6 Level 3 　 46.9 48.4 49.3 45.9 47.0 44.8 51.1

Empirical Models

It can be shown that the best estimates of these function values are

a(A2) = ηA2

−η, b(B1) = ηB1

−η, c(C3) = ηC3

−η, etc.

Also, it can be shown that the best estimate of the constant ηC is the grand

average η , i.e., ηC = η .

Therefore,

η(A2,B1,C3,...) = ηC + a(A2) + b(B1) + c(C3) + ...= η + (ηA2

−η) + (ηB1−η) + (ηC3

−η) + ...In general,

η(Ai ,Bj ,Ck ,...) = η + (ηAi

−η) + (ηBj−η) + (ηCk

−η) + ...

Video: https://youtu.be/UH6aWD5Dmkg

49

Grand average ηη = 46.9

　 A B C D E F G HLevel 1 48.3 47.0 45.4 45.0 47.8 46.8 49.2 43.1 Level 2 45.6 46.9 47.0 46.4 47.1 46.9 46.8 46.6 Level 3 　 46.9 48.4 49.3 45.9 47.0 44.8 51.1

Empirical ModelsSummary

Taguchi Methods use an additive model:

η(A,B,C,...) = ηC + a(A) + b(B) + c(C) + ...

The model can be evaluated using the response data

η(Ai ,Bj ,Ck ,...) = η + (ηAi

−η) + (ηBj−η) + (ηCk

−η) + ...

The additive model assumes that control factors are not coupled, i.e., they are

independent one another, no interactions among them, the effects are synergetic.

When the effect of a factor depends on another factor's level, we say there exists

interaction between the two factors.

Video: https://youtu.be/vhi20qxl1Qo

50

Exp. A B AxB y1 1 1 1 02 1 2 2 503 2 1 2 304 2 2 1 80

Level 1 25 15 40 AveLevel 2 55 65 40 40

0

50

30

80

0

20

40

60

80

100

B1 B2

A1

A2

　 B1 B2A1 0 50A2 30 80

Example: Weight LiftingWithout Interactions

[1] The effects of A is independent of B,

and vice versa.

η(A2,B2) = η + (ηA2−η) + (ηB2

−η)

= 40 + (55 − 40) + (65 − 40)= 80

[2] The additive model accurately predicts the

behaviors.

15

65

25

55

40

Video: https://youtu.be/1a84SYtlIkk

51

0

50

30

95

0

20

40

60

80

100

B1 B2

A1

A2

15

72.5

25

62.5

43.75

　 B1 B2A1 0 50A2 30 95

Exp. A B AxB y1 1 1 1 02 1 2 2 503 2 1 2 304 2 2 1 95

Level 1 25.0 15.0 47.5 AveLevel 2 62.5 72.5 40.0 43.75

[1] The effects of A depends on B, and vice versa.

Example: Weight LiftingWith Interactions

η(A2,B2) = η + (ηA2−η) + (ηB2

−η)

= 43.75 + (62.5 − 43.75) + (72.5 − 43.75)= 91.25

[2] The additive model fails to predicts the

behaviors.

η(A2,B2) = η + (ηA2B2−η)

= 43.75 + (95 − 43.75)= 95

[3] The modified model successfully predicts the

behaviors.

Video: https://youtu.be/j_LlwBalq-4

52

Case Study: Design of a Brake AssemblyAre there interactions among control factors?

η(A1,C2,D2,G2,H2) = η + (ηA1−η)+ (ηC2

−η)+ (ηD2−η)+ (ηG2

−η)+ (ηH2−η)

= 46.9 + (48.3 − 46.9)+ (47.0 − 46.9)+ (46.4 − 46.9)+ (46.8 − 46.9)+ (46.6 − 46.9)= 46.9 +1.4 + 0.1− 0.5 − 0.1− 0.3= 47.3

η(A1,C3,D3,G1,H3) = η + (ηA1−η)+ (ηC3

−η)+ (ηD3−η)+ (ηG1

−η)+ (ηH3−η)

= 46.9 + (48.3 − 46.9)+ (48.4 − 46.9)+ (49.3 − 46.9)+ (49.2− 46.9)+ (51.1− 46.9)= 46.9 +1.4 +1.5 + 2.4 + 2.3 + 4.2= 58.6

Video: https://youtu.be/47auRR2D_1M

53

Orthogonal ArraysOverview

Full-Factorial OA's

2-level: L4(23), L8(27), L16(215), L32(231)

3-level: L9(34), L27(313)

4-level: L16(45)

5-level: L25(56)

Distributed Interactions OA's

2-level: L12(211)

3-level: L18(21×37), L36(23×313), L36(211×312), L54(21×325)

4-level: L32(21×49)

5-level: L50(21×511)

Note: These OA's can be downloaded from http://myweb.ncku.edu.tw/~hhlee/Myweb_at_NCKU/Taguchi4.html

L36(211×312)

Video: https://youtu.be/qkkkEmVLHvo

54

Orthogonal ArraysDegrees of Freedom (DOF)

The DOF of a group of data is the number

of independent pieces of information it

provides.

In L36(211×312), with 36 experiments,

provides at most 36 independent pieces of

information, e.g.,

The grand average counts 1 dof.

Each two-level column takes 1 dof.

Each Three-level column takes 2 dof's.

A = 178 B = 171 C = 167

A − B = 7

B −C = 4

A + B +C = 516 A + B − 2C = 15

[1] There are 3 dof's in this group of data.

[2] 1+11× (2−1)+12× (3 −1) = 36

　 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23Level 1Level 2Level 3 　

Grand Average

Video: https://youtu.be/6dUyjD6u7p8

55

Orthogonal ArraysConfounding

Exp. 1 2 31 1 1 12 1 2 23 2 1 24 2 2 1

Exp. 1 2 31 -1 -1 -12 -1 +1 +13 +1 -1 +14 +1 +1 -1

3 = −1× 21= −2 × 32 = −3 ×1

Exp. A B AxB1 1 1 12 1 2 23 2 1 24 2 2 1

Exp. A B C1 1 1 12 1 2 23 2 1 24 2 2 1

[2] The effect of C is confounded with the

interaction AxB.

[1] The effects of A, B and the interaction

AxB can be evaluated

respectively.

Video: https://youtu.be/q5h7sb5Q_jw

56

Orthogonal ArraysResolution

[1] Interactions for L4(23), L8(27),

L16(215), and L32(231).

Exp. 1 2 3 4 5 6 71 1 1 1 1 1 1 12 1 1 1 2 2 2 23 1 2 2 1 1 2 24 1 2 2 2 2 1 15 2 1 2 1 2 1 26 2 1 2 2 1 2 17 2 2 1 1 2 2 18 2 2 1 2 1 1 2

�

RResolutiion IVA B 　 C 　 　 D

L8(27) 　 　 AxB 　 AxC BxC 　　 　 C×D 　 B×D A×D 　

Column 1 2 3 4 5 6 7

Resolution V

L8(27) A B CL8(27) AxB AxC BxCColumn 1 2 3 4 5 6 7

RResolutiion IIIA B C D E F G

L8(27) B×C A×C A×B B×F A×D B×D A×FL8(27) D×E D×F D×G C×G B×G C×E B×EF×G E×G E×F 　 C×F 　 C×D

Column 1 2 3 4 5 6 7[2] Linear graphs are so designed such that when you fill all "dots" with control factors, you will achieve resolution IV.

Video: https://youtu.be/7lbuhy7aBZk

57

Orthogonal ArraysL9(34), L27(313), etc.

�

Interactions for L9(34), L27(313).

In general, interactions between an

N-level factor and an M-level factor

needs (N-1)x(M-1) dof's.

Video: https://youtu.be/oBz74MFDn0A

58

Orthogonal ArraysConfiguration of Control Factors

Video: https://youtu.be/f4_hiNRZLxY

59

Orthogonal ArraysDistributed Interactions OA's

Name of orthogonal array L12(211) L18(21×37) L36(23×313) L36(211×312) L54(21×325) L32(21×49) L50(21×511)

Total DOF's 12 18 36 36 54 32 50

DOF's occupied by columns

2x11= 11

1+2x7= 15

3+2x13= 29

11+2x12= 35

1+2x25= 51

1+3x9= 28

1+4x11= 45

DOF's for grand average 1 1 1 1 1 1 1

Remaining DOF's 0 2 6 0 2 3 4

They are mix-level orthogonal arrays, except L12(211).

They are mainly used to evaluate factor effects; they can evaluate very few

interactions.

With these OA's, it is possible to achieve Resolution III+ by conducting

Resolution III experiments.

Video: https://youtu.be/4Mmba41HpmE

60

　 A B C D E F G HExp. 1 2 3 4 5 6 7 8 SN

1 1 1 1 1 1 1 1 1 44.0 2 1 1 2 2 2 2 2 2 47.4 3 1 1 3 3 3 3 3 3 53.2 4 1 2 1 1 2 2 3 3 46.9 5 1 2 2 2 3 3 1 1 45.3 6 1 2 3 3 1 1 2 2 52.4 7 1 3 1 2 1 3 2 3 51.4 8 1 3 2 3 2 1 3 1 45.0 9 1 3 3 1 3 2 1 2 48.9

10 2 1 1 3 3 2 2 1 41.6 11 2 1 2 1 1 3 3 2 42.4 12 2 1 3 2 2 1 1 3 53.3 13 2 2 1 2 3 1 3 2 40.0 14 2 2 2 3 1 2 1 3 55.3 15 2 2 3 1 2 3 2 1 41.5 16 2 3 1 3 2 3 1 2 48.4 17 2 3 2 1 3 1 2 3 46.3 18 2 3 3 2 1 2 3 1 41.2

Orthogonal ArraysL18(211x37): The most widely used OA

�

43.0

44.0

45.0

46.0

47.0

48.0

49.0

B1 B2 B3

A1 A2

B1 B2 B3

A1 48.20 48.20 48.43A2 45.77 45.60 45.30

The remaining 2 dof's can be used to

evaluate the interaction between the

factors occupying the first two columns.

Video: https://youtu.be/Vy2POI39MeU

61

More Ideal Functions and SN Ratios

Operating Window

SN = −10log

ylower , i2

i=1

n

∑n

−10log

1yupper , i

2i=1

n

∑n

Reference-Point Proportional

y − yo = β M −Mo( )

Double Signals, e.g.,

y = βM *M , y = β M

M *

Nonlinear Characteristics, e.g.,

y = e−βM

Free Functions

y lower yupper

Spring force

Operating Window

Video: https://youtu.be/HqW5xtuAT8U

62

37 38 39 40 41 42 43 44 45 46 47

A1 A2 B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3 G1 G2 G3 H1 H2 H3

Analysis of Variance (ANOVA)Purpose of ANOVA

Experimental Error

Significance test of factors

Errors of data

Comparison between the

experimental and predicted

values

Source SS DOF Var F ConfidenceA 33.1 1 33.1 22.19 99.8%B PooleddC 27.6 2 13.8 9.27 99.2%D 58.3 2 29.1 19.56 99.9%E PooleddF PooleddG 58.7 2 29.3 19.69 99.9%H 192.0 2 96.0 64.40 100.0%

Others PooleddError 11.9 8 1.5 S = 1.22 Total 381.5 17

Video: https://youtu.be/4s2lZRYBR4Y