SIX SIGMA

. ?

1. SIX-SIGMA ?

6 ( : Process) .

6

Tool.

(Ideal Process)

Improvement

Avenue(6)

Q P

Q P

Q P

2. SIXMA (PPM)

6 .

(Sigma) % CpkPPM

2

3

4

5

6

95.45%

99.73%

99.994%

99.99994%

99.9999998%

45,000PPM

2,700PPM

63PPM

0.6PPM

2PPB

Cpk = 0.67

Cpk = 1.00

Cpk = 1.33

Cpk = 1.67

Cpk = 2.00

*PPB : Parts Per Billion( )*6 : 6-Sigma (2PPB = 0.002PPM)

*DPU : Defects Per Unit ( = /)

*DPMO : Defects Per Million Opportunities

(100 = /TOP)

*TOP : Total Opportunities = Units op/Unit

( = )

(+1.5 SHIFT )

(K = 0 )

3. PPM

PPM

() TV

PPM .

, TV PPM ?

TV 1 = 500EA

(AQL) = 10PPM

TV (, TV = 0% )

, TV

= 100 1 .

10

1,000,000( ) 500 = 1/200

World Class Car = 1,000 DPTV/Year Warranty (35PPM)

1 = 15,000, AQL = 10 DPTV

,

World Class Car Level 1,000 DPTV ,

Maker = (70PPM/2) = 35PPM

, Cpk = 1.33 .

10

1,000P=( ) 15,000 = 150/

10DPTV

150( ) = 0.067DPTV 70PPM

4. () (m- )

6 ,

, ,

,

.

A-Type B-Type C-Type

?

B-Type ?

C-Type ?

Spec : m A-Type B-Type C-Type

(m-X bar)

5. (Statistical) ?

, ( = Sample)

( = Population) .

(Statistics) .

6 .

.

, .

UCL

CL

LCL

,

,

(X bar) (R) ,

, (=Plot) .

UCL(Upper Control Limit = )

CL(Center Line = )

LCL(Lower Control Limit = )

(Su, SL or USL, LSL)

6. (Process) ? (6 Process .)

(Output=, ) ,

(Input=4M) .

.

< >

*

()*

*/4M=Man+Machine+Material+Method

Process = ( = ) =

(, ) ,

Process .

, ( = = .)

*Six Sigma Process !

7. (Control) ?

4 .

< >

(Change)

/

(Observe and

or Measure)

(Investigate)

(Decide)

(Compare)

.

.* ., . * .

8. Six Sigma (MAIC-CYCLE)

ControlMeasure

(*Define )

Improve Analyze

MAIC

(Deming-cycle)

SPC // /

Improve Quality Increase Productivity Reduce Cost

Lower Price More Competitive Capture More Market

Share Create More Jobs

9. (QE) Six sigma

(, Taguchi )

, (Loss)

Quality Engineering

OFF-Line-QC ON-Line-QC

(Robust-Design Of

Experiments)

(SPC)

SIX SIGMA TOOL

. Data ( )

1.

* N=(//)* (/)

* n

DATA

*

Action

Tool :

* : , * : X bar, s,* : Cp, Cpk, Z, DPO, DPMO, etc

* : Histogram* : Control-Chart* : Six Sigma

* N = n =

* RS = Random Sampling ( )

* M = Measurement () T = Testing ()

RS M

T

2. (MSA) .

SPC , (Gage)

(Measurement Method) , 6

.

(Measurement System analysis)

(, Bias) , .

, . . , .

, , .

, .

1.

(Accuracy)

2.

(Repeatability or

Precision)

3.

(Stability)

4.

(Reproducibility)

* R&R .

* Six Sigma Measure .

3. .

(Random Sampling)

*, .

*()=

( = = Stratified Sampling)

* , .

( = = Cluster Sampling)

*.* .

,, ., ,

.

4. Pareto()

Vital - Few ( = )Trivial - Many ( = )

Vital-Few

Trivial-Many

* Paretos Diagram J.M.Juran .

* QC 7 Tools .

* , ABC

5. ( )

Sample

< Population Parameters > < Sample Statistics >

* Inferences(or Estimation)

S or

1) () =

* N= * n=

= NX

X bar = n

X

2) () =

(Sum Of Squares) = = ()

S = (Xi - X bar) 2 = Xi 2 - ( Xi) 2 / n

= s2 = () = 2

* Sample Mean Of Squares

s2 = nS

* Unbiased Variance

2 = = n-1S

= s

* Sample Standard Deviation

s =

* True Standard Deviation

= S2

(Unbiased Estimator) =

(Degrees Of Freedom) = = (n-1)* : (Xi - X bar) = 0 *

= () = 25

6.

DATA : 25, 40, 35, 10, 17, 15 ( )

Xi Xi2

25625

401600

351225

10100

17289

15225

xi = 142Xi2 = 4064

(X bar)

X bar = 6

142= 23.67()

(S)

S = 4,064 -1422

= 703.336

(s)

s = 703.336

= 10.83 ()

()

= 703.335

= 11.86 ()

* 6

*

* 6

4

. Histogram

1.

.

* (Spray or Dispersion)

.

*(Distribution)

.

* (Spray or Dispersion)

* .

* (Normality-Study) .

* N-Study ( Six Sigma measure )

Location Spread Shape

2. Histogram

(Freqency Distribution)

KS A 3001 ,

,

(, Class) ,

.

Histogram,

() .

Histogram

.

.

() () .

* (Normality-Study)

() () .

* = X bar =

(6) .

* Cp, Cpk, Cpu, Cpl, PP%, Ppk .. .

Six Sigma Z-value .

< >

1

2

3

4

5

6

7

8

9

No (Class) (Xi) Check fi

20.5~25.5

25.5~30.5

30.5~35.5

35.5~40.5

40.5~45.5

45.5~50.5

50.5~55.5

55.5~60.5

60.5~65.5

25

28

33

38

43

48

53

58

63

4

8

15

18

24

17

10

3

1

100

< Histogram >

0

5

10

15

20

25

20.5 25.5 30.5 35.5 40.5 45.5 50.5 55.5 60.5 65.5

sL = 25.0 m = 37.5 Su = 50.0

< Histogram >

* *

*

*

* , *

* , 2

*

*

* 50 , 100 , .

* A ?

. (Normal-Distribution)

1.

.

(Parttern)

.

,

, .

, ?

(pdf =Probability Density Function)

f(X) = 2

122 ~ N (, 2)

-e(- )2

100, 5 ?

f(X) = 5 2

1 (X-100)2

~ N (100, 52)

=0.5

,

,

=1.0

=1.5

=-2 = 0 = 2

= X bar(Mean)=X median = X mode

-e50

2. (Standard Normal Distribution)

U = p.d.f.,

X-

f(U) = 2

12U2

~ N (0, 12)

3 0.27%. (1 = )

-3 -2 -1 +1 +2 +3

68.26%

95.44%

99.73%

,

68.26% 1

95.44% 2

99.73% 3

. Minitab u Z

-e

X-Z = U = , OK!

N(1,4), Pr(1X7) ?

() Pr(1X7)= 0.5 - Pr(X7)= 0.5 - Pr(U )= 0.5 - Pr(U3)= 0.5 - 0.0013 = 0.4987

27-1

Pr ?

0.0013

1 7

, Pr(-3X7) ?

() Pr(1X7)= 1 - - = 1 - - Pr(X-3)= 1 - - Pr(U )= 1 - - 0.0228= 1- 0.0013 - 0.0228 = 0.9759

Pr ?

=0.0013

1 7

2-3-1

0.0228=

-3

, SL = 3, Su = 7,

() = + = 0.0013 + 0.0228

(, )

(Normality - Study)

3. (U Pr )

Pr ?

O U

N(0,1) Pr = 2

1U e 2

U 2 du

U

0.0*0.1*0.2*0.3*0.4*0.5*

0.6*0.7*0.8*0.9*1.0*

1.1*1.2*1.3*1.4*1.5*

1.6*1.7*1.8*1.9*2.0*

2.1*2.2*2.3*2.4*2.5*

2.6*2.7*2.8*2.9*3.0*

.5000

.4602

.4207

.3821

.3446

.3085

.2743

.2420

.2119

.1841

.1587

.1357

.1151

.0968

.0808

.0668

.0548

.0446

.0359

.0287

.0228

.0179

.0139

.0107

.0082

.0062

.0047

.0035

.0026

.0019

.0013

.4960

.4562

.4168

.3783

.3409

.3050

.2707

.2389

.2090

.1814

.1562

.1335

.1131

.0951

.0793

.0655

.0537

.0436

.0351

.0281

.0222

.0174

.0136

.0104

.0080

.0060

.0045

.0034

.0025

.0018

.0013

.4930

.4522

.4129

.3745

.3372

.3015

.2676

.2358

.2061

.1788

.1539

.1314

.1112

.0934

.0778

.0643

.0526

.0427

.0344

.0274

.0217

.0170

.0132

.0102

.0078

.0059

.0044

.0033

.0024

.0018

.0013

.4880

.4483

.4090

.3707

.3336

.2981

.2643

.2327

.2033

.1762

.1515

.1292

.1093

.0918

.0764

.0630

.0516

.0418

.0336

.0268

.0212

.0166

.0129

.0099

.0075

.0057

.0043

.0032

.0023

.0017

.0012

.4840

.4443

.4052

.3669

.3300

.2946

.2611

.2296

.2005

.1736

.1492

.1271

.1075

.0901

.0749

.0618

.0505

.0409

.0329

.0262

.0207

.0162

.0125

.0096

.0073

.0055

.0041

.0031

.0023

.0016

.0012

.4801

.4404

.4013

.3632

.3264

.2912

.2578

.2266

.1977

.1711

.1469

.1251

.1056

.0885

.0735

.0606

.0495

.0401

.0322

.0256

.0202

.0158

.0122

.0094

.0071

.0054

.0040

.0030

.0022

.0016

.0011

.4761

.4364

.3974

.3594

.3228

.2877

.2546

.2236

.1949

.1685

.1446

.1230

.1038

.0869

.0721

.0594

.0485

.0392

.0314

.0250

.0197

.0154

.0119

.0091

.0069

.0052

.0039

.0029

.0021

.0015

.0011

.4721

.4325

.3936

.3557

.3192

.2843

.2514

.2206

.1922

.1660

.1423

.1210

.1020

.0853

.0708

.0582

.0475

.0384

.0307

.0244

.0192

.0150

.0116

.0089

.0068

.0051

.0038

.0028

.0021

.0015

.0011

.4681

.4286

.3897

.3520

.3156

.2810

.2483

.2177

.1891

.1635

.1401

.1190

.1003

.0838

.0694

.0571

.0465

.0375

.0301

.0239

.0188

.0146

.0113

.0087

.0066

.0049

.0037

.0027

.0020

.0014

.0010

.4641

.4247

.3859

.3483

.3121

.2776

.2451

.2148

.1867

.1611

.1379

.1170

.0985

.0823

.0681

.0559

.0455

.0367

.0294

.0233

.0183

.0143

.0110

.0084

.0064

.0048

.0036

.0026

.0019

.0114

.0010

*=0 1 2 3 4 5 6 7 8 9

-

4.

Color-TV A =50(), =8()

. SL=45, SU=62() ,

?

()Pr(X62) + Pr(X45)= + = Pr(U + Pr(U )

= Pr(U1.50)+Pr(U-0.63)= 0.0668+0.2643 = 0.3311

( 33.11%)O U

Wafer =3.10(), =0.02()

, 3.10 0.0392() ,

?

()

862-50

845-50

Pr(XSU) + Pr(XSL) = Pr(U )

= Pr(U1.96) 2 = 0.025 2 = 0.05 = 5%

3.1392-3.100.02

2

. (6) Cp, Cpk

1. (Process Capability = 6 SIGMA

6 SIGMA.

J.M.Juran ,

,

,

(Natural Tolerance)

.

,

= 6 Sigma.

()

()

*4M

2. (Process Capability Index)

= Cp, Cpk, Cpu, CpL

1) (m) = ( ) m=X bar ( ) = Cp

6

SL m SU

6

6

Cp 1.00

Cp =1.00

Cp 1.00

T

Cp =6T

(, T=SU - SL = = R bar/d2 )

mX bar ( ) = Cpk

(T/2)

(m-Xbar)

SL m X bar Su

() = Bias = K | m-X bar | / ( ) Cpk

Cpk = (1-K) Cp = (1-K)

2T

6T

--- Cpk

Cp = ... K

m X bar(m=X bar) Cpk.

2)

Su = ()

X bar Su

(Su-Xbar) Cpu = (Su-X bar)/3

SL = ()

X barSL

(Xbar-SL) CpL = (X bar-SL)/3

3

3

--- Cpk

* Capability Ratio = CR = Cp =

* Performance Percent = PP % = Cp = %

(75% Rule !)

3. (Cp, Cpk, PP%) ()

Cp or Cpk PP% /

Cp 1.67 59.9%

* .. , .

11.67 Cp1.33

75.2%* ..

21.33 Cp1.00

100%

* .

.Cp 1 ,

.

31.00 Cp0.67

149.3%

* .. , , .

4 0.67 Cp149.3%

* . . , ,

. .

* +1.5 Shift

4.

* User,

*

4M

/

Action/

() () ,

Action

()

,

,

()

()

,

/,

Action

5. Cpk (*PPM )

(Cp PPM 2)

0.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

.60

.61

.62

.63

.64

184060

176186

168528

161087

153864

146859

140071

133499

127143

121000

115070

109348

103835

98532

93417

88508

83793

79270

74934

70781

66807

63008

59380

55917

52616

49471

46479

43633

40930

38364

35930

33625

31443

29378

27429

0.65

.66

.67

.68

.69

.70

.71

.72

.73

.74

.75

.76

.77

.78

.79

.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

25588

23852

22216

20675

19226

17865

16586

15386

14262

13209

12225

11304

10444

9642

8894

8198

7549

6947

6387

5868

5386

4940

4527

4145

3792

3467

3167

2890

2635

2401

2186

1988

1807

1641

1489

1.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

.20

.21

.22

.23

.24

.25

.26

.27

.28

.29

.30

.31

.32

.33

.34

1350

1223

1107

1001

904

816

736

664

598

538

483

434

390

350

313

280

251

240

200

179

159

142

126

112

100

89

79

70

62

55

48

43

38

33

29

1.35

.36

.37

.38

.39

.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

26

23

20

17

15

13

12

10

9

8

6

6

5

5

4

3

3

3

2

2

2

1

1

1

1

1

1

1

1

0

0

-

-

-

-

Cpk PPMCpk PPMCpk PPMCpk PPM

*PPM(Parts Per Million) = DPM(Defects Per Million)

6. Cp, Cpk

() Cp, Cpk , ?

(, 500 10)

bar = 500 (=2.4) bar = 495 (=2.4)

()

Cp =6T =

62.420 = 1.39

Cp = 1.39 ,

( m=500) X bar=495

() K .

2T

220* K = | m-X bar | / ( ) = | 500-495 | / ( ) = 0.5

* Cpk = (1-K) Cp = (1-0.5) 1.39 = 0.695 0.7

SL=490 m=500 =X bar SU=510 SL=490 X bar=495 m=500 SU=510

()

( x bar m )

7. Cp, Cpk ( )

X bar m

SL m SLSU m SU SL m SU

Cpk = 0.5 Cpk = 1.0 Cpk = 1.5

SL m SU SL m SU SL m SU

Cp = 0.5 Cp = 1.0 Cp = 1.5

, .

(m-x bar)

8. 6 SIGMA

, 6 2 6

. ,

.

1.5

* = 2PPB

*Parts Per Million

x bar

-6 -5-4 -3 -2 -1 1 2 3 4 5 6

SL Su

* 1.5 * = 3.4PPM

x bar

-6 -5-4 -3 -2 -1 1 2 3 4 5 6

SL Su

-1.5 +1.5

0

9. 4

6 ( ), 8

.

1.5 , 6210PPM

.

34

* = 63 PPM * = 6,210 PPM

x bar

-4-3 3 4

SL SU-4 -3 3 4

SL SU

-1.5 +1.5

6 12

. * Cp=2.00

Customer , , .

10. ()

() X A n=100

. . , A

1510().

No

1

2

3

4

5

6

7

8

9

10.5 ~ 12.5

12.5 ~ 14.5

14.5 ~ 16.5

16.5 ~ 18.5

18.5 ~ 20.5

20.5 ~ 22.5

22.5 ~ 24.5

24.5 ~ 26.5

26.5 ~ 28.5

11.5

13.5

15.5

17.5

19.5

21.5

23.5

25.5

27.5

2

8

14

20

23

15

10

6

2

?

?

%

? ( )

Cp Cpk , ?

( )

.

No

1

2

3

4

5

6

7

8

9

10.5 ~ 12.5

12.5 ~ 14.5

14.5 ~ 16.5

16.5 ~ 18.5

18.5 ~ 20.5

20.5 ~ 22.5

22.5 ~ 24.5

24.5 ~ 26.5

26.5 ~ 28.5

11.5

13.5

15.5

17.5

19.5

21.5

23.5

25.5

27.5

2

8

14

20

23

15

10

6

2

Xi fi Ui fiUi fiUi2

-4

-3

-2

-1

0

1

2

3

4

-8

-24

-28

-20

0

15

20

18

8

32

72

56

20

0

15

40

54

32

100 -19 321()

= X bar = 19.5 + 2 = 19.12()

= = 2 = 3.58 ()

100(-19)

9999

321 (-19)100

Histogram (SL & SU ) :

SL=5 M=15 SU=25

27.511.5

X bar =19.12

2

,

, ( )

Su=25 , 5.05% .

Cp Cpk

* 0.67 Cp 1.00 ,

.

* m=15 X bar=19.12 ,

Cpk .

* K = | m-X bar| / ( ) = 15-19.12 | / ( ) = 0.412

* Cpk = (1-K) Cp = (1-0.412) 0.931 0.55

* , , .

X bar Su ,

SN(Near) ,

Cpk =

*

* K ,

. -D.J.Lee

U =

X- = 3.58

25-19.12 1.64 Pr = 0.0505

Cp = 6SU-SL =

63.5825-5

0.931

2T

220

3SN- =

33.5825-19.12

0.55

1945 ,

.

,

.

*

(Taguchi Genichi),

** Taguchi, (NTT) (ECL)

1950~1960 ,

.

*** 1962 Deming .

Robust ?

()(), ,Strong

. Taguchi()

1. Deming() Taguchi

, !

!

* = = =

!

* = + + +

() ( )

( )

+ +

+

+

=

2. Taguchi

1979 Sony-USA Sony-JAPAN TV

. (* .)

TV

.

* , ?

2) 1979 4 17 Asahi (TV )

1)

3) ,

4)

Sony-JAPANSony-USA

m-5 m m+5

Sony-JAPAN = m, = 5/3 0.3% > > >

Sony-USA m 5 0.0%

3. !

m- m+ m- m+

* * 2

2)

1) Old New

3) L(y) = k { 2 + ( m - x bar )2 }

(Norminal-the-best Type Characteristics)

: , m

(Smaller-the-better Type Characteristics)

: , , Su

(Larger-the-better Type Characteristics)

: , , SL

, L(y)

- Old- - New-

L(y) L(y)

4. Taguchi (Quality Loss Function)

m m

FMC W.H.Moor Weather-Strip

Weather-Strip () : Y = 20 4(mm)

Weather-Strip Maker 3 :

Front Door Weather-Strip Door Weather-Strip ()

1) Cpk

20 4(mm)

Maker X bar (mm) (mm) Cpk

20.0 1.333 1.00251.00

18.0 0.667 0.99951.00

17.2 0.400 0.99991.00

Weather-Strip Maker 3 Cp, Cpk

16.0 20.0 24. 0m- m m+

( )CpkCpkbarXmk

Cpm

=

=

=

1

62

5. Cpk ? (Ford-Motor )

2) FMC 250,000 (/Year) , ($50/).

(1)

Maker = 250,000 0.00270 50$ = 33,750 $ Maker = 250,000 0.00135 50$ = 16,875 $ * Maker = 250,000 0.00135 50$ = 16,875 $ *

(2) L(Y) = K { 2 + ( X bar - m )2 }

Maker = L1 = 2.0 { 1.3332 + (20-20.0)2 } = 3.55$/ *Maker = L2 = 2.0 { 0.6672 + (20-18.0)2 } = 8.89$/Maker = L3= 2.0 { 0.4002 + (20-17.2)2 } = 16.00$/

(3)

Maker X bar Cpk Taguchi

20.0 1.333 1.00 0.270% 33,750$ 887,500$ *

18.0 0.667 1.00 0.135% 16,875$ 2,222,500$ *

17.2 0.400 1.00 0.135% 16,875$ 4,000,000$ *

L(y) 250,000

* , K=2

* ?