analysis of clements' method for capability

10
55 METALLURGY AND FOUNDRY ENGINEERING Vol. 35, 2009, No. 1 * Ph.D.: Faculty of Metals Engineering and Industrial Computer Sciences, AGH University of Science and Technology, Kraków, Poland; e-mail [email protected] Andrzej Czarski* ANALYSIS OF CLEMENTS METHOD FOR CAPABILITY INDICES ESTIMATION 1. INTRODUCTION In a practical approach the statistical methods in quality management are first of all: statistical process control (SPC), measurement system analysis (MSA), acceptance sam- pling techniques and process and product reliability assessment. Statistical methods take particular place in a process improvement; in this scope the following statistical instruments are applied: hypotheses and test procedures, analysis of variance (ANOVA), regression and correlation analysis, design of experiment (DOE) etc. [1–4]. One of the fundamental tasks of SPC is the assessment of capability of a process/ ma- chine relating to client’s expectations. In this scope many different capability indices are applied in practice, e.g. Cp, Cpk, Pp, Ppk, Cm, Cmk, Tp, Tpk [5– 10]. This work concerns the comparative analysis of capability indices determination in case of the distribution unlike the normal one. 2. CAPABILITY PROCESS ESTIMATION IN CASE OF DISTRIBUTION UNLIKE THE NORMAL ONE It is stressed widely that SPC methods are situated in an “avoiding loss” convention – information achieved by these methods we use “on-line” in order to assure a desirable performance of the process (instead of concentrate on process effects that not always are desirable and expected, we focus on the process that generates these results!). In case of distributions unlike the normal a natural tolerance – i.e. the range in which practically all possible realization of the process should be contained – we define as [6, 8, 10]: T n = x 0.99865 x 0.00135 (1)

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In a practical approach the statistical methods in quality management are first of all:statistical process control (SPC), measurement system analysis (MSA), acceptance sam-pling techniques and process and product reliability assessment. Statistical methods takeparticular place in a process improvement; in this scope the following statistical instrumentsare applied: hypotheses and test procedures, analysis of variance (ANOVA), regression andcorrelation analysis, design of experiment (DOE) etc. [1–4].

One of the fundamental tasks of SPC is the assessment of capability of a process/ ma-chine relating to client’s expectations. In this scope many different capability indices areapplied in practice, e.g. Cp, Cpk, Pp, Ppk, Cm, Cmk, Tp, Tpk [5– 10].

This work concerns the comparative analysis of capability indices determination incase of the distribution unlike the normal one.

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It is stressed widely that SPC methods are situated in an “avoiding loss” convention –information achieved by these methods we use “on-line” in order to assure a desirableperformance of the process (instead of concentrate on process effects that not always aredesirable and expected, we focus on the process that generates these results!).

In case of distributions unlike the normal a natural tolerance – i.e. the range in whichpractically all possible realization of the process should be contained – we define as [6, 8, 10]:

Tn = x0.99865 – x0.00135 (1)

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where:x0.00135 – value meeting the condition: P(x < x0.00135) = 0.00135,x0.99865 – value meeting the condition P(x < x0.99865) = 0.99865 (i.e. P(x >= x0.99865) =

= 0.00135) (Fig. 1).

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Let’s give attention that the definition (1) guarantees exactly the same as the 6s range(s – standard deviation) in case of a normal distribution, i.e. within the range determined bya natural tolerance practically all possible realizations of the process should be contained,and participation of the realization outside the range determined by natural tolerance limitsequals 0.0027 (i.e. 2700 ppm).

Additionally, let x0.5 means the median (P(x < x0.5) = 0.5) which, in case of an asym-metric distribution, will be a measure of midpoint of grouping of values (of course, in caseof symmetric distributions e.g. a normal distribution there is the equality x0.5 = ,x where:x – mean value).

Based on the values defined above, i.e. x0.00135, x0.5, x0.99865 process capability indicesCp, Cpk we define as below (Tab. 1) [6, 8].

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Specification limits potential capability Cp real capability Cpk

double-sided specification limits 0 ,9 9 8 6 5 0 ,0 0 1 3 5

U S L L S L

x x

−−

(2) 0,5 0,5

0,5 0,00135 0,99865 0,5min ;

x LSL USL x

x x x x

⎛ ⎞− −⎜ ⎟⎜ ⎟− −⎝ ⎠

(3)

lower specification limit only

N / A 0 ,5

0 ,5 0 ,0 0 1 3 5

x L S L

x x

−−

(4)

upper specification limit only

N / A 0 ,5

0 ,9 9 8 6 5 0 ,5

U S L x

x x

−−

(5)

Symbols: USL – upper specification limit LSL – lower specification limit N / A – not available

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There are two methods of determination of the values x0.00135, x0.5, x0.99865 [6, 10]:

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0.00135

( ) 0.00135x

f x dx−∞

=∫ (6)

0.5

( ) 0.5x

f x dx−∞

=∫ (7)

0.99865

( ) 0.99865x

f x dx−∞

=∫ (8)

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In order to determine the capability indices (Table 1, formulae 2–5) it is necessary todetermine the values x0.00135, x0.5, x0.99865. For that purpose Clements proposed a methodbasing on the Pearson curves [6, 8].

The course of proceeding in the Clements’ method is the following:

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0.00135 0.00135x x k s= − ⋅ (9)

0.99865 0.99865x x k s= + ⋅ (10)

0.50.5

0.5

; 0

; 0

x k s Skx

x k s Sk

− ⋅ >⎧⎪= ⎨+ ⋅ <⎪⎩

(11)

where: k0,00135, k0,5, k0,99865 – indices depending on skewness and kurtosis; read from thetables, indices Cp, Cpk are determined according to adequate expressions placedin Table 1 (formulae 2–5).

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The aim of testing was the evaluation of accuracy of the Clements’ method. The proce-dure was as follows:

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In the analysis carried out the following theoretical distributions have been used:

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( ) ; 0xf x e x−λ= λ ≥ 8;?9

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( )2

2

ln1( ) exp ; 0

2 2

x mf x x

x

⎡ ⎤−⎢ ⎥= − ≥⎢ ⎥σ π σ⎣ ⎦

8;<9

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( )1 exp /( ) ; 0

( )

kk

x bf x x x

k b

− −= ≥

Γ8;@9

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( ) ( ) 11( ) 1 ; [0, 1]( ) ( )

f x x x xβ−α−Γ α + β= − ∈

Γ α Γ β8;�9

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1( / )( ) ; 0

kk

xk xf x e x

−− λ⎛ ⎞= ≥⎜ ⎟λ λ⎝ ⎠

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Density probability functions of selected distributions for the assumed parameters arepresented in Figures 2–7. The fundamental description parameters are in Table 2.

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Descriptive Statistics Distribution Parameters mean

value median

standard deviation

skewness kurtosis

λ=1 1.0 0.6931 1.0 2 6

λ=0.5 1.0 1.3863 2.0 2 6 Exponential

λ=0.333 3.0 2.0794 3.0 2 6

m=0.25; σ=0.2 1.2840 0.2646 0.2646 0.6143 0.6784

m=0.25; σ=0.3 1.2840 0.4122 0.4122 0.9495 1.6449 Lognormal

m=0.25; σ=0.4 1.2840 0.5794 0.5794 1.3219 3.2600

b=0.5; k=2 1.0 0.83917 0.7071 1.4142 3

b=0.5; k=3 1.5 1.33703 0.8660 1.1547 2

b=0.5; k=4 2.0 1.83603 1.0 1.0 1.5

b=0.5; k=5 2.5 2.33545 1.1180 0.8944 1.2

b=0.5; k=6 3.0 2.88351 1.2247 0.8165 1

Gamma

b=0.5; k=10 5.0 4.83436 1.5811 0.6324 0.6

α=1; β=2.5 0.2857 0.24214 0.2130 0.7318 –0.2434

α=2; β=2.5 0.4444 0.43555 0.2119 0.1614 –07661

α=3; β=2.5 0.5454 0.55133 0.1953 –0.1241 –0.6855

α=4; β=2.5 0.6154 0.62787 0.1776 –0.3056 –0.5062

α=5; β=2.5 0.6667 0.68214 0.1617 –0.4340 –0.3158

Beta

α=6; β=2.5 0.7059 0.72262 0.1478 –0.5305 –0.1362

λ=1; k=1 1.0 0.6931 1.0 2 6

λ=1; k=1.5 0.9027 0.7832 0.6129 1.0720 1.3904

λ=1; k=3 0.8930 0.8850 0.3245 0.1681 –0.2705

λ=3; k=20 2.9205 2.9455 0.1810 –0.8680 1.2672

λ=3; k=30 2.9455 2.9636 0.1230 –0.9531 1.5841

Weibull

λ=3; k=40 2.9585 2.9726 0.0932 –0.9975 1.7630

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The

oret

ical

val

ue

Acc

ordi

ng to

Cle

men

ts

Dis

trib

utio

n Pa

ram

eter

s x 0

.001

35

x 0.5

x 0

.998

65

x 0.0

0135

x 0

.5

x 0.9

9865

m

1 m

2 m

3

λ =1

0.00

135

0.69

315

6.60

765

0.00

1 0.

693

6.60

8 0.

9997

0.

9999

0.

9999

λ=0.

5 0.

0027

0 1.

3862

9 13

.215

3 0.

002

1.38

6 13

.216

0.

9997

0.

9999

0.

9999

E

xpon

enti

al

λ=0.

333

0.00

405

2.07

944

19.8

23

0.00

3 2.

079

19.8

24

0.99

97

0.99

99

0.99

99

m=

0.25

; σ=

0.2

0.57

695

1.28

402

2.58

569

0.70

613

1.28

40

2.33

89

1.22

36

1.23

03

1.23

39

m=

0.25

; σ=

0.3

0.63

763

1.28

402

2.85

763

0.52

737

1.28

38

3.15

63

0.85

45

0.84

44

0.84

04

Log

norm

al

m=

0.25

; σ=

0.4

0.70

469

1.28

402

3.15

817

0.40

342

1.28

34

4.25

81

0.65

83

0.63

65

0.63

00

b=0.

5; k

=2

0.02

644

0.83

917

4.45

01

0.02

499

0.83

89

4.44

83

0.99

85

1.00

01

1.00

04

b=0.

5; k

=3

0.10

584

1.33

703

5.43

48

0.10

377

1.33

63

5.43

15

0.99

89

1.00

02

1.00

06

b=0.

5; k

=4

0.23

265

1.83

603

6.34

02

0.23

3 1.

836

6.33

8 1.

0002

1.

0004

1.

0005

b=0.

5; k

=5

0.39

594

2.33

545

7.19

62

0.38

66

2.33

61

7.20

38

0.99

49

0.99

75

0.99

86

b=0.

5; k

=6

0.58

749

2.88

351

8.01

74

0.58

813

2.83

49

8.01

47

1.02

19

1.00

04

0.99

11

Gam

ma

b=0.

5; k

=10

1.

5421

3 4.

8343

6 11

.087

9 1.

5448

8 4.

8338

11

.084

1.

0010

1.

0007

1.

0005

α=1;

β=

2.5

0.00

054

0.24

214

0.92

886

–0.0

003

0.24

15

0.92

71

0.99

89

1.00

10

1.00

17

α =2;

β=

2.5

0.01

772

0.43

555

0.95

635

0.01

832

0.43

55

0.95

52

1.00

16

1.00

19

1.00

21

α=3;

β=

2.5

0.06

041

0.55

133

0.96

826

0.06

153

0.55

14

0.96

76

1.00

21

1.00

19

1.00

17

α=4;

β=

2.5

0.11

465

0.62

787

0.97

501

0.11

558

0.62

79

0.97

48

1.00

17

1.00

13

1.00

07

α=5;

β=

2.5

0.17

067

0.68

214

0.97

938

0.17

199

0.68

23

0.97

91

1.00

23

1.00

20

1.00

14

Bet

a

α=6;

β=

2.5

0.22

412

0.72

262

0.98

244

0.22

529

0.72

28

0.98

23

1.00

21

1.00

17

1.00

10

λ=1;

k=

1 0.

0013

5 0.

6931

5 6.

6076

5 0.

001

0.69

3 6.

608

0.99

97

0.99

99

0.99

99

λ=1;

k=

1.5

0.00

203

0.78

322

3.52

126

–0.0

308

0.78

24

3.51

16

0.96

06

0.99

34

1.00

32

λ=1;

k=

3 0.

0040

5 0.

8850

0 1.

8765

0.

0782

3 0.

8825

1.

8541

1.

0953

1.

0544

1.

0205

λ=3;

k=

20

2.15

602

2.94

552

3.29

704

2.15

439

2.94

53

3.28

14

0.99

82

1.01

24

1.04

59

λ =3;

k=

30

2.40

7 2.

9635

7 3.

1948

9 2.

4072

1 2.

9636

3.

1839

1.

0004

1.

0144

1.

0497

Wei

bull

λ=3;

k=

40

2.54

324

2.97

264

3.14

501

2.54

355

2.97

27

3.13

67

1.00

06

1.01

44

1.05

05

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Values x0.00135, x0.5, x0.99865 determined theoretically for different distributions anddifferent parameters (according to formulae 6–8) and on the basis of the Clements’ method(according to formulae 9–11) are presented in Table 3.

Because these values serve to determine the capability indices Cp, Cpk (Tab. 1), addi-tionally – in this context – the following consistency measures have been defined:

0.5 0.001351

0.5 0.00135

( ) ( )

( ) ( )

x theor x theorm

x Clements x Clements

−=

−(17)

0.99865 0.001352

0.99865 0.00135

( ) ( )

( ) ( )

x theor x theorm

x Clements x Clements

−=

−(18)

0.99865 0.53

0.99865 0.5

( ) ( )

( ) ( )

x theor x theorm

x Clements x Clements

−=

−(19)

In the meaning of these total consistency measures it is the value 1.The values of measures m1, m2, m3 for different distributions are in Table 3.

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With the exception of the lognormal distribution the results achieved during investiga-tion confirm a very good accuracy of the Clements’ method. A very good conformity ofvalues x0.00135, x0.5, x0.99865 determined theoretically and with the Clements’ method, aswell as defined consistency measures (Tab. 3) indicate the truth of this statement.

In case of the lognormal distribution the operating of the Clements’ method is worse(Tab. 3).

The comparative analysis performed should enable the Clements’ method users its op-timal application.

Acknowledgements

The financial support from the Polish Ministry of Science and Higher Education, con-tract AGH no.10.10.110.797 is gratefully acknowledged.

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