Estimating capability index C-pk for processes with asymmetric tolerances

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Estimating capability index c

pk

for processes with

asymmetric tolerances

W. L. Pearn a & G. H. Lin b a

Department of Industrial Engineering & Management , National Chiao Tung University , Hsinchu, Taiwan, ROC

b

Department of Telecommunication Engineering , National Penghu Institute of Technology , Penghu, Taiwan, ROC

Published online: 23 Jul 2008.

To cite this article: W. L. Pearn & G. H. Lin (2000) Estimating capability index cpk for processes with asymmetric tolerances, Communications in Statistics - Theory and Methods, 29:11, 2593-2604, DOI: 10.1080/03610920008832625 To link to this article: http://dx.doi.org/10.1080/03610920008832625

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COMMUN. STATIST.-THEORY METH., 29(1 I), 2593-2604 (2000)

ESTIMATING

CAPABILITY INDEX C,,

FOR PROCESSES

WITH

ASYMMETRIC TOLERANCES

W. L. Pearn

Department of Industrial Engineering & Management National Chiao Tung University

Hsinchu, Taiwan ROC G. H. Lin

Department of Teleconununication Engineering National Penghu Institute of Technology

Penghu, Taiwan ROC

Keywords:

process capability index; specification limit; process mean; process standard deviation.

ABSTRACT

Pearn and Chen (1996) considered the process capability index C,,, and investigated the statistical properties of its natural estimator under various process

conditions. Their investigation, however, was restricted to processes with

symmetric tolerances. Recently, Pearn and Chen (1998) considered a

generalization of C,,, referred to as c i k , to cover processes with asymmetric tolerances. They investigated the statistical properties of the natural estimator of Cik

,

and obtained the exact formulae for the expected value and variance. In this paper, we consider a new estimator of c;,, assuming the knowledge on

P(p 2 T) = p is available, where 0 5 p 2 1 , which can be obtained from historical information of a stable process. We obtain the exact distribution of the

Copyright O 2000 by Marcel Dekker, Inc.

2594 PEARN AND LIN new estimator assuming the process characteristic follows the normal distribution. We show that the new estimator is consistent, asymptotically unbiased, which converges to a mixture of two normal distributions. We also show that by adding suitable correction factors to the new estimator, we may obtain the UMVUE and the MLE of the generalization

cik

.

1. INTRODUCTION

Process capability indices C, and C,,,, have been widely used in the

manufacturing industry providing numerical measures on process potential and performance. Application examples include the manufacturing of semiconductor

products (Hoskins et al. (1988)), headtgimbals assembly for memory storage

systems (Rado (1989)), jet-turbine engine components (Hubele et a/. (1991)), flip- chips and chip-on-boards (Noguera and Nielson (1992)), rubber surrounds (Pearn and Kotz (1994)), wood products (Lyth and Rabiej (1995)), audio-speaker drivers (Chen and Pearn (1997)) electrolytic capacitors (Pearn and Chen (1997)) and many others. The two indices C, and C,, have been defined as the following (Kane (1 986)):

C, = USL - LSL

6s

where USL and LSL are the upper and the lower specification limits, p and o are the process mean and the process standard deviation. The natural estimators of the two indices can be expressed as:

-

USL-LSL

C, =

6 s

-

where X = (C:=,X,) / n and S = {(n -

I)-IC''',(X,

-

X)'J''~

are conventional

estimators of p and (3. We note that C,, can be rewritten as

PROCESS CAPABILITY INDEX Cpk 2595 Cpk ={d-b-m1}/3o, where d = ( U S L - L S L ) / 2 , and m = ( U S L + L S L ) / 2 .

Thus,

e,

and

ep,

may be expressed as:

A d

-

I

-

{

i F ; m i h p

C = - and Cpk =

-

3 s ' 3 s

Under normality assumption,

kp

is distributed as (n-1)'12Cp(X,!,), and

n1l2lX-mllo is distributed as the folded normal distribution with parameter n " 2 1 p - m / / ~ . Thus,

tPh

is a convolution of

xi!,

and the folded normal

distributions (Peam et al. (1992)). If the knowledge on P(p 2 m) = p is

available, where 0 5 p 2 1 , Peam and Chen (1 996) proposed a Bayesian-like estimator which is more reliable (with smaller variance) than the natural estimator.

2.

Tl3X GENERALIZATION

c*,

The indices C, and Cpk are appropriate for processes with symmetric

tolerances, but have been shown to be inappropriate for processes with asymmetric tolerances. For processes with asymmetric tolerances, Peam and Chen (1998) considered a generalization of Cpk, referred to as c ; ~ , which is defined as :

where d' = min {d,, d,}, A' = max {d*(p

- T)/d,, d * ( ~

- p )/dl }, d, = USL

- T,

and dl = T - LSL. Clearly, if T = m (symmetric case) then d' = d, A * = ( p

- m

1

and the generalization

cIpk

reduces to the original index Cpk. The factors d* and

A* ensure that the generalization

c ; ~

obtain its maximal value at T (process is on-target in this case) regardless of whether the tolerances are symmetric or asymmetric.

The natural estimator of

c',~

can be obtained by replacing p and o by

X

and S as defined earlier. Thus the natural estimator can be expressed as the following, where

A'

= max {da(X -~)/d,, d * ( ~ -

X

)/dl). Pearn and Chen (1 998)

2596 PEARN AND L N

investigated the statistical properties of the natural estimator of C;k, and obtained the exact formulae for the expected value and variance.

3. A

NEW

ESTIMATOR OF Ck

If the knowledge on P(p 2 T) = p is available, where

0

<

p 2 1 , then we consider the following new estimator

e*,,

.

The knowledge on the probability P(p 2 T j = p may be obtained from historical information of a process that is demonstrably stable.

where

x*

= max { ( X -~j~,,(p)d'/d,, ( 2 -T)I,,(p)d*/d, }, I,,(.) is the indicator function defined as I,,(p) = 1 if p E

B*,

and I,,(p) = -1 if p E B*, with

B*

= { p

/

p 2 T } . In the following, we show that if the process characteristic follows the normal distribution N ( p , o') then the new estimator

C;,

is distributed as t,_, (6 ') , a non-central t distribution with n -1 degrees of freedom and non-centrality parameter 6 ' = 3 m i k .

Theorem

1. If the process characteristic follows the normal distribution, then 3n1"Eik is distributed as t,-,(6 * ) , a non-central t distribution with n - 1 degrees of freedom and non-centrality parameter 6 * = 3 j n c i k .

Proof:

Case 1 : If USL - T > T - LSL, then E',k ={d, -

(X

-T) I,. (P) }/(3Sj,

Case 2: If USL - T < T - LSL, then E * , k ={d,

-

(X

-T)

I ~ . (p) )1(3S),

-*

s

"

- -

1 P

s

" - p - 1 .P

(X

- PY,. (p)

Cpk--C =

--

=

-

(3 pk 3s (3 30 30

Case 3: If USL - T = T - LSL, then CIpk reduces to Cpk ,

PROCESS CAPABILITY INDEX Cpk 2597

C I p k = {d -

(X

- m)IB. (p)}/(3S), and

Thus, for all three cases, and for all x E R ,

Under normality assumption, and S2 are mutually independent. Therefore,

Z = 3 n 1 i 2 ~ ~ a ~ / o and W = (n -1)s' l o 2 are also mutually independent, with Z distributed as ~ ( 6 * , I), a normal distribution with mean 6 * = 3 k ; k , and W distributed as

Xi_,

,

a chi-squared distribution with n - 1 degrees of freedom. Thus 3n1'2Eik = ZI is distributed as t,-I (6 * )

,

a non-central t distribution with n - 1 degrees of freedom and non-centrality parameter 6 ' =

3&ciL.

The r-th moment (about zero), therefore, is:

By setting r = 1, and r = 2 , we may obtain the expected value E(?;~), and the variance var(?;,),

2598 PEARN AND LIN

Table

1.

Values

of MSE(E;,) for n = 5 ( 5 ) 100 with

c;,

= 1. Sample

size MSE Sample size MSE

Therefore, the mean-squared error M S E ( ~ ; , ) can be obtained as:

It can be shown that the coefficient of E(C;~), bn-, = [2/(n - l)]li2 x T[ (n- 1),/2]/T[(n - 2)/2]< 1 for all n, which in fact converges to 1 as n approaches to infinity. Thus, the estimator

Eik

is biased, which over-estimates the actual

value of C;, . Table 1 displays various values of MSE(~;,) for sample sizes n

= 5(5)100 under the condition of ~ i = k1. For sample size n

>

65, the mean-

squared error is negligibly small (less than 0.01).

PROCESS CAPABILITY INDEX Cllk 2599

Theorem

2. If the process characteristic follows the normal distribution, then (a)

eik

is asymptotically unbiased, (b)

eik

is consistent.

Proof

(a) By Stirling's formula, it is easy to show that b,-I converges to 1. Therefore, E(?;~) = cik/bn-] converges to Cik. Hence, the estimator

E',,

is

asymptotically unbiased. (b) Since

X

converges to p in probability, and S

converges to o in probability, then E i k must converges to

cik

in probability.

Therefore, the estimator

Eik

is consistent.

Theorem

3. If the process characteristic follows the normal distribution, then nIi2 (EiA - c ; ~ ) converges to the following in distribution:

(a) p . N(0, [a;,12)

+

(1 - p) . N(0, [o;,J2), if USL - T > T - LSL ,

(b) p . N(0, [o:,l2)

+

(1 - p)

.

N(0, [o:,]') , if USL - T <

T

- LSL

,

(c) p

.

N(0, [o:,~]~)

+

(1

-

p)

.

N(0, [oi212), if USL - T = T - LSL , where

Proof:

See Appendix.

Theorem

4. If the process characteristic follows the normal distribution, then (a) b,.,e6L is the UMVUE of c i k , (b)

c , . ~ E ~ ~

is the MLE of Cik, where

b,_, =

r

[(n - 1)/2]{

r

[(n -2)/2])-'[2/(n -I)]'/~

,

and c,., = [d(n

-

1)]'/*

.

Proof:

(a) Since E[E& = ~ i ~ / b , - ~ , then b,.& is an unbiased estimator of c ; ~ . Since the unbiased estimator b,.,?; is based on the complete and sufficient statistics

(x

S 2 ) only, then by Lehrnann-Scheffe's theorem, b,.,E',, is the

UMVUE of

c',,

.

(b) We first note that the statistic

(x

[(n -l)/n]S2) is the MLE of@, 0 2 ) . By the invariance property of the MLE, c,.,EiA is the MLE of c i k , where c,., =

[n/(n - 1)]1'2

.

2600 PEARN AND LIN Applying the Slutsky's theorem (Arnold (1990)), it is straightforward to

show that the UMVUE bn.,Eik, and the MLE c,.,Ei, both converge to the

mixture of two normal distributions as stated in Theorem 3.

4. CONCLUSIONS

Pearn and Chen (1998) proposed a generalization of C,k, referred to as Cik,

for processes with asymmetric tolerances. They investigated the statistical

properties of the natural estimator of c i k , and obtained the exact formulae for the expected value and variance. In this paper, we considered a new estimator of Cikr assuming the knowledge on P(p 2 T) = p is available, where 0

r

p 5 1 , which can be obtained from historical information of a stable process. We showed that under normality assumption the new estimator is distributed as t,_i(6 * ) , a non-central t distribution with n

-

1 degrees of freedom and non- centrality parameter 6 * =

3&cik.

We also showed that the new estimator is consistent, asymptotically unbiased, which converges to a mixture of two normal distributions. In addition, we showed that by adding suitable correction factors

to the new estimator, we may obtain the UMVUE and the MLE of the

generalization

c*,~

.

APPENDIX

Theorem 3. If the process characteristic follows the normal distribution, then &(Eik - cBk) converges to the following in distribution:

(a) p.N(O, [ O ; , ] ' ) + ( ~ - ~ ) . N ( O , [u;,12),if U S L - T > T - L S L , (b) p,N(O, [ o l 1 l 2 )

+

(1 - p ) . ~ ( O , [o:,12), if USL - T < T - LSL ,

(c) p .N(O, [oil]')

+

(1 -p) .N(O, [o:,l2), if USL - T = T - LSL , with

1 1 d - ( p - m )

PROCESS CAPABILITY INDEX Cpk

Proof

(a) If USL - T > T - LSL

,

then

p*

- d t - ( Z - ~ )

, for p

>

T and

E',,

= d,

+

(TT-T)

pk - ,for p < T .

3 s 3 S

dl -(x-T)

(1) For . p > T

,

we define gll (x, y) =

,

x

> T, y > 0 . Then, 3JS;

&(Eik -

c;t)

= l/;;{gll(X, s*) - gtl(p, 02)} converges to N(O, [o;,12) in distribution. with

d, + ( x - T )

(2) For p < T , we define g,,

(x,

y) = , x < T, y > 0 . Then, 3 f i

Z/;;(Eik -

c ; ~ )

= I/f;{gi2(X, s*)

-

gi2 (p, 0 *)) converges to N(0, [o;,12)

in distribution, with

(3) For p = T

,

we have

&(eiI.

-

c',~) = - l/;;(T(- p ) - dl&(s2 -0 2,

3 s 30 ( 0 +S)S

Since

&(X

- p,

s2

- 0 2 ) converges in distribution to (V, W)

,

a bivariate normal distribution N((0, 0), C)

,

with variance-covariance matrix

2=[0' 0 2~~ 0 1 , a n d

/-l

3 s ' - 30(o

+

S)S converges to

(-

--,

1

-

&)

2602 PEARN AND LM

in probability, then &(Cik - c ; ~ ) converges to Y, in distribution, where

1 d

Y, = --V -1 W is distributed as a normal distribution with mean

30 603

1 d

EY, = -- EV -

4

EW = 0 , and variance

30 60'

Hence, if USL - T > T - LSL , then &(eik - c i k ) converges to ~ { p 2 T)

.

N(O; [o;,J2) -t ~ ( p < T). N(0, [0;,J2) in distribution, where

(b) If USL - T < T - LSL , then

-.

d, -(X-T)

-

d , + ( X - T )

Cpk = , for p

r

T and C;, = .for p < T .

3 s 3 s '

d u - ( x - T )

Applying the same technique used in (a) with g,, (x, y) = f o ~

3 h

d, + ( x - T ) 1 d

p > T , gU2(x,y)= for p < T , a n d Y, =--V-+W

3 5 30 60'

for p = T . Then, &(e',k - C;k) converges to N(0, [o:, J 2 ) and N(0, [ c / ~ J 2 ) in distribution for p 2 T and p < T respectively.

Therefore, if USL - T < T - LSL , then

A(?>

-

c ; ~ )

converges to ~ ( p 2 T}.N(O, [ o t , l 2 ) + ~ { p < T}.N(O, [0:,J2) in distribution, where

(c) If USL - T = T - LSL ( T = m in this case), then

PROCESS CAPABILITY INDEX CDk

- d - ( X - m )

-

d + ( X - m )

pk -

,

for p 2 m and Cpk =

,

for p < m

3 s 3 s

d - ( x - m )

Applying the same technique used in (a) with g,, (x, y) = for

3 h

d + ( x - m ) 1 d

p > m , gm2(x,y)= for p < m , a n d

Y,,,

=--V--W

3& 30 60' for p = m . Then, &(e;k - C i k ) converges to N(0, [0',,,12) and

N(0, [cY:,,]~) in distribution for p 2 m and p < m respectively.

Therefore, if USL - T = T - LSL

,

we have

&(?ik

- c i k ) converges to P ( ~ > ~ ) . N ( O , [ o ; , , l 2 ) + ~ { p <m}.N(O, [0*,,12) indistribution, where

ACKNOWLEDGEMENTS

The authors would like to thank the anonymous referees for their careful reading and helpful comments which improved the paper. The authors would also like to thank the editor Professor William Smith for his time and patience on editing the paper.

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