A Bayesian approach based on multiple samples for measuring process performance with incapability index

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Int. J. Production Economics 106 (2007) 506–512

A Bayesian approach based on multiple samples for measuring

process performance with incapability index

Gu-Hong Lin

Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, ROC Received 14 October 2004; accepted 29 June 2006

Available online 15 September 2006

Abstract

Process incapability index, which provides an uncontaminated separation between information concerning the process

accuracy and the process precision, has been proposed to the manufacturing industry for measuring process performance.

Investigations concerning the estimated incapability index have focused on single sample in existing quality and statistical

literatures. However, contributions based on multiple samples have been comparatively neglected. In this paper,

investigations based on multiple samples are considered for normally distributed processes. A Bayesian approach to obtain

an upper bound for the incapability index is proposed. A computational program using Maple software is proposed to

evaluate critical values required to ensure the posterior probability reaching a certain desirable level for the incapability

index. A practical example is also provided to illustrate how the proposed reliable Bayesian procedure may be applied in

process capability assessment.

r

2006 Elsevier B.V. All rights reserved.

Keywords: Bayesian; Incapability index; Multiple samples

1. Introduction

Process capability indices, whose purpose is to

provide numerical measures on whether or not a

manufacturing process is capable of reproducing

items satisfying the quality requirements preset by

the customers, the product designers, have received

substantial research attention in the quality control

and statistical literature. The three basic capability

indices C

p

, C

a

and C

pk

, have been deﬁned as (

Kane,

1986

;

Pearn et al., 1998

;

Lin, 2006a

):

C

p

¼

USL LSL

6s

,

(1)

C

a

¼

1 m m

d

,

(2)

C

pk

¼

min

USL m

3s

;

m LSL

3s

,

(3)

where USL and LSL are the upper and lower

speciﬁcation limits preset by the customers, the

product designers, m is the process mean, s is the

process standard deviation, m ¼ ðUSL þ LSLÞ=2

and d ¼ ðUSL LSLÞ=2 are the mid-point and half

length of the speciﬁcation interval, respectively.

The index C

p

reﬂects only the magnitude of the

process variation relative to the speciﬁcation

toler-ance and, therefore, is used to measure process

potential. The index C

a

measures the degree of

process centering (the ability to cluster around the

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exp

1

2 X

m i¼1

X

n j¼1

x

ij

x

s

2

"

#)

d

1

2 X

m i¼1

X

n j¼1

x

ij

x

s

2

"

#

¼

ffiffiffiffiffiffiffiffiffi

p

2mn

r

_X

m i¼1

X

n j¼1

x

ij

x

ffiffiffi

2 p

2

"

#

ðmn1Þ=2

G

mn 1

2 .

Therefore, the posterior distribution of y ¼ ðm; sÞ

given x is

f ðyjxÞ ¼

hðy x

j Þ

k

¼

2 exp½ðs

2

_b

_Þ

1

s

mn

_Gða

_Þðb

_Þ

a

ffiffiffiffiffiffiffiffiffiffi

mn

2ps

2

r

exp

mn

2 _{m x}

s

2

"

#

,

where x ¼ fx

11

; x

12

; . . . ; x

mn

g, N

omoN, 0o

soN, a

_{¼ ðmn 1Þ=2, b}

_¼

2½ðmnÞS

2_mn

1

¼

2=

S

m_i¼1

S

n_j¼1

x

ij

x

2

h

i

.

Appendix B

Derivation of expression (8): The posterior

probability

p

¼

PrfC

pp

oC

0

jxg

¼

Prf½ðm TÞ

2

þ

s

2

=D

2

oC

0

jxg

¼

Prfs

2

þ ðm T Þ

2

oðD

ffiffiffiffiffiffi

C

0

p

Þ

2

jxg

¼

Z

b 0

Z

TþgðsÞ TgðsÞ

f ðm; s x

j Þ

dm ds

¼

Z

b 0

2 exp½ðs

2

_b

Þ

1

s

mn

_Gða

_Þðb

Þ

a

ffiffiffiffiffiffiffiffiffiffi

mn

2ps

2

r

(

Z

T þgðsÞ T gðsÞ

exp

mn

2 _{m x}

s

2

"

#

dm

)

ds

¼

Z

b 0

2 exp½ðs

2

_b

_Þ

1

s

mn

_Gða

_Þðb

Þ

a

F

ffiffiffiffiffiffiffi

mn

p

½T x þ gðsÞ

s

F

ffiffiffiffiffiffiffi

mn

p

½T x gðsÞ

s

ds,

where b ¼ D

ffiffiffiffiffiffi

C

0

p

, gðsÞ ¼

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b

2

s

2

_{, a}

_{¼ ðmn}

1Þ=2, b

¼

2½ðmnÞS

2_mn

1

, F is the cumulative

dis-tribution function of the standard normal

distribu-tion N(0, 1).

Let

y

_¼

_b

0

s,

b

0

¼

2=½S

m_i¼1

S

n_j¼1

ðx

ij

T Þ

2

,

g

_¼

_{1 þ ðd}

Þ

2

,

d

¼

x T

=s

mn

,

b

1

ðy

Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðd

Þ

2

=½y

_ð1þðd

Þ

2

Þ

q

, b

₂

ðy

_Þ¼

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

_mnf½1=ðt

_y

_Þ

₁

_g,

t

_¼

_{mn ~}

_C

pp

=ð2C

0

Þ, then the posterior probability

based on C

pp

with multiple samples can be

expressed as

p

_¼

Z

1=t 0

F b

1

ðy

Þ þ

b

2

ðy

Þ

F b

1

ðy

Þ

b

2

ðy

Þ

ðg

_Þ

a

_ðy

_Þ

aþ1

_Gða

_Þ

exp

1 g

_y

dy

_.

Appendix C

Digits: ¼ 6: p

0

: ¼ 0.95: c

0

: ¼ 1: m: ¼ 25: n: ¼ 5:

delta: ¼ 0.4137:

t: ¼ Cpp-(mnCpp)/(2*c

0

):

b1: ¼ y-(2delta^2/(y(1+delta^2)))^0.5:

b2: ¼ (Cpp,

y)-(mn(2c0/(mnCppy)1))

^0.5:

c1: ¼ (Cpp, y)

-b1(y)+b2(Cpp, y):

c2: ¼ (Cpp, y)

-b1(y)b2(Cpp, y):

f1: ¼ (Cpp, y)

-(1+erf(c1(Cpp, y)*2^(0.5)))/

2(1+erf(c2(Cpp, y)*2^(0.5)))/2:

f2: ¼ (Cpp,

y)

-exp(1/(y(1+delta^2)))

(1+delta^2)^((mn1)/2)y^((m*n+1)/2)/

GAMMA((m*n1)/2):

f: ¼ (Cpp, y)

-f1(Cpp, y)*f2(Cpp, y):

pstar: ¼ Cpp-int(f(Cpp, y), y ¼ 0..(2*c

0

)/(mn

Cpp)):

cofpstar: ¼ proc(r::numeric)

local Cpp;

Cpp: ¼ r;

if

evalf(pstar(Cpp))

o ¼ p

0

and

evalf(pstar

(Cpp0.0001))4p

0

then Cpp

elif evalf(pstar(Cpp))

op

0

and evalf(pstar(Cpp

0.0001))

op

0

then Cpp: ¼ Cpp0.0001: cofpstar(Cpp)

else Cpp: ¼ Cpp+0.0001: cofpstar(Cpp)

end if

end proc:

evalf(cofpstar(0.7938));

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A Bayesian approach based on multiple samples for measuring process performance with incapability index