FLEXIBLE PROCESS CAPABILITY INDICES
Norman L. Johnson
Samuel Kotz
Department of Statistics
University of North Carolina
Dept. of Management Science
and Statistics
University of Maryland
Chape I Hi 11. NC
27599-3260
College Park. MD
20742
W.L. Pearn
Dept. of Industrial Engineering
and Management
National Chiao Tung Universi ty
Hs inchu. Taiwan 300 50
Republic of China
Key Words and. Phrases:
asymmetry; Oti-squared distribution;
mixture distributions; quality control
ABSTRACT
A new process capability index (Pel)
is proposed.
which
takes into account possible asymmetry in the distribution of the
measured process characteristic
(X).
The distribution of a
natural estimator of this index is investigated.
INTRODUCTION
Interest in the use of process capability indices (Pels) has
been increasing in recent years. at an accelerating rate (see.
e.g.. Kane (l986)}.
Many different views have been expressed
(e.g. Beazley and Marcucci (l9B8). Bissell (l990). Boyles (l99l).
Kitska
(l99l).
applicability.
Spiring
There
sampling distributions
(l99l)}
have
for
on
been
their
usefulness
difficulties
in
and
obtaining
estimators of
some Pels. though.
happily. these have now been largely overcome. at least on the
assumption of normality of the process distribution (e.g. Chan et
al.
(l9B8). Chen and Owen (l989). Zhang et al.
(lOOl). Kotz and Johnson (l992). Pearn et al.
have
also
been
objections
to
the
(l990). Boyles
(lOO2)}.
widespread
use
There
of
this
assumption.
both for
establishing relationship wi th expected
proportion of nonconforming (NC) product. and also. even when
this is regarded as, irrelevant (a view not shared by the present
authors). for consideration of sampling distributions ..
In
the present paper we present new PCIs.
specifically
designed to make some allowance for possible aSYmmetry in the
process
distribution -
in particular
for
difference between
variabili ty of the measured characteristic (X) for values less
than and greater than a target value (T).
Al though our proposals
are aimed at producing PCIs which can be used for aSYmmetric
distributions we will base our analysis of sampling distribution
on assumptions
of
normali ty.
Indeed we will
restrict
our
detailed discussion to further. assuming that the process mean of
X
(~)
is equal to the target value. T.
We will. however. present
the structure of relevant distributions for more general cases.
PROCESS CAPABILITY INDICES
We first present. for reference purposes. a summary of some
current PCIs.
We use the follOWing notation:
USL
upper specification limit
LSL
lower specification limit
d
= 2'1
(USL-LSL)
T:
a target value for X
~.a
mean and standard deviation respectively. of the population
distribution of the measured characteristic (X)
X •...• X : random variables representing results of measurements
1
n
of X on a random sample of size n
-
X= n
-1
n
! X.;
i=1 1
V = (n-1)
-1
An ear ly PCI was
n
!
i=1
_-*
V = n
-1
S;
Cp =
A natural estimator of C
p
USL-LSL
6a
=
is C
P
d
= 3a
~ with ~
= -~
y-- .
This index
:b
does not take account of the values of Jl or T.
The index
- min(USL-Jl. Jl-LSL) _ 2::.JJl~(USL+LSL)
Cpk 3a
3a
takes Jl into account but not T.
C pk -
L
A natural estimator of C
pk
2::.JX~(USL+LSL)
:b
is
l.
The index
C
_
USL-LSL
_
d
6{E[(X-T)2]}~ - 3{a2+(T-Jl)2}~
pm -
takes T into account.
A natural estimator of Cpm is
,.
d
Cpm
= 3K
It is a defect of Cpm that it does not distinguish between T-Jl =
6 and T-Jl = -6.
Unless T =
~(USL+LSL)
this can lead to products
wi th expected proportion NC over 50% and less than 0.3% having
identical values for Cpm ' (Take T
= t [3
x USL+LSL] and 6
= ~ d.)
The PCI
_ min(USL-Jl. Jl-LSL)
Cpmk 2
2 ~
3{a +(T-Jl) }
with natural estimator
,.
C
- ~X~(USL+LSL)
pmk-
3K
L
.
introduced by Pearn et al. (1992). combines features of C and
pk
C
pm
These
generation'
latter
two
indices can be regarded as
indices and Cpmk as a
following C
p
. third generation'
. second
index
However none of these Pel's attempts to take into account
possible asymmetry in the distribution of X. In the next section
we suggest a. way in which this might be done, and in the
following section study the distribution of the new index under
some special conditions.
A FLEXIBLE Pel
We will construct a Pel taking into account possible
differences in variability of X for values above and below the
target value T.
As one-sided Pels we could use
1
2
aI.kp = (USL-T)/{EX>T[(X-T) ]}
J
3v2
and
1
2
~
(1.1)
~
CL.kp = (T-LSL)/{E'x"T[(X-T) ]} ,
J
3v2
~
(1.2)
where
EX>T[(X-T)2] = E[(X-T)2IX > T] Pr[X > T]
(2.1)
and
E'x<T[(X-T)2] = E[(X-T)2IX < T] Pr[X < T]
(2.2)
(We assume Pr[X=T] = 0).
The multiplier 1/(3v2) while the earlier Pel's use ~ arises
from the fact that for a symmetrical distribution with variance
2
a and expected value T we would have
2
2
1 2
(3)
EX>T[(X-T) ] = EX<T[(X-T) ] = 2 a ,
Finally we define
Cjkp = min(aI jkp , CL jkp )
1
. [
USL-T
= 3v2 mIn {EX>T[(X-T)2]}~ ,
Note that if we have T =
2'1 (USL
T-LSL
{E'x<T[(X-T)2]}~
]
(4)
+ LSL) so that USL-T = T-LSL = d
then
".
FSflMATION OF Cjkp
A natural estimator of Cjkp is
- -1- mi n [USL-T
-;;;::::;;::::;: T-LSL],
"
C
3v2
jkp -
where S+
=~
2
>T (Xi-T) ; S_
i
(6)
J{S+/n)' vSTn
=~
2
<T{Xi-T) .
i
Our analysis will be based on the very reasonable condition
LSL
<T
< USL.
"
We can express Cjkp as
1
1
d v h . [d- {USL-Tl d- {T-LSLl],
C jkp = {3/2)0 mIn
,
~o
"
~
vS+/o
where
0
(6)'
/0
is an arbi trary constant.
To study the distribution of Cjkp it will be convenient to
consider the statistic
"
n
d 2 "_2
D
where a 1
= 18
((j')
_rUSLd- T]-2
-l
The
Cjkp
= max{a1
d
_ [T-LSL]-2
d
.
an
0
-2-2
, ~ S_ 0 ) ,
(7)
~ -
distribution
complicated.
S+
of
"
D
will
We will discuss a
be,
special
in
general,
case
qui te
in which
the
distribution of X is, indeed, normal wi th expected value T and
variance
0
2
.
Al though this is not,
in fact, an asytlllletrical
distribution, consideration of this case can provide an initial
point of reference.
Later we will indicate ways in which our
resul ts can be extended to somewhat broader situations - though
not as broad as we would wish.
With the stated assumptions, we know that
the distribution of {X i -T)2 is that of x~ 0 . (We use the
symbol 'X2 . to denote 'X2 with v degrees of freedom').
v
2
(i)
(ii)
This
is also
the
whether Xi > T or Xi < T.
conditional
distribution of
2
(X.-T)
,
I
(i i i)
The number, K,
of X' s which exceed T has a binomial
n,~.
distribution with parameters
(iv)
(Denoted
Bin(n,~».
Given K. the conditional distributions of S+ a
2
2
are those of '<K' '<n-K respec t i ve ly , and S+ a
IIRltually independent.
~
-2
Hence,
and S_ a
and S
a
~
-2
are
And also.
2
2
The distribution of H = S+/(S+ + S_) is that of '<j(I('<K +
(v)
~-K) which is Beta(~. ~(n-K» so that the density function of H
is
fH(h)
= {B(~.
~(n-K»}-1 h~-I(I_h)~(n-K)-1 (0 < h < 1)
for K = 1.2•... n-1. and H and S+ + S
(8)
are IIRltually independent
and further
(vi)
The conditional distributions of S+ a
2
-2 and S_ a -2 ,given K
2
and H. are those of H '<n and (I-H) '<n' respec tive ly .
From (7)
....
{a1 S+ a-
D=
a 2 S_ a
2
for
S+/SI
> ~/al
for
S+/S_
< ~/al
(9)
-2
....
SoD is distributed as
aj H
~
2£or 1V(1-H)
{ a (I-H)'<n
2
for
H
> ":fa j
Le.
< ~/(al+~)
....
The overall distribution of D can be represented as
....
D
{a1 H for H > b } 2
a-(I-H) for H < b '<n
~
where the symbol
A
A
H
Beta (~. ~(n-K»
A
K
Bin(n.~). (10)
means 'mixed with respect to yo having the
y
distribution that follows, and b
= a2/(al~).
Without loss of generali ty, we will assume that USL-T
T-LSL.
Then.
variance a 2 ,
~
for a sytlllletrical distribution wi th mean T and
C
_ ~ USL-T _ ~ _1_
jkp - 3a d
- 3a ..Ja
(ll)
1
and -1+ -1= 2.
';';:1
~
~_Jn
Hence
C
jkp
-
a A~
2
l
(12)
D
and
A
E[~n
=
F2
al
t
E[D~r].
~r
E[~
Now, from (a), (9) and (10), (noting that
-~r
{z-
-1
r(~r}}
(13)
J =
r(~(n-r}}},
n
[~r
n1
(k )
E[D~rJ = r(~(n-r» L
+ ~
I
.
n
~r r(~} 2 ~
k=l B(~, ~(n-k})
A
{a~r B1_b(~(n-k}, ~(k-r})
+
a~r ~(~, ~(n-k-r}}}
+
a~r],
whence
_ n~r
-
r(~(n-r»
2n +r r(~}
{B l _b (Il(n-k). Il(k-r» +
¥
where B¥(u 1 ,u2 } = f 0 y
[~t +
~tr ~ (Ilk.
u1-1
(l-y)
u2-1
Il(n-k-r»}
J.
(14)
dyand B(u 1 ,u2 } = B1 (u 1 ,u2 }·
In the next section, we will present numerical ¥alues for
A
the mean and ¥ariance of Cjkp/C jkp for certain ¥alues of the
parameters, and comment thereon.
NUMERICAL RESULTS
Table 1 presents numerical ¥alues of
A
E[Cjkp/CjkpJ
A
and Var(Cjkp/C jkp }
for se¥eral ¥alues of (USL-T}/d, and n = 10, 20, 30.
...
...
Table 1. Values of E=E[Cjkp/C jkp ] and V=Var(Cjkp/C jkp ) when J,L=T
USL-T
d
n
1.0
0.9
0.7
0.8
0.6
0.5
0.4
10
E
V
0.9210
0.0615
1.0075
0.0782
1.0814
0.1054
1.1420
0.1440
1.1900
0.1930
1.2269
0.2504
1.2550
0.3154
20
E
V
0.9191
0.0290
0.9989
0.0387
1.0536
0.0561
1.0864
0.0775
1.1036
0.0974
1.1116
0.1128
1.1151
0.1232
30
E
V
0.9245
0.0194
1.0001
0.0264
1.0431
0.0396
1.0623
0.0519
1.0691
0.0597
1.0711
0.0634
1.0715
0.0648
It
is
instructive
to
...
...compare
these values
with similar
quanti ties for C k/C k and C IC • in Kotz and Johnson (1992)
p
p
pm pm
and Pearn et al. (1992) respectively.
(Values available for the
second author. on request.)
As to Table 1 itself. we have the following comments.
general the estimator C
T
=
~(USL+LSL)
but
is biased.
jkp
increases
as
In
The bias is negative when
(USL-T)/d
decreases.
The
increase is quite substantial when (USL-T)/d is as small as 0.4 .
...
As
might
expec ted
be
increases -
the
var iance
of
Cjkp
decreases
n
it increases as (USL-T)/d decreases, as the target
value gets nearer to the upper specification limit.
also.
as
decreases
as
n
increases:
this
effect
is
The bias,
particularly
noticeable for smaller values of (USL-T)/d.
EXTENSIONS
Since
the
index C
jkp
is
intended
to make allowance
for
asymmetry in the distribution of the process characteristic X it
would be of interest to learn something of the distribution of
...
the estimator C
jkp
under such condi tions.
The analysis of the
preceding section can be extended to certain kinds of asymmetry
of distribution of X, although they are not. unfortunately, of
what may be regarded as common kinds of asymmetry.
We note two ways in which this may be done.
(i)
If the population density function of X is
1
~g(x;T,al)
(15)
+ g(-x;-T,a2 )]
where
for x
~
T
for x
< T,
A
then the distribution of D is as in (10) wi th a
aj(a/a j )
(i i)
2
(j
j
replaced by
= 1,2).
The number of Xi's exceeding T can have a binomial
distribution with parameters n, p with 0
< p < 1.
If (i) and (ii) are combined, the density function of X will
be
(16)
For either (i) or (ii)
the expected value of X is not,
general. T for density functions (15) or (16).
in
(The distribution
of X is a mixture of a half-normal and a negative half-normal.)
However.
the conditional distributions of ~i)T(Xi-T)2/a~ and
~.<T(Xi-T)2/a~. given
1
K are still
~~
and
~-K'
respectively.
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Appl. Statist .. 39. 331-340.
Boyles. R.A. (1991) The Taguchi capability index. J. Qual.
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Chan. L.K .• Cheng. S. W. and Spiring. F.A. (1988)
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Chou. Y.M. and Owen. D.B. (1989) On the distribution of the
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Kitska. D. (1991)
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Quality Progress. 24(3). 8.
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..... '-'>.;
"~