PROC CAPABILITY and General Statements |
Specialized Capability Indices
This section describes a number of specialized capability
indices which you can request with the
SPECIALINDICES option in the PROC CAPABILITY
statement.
The Index k
The process capability index k (also denoted by K)
is computed as
where m = (1/2)(USL + LSL) is the
midpoint of the specification limits, is the
sample mean, USL is the upper specification limit,
and LSL is the lower specification limit.
The formula for k used here is given by Kane (1986).
Note that k is sometimes computed without taking the
absolute value of in the numerator.
See Wadsworth et al. (1986).
If you do not specify the upper and lower limits in the
SPEC statement or the SPEC= data set, then k is assigned
a missing value.
Boyles' Index Cpm+
Boyles (1992) proposed
the process capability index Cpm+
which is defined as
-
Cpm+ = (1/3) [ [( EX<T [ (X-T)2 ] )/( (T - LSL)2 )] + [( EX>T [ (X-T)2 ] )/( (USL - T)2 )] ] -1/2
He proposed this index as a modification of
Cpm for use when
.The quantities
and
are referred to as semivariances.
Kotz and Johnson (1993) point out that
if T = (LSL + USL ) / 2,
then
Cpm+ = Cpm.
Kotz and Johnson (1993)
suggest that a natural estimator for Cpm+ is
Note that this index is not defined when either of the
specification limits is equal to the target T.
Refer to Section 3.5 of
Kotz and Johnson (1993)
for further details.
The Index Cjkp
Johnson et al. (1992)
introduced a so-called
"flexible"
process capability index
which takes into account possible differences in variability
above and below the target T.
They
defined this index as
where d = ( USL - LSL ) / 2.
A natural estimator of this index is
For further details, refer to Section 4.4 of Kotz and Johnson (1993).
The Indices Cpm(a)
The class of capability indices
Cpm(a), indexed by the parameter a
(a>0) allows flexibility
in choosing between the relative importance of
variability and deviation of the mean from
the target value T.
The class defined as
where .The motivation for this definition is that
if is small, then
A natural estimator of Cpm(a) is
where
d = ( USL - LSL ) / 2.
You can specify the value of a with the
CPMA= option in
the PROC CAPABILITY statement.
By default, a=0.5.
This index is not recommended for situation in which
the target T is not equal to the
midpoint of the specification limits.
For additional details, refer to Section 3.7 of
Kotz and Johnson (1993).
The Index Cp(5.15)
Johnson et al. (1992)
suggest the class of process capability indices
defined as
where is chosen so that the proportion of
conforming items is robust with respect to the
shape of the process distribution.
In particular,
Kotz and Johnson (1993) recommend use of
which is estimated as
For details, refer to Section 4.3.2 of
Kotz and Johnson (1993).
The Index Cpk(5.15)
Similarly,
Kotz and Johnson (1993) recommend use of
the robust capability index
where d = ( USL - LSL ) / 2.
This index
is estimated as
For details, refer to Section 4.3.2 of
Kotz and Johnson (1993).
The Index Cpmk
Pearn et al. (1992) proposed the index
Cpmk
where m = ( LCL + UCL) / 2.
A natural estimator
for
Cpmk is
where
m = ( USL + LSL ) / 2.
For further details, refer to Section 3.6 of Kotz and Johnson (1993).
Wright's Index Cs
Wright (1995) defines
the capability index
where
.A natural estimator of Cs is
where
c4 is
an unbiasing constant for the
sample standard deviation,
and
b3 is a measure of skewness.
Wright (1995)
shows that
Cs
compares favorably with
Cpmk
even when skewness is not present,
and he advocates the use of
Cs
for monitoring near-normal processes
when loss of capability typically leads to asymmetry.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.