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302 Part IV: Quality Assurance
were at the center of the specification target. In the above case, however, the process
is producing 1350 nonconforming ppm because the process mean is off center
by three s, a shift larger than 1.5s, which would be tolerable only in a Six Sigma
quality process. For discussion on other indices see Gupta and Walker (2007b).
Process Capability Indices: P p and P pk
The capability indices we have studied so far have one characteristic in common:
each one of them is used when the process is stable. In 1991 the American Automotive
Industry Group (AAIG), consisting of representatives from Ford Motor
Company, General Motors Corporation, American Daimler/Chrysler Corporation,
and the American Society for Quality, was founded. This group standardized
the supplier quality reporting procedure for their industry and they advocated the
use of two sets of process capability indices, one consisting of C p and C pk , and
the other consisting of P p and P pk . Furthermore, they advised using C p and C pk
when the process is stable and P p and P pk when the process is not stable, where the
two sets of indices P p , P pk and C p , C pk are defined in exactly the same manner and
only differ in their estimates. Thus, we have
Pˆ
p =
USL LSL
6sˆ
s
(19.71)
Cˆ
p =
USL LSL
(19.72)
6s R d
ˆ
/ 2
and
Part IV.B.5
Pˆ
= USL X XLSL
min ,
pk
3sˆ
3sˆ
Cˆ
= USL X XLSL
min , .
pk
ˆ ˆ
3s
3s
R/ d
R/
d
s
s
2 2
(19.73)
(19.74)
In other words, they differ only in the method by which one estimates the standard
deviation. The estimate sˆ s is the usual standard deviation
n m
2
( ij )
i=
1 j=
1
( 1)
s= x x / mn
and sˆ R– /d2 = R – /d 2 is an estimate obtained using the subgroup ranges R i , i = 1, 2,
. . . , m and the corresponding value of d 2 . Note that the estimate sˆ s measures
the process variability by using the variability within and between subgroups
whereas sˆ R– /d2 uses only the variability within the subgroups. When the process is
stable, there is very little variability between the subgroups. Thus, when the process
is stable the two sets of indices are essentially the same because the estimates