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296 Part IV: Quality Assurance
is under control, the percentage of the product produced by the process with its
quality characteristic falling within the interval (m – 3s, m + 3s) is approximately
99.73 percent. As noted earlier, a process capability index is nothing but the comparison
between what a process is expected to produce and what it is actually producing.
Thus, we now define a process capability index (PCI) C p , one of the first
five indices used by the Japanese and proposed by Juran et al. (1974), as follows:
C
p =
USL LSL
UNTL LNTL
USL LSL
=
( m + 3s) m 3s
USL
LSL
=
.
6s
( )
(19.62)
Part IV.B.5
Note that the numerator in (19.62) is the desired range of the process quality
charac teristic whereas the denominator is the actual range of the process quality
char acteristic. From this definition, we see that a process can produce product
of desired quality and the process is capable only if the range in the numerator is
at least as large as that in the denominator. In other words, the process is capable
only if C p 1, and larger values of C p are indicative of a process being more capable.
For a 6s process C p = 2. A predictable process, which is normally distributed
with C p = 1 and its mean located at the center of the specification limits, usually
known as the target value of the process characteristic, is expected to produce 0.27
percent nonconforming units. Montgomery (2005) has given a comprehensive list
of values of the process capability index (C p ) and associated process nonconforming
for one-sided specifications and two-sided specifications.
Since C p is very easy to calculate, it is used very widely in industry. However,
the fact that it does not take into consideration the position of the process
mean is a major drawback. A process could be incapable even if the value of C p is
large (> 1). That is, a process could produce 100 percent defectives, for example, if
the process mean falls outside the specification limits and is far from the target
value. Furthermore, note that the value of C p will become even larger if the value
of the process standard deviation s decreases while the process mean moves away
from the target value.
Note that the numerator in (19.62) is always known, but the denominator is
usually unknown. This is due to the fact that in almost all practical applications,
the process standard deviation s is unknown. Thus, in order to calculate C p we
must replace s in (19.62) by its estimator s. Now we know that s can be estimated
either by the sample standard deviation s or by R – /d 2 . However, remember that the
estimate R – /d 2 is normally used only when the process is under control and
the sample size is less than 10. Thus, an estimated value of C p is given by
Cˆ = USL LSL
.
p
6s
(19.63)
We illustrate the computation of Ĉ p with the following example.