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OTE/SPH
OTE/SPH
August 31, 2006
3:7
JWBK119-21
Char Count= 0
328 Establishing Cumulative Conformance Count Charts
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In the scheme proposed by Tang and Cheong, ¯p is updated sequentially and the
control limits are revised so that the ARL 0 of the chart not only can be kept to the
desired value, but also is the peak of the ARL curve. This can be done by making use
of the same adjustment given in equations (21.5) and (21.6). The control limits are
ln (φ m /2)
UCL = × γ φ m , (21.10)
ln (1 − (m − 1) / (N m − 1))
ln (1 − φ m /2)

LCL = γ φ m + 1, (21.11)
ln (1 − (m − 1) / (N m − 1))
where φ m can be obtained by solving

−1
∞
ln(φ m /2)γ φm /ln(1− ¯p) ln(1−φ m /2)γ φm /ln(1− ¯p)
(1 − ¯p) − (1 − ¯p) + 1
n=m

n − 1 m n−m
× ¯ p (1 − ¯p) = ARL 0 , (21.12)
m − 1
given by (21.4).
with ¯p given by equation (21.9) and γ φ m
Using similar steps to those proceeding (21.8), φ m is shown to be independent of ¯p.
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Tang and Cheong showed that, as m increases, the sensitivity of the chart improves
and approaches that with known process parameter. Here, the index ρ, reflecting the
shift in p, is used; that is, p = ρp 0 . When ρ = 1, the process is in statistical control; if
ρ> 1, the process is deteriorating; and ρ< 1 indicates process improvement.
To obtain the ARL curves for different m using the proposed scheme, equation
(21.12) can be re-written as

−1

∞
ln(φ m /2)γ φ m /ln(1− ¯p) ln(1−φ m /2)γ φ m /ln(1− ¯p)
(1 − ¯p) − (1 − ¯p) + 1
n=m

n − 1 m n−m
× ¯ p (1 − ¯p) = ARL m , (21.13)
m − 1
where ARL m is the ARL with ¯p estimated after observing m nonconforming items.
Figure 21.1 depicts the ARL curves of the CCC scheme under sequential estimation,
using m = 5, 10, 30 and 50, for ARL 0 = 370 and p 0 = 500 ppm.

21.3.2 Conventional estimation scheme
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On the other hand, Yang et al. considered using the conventional estimator,
D n
ˆ p = , (21.14)
n
wherenistheinitialsamplesize,fixedapriori,and D n isthenumberofnonconforming
items among n items sampled.