Page 341 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

OTE/SPH
OTE/SPH
August 31, 2006
3:7
JWBK119-21
Char Count= 0
326 Establishing Cumulative Conformance Count Charts
to be very useful in monitoring p, the process fraction nonconforming, for high-
quality processes. Most studies of CCC charts assume that p is known. 1--4 However,
in practice, p is not likely to be known before implementing the chart and will need
6
5
to be estimated. Though Tang and Cheong and Yang et al. examine the effects of
parameter estimation on the CCC chart, a systematic framework for establishing the
CCC chart under different ways of estimating p has not been presented. For example,
5
to implement the CCC chart using Tang and Cheong’s scheme, one must know when
to stop updating the estimate of p so that the control limits are not affected by data
6
from processes with drift. If Yang et al.’s scheme is used, the initial sample size must
be specified.
In the following, the statistical properties of CCC charts and some preliminary
results on the effects of parameter estimation are revisited. This is followed by guide-
lines for establishing CCC charts for schemes proposed by both Tang and Cheong 5
6
and Yang et al. Numerical examples for designing CCC charts are given for both
cases when the process fraction nonconforming, p, is given or unknown to illustrate
the applicability of the proposed guidelines.

21.2 BASIC PROPERTIES OF THE CCC CHART
The CCC chart tracks the number of conforming items produced before a noncon-
forming one is produced. As a result, the probability of the nth item being the first
nonconforming item to be discovered is given by
P {X = n} = (1 − p) n−1 p, n = 1, 2,..., (21.1)
which is a geometric distribution with parameter p. It is assumed that inspections
are carried out sequentially and the above random variables are independently and
identically distributed.
In CCC charts, the control limits are determined based on the probability limits
from the geometric model shown in equation (21.1). For a given probability of type I
error, α, the two-sided control limits are given by

ln α/2
UCL = , (21.2)
ln (1 − p 0 )
ln (1 − α/2)
LCL = + 1, (21.3)
ln (1 − p 0 )

5
where p 0 is the in-control fraction nonconforming. The resulting average run length
(ARL), from the above limits, initially increases when the process starts to deteriorate
(p increases) and decreases after attaining a maximum point at p > p 0 . This is a
7
common problem for data having a skewed distribution, and renders the CCC chart
rather insensitive in detecting an increase in p and may lead to the misinterpretation
that the process is well in control, or has been improved.
7
To overcome the problem mentioned above, Xie et al. showed that, for a given p 0
and initial type I error rate φ, an adjustment factor, γ φ , given by
ln [ln(1 − φ/2)/ ln(φ/2)]
γ φ = (21.4)
ln [(φ/2)/ ln(1 − φ/2)]