Page 347 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

OTE/SPH
OTE/SPH
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JWBK119-21
332 Establishing Cumulative Conformance Count Charts
where the values of ARL m can be calculated from equation (21.13) with respective
values of ρ and m, and ARL ∞ is the out-of-control ARL with respect to ρ for a given
p 0 , given by
1
ARL ∞ = UCL LCL−1 . (21.20)
(1 − ρp 0 ) + 1 − (1 − ρp 0 )
The choice of R s reflects the user’s desirability of having a CCC chart that performs
close to the ideal case. When the ratio R s → 1, the performance of the scheme is
comparable with that of a known parameter CCC chart.
For a given ρ and ARL 0 , Table 21.3 gives the corresponding m required, m * ,in
achieving the desired R s . For example, if the user wishes to establish a CCC chart with
ARL 0 set at 200, and the required ARL performance at ρ ≥ 2 is given by R s = 1.150,
then, from Table 21.3, one should keep updating the estimate of p and the control
limits up to m = 19, as long as the suspension rule described below is not violated.

21.4.1.2 Suspension of sequential update
The updating must be done judiciously to avoid using the contaminated data from
processes with drift which will affect the sensitivity of the control scheme. To mitigate
this problem, a more stringent criterion than the usual out-of-control rule is imposed
for the suspension of sequential estimation.
Two decision lines corresponding to the 5th and 95th percentile points are used to
create a warning zone. Figure 21.3 shows the warning zones of the CCC chart (the
unshaded areas in the chart). The center line of the graph is the 50th percentile of the
cumulative distribution function (CDF) and is given by
ln 0.5 0.693
CL = + 1 =− + 1, (21.21)
ln (1 − ¯p) ln (1 − ¯p)
and the two decision lines are given by
ln (0.05) ln(1 − 0.05)
UDL = and LDL = + 1.
ln (1 − ¯p) ln(1 − ¯p)
The updating is suspended after observing any one of the following:

1. One point beyond the shaded area, which will occur with probability 0.05.
2. Four consecutive points on one side of the center line, which will occur with prob-
ability 0.0625.

These simple rules provide adequate protection against using contaminated data
for updating. Following the suspension, process monitoring should be continued
without revising the control limits. A new sequential estimate of p is then initiated
until the same value of m before suspension occurs. If there is no significant difference
between the two estimates, all data can then be combined to give the estimate of p and
updating is continued until m = m * provided that the process remains in control. If
another suspension criterion is observed during this period, the same decision process
is repeated.