Page 390 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

OTE/SPH
OTE/SPH
August 31, 2006
Char Count= 0
3:8
JWBK119-24
Fast Initial Response 375
V-mask scheme, one may think of using (0, FIR) as the first point to be plotted in the
+
chart and the next point being CUSUM = y 1 + FIR. This is only equivalent to shifting
1
the CUSUM plot by a magnitude given by the FIR. Thus, in using this approach, we
+
can detect a signal only when CUSUM exceeds h + k, which is precisely equivalent
1
to not implementing an FIR. By analogy, the FIR value must be added only to the first
process observation and the point at the origin (0,0) still represents a point that must
be within the arms of the mask when the process is in control. Thus in implementing
the FIR feature under the V-mask scheme, it is only the first CUSUM value that is
altered, that is, by adding or subtracting the FIR value. This means that for a two-
sided monitoring scheme we must calculate the following CUSUMs:
⎧
0, i = 0,
⎪
y 1 + FIR, i = 1,
⎪
⎨
+
CUSUM = i (24.5)
i

y i + FIR, i > 1,
⎪
⎪
⎩
j=1
and
⎧
0, i = 0,
⎪
y 1 − FIR, i = 1,
⎪
⎨
−
CUSUM = i (24.6)
i

y i −FIR, i > 1.
⎪
⎪
⎩
j=1
Since equation (24.5), which adds the FIR value to the first observation, is meant to
rapidly detect a process mean above the target value, we must use the lower arm of
+
the mask for CUSUM . Similarly, for CUSUM , we must use the upper portion of the
−
i i
mask.
24.2.2.1 Numerical Illustration
Here, by way illustration, the one-side CUSUM example given by Lucas and Crosier 11
is replicated. Table 24.1 gives the initially-out-of-control case, in which the mean of
the observations is 1.36 while the desired mean is zero.
Using the data in Table 24.1, we detected the out-of-control condition at time i =
7 when the tabular CUSUM value (k = 0.5) exceeded the decision interval, h = 5.
Similarly, at i = 7, the V-mask CUSUM depicts points that lie outside the mask arms
(see Figure 24.3).
On the other hand, by using the FIR feature, one can expect a speedier detection of
change when the process starts in an out-of-control state. For the data in Table 24.1,
the tabular FIR CUSUM detects the out-of-control condition one time period earlier
(i.e. at i = 3). This tabular FIR CUSUM can also be represented in V-mask form using
equations (24.5) and (24.6). Both the tabular and the V-mask form of the FIR CUSUM
are displayed in Figure 24.4. In both FIR schemes, we use the control chart parameters
given by Lucas and Crosier: 11 h = 5.0, k = 0.5, and the recommended FIR value of
h/2 = 2.5.