Page 203 - Special Topic Session (STS) - Volume 3

STS540 Zhi Lin C. et al.
samples. The GR scheme issues an OC signal for the first occurrence of CRL1 < LG
or two successive CRLi < LG and CRLi+1 < LG, for i = 2, 3, .... Here, CRLi is the i-th
CRL value and LG is the LCL of the extended CRL sub-chart. For this reason, the
GR scheme can signal at any sample except the second sample, i.e. i = 2. The
performance of the GR scheme can be measured by using the ATS criterion, given
by (Gadre and Rattihalli, 2004)
ATS () = x 1 (1)
() [1−(1−())] 2
where P () is the probability for nonconforming sample to happen under shift
size δ, i.e.
P(δ ) = 1 −  (k − δ√ ) +  (−k − δ√ ) (2)

2.2 The SSGR scheme
The SSGR scheme is proposed as an extension of the GR scheme with side-
sensitivity. Similar to the GR scheme, the SSGR scheme also issues an OC signal
if CRL1 < LSSG or two successive CRL i < LSSG and CRL i+1 < LssG, for i = 2, 3, ..., but the

difference is that the SSGR scheme needs to satisfy another condition, i.e. the two
̅
consecutive samples contributing to two successive CRLs have to be plotted on
the same side of the target mean. The ATS of the SSGR scheme is given by (Gadre
and Rattihalli, 2007)
1− (1−) 2
ATS () = x (3)
() [1+(1−)(−2)] 2

where P() is given in (2), = 1 −  (k−δ√ ) and A = 1 — [1 — P()] .
L
()

3. Optimization design of the GR and SSGR schemes
In this section, we present the optimization design of the GR and SSGR
schemes. For the optimization design, we need to find the optimal parameters
(k, LG) and (k, LssG) of the GR and SSGR schemes, respectively, such that the OC
ATS (ATS1) is minimized, where the IC average run length (ARL0) is attained at
370 so that a low false alarm rate is achieved. We present the optimization
procedure written in the ScicosLab software (www.scicoslab.org) to search for
the optimal parameters (k, LG) of the GR scheme as follows:
Step 1. Specify the desired n, δ and ARL0.
Step 2. Initialize LG as 1.
Step 3. Use nonlinear equation solver to solve (1) by searching for k such
that ATS() = ATS0 is satisfied, where ATS0 = n x ARL0 is the desired
IC ATS value. Then, calculate the ATS1 value using the present
parameters (k, LG).
Step 4. Increase LG by 1 and go back to Step 3 if LG = 1 or there is a
reduction in the ATS1 value. Otherwise, go to the next step.
Step 5. Choose the parameters (k, LG) that result in the minimum ATS1
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