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STS540 Zhi Lin C. et al.
value as the optimal parameters.
Note that the optimal parameters (k, LssG) of the SSGR scheme can also be
obtained using the above optimization procedure by replacing LG with LssG
and (1) with (3).

4. A Comparison between the GR and SSGR schemes based on SDTS
We compare the performance between the optimal GR and SSGR schemes
in terms of SDTS in this section. Note that we use the Statistical Analysis
System (SAS) software to compute the SDTSs based on the Monte-Carlo
simulation procedure. We utilize the same simulation procedure as shown in
Yew et al. (2016) to compute the ATS for the GR and SSGR schemes. Here, we
repeat the simulation for 10000 trials and the average of the 10000 trials is the
ATS value. Similarly, the SDTS for the GR and SSGR schemes can be computed
based on the standard deviation of these 10000 trials.
To have a comprehensive comparison between the GR and SSGR schemes,
we considered the input parameters combination of n ∈ {3, 5, 7}, opt ∈ {0.5,
1.0, 1.5} and ARL0 ∈ [370, 500). Note that opt represents the optimal mean
shift size where a prompt detection is desired. In practical situations, small to
moderate sample sizes are generally recommended to reduce the cost of
sampling. Here, the combinations of opt are considered as in Lee et al. (2013).
To have a correct implementation of the GR and SSGR schemes, the
practitioners are encouraged to investigate the input parameters combination
based on their respective needs.
Then, based on the combination of (n, opt, ARL0 input parameters, we
determine the optimal parameters, i.e. (k, LG) and (k, LssG) of the GR and SSGR
schemes, respectively, using the optimization procedure in Section 3. We
present these optimal parameters in Table 1. Note that these optimal
parameters are chosen to minimize the ATS1 for respective n and opt, such
that the desired ARL0 is attained. From Table 1, we observe that the optimal
parameters (k, LG) and (k, LssG) of the GR and SSGR schemes, respectively,
generally decrease or remain the same as n and opt increase. However, when
ARL0 increases from 370 to 500, the optimal parameter k will increase; whereas
the optimal parameter LG or LssG will either increase or remain the same.
The optimal parameters (k, LG) and (k, LssG) presented in Table 1 are applied
to compute the SDTSs of the GR and SSGR schemes, respectively, and the
results are shown in Tables 2 to 4. We compare and study the SDTSs of the
optimal GR and SSGR schemes for different mean shifts sizes, i.e. {0, 0.25,
0.50, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0}. We say that the process is IC when S = 0,
whereas the process is OC when > 0. Note that in Tables 2 to 4, the SDTSG
and SDTSssG, respectively, denote the SDTSs of the GR and SSGR schemes.
4.1 Performance comparison between the GR and SSGR schemes (IC SDTS):

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