The potential areas that are associated to the scope of this thesis are suggested in this section for future research. From our study, the construction procedures of the synthetic EWMA median chart are introduced under the basic ranked set sampling (RSS) scheme. The data sampling characteristics such as cost-effective, efficient and time-savings often take into consideration by practitioners in manufacturing production process during the selection of control chart. These characteristics based new RSS scheme such as paired double ranked set sampling, extreme paired double ranked set sampling and quartile paired double ranked set sampling are utilized by Noor-ul-Amin et al. (2021) to the creation of some new control charts for monitoring the process mean. They revealed that the incorporation of the proposed control charting structures with the new RSS scheme is more efficacious than other control charting structures under single RSS scheme. The studies completed by Noor-ul-Amin et al. (2021) can be extended to the application of the new RSS scheme as a sampling criterion in the construction of the median chart, synthetic chart and Synthetic EWMA
67
chart in the future. Additionally, another new RSS scheme such as neoteric ranked set sampling (NRSS) has been recommended by Zamanzade and Al‐
Omari (2016) in the valuation of the population mean and variance. This is because the researchers have successfully verified that the performance of NRSS is better than other existing sampling method regardless the type of the underlying distribution of the variables of interest. Koyuncu and Karagoz (2018) and Karagoz and Koyuncu (2019) effectively employed NRSS in the formation of the control charting structures for mean and range of bivariate asymmetric distributions. The studies illustrated that the NRSS method can be expanded in the creation of the synthetic chart and Synthetic EWMA chart.
The underlying distribution of a process is commonly non-normal or undiscovered in the real world. Hence, the researchers such as Abid et al. (2017), Haq (2019), and Abbas et al. (2020b) have suggested the nonparametric control charts like those based on the sign, signed-rank, Wilcoxon signed-rank and nonparametric goodness-of-fit. The findings from Abid et al. (2017) revealed that the suggested nonparametric Cumulative Sum (CUSUM) sign control chart under RSS surpass the nonparametric CUSUM sign chart under SRS with regards to the efficiency of the process mean monitoring. Besides that, the combination of the EWMA sign chart and CRL chart to create synthetic EWMA sign chart are introduced by Haq (2019) for process target deviations examining and the researcher proved that the sensitivity of the suggested chart in determining process shifts is greater than the classical counterparts. Additionally, to examine the process location efficiently, Abbas et al. (2020b) introduced a nonparametric double exponentially weighted moving average (DEWMA) chart
68
based on Wilcoxon signed rank statistic using SRS and RSS schemes. In spite of that, to the best of the author’s information, nonparametric synthetic charts based on RSS scheme have not been proposed yet. Therefore, the attention for the future research may be given on the univariate and multivariate nonparametric synthetic charts proposal using different types of RSS scheme available.
Conventionally, when the mean of a parameter is constant and the process variance is independent of the mean, the traditional Shewhart R or S charts are utilized to examine the process variability. However, the mean may be fluctuating, and the standard deviation may not be independent with mean in practice. For instance, the situation commonly occurs when the practitioners are planning to examine those process outputs which may alter occasionally. While the coefficient of variation (CV) is always constant in a process, the integration of CV to control charts are frequently utilized to tackle and deal with the problematic circumstances. For example, Abbasi et al. (2019) illustrated that the detection ability of the CV charts under numerous ranked set sampling schemes such as RSS, median RSS (MRSS) or extreme RSS (ERSS) is higher than the existing CV chart based on SRS. Besides that, the study from Noor-ul-Amin and Riaz (2020) on the proposed Exponentially Weighted Moving Coefficient of variation control chart under RSS scheme proved that it is more effective than the other CV charts under the log-normal transformation at each shift irrespective with the sample sizes. Moreover, a synthetic chart to examine the multivariate coefficient of variation is suggested by Khaw et al. (2019) when more than one quality variable is involved in the process. Since research studies
69
on the univariate and multivariate synthetic charts and synthetic exponentially weighted moving coefficient of variation chart based on RSS scheme are still deficient, future studies on this area of research could be highlighted.
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76 Appendix A
SAS Program to obtain the (n, λ, L, K) parameters when sample size, n=3
data ewma_syn_median2;
mu=0;
delta=0; /*no shift to find the in-control ARL of 370*/
sigma=1;
n=3; /*sample size selected is 3*/
lambda=0.05;/*smoothing constant 0.05,0.25,0.50,0.75*/
constant=0.4255; /*the parameter K to be found*/
L=20; /*lower limits of CRL 1,5,10,20,50*/
m=10000; /*subgroup size*/
Do p=1 to 10000;
array data_array{10000,3};
Do r=1 to m;
array u[3] u1-u3;
Do j=1 to 3;
Do i=1 to dim(u);
u[i]=mu+sigma*rannor(33333);
end;
data_array{r,j}=smallest(j,of u[*]);
end;
end;
Xsum=0;
Do a=1 to m;
Xsum=Xsum+data_array{a,2};
end;
Xbar=Xsum/m;
Xsumsq=0;
Do c=1 to m;
Do d=1 to n;
Xsumsq=Xsumsq+(data_array{c,d}-Xbar)**2;
end;
end;
Xrsssq=Xsumsq/(n*m-1);
Xvar1=Xrsssq/n;
array column1(10000);
Do a=1 to dim(column1);
column1(a)=data_array{a,1};
77 end;
column1med1=smallest(5000,of column1[*]);
column1med2=smallest(5001,of column1[*]);
column1med=(column1med1+column1med2)/2;
array column2(10000);
Do a=1 to dim(column2);
column2(a)=data_array{a,2};
end;
column2med1=smallest(5000,of column2[*]);
column2med2=smallest(5001,of column2[*]);
column2med=(column2med1+column2med2)/2;
array column3(10000);
Do a=1 to dim(column3);
column3(a)=data_array{a,3};
end;
column3med1=smallest(5000,of column3[*]);
column3med2=smallest(5001,of column3[*]);
column3med=(column3med1+column3med2)/2;
columnmedian=(column1med+column2med+column3med)/n;
Xcol1sum=(column1med-columnmedian)**2;
Xcol2sum=(column2med-columnmedian)**2;
Xcol3sum=(column3med-columnmedian)**2;
Xcolsq=Xcol1sum+Xcol2sum+Xcol3sum;
Xvar2=Xcolsq/(n*n);
Xvar=Xvar1-Xvar2;
Xstdev=sqrt(Xvar);
muhat=Xbar;
sigmahat=Xstdev;
LCL=muhat-constant*sigmahat;
UCL=muhat+constant*sigmahat;
CRL=0;Z=muhat;c=1;
array crl_array{50000,3};
Do k=1 to 50000;
array t[3];
Do a=1 to 3;
Do b=1 to dim(t);
t[b]=(delta/sqrt(n))+sigma*rannor(33333);
end;
crl_array{k,a}=smallest(a,of t[*]);
end;
Xmedcrl=crl_array{k,2};
Z=lambda*Xmedcrl+(1-lambda)*Z;
CRL=CRL+1;
If (Z>UCL) or (Z<LCL) then do;
78 c=0;
end;
else do;
c=1;
end;
If (c=0) and (CRL<=L) then do;
RL=k;
output;k=50001;
end;
If (c=0)and (CRL>L) then do;
Z=muhat;CRL=0;c=1;
end;
end;
end;
run;
proc univariate;
var RL;
output pctlpts=5 10 20 25 30 40 50 60 70 75 80 90 95 pctlpre=p;
proc print;
run;
79 Appendix B
SAS Program to obtain the (n, λ, L, K) parameters when sample size, n=5
data ewma_syn_median2;
mu=0;
delta=0; /*no shift to find the in-control ARL of 370*/
sigma=1;
n=5; /*sample size selected is 5*/
lambda=0.75; /*smoothing constant 0.05,0.25,0.50,0.75*/
constant=3.3058; /*the parameter K to be found*/
L=5; /*lower limits of CRL 1,5,10,20,50*/
m=10000; /*subgroup size*/
Do p=1 to 10000;
array data_array{10000,5};
Do r=1 to m;
array u[5] u1-u5;
Do j=1 to 5;
Do i=1 to dim(u);
u[i]=mu+sigma*rannor(33333);
end;
data_array{r,j}=smallest(j,of u[*]);
end;
end;
Xsum=0;
Do a=1 to m;
Xsum=Xsum+data_array{a,3};
end;
Xbar=Xsum/m;
Xsumsq=0;
Do c=1 to m;
Do d=1 to n;
Xsumsq=Xsumsq+(data_array{c,d}-Xbar)**2;
end;
end;
Xrsssq=Xsumsq/(n*m-1);
Xvar1=Xrsssq/n;
array column1(10000);
Do a=1 to dim(column1);
column1(a)=data_array{a,1};
80 end;
column1med1=smallest(5000,of column1[*]);
column1med2=smallest(5001,of column1[*]);
column1med=(column1med1+column1med2)/2;
array column2(10000);
Do a=1 to dim(column2);
column2(a)=data_array{a,2};
end;
column2med1=smallest(5000,of column2[*]);
column2med2=smallest(5001,of column2[*]);
column2med=(column2med1+column2med2)/2;
array column3(10000);
Do a=1 to dim(column3);
column3(a)=data_array{a,3};
end;
column3med1=smallest(5000,of column3[*]);
column3med2=smallest(5001,of column3[*]);
column3med=(column3med1+column3med2)/2;
array column4(10000);
Do a=1 to dim(column4);
column4(a)=data_array{a,4};
end;
column4med1=smallest(5000,of column4[*]);
column4med2=smallest(5001,of column4[*]);
column4med=(column4med1+column4med2)/2;
array column5(10000);
Do a=1 to dim(column5);
column5(a)=data_array{a,5};
end;
column5med1=smallest(5000,of column5[*]);
column5med2=smallest(5001,of column5[*]);
column5med=(column5med1+column5med2)/2;
columnmedian=(column1med+column2med+column3med+column4med+colu mn5med)/n;
Xcol1sum=(column1med-columnmedian)**2;
Xcol2sum=(column2med-columnmedian)**2;
Xcol3sum=(column3med-columnmedian)**2;
Xcol4sum=(column4med-columnmedian)**2;
Xcol5sum=(column5med-columnmedian)**2;
Xcolsq=Xcol1sum+Xcol2sum+Xcol3sum+Xcol4sum+Xcol5sum;
Xvar2=Xcolsq/(n*n);
Xvar=Xvar1-Xvar2;
Xstdev=sqrt(Xvar);
muhat=Xbar;
sigmahat=Xstdev;
81 LCL=muhat-constant*sigmahat;
UCL=muhat+constant*sigmahat;
CRL=0;Z=muhat;c=1;
array crl_array{50000,5};
Do k=1 to 50000;
array t[5];
Do a=1 to 5;
Do b=1 to dim(t);
t[b]=(delta/sqrt(n))+sigma*rannor(33333);
end;
crl_array{k,a}=smallest(a,of t[*]);
end;
Xmedcrl=crl_array{k,3};
Z=lambda*Xmedcrl+(1-lambda)*Z;
CRL=CRL+1;
If (Z>UCL) or (Z<LCL) then do;
c=0;
end;
else do;
c=1;
end;
If (c=0) and (CRL<=L) then do;
RL=k;
output;k=50001;
end;
If (c=0)and (CRL>L) then do;
Z=muhat;CRL=0;c=1;
end;
end;
end;
run;
proc univariate;
var RL;
output pctlpts=5 10 20 25 30 40 50 60 70 75 80 90 95 pctlpre=p;
proc print;
run;
82 Appendix C
SAS Program to obtain the (n, λ, L, K) parameters when sample size, n=9
data ewma_syn_median2;
mu=0;
delta=0; /*no shift to find the in-control ARL of 370*/
sigma=1;
n=9; /*sample size selected is 9*/
lambda=0.05; /*smoothing constant 0.05,0.25,0.50,0.75*/
constant=0.5633; /*the parameter K to be found*/
L=10; /*lower limits of CRL 1,5,10,20,50*/
m=10000;/*subgroup size*/
Do p=1 to 10000;
array data_array{10000,9};
Do r=1 to m;
array u[9] u1-u9;
Do j=1 to 9;
Do i=1 to dim(u);
u[i]=mu+sigma*rannor(33333);
end;
data_array{r,j}=smallest(j,of u[*]);
end;
end;
Xsum=0;
Do a=1 to m;
Xsum=Xsum+data_array{a,5};
end;
Xbar=Xsum/m;
Xsumsq=0;
Do c=1 to m;
Do d=1 to n;
Xsumsq=Xsumsq+(data_array{c,d}-Xbar)**2;
end;
end;
Xrsssq=Xsumsq/(n*m-1);
Xvar1=Xrsssq/n;
array column1(10000);
Do a=1 to dim(column1);
column1(a)=data_array{a,1};
83 end;
column1med1=smallest(5000,of column1[*]);
column1med2=smallest(5001,of column1[*]);
column1med=(column1med1+column1med2)/2;
array column2(10000);
Do a=1 to dim(column2);
column2(a)=data_array{a,2};
end;
column2med1=smallest(5000,of column2[*]);
column2med2=smallest(5001,of column2[*]);
column2med=(column2med1+column2med2)/2;
array column3(10000);
Do a=1 to dim(column3);
column3(a)=data_array{a,3};
end;
column3med1=smallest(5000,of column3[*]);
column3med2=smallest(5001,of column3[*]);
column3med=(column3med1+column3med2)/2;
array column4(10000);
Do a=1 to dim(column4);
column4(a)=data_array{a,4};
end;
column4med1=smallest(5000,of column4[*]);
column4med2=smallest(5001,of column4[*]);
column4med=(column4med1+column4med2)/2;
array column5(10000);
Do a=1 to dim(column5);
column5(a)=data_array{a,5};
end;
column5med1=smallest(5000,of column5[*]);
column5med2=smallest(5001,of column5[*]);
column5med=(column5med1+column5med2)/2;
array column6(10000);
Do a=1 to dim(column6);
column6(a)=data_array{a,6};
end;
column6med1=smallest(5000,of column6[*]);
column6med2=smallest(5001,of column6[*]);
column6med=(column6med1+column6med2)/2;
array column7(10000);
Do a=1 to dim(column7);
column7(a)=data_array{a,7};
end;
column7med1=smallest(5000,of column7[*]);
84 column7med2=smallest(5001,of column7[*]);
column7med=(column7med1+column7med2)/2;
array column8(10000);
Do a=1 to dim(column8);
column8(a)=data_array{a,8};
end;
column8med1=smallest(5000,of column8[*]);
column8med2=smallest(5001,of column8[*]);
column8med=(column8med1+column8med2)/2;
array column9(10000);
Do a=1 to dim(column9);
column9(a)=data_array{a,9};
end;
column9med1=smallest(5000,of column9[*]);
column9med2=smallest(5001,of column9[*]);
column9med=(column9med1+column9med2)/2;
columnmedian=(column1med+column2med+column3med+column4med+colu mn5med+column6med+column7med+column8med+column9med)/n;
Xcol1sum=(column1med-columnmedian)**2;
Xcol2sum=(column2med-columnmedian)**2;
Xcol3sum=(column3med-columnmedian)**2;
Xcol4sum=(column4med-columnmedian)**2;
Xcol5sum=(column5med-columnmedian)**2;
Xcol6sum=(column6med-columnmedian)**2;
Xcol7sum=(column7med-columnmedian)**2;
Xcol8sum=(column8med-columnmedian)**2;
Xcol9sum=(column9med-columnmedian)**2;
Xcolsq=Xcol1sum+Xcol2sum+Xcol3sum+Xcol4sum+Xcol5sum+Xcol6sum+
Xcol7sum+Xcol8sum+Xcol9sum;
Xvar2=Xcolsq/(n*n);
Xvar=Xvar1-Xvar2;
Xstdev=sqrt(Xvar);
muhat=Xbar;
sigmahat=Xstdev;
LCL=muhat-constant*sigmahat;
UCL=muhat+constant*sigmahat;
CRL=0;Z=muhat;c=1;
array crl_array{50000,9};
Do k=1 to 50000;
array t[9];
Do a=1 to 9;
Do b=1 to dim(t);
t[b]=(delta/sqrt(n))+sigma*rannor(33333);