This study presents the design of a modified attributed control chart based on a double sampling (DS)
A control chart is a statistical analysis tool used to monitor processes through time. It can also identify changes or trends that could indicate a potential problem. Control charts are used to control quality in production to ensure consistency and help identify areas for improvement. The concept of statistical process control (SPC) was introduced by Walter A. Shewhart during the 1920s. One of his key contributions to SPC was the development of the control chart, which is a tool used to monitor and control a process over time. Shewhart’s control chart revolutionized the field of quality control by providing a way to monitor processes in realtime and make datadriven decisions to improve quality according to Montgomery [
The
There are also different techniques in sampling plans proposed by many researchers to improve processes to be more efficient. One popular acceptance sampling technique is MDS sampling proposed by Wortham et al. [
During this study, the modified attributed
The Weibull distribution is often employed in statistical quality control studies [
Authors/Year  Types of chart  Topics 

Aslam et al. [ 
Proposed control chart when the lifetime of the product follows a Weibull distribution based on the number of failure items in a truncated life test.  
Akhundjanov et al. [ 
Moving range EWMA chart  Presented a control chart for monitoring shifts in the Weibull shape parameters. 
Faraz et al. [ 
Proposed control charts for monitoring individual or joint shifts in the scale and shape parameters of a Weibull distributed process.  
Aslam [ 
Mixed EWMACUSUM chart  Proposed a mixed control chart combining CUSUM and EWMA statistics by assuming that the quality characteristic of interest follows a Weibull distribution. 
Aslam et al. [ 
Presented a control chart using accelerated hybrid censoring logic for the monitoring of defective items whose lifetime follows a Weibull distribution.  
Arif et al. [ 
EWMA 
Designed the attribute control chart based on the number of failures under a time truncated life test when the lifetime of the product follows a Weibull distribution. 
Balamurali et al. [ 
Designed a control chart using MDS sampling for monitoring the mean life of the products when the lifetime follows a Weibull distribution based on a time truncated life test.  
Huwang et al. [ 
new EWMA chart  Developed an EWMA chart for monitoring the shape parameters of a Weibull process. 
Aslam et al. [ 
Designed a control chart for monitoring the mean lifetime of the products following a Weibull distribution under an accelerated life test.  
Khan et al. [ 
Moving average EWMA chart  Presented a control chart to monitor the number of defective counts before the specified time which follows a Weibull distribution. 
Let
The value of
If the process mean is the same as the target mean, or
From
The following section presents the modified DS
Specify the limits indicated as
The initial sample, of size
In Stage 1 (see
If
If
If
In Stage 2 (see
Let
The developed control chart monitors the mean life of products under a timetruncated life test when the lifetime of the product follows a Weibull distribution.
The developed control chart is constructed based on a double sampling
At the start of the process, the process is assumed to fit the incontrol region, that is
The process mean may be shifted to the outofcontrol region, that is
In this study, the genetic algorithm (GA) with the R program is used to find the optimal parameters.
Therefore, the control limits for Stages 1 and 2 are shown as follows:
Based on the developed control chart, the probability that the process will be considered incontrol at Stage 1 is indicated by
The probability declares that the process is incontrol at Stage 2 when given that
According to the modified DS
The probability of declaring that a process is incontrol when it is actually incontrol
Moreover, the probability of declaring that a process is incontrol when it is actually outofcontrol
The
The
Additionally, the average sample size
In this section, we used the following optimization problem to obtain the optimal parameters for constructing the modified DS
Assign the values of
Find out the values of the optimal parameters
Calculate the
The pseudocode of the modified DS
The following section presents the assessment of the performance achieved by the modified DS
When
When
It was found that if
As
The findings also indicate a decrease in the values of
1.0  200.64  200.06  370.05  370.56 
0.9  192.30  188.11  287.15  279.50 
0.8  147.12  139.63  193.50  183.59 
0.7  117.64  77.02  174.84  163.80 
0.6  104.70  66.93  66.80  47.31 
0.5  68.77  47.24  57.53  41.77 
0.4  26.67  1.00  7.69  1.00 
0.3  1.00  1.00  1.00  1.00 
0.2  1.00  1.00  1.00  1.00 
0.1  1.00  1.00  1.00  1.00 
1.0  200.99  200.00  370.60  370.65 
0.9  189.59  182.50  229.10  366.45 
0.8  162.19  145.83  223.50  279.75 
0.7  150.78  120.10  191.75  269.52 
0.6  119.16  66.54  97.44  190.36 
0.5  79.90  24.00  91.63  14.63 
0.4  37.82  1.00  28.55  1.00 
0.3  1.00  1.00  1.00  1.00 
0.2  1.00  1.00  1.00  1.00 
0.1  1.00  1.00  1.00  1.00 
1.0  200.00  201.03  200.00  200.00  200.51  200.11 
0.9  155.54  120.62  187.09  182.78  177.29  189.15 
0.8  139.24  116.09  181.36  176.15  175.75  186.23 
0.7  63.16  51.22  155.43  144.41  127.21  162.33 
0.6  37.56  33.37  97.44  97.01  72.46  137.44 
0.5  18.63  18.26  47.27  19.37  17.31  80.04 
0.4  5.35  5.27  25.98  1.00  1.00  25.78 
0.3  1.00  1.00  1.00  1.00  1.00  1.00 
0.2  1.00  1.00  1.00  1.00  1.00  1.00 
0.1  1.00  1.00  1.00  1.00  1.00  1.00 
1.0  370.14  370.32  370.08  370.00  370.00  370.01 
0.9  209.53  129.45  323.20  252.02  222.94  299.01 
0.8  174.31  111.80  292.22  153.04  131.05  198.91 
0.7  161.51  104.96  165.76  125.55  121.13  145.51 
0.6  80.82  80.66  139.42  52.39  20.79  55.90 
0.5  72.74  72.59  107.90  38.36  11.22  38.63 
0.4  11.46  5.74  36.76  1.00  1.00  1.00 
0.3  1.00  1.00  1.00  1.00  1.00  1.00 
0.2  1.00  1.00  1.00  1.00  1.00  1.00 
0.1  1.00  1.00  1.00  1.00  1.00  1.00 
In
In some cases, the parameters presented in
Model parameter  Level 1  Level 2  Model parameter  Level 1  Level 2 

2  3  200  370  
0.4  0.8  50  100  
3  6 
The data shown in
Trial  Model parameters  Solution  

1  0.4  3  2  200  50  24  51  2.8725  4.7133  1.0547  2  24.24  207.40 
2  0.4  3  2  200  100  45  130  3.0150  3.3530  1.0102  2  45.43  201.86 
3  0.4  3  2  370  50  42  60  3.0074  3.7802  3.1968  2  42.11  375.52 
4  0.4  3  2  370  100  42  128  2.9870  3.2854  2.2429  2  42.00  375.08 
5  0.4  3  3  200  50  36  60  3.2161  4.3097  2.3222  2  36.23  208.29 
6  0.4  3  3  200  100  60  129  3.3091  4.8395  3.8636  2  60.64  200.50 
7  0.4  3  3  370  50  7  57  3.3711  4.9495  1.9035  2  7.15  370.50 
8  0.4  3  3  370  100  7  113  3.3815  3.6287  2.7589  1  7.00  370.02 
9  0.4  6  2  200  50  29  65  2.8335  3.7658  1.2450  5  29.24  205.39 
10  0.4  6  2  200  100  45  118  3.0755  3.9633  1.1358  5  45.53  201.95 
11  0.4  6  2  370  50  22  65  3.1862  4.2079  1.1871  4  22.14  384.23 
12  0.4  6  2  370  100  22  150  3.0091  4.8920  1.2810  3  22.39  384.56 
13  0.4  6  3  200  50  36  72  3.0444  4.4546  1.1038  4  36.34  208.10 
14  0.4  6  3  200  100  36  111  2.9094  3.3812  2.9678  3  36.00  207.97 
15  0.4  6  3  370  50  7  61  3.9567  4.9899  2.2981  5  7.16  370.67 
16  0.4  6  3  370  100  79  143  3.3983  4.1036  1.2079  4  79.35  375.52 
17  0.8  3  2  200  50  17  66  2.8058  3.3188  1.1593  2  17.12  219.23 
18  0.8  3  2  200  100  44  122  2.8691  3.2644  1.0168  2  44.21  205.15 
19  0.8  3  2  370  50  18  68  2.9345  3.9863  1.0471  2  18.07  383.49 
20  0.8  3  2  370  100  24  126  3.0376  3.3569  1.1195  2  24.18  376.90 
21  0.8  3  3  200  50  11  72  2.8742  3.6765  1.0103  2  11.30  205.84 
22  0.8  3  3  200  100  11  109  2.8671  3.4283  1.0517  2  11.46  205.89 
23  0.8  3  3  370  50  39  89  3.0340  4.2022  1.1309  1  39.18  372.60 
24  0.8  3  3  370  100  64  133  3.0573  3.7718  1.0453  2  64.25  372.17 
25  0.8  6  2  200  50  17  62  2.8258  3.5193  1.0720  4  17.11  219.23 
26  0.8  6  2  200  100  71  116  2.8970  4.9059  1.4949  4  71.35  203.28 
27  0.8  6  2  370  50  35  69  2.9698  4.2856  1.1803  4  35.11  372.27 
28  0.8  6  2  370  100  86  120  3.0262  3.1524  1.0841  5  86.10  370.53 
29  0.8  6  3  200  50  11  60  3.0147  3.7385  1.1034  3  11.29  205.89 
30  0.8  6  3  200  100  35  134  2.8122  4.2819  1.5467  4  35.39  203.67 
31  0.8  6  3  370  50  5  63  2.9077  4.5687  3.2154  3  5.17  376.70 
32  0.8  6  3  370  100  37  118  2.9835  3.5130  1.5543  2  37.26  376.36 
(a) Table of ANOVA  

Source  DF  SS  MS  Fvalue  
Regression  1  28800  28800.0  286.28  0.000 
Residual  30  3018  100.6  
Total  31  31818  
(b) Table of regression coefficients  
Independent variable  Coefficients  Std. error  Tvalue  VIF  
Constant  5.00  5.61  0.89  0.380  
1.20  0.0709  16.92  0.000  1.00  
Adjusted 
(a) Table of ANOVA  

Source  DF  SS  MS  Fvalue  
Regression  3  0.9427  0.31422  11.46  0.000 
Residual  28  0.7678  0.02742  
Total  31  1.7105  
(b) Table of regression coefficients  
Independent variable  Coefficients  Std. error  Tvalue  VIF  
Constant  2.639  0.199  13.26  0.000  
−0.571  0.146  −3.90  0.001  1.00  
0.1741  0.0585  2.97  0.006  1.00  
0.001105  0.000344  3.21  0.003  1.00  
Adjusted 
(a) Table of ANOVA  

Source  DF  SS  MS  Fvalue  
Regression  2  34.000  17.0000  42.87  0.000 
Residual  29  11.500  0.3966  
Total  31  45.500  
(b) Table of regression coefficients  
Independent variable  Coefficients  Std. error  Tvalue  VIF  
Constant  1.125  0.659  1.71  0.098  
0.6667  0.0742  8.98  0.000  1.00  
−0.500  0.223  −2.25  0.033  1.00  
Adjusted 
(a) Table of ANOVA  

Source  DF  SS  MS  Fvalue  
Regression  3  227599  75866  3664.48  0.000 
Residual  28  580  21  
Total  31  228179  
(b) Table of regression coefficients  
Independent variable  Coefficients  Std. error  Tvalue  VIF  
Constant  38.9  11.1  3.49  0.002  
1.0019  0.0104  96.73  0.000  1.20  
−0.0699  0.0322  −2.17  0.038  1.00  
−9.19  3.81  −2.41  0.023  1.20  
Adjusted 
(a) Table of ANOVA  

Source  DF  SS  MS  Fvalue  
Regression  4  14189  3547.25  328444.48  0.000 
Residual  27  0.3  0.01  
Total  31  14189.3  
(b) Table of regression coefficients  
Independent variable  Coefficients  Std. error  Tvalue  VIF  
Constant  −0.235  0.162  −1.46  0.157  
−0.00099  0.000219  −4.54  0.000  1.02  
1.00014  0.00106  942.38  0.000  1.47  
0.00249  0.000726  3.43  0.002  1.55  
0.1283  0.0335  3.83  0.001  1.07  
Adjusted 
In
Comparisons are drawn in this section between the performance of the modified DS
Developed control chart  EWMA 
Developed control chart  EWMA 


1  205.04  200.51  201.64  200.73  370.09  370.75  377.84  370.19 
0.9  25.13  28.52  78.75  54.34  14.25  48.61  79.85  79.37 
0.8  3.25  4.58  12.11  10.10  3.12  5.72  13.78  12.10 
0.7  1.25  2.05  3.08  2.69  1.13  1.85  3.12  2.82 
0.6  1.00  1.58  1.94  1.21  1.00  1.32  1.56  1.20 
0.5  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00 
0.4  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00 
0.3  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00 
0.2  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00 
0.1  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00 
The following section presents the implementation of a modified DS
First of all, the dataset must be examined to determine whether a Weibull distribution is applicable. To check the goodness of fit, the Kolmogorov–Smirnov (KS) test was used, while the unknown parameters were estimated using the maximum likelihood method. The result for the KS test is 0.11212, giving a
Sub group  First sample ( 
Sub group  First sample ( 
Sub group  First sample ( 
Sub group  First sample ( 
Second sample ( 


1  10  11  12  21  12  31  11  
2  7  12  11  22  10  32  14  
3  10  13  4  23  11  33  14  
4  12  14  10  24  12  34  10  
5  16  15  14  25  13  35  19  31  50 
6  11  16  16  26  10  36  15  
7  12  17  6  27  15  37  11  
8  7  18  10  28  11  38  12  
9  15  19  11  29  11  39  18  27  45 
10  11  20  7  30  6  40  13 
The design of a novel attributed control chart is achieved through the application of the DS
The authors are highly grateful to the reviewers and editors for taking the time to make their comments and suggestions very helpful to the paper.
This research was supported by the Science, Research and Innovation Promotion Funding (TSRI) (Grant No. FRB660012/0168). This research block grants was managed under Rajamangala University of Technology Thanyaburi (FRB66E0646O.4).
The authors confirm contribution to the paper as follows: study conception and design: W. Bamrungsetthapong, P. Charongrattanasakul; data collection: P. Charongrattanasakul; analysis and interpretation of results: W. Bamrungsetthapong, P. Charongrattanasakul; draft manuscript preparation: W. Bamrungsetthapong. All authors reviewed the results and approved the final version of the manuscript.
The data used in this article are freely available in the cited references.
The authors declare that they have no conflicts of interest to report regarding the present study.