Improved Nonparametric Control Chart Based on Ranked Set Sampling with Application of Chemical Data Modelling

Rasheed, Zahid; Khan, Majid; Abiodun, Nafiu Lukman; Anwar, Syed Masroor; Khalaf, G.; Abbasi, Saddam Akber

doi:https://doi.org/10.1155/2022/7350204

Mathematical Problems in Engineering

On this page

Abstract Introduction Data Availability Conflicts of Interest References Copyright Related Articles

Special Issue

Multivariate and Big Data Modeling and Related Issues

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 7350204 | https://doi.org/10.1155/2022/7350204

Improved Nonparametric Control Chart Based on Ranked Set Sampling with Application of Chemical Data Modelling

Zahid Rasheed,^1,2Majid Khan,³Nafiu Lukman Abiodun,⁴Syed Masroor Anwar,⁵G. Khalaf,⁶and Saddam Akber Abbasi^7,8

Academic Editor: Adrian Neagu

Received09 Feb 2022

Revised27 Apr 2022

Accepted04 May 2022

Published13 Jun 2022

Abstract

The statistical performance of parametric control charts is questionable when the underlying process does not follow any specified probability distribution. Nonparametric control charts are the best substitute for this situation. On the other hand, the ranked set sampling technique is preferred over the simple random sampling technique because it reduces the variability of process parameters and improves the control chart’s performance. This study aims to offer a nonparametric double homogeneously weighted moving average control chart under Wilcoxon signed-rank test considering the ranked set sampling technique (regarded as ), to further enhance the process location monitoring. The proposed chart’s run-length performance is compared with competing control charts, such as DEWMA-, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and control charts. The comparison revealed that the proposed control chart outperformed the other competing control charts, particularly for small to moderate shifts in process location. Finally, a real-life application is also offered for quality practitioners to show the strength of the proposed control chart.

1. Introduction

A quality assurance system offers a variety of management approaches that save time and money while producing a high-quality final product. These approaches are commonly utilized in the manufacturing process to detect irregularities and improve product quality. Statistical process control (SPC) is a key aspect in detecting abnormalities of ultimate products. SPC techniques are widely used in industrial applications, biological sciences, environmental studies, and healthcare departments to monitor the ongoing processes. The quality of the products is influenced by unnatural variation. The existence of unnatural variations causes the shift in process parameters (location and/or dispersion). Control charts are popular tools in SPC that helps in identifying the shifts in process parameters. Usually, control charts are generally classified as memoryless and memory-type charts based on their design structure. Shewhart [1] introduced the first memoryless chart known as the Shewhart chart, whereas Page [2] and Roberts [3] introduced the concept of memory charts, known as a cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts, respectively.

Classical parametric control charts are usually used when an ongoing process follows a predefined probability distribution. The ongoing process may not follow a specific distribution, or the distribution of the ongoing process may be in doubt. Nonparametric (NP) control charts are a reliable substitute for parametric control charts to handle such scenarios. NP control charts are convenient because their in-control (IC) run-length (RL) distribution is the same for all continuous distributions. The sign (SN) and Wilcoxon signed-rank (SR) are two well-known NP techniques practiced in statistical process monitoring (SPM) with control charts. Likewise, simple random sampling (SRS) and ranked set sampling (RSS) techniques are quite often used in SPM with both parametric and NP control charts to observe the data of the ongoing processes [4]. RSS is recommended in the SPM literature as it decreases variability and improves the efficiency of the associated control charts [5, 6]. For instance, hazardous waste sites with varying levels of contamination can be classified visually based on soil staining, whereas actual statistics of toxic chemicals and quantification of their ecological impact are prohibitively expensive.

Numerous researchers introduced a wide range of NP control charts using different NP statistics. Amin and Searcy [7] introduced an efficient NP EWMA-SR control chart to monitor the process location shift efficiently. Similarly, Bakir [8] presented NP Shewhart-SR control chart for process location monitoring. Subsequently, Yang, et al. [9] offered an EWMA-SN control chart for process location. Moreover, Graham, et al. [10] and Graham et al. [11] designed an NP EWMA-SN control chart based on a single observation and NP EWMA-SR control chart, respectively, for monitoring shifts in process location. Likewise, Lu [12] and Tsai et al. [13] presented enhanced NP EWMA control charts based on SN statistics and used RSS techniques in conjunction with NP control charts. Eventually, Chakraborty et al. [14] and Lu [15] developed a generally weighted moving average-SR (GWMA-SR) and SN statistic-based NP double GWMA control charts for process proportion, respectively, to improve the shift detection ability. Furthermore, Chakraborti and Graham [16] reviewed some latest development in both univariate and multivariate NP control charts. Also, Rasheed et al. [17] and Rasheed et al. [18] advocated RSS-based parametric and NP control charts for efficiently monitoring the process mean, respectively.

The recently introduced homogeneously weighted moving average (HWMA) control chart by Abbas [19] is efficient for monitoring process location. Adegoke et al. [20] provided an auxiliary information-based (AIB) HWMA control chart for process location more efficient than the HWMA control chart. Later, Anwar et al. [21] extended the AIB-HWMA and suggested the AIB-DHWMA control chart for improved process location monitoring. Based on a thorough literature review, it is observed that no one has developed an NP DHWMA control chart under SR along with RSS methodology to date. This is a research gap that needs to be explored. So this study proposes an NP DHWMA-SR control chart under RSS () for monitoring shifts in process location for continuous and symmetric distribution. The study’s main goal is to propose a control chart for detecting small and moderate shifts in process location more quickly because small changes in process parameters can have a significant financial impact on an organization’s operations. The run-length () characteristics of the proposed control chart, including average run length (), median run length (), and standard deviation of run-length (), are obtained using various distributions like normal, student’s t, contaminated normal (CN), Laplace, and logistic distributions. The performance of proposed control chart is decided by providing a valid comparison with other existing control charts such as DEWMA-, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and control charts.

The rest of the paper is organized as follows: Section 2 presents the design structure of the competing and the proposed control chart. Similarly, Section 3 describes the proposed control chart’s IC and out-of-control (OOC) performance. Likewise, Section 4 contains a comparative study of the proposed control chart, whereas Section 5 provides a real-life application. Finally, concluding remarks are given in Section 6.

2. Competing and Proposed Control Charts

This section explains the design structure of the competing and the proposed control charts. These competing control charts are DEWMA-, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and . More detail is provided in the following subsections.

2.1. NPRDEWMA-SR Control Chart

Abbas et al. [4] presented an NP double EWMA SR under the RSS (NPRDEWMA-SR) control chart that outperforms the NPREWMA-SR control chart in terms of shift detection in process location. The plotting statistics of the NPRDEWMA-SR control chart are given as follows:where is a smoothing constant. The control limits of the NPRDEWMA-SR control chart can be designed as follows:

The process goes OOC when ; otherwise, it will remain an IC state.

2.2. DHWMA Control Chart

Abid et al. [22] developed a DHWMA control chart that detects shifts more efficiently than the HWMA control chart. The DHWMA control chart plotting statistic is given as follows:where and are the mean of and mean of the previous samples, respectively. The control limits of the DHWMA control chart based on this and are defined as follows:

The process remain is IC if ; otherwise, it goes OOC.

2.3. Proposed Control Chart

Various researchers like Kim and Kim [23], Abid et al. [24], and Abbas et al. [4] used the following RSS-based Wilcoxon signed-rank statistic to monitor shifts in process location:where symbolizes the process median, and denote the number of samples, observations, and cycles used in the RSS approach, respectively. The mean and variance of statistic are and , respectively, where r denotes the number of replications and can be defined as . The quantity is used to improve the efficiency of the control chart and can be defined as . The values of can be obtained by solving the following mathematical expression:

Abid et al. [24] have more information on the RSS approach and related terms. The methodology of the proposed control chart is defined as follows:where is the mean of of samples. The simplified form of the plotting statistic is

The control limits of the proposed control chart are

If or the underlying process is OOC; else, it is IC.

3. Implementation of the Proposed Control Chart

This section investigates performance metrics, the IC, and the OOC performances of the proposed control chart for monitoring shifts in process location. Subsection 3.1 provides the performance metrics of the proposed control chart. Likewise, the proposed control chart’s robustness, IC, and OOC performance are presented in Subsection 3.2.

3.1. Performance Metrics

is widely used to evaluate the performance of the control chart. The is the expected number of sample points before the first OOC signal from the control chart. The is categorized as IC () and out-of-control (). If a process is functioning in an in-control state, the needed to be large enough to avoid frequent false alarms. However, the should be small enough; it quickly detects the shift. It is necessary for better performance of a control chart; it should have a smaller as compared to other control charts at the fixed value of . In this study, we set to 370 and 500, with sample sizes () of 5 and 10. Monte Carlo simulations with 50,000 simulations in the R program are used to determine the characteristics. To examine the performance behavior of the proposed control chart, various values of (0.05, 0.10, 0.25, 0.50) and (0.025, 0.05, 0.075, 0.10, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00, 5.00) are used. The following algorithm is used for simulations:(i)To create samples from considered distributions, a finite loop is used.(ii)Specify the process parameters ().(iii)Draw a sample from a distribution used in this study.(iv)Determine the plotting statistics using equation (7).(v)Find and from equation (8).(vi)Plot the plotting statistic against and over .(vii)If or note this sample of statistic as an RL. For instance, at , if or record 105 as a first RL.(viii)Repeat steps (ii) through (vi) for 50,000 times and record RLs.(ix)Calculate from 50,000 and recorded RLs.(x); if not, adjust constant accordingly in step (ii) and repeat from (ii) to (ix) steps to obtain .(xi)To acquire values, draw a shifted sample from the considered distribution again, and repeat steps (ii) to (ix).

3.2. Robustness, IC, and OOC Performances of the Control Chart

This subsection highlights the proposed control chart’s robustness, IC, and OOC behavior when a process location is shifted. Table 1 shows the RL characteristics of the proposed control chart for location shift. These characteristics are assessed using normal and non-normal continuous symmetric distributions. The distributions used for this study are standard normal distribution, that is, ; double exponential or Laplace distribution, that is, ; heavy tail student’s distribution, that is, ; logistic distribution, that is, ; and contaminated normal (CN) distribution, which is the mixture of and . All these distributions were reparametrized with zero mean/median and unit variance for comparison purposes. For all symmetric continuous distributions, the IC RL characteristics of the NP control chart remain constant [4].

For comparison purposes, the same parameters are used as reported in numerous relevant articles. The measures are used to compare the proposed and competing control charts. Based on the research findings and sensitivity analysis, the following observations have been made.(i)The proposed control chart’s IC distribution looks remarkably similar for all distributions examined in this study. For example, at (0.05, 0.10, 0.25, 0.50) and , the for all investigated distributions (see Table 1).(ii)As the smoothing parameter reduces, the proposed control chart becomes more effective in detecting shifts. This illustrates that the proposed control chart is more sensitive to small smoothing parameters (see Figure 1).(iii)As the sample size increases, the proposed control chart’s shift detection ability for process location improves (see Figure 2).(iv)The Laplace distribution outperforms the other distributions in terms of OOC RL performance (see Figures 3).(v)The proposed control chart’s values increase as increases at a certain size of the shift. For instance, under normal distribution at , the , whereas when , the (see Table 1).(vi)The of the proposed control chart are smaller than those in the competing control charts with different shift sizes in process location (see Figure 4).(vii)The distribution of RL values is positively skewed, that is, (see Table 1).

4. Comparative Study

This section provides a comparative performance study in terms of the values of the proposed control chart for process location shifts. The proposed control chart is compared to the competing control charts, including DEWMA-, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and .

4.1. Proposed versus DEWMA- Control Chart

The profile demonstrates that the proposed control chart outperforms the DEWMA- control chart. For example, under normal distribution, at , and , the values of the proposed control chart are , whereas the values of the DEWMA- control charts are (see Tables 1 and 2). Similarly, when we consider Laplace distribution for comparison, we observed the same behavior in the proposed control chart. For instance, at , and , the values of the proposed control chart are , while the values of DEWMA- control chart are (see Figure 4 and Tables 1 and 2).

4.2. Proposed versus NPDEWMA-SR Control Chart

The proposed control chart is more sensitive than the NPDEWMA-SR control chart for all combinations of and . For instance, in case of logistic distribution, when , and 0.025, 0.05, 0.075, 0.10, 0.25, 0.50, 0.75, the values of the proposed and NPDEWMA-SR control charts are 30.34, 12.03, 7.21, 5.21, 1.82, 1.04, and 1.00 and 280.87, 123.83, 66.59, 42.46, 9.40, 3.08, and 1.734, respectively (see Tables 1 and 3). The supremacy of the proposed control chart to the NPDEWMA-SR can also be seen in Figure 4. Likewise, in the scenario of distribution comparison, we noted the same behavior of the proposed control chart. As an illustration, at , and , the values of the proposed control chart are , respectively, while the values of the NPDEWMA-SR control chart are (see Tables 1 and 3).

4.3. Proposed versus NPRDEWMA-SR Control Chart

The proposed control chart performs better than the NPRDEWMA-SR control chart. For instance, using the distribution with , the values of the proposed and the NPRDEWMA-SR control charts are and , respectively (see Tables 1 and 4). The proposed charts’ efficiency may also be observed in the case of CN distribution. For example, at , the values for the proposed control chart are whereas the values for the NPRDEWMA-SR control chart are (see Figure 4 and Tables 1 and 4).

4.4. Proposed versus DHWMA Control Chart

The study reveals that the proposed control chart outperforms the DHWMA control chart for all combinations of (see Tables 1 and 5). As an illustration, for normal distribution, at , the values of the and DHWMA control charts are and , respectively (see Tables 1 and 5). Figure 4 also shows the superiority of the control chart over the DHWMA control chart. The results show that the control chart is superior to the DHWMA control chart for monitoring process location shifts.

4.5. Proposed versus Control Chart

The findings show that the proposed control chart is better than the control chart in terms of OOC performance. For example, when we examine the distribution at , the values of the proposed and control charts are and , respectively (see Figure 4 and Tables 1 and 6). Likewise, under Laplace distribution, when , the values of the proposed control chart are , while the values for control chart are , respectively (see Tables 1 and 6). The statistics show that the proposed structure works effectively for all distributions when compared to the control chart

5. Illustrative Example

The proposed charts’ implementation is generally associated with industrial processes and finished products, and it can be adapted to different of many other fields such as medicine, planning, financial reporting, neutrosophic statistics, and so on. This section provides a real-life application of the non-isothermal continuous stirred tank reactor (CSTR) process to demonstrate the applicability of the proposed control chart. This real-life data was originally proposed by Marlin and Marlin [25], and has since been widely used as a standard in fault diagnosis, for instance, Xiangrong et al. [26], Ridwan et al. [27], Adegoke et al. [20], and many more. The CSTR process has nine different variables, one of which we choose as the variable of interest () represents the output temperature.

The data initially consists of 1,000 observations, with the first 600 occurring when the process was in an IC condition. The phase I sample’s parameters are as follows: and . We used the RSS approach to generate 40 paired observations of size = 5 from a normal distribution. After the 24th sample, a shift in the process location is introduced following Anwar et al. [28]. The parameters of the proposed and NPRDHWMA-SR control charts used for real-life analysis are , and , respectively. Figure 5 indicates that the proposed control chart triggers the first OOC signal at sample number 25, while the NPRDEWMA-SR control chart detects the first OOC point at sample number 29. Similarly, the proposed control chart detects overall 16 OOC points, whereas the NPRDEWMA-SR control chart detects 12 OOC points (see Table 7 and Figure 5).

6. Summary, Conclusions, and Future Recommendations

Usually, control charts are used to monitor the process parameters (location and/or dispersion) when the quality characteristic follows the specific distribution. When this assumption is not fulfilled, the nonparametric (NP) control charts are used to handle this situation. On the other hand, the double homogeneously weighted moving average (DHWMA) is the advanced version of the double exponentially weighted moving average (DEWMA) control chart for process location monitoring. Similarly, the ranked set sampling (RSS) technique is more efficient than simple random sampling (SRS). So this study combines the NP DHWMA control chart and RSS scheme and presents an NPDHWMA Wilcoxon signed-rank control chart under the RSS technique (denoted by ) for enhanced monitoring of process location shifts. The performance of the proposed control chart is investigated in terms of . The results revealed that the proposed control chart performs better than the competing control charts such as DEWMA-, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and . Moreover, a real-life application is also offered to show the proposed control chart’s applicability in practice. This study is carried out where the process variable follows the univariate distributions. However, the proposed charting scheme can be used to enhance the monitoring of high-quality processes [29, 30], time-between-events [31], multivariate processes [32], and neutrosophic statistics [33, 34] scenarios.

Data Availability

The data used in the real-life application can be obtained from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

W. A. ShewhartD. Van Nostrand, Economic Control of Quality Manufactured Product, American Society for Quality Control in 1980, Milwaukee, WI, 1931.
E. S. Page, “Continuous inspection schemes,” Biometrika, vol. 41, no. 1-2, pp. 100–115, 1954.
View at: Publisher Site | Google Scholar
S. W. Roberts, “Control chart tests based on geometric moving averages,” Technometrics, vol. 1, no. 3, pp. 239–250, 1959.
View at: Publisher Site | Google Scholar
Z. Abbas, H. Z. Nazir, M. Abid, N. Akhtar, and M. Riaz, “Enhanced nonparametric control charts under simple and ranked set sampling schemes,” Transactions of the Institute of Measurement and Control, vol. 42, no. 14, pp. 2744–2759, 2020.
View at: Publisher Site | Google Scholar
A. Haq, J. Brown, E. Moltchanova, and A. I. Al-Omari, “Effect of measurement error on exponentially weighted moving average control charts under ranked set sampling schemes,” Journal of Statistical Computation and Simulation, vol. 85, no. 6, pp. 1224–1246, 2015.
View at: Publisher Site | Google Scholar
M. Noor-ul-Amin and M. Tayyab, “Enhancing the performance of exponential weighted moving average control chart using paired double ranked set sampling,” Journal of Statistical Computation and Simulation, vol. 90, no. 6, pp. 1118–1130, 2020.
View at: Publisher Site | Google Scholar
R. W. Amin and A. J. Searcy, “A nonparametric exponentially weighted moving average control scheme,” Communications in Statistics - Simulation and Computation, vol. 20, no. 4, pp. 1049–1072, 1991.
View at: Publisher Site | Google Scholar
S. T. Bakir, “A distribution-free Shewhart quality control chart based on signed-ranks,” Quality Engineering, vol. 16, no. 4, pp. 613–623, 2004.
View at: Publisher Site | Google Scholar
S.-F. Yang, J.-S. Lin, and S. W. Cheng, “A new nonparametric EWMA sign control chart,” Expert Systems with Applications, vol. 38, no. 5, pp. 6239–6243, 2011.
View at: Publisher Site | Google Scholar
M. Graham, S. Chakraborti, and S. Human, “A nonparametric EWMA sign chart for location based on individual measurements,” Quality Engineering, vol. 23, pp. 227–241, 2011.
View at: Google Scholar
M. Graham, S. Chakraborti, and S. Human, “A nonparametric exponentially weighted moving average signed-rank chart for monitoring location,” Computational Statistics & Data Analysis, vol. 55, pp. 2490–2503, 2011.
View at: Google Scholar
S.-L. Lu, “An extended nonparametric exponentially weighted moving average sign control chart,” Quality and Reliability Engineering International, vol. 31, no. 1, pp. 3–13, 2015.
View at: Publisher Site | Google Scholar
C. F. Tsai, S. L. Lu, and C. J. Huang, “Design OF an extended nonparametric EWMA sign chart,” International Journal of Industrial Engineering: Theory, Applications and Practice, vol. 22, no. 6, pp. 705–716, 2015.
View at: Google Scholar
N. Chakraborty, S. Chakraborti, S. W. Human, and N. Balakrishnan, “A generally weighted moving average signed-rank control chart,” Quality and Reliability Engineering International, vol. 32, 2016.
View at: Publisher Site | Google Scholar
S.-L. Lu, “Non parametric double generally weighted moving average sign charts based on process proportion,” Communications in Statistics - Theory and Methods, vol. 47, no. 11, pp. 2684–2700, 2018.
View at: Publisher Site | Google Scholar
S. Chakraborti and M. Graham, “Nonparametric (distribution-free) control charts: an updated overview and some results,” Quality Engineering, vol. 31, pp. 1–22, 2019.
View at: Google Scholar
Z. Rasheed, H. Zhang, S. M. Anwar, and B. Zaman, “Homogeneously mixed memory charts with application in the substrate production process,” Mathematical Problems in Engineering, vol. 2021, Article ID 2582210, 2021.
View at: Publisher Site | Google Scholar
Z. Rasheed, H. Zhang, M. Arslan et al., “An efficient robust nonparametric triple EWMA Wilcoxon signed-rank control chart for process location,” Mathematical Problems in Engineering, vol. 2021, Article ID 2570198, 2021.
View at: Publisher Site | Google Scholar
N. Abbas, “Homogeneously weighted moving average control chart with an application in substrate manufacturing process,” Computers & Industrial Engineering, vol. 120, pp. 460–470, 2018.
View at: Publisher Site | Google Scholar
N. A. Adegoke, A. N. H. Smith, M. J. Anderson, R. A. Sanusi, and M. D. M. Pawley, “Efficient homogeneously weighted moving average chart for monitoring process mean using an auxiliary variable,” IEEE Access, vol. 7, pp. 94021–94032, 2019.
View at: Publisher Site | Google Scholar
S. M. Anwar, M. Aslam, B. Zaman, and M. Riaz, “An enhanced double homogeneously weighted moving average control chart to monitor process location with application in automobile field,” Quality and Reliability Engineering International, vol. 38, 2021.
View at: Publisher Site | Google Scholar
M. Abid, A. Shabbir, H. Z. Nazir, R. A. K. Sherwani, and M. Riaz, “A double homogeneously weighted moving average control chart for monitoring of the process mean,” Quality and Reliability Engineering International, vol. 38, 2020.
View at: Publisher Site | Google Scholar
D. H. Kim and Y. C. Kim, “Wilcoxon signed rank test using ranked-set sample,” Korean Journal of Computational & Applied Mathematics, vol. 3, no. 2, pp. 235–243, 1996.
View at: Publisher Site | Google Scholar
M. Abid, H. Z. Nazir, M. Riaz, and Z. Lin, “An efficient nonparametric EWMA Wilcoxon signed-rank chart for monitoring location,” Quality and Reliability Engineering International, vol. 33, no. 3, pp. 669–685, 2017.
View at: Publisher Site | Google Scholar
T. E. Marlin and T. Marlin, Process Control: Controling processes and control systems for dynamic performance, MC Master University, Canada, 2000.
S. Xiangrong, L. Yan, F. ZhengShun, and L. Jun, “A multivariable statistical process monitoring method based on multiscale analysis and principal curves,” Int. J. Innov. Comput. Inf. Control., vol. 9, pp. 1781–1800, 2013.
View at: Google Scholar
A. S. Ridwan, R. Muhammad, A. A. Nurudeen, and X. Min, “An EWMA monitoring scheme with a single auxiliary variable for industrial processes,” Computers & Industrial Engineering, vol. 114, pp. 1–10, 2017.
View at: Google Scholar
S. M. Anwar, M. Aslam, B. Zaman, and M. Riaz, “Mixed memory control chart based on auxiliary information for simultaneously monitoring of process parameters: an application in glass field,” Computers & Industrial Engineering, vol. 156, Article ID 107284, 2021.
View at: Publisher Site | Google Scholar
T. Mahmood and M. Xie, “Models and monitoring of zero‐inflated processes: the past and current trends,” Quality and Reliability Engineering International, vol. 35, no. 8, pp. 2540–2557, 2019.
View at: Publisher Site | Google Scholar
T. Mahmood, “Generalized linear model based monitoring methods for high‐yield processes,” Quality and Reliability Engineering International, vol. 36, no. 5, pp. 1570–1591, 2020.
View at: Publisher Site | Google Scholar
I. M. Zwetsloot, T. Mahmood, and W. H. Woodall, “Multivariate time-between-events monitoring: an overview and some overlooked underlying complexities,” Quality Engineering, vol. 33, 2020.
View at: Google Scholar
J. O. Ajadi, K. Hung, M. Riaz, N. A. Ajadi, and T. Mahmood, “On the multivariate progressive control chart for effective monitoring of covariance matrix,” Journal of Quality Reliability Engineering International, 2021.
View at: Publisher Site | Google Scholar
M. Aslam, O. H. Arif, and R. A. K. Sherwani, “New diagnosis test under the neutrosophic statistics: an application to diabetic patients,” BioMed Research International, vol. 2020, Article ID 2086185, pp. 1–7, 2020.
View at: Publisher Site | Google Scholar
M. Albassam, N. Khan, and M. Aslam, “Neutrosophic D’agostino test of normality: an application to water data,” Journal of Mathematics, vol. 2021, Article ID 5582102, 2021.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Zahid Rasheed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

418

Downloads

367

Citations