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

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China Department of Mathematics, Women University of Azad Jammu and Kashmir, Bagh, AJK, Pakistan Department of Mathematics and Statistics, Riphah International University Islamabad, Islamabad, Pakistan Department of Economics and Statistics, Faculty of Economics and Management Sciences, Kabale University, Kabale, Uganda Department of Statistics, University of Azad Jammu and Kashmir, Muzaarabad, Pakistan Department of Computer Engineering Techniques, Mazaya University College, Nasiriyah, Iraq Statistics Program, Department of Mathematics, Statistics, and Physics, College of Arts and Science, Qatar University, Doha 2713, Qatar Statistical Consulting Unit, College of Arts and Science, Qatar University, Doha 2713, Qatar


Introduction
A quality assurance system o ers a variety of management approaches that save time and money while producing a high-quality nal product. ese 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. e quality of the products is in uenced by unnatural variation. e 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 classi ed as memoryless and memory-type charts based on their design structure. Shewhart [1] introduced the rst 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. e 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 incontrol (IC) run-length (RL) distribution is the same for all continuous distributions. e 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. e 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. is is a research gap that needs to be explored. So this study proposes an NP DHWMA-SR control chart under RSS (NPDHWMA RSS ) for monitoring shifts in process location for continuous and symmetric distribution. e 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. e run-length (RL) characteristics of the proposed control chart, including average run length (ARL), median run length (MDRL), and standard deviation of run-length (SDRL), are obtained using various distributions like normal, student's t, contaminated normal (CN), Laplace, and logistic distributions. e performance of proposed NPDHWMA RSS control chart is decided by providing a valid comparison with other existing control charts such as DEWMA-X, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and NPHWMA RSS control charts. e 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.

Competing and Proposed Control Charts
is section explains the design structure of the competing and the proposed control charts. ese competing control charts are DEWMA-X, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and NPHWMA RSS . More detail is provided in the following subsections.

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. e plotting statistics of the NPRDEWMA-SR control chart are given as follows: where λ ∈ (0, 1] is a smoothing constant. e control limits of the NPRDEWMA-SR control chart can be designed as follows: 2 Mathematical Problems in Engineering e process goes OOC when DE (SR RSS )t > UCL (NPRDEWMA − SR)t or DE (SR RSS )t < LCL (NPRDEWMA− SR)t ; otherwise, it will remain an IC state.

DHWMA Control Chart.
Abid et al. [22] developed a DHWMA control chart that detects shifts more efficiently than the HWMA control chart. e DHWMA control chart plotting statistic is given as follows: where X t and X t− 1 are the mean of t th and mean of the previous t − 1 samples, respectively. e control limits of the DHWMA control chart based on this E(DH t ) and Var(DH t ) are defined as follows: e process remain is IC if LCL (DH WMA)t < DH t < UCL (DHWMA)t ; otherwise, it goes OOC.

Proposed NPDHWMA RSS Control Chart.
Various researchers like Kim and Kim [23], Abid et al. [24], and Abbas et al. [4] used the following RSS-based Wilcoxon signedrank statistic to monitor shifts in process location: where θ 0 symbolizes the process median, and t, j, and h denote the number of samples, observations, and cycles used in the RSS approach, respectively. e mean and variance of SR RSS t statistic are E(SR RSS t ) � 0 and Var(SR RSS t ) � (r(r + 1)(2r + 1)/6)ϖ 2 0 , respectively, where r denotes the number of replications and can be defined as r � nm. e quantity ϖ 2 0 is used to improve the efficiency of the control chart and can be defined as can be obtained by solving the following mathematical expression: Abid et al. [24] have more information on the RSS approach and related terms. e methodology of the proposed NPDHWMA RSS control chart is defined as follows: where SR (RSS)t− 1 is the mean of SR (RSS) of t − 1 samples. e simplified form of the plotting statistic NP DH t is e control limits of the proposed NPDHWMA RSS control chart are Mathematical Problems in Engineering the underlying process is OOC; else, it is IC.

Implementation of the Proposed Control Chart
is section investigates performance metrics, the IC, and the OOC performances of the proposed NPDHWMA RSS control chart for monitoring shifts in process location. Subsection 3.1 provides the performance metrics of the proposed NPDHWMA RSS control chart. Likewise, the proposed control chart's robustness, IC, and OOC performance are presented in Subsection 3.2.

Performance Metrics.
ARL is widely used to evaluate the performance of the control chart. e ARL is the expected number of sample points before the first OOC signal from the control chart. e ARL is categorized as IC ARL (ARL 0 ) and out-of-control ARL (ARL 1 ). If a process is functioning in an in-control state, the ARL 0 needed to be large enough to avoid frequent false alarms. However, the ARL 1 should be small enough; it quickly detects the shift. It is necessary for better performance of a control chart; it should have a smaller ARL 1 as compared to other control charts at the fixed value of ARL 0 . In this study, we set ARL 0 to 370 and 500, with sample sizes (n) of 5 and 10. Monte Carlo simulations with 50,000 simulations in the R program are used to determine the RL characteristics. To examine the performance behavior of the proposed NPDHWMA RSS control chart, various values of λ ∈ (0.05, 0.10, 0.25, 0.50) and δ ∈ (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. e following algorithm is used for simulations: (i) To create samples from considered distributions, a finite loop is used. (ii) Specify the process parameters (λ and L).
(iii) Draw a sample from a distribution used in this study. (iv) Determine the NPDH t plotting statistics using equation (7). (v) Find LCL (NPDHWMA RSS )t and UCL (NPDHWMA RSS )t from equation (8).

Robustness, IC, and OOC Performances of the NPDHWMA RSS Control Chart.
is subsection highlights the proposed NPDHWMA RSS control chart's robustness, IC, and OOC behavior when a process location is shifted. Table 1 shows the RL characteristics of the proposed NPDHWMA RSS control chart for location shift. ese characteristics are assessed using normal and non-normal continuous symmetric distributions. e distributions used for this study are standard normal distribution, that is, N(0, 1); double exponential or Laplace distribution, that is, )); heavy tail student's t distribution, that is, t(υ); logistic distribution, that is, (0, ( � 3 √ /π)); and contaminated normal (CN) distribution, which is the mixture of N(0, σ 2 0 ) and N(0, σ 2 1 ). 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. e ARL 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) e proposed control chart's IC RL distribution looks remarkably similar for all distributions examined in this study. For example, at λ � (0.05, 0.10, 0.25, 0.50) and n � 5, 10, the ARL 0 � 370, 500 for all investigated distributions (see Table 1). (ii) As the smoothing parameter reduces, the proposed control chart becomes more effective in detecting shifts.
is illustrates that the proposed NPDHWMA RSS control chart is more sensitive to small smoothing parameters (see Figure 1). (iii) As the sample size increases, the proposed NPDHWMA RSS control chart's shift detection ability for process location improves (see Figure 2). (iv) e Laplace distribution outperforms the other distributions in terms of OOC RL performance (see  Table 1). (vi) e ARL 1 of the proposed NPDHWMA RSS control chart are smaller than those in the competing control charts with different shift sizes in process location (see Figure 4). (vii) e distribution of RL values is positively skewed, that is, ARL > MRDL (see Table 1).

Comparative Study
is section provides a comparative performance study in terms of the ARL values of the proposed NPDHWMA RSS control chart for process location shifts. e proposed NPDHWMA RSS control chart is compared to the competing control charts, including DEWMA-X, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and NPHWMA RSS .

Proposed versus NPDEWMA-SR Control Chart.
e proposed NPDHWMA RSS control chart is more sensitive than the NPDEWMA-SR control chart for all combinations of δ and λ. For instance, in case of logistic distribution, when n � 10, λ � 0.05, and δ � 0.025, 0.05, 0.075, 0.10, 0. 25 Tables 1 and 3). e supremacy of the proposed NPDHWMA RSS control chart to the NPDEWMA-SR can also be seen in Figure 4. Likewise, in the scenario of t (8) distribution comparison, we noted the same behavior of the proposed NPDHWMA RSS control chart. As an illustration, at λ � 0.05, n � 10, and δ � 0.05, 0.10, 0.50, the ARL 1 values of the proposed NPDHWMA RSS control chart are 14.81, 6.19, 1.09, respectively, while the ARL 1 values of the NPDEWMA-SR control chart are 123.93, 42.69, 3.05 (see Tables 1 and 3).

Proposed versus NPRDEWMA-SR Control Chart.
e proposed NPDHWMA RSS control chart performs better than the NPRDEWMA-SR control chart. For instance, using the t (4) distribution with n � 10, λ � 0.05, and δ � 0.05, 0.50, the ARL 1 values of the proposed NPDHWMA RSS and the NPRDEWMA-SR control charts are 16.39, 1.14, and 49.00, 1.28, respectively (see Tables 1 and  4). e proposed NPDHWMA RSS charts' efficiency may also be observed in the case of CN distribution. For example, at  Figure 4 and Tables 1 and 4).

Proposed versus DHWMA Control Chart.
e ARL study reveals that the proposed NPDHWMA RSS control chart outperforms the DHWMA control chart for all combinations of η and δ (see Tables 1 and 5). As an illustration, for normal distribution, at λ � 0.05, n � 10, and δ � 0.075, 0.10, 0.25, the ARL 1 values of the NPDHWMA RSS and DHWMA control charts are 7.96, 5.65, 1.94 and 21.24, 14.81, 4.63, respectively (see Tables 1 and 5). Figure 4 also shows the superiority of the NPDHWMA RSS control chart over the DHWMA control chart. e results show that the NPDHWMA RSS control chart is superior to the DHWMA control chart for monitoring process location shifts.

Proposed versus NPHWMA RSS Control
Chart. e findings show that the proposed NPDHWMA RSS control chart is better than the NPHWMA RSS control chart in terms of OOC performance. For example, when we examine the t (8)

Illustrative Example
e 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. is section provides a real-life application of the nonisothermal continuous stirred tank reactor (CSTR) process to demonstrate the applicability of the proposed NPDHWMA RSS control chart. is 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. e CSTR process has nine different variables, one of which we choose as the variable of interest (X) represents the output temperature. e data initially consists of 1,000 observations, with the first 600 occurring when the process was in an IC condition. e phase I sample's parameters are as follows: μ Y � 368.2328, μ X � 369.8789, σ 2 Y � 0.2185915, σ 2 X � 0.3180327, and ρ YX � 0.08974039. We used the RSS approach to generate 40 paired observations of size n � 5 from a normal distribution. After the 24th sample, a shift in the process location is introduced following Anwar et al. [28]. e parameters of the proposed and NPRDHWMA-SR control charts used for real-life analysis are L � 1.535, λ � 0.10, ARL 0 � 500, and L � 2.117, λ � 0.10, ARL 0 � 500, respectively. Figure 5 indicates that the proposed NPDHWMA RSS 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 NPDHWMA RSS control chart detects overall 16 OOC points, whereas the NPRDEWMA-SR control chart detects 12 OOC points (see Table 7 and Figure 5).

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 NPDHWMA RSS ) for enhanced monitoring of process location shifts. e performance of the proposed control chart is investigated in terms of ARL, MDRL, and SDRL. e results revealed that the proposed control chart performs better than the competing control charts such as DEWMA-X, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and NPHWMA RSS . Moreover, a real-life application is also offered to show the proposed control chart's applicability in practice. is study is carried out where the process variable follows the univariate distributions. However, the proposed NPDHWMA RSS 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.