An Efficient Robust Nonparametric Triple EWMA Wilcoxon Signed-Rank Control Chart for Process Location

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 School of Statistics, Shanxi University of Finance and Economics, Taiyuan, China Department of Mathematics, College of Science, University of Hafr Al Batin, Hafr Al Batin, Saudi Arabia Department of Statistics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan Department of Statistics, Government College University Faisalabad, Faisalabad, Pakistan Statistics Program, Department of Mathematics, Statistics and Physics, College of Arts and Science, Qatar University, Doha 2713, Qatar


Introduction
Variations are an essential part of every manufacturing and service process, and these variations can be categorized as common and special cause variations. e common cause variations are harmless, and the mechanism or process that operates under these variations is called in-control (IC). However, the special cause variations affect the product quality, and the process that runs in these variations is referred to as out-of-control (OOC). Statistical process control (SPC) tool (e.g., cause-and-effect diagram, check sheet, control charts, histogram, Pareto chart, scatter diagram, and stratification) kit is famous for monitoring the shift (i.e., special cause variations) in the process parameters (location and/or dispersion). Compared to other SPC tools, the control charts got special attention because they are competent and easy to implement to detect the shift in the process parameters. Furthermore, the control charts can be classified into memoryless and memory-type depending on their design structures. Shewhart [1] introduced control charts known as the Shewhart control charts; these control charts are also named memoryless-type control charts. Shewhart control charts only use current information to detect a large shift quickly in the process parameters. Later on, Page [2] and Roberts [3] proposed cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts, respectively; these control charts are also known as memory-type. Memory-type control charts are famous for identifying a small-to-moderate shift in the process parameters.
Generally, for classical parametric control charts (i.e., Shewhart, CUSUM, and EWMA), the underlying process characteristics follow a normal distribution or any other specified probability distribution to identify a shift in the process parameters. Occasionally, there may be a situation when the underlying process characteristics do not follow any specific distribution or distribution of the underlying process characteristics is in doubt. In this case, nonparametric (NP) control charts are the robust substitute for parametric control charts. ese control charts are convenient because their IC run length (RL) distribution is similar for all continuous distributions. e sign (SN) and Wilcoxon signed-rank (SR) are well-known NP techniques. e SN technique only required the assumption of continuity, whereas the SR method required symmetry assumption as well [4]. Bakir and Reynolds [5] introduced an NP CUSUM control chart (NPCUSUM-SR) based on the signed-rank statistic for the process location. Similarly, Amin and Searcy [6] developed an NP EWMA-SR control chart by combining the SR technique with the EWMA control chart to monitor the process location shift effectively. In the same lines, Bakir [7] presented the SR-based Shewhart control chart. After that, Balakrishnan et al. [8] suggested an NP Shewhart control chart using runs rules to make it statistically sensitive. Correspondingly, Yang and Cheng [9] and Yang et al. [10] introduced CUSUM-SN and EWMA-SN control charts, respectively, for effective process location monitoring. In the same way, Graham et al. [11] presented the design structure of the single observation-based NP EWMA control chart. Furthermore, Chakraborty et al. [12] developed the GWMA-SR control chart to improve the EWMA-SR control chart's ability to detect small shifts. Later on, Raza et al. [13] introduced the double NP EWMA-SR based (DEWMA-SR) control chart to monitor shift and showed that it is more sensitive than the EWMA-SR control chart. Likewise, Malela-Majika [14] developed new distribution-free control charts based on the Wilcoxon rank-sum test for efficiently monitoring the process location. Also, He et al. [15] designed an NP multivariate control chart for the time between events and amplitude data. is control chart is used to track the time intervals' location shifts and the amplitudes of an event.
In SPC, sampling techniques such as simple random sampling (SRS) and perfect ranked set sampling (RSS) are well known and are commonly used with SN and SR techniques to observe the underlying process data. Furthermore, both SRS and RSS sampling techniques are widely used with parametric and NP control charts. RSS is a necessary and valuable statistical technique commonly used in statistical quality control when the precise measurement of a selected unit is either difficult or prohibitively expensive and time-consuming [16]. It is self-evident that RSS efficiency is dependent on the precision with which the randomly selected units are defined. Errors in the ranking have a negative impact on the estimator's efficiency and result in imprecise estimates. Dell and Clutter [17] investigated the effect of error and imperfect ranking on mean estimator performance. ey presented that even when imperfect rankings are used, the RSS mean estimator remains unbiased and outperforms the SRS mean estimator. However, the performance of the RSS remains superior to that of the imperfect RSS. Visual inspection of the study variable or based on the auxiliary variable may easily lead to ranking a small set of the selected units [18]. For example, hazardous waste sites with different contamination levels can be ranked by a visual inspection of soil discoloration, when the actual measurements of toxic chemicals and quantifying their environmental impact are very costly. In this regard, Salazar and Sinha [19] introduced RSS with control charts to monitor the process location shift. ey demonstrated that the RSS-based control charts outperformed against SRSbased control charts. Similarly, Muttlak and Al-Sabah [20] and Abujiya and Muttlak [21] presented an RSS-based (median ranked set sampling and extreme ranked set sampling) and double RSS (DRSS) based control charts, respectively, to monitor the shift in the process location. Later, Al-Omari and Haq [22] proposed a Shewhart-type control chart based on the DRSS technique.
Haq et al. [23] suggested a maximum EWMA (Max-EWMA) control chart based on order DRSS (ODRSS) and order imperfect DRSS (OIDRSS) sampling techniques. In addition, they [24] also developed a new synthetic control chart for process location and dispersion using different RSS techniques. Furthermore, based on RSS, Haq and Khoo [25] developed synthetic EWMA and synthetic CUSUM control charts to monitor the process location. Recently, Abbas et al. [26] introduced an NP DEWMA-SR control chart using the RSS technique for process location monitoring, labelled as DEWMA − SR RSS control chart.
Similarly, in the SPC, modifications and enhancements are continually practiced to enhance the performance of the memory-type control charts. For example, Shamma and Shamma [27] extended the classical EWMA control chart and suggested a double EWMA (DEWMA) control chart for the process location. e DEWMA control chart is more responsive than the classical EWMA control chart. Anwar et al. [28] and Aslam and Anwar [29] introduced modified-EWMA control charts, respectively, in the presence of auxiliary information and Bayesian methodology. Similarly, Anwar et al. [30] introduced auxiliary information-based (AIB) combined MEC for the simultaneous monitoring of process location and dispersion.
Recently, Chatterjee et al. [31] proposed a TEWMA control chart to monitor process dispersion shift. Later on, Alevizakos et al. [32] and Alevizakos et al. [4] suggested an NP TEWMA-SN (TEWMA-SN) and NP TEWMA-SR (TEWMA-SR) control charts, respectively. Furthermore, Alevizakos et al. [33] extended EWMA and DEWMA control charts and proposed a triple EWMA (TEWMA) control chart further to enhance the efficiency of the EWMA control chart. e TEWMA control chart is more responsive than the EWMA and DEWMA control charts to monitor small-to-moderate shifts in process location.

Mathematical Problems in Engineering
As discussed earlier, the SRS-based TEWMA-SN and TEWMA-SR control charts are well known for timely monitoring of the quality characteristics easily available. Still, when selecting quality characteristics is either difficult or expensive and time-consuming, these SRS-based control charts fail to monitor the process efficiently. To monitor such quality characteristics, this study suggested an NP triple EWMA-SR control chart with the RSS technique (TEWMA − SR RSS ) for effectively monitoring the process location of a continuous and symmetric distribution. Various processing environments like normal, Contaminated normal (CN), Laplace, Student's t, and Logistic are used to measure the RL characteristics of the proposed control chart. e average run length (ARL), the median of the run length (MDRL), and the standard deviation of the run length (SDRL) are assessed to determine the efficiency of the proposed control chart against other control charts. e sample points before the control chart signal are referred to as RL, and the RL's expected values are referred to as ARL. ARL 0 is the ARL of the IC process while ARL 1 is the ARL of the OOC process. A control chart is considered to be efficient when it has smaller ARL 1 values as compared to its contestant at different shifts. e remainder of the article is as follows: Section 2 presents the design structures of existing and the proposed control charts. Similarly, Section 3 offers the implementation of the proposed control chart. Also, the comparative study is given in Section 4, while Section 5 provides an illustrative example related to the proposed control chart. Finally, Section 6 offers the summary, conclusions, and recommendations of the study.

Existing and Proposed Methods
is section explains the design structures of some existing and the proposed control charts. For example, Section 2.1 presents the RSS-based Wilcoxon signed-rank statistic. Similarly, methodologies of the NP EWMA SR with RSS (EWMA − SR RSS ) and DEWMA − SR RSS control charts are given in Sections 2.2 and 2.3, respectively. Finally, the design structure of the proposed TEWMA − SR RSS control chart is provided in Section 2.4.

EWMA − SR RSS Control Chart.
e EWMA − SR RSS control chart is more sensitive for detecting shifts in the process location/median than the NPEWMA-SR control chart. e plotting statistic of the EWMA − SR RSS control chart based on equation (1) is defined as where λ ∈ (0, 1] is a smoothing constant and the initial value of the E q is equal to 0. e mean and variance of the statistic E q are E(E q ) � 0 and var(E q ) � (λ/(2 − λ))(1 − (1 − λ) 2q ) (r(r + 1)(2r + 1)/6)ω 2 0 , respectively. e variance of E q is reduced to (λ/(2 − λ))(r(r + 1)(2r + 1)/6)ω 2 0 , when the term (1 − (1 − λ) 2q ) approaches to unity. e control limits of the EWMA − SR RSS control chart (see Table 1) are defined below: where L is the width of the control limits, which helps to determine ARL 0 . e process will be OOC if any E q > UCL EWMA− SR RSS or E q < LCL EWMA− SR RSS ; otherwise, it will remain in an IC state.

DEWMA − SR RSS Control Chart.
Abbas et al. [26] introduced the following plotting statistics of the DEWMA − SR RSS control chart: Mathematical Problems in Engineering 3 e mean and variance of DE q (DEWMA − SR RSS control chart) for IC process are E(DE q ) � 0 and var(DE q ) � ((r(r + 1)(2r + 1)/6)ω 2 0 )λ 4 ((1 + λ 2 − (q 2 + 2q + 1)λ 2q + (2 e control limits of the DEWMA − SR RSS control chart can be designed as where L is the control chart width and the initial value of the DE q � 0. e process will be OOC when LCL DEWMA− SR RSS > DE q or UCL DEWMA− SR RSS < DE q ; otherwise, it will remain in an IC state.

Proposed TEWMA − SR RSS Control Chart.
is section offers a proposed control chart structure. In more detail, the proposed TEWMA − SR RSS control chart is designed for monitoring the process location shift. e plotting statistics of the proposed TEWMA − SR RSS control chart are defined as e initial values of E 0 , DE 0 , and TE 0 are all equal to 0 (zero). e mean and variance of the statistic TE q under IC state are E(TE q ) � 0 and respectively, where θ � (1 − λ) 2 . e time-varying control limits of the proposed TEWMA − SR RSS control chart are specified as where L is the width of the control limits. For large values of q, we have var(TE q ) � [(6(1 − λ) 6 ](n(n + 1)(2n + 1)/6)ω 2 0 and the asymptotic control limits become e process goes OOC if any TE q > UCL TEWMA− SR RSS or TE q < LCL TEWMA− SR RSS . However, if LCL TEWMA− SR RSS < TE q < UCL TEWMA− SR RSS , the process will be in IC state.

Implementation of the Proposed TEWMA − SR RSS Control Chart
is section provides the implementation of the proposed TEWMA − SR RSS control chart. Section 3.1 describes the choice of design parameters. Similarly, Section 3.2 presents the evaluation of the proposed control chart. e control chart width L is chosen with various values of λ, such that the prespecified ARL 0 is obtained. e best subgroup size is between 5 and 10, depending on ARL 0 and shift size [5]. Different RL properties are assessed to study the performance behavior of the proposed TEWMA − SR RSS control chart with λ ∈ (0.05, 0.10, 0.25, 0.50) and n ∈ (5, 10). ARL 0 is assumed to be 370 for this study.

Evaluation.
is section provides the RL characteristics of the different distributions (normal and non-normal distributions) for IC robustness and IC performance of the proposed control chart. e proposed TEWMA − SR RSS control chart's RL features are evaluated using normal and non-normal distributions.

Evaluation of the Run-Length Distribution.
e performance of the proposed control chart can be assessed in both normal and non-normal symmetric distributions. e distributions used in this analysis are as follows: (i) Standard normal distribution, i.e., N(0, 1) (ii) Double exponential (Laplace) distribution, i.e., DE(0, (1/ � 2 √ )) (iii) Heavy tail Student's t distribution t v , with the degree of freedom (v) � 4 and 8 (iv) e logistic distribution, i.e., LOG(0, ( For comparison purpose, all the considered distributions in this analysis were reparametrized with mean zero and unit standard deviation. e probability density functions of the distributions used in this study are shown in Table 2.

Monte Carlo Simulation.
e Monte Carlo simulation is used to obtain the RL characteristics of the proposed control chart. e simulation algorithm is developed in R software to compute the RL characteristics. 10 [4]. e simulated results of the proposed control chart with different combinations of λ and L to obtain the nominal ARL 0 � 370 are shown in Tables 3 and 4. e IC RL distribution of the proposed TEWMA − SR RSS control chart seems to be similar for all distributions used in this study.
e RL distribution is positively skewed when ARL > MDRL.

Out-of-Control
Performance. OOC performance of the proposed control chart can be discussed under perfect RSS and imperfect RSS. e OOC control chart's efficiency is primarily compared, and it demonstrates how sensitive the control chart is to detect shifts in process parameters. Tables 3 and 4 show the OOC performance of the proposed TEWMA − SR RSS control chart for the various distributions evaluated in this study. For small and moderate shifts, the OOC performance of the proposed TEWMA − SR RSS control chart increases as n increases; i.e., ARL 1 values decrease by increasing n. e proposed scheme's RL values reduce with the rise in n and m values. e OOC RL performance of the Mathematical Problems in Engineering Laplace distribution is higher as compared to the other distributions used in this study (see Figures 1 and 2). e proposed control chart's ability to detect shifts deteriorates when it occurs later for small values of λ (λ � 0.05 or λ � 0.10), whereas this trend is noticed for large values of λ (λ � 0.25 or λ � 0.50) for large shifts (see Figures 3 and 4). e proposed control chart's OOC performance under imperfect RSS is also determined to illustrate the superiority of perfect RSS or generally RSS over imperfect RSS (see Tables  5-7). For example, under normal distribution, at λ � 0.05, n � 5, and δ � 0.025, 0.05, 0.075, and 0. 10 Tables 3 and 5). In general, the proposed control chart under RSS is very sensitive to detect small and moderate shifts in the process location.
In terms of OOC performance at a specific shift (δ), the proposed TEWMA − SR RSS control chart outperforms all other control charts. For instance, at δ � 0.05, λ � 0.25, and n � 5, the ARL 1 values of the proposed TEWMA − SR RSS , TEWMA-SR, TEWMA-SN, TEWMA − X, and DEWMA − SR RSS control charts are 151.84, 251.08, 341.92, 337.05, and 167.12, respectively. e ARL is used as a metric to compare the proposed and existing control charts. Some findings are presented in the percentage decreasing in ARL � ((ARL 0 − ARL 1 )/ARL 0 ) × 100%. Under normal setup, the TEWMA − SR RSS control chart displaying a 3 percent improvement in the process median minimizes the ARL by 54.56 percent for n � 5, λ � 0.05 , and ARL 0 � 370, whereas at the same shift (δ), the proposed control chart reduced the ARL by 80.55 percent for n � 10, λ � 0.05 , and ARL 0 � 370 (see Tables 3  and 4).

Comparative Study
is section provides the proposed control chart's comparisons to several existing control charts, which includes DEWMA − SR RSS [26], TEWMA − X [33], TEWMA-SN [32], and TEWMA-SR [4] control charts. More detail is provided in the following sections.

Proposed versus TEWMA-SR Control Chart.
e proposed TEWMA − SR RSS control chart outperforms the TEWMA-SR control chart. For instance, at n � 5 and λ � 0.05 with δ � 5 percent shift in the Logistic distribution, the proposed TEWMA − SR RSS control chart reduces the ARL 1 value by almost 78.28 percent, whereas the TEWMA-SR control chart reduces the ARL 1 value by 58.74 percent (see Table 8). Similarly, under t (4) distribution, when n � 10 and λ � 0.10 along with δ � 5 percent, the ARL 1 values of the proposed TEWMA − SR RSS control chart are reduced by 86.46 percent, while the TEWMA-SR control chart reduces the values by 72.30 percent (see Tables 4 and 9). Figures 5 and 6 highlight the proposed TEWMA − SR RSS control chart superiority over the TEWMA-SR control chart. It is worth mentioning that, for small shifts, the proposed TEWMA − SR RSS control chart outperforms the TEWMA-SR control chart for different combinations of λ and n.

Proposed versus TEWMA-SN Control Chart.
e RL features of the TEWMA-SN control chart for the process location are listed in Table 10.
e proposed TEWMA − SR RSS control chart is more efficient than the TEWMA-SN control chart. For example, under Laplace distribution, the proposed TEWMA − SR RSS control chart with n � 10 and λ � 0.05 with δ � 10 percent shows OOC situation almost after 7.75 observations, whereas the TEWMA-SN control chart gives OOC state nearly after 25.67 observations (see Tables 4 and 10). Similarly, in the t (8) distribution scenario, at n � 10 and λ � 0.05 with a shift of 5 percent, the TEWMA-SN control chart decreases the ARL 1  Table 8). Furthermore, at n � 5 and λ � 0.05 with 10 percent shift in process location under the Logistic distribution, the TEWMA-SN control chart reduces the ARL 1 by 79.95 percent and the proposed TEWMA − SR RSS control chart reduces the ARL 1 by 91.79 percent (see Table 8). e proposed TEWMA − SR RSS control chart efficiency over the TEWMA-SN control chart can be seen in Figure 6. In short, the proposed TEWMA − SR RSS control chart is quite sensitive under normal and non-normal distributions for detecting small to moderate shifts in process location against the TEWMA-SN control chart.

Proposed versus TEWMA − X Control Chart.
e TEWMA − SR RSS control chart outperforms the TEWMA − X control chart in terms of early shift detection ability. For instance, using the Laplace distribution, the TEWMA − X control chart with n � 5 and λ � 0.25 for δ � 5 percent gives OOC signal after 238.73 observations, whereas the proposed TEWMA − SR RSS control chart shows the OOC signal after 105.76 observations (see Table 8).
e superiority of the proposed control chart can also be seen in the case of outliers under CN distribution. For example, the proposed TEWMA − SR RSS control chart with n � 10 and λ � 0.10 for δ � 10 percent produces OOC signal after 14.61 observations, while the TEWMA − X control chart shows OOC signal after 43.39 observations (see Tables 4 and 11). e performance of the proposed control chart over the TEWMA − X control chart is depicted in Figures 7 and 8. In brief, the proposed TEWMA − SR RSS control chart is quite sensitive for detecting process location shifts relative to the TEWMA − X control chart.

Proposed versus DEWMA − SR RSS Control
Chart. e proposed TEWMA − SR RSS control chart provides better performance than the DEWMA − SR RSS control chart. For example, under a normal environment, at n � 10 and λ � 0.50 for δ � 2.5 percent, the proposed TEWMA − SR RSS control chart reduces the ARL 1 by 44.92 percent, while the DEWMA − SR RSS control chart reduces the ARL 1 by 39.99 percent (see Table 8). Similarly, when we examine the Logistic distribution at n � 5 and λ � 0.25 for δ � 3 percent, the proposed TEWMA − SR RSS control chart decreases the ARL 1 by 34.59 percent and the DEWMA − SR RSS control

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Mathematical Problems in Engineering  chart decreases the ARL 1 by 30.57 percent (see Tables 3 and  12). Furthermore, under a CN environment, at n � 10 and λ � 0.50 with δ � 7.5 percent, the proposed TEWMA − SR RSS control chart gives OOC signal after 45.28 observations, whereas the TEWMA − SR RSS control chart does after 50.84 observations (see Tables 4 and 13). e comprehensive RL features of the DEWMA − SR RSS control chart for process location shift under selected distributions are shown in Tables 12 and 13. e TEWMA − SR RSS control chart outperforms the DEWMA − SR RSS control chart, as shown in Figures 9 and 10. Results indicate that the proposed structure works effectively in all environments relative to the DEWMA − SR RSS control chart.

Illustrative Example
is section demonstrates practical application to implement the proposed TEWMA − SR RSS control chart in practice. For this purpose, the proposed TEWMA − SR RSS with TEWMA-SR and DEWMA − SR RSS control charts are considered to monitor the process location shift. For the practical implementation of the proposed TEWMA − SR RSS control chart, we assumed the data set used by Muttlak and Al-Sabah [20]. ese data were also used by Abid et al. [37] for the execution of the EWMA − SR RSS control chart. e collected data were about the Pepsi Cola Company in AL-Khobar, Saudi Arabia. e company uses different production lines, like a line used for filling soft drinks in bottles is referred to as an interest production and the volume of drink in the bottle is referred to as a variable of interest. ey used the RSS method to hold the appropriate volume of soft drinks in bottles during the filling process. e RSS technique is used to collect 27 ranked set samples by repeating the two cycles four times, with each cycle having a size of n � 3. To compare the proposed TEWMA − SR RSS control chart with TEWMA-SR and DEWMA − SR RSS control charts, 27 samples with r � 12 are collected using RSS. e design parameters (L, λ) of the DEWMA − SR RSS , TEWMA-SR, and TEWMA − SR RSS are (1.578, 0.05), (1.75, 0.05), and (1.305, 0.05), respectively, at ARL 0 � 370. Comparison of the proposed TEWMA − SR RSS control chart with the TEWMA-SR and DEWMA − SR RSS control charts can be visualized in Figures 11-13. e DEWMA − SR RSS control chart shows 4 OOC signals (at sample numbers [24][25][26][27], whereas the TEWMA-SR control chart shows no OOC signals. In contrast, the proposed TEWMA − SR RSS control chart shows 7 OOC signals (at sample numbers 21-27) in the process location parameter. ese findings indicate that the proposed TEWMA − SR RSS has improved performance for the monitoring of the process location as compared to DEWMA − SR RSS and TEWMA-SR control charts. Hence, the proposed TEWMA − SR RSS control chart under RSS has a better ability to detect shifts in process location.

Summary, Conclusions, and Recommendations
e ranked set sampling (RSS) technique is preferred over the simple random sampling (SRS) for the processes when the estimations are destructive or costly, and the ranking of      observational data is comparatively simple. Similarly, the nonparametric (NP) control charts are very useful to monitor shifts in the process parameters when the distribution of the underlying process is questionable or unknown. So, this study proposes an NP triple exponentially weighted moving average (TEWMA) Wilcoxon signed-rank (SR) control chart under the RSS technique (represented as TEWMA − SR RSS control chart) using different continuous symmetric distributions.
e Monte Carlo simulation method is used to obtain the numerical results of the proposed TEWMA − SR RSS control chart along with other existing control charts. e performance of TEWMA − SR RSS control chart is substantially better than the SRS-based NP TEWMA-SR, TEWMA sign, TEWMA − X, and double EWMA-SR (DEWMA − SR RSS ) control charts. A real-life example is also provided to demonstrate how the proposed control chart can be used in practice. erefore, to adopt a robust and efficient control chart, the proposed control chart gives an alternate choice to quality practitioners. e proposed work can be extended to the multivariate scenario.