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

Authors: Zahid Rasheed; Hongying Zhang; Muhammad Arslan; Babar Zaman; Syed Masroor Anwar; Muhammad Abid; Saddam Akber Abbasi
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

The nonparametric (NP) control charts are famous for detecting a shift in the process parameters (location and/or dispersion) when the underlying process characteristic does not follow the distributional assumptions. Similarly, when the cost of estimations is very high and the ranking of observational is relatively simple, the ranked set sampling (RSS) technique is preferred over the simple random sampling (SRS) technique. On the other hand, the NP triple exponentially weighted moving average (EWMA) control chart based on SRS is superior to the NP EWMA and NP double EWMA (NP DEWMA) based on the SRS technique to detect a shift in the process location. This study designed an advanced form of NP TEWMA Wilcoxon signed-rank based on RSS, denoted asTEWMA−SRRSS control chart to identify a shift in the process location parameter. The Monte Carlo simulation method is used to assess the performance of the proposed TEWMA−SRRSS control chart along with SRS-based NP TEWMA (TEWMA-SR), SRS-based NP TEWMA sign (TEWMA-SN), SRS-based TEWMA−X¯, and RSS-based NP DEWMA-SR DEWMA−SRRSS control charts. The study shows that the proposed TEWMA−SRRSS control chart is more efficient in identifying shifts (especially in small shifts) in the process location than the existing control charts. Finally, a real-life application is also provided for the practical implementation of the proposed TEWMA−SRRSS control chart.


---

## Body

## 1. Introduction

Variations are an essential part of every manufacturing and service process, and these variations can be categorized as common and special cause variations. The 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. These control charts are convenient because their IC run length (RL) distribution is similar for all continuous distributions. The sign (SN) and Wilcoxon signed-rank (SR) are well-known NP techniques. The 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. This 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. They 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. They demonstrated that the RSS-based control charts outperformed against SRS-based 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 (MaxEWMA) 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−SRRSS 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. The 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. The TEWMA control chart is more responsive than the EWMA and DEWMA control charts to monitor small-to-moderate shifts in process location.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 techniqueTEWMA−SRRSS 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. The 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. The sample points before the control chart signal are referred to as RL, and the RL’s expected values are referred to as ARL. ARL0 is the ARL of the IC process while ARL1 is the ARL of the OOC process. A control chart is considered to be efficient when it has smaller ARL1 values as compared to its contestant at different shifts.The remainder of the article is as follows: Section2 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.


## 2. Existing and Proposed Methods

This section explains the design structures of some existing and the proposed control charts. For example, Section2.1 presents the RSS-based Wilcoxon signed-rank statistic. Similarly, methodologies of the NP EWMA SR with RSS EWMA−SRRSS and DEWMA−SRRSS control charts are given in Sections 2.2 and 2.3, respectively. Finally, the design structure of the proposed TEWMA−SRRSS control chart is provided in Section 2.4.


### 2.1. RSS-Based Wilcoxon Signed-Rank Statistic

Mclntyre [16] gives the idea of the RSS technique for data collection. Suppose RSS of size n is taken from a quality variable Xqjh, where q=1,2,3,…,j=1,2,3,…,n, and h=1,2,3,…,m. q denotes each sample, j denotes the number of observations, and h denotes the number of cycles used in the RSS technique. r=nm is the number of iterations used in the RSS technique with a sample of size n and m number of cycles. Let Rqjh+=Xqjh−φ0, where Rqjh+ denote the ranks of absolute deviations of RSS values Xqjh from median φ0. According to Kim and Kim [34], the statistic of RSS under SR can be written as(1)SRRSSq=∑j=1n∑h=1msignXqjh−φ0Rqjh+,where(2)signXqjh−φ0=1,ifXqjh−φ0>0,0,ifXqjh−φ0=0,−1,ifXqjh−φ0<0.The mean and variance of the statisticSRRSS are ESRRSS=0 and varSRRSS=rr+12r+1/6ω02, respectively [34]. The ω02 can be defined as follows:(3)ω02=1−4n∑j=1nFk0−122,where the values of Fk0 can be obtained by solving the following expression for j=1,2,…,n:(4)Fk0=r!j−1!r−j!∫−∝0Ftj−11−Ftr−jftdt,whereas ω02 is used to enhance the performance of the control chart. The values of ω02 at different sample sizes taken from [35] are given as n,ω02= (2, 0.750), (3, 0.625), (4, 0.547), (5, 0.490), (6, 0.451), (7, 0.416), (8, 0.393), (9, 0.371), and (10, 0.352).


### 2.2.EWMA−SRRSS Control Chart

TheEWMA−SRRSS control chart is more sensitive for detecting shifts in the process location/median than the NPEWMA-SR control chart. The plotting statistic of the EWMA−SRRSS control chart based on equation (1) is defined as(5)Eq=λSRRSSq+1−λEq−1,where λ∈0,1 is a smoothing constant and the initial value of the Eq is equal to 0. The mean and variance of the statistic Eq are EEq=0 and varEq=λ/2−λ1−1−λ2qrr+12r+1/6ω02, respectively. The variance of Eq is reduced to λ/2−λrr+12r+1/6ω02, when the term 1−1−λ2q approaches to unity. The control limits of the EWMA−SRRSS control chart (see Table 1) are defined below:(6)UCLEWMA−SRRSS=+Lλ2−λrr+12r+16ω02LCLEWMA−SRRSS=−Lλ2−λrr+12r+16ω02,where L is the width of the control limits, which helps to determine ARL0. The process will be OOC if any Eq>UCLEWMA−SRRSS or Eq<LCLEWMA−SRRSS; otherwise, it will remain in an IC state.


### 2.3.DEWMA−SRRSS Control Chart

Abbas et al. [26] introduced the following plotting statistics of the DEWMA−SRRSS control chart:(7)Eq=λSRRSSq+1−λEq−1DEq=λEq+1−λDEq−1,The mean and variance ofDEq (DEWMA−SRRSS control chart) for IC process are EDEq=0 and varDEq=rr+12r+1/6ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q2/1−λ23, respectively. The control limits of the DEWMA−SRRSS control chart can be designed as(8)LCLDEWMA−SRRSS=−Lrr+12r+16ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q21−λ23UCLDEWMA−SRRSS=+Lrr+12r+16ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q21−λ23,where L is the control chart width and the initial value of the DEq=0. The process will be OOC when LCLDEWMA−SRRSS>DEq or UCLDEWMA−SRRSS<DEq; otherwise, it will remain in an IC state.


### 2.4. ProposedTEWMA−SRRSS Control Chart

This section offers a proposed control chart structure. In more detail, the proposedTEWMA−SRRSS control chart is designed for monitoring the process location shift. The plotting statistics of the proposed TEWMA−SRRSS control chart are defined as(9)Eq=λSRRSSq+1−λEq−1DEq=λEq+1−λDEq−1TEq=λDEq+1−λTEq−1,The initial values ofE0,DE0,andTE0 are all equal to 0 (zero). The mean and variance of the statistic TEq under IC state are ETEq=0 and(10)varTEq=θ3λ64−qq2−1q−2θq−31−θ−4qq2−1θq−21−θ2−12qq+1θq−11−θ3−24q+1θq1−θ4+241−θq+11−θ5+2θ2λ6−qq2−1θq−21−θ−3qq+1θq−11−θ2−6q+1θq1−θ3+61−θq+11−θ4+7θλ62−qq+1θq−11−θ−2q+1θq1−θ2+21−θq+11−θ3+λ61−θq+11−θ2−q+1θq1−θnn+12n+16ω02,respectively, where θ=1−λ2. The time-varying control limits of the proposed TEWMA−SRRSS control chart are specified as(11)LCLTEWMA−SRRSSq=−LvarTEqUCLTEWMA−SRRSSq=+LvarTEq,where L is the width of the control limits. For large values of q, we have varTEq=61−λ6λ/2−λ5+121−λ4λ2/2−λ4+71−λ2λ3/2−λ3+λ4/2−λ2nn+12n+1/6ω02 and the asymptotic control limits become(12)LCLTEWMA−SRRSS=−Lσ61−λ6λ2−λ5+121−λ4λ22−λ4+71−λ2λ32−λ3+λ42−λ2nn+12n+16ω02UCLTEWMA−SRRSS=+Lσ61−λ6λ2−λ5+121−λ4λ22−λ4+71−λ2λ32−λ3+λ42−λ2nn+12n+16ω02,The process goes OOC if anyTEq>UCLTEWMA−SRRSSorTEq<LCLTEWMA−SRRSS. However, if LCLTEWMA−SRRSS<TEq<UCLTEWMA−SRRSS, the process will be in IC state.


## 2.1. RSS-Based Wilcoxon Signed-Rank Statistic

Mclntyre [16] gives the idea of the RSS technique for data collection. Suppose RSS of size n is taken from a quality variable Xqjh, where q=1,2,3,…,j=1,2,3,…,n, and h=1,2,3,…,m. q denotes each sample, j denotes the number of observations, and h denotes the number of cycles used in the RSS technique. r=nm is the number of iterations used in the RSS technique with a sample of size n and m number of cycles. Let Rqjh+=Xqjh−φ0, where Rqjh+ denote the ranks of absolute deviations of RSS values Xqjh from median φ0. According to Kim and Kim [34], the statistic of RSS under SR can be written as(1)SRRSSq=∑j=1n∑h=1msignXqjh−φ0Rqjh+,where(2)signXqjh−φ0=1,ifXqjh−φ0>0,0,ifXqjh−φ0=0,−1,ifXqjh−φ0<0.The mean and variance of the statisticSRRSS are ESRRSS=0 and varSRRSS=rr+12r+1/6ω02, respectively [34]. The ω02 can be defined as follows:(3)ω02=1−4n∑j=1nFk0−122,where the values of Fk0 can be obtained by solving the following expression for j=1,2,…,n:(4)Fk0=r!j−1!r−j!∫−∝0Ftj−11−Ftr−jftdt,whereas ω02 is used to enhance the performance of the control chart. The values of ω02 at different sample sizes taken from [35] are given as n,ω02= (2, 0.750), (3, 0.625), (4, 0.547), (5, 0.490), (6, 0.451), (7, 0.416), (8, 0.393), (9, 0.371), and (10, 0.352).


## 2.2.EWMA−SRRSS Control Chart

TheEWMA−SRRSS control chart is more sensitive for detecting shifts in the process location/median than the NPEWMA-SR control chart. The plotting statistic of the EWMA−SRRSS control chart based on equation (1) is defined as(5)Eq=λSRRSSq+1−λEq−1,where λ∈0,1 is a smoothing constant and the initial value of the Eq is equal to 0. The mean and variance of the statistic Eq are EEq=0 and varEq=λ/2−λ1−1−λ2qrr+12r+1/6ω02, respectively. The variance of Eq is reduced to λ/2−λrr+12r+1/6ω02, when the term 1−1−λ2q approaches to unity. The control limits of the EWMA−SRRSS control chart (see Table 1) are defined below:(6)UCLEWMA−SRRSS=+Lλ2−λrr+12r+16ω02LCLEWMA−SRRSS=−Lλ2−λrr+12r+16ω02,where L is the width of the control limits, which helps to determine ARL0. The process will be OOC if any Eq>UCLEWMA−SRRSS or Eq<LCLEWMA−SRRSS; otherwise, it will remain in an IC state.


## 2.3.DEWMA−SRRSS Control Chart

Abbas et al. [26] introduced the following plotting statistics of the DEWMA−SRRSS control chart:(7)Eq=λSRRSSq+1−λEq−1DEq=λEq+1−λDEq−1,The mean and variance ofDEq (DEWMA−SRRSS control chart) for IC process are EDEq=0 and varDEq=rr+12r+1/6ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q2/1−λ23, respectively. The control limits of the DEWMA−SRRSS control chart can be designed as(8)LCLDEWMA−SRRSS=−Lrr+12r+16ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q21−λ23UCLDEWMA−SRRSS=+Lrr+12r+16ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q21−λ23,where L is the control chart width and the initial value of the DEq=0. The process will be OOC when LCLDEWMA−SRRSS>DEq or UCLDEWMA−SRRSS<DEq; otherwise, it will remain in an IC state.


## 2.4. ProposedTEWMA−SRRSS Control Chart

This section offers a proposed control chart structure. In more detail, the proposedTEWMA−SRRSS control chart is designed for monitoring the process location shift. The plotting statistics of the proposed TEWMA−SRRSS control chart are defined as(9)Eq=λSRRSSq+1−λEq−1DEq=λEq+1−λDEq−1TEq=λDEq+1−λTEq−1,The initial values ofE0,DE0,andTE0 are all equal to 0 (zero). The mean and variance of the statistic TEq under IC state are ETEq=0 and(10)varTEq=θ3λ64−qq2−1q−2θq−31−θ−4qq2−1θq−21−θ2−12qq+1θq−11−θ3−24q+1θq1−θ4+241−θq+11−θ5+2θ2λ6−qq2−1θq−21−θ−3qq+1θq−11−θ2−6q+1θq1−θ3+61−θq+11−θ4+7θλ62−qq+1θq−11−θ−2q+1θq1−θ2+21−θq+11−θ3+λ61−θq+11−θ2−q+1θq1−θnn+12n+16ω02,respectively, where θ=1−λ2. The time-varying control limits of the proposed TEWMA−SRRSS control chart are specified as(11)LCLTEWMA−SRRSSq=−LvarTEqUCLTEWMA−SRRSSq=+LvarTEq,where L is the width of the control limits. For large values of q, we have varTEq=61−λ6λ/2−λ5+121−λ4λ2/2−λ4+71−λ2λ3/2−λ3+λ4/2−λ2nn+12n+1/6ω02 and the asymptotic control limits become(12)LCLTEWMA−SRRSS=−Lσ61−λ6λ2−λ5+121−λ4λ22−λ4+71−λ2λ32−λ3+λ42−λ2nn+12n+16ω02UCLTEWMA−SRRSS=+Lσ61−λ6λ2−λ5+121−λ4λ22−λ4+71−λ2λ32−λ3+λ42−λ2nn+12n+16ω02,The process goes OOC if anyTEq>UCLTEWMA−SRRSSorTEq<LCLTEWMA−SRRSS. However, if LCLTEWMA−SRRSS<TEq<UCLTEWMA−SRRSS, the process will be in IC state.


## 3. Implementation of the ProposedTEWMA−SRRSS Control Chart

This section provides the implementation of the proposedTEWMA−SRRSS control chart. Section 3.1 describes the choice of design parameters. Similarly, Section 3.2 presents the evaluation of the proposed control chart.


### 3.1. Choice of Design Parameters

λ and L are the design parameters of the proposed TEWMA−SRRSS control chart. The control chart width L is chosen with various values of λ, such that the prespecified ARL0 is obtained. The best subgroup size is between 5 and 10, depending on ARL0 and shift size [5]. Different RL properties are assessed to study the performance behavior of the proposed TEWMA−SRRSS control chart with λ∈0.05,0.10,0.25,0.50 and n∈5,10. ARL0 is assumed to be 370 for this study.


### 3.2. Evaluation

This section provides theRL characteristics of the different distributions (normal and non-normal distributions) for IC robustness and IC performance of the proposed control chart. The proposed TEWMA−SRRSS control chart’s RL features are evaluated using normal and non-normal distributions.


#### 3.2.1. Evaluation of the Run-Length Distribution

The performance of the proposed control chart can be assessed in both normal and non-normal symmetric distributions. The distributions used in this analysis are as follows:(i)
Standard normal distribution, i.e.,N0,1(ii)
Double exponential (Laplace) distribution, i.e.,DE0,1/2(iii)
Heavy tail Student’st distribution tv, with the degree of freedom v=4and8(iv)
The logistic distribution, i.e.,LOG0,3/π(v)
Contaminated normal (CN) distribution, which is a mixture ofN0,σ12 and N0,σ02, i.e., 1−αN0,σ12+αN0,σ02, with α such that 1−ασ12+ασ02=1, where α=0.05 and σ1/σ0=2. For comparison purpose, all the considered distributions in this analysis were reparametrized with mean zero and unit standard deviation. The probability density functions of the distributions used in this study are shown in Table 2.Table 1
Details about the control charts used in this study.
Control chart’s nameNotationPlotting statisticsControl limits1NP EWMA under RSSEWMA−SRRSSEq=λSRRSSq+1−λEq−1UCLEWMA−SRRSS=+Lλ/2−λrr+12r+1/6ω02LCLEWMA−SRRSS=−Lλ/2−λrr+12r+1/6ω022NP DEWMA under RSSDEWMA−SRRSSEq=λSRRSSq+1−λEq−1,DEq=λEq+1−λDEq−1LCLDEWMA−SRRSS=−Lrr+12r+1/6ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q2/1−λ23UCLDEWMA−SRRSS=+Lrr+12r+1/6ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q2/1−λ233NP TEWMA under RSSTEWMA−SRRSSEq=λSRRSSq+1−λEq−1,DEq=λEq+1−λDEq−1,TEq=λDEq+1−λTEq−1LCLTEWMA−SRRSS=−Lσ61−λ6λ/2−λ5+121−λ4λ2/2−λ4+71−λ2λ3/2−λ3+λ4/2−λ2nn+12n+1/6ω02UCLTEWMA−SRRSS=+Lσ61−λ6λ/2−λ5+121−λ4λ2/2−λ4+71−λ2λ3/2−λ3+λ4/2−λ2nn+12n+1/6ω02


#### 3.2.2. Monte Carlo Simulation

The Monte Carlo simulation is used to obtain theRL characteristics of the proposed control chart. The simulation algorithm is developed in R software to compute the RL characteristics. 104 random samples of the size n>1 from any distribution covered by the study for a shift δ were generated. The simulation algorithm to determine the nominal values of ARL0 under various distributions can be explained in the following steps [36]:(a)
An finite loop is practiced to generate samples from different distributions(b)
Select a particular value forλ(c)
Determine theTEq plotting statistic offered in equation (9)(d)
Let us assumeL=3(e)
Calculate theUCLTEWMA−SRRSSandLCLTEWMA−SRRSS from equation (11)(f)
Plot theTEq statistic against control limits in step (e) over q(g)
If theTEq>UCLTEWMA−SRRSSorTEq<LCLTEWMA−SRRSS, record the sample number of TEq statistic as an RL. For example, at q=255, if TEq>UCLTEWMA−SRRSSorTEq<LCLTEWMA−SRRSS record 255 as a first RL.(h)
Repeat steps from (a) to (g) for 105 times and record RLs.(i)
Compute the average of105 times noted RLs, which is called ARL0.(j)
IfARL0=400; otherwise, adjust L (i.e., L<3orL>3) constant accordingly in step (d) and repeat steps from (a) to (i) to obtain ARL0=400.(k)
To compute theARL1 values, again generate samples from relevant distribution and repeat steps from (b) to (i).


#### 3.2.3. In-Control Robustness of the Proposed Control Chart

The EWMA andDEWMA−X¯ control charts have the limitation that their IC RL features do not remain the same for all continuous distributions. The IC RL characteristics of the NP control chart’s characteristics remain the same for all symmetric continuous distributions [4]. The simulated results of the proposed control chart with different combinations of λandL to obtain the nominal ARL0=370 are shown in Tables 3 and 4. The IC RL distribution of the proposed TEWMA−SRRSS control chart seems to be similar for all distributions used in this study. The RL distribution is positively skewed when ARL>MDRL.Table 2
PDFs of the continuous distributions used for this study.
(i)Standard normalfX=e−x2/2/2π, where X∈ℜ,M0=0,andσ2=1(ii)Student’stυfX=Γυ+1/2/Γυ/2υπ1+X2/υυ+1/2, where X∈ℜ,M0=0,andσ2=υ/υ−2 and υ=4,8 are taken(iii)LogisticfX=e−πX/3/3/π1+e−πX/32, where X∈ℜ,M0=0,andσ2=3/π2(iv)LaplacefX=1/2e−X, where X∈ℜ,M0=0,andσ2=1/2(v)Contaminated normal (CN)fX=0.95e−X2/2/2π+0.05e−X2/2σ02/σ02π, where X∈ℜ,M0=0,andσ2=0.95+0.05σ02Table 3
RL characteristics of the proposedTEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=5.
δλ,L0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.585)N(0, 1)ARL371.11206.77168.6292.6750.5333.9119.1212.027.942.351.371.081.011.00MDRL218.00131.00111.0070.0041.0028.0017.009.007.002.001.001.001.001.00SDRL453.10234.01183.7090.3349.0629.1515.7210.546.621.770.750.310.080.00CNARL369.92212.31164.0494.3151.9335.6319.3712.068.352.511.461.141.011.00MDRL217.50139.00113.0065.0040.0027.5016.009.006.002.001.001.001.001.00SDRL441.07235.19179.2897.3550.1533.3516.1610.777.502.080.870.430.130.04LogisticARL371.69189.29156.5180.7249.3730.4916.2110.277.202.181.311.101.011.00MDRL230.50118.00112.0062.0041.0025.0013.008.006.001.001.001.001.001.00SDRL429.72216.77159.5083.6344.7026.4214.268.946.391.610.660.330.110.03LaplaceARL371.65158.45123.5963.4934.1822.5412.257.525.482.011.321.121.031.00MDRL237.00106.5087.0050.0028.0018.0010.005.004.001.001.001.001.001.00SDRL428.92173.36126.8360.3330.9220.2910.726.894.851.500.660.380.170.06t4ARL369.29215.22187.04111.3462.0041.6523.5515.3910.433.341.791.361.091.02MDRL232.50126.50123.0083.0047.0033.0020.0012.008.002.001.001.001.001.00SDRL423.91254.75210.06111.4059.0239.9120.4512.909.582.791.310.700.340.17t8ARL370.51202.75174.3799.3658.9237.6420.0013.338.892.801.581.201.031.01MDRL231.00128.00118.0070.0045.0033.0015.0010.007.002.001.001.001.001.00SDRL455.11247.02189.65102.2255.9732.3117.5011.807.862.401.030.540.170.12(0.10, 1.837)N(0, 1)ARL370.58239.49200.36107.4457.6038.6621.5813.269.193.091.751.281.031.00MDRL241.50164.00143.0075.5044.0032.0019.0012.008.002.001.001.001.001.00SDRL399.98252.12200.34104.0952.2931.4716.689.586.522.100.970.540.180.03CNARL371.69222.24189.81110.7164.1940.5821.0213.659.673.281.791.321.051.01MDRL240.50155.00136.5075.0048.0034.0018.0012.008.003.002.001.001.001.00SDRL400.58233.16189.80111.1460.2732.0916.179.936.982.331.020.600.220.08LogisticARL369.89220.35179.1597.8552.1735.7718.2611.258.242.731.671.251.041.01MDRL257.50150.00124.0072.0040.0029.0016.0010.007.002.001.001.001.001.00SDRL379.94229.75177.9490.9947.5729.9613.908.425.941.800.890.520.200.07LaplaceARL369.00179.37145.5972.1539.0025.6214.368.836.402.421.521.271.051.01MDRL257.50117.50103.0054.0031.0022.0013.008.005.002.001.001.001.001.00SDRL379.94191.51141.8165.4633.7819.3910.666.284.771.580.830.560.230.10t4ARL369.26257.35229.58142.1981.2548.9026.1116.6812.293.982.271.601.161.04MDRL239.00166.50154.00102.0059.5038.0023.0014.0011.003.002.001.001.001.00SDRL382.67289.42240.47134.4775.2542.7319.5612.468.782.781.430.900.430.22t8ARL370.13241.63203.62125.9067.9944.7622.6315.9210.573.421.921.381.061.01MDRL250.00169.50139.0089.5052.0037.0020.0014.009.003.002.001.001.001.00SDRL417.20260.53202.72129.4859.3836.4417.1311.117.522.421.140.620.270.08(0.25, 2.204)N(0, 1)ARL370.74284.31253.51151.8488.7854.5724.9215.5710.383.592.091.441.071.01MDRL267.50199.50167.00108.5064.0038.5019.0013.009.003.002.001.001.001.00SDRL369.16291.85267.11151.3588.6650.8020.1711.016.992.131.060.630.260.08CNARL371.10293.39250.56154.4888.7957.9427.4515.7411.223.782.121.581.131.02MDRL273.00214.50172.50109.5064.0042.0021.0013.0010.003.502.001.001.001.00SDRL366.10279.85252.76141.0183.9053.5423.5911.447.552.101.080.750.370.15LogisticARL370.38271.57242.25137.5877.8747.2022.3413.599.283.231.951.441.081.02MDRL265.50180.50169.50103.0058.0036.0018.5012.008.003.002.001.001.001.00SDRL360.62282.24242.01135.3869.5640.4416.299.515.851.871.020.650.280.13LaplaceARL370.38230.11199.04105.7652.4430.7215.0810.527.072.821.831.351.101.02MDRL265.50156.50144.0075.0038.5023.0013.009.006.003.002.001.001.001.00SDRL360.62223.94196.34100.8147.6726.9711.197.144.411.680.970.650.330.17t4ARL371.04304.19282.63193.67113.3872.9632.7819.3713.044.522.521.851.281.11MDRL251.00208.50200.00136.5080.0051.0025.0015.0011.004.002.002.001.001.00SDRL367.97313.54280.19186.75112.5370.9227.1616.718.952.581.480.980.550.36t8ARL370.07116.22108.7681.7354.0536.0819.7012.128.853.081.891.351.071.01MDRL261.5082.5074.0060.5039.0026.0016.0011.008.003.002.001.001.001.00SDRL366.37115.99114.0880.4953.3236.9016.959.096.201.921.040.590.260.11(0.50, 2.491)N(0, 1)ARL371.08311.27311.97210.28137.4187.0238.2422.1712.963.722.121.471.071.00MDRL252.50207.50204.50147.5097.0063.0028.0016.0010.003.002.001.001.001.00SDRL381.21322.22323.87205.59134.8182.8635.6119.9510.582.131.090.670.260.04CNARL369.07317.34298.10218.94142.9391.0441.2922.5614.443.952.291.541.121.02MDRL254.00223.00210.00150.0099.0065.0030.0017.0011.004.002.001.001.001.00SDRL371.98317.49304.17207.28141.4286.4138.0020.7111.922.381.170.740.350.13LogisticARL371.67305.18284.01191.15116.2372.5434.4416.8111.033.481.981.391.081.02MDRL257.50201.00199.00129.5087.0052.0026.0013.009.003.002.001.001.001.00SDRL356.53305.98274.70187.91113.6370.1931.7713.288.811.811.070.620.300.13LaplaceARL371.67274.31254.24151.9081.5152.0522.6712.598.142.821.881.411.111.02MDRL257.50191.50179.00112.0060.0036.0017.0010.007.003.002.001.001.001.00SDRL356.53269.48255.88148.1377.5249.7019.7610.156.241.721.010.660.350.13t4ARL371.87322.54311.03239.91161.54109.6652.4528.8718.264.662.751.921.281.12MDRL250.00210.50220.50167.00110.0076.5036.0020.0014.004.003.002.001.001.00SDRL384.56349.01284.83233.04160.25106.9652.0326.9615.062.911.561.030.580.38t8ARL371.82320.55293.03225.65143.3995.9746.4624.4815.524.192.371.651.181.04MDRL260.00221.00203.00158.00103.0067.0034.0018.0012.004.002.001.001.001.00SDRL367.33335.03302.40221.66139.5696.2042.6320.9912.362.511.290.820.420.20


#### 3.2.4. Out-of-Control Performance

OOC performance of the proposed control chart can be discussed under perfect RSS and imperfect RSS. The OOC control chart’s efficiency is primarily compared, and it demonstrates how sensitive the control chart is to detect shifts in process parameters. Tables3 and 4 show the OOC performance of the proposed TEWMA−SRRSS control chart for the various distributions evaluated in this study. For small and moderate shifts, the OOC performance of the proposed TEWMA−SRRSS control chart increases as n increases; i.e., ARL1 values decrease by increasing n. The proposed scheme’s RL values reduce with the rise in n and m values. The OOC RL performance of the Laplace distribution is higher as compared to the other distributions used in this study (see Figures 1 and 2). The 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). The 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,and0.10, the ARL values of the proposed control chart under RSS are 206.77,92.67,50.53,and33.9, whereas the ARL values of the proposed control chart under imperfect RSS are 295.71,178.64,111.43,and73.63 (see 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.Figure 1
ARL behavior of various distributions using the proposed TEWMA−SRRSS control chart when n=5, λ=0.05, and ARL0=370.Figure 2
ARL behavior of various distributions using the proposedTEWMA−SRRSS control chart when n=10, λ=0.05, and ARL0=370.Figure 3
ARL comparisons of the proposed TEWMA−SRRSS control chart for different values of λ when n=5 and ARL0=370.Figure 4
ARL comparisons of the proposed TEWMA−SRRSS control chart for different values of λ when n=10 and ARL0=370.Table 4
RL characteristics of the proposedTEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=10.
δλ,L0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.3635)N(0, 1)ARL369.6196.0171.8634.9919.8912.626.243.752.471.061.001.001.001.00MDRL240.067.054.029.016.010.05.03.02.01.01.01.01.01.0SDRL418.41101.6470.4930.6016.9610.995.503.191.950.260.000.000.000.00CNARL369.5498.5477.9937.5220.2113.226.203.852.651.111.011.001.001.00MDRL244.0069.0061.0032.0016.0011.005.003.002.001.001.001.001.001.00SDRL429.63102.0574.3631.9317.2211.365.243.252.180.370.070.000.000.00LogisticARL370.0485.1065.6030.6316.5310.775.083.092.191.061.001.001.001.00MDRL215.5064.0053.0025.0013.008.004.002.002.001.001.001.001.001.00SDRL451.5583.9561.1727.0714.169.514.442.541.640.260.040.000.000.00LaplaceARL371.0465.8251.9523.6812.327.754.032.361.741.041.001.001.001.00MDRL218.0055.0042.0019.009.006.003.002.001.001.001.001.001.001.00SDRL452.2961.7346.0920.9911.016.813.501.901.190.210.000.000.000.00t4ARL370.73116.7991.8743.4924.3615.907.424.853.291.231.021.001.001.00MDRL228.5084.0069.0035.0020.0013.005.004.002.001.001.001.001.001.00SDRL421.19125.8592.9840.0120.1613.586.694.042.710.540.140.000.000.00t8ARL369.67107.7585.8240.1523.7114.787.084.292.871.111.011.001.001.00MDRL225.0077.0065.0033.0020.0012.005.003.002.001.001.001.001.001.00SDRL441.35113.1583.9236.9018.6312.456.153.582.350.390.090.000.000.00(0.10, 1.57)N(0, 1)ARL369.43109.5284.4839.8021.0813.556.824.302.871.131.001.001.001.00MDRL242.5076.0061.0033.0018.0012.006.004.002.001.001.001.001.001.00SDRL401.47109.1979.7832.9916.7510.215.303.262.140.390.040.000.000.00CNARL369.18117.3391.8741.3223.1914.617.154.583.141.171.011.001.001.00MDRL244.5084.0068.0032.5021.0013.006.004.003.001.001.001.001.001.00SDRL401.76115.0087.6636.2716.7910.755.593.482.300.460.070.000.000.00LogisticARL369.92104.6673.8935.3419.3312.306.063.732.461.091.001.001.001.00MDRL243.0077.0054.0029.5017.0010.005.003.002.001.001.001.001.001.00SDRL419.0299.9371.9429.9014.519.214.562.781.760.330.060.000.000.00LaplaceARL369.9280.3057.7025.0813.958.934.652.882.011.081.001.001.001.00MDRL243.0060.0045.0022.0012.007.504.002.001.001.001.001.001.001.00SDRL419.0273.0352.3419.7110.636.763.662.161.380.290.070.000.000.00t4ARL370.86137.04109.4550.2125.9117.079.225.673.941.331.041.001.001.00MDRL251.0098.0080.0039.0023.0015.008.005.003.001.001.001.001.001.00SDRL388.38138.72108.9243.3119.8412.636.934.303.070.660.200.050.000.00t8ARL370.56126.76101.5448.4523.0115.848.165.043.331.211.011.001.001.00MDRL257.5087.0068.5038.0020.0014.007.004.003.001.001.001.001.001.00SDRL387.66130.74104.0141.8417.8811.166.143.682.390.530.100.000.000.00(0.25, 1.867)N(0, 1)ARL370.25148.30114.5053.8726.4615.297.574.943.561.251.021.001.001.00MDRL271.50103.0078.0040.0020.0013.007.004.003.001.001.001.001.001.00SDRL353.98143.90113.0449.0822.9411.385.062.972.090.520.130.000.000.00CNARL371.05154.57129.1059.2326.8116.477.955.343.661.301.021.001.001.00MDRL255.50105.5090.0044.0021.0014.007.005.003.001.001.001.001.001.00SDRL383.00152.56124.9556.4222.5412.564.993.362.090.560.140.000.000.00LogisticARL369.80142.12110.9745.0022.0113.106.864.233.031.201.011.001.001.00MDRL249.0097.0078.0032.0018.0011.006.004.003.001.001.001.001.001.00SDRL402.95147.38102.8141.4516.839.214.222.721.850.460.090.030.000.00LaplaceARL369.80111.9278.3833.4616.679.765.283.302.501.141.021.001.001.00MDRL249.0081.0054.0026.0014.009.005.003.002.001.001.001.001.001.00SDRL402.95105.9575.9828.8112.946.803.232.041.500.370.140.000.000.00t4ARL368.97186.35140.2371.3834.8619.439.786.194.221.561.091.011.001.00MDRL262.00131.0097.5050.0026.0016.009.006.004.001.001.001.001.001.00SDRL359.24181.23140.4467.1629.6714.586.514.052.630.790.300.110.000.00t8ARL371.13177.20148.4962.0930.5017.809.015.813.751.411.041.001.001.00MDRL259.00125.00101.5044.5023.0015.008.005.003.001.001.001.001.001.00SDRL367.57173.08155.7354.4625.6413.555.733.452.210.690.200.050.000.00(0.50, 2.1185)N(0, 1)ARL370.25203.92181.5381.6939.7321.858.995.163.641.351.021.001.001.00MDRL261.00144.00131.5055.0029.0016.007.005.003.001.001.001.001.001.00SDRL361.13203.76174.2781.5636.7919.276.493.352.030.560.150.000.000.00CNARL371.05220.20176.0189.5145.2823.669.175.403.821.401.051.001.001.00MDRL266.50144.00129.5063.0033.0017.007.005.003.001.001.001.001.001.00SDRL355.54224.64167.9786.1144.0920.666.813.402.190.590.210.040.000.00LogisticARL369.71207.36144.9472.6833.5718.757.774.623.281.261.011.001.001.00MDRL262.00140.50107.5053.0025.0015.007.004.003.001.001.001.001.001.00SDRL389.82220.88139.7969.4231.0815.695.482.891.810.490.120.040.000.00LaplaceARL369.71165.97118.1147.0121.8412.145.653.502.551.221.021.001.001.00MDRL262.00112.0081.5033.5016.009.005.003.002.001.001.001.001.001.00SDRL389.82177.25114.3642.6419.129.663.582.001.380.450.140.050.000.00t4ARL369.17242.08202.82104.2055.1229.4412.567.094.641.641.121.031.001.00MDRL261.50172.00141.0076.0038.0022.009.006.004.001.001.001.001.001.00SDRL359.03241.26199.81101.7154.0826.6110.824.732.730.790.340.160.040.00t8ARL369.74240.61203.2299.4948.1326.5510.965.744.131.561.061.011.001.00MDRL253.00173.00138.0067.0034.5020.009.005.004.001.001.001.001.001.00SDRL366.34229.90209.30100.8844.5024.288.713.802.320.740.240.080.000.00Table 5
ARL measures of the proposed control chart with imperfect ranked set sampling under normal distribution atn=5 and ARL0=370.
δλ,Lγ00.0250.030.050.0750.10.150.20.250.50.7511.520.05, 1.7930.30ARL373.21295.71274.87178.64111.4373.6339.9424.5318.506.032.972.021.341.12MDRL242.50192.50184.50118.5080.5059.0033.0020.0015.005.002.002.001.001.00SDRL437.59321.50293.65193.69114.1471.0035.3620.8315.185.012.311.280.560.320.50ARL370.86282.37265.80154.96101.6467.1737.9023.5316.465.172.631.661.131.02MDRL218.50174.50158.50106.5073.0053.0032.5020.0013.004.002.001.001.001.00SDRL454.57327.88315.97169.82104.7964.6132.5819.6514.334.482.191.130.400.160.70ARL372.89266.56243.76140.6688.9059.9633.1819.7715.024.542.291.591.091.01MDRL242.00169.50153.5095.5066.0049.0027.0016.0012.004.002.001.001.001.00SDRL452.69294.47298.29155.5887.2755.6128.4617.1612.793.711.730.980.320.090.90ARL372.08243.06209.25121.6566.3344.5424.9216.1211.253.481.831.321.041.00MDRL228.00160.00138.5084.5048.0036.0020.0013.009.003.001.001.001.001.00SDRL459.42272.24231.19127.6764.0138.8721.6713.489.352.781.190.630.190.040.1, 2.0690.30ARL373.72313.05303.81207.33131.8581.3646.5530.3721.237.283.952.732.092.01MDRL265.50213.00219.50139.5093.0059.0038.0025.0019.006.003.002.002.002.00SDRL405.22325.15300.95227.28122.9176.2239.1322.7814.844.852.351.240.360.110.50ARL372.77292.15277.20186.86116.4476.8040.3526.4719.166.373.232.201.361.13MDRL242.50199.00184.50129.0084.0057.0034.5023.0017.006.003.002.001.001.00SDRL421.10321.22303.40190.83118.8772.9032.4820.8914.434.442.201.380.660.350.70ARL368.36291.14266.67170.19111.4363.5736.2221.4816.135.392.862.011.341.11MDRL239.50194.00180.00116.5080.0050.0031.0019.0014.005.002.002.001.001.00SDRL388.46310.09292.56174.63107.9556.5428.7816.1111.853.851.881.120.520.310.90ARL370.71266.00243.61139.1078.5953.2727.1517.9912.304.012.141.521.091.01MDRL245.00171.00163.0099.0059.0042.0023.0016.0010.003.002.001.001.001.00SDRL383.69280.70257.04138.7171.7444.7721.0312.929.132.851.360.750.290.110.25, 2.0450.30ARL372.80344.43325.65261.86168.50121.6963.3039.5924.327.794.473.172.242.03MDRL250.00230.00236.00177.50119.0089.5046.0029.0019.007.004.003.002.002.00SDRL380.99347.62312.31267.28160.73118.3657.8834.3418.424.262.171.320.520.180.50ARL371.94337.96304.40236.55159.02107.1356.4533.3223.407.054.152.952.162.02MDRL249.50240.50196.50156.50114.0073.0041.0026.0019.006.004.003.002.002.00SDRL378.92325.82317.83241.56150.45106.1553.8728.0417.343.982.021.190.420.130.70ARL372.26333.52317.06227.03147.7495.5649.1927.6618.956.123.622.602.072.00MDRL247.00232.50207.50153.0099.0070.0036.0021.0016.006.003.002.002.002.00SDRL389.33325.75346.54235.50152.8288.0143.3822.6113.863.411.690.870.290.040.90ARL368.96319.23277.69185.12114.7470.3234.7819.4813.914.632.681.901.331.11MDRL243.00214.00199.00128.5081.5051.0027.0016.0012.004.002.002.001.001.00SDRL377.52323.88280.01183.71114.7763.4529.2414.789.942.601.440.840.480.310.50, 2.7250.30ARL370.94333.83350.08292.49225.25175.1694.5158.6037.479.344.733.472.552.23MDRL255.00220.00237.00210.50158.00113.0068.5042.0027.008.004.003.002.002.00SDRL363.45345.01342.11276.99238.32183.4387.7354.9433.296.502.091.230.600.420.50ARL368.41336.70312.65270.29210.88154.2083.5652.1534.798.344.353.062.252.05MDRL253.50237.00220.50198.00147.00111.0057.0035.0025.007.004.003.002.002.00SDRL351.77342.73305.48265.20208.49153.4082.9050.5531.975.642.041.100.460.220.70ARL370.94364.34336.00280.24203.91152.0976.8345.7128.226.933.832.842.162.02MDRL246.50273.50234.00193.50144.50102.0054.5035.0021.006.004.003.002.002.00SDRL378.51357.52335.50276.70200.57156.7876.0939.6125.804.371.580.900.390.150.90ARL369.05349.47318.40251.53170.83111.8355.2432.5918.515.092.972.342.022.00MDRL271.00260.00221.00173.00126.0078.0038.5024.0014.005.003.002.002.002.00SDRL354.84328.59314.95247.96161.02106.9351.7930.1915.072.721.130.630.150.03Table 6
ARL measures of the proposed control chart with imperfect ranked set sampling under normal distribution atn=10 and ARL0=370.
δλ,Lγ00.0250.030.050.0750.10.150.20.250.50.7511.520.05, 1.760.30ARL373.11218.89192.5299.2260.6139.7820.6413.469.842.931.561.171.011.00MDRL204.50146.00131.0078.0048.0034.0016.0010.508.002.001.001.001.001.00SDRL454.49241.25198.8097.0457.9233.9718.1611.818.362.441.000.440.090.000.50ARL373.21207.83168.2298.2457.0737.6419.4812.868.892.621.381.121.001.00MDRL216.50128.00112.0075.5048.0030.0016.009.007.002.001.001.001.001.00SDRL477.82243.21178.19100.5550.7833.8816.6511.207.602.070.810.380.050.000.70ARL372.35178.19155.3876.4845.1329.6316.6210.076.782.011.271.041.001.00MDRL231.00118.00105.5059.0037.0024.5013.008.005.001.001.001.001.001.00SDRL454.56199.24164.8375.3039.1825.7514.878.915.751.480.630.190.000.000.90ARL374.41130.95112.5655.2330.9920.1710.806.543.991.431.041.001.001.00MDRL255.0093.5086.5044.0025.0017.009.005.003.001.001.001.001.001.00SDRL430.14142.14109.6550.6627.5416.648.995.683.470.790.190.050.000.000.10, 2.0450.30ARL371.60251.30203.89117.6967.6744.8922.9215.1910.753.551.821.241.011.00MDRL234.50180.50142.5086.0049.0037.0020.0013.5010.003.001.001.001.001.00SDRL414.94261.17210.13124.0461.2537.3717.8711.198.042.541.140.560.090.030.50ARL368.71236.57205.49118.9664.6540.9821.8614.369.633.091.671.171.011.00MDRL236.00152.00146.5082.5050.0035.0019.0013.009.002.001.001.001.001.00SDRL395.36259.02207.69120.2257.5334.1117.0610.427.002.291.040.440.070.030.70ARL369.42220.71182.6692.7751.6033.8517.3811.207.962.461.371.081.001.00MDRL249.00157.00122.0068.0041.0028.0015.0010.007.002.001.001.001.001.00SDRL380.61230.25186.9085.0744.6529.4212.758.396.011.750.720.300.050.000.90ARL368.55154.11128.5763.3034.4121.6211.697.514.951.541.071.011.001.00MDRL244.50102.0097.5049.0028.0018.0010.007.004.001.001.001.001.001.00SDRL376.02169.42124.1957.5727.8816.539.135.293.810.920.310.090.000.000.25, 2.4450.30ARL368.15291.14249.72167.77103.4464.4131.1019.1412.314.272.321.601.091.01MDRL257.50201.00176.50118.0072.0048.0024.0016.0011.004.002.001.001.001.00SDRL368.63304.07269.34162.29100.2258.6525.5014.848.822.411.230.760.290.090.50ARL371.23299.87272.71172.55100.0059.6228.4615.8311.113.882.091.471.051.00MDRL260.00210.00179.00131.0073.0041.0022.0013.0010.003.002.001.001.001.00SDRL364.66296.55284.84157.2193.6057.6622.1811.107.062.211.100.610.220.040.70ARL373.52261.53224.48128.5774.5146.1121.3912.858.932.891.681.221.001.00MDRL256.00187.50152.0093.0053.0033.0017.0011.008.003.001.001.001.001.00SDRL373.99261.14230.15125.3469.3842.7217.558.425.751.640.820.470.050.000.90ARL368.26204.11178.2188.5745.6025.6913.018.015.781.951.221.031.001.00MDRL244.50147.50132.5062.0033.0019.0011.007.005.002.001.001.001.001.00SDRL372.36199.39164.4187.9041.7022.399.145.093.451.060.470.160.000.000.50, 2.7640.30ARL371.28308.77280.73225.25146.06101.0546.5225.4616.154.492.531.911.321.06MDRL272.50226.50205.00153.00106.0067.0032.0018.0012.004.002.002.001.001.00SDRL340.94288.19268.66217.69139.07100.3941.5722.7713.142.431.050.710.470.240.50ARL369.52329.86318.54231.50150.5988.1745.1523.4414.673.972.311.751.231.03MDRL248.00231.50213.50164.00117.0061.0033.0017.0011.004.002.002.001.001.00SDRL368.35328.72329.20220.58138.5089.2741.2319.8412.322.020.950.610.420.170.70ARL367.86299.83271.13188.83122.4276.2031.6417.5811.073.281.951.401.051.00MDRL243.00213.50176.50131.0087.0054.0023.0014.009.003.002.001.001.001.00SDRL333.88292.66272.78192.12119.0573.7528.3314.718.811.670.790.550.210.030.90ARL367.95247.37232.26139.1973.1740.6817.759.366.102.151.321.051.001.00MDRL261.50180.00165.5094.0051.0029.0013.007.005.002.001.001.001.001.00SDRL356.50243.31219.41138.1772.3237.5314.597.173.930.990.510.210.000.00In terms of OOC performance at a specific shiftδ, the proposed TEWMA−SRRSS control chart outperforms all other control charts. For instance, at δ=0.05,λ=0.25, and n=5, the ARL1 values of the proposed TEWMA−SRRSS, TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts are 151.84, 251.08, 341.92, 337.05, and 167.12, respectively. The ARL is used as a metric to compare the proposed and existing control charts. Some findings are presented in the percentage decreasing in ARL=ARL0−ARL1/ARL0×100%. Under normal setup, the TEWMA−SRRSS control chart displaying a 3 percent improvement in the process median minimizes the ARL by 54.56 percent for n=5,λ=0.05, andARL0=370, whereas at the same shift δ, the proposed control chart reduced the ARL by 80.55 percent for n=10,λ=0.05, andARL0=370 (see Tables 3 and 4).


## 3.1. Choice of Design Parameters

λ and L are the design parameters of the proposed TEWMA−SRRSS control chart. The control chart width L is chosen with various values of λ, such that the prespecified ARL0 is obtained. The best subgroup size is between 5 and 10, depending on ARL0 and shift size [5]. Different RL properties are assessed to study the performance behavior of the proposed TEWMA−SRRSS control chart with λ∈0.05,0.10,0.25,0.50 and n∈5,10. ARL0 is assumed to be 370 for this study.


## 3.2. Evaluation

This section provides theRL characteristics of the different distributions (normal and non-normal distributions) for IC robustness and IC performance of the proposed control chart. The proposed TEWMA−SRRSS control chart’s RL features are evaluated using normal and non-normal distributions.


### 3.2.1. Evaluation of the Run-Length Distribution

The performance of the proposed control chart can be assessed in both normal and non-normal symmetric distributions. The distributions used in this analysis are as follows:(i)
Standard normal distribution, i.e.,N0,1(ii)
Double exponential (Laplace) distribution, i.e.,DE0,1/2(iii)
Heavy tail Student’st distribution tv, with the degree of freedom v=4and8(iv)
The logistic distribution, i.e.,LOG0,3/π(v)
Contaminated normal (CN) distribution, which is a mixture ofN0,σ12 and N0,σ02, i.e., 1−αN0,σ12+αN0,σ02, with α such that 1−ασ12+ασ02=1, where α=0.05 and σ1/σ0=2. For comparison purpose, all the considered distributions in this analysis were reparametrized with mean zero and unit standard deviation. The probability density functions of the distributions used in this study are shown in Table 2.Table 1
Details about the control charts used in this study.
Control chart’s nameNotationPlotting statisticsControl limits1NP EWMA under RSSEWMA−SRRSSEq=λSRRSSq+1−λEq−1UCLEWMA−SRRSS=+Lλ/2−λrr+12r+1/6ω02LCLEWMA−SRRSS=−Lλ/2−λrr+12r+1/6ω022NP DEWMA under RSSDEWMA−SRRSSEq=λSRRSSq+1−λEq−1,DEq=λEq+1−λDEq−1LCLDEWMA−SRRSS=−Lrr+12r+1/6ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q2/1−λ23UCLDEWMA−SRRSS=+Lrr+12r+1/6ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q2/1−λ233NP TEWMA under RSSTEWMA−SRRSSEq=λSRRSSq+1−λEq−1,DEq=λEq+1−λDEq−1,TEq=λDEq+1−λTEq−1LCLTEWMA−SRRSS=−Lσ61−λ6λ/2−λ5+121−λ4λ2/2−λ4+71−λ2λ3/2−λ3+λ4/2−λ2nn+12n+1/6ω02UCLTEWMA−SRRSS=+Lσ61−λ6λ/2−λ5+121−λ4λ2/2−λ4+71−λ2λ3/2−λ3+λ4/2−λ2nn+12n+1/6ω02


### 3.2.2. Monte Carlo Simulation

The Monte Carlo simulation is used to obtain theRL characteristics of the proposed control chart. The simulation algorithm is developed in R software to compute the RL characteristics. 104 random samples of the size n>1 from any distribution covered by the study for a shift δ were generated. The simulation algorithm to determine the nominal values of ARL0 under various distributions can be explained in the following steps [36]:(a)
An finite loop is practiced to generate samples from different distributions(b)
Select a particular value forλ(c)
Determine theTEq plotting statistic offered in equation (9)(d)
Let us assumeL=3(e)
Calculate theUCLTEWMA−SRRSSandLCLTEWMA−SRRSS from equation (11)(f)
Plot theTEq statistic against control limits in step (e) over q(g)
If theTEq>UCLTEWMA−SRRSSorTEq<LCLTEWMA−SRRSS, record the sample number of TEq statistic as an RL. For example, at q=255, if TEq>UCLTEWMA−SRRSSorTEq<LCLTEWMA−SRRSS record 255 as a first RL.(h)
Repeat steps from (a) to (g) for 105 times and record RLs.(i)
Compute the average of105 times noted RLs, which is called ARL0.(j)
IfARL0=400; otherwise, adjust L (i.e., L<3orL>3) constant accordingly in step (d) and repeat steps from (a) to (i) to obtain ARL0=400.(k)
To compute theARL1 values, again generate samples from relevant distribution and repeat steps from (b) to (i).


### 3.2.3. In-Control Robustness of the Proposed Control Chart

The EWMA andDEWMA−X¯ control charts have the limitation that their IC RL features do not remain the same for all continuous distributions. The IC RL characteristics of the NP control chart’s characteristics remain the same for all symmetric continuous distributions [4]. The simulated results of the proposed control chart with different combinations of λandL to obtain the nominal ARL0=370 are shown in Tables 3 and 4. The IC RL distribution of the proposed TEWMA−SRRSS control chart seems to be similar for all distributions used in this study. The RL distribution is positively skewed when ARL>MDRL.Table 2
PDFs of the continuous distributions used for this study.
(i)Standard normalfX=e−x2/2/2π, where X∈ℜ,M0=0,andσ2=1(ii)Student’stυfX=Γυ+1/2/Γυ/2υπ1+X2/υυ+1/2, where X∈ℜ,M0=0,andσ2=υ/υ−2 and υ=4,8 are taken(iii)LogisticfX=e−πX/3/3/π1+e−πX/32, where X∈ℜ,M0=0,andσ2=3/π2(iv)LaplacefX=1/2e−X, where X∈ℜ,M0=0,andσ2=1/2(v)Contaminated normal (CN)fX=0.95e−X2/2/2π+0.05e−X2/2σ02/σ02π, where X∈ℜ,M0=0,andσ2=0.95+0.05σ02Table 3
RL characteristics of the proposedTEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=5.
δλ,L0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.585)N(0, 1)ARL371.11206.77168.6292.6750.5333.9119.1212.027.942.351.371.081.011.00MDRL218.00131.00111.0070.0041.0028.0017.009.007.002.001.001.001.001.00SDRL453.10234.01183.7090.3349.0629.1515.7210.546.621.770.750.310.080.00CNARL369.92212.31164.0494.3151.9335.6319.3712.068.352.511.461.141.011.00MDRL217.50139.00113.0065.0040.0027.5016.009.006.002.001.001.001.001.00SDRL441.07235.19179.2897.3550.1533.3516.1610.777.502.080.870.430.130.04LogisticARL371.69189.29156.5180.7249.3730.4916.2110.277.202.181.311.101.011.00MDRL230.50118.00112.0062.0041.0025.0013.008.006.001.001.001.001.001.00SDRL429.72216.77159.5083.6344.7026.4214.268.946.391.610.660.330.110.03LaplaceARL371.65158.45123.5963.4934.1822.5412.257.525.482.011.321.121.031.00MDRL237.00106.5087.0050.0028.0018.0010.005.004.001.001.001.001.001.00SDRL428.92173.36126.8360.3330.9220.2910.726.894.851.500.660.380.170.06t4ARL369.29215.22187.04111.3462.0041.6523.5515.3910.433.341.791.361.091.02MDRL232.50126.50123.0083.0047.0033.0020.0012.008.002.001.001.001.001.00SDRL423.91254.75210.06111.4059.0239.9120.4512.909.582.791.310.700.340.17t8ARL370.51202.75174.3799.3658.9237.6420.0013.338.892.801.581.201.031.01MDRL231.00128.00118.0070.0045.0033.0015.0010.007.002.001.001.001.001.00SDRL455.11247.02189.65102.2255.9732.3117.5011.807.862.401.030.540.170.12(0.10, 1.837)N(0, 1)ARL370.58239.49200.36107.4457.6038.6621.5813.269.193.091.751.281.031.00MDRL241.50164.00143.0075.5044.0032.0019.0012.008.002.001.001.001.001.00SDRL399.98252.12200.34104.0952.2931.4716.689.586.522.100.970.540.180.03CNARL371.69222.24189.81110.7164.1940.5821.0213.659.673.281.791.321.051.01MDRL240.50155.00136.5075.0048.0034.0018.0012.008.003.002.001.001.001.00SDRL400.58233.16189.80111.1460.2732.0916.179.936.982.331.020.600.220.08LogisticARL369.89220.35179.1597.8552.1735.7718.2611.258.242.731.671.251.041.01MDRL257.50150.00124.0072.0040.0029.0016.0010.007.002.001.001.001.001.00SDRL379.94229.75177.9490.9947.5729.9613.908.425.941.800.890.520.200.07LaplaceARL369.00179.37145.5972.1539.0025.6214.368.836.402.421.521.271.051.01MDRL257.50117.50103.0054.0031.0022.0013.008.005.002.001.001.001.001.00SDRL379.94191.51141.8165.4633.7819.3910.666.284.771.580.830.560.230.10t4ARL369.26257.35229.58142.1981.2548.9026.1116.6812.293.982.271.601.161.04MDRL239.00166.50154.00102.0059.5038.0023.0014.0011.003.002.001.001.001.00SDRL382.67289.42240.47134.4775.2542.7319.5612.468.782.781.430.900.430.22t8ARL370.13241.63203.62125.9067.9944.7622.6315.9210.573.421.921.381.061.01MDRL250.00169.50139.0089.5052.0037.0020.0014.009.003.002.001.001.001.00SDRL417.20260.53202.72129.4859.3836.4417.1311.117.522.421.140.620.270.08(0.25, 2.204)N(0, 1)ARL370.74284.31253.51151.8488.7854.5724.9215.5710.383.592.091.441.071.01MDRL267.50199.50167.00108.5064.0038.5019.0013.009.003.002.001.001.001.00SDRL369.16291.85267.11151.3588.6650.8020.1711.016.992.131.060.630.260.08CNARL371.10293.39250.56154.4888.7957.9427.4515.7411.223.782.121.581.131.02MDRL273.00214.50172.50109.5064.0042.0021.0013.0010.003.502.001.001.001.00SDRL366.10279.85252.76141.0183.9053.5423.5911.447.552.101.080.750.370.15LogisticARL370.38271.57242.25137.5877.8747.2022.3413.599.283.231.951.441.081.02MDRL265.50180.50169.50103.0058.0036.0018.5012.008.003.002.001.001.001.00SDRL360.62282.24242.01135.3869.5640.4416.299.515.851.871.020.650.280.13LaplaceARL370.38230.11199.04105.7652.4430.7215.0810.527.072.821.831.351.101.02MDRL265.50156.50144.0075.0038.5023.0013.009.006.003.002.001.001.001.00SDRL360.62223.94196.34100.8147.6726.9711.197.144.411.680.970.650.330.17t4ARL371.04304.19282.63193.67113.3872.9632.7819.3713.044.522.521.851.281.11MDRL251.00208.50200.00136.5080.0051.0025.0015.0011.004.002.002.001.001.00SDRL367.97313.54280.19186.75112.5370.9227.1616.718.952.581.480.980.550.36t8ARL370.07116.22108.7681.7354.0536.0819.7012.128.853.081.891.351.071.01MDRL261.5082.5074.0060.5039.0026.0016.0011.008.003.002.001.001.001.00SDRL366.37115.99114.0880.4953.3236.9016.959.096.201.921.040.590.260.11(0.50, 2.491)N(0, 1)ARL371.08311.27311.97210.28137.4187.0238.2422.1712.963.722.121.471.071.00MDRL252.50207.50204.50147.5097.0063.0028.0016.0010.003.002.001.001.001.00SDRL381.21322.22323.87205.59134.8182.8635.6119.9510.582.131.090.670.260.04CNARL369.07317.34298.10218.94142.9391.0441.2922.5614.443.952.291.541.121.02MDRL254.00223.00210.00150.0099.0065.0030.0017.0011.004.002.001.001.001.00SDRL371.98317.49304.17207.28141.4286.4138.0020.7111.922.381.170.740.350.13LogisticARL371.67305.18284.01191.15116.2372.5434.4416.8111.033.481.981.391.081.02MDRL257.50201.00199.00129.5087.0052.0026.0013.009.003.002.001.001.001.00SDRL356.53305.98274.70187.91113.6370.1931.7713.288.811.811.070.620.300.13LaplaceARL371.67274.31254.24151.9081.5152.0522.6712.598.142.821.881.411.111.02MDRL257.50191.50179.00112.0060.0036.0017.0010.007.003.002.001.001.001.00SDRL356.53269.48255.88148.1377.5249.7019.7610.156.241.721.010.660.350.13t4ARL371.87322.54311.03239.91161.54109.6652.4528.8718.264.662.751.921.281.12MDRL250.00210.50220.50167.00110.0076.5036.0020.0014.004.003.002.001.001.00SDRL384.56349.01284.83233.04160.25106.9652.0326.9615.062.911.561.030.580.38t8ARL371.82320.55293.03225.65143.3995.9746.4624.4815.524.192.371.651.181.04MDRL260.00221.00203.00158.00103.0067.0034.0018.0012.004.002.001.001.001.00SDRL367.33335.03302.40221.66139.5696.2042.6320.9912.362.511.290.820.420.20


### 3.2.4. Out-of-Control Performance

OOC performance of the proposed control chart can be discussed under perfect RSS and imperfect RSS. The OOC control chart’s efficiency is primarily compared, and it demonstrates how sensitive the control chart is to detect shifts in process parameters. Tables3 and 4 show the OOC performance of the proposed TEWMA−SRRSS control chart for the various distributions evaluated in this study. For small and moderate shifts, the OOC performance of the proposed TEWMA−SRRSS control chart increases as n increases; i.e., ARL1 values decrease by increasing n. The proposed scheme’s RL values reduce with the rise in n and m values. The OOC RL performance of the Laplace distribution is higher as compared to the other distributions used in this study (see Figures 1 and 2). The 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). The 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,and0.10, the ARL values of the proposed control chart under RSS are 206.77,92.67,50.53,and33.9, whereas the ARL values of the proposed control chart under imperfect RSS are 295.71,178.64,111.43,and73.63 (see 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.Figure 1
ARL behavior of various distributions using the proposed TEWMA−SRRSS control chart when n=5, λ=0.05, and ARL0=370.Figure 2
ARL behavior of various distributions using the proposedTEWMA−SRRSS control chart when n=10, λ=0.05, and ARL0=370.Figure 3
ARL comparisons of the proposed TEWMA−SRRSS control chart for different values of λ when n=5 and ARL0=370.Figure 4
ARL comparisons of the proposed TEWMA−SRRSS control chart for different values of λ when n=10 and ARL0=370.Table 4
RL characteristics of the proposedTEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=10.
δλ,L0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.3635)N(0, 1)ARL369.6196.0171.8634.9919.8912.626.243.752.471.061.001.001.001.00MDRL240.067.054.029.016.010.05.03.02.01.01.01.01.01.0SDRL418.41101.6470.4930.6016.9610.995.503.191.950.260.000.000.000.00CNARL369.5498.5477.9937.5220.2113.226.203.852.651.111.011.001.001.00MDRL244.0069.0061.0032.0016.0011.005.003.002.001.001.001.001.001.00SDRL429.63102.0574.3631.9317.2211.365.243.252.180.370.070.000.000.00LogisticARL370.0485.1065.6030.6316.5310.775.083.092.191.061.001.001.001.00MDRL215.5064.0053.0025.0013.008.004.002.002.001.001.001.001.001.00SDRL451.5583.9561.1727.0714.169.514.442.541.640.260.040.000.000.00LaplaceARL371.0465.8251.9523.6812.327.754.032.361.741.041.001.001.001.00MDRL218.0055.0042.0019.009.006.003.002.001.001.001.001.001.001.00SDRL452.2961.7346.0920.9911.016.813.501.901.190.210.000.000.000.00t4ARL370.73116.7991.8743.4924.3615.907.424.853.291.231.021.001.001.00MDRL228.5084.0069.0035.0020.0013.005.004.002.001.001.001.001.001.00SDRL421.19125.8592.9840.0120.1613.586.694.042.710.540.140.000.000.00t8ARL369.67107.7585.8240.1523.7114.787.084.292.871.111.011.001.001.00MDRL225.0077.0065.0033.0020.0012.005.003.002.001.001.001.001.001.00SDRL441.35113.1583.9236.9018.6312.456.153.582.350.390.090.000.000.00(0.10, 1.57)N(0, 1)ARL369.43109.5284.4839.8021.0813.556.824.302.871.131.001.001.001.00MDRL242.5076.0061.0033.0018.0012.006.004.002.001.001.001.001.001.00SDRL401.47109.1979.7832.9916.7510.215.303.262.140.390.040.000.000.00CNARL369.18117.3391.8741.3223.1914.617.154.583.141.171.011.001.001.00MDRL244.5084.0068.0032.5021.0013.006.004.003.001.001.001.001.001.00SDRL401.76115.0087.6636.2716.7910.755.593.482.300.460.070.000.000.00LogisticARL369.92104.6673.8935.3419.3312.306.063.732.461.091.001.001.001.00MDRL243.0077.0054.0029.5017.0010.005.003.002.001.001.001.001.001.00SDRL419.0299.9371.9429.9014.519.214.562.781.760.330.060.000.000.00LaplaceARL369.9280.3057.7025.0813.958.934.652.882.011.081.001.001.001.00MDRL243.0060.0045.0022.0012.007.504.002.001.001.001.001.001.001.00SDRL419.0273.0352.3419.7110.636.763.662.161.380.290.070.000.000.00t4ARL370.86137.04109.4550.2125.9117.079.225.673.941.331.041.001.001.00MDRL251.0098.0080.0039.0023.0015.008.005.003.001.001.001.001.001.00SDRL388.38138.72108.9243.3119.8412.636.934.303.070.660.200.050.000.00t8ARL370.56126.76101.5448.4523.0115.848.165.043.331.211.011.001.001.00MDRL257.5087.0068.5038.0020.0014.007.004.003.001.001.001.001.001.00SDRL387.66130.74104.0141.8417.8811.166.143.682.390.530.100.000.000.00(0.25, 1.867)N(0, 1)ARL370.25148.30114.5053.8726.4615.297.574.943.561.251.021.001.001.00MDRL271.50103.0078.0040.0020.0013.007.004.003.001.001.001.001.001.00SDRL353.98143.90113.0449.0822.9411.385.062.972.090.520.130.000.000.00CNARL371.05154.57129.1059.2326.8116.477.955.343.661.301.021.001.001.00MDRL255.50105.5090.0044.0021.0014.007.005.003.001.001.001.001.001.00SDRL383.00152.56124.9556.4222.5412.564.993.362.090.560.140.000.000.00LogisticARL369.80142.12110.9745.0022.0113.106.864.233.031.201.011.001.001.00MDRL249.0097.0078.0032.0018.0011.006.004.003.001.001.001.001.001.00SDRL402.95147.38102.8141.4516.839.214.222.721.850.460.090.030.000.00LaplaceARL369.80111.9278.3833.4616.679.765.283.302.501.141.021.001.001.00MDRL249.0081.0054.0026.0014.009.005.003.002.001.001.001.001.001.00SDRL402.95105.9575.9828.8112.946.803.232.041.500.370.140.000.000.00t4ARL368.97186.35140.2371.3834.8619.439.786.194.221.561.091.011.001.00MDRL262.00131.0097.5050.0026.0016.009.006.004.001.001.001.001.001.00SDRL359.24181.23140.4467.1629.6714.586.514.052.630.790.300.110.000.00t8ARL371.13177.20148.4962.0930.5017.809.015.813.751.411.041.001.001.00MDRL259.00125.00101.5044.5023.0015.008.005.003.001.001.001.001.001.00SDRL367.57173.08155.7354.4625.6413.555.733.452.210.690.200.050.000.00(0.50, 2.1185)N(0, 1)ARL370.25203.92181.5381.6939.7321.858.995.163.641.351.021.001.001.00MDRL261.00144.00131.5055.0029.0016.007.005.003.001.001.001.001.001.00SDRL361.13203.76174.2781.5636.7919.276.493.352.030.560.150.000.000.00CNARL371.05220.20176.0189.5145.2823.669.175.403.821.401.051.001.001.00MDRL266.50144.00129.5063.0033.0017.007.005.003.001.001.001.001.001.00SDRL355.54224.64167.9786.1144.0920.666.813.402.190.590.210.040.000.00LogisticARL369.71207.36144.9472.6833.5718.757.774.623.281.261.011.001.001.00MDRL262.00140.50107.5053.0025.0015.007.004.003.001.001.001.001.001.00SDRL389.82220.88139.7969.4231.0815.695.482.891.810.490.120.040.000.00LaplaceARL369.71165.97118.1147.0121.8412.145.653.502.551.221.021.001.001.00MDRL262.00112.0081.5033.5016.009.005.003.002.001.001.001.001.001.00SDRL389.82177.25114.3642.6419.129.663.582.001.380.450.140.050.000.00t4ARL369.17242.08202.82104.2055.1229.4412.567.094.641.641.121.031.001.00MDRL261.50172.00141.0076.0038.0022.009.006.004.001.001.001.001.001.00SDRL359.03241.26199.81101.7154.0826.6110.824.732.730.790.340.160.040.00t8ARL369.74240.61203.2299.4948.1326.5510.965.744.131.561.061.011.001.00MDRL253.00173.00138.0067.0034.5020.009.005.004.001.001.001.001.001.00SDRL366.34229.90209.30100.8844.5024.288.713.802.320.740.240.080.000.00Table 5
ARL measures of the proposed control chart with imperfect ranked set sampling under normal distribution atn=5 and ARL0=370.
δλ,Lγ00.0250.030.050.0750.10.150.20.250.50.7511.520.05, 1.7930.30ARL373.21295.71274.87178.64111.4373.6339.9424.5318.506.032.972.021.341.12MDRL242.50192.50184.50118.5080.5059.0033.0020.0015.005.002.002.001.001.00SDRL437.59321.50293.65193.69114.1471.0035.3620.8315.185.012.311.280.560.320.50ARL370.86282.37265.80154.96101.6467.1737.9023.5316.465.172.631.661.131.02MDRL218.50174.50158.50106.5073.0053.0032.5020.0013.004.002.001.001.001.00SDRL454.57327.88315.97169.82104.7964.6132.5819.6514.334.482.191.130.400.160.70ARL372.89266.56243.76140.6688.9059.9633.1819.7715.024.542.291.591.091.01MDRL242.00169.50153.5095.5066.0049.0027.0016.0012.004.002.001.001.001.00SDRL452.69294.47298.29155.5887.2755.6128.4617.1612.793.711.730.980.320.090.90ARL372.08243.06209.25121.6566.3344.5424.9216.1211.253.481.831.321.041.00MDRL228.00160.00138.5084.5048.0036.0020.0013.009.003.001.001.001.001.00SDRL459.42272.24231.19127.6764.0138.8721.6713.489.352.781.190.630.190.040.1, 2.0690.30ARL373.72313.05303.81207.33131.8581.3646.5530.3721.237.283.952.732.092.01MDRL265.50213.00219.50139.5093.0059.0038.0025.0019.006.003.002.002.002.00SDRL405.22325.15300.95227.28122.9176.2239.1322.7814.844.852.351.240.360.110.50ARL372.77292.15277.20186.86116.4476.8040.3526.4719.166.373.232.201.361.13MDRL242.50199.00184.50129.0084.0057.0034.5023.0017.006.003.002.001.001.00SDRL421.10321.22303.40190.83118.8772.9032.4820.8914.434.442.201.380.660.350.70ARL368.36291.14266.67170.19111.4363.5736.2221.4816.135.392.862.011.341.11MDRL239.50194.00180.00116.5080.0050.0031.0019.0014.005.002.002.001.001.00SDRL388.46310.09292.56174.63107.9556.5428.7816.1111.853.851.881.120.520.310.90ARL370.71266.00243.61139.1078.5953.2727.1517.9912.304.012.141.521.091.01MDRL245.00171.00163.0099.0059.0042.0023.0016.0010.003.002.001.001.001.00SDRL383.69280.70257.04138.7171.7444.7721.0312.929.132.851.360.750.290.110.25, 2.0450.30ARL372.80344.43325.65261.86168.50121.6963.3039.5924.327.794.473.172.242.03MDRL250.00230.00236.00177.50119.0089.5046.0029.0019.007.004.003.002.002.00SDRL380.99347.62312.31267.28160.73118.3657.8834.3418.424.262.171.320.520.180.50ARL371.94337.96304.40236.55159.02107.1356.4533.3223.407.054.152.952.162.02MDRL249.50240.50196.50156.50114.0073.0041.0026.0019.006.004.003.002.002.00SDRL378.92325.82317.83241.56150.45106.1553.8728.0417.343.982.021.190.420.130.70ARL372.26333.52317.06227.03147.7495.5649.1927.6618.956.123.622.602.072.00MDRL247.00232.50207.50153.0099.0070.0036.0021.0016.006.003.002.002.002.00SDRL389.33325.75346.54235.50152.8288.0143.3822.6113.863.411.690.870.290.040.90ARL368.96319.23277.69185.12114.7470.3234.7819.4813.914.632.681.901.331.11MDRL243.00214.00199.00128.5081.5051.0027.0016.0012.004.002.002.001.001.00SDRL377.52323.88280.01183.71114.7763.4529.2414.789.942.601.440.840.480.310.50, 2.7250.30ARL370.94333.83350.08292.49225.25175.1694.5158.6037.479.344.733.472.552.23MDRL255.00220.00237.00210.50158.00113.0068.5042.0027.008.004.003.002.002.00SDRL363.45345.01342.11276.99238.32183.4387.7354.9433.296.502.091.230.600.420.50ARL368.41336.70312.65270.29210.88154.2083.5652.1534.798.344.353.062.252.05MDRL253.50237.00220.50198.00147.00111.0057.0035.0025.007.004.003.002.002.00SDRL351.77342.73305.48265.20208.49153.4082.9050.5531.975.642.041.100.460.220.70ARL370.94364.34336.00280.24203.91152.0976.8345.7128.226.933.832.842.162.02MDRL246.50273.50234.00193.50144.50102.0054.5035.0021.006.004.003.002.002.00SDRL378.51357.52335.50276.70200.57156.7876.0939.6125.804.371.580.900.390.150.90ARL369.05349.47318.40251.53170.83111.8355.2432.5918.515.092.972.342.022.00MDRL271.00260.00221.00173.00126.0078.0038.5024.0014.005.003.002.002.002.00SDRL354.84328.59314.95247.96161.02106.9351.7930.1915.072.721.130.630.150.03Table 6
ARL measures of the proposed control chart with imperfect ranked set sampling under normal distribution atn=10 and ARL0=370.
δλ,Lγ00.0250.030.050.0750.10.150.20.250.50.7511.520.05, 1.760.30ARL373.11218.89192.5299.2260.6139.7820.6413.469.842.931.561.171.011.00MDRL204.50146.00131.0078.0048.0034.0016.0010.508.002.001.001.001.001.00SDRL454.49241.25198.8097.0457.9233.9718.1611.818.362.441.000.440.090.000.50ARL373.21207.83168.2298.2457.0737.6419.4812.868.892.621.381.121.001.00MDRL216.50128.00112.0075.5048.0030.0016.009.007.002.001.001.001.001.00SDRL477.82243.21178.19100.5550.7833.8816.6511.207.602.070.810.380.050.000.70ARL372.35178.19155.3876.4845.1329.6316.6210.076.782.011.271.041.001.00MDRL231.00118.00105.5059.0037.0024.5013.008.005.001.001.001.001.001.00SDRL454.56199.24164.8375.3039.1825.7514.878.915.751.480.630.190.000.000.90ARL374.41130.95112.5655.2330.9920.1710.806.543.991.431.041.001.001.00MDRL255.0093.5086.5044.0025.0017.009.005.003.001.001.001.001.001.00SDRL430.14142.14109.6550.6627.5416.648.995.683.470.790.190.050.000.000.10, 2.0450.30ARL371.60251.30203.89117.6967.6744.8922.9215.1910.753.551.821.241.011.00MDRL234.50180.50142.5086.0049.0037.0020.0013.5010.003.001.001.001.001.00SDRL414.94261.17210.13124.0461.2537.3717.8711.198.042.541.140.560.090.030.50ARL368.71236.57205.49118.9664.6540.9821.8614.369.633.091.671.171.011.00MDRL236.00152.00146.5082.5050.0035.0019.0013.009.002.001.001.001.001.00SDRL395.36259.02207.69120.2257.5334.1117.0610.427.002.291.040.440.070.030.70ARL369.42220.71182.6692.7751.6033.8517.3811.207.962.461.371.081.001.00MDRL249.00157.00122.0068.0041.0028.0015.0010.007.002.001.001.001.001.00SDRL380.61230.25186.9085.0744.6529.4212.758.396.011.750.720.300.050.000.90ARL368.55154.11128.5763.3034.4121.6211.697.514.951.541.071.011.001.00MDRL244.50102.0097.5049.0028.0018.0010.007.004.001.001.001.001.001.00SDRL376.02169.42124.1957.5727.8816.539.135.293.810.920.310.090.000.000.25, 2.4450.30ARL368.15291.14249.72167.77103.4464.4131.1019.1412.314.272.321.601.091.01MDRL257.50201.00176.50118.0072.0048.0024.0016.0011.004.002.001.001.001.00SDRL368.63304.07269.34162.29100.2258.6525.5014.848.822.411.230.760.290.090.50ARL371.23299.87272.71172.55100.0059.6228.4615.8311.113.882.091.471.051.00MDRL260.00210.00179.00131.0073.0041.0022.0013.0010.003.002.001.001.001.00SDRL364.66296.55284.84157.2193.6057.6622.1811.107.062.211.100.610.220.040.70ARL373.52261.53224.48128.5774.5146.1121.3912.858.932.891.681.221.001.00MDRL256.00187.50152.0093.0053.0033.0017.0011.008.003.001.001.001.001.00SDRL373.99261.14230.15125.3469.3842.7217.558.425.751.640.820.470.050.000.90ARL368.26204.11178.2188.5745.6025.6913.018.015.781.951.221.031.001.00MDRL244.50147.50132.5062.0033.0019.0011.007.005.002.001.001.001.001.00SDRL372.36199.39164.4187.9041.7022.399.145.093.451.060.470.160.000.000.50, 2.7640.30ARL371.28308.77280.73225.25146.06101.0546.5225.4616.154.492.531.911.321.06MDRL272.50226.50205.00153.00106.0067.0032.0018.0012.004.002.002.001.001.00SDRL340.94288.19268.66217.69139.07100.3941.5722.7713.142.431.050.710.470.240.50ARL369.52329.86318.54231.50150.5988.1745.1523.4414.673.972.311.751.231.03MDRL248.00231.50213.50164.00117.0061.0033.0017.0011.004.002.002.001.001.00SDRL368.35328.72329.20220.58138.5089.2741.2319.8412.322.020.950.610.420.170.70ARL367.86299.83271.13188.83122.4276.2031.6417.5811.073.281.951.401.051.00MDRL243.00213.50176.50131.0087.0054.0023.0014.009.003.002.001.001.001.00SDRL333.88292.66272.78192.12119.0573.7528.3314.718.811.670.790.550.210.030.90ARL367.95247.37232.26139.1973.1740.6817.759.366.102.151.321.051.001.00MDRL261.50180.00165.5094.0051.0029.0013.007.005.002.001.001.001.001.00SDRL356.50243.31219.41138.1772.3237.5314.597.173.930.990.510.210.000.00In terms of OOC performance at a specific shiftδ, the proposed TEWMA−SRRSS control chart outperforms all other control charts. For instance, at δ=0.05,λ=0.25, and n=5, the ARL1 values of the proposed TEWMA−SRRSS, TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts are 151.84, 251.08, 341.92, 337.05, and 167.12, respectively. The ARL is used as a metric to compare the proposed and existing control charts. Some findings are presented in the percentage decreasing in ARL=ARL0−ARL1/ARL0×100%. Under normal setup, the TEWMA−SRRSS control chart displaying a 3 percent improvement in the process median minimizes the ARL by 54.56 percent for n=5,λ=0.05, andARL0=370, whereas at the same shift δ, the proposed control chart reduced the ARL by 80.55 percent for n=10,λ=0.05, andARL0=370 (see Tables 3 and 4).


## 3.2.1. Evaluation of the Run-Length Distribution

The performance of the proposed control chart can be assessed in both normal and non-normal symmetric distributions. The distributions used in this analysis are as follows:(i)
Standard normal distribution, i.e.,N0,1(ii)
Double exponential (Laplace) distribution, i.e.,DE0,1/2(iii)
Heavy tail Student’st distribution tv, with the degree of freedom v=4and8(iv)
The logistic distribution, i.e.,LOG0,3/π(v)
Contaminated normal (CN) distribution, which is a mixture ofN0,σ12 and N0,σ02, i.e., 1−αN0,σ12+αN0,σ02, with α such that 1−ασ12+ασ02=1, where α=0.05 and σ1/σ0=2. For comparison purpose, all the considered distributions in this analysis were reparametrized with mean zero and unit standard deviation. The probability density functions of the distributions used in this study are shown in Table 2.Table 1
Details about the control charts used in this study.
Control chart’s nameNotationPlotting statisticsControl limits1NP EWMA under RSSEWMA−SRRSSEq=λSRRSSq+1−λEq−1UCLEWMA−SRRSS=+Lλ/2−λrr+12r+1/6ω02LCLEWMA−SRRSS=−Lλ/2−λrr+12r+1/6ω022NP DEWMA under RSSDEWMA−SRRSSEq=λSRRSSq+1−λEq−1,DEq=λEq+1−λDEq−1LCLDEWMA−SRRSS=−Lrr+12r+1/6ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q2/1−λ23UCLDEWMA−SRRSS=+Lrr+12r+1/6ω02λ41+λ2−q2+2q+1λ2q+2q2+2q−1λ2q+2−q2/1−λ233NP TEWMA under RSSTEWMA−SRRSSEq=λSRRSSq+1−λEq−1,DEq=λEq+1−λDEq−1,TEq=λDEq+1−λTEq−1LCLTEWMA−SRRSS=−Lσ61−λ6λ/2−λ5+121−λ4λ2/2−λ4+71−λ2λ3/2−λ3+λ4/2−λ2nn+12n+1/6ω02UCLTEWMA−SRRSS=+Lσ61−λ6λ/2−λ5+121−λ4λ2/2−λ4+71−λ2λ3/2−λ3+λ4/2−λ2nn+12n+1/6ω02


## 3.2.2. Monte Carlo Simulation

The Monte Carlo simulation is used to obtain theRL characteristics of the proposed control chart. The simulation algorithm is developed in R software to compute the RL characteristics. 104 random samples of the size n>1 from any distribution covered by the study for a shift δ were generated. The simulation algorithm to determine the nominal values of ARL0 under various distributions can be explained in the following steps [36]:(a)
An finite loop is practiced to generate samples from different distributions(b)
Select a particular value forλ(c)
Determine theTEq plotting statistic offered in equation (9)(d)
Let us assumeL=3(e)
Calculate theUCLTEWMA−SRRSSandLCLTEWMA−SRRSS from equation (11)(f)
Plot theTEq statistic against control limits in step (e) over q(g)
If theTEq>UCLTEWMA−SRRSSorTEq<LCLTEWMA−SRRSS, record the sample number of TEq statistic as an RL. For example, at q=255, if TEq>UCLTEWMA−SRRSSorTEq<LCLTEWMA−SRRSS record 255 as a first RL.(h)
Repeat steps from (a) to (g) for 105 times and record RLs.(i)
Compute the average of105 times noted RLs, which is called ARL0.(j)
IfARL0=400; otherwise, adjust L (i.e., L<3orL>3) constant accordingly in step (d) and repeat steps from (a) to (i) to obtain ARL0=400.(k)
To compute theARL1 values, again generate samples from relevant distribution and repeat steps from (b) to (i).


## 3.2.3. In-Control Robustness of the Proposed Control Chart

The EWMA andDEWMA−X¯ control charts have the limitation that their IC RL features do not remain the same for all continuous distributions. The IC RL characteristics of the NP control chart’s characteristics remain the same for all symmetric continuous distributions [4]. The simulated results of the proposed control chart with different combinations of λandL to obtain the nominal ARL0=370 are shown in Tables 3 and 4. The IC RL distribution of the proposed TEWMA−SRRSS control chart seems to be similar for all distributions used in this study. The RL distribution is positively skewed when ARL>MDRL.Table 2
PDFs of the continuous distributions used for this study.
(i)Standard normalfX=e−x2/2/2π, where X∈ℜ,M0=0,andσ2=1(ii)Student’stυfX=Γυ+1/2/Γυ/2υπ1+X2/υυ+1/2, where X∈ℜ,M0=0,andσ2=υ/υ−2 and υ=4,8 are taken(iii)LogisticfX=e−πX/3/3/π1+e−πX/32, where X∈ℜ,M0=0,andσ2=3/π2(iv)LaplacefX=1/2e−X, where X∈ℜ,M0=0,andσ2=1/2(v)Contaminated normal (CN)fX=0.95e−X2/2/2π+0.05e−X2/2σ02/σ02π, where X∈ℜ,M0=0,andσ2=0.95+0.05σ02Table 3
RL characteristics of the proposedTEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=5.
δλ,L0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.585)N(0, 1)ARL371.11206.77168.6292.6750.5333.9119.1212.027.942.351.371.081.011.00MDRL218.00131.00111.0070.0041.0028.0017.009.007.002.001.001.001.001.00SDRL453.10234.01183.7090.3349.0629.1515.7210.546.621.770.750.310.080.00CNARL369.92212.31164.0494.3151.9335.6319.3712.068.352.511.461.141.011.00MDRL217.50139.00113.0065.0040.0027.5016.009.006.002.001.001.001.001.00SDRL441.07235.19179.2897.3550.1533.3516.1610.777.502.080.870.430.130.04LogisticARL371.69189.29156.5180.7249.3730.4916.2110.277.202.181.311.101.011.00MDRL230.50118.00112.0062.0041.0025.0013.008.006.001.001.001.001.001.00SDRL429.72216.77159.5083.6344.7026.4214.268.946.391.610.660.330.110.03LaplaceARL371.65158.45123.5963.4934.1822.5412.257.525.482.011.321.121.031.00MDRL237.00106.5087.0050.0028.0018.0010.005.004.001.001.001.001.001.00SDRL428.92173.36126.8360.3330.9220.2910.726.894.851.500.660.380.170.06t4ARL369.29215.22187.04111.3462.0041.6523.5515.3910.433.341.791.361.091.02MDRL232.50126.50123.0083.0047.0033.0020.0012.008.002.001.001.001.001.00SDRL423.91254.75210.06111.4059.0239.9120.4512.909.582.791.310.700.340.17t8ARL370.51202.75174.3799.3658.9237.6420.0013.338.892.801.581.201.031.01MDRL231.00128.00118.0070.0045.0033.0015.0010.007.002.001.001.001.001.00SDRL455.11247.02189.65102.2255.9732.3117.5011.807.862.401.030.540.170.12(0.10, 1.837)N(0, 1)ARL370.58239.49200.36107.4457.6038.6621.5813.269.193.091.751.281.031.00MDRL241.50164.00143.0075.5044.0032.0019.0012.008.002.001.001.001.001.00SDRL399.98252.12200.34104.0952.2931.4716.689.586.522.100.970.540.180.03CNARL371.69222.24189.81110.7164.1940.5821.0213.659.673.281.791.321.051.01MDRL240.50155.00136.5075.0048.0034.0018.0012.008.003.002.001.001.001.00SDRL400.58233.16189.80111.1460.2732.0916.179.936.982.331.020.600.220.08LogisticARL369.89220.35179.1597.8552.1735.7718.2611.258.242.731.671.251.041.01MDRL257.50150.00124.0072.0040.0029.0016.0010.007.002.001.001.001.001.00SDRL379.94229.75177.9490.9947.5729.9613.908.425.941.800.890.520.200.07LaplaceARL369.00179.37145.5972.1539.0025.6214.368.836.402.421.521.271.051.01MDRL257.50117.50103.0054.0031.0022.0013.008.005.002.001.001.001.001.00SDRL379.94191.51141.8165.4633.7819.3910.666.284.771.580.830.560.230.10t4ARL369.26257.35229.58142.1981.2548.9026.1116.6812.293.982.271.601.161.04MDRL239.00166.50154.00102.0059.5038.0023.0014.0011.003.002.001.001.001.00SDRL382.67289.42240.47134.4775.2542.7319.5612.468.782.781.430.900.430.22t8ARL370.13241.63203.62125.9067.9944.7622.6315.9210.573.421.921.381.061.01MDRL250.00169.50139.0089.5052.0037.0020.0014.009.003.002.001.001.001.00SDRL417.20260.53202.72129.4859.3836.4417.1311.117.522.421.140.620.270.08(0.25, 2.204)N(0, 1)ARL370.74284.31253.51151.8488.7854.5724.9215.5710.383.592.091.441.071.01MDRL267.50199.50167.00108.5064.0038.5019.0013.009.003.002.001.001.001.00SDRL369.16291.85267.11151.3588.6650.8020.1711.016.992.131.060.630.260.08CNARL371.10293.39250.56154.4888.7957.9427.4515.7411.223.782.121.581.131.02MDRL273.00214.50172.50109.5064.0042.0021.0013.0010.003.502.001.001.001.00SDRL366.10279.85252.76141.0183.9053.5423.5911.447.552.101.080.750.370.15LogisticARL370.38271.57242.25137.5877.8747.2022.3413.599.283.231.951.441.081.02MDRL265.50180.50169.50103.0058.0036.0018.5012.008.003.002.001.001.001.00SDRL360.62282.24242.01135.3869.5640.4416.299.515.851.871.020.650.280.13LaplaceARL370.38230.11199.04105.7652.4430.7215.0810.527.072.821.831.351.101.02MDRL265.50156.50144.0075.0038.5023.0013.009.006.003.002.001.001.001.00SDRL360.62223.94196.34100.8147.6726.9711.197.144.411.680.970.650.330.17t4ARL371.04304.19282.63193.67113.3872.9632.7819.3713.044.522.521.851.281.11MDRL251.00208.50200.00136.5080.0051.0025.0015.0011.004.002.002.001.001.00SDRL367.97313.54280.19186.75112.5370.9227.1616.718.952.581.480.980.550.36t8ARL370.07116.22108.7681.7354.0536.0819.7012.128.853.081.891.351.071.01MDRL261.5082.5074.0060.5039.0026.0016.0011.008.003.002.001.001.001.00SDRL366.37115.99114.0880.4953.3236.9016.959.096.201.921.040.590.260.11(0.50, 2.491)N(0, 1)ARL371.08311.27311.97210.28137.4187.0238.2422.1712.963.722.121.471.071.00MDRL252.50207.50204.50147.5097.0063.0028.0016.0010.003.002.001.001.001.00SDRL381.21322.22323.87205.59134.8182.8635.6119.9510.582.131.090.670.260.04CNARL369.07317.34298.10218.94142.9391.0441.2922.5614.443.952.291.541.121.02MDRL254.00223.00210.00150.0099.0065.0030.0017.0011.004.002.001.001.001.00SDRL371.98317.49304.17207.28141.4286.4138.0020.7111.922.381.170.740.350.13LogisticARL371.67305.18284.01191.15116.2372.5434.4416.8111.033.481.981.391.081.02MDRL257.50201.00199.00129.5087.0052.0026.0013.009.003.002.001.001.001.00SDRL356.53305.98274.70187.91113.6370.1931.7713.288.811.811.070.620.300.13LaplaceARL371.67274.31254.24151.9081.5152.0522.6712.598.142.821.881.411.111.02MDRL257.50191.50179.00112.0060.0036.0017.0010.007.003.002.001.001.001.00SDRL356.53269.48255.88148.1377.5249.7019.7610.156.241.721.010.660.350.13t4ARL371.87322.54311.03239.91161.54109.6652.4528.8718.264.662.751.921.281.12MDRL250.00210.50220.50167.00110.0076.5036.0020.0014.004.003.002.001.001.00SDRL384.56349.01284.83233.04160.25106.9652.0326.9615.062.911.561.030.580.38t8ARL371.82320.55293.03225.65143.3995.9746.4624.4815.524.192.371.651.181.04MDRL260.00221.00203.00158.00103.0067.0034.0018.0012.004.002.001.001.001.00SDRL367.33335.03302.40221.66139.5696.2042.6320.9912.362.511.290.820.420.20


## 3.2.4. Out-of-Control Performance

OOC performance of the proposed control chart can be discussed under perfect RSS and imperfect RSS. The OOC control chart’s efficiency is primarily compared, and it demonstrates how sensitive the control chart is to detect shifts in process parameters. Tables3 and 4 show the OOC performance of the proposed TEWMA−SRRSS control chart for the various distributions evaluated in this study. For small and moderate shifts, the OOC performance of the proposed TEWMA−SRRSS control chart increases as n increases; i.e., ARL1 values decrease by increasing n. The proposed scheme’s RL values reduce with the rise in n and m values. The OOC RL performance of the Laplace distribution is higher as compared to the other distributions used in this study (see Figures 1 and 2). The 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). The 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,and0.10, the ARL values of the proposed control chart under RSS are 206.77,92.67,50.53,and33.9, whereas the ARL values of the proposed control chart under imperfect RSS are 295.71,178.64,111.43,and73.63 (see 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.Figure 1
ARL behavior of various distributions using the proposed TEWMA−SRRSS control chart when n=5, λ=0.05, and ARL0=370.Figure 2
ARL behavior of various distributions using the proposedTEWMA−SRRSS control chart when n=10, λ=0.05, and ARL0=370.Figure 3
ARL comparisons of the proposed TEWMA−SRRSS control chart for different values of λ when n=5 and ARL0=370.Figure 4
ARL comparisons of the proposed TEWMA−SRRSS control chart for different values of λ when n=10 and ARL0=370.Table 4
RL characteristics of the proposedTEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=10.
δλ,L0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.3635)N(0, 1)ARL369.6196.0171.8634.9919.8912.626.243.752.471.061.001.001.001.00MDRL240.067.054.029.016.010.05.03.02.01.01.01.01.01.0SDRL418.41101.6470.4930.6016.9610.995.503.191.950.260.000.000.000.00CNARL369.5498.5477.9937.5220.2113.226.203.852.651.111.011.001.001.00MDRL244.0069.0061.0032.0016.0011.005.003.002.001.001.001.001.001.00SDRL429.63102.0574.3631.9317.2211.365.243.252.180.370.070.000.000.00LogisticARL370.0485.1065.6030.6316.5310.775.083.092.191.061.001.001.001.00MDRL215.5064.0053.0025.0013.008.004.002.002.001.001.001.001.001.00SDRL451.5583.9561.1727.0714.169.514.442.541.640.260.040.000.000.00LaplaceARL371.0465.8251.9523.6812.327.754.032.361.741.041.001.001.001.00MDRL218.0055.0042.0019.009.006.003.002.001.001.001.001.001.001.00SDRL452.2961.7346.0920.9911.016.813.501.901.190.210.000.000.000.00t4ARL370.73116.7991.8743.4924.3615.907.424.853.291.231.021.001.001.00MDRL228.5084.0069.0035.0020.0013.005.004.002.001.001.001.001.001.00SDRL421.19125.8592.9840.0120.1613.586.694.042.710.540.140.000.000.00t8ARL369.67107.7585.8240.1523.7114.787.084.292.871.111.011.001.001.00MDRL225.0077.0065.0033.0020.0012.005.003.002.001.001.001.001.001.00SDRL441.35113.1583.9236.9018.6312.456.153.582.350.390.090.000.000.00(0.10, 1.57)N(0, 1)ARL369.43109.5284.4839.8021.0813.556.824.302.871.131.001.001.001.00MDRL242.5076.0061.0033.0018.0012.006.004.002.001.001.001.001.001.00SDRL401.47109.1979.7832.9916.7510.215.303.262.140.390.040.000.000.00CNARL369.18117.3391.8741.3223.1914.617.154.583.141.171.011.001.001.00MDRL244.5084.0068.0032.5021.0013.006.004.003.001.001.001.001.001.00SDRL401.76115.0087.6636.2716.7910.755.593.482.300.460.070.000.000.00LogisticARL369.92104.6673.8935.3419.3312.306.063.732.461.091.001.001.001.00MDRL243.0077.0054.0029.5017.0010.005.003.002.001.001.001.001.001.00SDRL419.0299.9371.9429.9014.519.214.562.781.760.330.060.000.000.00LaplaceARL369.9280.3057.7025.0813.958.934.652.882.011.081.001.001.001.00MDRL243.0060.0045.0022.0012.007.504.002.001.001.001.001.001.001.00SDRL419.0273.0352.3419.7110.636.763.662.161.380.290.070.000.000.00t4ARL370.86137.04109.4550.2125.9117.079.225.673.941.331.041.001.001.00MDRL251.0098.0080.0039.0023.0015.008.005.003.001.001.001.001.001.00SDRL388.38138.72108.9243.3119.8412.636.934.303.070.660.200.050.000.00t8ARL370.56126.76101.5448.4523.0115.848.165.043.331.211.011.001.001.00MDRL257.5087.0068.5038.0020.0014.007.004.003.001.001.001.001.001.00SDRL387.66130.74104.0141.8417.8811.166.143.682.390.530.100.000.000.00(0.25, 1.867)N(0, 1)ARL370.25148.30114.5053.8726.4615.297.574.943.561.251.021.001.001.00MDRL271.50103.0078.0040.0020.0013.007.004.003.001.001.001.001.001.00SDRL353.98143.90113.0449.0822.9411.385.062.972.090.520.130.000.000.00CNARL371.05154.57129.1059.2326.8116.477.955.343.661.301.021.001.001.00MDRL255.50105.5090.0044.0021.0014.007.005.003.001.001.001.001.001.00SDRL383.00152.56124.9556.4222.5412.564.993.362.090.560.140.000.000.00LogisticARL369.80142.12110.9745.0022.0113.106.864.233.031.201.011.001.001.00MDRL249.0097.0078.0032.0018.0011.006.004.003.001.001.001.001.001.00SDRL402.95147.38102.8141.4516.839.214.222.721.850.460.090.030.000.00LaplaceARL369.80111.9278.3833.4616.679.765.283.302.501.141.021.001.001.00MDRL249.0081.0054.0026.0014.009.005.003.002.001.001.001.001.001.00SDRL402.95105.9575.9828.8112.946.803.232.041.500.370.140.000.000.00t4ARL368.97186.35140.2371.3834.8619.439.786.194.221.561.091.011.001.00MDRL262.00131.0097.5050.0026.0016.009.006.004.001.001.001.001.001.00SDRL359.24181.23140.4467.1629.6714.586.514.052.630.790.300.110.000.00t8ARL371.13177.20148.4962.0930.5017.809.015.813.751.411.041.001.001.00MDRL259.00125.00101.5044.5023.0015.008.005.003.001.001.001.001.001.00SDRL367.57173.08155.7354.4625.6413.555.733.452.210.690.200.050.000.00(0.50, 2.1185)N(0, 1)ARL370.25203.92181.5381.6939.7321.858.995.163.641.351.021.001.001.00MDRL261.00144.00131.5055.0029.0016.007.005.003.001.001.001.001.001.00SDRL361.13203.76174.2781.5636.7919.276.493.352.030.560.150.000.000.00CNARL371.05220.20176.0189.5145.2823.669.175.403.821.401.051.001.001.00MDRL266.50144.00129.5063.0033.0017.007.005.003.001.001.001.001.001.00SDRL355.54224.64167.9786.1144.0920.666.813.402.190.590.210.040.000.00LogisticARL369.71207.36144.9472.6833.5718.757.774.623.281.261.011.001.001.00MDRL262.00140.50107.5053.0025.0015.007.004.003.001.001.001.001.001.00SDRL389.82220.88139.7969.4231.0815.695.482.891.810.490.120.040.000.00LaplaceARL369.71165.97118.1147.0121.8412.145.653.502.551.221.021.001.001.00MDRL262.00112.0081.5033.5016.009.005.003.002.001.001.001.001.001.00SDRL389.82177.25114.3642.6419.129.663.582.001.380.450.140.050.000.00t4ARL369.17242.08202.82104.2055.1229.4412.567.094.641.641.121.031.001.00MDRL261.50172.00141.0076.0038.0022.009.006.004.001.001.001.001.001.00SDRL359.03241.26199.81101.7154.0826.6110.824.732.730.790.340.160.040.00t8ARL369.74240.61203.2299.4948.1326.5510.965.744.131.561.061.011.001.00MDRL253.00173.00138.0067.0034.5020.009.005.004.001.001.001.001.001.00SDRL366.34229.90209.30100.8844.5024.288.713.802.320.740.240.080.000.00Table 5
ARL measures of the proposed control chart with imperfect ranked set sampling under normal distribution atn=5 and ARL0=370.
δλ,Lγ00.0250.030.050.0750.10.150.20.250.50.7511.520.05, 1.7930.30ARL373.21295.71274.87178.64111.4373.6339.9424.5318.506.032.972.021.341.12MDRL242.50192.50184.50118.5080.5059.0033.0020.0015.005.002.002.001.001.00SDRL437.59321.50293.65193.69114.1471.0035.3620.8315.185.012.311.280.560.320.50ARL370.86282.37265.80154.96101.6467.1737.9023.5316.465.172.631.661.131.02MDRL218.50174.50158.50106.5073.0053.0032.5020.0013.004.002.001.001.001.00SDRL454.57327.88315.97169.82104.7964.6132.5819.6514.334.482.191.130.400.160.70ARL372.89266.56243.76140.6688.9059.9633.1819.7715.024.542.291.591.091.01MDRL242.00169.50153.5095.5066.0049.0027.0016.0012.004.002.001.001.001.00SDRL452.69294.47298.29155.5887.2755.6128.4617.1612.793.711.730.980.320.090.90ARL372.08243.06209.25121.6566.3344.5424.9216.1211.253.481.831.321.041.00MDRL228.00160.00138.5084.5048.0036.0020.0013.009.003.001.001.001.001.00SDRL459.42272.24231.19127.6764.0138.8721.6713.489.352.781.190.630.190.040.1, 2.0690.30ARL373.72313.05303.81207.33131.8581.3646.5530.3721.237.283.952.732.092.01MDRL265.50213.00219.50139.5093.0059.0038.0025.0019.006.003.002.002.002.00SDRL405.22325.15300.95227.28122.9176.2239.1322.7814.844.852.351.240.360.110.50ARL372.77292.15277.20186.86116.4476.8040.3526.4719.166.373.232.201.361.13MDRL242.50199.00184.50129.0084.0057.0034.5023.0017.006.003.002.001.001.00SDRL421.10321.22303.40190.83118.8772.9032.4820.8914.434.442.201.380.660.350.70ARL368.36291.14266.67170.19111.4363.5736.2221.4816.135.392.862.011.341.11MDRL239.50194.00180.00116.5080.0050.0031.0019.0014.005.002.002.001.001.00SDRL388.46310.09292.56174.63107.9556.5428.7816.1111.853.851.881.120.520.310.90ARL370.71266.00243.61139.1078.5953.2727.1517.9912.304.012.141.521.091.01MDRL245.00171.00163.0099.0059.0042.0023.0016.0010.003.002.001.001.001.00SDRL383.69280.70257.04138.7171.7444.7721.0312.929.132.851.360.750.290.110.25, 2.0450.30ARL372.80344.43325.65261.86168.50121.6963.3039.5924.327.794.473.172.242.03MDRL250.00230.00236.00177.50119.0089.5046.0029.0019.007.004.003.002.002.00SDRL380.99347.62312.31267.28160.73118.3657.8834.3418.424.262.171.320.520.180.50ARL371.94337.96304.40236.55159.02107.1356.4533.3223.407.054.152.952.162.02MDRL249.50240.50196.50156.50114.0073.0041.0026.0019.006.004.003.002.002.00SDRL378.92325.82317.83241.56150.45106.1553.8728.0417.343.982.021.190.420.130.70ARL372.26333.52317.06227.03147.7495.5649.1927.6618.956.123.622.602.072.00MDRL247.00232.50207.50153.0099.0070.0036.0021.0016.006.003.002.002.002.00SDRL389.33325.75346.54235.50152.8288.0143.3822.6113.863.411.690.870.290.040.90ARL368.96319.23277.69185.12114.7470.3234.7819.4813.914.632.681.901.331.11MDRL243.00214.00199.00128.5081.5051.0027.0016.0012.004.002.002.001.001.00SDRL377.52323.88280.01183.71114.7763.4529.2414.789.942.601.440.840.480.310.50, 2.7250.30ARL370.94333.83350.08292.49225.25175.1694.5158.6037.479.344.733.472.552.23MDRL255.00220.00237.00210.50158.00113.0068.5042.0027.008.004.003.002.002.00SDRL363.45345.01342.11276.99238.32183.4387.7354.9433.296.502.091.230.600.420.50ARL368.41336.70312.65270.29210.88154.2083.5652.1534.798.344.353.062.252.05MDRL253.50237.00220.50198.00147.00111.0057.0035.0025.007.004.003.002.002.00SDRL351.77342.73305.48265.20208.49153.4082.9050.5531.975.642.041.100.460.220.70ARL370.94364.34336.00280.24203.91152.0976.8345.7128.226.933.832.842.162.02MDRL246.50273.50234.00193.50144.50102.0054.5035.0021.006.004.003.002.002.00SDRL378.51357.52335.50276.70200.57156.7876.0939.6125.804.371.580.900.390.150.90ARL369.05349.47318.40251.53170.83111.8355.2432.5918.515.092.972.342.022.00MDRL271.00260.00221.00173.00126.0078.0038.5024.0014.005.003.002.002.002.00SDRL354.84328.59314.95247.96161.02106.9351.7930.1915.072.721.130.630.150.03Table 6
ARL measures of the proposed control chart with imperfect ranked set sampling under normal distribution atn=10 and ARL0=370.
δλ,Lγ00.0250.030.050.0750.10.150.20.250.50.7511.520.05, 1.760.30ARL373.11218.89192.5299.2260.6139.7820.6413.469.842.931.561.171.011.00MDRL204.50146.00131.0078.0048.0034.0016.0010.508.002.001.001.001.001.00SDRL454.49241.25198.8097.0457.9233.9718.1611.818.362.441.000.440.090.000.50ARL373.21207.83168.2298.2457.0737.6419.4812.868.892.621.381.121.001.00MDRL216.50128.00112.0075.5048.0030.0016.009.007.002.001.001.001.001.00SDRL477.82243.21178.19100.5550.7833.8816.6511.207.602.070.810.380.050.000.70ARL372.35178.19155.3876.4845.1329.6316.6210.076.782.011.271.041.001.00MDRL231.00118.00105.5059.0037.0024.5013.008.005.001.001.001.001.001.00SDRL454.56199.24164.8375.3039.1825.7514.878.915.751.480.630.190.000.000.90ARL374.41130.95112.5655.2330.9920.1710.806.543.991.431.041.001.001.00MDRL255.0093.5086.5044.0025.0017.009.005.003.001.001.001.001.001.00SDRL430.14142.14109.6550.6627.5416.648.995.683.470.790.190.050.000.000.10, 2.0450.30ARL371.60251.30203.89117.6967.6744.8922.9215.1910.753.551.821.241.011.00MDRL234.50180.50142.5086.0049.0037.0020.0013.5010.003.001.001.001.001.00SDRL414.94261.17210.13124.0461.2537.3717.8711.198.042.541.140.560.090.030.50ARL368.71236.57205.49118.9664.6540.9821.8614.369.633.091.671.171.011.00MDRL236.00152.00146.5082.5050.0035.0019.0013.009.002.001.001.001.001.00SDRL395.36259.02207.69120.2257.5334.1117.0610.427.002.291.040.440.070.030.70ARL369.42220.71182.6692.7751.6033.8517.3811.207.962.461.371.081.001.00MDRL249.00157.00122.0068.0041.0028.0015.0010.007.002.001.001.001.001.00SDRL380.61230.25186.9085.0744.6529.4212.758.396.011.750.720.300.050.000.90ARL368.55154.11128.5763.3034.4121.6211.697.514.951.541.071.011.001.00MDRL244.50102.0097.5049.0028.0018.0010.007.004.001.001.001.001.001.00SDRL376.02169.42124.1957.5727.8816.539.135.293.810.920.310.090.000.000.25, 2.4450.30ARL368.15291.14249.72167.77103.4464.4131.1019.1412.314.272.321.601.091.01MDRL257.50201.00176.50118.0072.0048.0024.0016.0011.004.002.001.001.001.00SDRL368.63304.07269.34162.29100.2258.6525.5014.848.822.411.230.760.290.090.50ARL371.23299.87272.71172.55100.0059.6228.4615.8311.113.882.091.471.051.00MDRL260.00210.00179.00131.0073.0041.0022.0013.0010.003.002.001.001.001.00SDRL364.66296.55284.84157.2193.6057.6622.1811.107.062.211.100.610.220.040.70ARL373.52261.53224.48128.5774.5146.1121.3912.858.932.891.681.221.001.00MDRL256.00187.50152.0093.0053.0033.0017.0011.008.003.001.001.001.001.00SDRL373.99261.14230.15125.3469.3842.7217.558.425.751.640.820.470.050.000.90ARL368.26204.11178.2188.5745.6025.6913.018.015.781.951.221.031.001.00MDRL244.50147.50132.5062.0033.0019.0011.007.005.002.001.001.001.001.00SDRL372.36199.39164.4187.9041.7022.399.145.093.451.060.470.160.000.000.50, 2.7640.30ARL371.28308.77280.73225.25146.06101.0546.5225.4616.154.492.531.911.321.06MDRL272.50226.50205.00153.00106.0067.0032.0018.0012.004.002.002.001.001.00SDRL340.94288.19268.66217.69139.07100.3941.5722.7713.142.431.050.710.470.240.50ARL369.52329.86318.54231.50150.5988.1745.1523.4414.673.972.311.751.231.03MDRL248.00231.50213.50164.00117.0061.0033.0017.0011.004.002.002.001.001.00SDRL368.35328.72329.20220.58138.5089.2741.2319.8412.322.020.950.610.420.170.70ARL367.86299.83271.13188.83122.4276.2031.6417.5811.073.281.951.401.051.00MDRL243.00213.50176.50131.0087.0054.0023.0014.009.003.002.001.001.001.00SDRL333.88292.66272.78192.12119.0573.7528.3314.718.811.670.790.550.210.030.90ARL367.95247.37232.26139.1973.1740.6817.759.366.102.151.321.051.001.00MDRL261.50180.00165.5094.0051.0029.0013.007.005.002.001.001.001.001.00SDRL356.50243.31219.41138.1772.3237.5314.597.173.930.990.510.210.000.00In terms of OOC performance at a specific shiftδ, the proposed TEWMA−SRRSS control chart outperforms all other control charts. For instance, at δ=0.05,λ=0.25, and n=5, the ARL1 values of the proposed TEWMA−SRRSS, TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts are 151.84, 251.08, 341.92, 337.05, and 167.12, respectively. The ARL is used as a metric to compare the proposed and existing control charts. Some findings are presented in the percentage decreasing in ARL=ARL0−ARL1/ARL0×100%. Under normal setup, the TEWMA−SRRSS control chart displaying a 3 percent improvement in the process median minimizes the ARL by 54.56 percent for n=5,λ=0.05, andARL0=370, whereas at the same shift δ, the proposed control chart reduced the ARL by 80.55 percent for n=10,λ=0.05, andARL0=370 (see Tables 3 and 4).


## 4. Comparative Study

This section provides the proposed control chart’s comparisons to several existing control charts, which includesDEWMA−SRRSS [26], TEWMA−X¯ [33], TEWMA-SN [32], and TEWMA-SR [4] control charts. More detail is provided in the following sections.


### 4.1. Proposed versus TEWMA-SR Control Chart

The proposedTEWMA−SRRSS 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−SRRSS control chart reduces the ARL1 value by almost 78.28 percent, whereas the TEWMA-SR control chart reduces the ARL1 value by 58.74 percent (see Table 8). Similarly, under t4 distribution, when n=10 and λ=0.10 along with δ=5 percent, the ARL1 values of the proposed TEWMA−SRRSS 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−SRRSS control chart superiority over the TEWMA-SR control chart. It is worth mentioning that, for small shifts, the proposed TEWMA−SRRSS control chart outperforms the TEWMA-SR control chart for different combinations of λandn.Table 7
ARL measures of the proposed control chart with imperfect ranked set sampling undert-distribution (df = 4) at n=5 and ARL0=370.
δλ,Lγ00.0250.030.050.0750.100.150.200.250.500.751.001.502.000.05, 1.79450.30ARL372.32300.30267.46200.13126.6383.9950.8630.8323.707.944.202.671.711.34MDRL229.50189.50164.00137.5090.0063.0043.0025.0019.006.003.002.001.001.00SDRL440.89337.09303.15217.65133.3386.7343.2326.8820.066.383.411.971.040.630.50ARL373.06295.49275.52199.90112.1379.7244.2229.8820.366.823.692.371.481.17MDRL252.00180.00175.00134.5077.0062.0037.0024.0017.005.003.001.001.001.00SDRL430.16347.46322.23208.37123.5780.0440.8225.9817.175.983.051.920.960.510.70ARL372.74292.94265.49165.08103.5473.3639.5525.2318.636.203.222.121.351.15MDRL232.50193.00175.00104.5071.5057.0031.0020.0015.005.002.002.001.001.00SDRL445.33331.13295.51187.21111.5369.6836.4621.3115.625.142.631.530.720.440.90ARL373.19260.06233.04141.7384.4352.5630.2020.8015.074.662.491.721.211.07MDRL250.50175.00152.0092.0063.0041.0025.5016.5013.003.002.001.001.001.00SDRL504.40293.82266.34154.1584.3052.0425.5217.1412.444.001.841.110.560.300.10, 2.07050.30ARL372.42308.05295.79231.65145.63105.8457.3036.3225.029.235.173.372.442.14MDRL258.50211.50205.50156.50100.0076.0044.0030.0022.007.004.003.002.002.00SDRL393.33324.55292.76242.35144.67101.6053.0428.3918.496.673.471.910.910.480.50ARL372.70310.23288.24218.26136.0596.3749.9232.0622.748.144.412.921.751.37MDRL258.00194.50199.00145.0093.0069.0041.0028.0020.007.004.002.001.001.00SDRL387.69346.88301.66227.18141.2792.7644.8925.0617.636.163.292.081.120.700.70ARL372.71295.91285.55191.25123.5580.5846.9728.2819.627.113.812.551.621.35MDRL235.00203.50188.00130.0084.5059.0038.0024.0018.006.003.002.001.001.00SDRL406.83305.18308.32192.77126.2476.1839.8622.4414.285.122.621.640.890.610.90ARL373.16285.55255.73160.9697.7863.5834.7421.5516.015.352.831.941.321.14MDRL251.00198.50172.50108.5069.0049.0029.0019.0014.005.002.001.001.001.00SDRL421.97306.04267.54167.5298.1558.0128.4016.2212.173.681.961.230.650.420.25, 2.4640.30ARL368.53336.72331.86269.27196.21146.9381.0849.8131.5210.035.583.982.722.33MDRL261.50242.00225.50189.00131.00100.0057.0036.0024.009.005.004.002.002.00SDRL368.10336.93330.71269.50198.76144.1174.4945.8525.245.813.021.881.040.680.50ARL373.78347.10311.94270.53196.53139.7171.9842.8929.009.225.263.812.572.20MDRL279.50227.00210.00184.00138.0095.0054.5030.0023.008.005.003.002.002.00SDRL375.56371.42321.47276.45188.26137.4764.9040.3722.725.792.851.800.910.510.70ARL372.39324.77314.90250.82176.30120.3561.2036.6924.707.914.573.292.402.16MDRL257.00210.00213.00177.50121.0084.0045.0028.0020.007.004.003.002.002.00SDRL364.03337.33327.42255.45176.38111.5556.7630.3919.264.532.321.460.770.460.90ARL372.90311.54288.37207.47141.8791.5445.2525.9517.775.833.252.341.591.31MDRL258.00206.50203.00137.00101.0062.0033.0020.0014.005.003.002.001.001.00SDRL396.02306.39282.09215.36138.8098.0640.9222.4013.553.521.891.220.740.520.50, 2.740.30ARL372.39348.41370.85303.81261.66209.67122.9776.5950.6412.026.194.282.932.54MDRL257.00242.00257.00224.00179.00146.0090.0056.5036.009.005.004.003.002.00SDRL378.95348.15372.31291.54270.59208.75113.4771.5048.478.603.411.990.910.670.50ARL368.91357.44337.55291.16255.87186.62108.4670.6346.0010.975.523.902.642.29MDRL250.00237.50235.50208.00182.50134.0075.0053.0032.009.005.004.002.002.00SDRL375.15361.59327.06275.20240.18183.41102.2264.9041.907.813.201.670.850.580.70ARL367.89304.77311.64256.76217.63164.0191.4556.0237.319.034.763.412.482.19MDRL229.00215.00210.00182.50149.50113.0066.5040.0025.008.004.003.002.002.00SDRL323.61293.02319.15243.85217.08162.0886.3050.8136.106.052.481.400.700.440.90ARL371.53294.00283.18233.03175.14119.4064.2739.4224.286.183.482.451.661.31MDRL227.00189.00201.50159.00128.0079.5044.0028.0017.005.003.002.001.001.00SDRL332.40313.32276.24233.78170.76120.1461.6337.7823.134.122.001.310.770.54Table 8
ARL values of the proposedTEWMA-SRRSS, TEWMA-SR, TEWMA-SN, TEWMA-X¯, and DEWMA-SRRSS control charts for λ=0.05 under various distributions when ARL0=370..
δControl chartsLNormalt4t8LogisticLaplaceCNn = 50.00TEWMA-SRRSS1.585371.11369.03370.51371.69371.65369.92TEWMA-SR1.761370.21370.70370.50369.93369.93370.41TEWMA-SN1.755370.12371.34369.35370.63370.63371.45TEWMA-X¯1.761370.42384.87375.61369.18368.30420.27DEWMA-SRRSS1.742370.75369.1137.065369.58368.53371.090.05TEWMA-SRRSS1.58592.67111.3499.3680.7263.4994.31TEWMA-SR1.761170.21133.43154.45152.63117.81164.23TEWMA-SN1.755215.69147.36177.50173.25107.59190.75TEWMA-X¯1.761157.04159.27155.59156.45156.49164.35DEWMA-SRRSS1.74295.77114.90101.4084.8166.1397.510.10TEWMA-SRRSS1.58533.9141.6537.6430.4922.5435.63TEWMA-SR1.76171.1852.7863.2863.8347.1268.51TEWMA-SN1.755102.0759.5975.4774.3043.9784.11TEWMA-X¯1.76163.9264.9463.3464.5964.9465.32DEWMA-SRRSS1.74236.1245.0840.2733.1625.2337.700.25TEWMA-SRRSS1.5857.9410.438.897.205.488.35TEWMA-SR1.76118.9513.7516.6616.9312.6618.06TEWMA-SN1.75528.4615.8520.0920.0712.4422.93TEWMA-X¯1.76116.5716.5916.2416.5916.5916.57DEWMA-SRRSS1.7429.0312.0711.749.627.3210.560.50TEWMA-SRRSS1.5852.353.342.802.182.012.51TEWMA-SR1.7616.164.505.485.454.465.83TEWMA-SN1.7559.625.206.706.624.717.60TEWMA-X¯1.7615.074.995.025.105.075.00DEWMA-SRRSS1.7422.883.912.992.782.092.840.75TEWMA-SRRSS1.5851.371.791.581.311.321.46TEWMA-SR1.7613.152.492.862.882.572.98TEWMA-SN1.7554.882.813.473.452.803.83TEWMA-X¯1.7612.442.382.422.442.422.40DEWMA-SRRSS1.7421.851.991.891.731.791.721.00TEWMA-SRRSS1.5851.081.361.201.101.121.14TEWMA-SR1.7612.061.771.941.951.851.97TEWMA-SN1.7553.021.932.252.262.012.41TEWMA-X¯1.7611.541.501.531.531.521.52DEWMA-SRRSS1.7421.441.671.461.331.321.311.25TEWMA-SRRSS1.5851.011.091.031.011.031.01TEWMA-SR1.7611.581.441.531.541.511.52TEWMA-SN1.7552.141.521.691.701.611.74TEWMA-X¯1.7611.191.181.191.191.191.18DEWMA-SRRSS1.7421.031.191.091.041.071.051.50TEWMA-SRRSS1.5851.001.021.011.001.001.00TEWMA-SR1.7611.331.281.311.321.331.30TEWMA-SN1.7551.661.321.391.401.381.38TEWMA-X¯1.7611.061.061.071.061.061.05DEWMA-SRRSS1.7421.001.061.021.001.001.00n = 100.00TEWMA-SRRSS1.3635369.61370.73369.67371.04371.04369.54TEWMA-SR1.773370.71370.32371.50371.75371.75370.57TEWMA-SN1.774370.63369.08368.98370.48370.48371.49TEWMA-X¯1.761370.31378.00365.57366.34368.59425.74DEWMA-SRRSS1.503369.55369.84371.06371.38370.20370.520.05TEWMA-SRRSS1.363534.9943.4940.1530.6323.6837.52TEWMA-SR1.773111.1385.42102.17101.1177.70106.18TEWMA-SN1.774135.2894.83118.69115.9066.23129.65TEWMA-X¯1.761102.68105.01103.54102.70102.90105.26DEWMA-SRRSS1.50337.7747.8043.2233.1626.0239.680.10TEWMA-SRRSS1.363512.6215.9014.7810.777.7513.22TEWMA-SR1.77342.6931.6038.0538.3029.0640.28TEWMA-SN1.77453.1235.7545.5344.5625.6750.43TEWMA-X¯1.76139.0039.5839.0339.2939.2739.33DEWMA-SRRSS1.50314.5817.6815.6212.689.5015.230.25TEWMA-SRRSS1.36352.473.292.872.191.742.65TEWMA-SR1.77310.407.469.299.267.099.84TEWMA-SN1.77413.538.6111.3911.156.6512.78TEWMA-X¯1.7619.349.349.459.389.389.31DEWMA-SRRSS1.5033.544.173.712.821.982.990.50TEWMA-SRRSS1.36351.061.231.111.261.041.11TEWMA-SR1.7733.122.352.822.782.352.95TEWMA-SN1.7744.132.603.433.362.343.89TEWMA-X¯1.7612.722.692.722.692.692.67DEWMA-SRRSS1.5031.331.381.271.461.091.180.75TEWMA-SRRSS1.36351.001.021.011.001.001.01TEWMA-SR1.7731.621.391.531.521.431.55TEWMA-SN1.7742.021.451.751.731.441.92TEWMA-X¯1.7611.431.411.431.421.411.41DEWMA-SRRSS1.5031.001.051.041.011.001.011.00TEWMA-SRRSS1.36351.001.001.001.001.001.00TEWMA-SR1.7731.181.121.161.161.151.15TEWMA-SN1.7741.341.141.241.241.161.29TEWMA-X¯1.7611.101.091.101.091.091.09DEWMA-SRRSS1.5031.011.031.021.021.011.001.25TEWMA-SRRSS1.36351.001.001.001.001.001.00TEWMA-SR1.7731.041.041.051.041.051.03TEWMA-SN1.7741.111.041.081.081.061.09TEWMA-X¯1.7611.011.021.021.021.021.01DEWMA-SRRSS1.5031.001.011.001.001.011.001.50TEWMA-SRRSS1.36351.001.001.001.001.001.00TEWMA-SR1.7731.011.011.011.011.021.01TEWMA-SN1.7741.031.011.021.021.021.02TEWMA-X¯1.7611.001.001.001.001.001.00DEWMA-SRRSS1.5031.001.001.001.001.001.00Figure 5
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts for normal distribution when n = 5, λ = 0.05, and ARL0=370.Figure 6
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under t4 distribution when n=5, λ=0.05, and ARL0=370.


### 4.2. Proposed versus TEWMA-SN Control Chart

TheRL features of the TEWMA-SN control chart for the process location are listed in Table 10. The proposed TEWMA−SRRSS control chart is more efficient than the TEWMA-SN control chart. For example, under Laplace distribution, the proposed TEWMA−SRRSS control chart with n=10andλ=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 t8 distribution scenario, at n=10andλ=0.05 with a shift of 5 percent, the TEWMA-SN control chart decreases the ARL1 by 67.83 percent, while the proposed TEWMA−SRRSS control chart decreases the ARL1 by 89.13 percent (see Table 8).Table 9
RL characteristics of the TEWMA-SR control chart under different distribution whenARL0=370.
δλ,LDistr.0.000.050.100.250.500.751.001.251.50n = 5(0.10, 2.04)N(0, 1)ARL370.40202.4387.0821.417.414.032.792.302.10MDRL249.00139.0064.0019.006.003.002.002.002.00SDRL387.42207.0781.8915.835.072.471.310.710.39t4ARL371.00160.9362.4715.695.563.292.532.232.11MDRL252.00112.0049.0014.005.002.002.002.002.00SDRL389.09161.6854.7811.013.701.841.040.640.42t8ARL370.56185.1776.8918.896.653.712.682.282.11MDRL247.00128.0058.0017.006.003.002.002.001.00SDRL397.41189.5171.0513.664.542.191.190.690.41LogisticARL369.90183.2076.9619.076.633.722.692.282.12MDRL249.00125.0057.0017.006.003.002.002.002.00SDRL389.84185.3870.8813.764.512.201.210.700.42LaplaceARL369.90142.7155.4914.535.523.392.612.282.13MDRL249.00100.0043.0013.005.002.002.002.002.00SDRL389.84141.4548.4710.353.711.931.130.710.46CNARL369.40195.4483.3220.417.063.832.702.252.08MDRL250.00134.0062.0018.006.003.002.002.002.00SDRL386.73199.0477.8014.964.772.291.220.660.36(0.25, 2.435)N(0, 1)ARL370.04251.08123.7426.128.214.713.362.722.39MDRL255.00173.0087.0020.007.004.003.003.002.00SDRL371.26248.36120.3120.924.682.281.350.880.61t4ARL369.94208.6288.8417.896.293.892.982.562.34MDRL255.00145.0064.0015.006.003.003.002.002.00SDRL369.71207.5683.8812.733.361.781.140.810.62t8ARL370.11234.16110.1122.477.424.363.202.662.38MDRL254.00164.0078.0018.007.004.003.002.002.00SDRL370.95233.40105.2817.244.152.071.250.860.62LogisticARL370.34232.92109.1422.917.424.363.212.672.38MDRL254.00162.0077.0018.007.004.003.002.002.00SDRL374.21229.98104.6517.784.132.061.280.860.62LaplaceARL370.34191.5977.8816.526.233.983.092.642.40MDRL254.00134.0056.0014.006.004.003.002.002.00SDRL374.21188.3973.2511.803.371.871.230.870.66CNARL370.82244.07117.9724.677.864.523.252.652.35MDRL255.00168.0084.0019.007.004.003.002.002.00SDRL369.23243.23112.2019.614.412.141.280.830.57n = 10(0.10, 2.058)N(0, 1)ARL370.56135.1149.1911.813.791.961.341.111.03MDRL247.0094.0039.0010.003.002.001.001.001.00SDRL398.14136.8542.908.612.761.250.650.340.17t4ARL371.81102.9635.838.652.861.651.261.111.05MDRL248.0074.0030.007.002.001.001.001.001.00SDRL399.99100.5429.336.382.040.960.540.340.23t8ARL370.70123.1643.5310.623.421.841.311.121.04MDRL248.0087.0035.009.003.001.001.001.001.00SDRL400.75122.7536.857.712.461.140.620.360.21LogisticARL370.85122.5043.6010.573.401.831.311.121.04MDRL86.0035.009.003.001.001.001.001.001.00SDRL401.74123.0737.527.792.441.120.610.350.21LaplaceARL370.8593.1932.588.252.871.701.301.141.06MDRL247.0068.0027.007.002.001.001.001.001.00SDRL401.7490.8326.506.122.031.010.600.380.25CNARL370.25128.8246.6311.223.581.871.301.091.02MDRL248.0091.0037.0010.003.001.001.001.001.00SDRL400.04129.3940.588.262.611.170.600.310.15(0.25, 2.448)N(0, 1)ARL370.18180.5268.4013.214.442.451.691.331.14MDRL254.00126.0050.0011.004.002.002.001.001.00SDRL375.22180.1563.649.232.531.280.780.530.36t4ARL369.69143.2947.669.553.412.041.511.261.14MDRL255.00101.0035.009.003.002.001.001.001.00SDRL375.73142.4642.576.151.951.060.690.490.36t8ARL369.78168.2360.5211.794.052.301.631.311.15MDRL254.00117.0044.0010.004.002.001.001.001.00SDRL374.36166.3655.567.862.301.190.760.530.37LogisticARL370.66168.7060.2111.694.012.281.621.311.15MDRL255.00116.0043.0010.004.002.001.001.001.00SDRL376.79168.9855.647.932.291.180.750.530.37LaplaceARL370.66131.8243.169.073.432.101.571.311.16MDRL255.0092.0032.008.003.002.001.001.001.00SDRL376.79130.9938.695.861.971.100.740.530.39CNARL371.72175.2465.0512.484.222.351.631.291.12MDRL256.00123.0047.0011.004.002.001.001.001.00SDRL376.29173.6760.768.602.421.220.740.500.33Furthermore, atn=5andλ=0.05 with 10 percent shift in process location under the Logistic distribution, the TEWMA-SN control chart reduces the ARL1 by 79.95 percent and the proposed TEWMA−SRRSS control chart reduces the ARL1 by 91.79 percent (see Table 8). The proposed TEWMA−SRRSS control chart efficiency over the TEWMA-SN control chart can be seen in Figure 6. In short, the proposed TEWMA−SRRSS 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.


### 4.3. Proposed versusTEWMA−X¯ Control Chart

TheTEWMA−SRRSS 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−SRRSS control chart shows the OOC signal after 105.76 observations (see Table 8). The superiority of the proposed control chart can also be seen in the case of outliers under CN distribution. For example, the proposed TEWMA−SRRSS 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). The performance of the proposed control chart over the TEWMA−X¯ control chart is depicted in Figures 7 and 8. In brief, the proposed TEWMA−SRRSS control chart is quite sensitive for detecting process location shifts relative to the TEWMA−X¯ control chart.Table 10
RL characteristics of the TEWMA-SN control chart whenARL0=370.
δλL0.000.050.100.250.500.751.001.251.50n = 50.051.755ARL370.16300.16127.3665.3923.6312.297.373.341.76MDRL226.00189.0090.0052.0019.0010.006.003.001.00SDRL433.64343.98132.8261.4320.0710.436.222.641.150.102.057ARL370.66317.07150.5975.5125.5813.518.323.902.07MDRL244.00209.00104.0056.0022.0012.007.003.001.00SDRL403.07343.76154.9671.4420.3310.126.202.871.430.252.444ARL370.71341.92201.57108.1632.9015.619.404.712.86MDRL255.00236.00140.0077.0025.0013.008.004.002.00SDRL376.22344.07198.79103.5127.8711.135.812.491.160.502.744ARL370.00354.05254.57160.7352.5621.9411.655.153.10MDRL256.00244.00176.00112.0037.0016.009.005.003.00SDRL373.11356.39252.52158.1949.2918.878.652.691.100.752.807ARL369.67360.66294.33215.1088.4338.7919.316.743.34MDRL260.00249.00205.00149.0063.0027.0014.005.003.00SDRL367.82367.82358.70291.34213.8586.5036.7217.464.80n=100.051.774ARL370.65254.2080.0238.6313.346.563.771.681.09MDRL224.00157.0061.0031.0010.005.003.001.001.00SDRL441.63297.4780.6335.4311.915.933.371.240.360.102.047ARL370.66280.3697.5145.0315.257.844.722.191.30MDRL247.00189.0072.0036.0013.007.004.002.001.00SDRL395.10293.4893.1338.3511.105.683.381.370.550.252.454ARL370.39316.88137.4461.4417.268.435.162.441.35MDRL252.00219.0096.0044.0014.008.005.002.001.00SDRL379.72320.57135.9757.8913.235.503.251.520.660.502.761ARL370.52337.38189.4197.1624.8310.235.772.841.77MDRL254.00233.00133.0069.0018.008.005.003.002.00SDRL375.53337.71186.4994.3622.077.673.561.300.660.752.891ARL370.35351.34244.79148.5142.5115.957.763.111.79MDRL255.00243.00169.00103.0030.0012.006.003.002.00SDRL369.68351.31243.05148.0640.8414.166.061.700.70Figure 7
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under t8 distribution when n=5, λ=0.05, and ARL0=370.Figure 8
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under logistic distribution when n=5, λ=0.05, and ARL0=370.


### 4.4. Proposed versusDEWMA−SRRSS Control Chart

The proposedTEWMA−SRRSS control chart provides better performance than the DEWMA−SRRSS control chart. For example, under a normal environment, at n=10 and λ=0.50 for δ=2.5 percent, the proposed TEWMA−SRRSS control chart reduces the ARL1 by 44.92 percent, while the DEWMA−SRRSS control chart reduces the ARL1 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−SRRSS control chart decreases the ARL1 by 34.59 percent and the DEWMA−SRRSS control chart decreases the ARL1 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−SRRSS control chart gives OOC signal after 45.28 observations, whereas the TEWMA−SRRSS control chart does after 50.84 observations (see Tables 4 and 13). The comprehensive RL features of the DEWMA−SRRSS control chart for process location shift under selected distributions are shown in Tables 12 and 13. The TEWMA−SRRSS control chart outperforms the DEWMA−SRRSS control chart, as shown in Figures 9 and 10. Results indicate that the proposed structure works effectively in all environments relative to the DEWMA−SRRSS control chart.Table 11
RL characteristics of the TEWMA-X¯ control chart under different distributions when ARL0=370.
δλ,L0.000.050.100.250.500.751.001.251.50n = 5(0.10, 2.052)N(0, 1)ARL369.03307.06200.7265.8721.8311.547.174.793.55MDRL250.00202.00136.0049.0019.0010.006.004.003.00SDRL395.95328.82211.0760.9416.998.575.373.602.72t4ARL370.29186.9977.4518.305.802.861.751.261.10MDRL242.00124.0056.0016.005.002.001.001.001.00SDRL400.55194.0974.0413.244.362.151.180.600.35t8ARL369.13187.8475.7318.586.072.911.781.311.10MDRL241.50129.5056.0017.005.002.001.001.001.00SDRL401.65197.8671.4813.834.562.181.160.650.34LogisticARL369.25182.8276.0118.405.952.941.801.281.10MDRL251.00129.0057.0016.005.002.001.001.001.00SDRL395.31190.5571.1514.084.512.171.200.630.35LaplaceARL369.16181.9676.6318.555.952.851.781.311.10MDRL253.00130.0057.0016.005.002.001.001.001.00SDRL396.30191.0772.1314.254.472.071.170.640.33CNARL370.86184.2175.8418.455.882.841.761.301.10MDRL238.00127.5057.0016.005.002.001.001.001.00SDRL406.48187.5670.4013.784.482.111.140.650.35(0.25, 2.456)N(0, 1)ARL369.58337.05253.3192.0326.4512.908.045.494.12MDRL255.00237.00180.0063.0021.0011.007.005.004.00SDRL371.62344.14249.8990.2721.869.255.233.442.57t4ARL370.01244.15112.6822.796.563.352.091.501.20MDRL259.00163.0080.0018.006.003.002.001.001.00SDRL374.25244.05106.5718.024.022.031.240.790.47t8ARL369.15245.69113.5722.106.603.422.121.501.22MDRL252.00170.0079.0018.006.003.002.001.001.00SDRL378.00244.08113.2517.964.072.081.260.770.48LogisticARL370.81235.39109.9722.426.723.442.101.501.22MDRL259.50165.0076.0018.006.003.002.001.001.00SDRL379.06236.63106.0317.584.192.151.210.770.48LaplaceARL370.35238.73113.3522.416.523.392.111.501.21MDRL259.00165.0079.0018.006.003.002.001.001.00SDRL385.77241.88109.2917.633.992.111.240.770.49CNARL370.73240.89111.7221.686.623.292.111.521.21MDRL246.50171.0079.0017.006.003.002.001.001.00SDRL382.48237.76108.4617.444.192.041.250.790.48n = 10(0.10, 2.052)N(0, 1)ARL371.96311.91204.0264.0521.8711.436.994.933.50MDRL249.00206.00139.5048.0019.0010.006.004.003.00SDRL403.38328.93212.0260.2416.618.445.353.772.64t4ARL370.15128.1045.7710.673.171.601.151.041.01MDRL246.0091.0037.009.003.001.001.001.001.00SDRL397.13125.5039.697.952.350.990.440.220.11t8ARL369.95124.7944.4810.743.241.631.151.031.01MDRL257.0091.0036.009.003.001.001.001.001.00SDRL381.78120.8738.667.792.441.060.430.180.08LogisticARL370.40123.9644.6310.633.201.621.161.031.00MDRL242.0091.0037.009.002.001.001.001.001.00SDRL402.56121.8137.007.852.421.000.460.180.07LaplaceARL370.40124.1144.6210.643.181.611.161.031.00MDRL239.0091.0036.009.002.001.001.001.001.00SDRL399.49121.4338.267.942.421.000.450.190.06CNARL368.84123.6343.3910.443.181.641.161.031.00MDRL240.0085.5035.009.003.001.001.001.001.00SDRL397.07123.3536.987.952.321.020.440.170.06(0.25, 2.456)N(0, 1)ARL369.37330.89250.4091.1326.7312.707.735.474.09MDRL253.00226.00172.0064.0021.0011.007.005.004.00SDRL371.41339.08252.8788.3222.378.835.083.372.56t4ARL370.58179.7164.0712.033.761.911.291.071.01MDRL251.00122.0046.0010.003.002.001.001.001.00SDRL376.77186.9558.958.472.271.070.570.270.11t8ARL370.68171.9963.6011.623.761.931.261.081.01MDRL251.00123.0045.5010.003.002.001.001.001.00SDRL378.68167.4559.578.062.341.100.540.280.12LogisticARL370.68178.2462.8811.513.711.931.281.071.01MDRL244.50123.0046.0010.003.002.001.001.001.00SDRL382.00179.3159.267.902.341.100.560.270.10LaplaceARL369.07179.3163.9611.673.811.911.291.081.01MDRL241.00127.0047.0010.003.002.001.001.001.00SDRL387.13177.0659.188.242.301.090.580.280.12CNARL370.01165.6661.8811.623.721.941.281.061.01MDRL245.00115.0045.0010.003.002.001.001.001.00SDRL377.58167.7958.148.182.351.120.560.260.11Table 12
RL characteristics of theDEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=5.
δλ,L0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.742)N(0, 1)ARL370.75210.72169.5195.7750.5136.1218.8311.068.032.681.651.241.031.00MDRL242.50144.00111.5068.0039.0029.0016.009.006.002.002.001.001.001.00SDRL404.79229.03191.5096.9146.7931.0515.648.826.231.690.770.450.170.04CNARL371.09210.55162.7695.5151.9434.7017.8310.847.562.541.421.111.021.00MDRL221.00140.50110.5066.0037.5027.0014.009.006.002.001.001.001.001.00SDRL431.56229.54176.3299.1750.6531.6514.869.156.521.850.770.380.130.03LogisticARL369.58199.90156.3788.8148.9230.1615.779.916.622.381.531.231.031.00MDRL238.00134.00114.0069.0038.0024.0012.008.005.002.001.001.001.001.00SDRL412.81210.52162.9885.8543.4326.8612.727.955.271.460.710.440.190.03LaplaceARL368.58162.91122.1563.1334.1522.2311.657.405.322.041.491.221.041.01MDRL238.00114.5089.5049.0028.0019.009.006.004.002.001.001.001.001.00SDRL412.81162.22120.2958.1728.5518.039.335.694.091.180.680.450.210.10t4ARL369.11260.66221.13116.9068.7545.0824.5116.1412.073.311.941.491.151.06MDRL216.50164.00137.0077.0049.0032.0017.0011.008.003.002.001.001.001.00SDRL395.36262.45218.25119.4265.4736.6018.9011.467.962.241.070.690.380.25t8ARL370.65224.56205.5799.4060.7337.2720.8112.118.742.921.761.361.071.02MDRL250.00146.50139.0073.0046.0030.0017.0010.007.002.002.001.001.001.00SDRL423.83246.91220.3994.6857.4131.4017.189.486.981.980.900.600.270.15(0.10, 2.009)N(0, 1)ARL368.51248.23212.33122.6066.0339.8819.9612.358.793.041.711.271.031.00MDRL243.50172.50143.0086.0048.0032.0017.0011.008.003.001.001.001.001.00SDRL403.68249.20224.65119.5364.0833.6416.109.275.941.970.950.510.170.03CNARL371.13228.27192.20114.4964.7440.0521.1612.989.293.191.801.331.051.01MDRL255.00159.50140.5084.0049.0031.0017.0011.008.003.002.001.001.001.00SDRL391.82230.71186.98112.0359.5534.1216.859.906.452.040.960.550.230.07LogisticARL370.86227.50204.89106.7856.3435.2817.3811.278.292.611.641.221.031.01MDRL256.00159.00134.5077.5042.0027.0015.0010.008.002.001.001.001.001.00SDRL402.88233.86220.06106.8152.1329.4513.218.285.451.660.870.490.170.09LaplaceARL369.36189.52158.1579.3341.1525.4213.298.505.992.381.511.231.041.01MDRL263.00135.00111.5060.0033.0021.0011.007.005.002.001.001.001.001.00SDRL412.96194.09164.2874.5834.3522.2510.105.824.171.460.760.500.210.09t4ARL370.78292.00249.00149.6886.8858.7929.2217.8913.343.842.151.551.181.05MDRL223.50179.50161.0097.0058.0041.0022.0013.5010.003.002.001.001.001.00SDRL360.78286.72250.57141.8674.4848.3221.1611.958.112.491.350.800.450.22t8ARL371.53256.23227.76132.7070.1346.4222.9214.639.973.371.901.391.081.01MDRL256.50175.50156.5093.0051.0038.0020.0012.509.003.002.001.001.001.00SDRL390.44267.12235.60134.5366.1838.6317.3610.857.302.121.040.670.280.11(0.25, 2.362)N(0, 1)ARL370.58301.13273.03167.12102.3358.7030.5417.3710.933.472.041.471.081.01MDRL258.50199.50189.00110.0075.0043.0023.0014.009.003.002.001.001.001.00SDRL362.34306.15284.31169.2699.9354.4326.5113.307.401.990.990.650.270.08CNARL371.19294.66257.26177.09104.4963.7929.4417.4611.153.602.141.511.101.02MDRL274.50202.00176.50128.0072.0047.0021.0014.009.003.002.001.001.001.00SDRL353.65299.44256.86167.69105.0558.3526.8114.128.052.051.070.670.310.15LogisticARL371.17277.17257.67149.2683.1352.4523.7413.679.303.111.911.391.081.02MDRL275.00188.00176.00101.5060.0040.0018.0011.008.003.002.001.001.001.00SDRL348.81281.41258.33150.3481.1943.8120.299.886.421.750.990.620.290.14LaplaceARL371.11247.40222.46116.1159.4036.8616.149.957.102.721.681.361.111.02MDRL275.00162.50165.0085.0043.0027.0013.008.006.003.001.001.001.001.00SDRL348.81260.58212.18110.4055.1632.4912.307.194.631.480.860.620.330.15t4ARL369.52307.12297.61207.84132.8477.7537.6422.4214.384.452.561.791.281.10MDRL243.00211.00208.50150.0089.5057.0029.0017.0012.004.002.002.001.001.00SDRL376.56315.27279.56200.11139.0572.2031.4318.3410.672.661.440.890.550.34t8ARL369.32285.23271.09184.52109.2169.6131.6918.2511.773.822.251.621.151.05MDRL255.50196.50196.50130.0081.0052.0024.0015.0010.003.002.001.001.001.00SDRL354.63280.49266.28186.00102.3665.6327.8314.368.302.231.180.770.390.22(0.50, 2.601)N(0, 1)ARL371.14336.08330.95232.29171.68101.8750.1126.0216.064.192.381.791.311.11MDRL258.00240.00221.00164.50120.5073.0034.0018.0012.004.002.002.001.001.00SDRL380.72326.29339.71222.02164.6395.4549.0624.4613.482.441.020.650.470.31CNARL370.72316.65306.03239.43164.08103.6753.5328.8917.494.272.551.831.371.15MDRL251.00219.00209.00168.50115.0072.0039.0022.0013.004.002.002.001.001.00SDRL361.63317.66294.81225.10163.16101.6752.2826.5214.782.471.130.670.510.36LogisticARL369.36316.07300.44220.68133.2985.1840.4319.9513.483.672.201.721.331.12MDRL251.50217.50218.00148.0094.0056.0030.0014.0010.003.002.002.001.001.00SDRL357.96306.79298.34222.27128.5887.0736.9917.9811.632.050.960.680.470.33LaplaceARL371.17291.10261.49170.05102.2061.0026.5613.839.453.132.051.641.311.15MDRL252.50208.00190.50120.0076.0042.0019.0010.007.003.002.002.001.001.00SDRL363.24282.51261.00168.3597.9457.9524.3011.947.291.710.940.650.480.36t4ARL369.99341.67332.01268.76178.68128.8863.6436.7222.195.232.912.121.571.29MDRL256.50224.00226.00189.00125.0089.0048.0027.0016.004.003.002.002.001.00SDRL375.89347.85335.27261.40177.72130.4359.6834.0820.123.491.460.920.620.47t8ARL370.07326.33319.56233.66169.06118.7556.0531.4019.154.502.661.951.391.19MDRL234.50223.00222.00164.00123.0088.0041.0022.0014.004.002.002.001.001.00SDRL383.44327.84325.64228.44162.83113.1253.1629.2516.762.691.240.760.510.40Figure 9
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under Laplace distribution when n=5, λ=0.05, and ARL0=370.Figure 10
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under CN distribution when n=5, λ=0.05, and ARL0=370.Table 13
RL characteristics of theDEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=10.
δλ,LDistr.0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.503)N(0, 1)ARL369.5595.0177.7834.7719.0011.585.853.562.541.131.001.001.001.00MDRL236.0069.0059.0029.0016.0010.005.003.002.001.001.001.001.001.00SDRL408.6097.3274.5330.6315.459.084.412.541.820.370.030.000.000.00CNARL370.52105.6283.6336.6819.3912.236.143.882.621.151.011.001.001.00MDRL243.5077.0062.0028.0016.0010.005.003.002.001.001.001.001.001.00SDRL413.79106.3483.5632.1315.939.824.742.801.770.380.080.000.000.00LogisticARL371.3885.9266.9931.1615.769.685.133.102.221.081.011.001.001.00MDRL236.0065.0048.0026.0013.008.004.002.002.001.001.001.001.001.00SDRL443.4282.7967.5726.8412.957.643.922.281.420.280.070.000.000.00LaplaceARL370.2068.9651.9224.0212.467.503.872.521.851.061.001.001.001.00MDRL261.0052.0039.0020.0010.006.003.002.002.001.001.001.001.001.00SDRL467.2465.3850.6119.889.925.842.871.701.100.240.060.030.000.00t4ARL369.84121.67100.8344.8022.0714.687.144.703.171.281.041.001.001.00MDRL263.5088.0072.0037.0018.0012.006.004.002.001.001.001.001.001.00SDRL427.48121.9699.1039.3718.4012.185.713.492.340.540.210.050.000.00t8ARL371.06109.2883.6340.2220.9712.626.744.072.711.171.021.001.001.00MDRL236.5073.0059.0032.0018.0010.005.003.002.001.001.001.001.001.00SDRL416.69118.7884.2436.8517.3210.105.403.101.930.410.120.030.000.00(0.10, 1.709)N(0, 1)ARL368.73118.7292.4842.7820.9713.356.564.182.831.171.021.001.001.00MDRL247.5082.0067.0032.0017.5012.006.004.002.001.001.001.001.001.00SDRL366.84118.5090.4137.8816.379.304.532.741.820.420.120.000.000.00CNARL369.38124.70100.4245.8422.5013.726.884.382.961.211.011.001.001.00MDRL251.5086.0069.5036.0019.0012.006.004.002.001.001.001.001.001.00SDRL382.14120.4799.0739.7316.959.634.833.042.000.460.120.000.000.00LogisticARL370.94106.2983.5035.0617.8911.395.973.592.531.131.001.001.001.00MDRL245.5077.0059.0028.0015.0010.005.003.002.001.001.001.001.001.00SDRL405.08104.5881.3928.2913.938.143.972.441.610.350.040.000.000.00LaplaceARL369.3084.6661.2826.8513.358.474.442.872.151.101.011.001.001.00MDRL257.0062.0046.0022.0011.508.004.002.002.001.001.001.001.001.00SDRL448.5883.6357.5321.019.815.863.201.821.290.330.080.000.000.00t4ARL370.49145.27120.5453.2926.2316.738.485.353.601.371.051.011.001.00MDRL251.00104.0082.0040.0022.0015.007.005.003.001.001.001.001.001.00SDRL396.26140.85118.9050.6920.5412.506.073.612.520.600.220.080.000.00t8ARL371.30136.63106.1447.2023.8814.287.294.683.321.251.011.001.001.00MDRL258.0093.0077.0036.0020.0012.006.004.003.001.001.001.001.001.00SDRL383.52136.48105.0142.2419.1211.385.143.152.150.510.100.030.000.00(0.25, 1.995)N(0, 1)ARL368.94163.89125.8861.0228.9715.797.674.753.411.361.031.001.001.00MDRL261.50110.0086.0045.0020.0013.007.004.003.001.001.001.001.001.00SDRL361.42161.56121.8557.6926.8011.945.132.781.900.570.160.000.000.00CNARL369.14179.78142.1460.2330.0316.788.485.103.621.391.051.001.001.00MDRL267.00121.5099.5044.0023.0013.007.005.003.001.001.001.001.001.00SDRL370.83176.44139.8455.7026.5013.805.813.132.130.560.210.040.000.00LogisticARL371.19151.80119.4952.9522.9213.926.634.082.941.261.021.001.001.00MDRL248.00105.0085.0037.0017.5011.006.004.003.001.001.001.001.001.00SDRL383.41152.05116.7151.9718.6110.784.242.371.600.470.140.000.000.00LaplaceARL370.19121.1891.2136.8416.409.615.063.382.521.181.021.001.001.00MDRL248.0088.0062.0027.0013.008.005.003.002.001.001.001.001.001.00SDRL383.41116.5791.7630.9612.756.642.991.821.270.410.130.040.000.00t4ARL369.05196.86156.3578.6736.0520.409.985.874.221.601.131.021.001.00MDRL247.50140.00108.0059.0026.0015.009.005.004.001.001.001.001.001.00SDRL376.25193.47162.1070.8630.5017.246.793.742.440.720.340.150.030.00t8ARL370.83197.61155.4076.3932.2619.259.265.473.891.501.061.001.001.00MDRL273.00134.50108.0055.0023.0015.008.005.003.001.001.001.001.001.00SDRL361.65203.48149.0573.4829.8115.246.873.442.160.630.240.050.000.00(0.50, 2.205)N(0, 1)ARL370.24222.16196.7798.1547.6826.3010.285.803.801.471.041.001.001.00MDRL261.00164.00138.5069.0033.0019.008.005.003.001.001.001.001.001.00SDRL362.63215.71194.2397.3447.3823.668.333.792.160.600.200.030.000.00CNARL370.38228.32202.24102.6150.8428.1010.926.134.101.511.081.001.001.00MDRL270.00159.00147.5072.5036.0020.008.005.004.001.001.001.001.001.00SDRL362.46225.70192.8098.2744.7525.798.484.192.370.630.260.040.000.00LogisticARL371.47226.11167.0189.5040.3622.698.904.973.401.371.051.001.001.00MDRL263.50147.00121.0067.0028.0016.507.004.003.001.001.001.001.001.00SDRL381.73255.35168.8383.1940.2720.297.193.101.830.540.220.000.000.00LaplaceARL371.27192.89133.9760.9727.5814.456.253.762.701.271.041.001.001.00MDRL263.50125.5095.0040.0019.5011.005.003.002.001.001.001.001.001.00SDRL381.73222.77130.9560.0025.7712.074.272.111.420.470.190.060.000.00t4ARL370.71267.22213.56128.2564.6135.5014.427.455.031.751.191.041.001.00MDRL255.00186.00148.5092.0046.0025.0010.006.004.002.001.001.001.001.00SDRL386.36261.54218.84123.8861.6333.5212.575.343.180.780.390.180.030.00t8ARL370.13246.85218.05118.3058.2830.5412.456.784.401.591.101.011.001.00MDRL266.00169.00152.5085.0041.0022.009.005.004.001.001.001.001.001.00SDRL353.53244.73216.05116.5155.8628.0010.304.862.720.700.300.090.000.00


## 4.1. Proposed versus TEWMA-SR Control Chart

The proposedTEWMA−SRRSS 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−SRRSS control chart reduces the ARL1 value by almost 78.28 percent, whereas the TEWMA-SR control chart reduces the ARL1 value by 58.74 percent (see Table 8). Similarly, under t4 distribution, when n=10 and λ=0.10 along with δ=5 percent, the ARL1 values of the proposed TEWMA−SRRSS 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−SRRSS control chart superiority over the TEWMA-SR control chart. It is worth mentioning that, for small shifts, the proposed TEWMA−SRRSS control chart outperforms the TEWMA-SR control chart for different combinations of λandn.Table 7
ARL measures of the proposed control chart with imperfect ranked set sampling undert-distribution (df = 4) at n=5 and ARL0=370.
δλ,Lγ00.0250.030.050.0750.100.150.200.250.500.751.001.502.000.05, 1.79450.30ARL372.32300.30267.46200.13126.6383.9950.8630.8323.707.944.202.671.711.34MDRL229.50189.50164.00137.5090.0063.0043.0025.0019.006.003.002.001.001.00SDRL440.89337.09303.15217.65133.3386.7343.2326.8820.066.383.411.971.040.630.50ARL373.06295.49275.52199.90112.1379.7244.2229.8820.366.823.692.371.481.17MDRL252.00180.00175.00134.5077.0062.0037.0024.0017.005.003.001.001.001.00SDRL430.16347.46322.23208.37123.5780.0440.8225.9817.175.983.051.920.960.510.70ARL372.74292.94265.49165.08103.5473.3639.5525.2318.636.203.222.121.351.15MDRL232.50193.00175.00104.5071.5057.0031.0020.0015.005.002.002.001.001.00SDRL445.33331.13295.51187.21111.5369.6836.4621.3115.625.142.631.530.720.440.90ARL373.19260.06233.04141.7384.4352.5630.2020.8015.074.662.491.721.211.07MDRL250.50175.00152.0092.0063.0041.0025.5016.5013.003.002.001.001.001.00SDRL504.40293.82266.34154.1584.3052.0425.5217.1412.444.001.841.110.560.300.10, 2.07050.30ARL372.42308.05295.79231.65145.63105.8457.3036.3225.029.235.173.372.442.14MDRL258.50211.50205.50156.50100.0076.0044.0030.0022.007.004.003.002.002.00SDRL393.33324.55292.76242.35144.67101.6053.0428.3918.496.673.471.910.910.480.50ARL372.70310.23288.24218.26136.0596.3749.9232.0622.748.144.412.921.751.37MDRL258.00194.50199.00145.0093.0069.0041.0028.0020.007.004.002.001.001.00SDRL387.69346.88301.66227.18141.2792.7644.8925.0617.636.163.292.081.120.700.70ARL372.71295.91285.55191.25123.5580.5846.9728.2819.627.113.812.551.621.35MDRL235.00203.50188.00130.0084.5059.0038.0024.0018.006.003.002.001.001.00SDRL406.83305.18308.32192.77126.2476.1839.8622.4414.285.122.621.640.890.610.90ARL373.16285.55255.73160.9697.7863.5834.7421.5516.015.352.831.941.321.14MDRL251.00198.50172.50108.5069.0049.0029.0019.0014.005.002.001.001.001.00SDRL421.97306.04267.54167.5298.1558.0128.4016.2212.173.681.961.230.650.420.25, 2.4640.30ARL368.53336.72331.86269.27196.21146.9381.0849.8131.5210.035.583.982.722.33MDRL261.50242.00225.50189.00131.00100.0057.0036.0024.009.005.004.002.002.00SDRL368.10336.93330.71269.50198.76144.1174.4945.8525.245.813.021.881.040.680.50ARL373.78347.10311.94270.53196.53139.7171.9842.8929.009.225.263.812.572.20MDRL279.50227.00210.00184.00138.0095.0054.5030.0023.008.005.003.002.002.00SDRL375.56371.42321.47276.45188.26137.4764.9040.3722.725.792.851.800.910.510.70ARL372.39324.77314.90250.82176.30120.3561.2036.6924.707.914.573.292.402.16MDRL257.00210.00213.00177.50121.0084.0045.0028.0020.007.004.003.002.002.00SDRL364.03337.33327.42255.45176.38111.5556.7630.3919.264.532.321.460.770.460.90ARL372.90311.54288.37207.47141.8791.5445.2525.9517.775.833.252.341.591.31MDRL258.00206.50203.00137.00101.0062.0033.0020.0014.005.003.002.001.001.00SDRL396.02306.39282.09215.36138.8098.0640.9222.4013.553.521.891.220.740.520.50, 2.740.30ARL372.39348.41370.85303.81261.66209.67122.9776.5950.6412.026.194.282.932.54MDRL257.00242.00257.00224.00179.00146.0090.0056.5036.009.005.004.003.002.00SDRL378.95348.15372.31291.54270.59208.75113.4771.5048.478.603.411.990.910.670.50ARL368.91357.44337.55291.16255.87186.62108.4670.6346.0010.975.523.902.642.29MDRL250.00237.50235.50208.00182.50134.0075.0053.0032.009.005.004.002.002.00SDRL375.15361.59327.06275.20240.18183.41102.2264.9041.907.813.201.670.850.580.70ARL367.89304.77311.64256.76217.63164.0191.4556.0237.319.034.763.412.482.19MDRL229.00215.00210.00182.50149.50113.0066.5040.0025.008.004.003.002.002.00SDRL323.61293.02319.15243.85217.08162.0886.3050.8136.106.052.481.400.700.440.90ARL371.53294.00283.18233.03175.14119.4064.2739.4224.286.183.482.451.661.31MDRL227.00189.00201.50159.00128.0079.5044.0028.0017.005.003.002.001.001.00SDRL332.40313.32276.24233.78170.76120.1461.6337.7823.134.122.001.310.770.54Table 8
ARL values of the proposedTEWMA-SRRSS, TEWMA-SR, TEWMA-SN, TEWMA-X¯, and DEWMA-SRRSS control charts for λ=0.05 under various distributions when ARL0=370..
δControl chartsLNormalt4t8LogisticLaplaceCNn = 50.00TEWMA-SRRSS1.585371.11369.03370.51371.69371.65369.92TEWMA-SR1.761370.21370.70370.50369.93369.93370.41TEWMA-SN1.755370.12371.34369.35370.63370.63371.45TEWMA-X¯1.761370.42384.87375.61369.18368.30420.27DEWMA-SRRSS1.742370.75369.1137.065369.58368.53371.090.05TEWMA-SRRSS1.58592.67111.3499.3680.7263.4994.31TEWMA-SR1.761170.21133.43154.45152.63117.81164.23TEWMA-SN1.755215.69147.36177.50173.25107.59190.75TEWMA-X¯1.761157.04159.27155.59156.45156.49164.35DEWMA-SRRSS1.74295.77114.90101.4084.8166.1397.510.10TEWMA-SRRSS1.58533.9141.6537.6430.4922.5435.63TEWMA-SR1.76171.1852.7863.2863.8347.1268.51TEWMA-SN1.755102.0759.5975.4774.3043.9784.11TEWMA-X¯1.76163.9264.9463.3464.5964.9465.32DEWMA-SRRSS1.74236.1245.0840.2733.1625.2337.700.25TEWMA-SRRSS1.5857.9410.438.897.205.488.35TEWMA-SR1.76118.9513.7516.6616.9312.6618.06TEWMA-SN1.75528.4615.8520.0920.0712.4422.93TEWMA-X¯1.76116.5716.5916.2416.5916.5916.57DEWMA-SRRSS1.7429.0312.0711.749.627.3210.560.50TEWMA-SRRSS1.5852.353.342.802.182.012.51TEWMA-SR1.7616.164.505.485.454.465.83TEWMA-SN1.7559.625.206.706.624.717.60TEWMA-X¯1.7615.074.995.025.105.075.00DEWMA-SRRSS1.7422.883.912.992.782.092.840.75TEWMA-SRRSS1.5851.371.791.581.311.321.46TEWMA-SR1.7613.152.492.862.882.572.98TEWMA-SN1.7554.882.813.473.452.803.83TEWMA-X¯1.7612.442.382.422.442.422.40DEWMA-SRRSS1.7421.851.991.891.731.791.721.00TEWMA-SRRSS1.5851.081.361.201.101.121.14TEWMA-SR1.7612.061.771.941.951.851.97TEWMA-SN1.7553.021.932.252.262.012.41TEWMA-X¯1.7611.541.501.531.531.521.52DEWMA-SRRSS1.7421.441.671.461.331.321.311.25TEWMA-SRRSS1.5851.011.091.031.011.031.01TEWMA-SR1.7611.581.441.531.541.511.52TEWMA-SN1.7552.141.521.691.701.611.74TEWMA-X¯1.7611.191.181.191.191.191.18DEWMA-SRRSS1.7421.031.191.091.041.071.051.50TEWMA-SRRSS1.5851.001.021.011.001.001.00TEWMA-SR1.7611.331.281.311.321.331.30TEWMA-SN1.7551.661.321.391.401.381.38TEWMA-X¯1.7611.061.061.071.061.061.05DEWMA-SRRSS1.7421.001.061.021.001.001.00n = 100.00TEWMA-SRRSS1.3635369.61370.73369.67371.04371.04369.54TEWMA-SR1.773370.71370.32371.50371.75371.75370.57TEWMA-SN1.774370.63369.08368.98370.48370.48371.49TEWMA-X¯1.761370.31378.00365.57366.34368.59425.74DEWMA-SRRSS1.503369.55369.84371.06371.38370.20370.520.05TEWMA-SRRSS1.363534.9943.4940.1530.6323.6837.52TEWMA-SR1.773111.1385.42102.17101.1177.70106.18TEWMA-SN1.774135.2894.83118.69115.9066.23129.65TEWMA-X¯1.761102.68105.01103.54102.70102.90105.26DEWMA-SRRSS1.50337.7747.8043.2233.1626.0239.680.10TEWMA-SRRSS1.363512.6215.9014.7810.777.7513.22TEWMA-SR1.77342.6931.6038.0538.3029.0640.28TEWMA-SN1.77453.1235.7545.5344.5625.6750.43TEWMA-X¯1.76139.0039.5839.0339.2939.2739.33DEWMA-SRRSS1.50314.5817.6815.6212.689.5015.230.25TEWMA-SRRSS1.36352.473.292.872.191.742.65TEWMA-SR1.77310.407.469.299.267.099.84TEWMA-SN1.77413.538.6111.3911.156.6512.78TEWMA-X¯1.7619.349.349.459.389.389.31DEWMA-SRRSS1.5033.544.173.712.821.982.990.50TEWMA-SRRSS1.36351.061.231.111.261.041.11TEWMA-SR1.7733.122.352.822.782.352.95TEWMA-SN1.7744.132.603.433.362.343.89TEWMA-X¯1.7612.722.692.722.692.692.67DEWMA-SRRSS1.5031.331.381.271.461.091.180.75TEWMA-SRRSS1.36351.001.021.011.001.001.01TEWMA-SR1.7731.621.391.531.521.431.55TEWMA-SN1.7742.021.451.751.731.441.92TEWMA-X¯1.7611.431.411.431.421.411.41DEWMA-SRRSS1.5031.001.051.041.011.001.011.00TEWMA-SRRSS1.36351.001.001.001.001.001.00TEWMA-SR1.7731.181.121.161.161.151.15TEWMA-SN1.7741.341.141.241.241.161.29TEWMA-X¯1.7611.101.091.101.091.091.09DEWMA-SRRSS1.5031.011.031.021.021.011.001.25TEWMA-SRRSS1.36351.001.001.001.001.001.00TEWMA-SR1.7731.041.041.051.041.051.03TEWMA-SN1.7741.111.041.081.081.061.09TEWMA-X¯1.7611.011.021.021.021.021.01DEWMA-SRRSS1.5031.001.011.001.001.011.001.50TEWMA-SRRSS1.36351.001.001.001.001.001.00TEWMA-SR1.7731.011.011.011.011.021.01TEWMA-SN1.7741.031.011.021.021.021.02TEWMA-X¯1.7611.001.001.001.001.001.00DEWMA-SRRSS1.5031.001.001.001.001.001.00Figure 5
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts for normal distribution when n = 5, λ = 0.05, and ARL0=370.Figure 6
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under t4 distribution when n=5, λ=0.05, and ARL0=370.


## 4.2. Proposed versus TEWMA-SN Control Chart

TheRL features of the TEWMA-SN control chart for the process location are listed in Table 10. The proposed TEWMA−SRRSS control chart is more efficient than the TEWMA-SN control chart. For example, under Laplace distribution, the proposed TEWMA−SRRSS control chart with n=10andλ=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 t8 distribution scenario, at n=10andλ=0.05 with a shift of 5 percent, the TEWMA-SN control chart decreases the ARL1 by 67.83 percent, while the proposed TEWMA−SRRSS control chart decreases the ARL1 by 89.13 percent (see Table 8).Table 9
RL characteristics of the TEWMA-SR control chart under different distribution whenARL0=370.
δλ,LDistr.0.000.050.100.250.500.751.001.251.50n = 5(0.10, 2.04)N(0, 1)ARL370.40202.4387.0821.417.414.032.792.302.10MDRL249.00139.0064.0019.006.003.002.002.002.00SDRL387.42207.0781.8915.835.072.471.310.710.39t4ARL371.00160.9362.4715.695.563.292.532.232.11MDRL252.00112.0049.0014.005.002.002.002.002.00SDRL389.09161.6854.7811.013.701.841.040.640.42t8ARL370.56185.1776.8918.896.653.712.682.282.11MDRL247.00128.0058.0017.006.003.002.002.001.00SDRL397.41189.5171.0513.664.542.191.190.690.41LogisticARL369.90183.2076.9619.076.633.722.692.282.12MDRL249.00125.0057.0017.006.003.002.002.002.00SDRL389.84185.3870.8813.764.512.201.210.700.42LaplaceARL369.90142.7155.4914.535.523.392.612.282.13MDRL249.00100.0043.0013.005.002.002.002.002.00SDRL389.84141.4548.4710.353.711.931.130.710.46CNARL369.40195.4483.3220.417.063.832.702.252.08MDRL250.00134.0062.0018.006.003.002.002.002.00SDRL386.73199.0477.8014.964.772.291.220.660.36(0.25, 2.435)N(0, 1)ARL370.04251.08123.7426.128.214.713.362.722.39MDRL255.00173.0087.0020.007.004.003.003.002.00SDRL371.26248.36120.3120.924.682.281.350.880.61t4ARL369.94208.6288.8417.896.293.892.982.562.34MDRL255.00145.0064.0015.006.003.003.002.002.00SDRL369.71207.5683.8812.733.361.781.140.810.62t8ARL370.11234.16110.1122.477.424.363.202.662.38MDRL254.00164.0078.0018.007.004.003.002.002.00SDRL370.95233.40105.2817.244.152.071.250.860.62LogisticARL370.34232.92109.1422.917.424.363.212.672.38MDRL254.00162.0077.0018.007.004.003.002.002.00SDRL374.21229.98104.6517.784.132.061.280.860.62LaplaceARL370.34191.5977.8816.526.233.983.092.642.40MDRL254.00134.0056.0014.006.004.003.002.002.00SDRL374.21188.3973.2511.803.371.871.230.870.66CNARL370.82244.07117.9724.677.864.523.252.652.35MDRL255.00168.0084.0019.007.004.003.002.002.00SDRL369.23243.23112.2019.614.412.141.280.830.57n = 10(0.10, 2.058)N(0, 1)ARL370.56135.1149.1911.813.791.961.341.111.03MDRL247.0094.0039.0010.003.002.001.001.001.00SDRL398.14136.8542.908.612.761.250.650.340.17t4ARL371.81102.9635.838.652.861.651.261.111.05MDRL248.0074.0030.007.002.001.001.001.001.00SDRL399.99100.5429.336.382.040.960.540.340.23t8ARL370.70123.1643.5310.623.421.841.311.121.04MDRL248.0087.0035.009.003.001.001.001.001.00SDRL400.75122.7536.857.712.461.140.620.360.21LogisticARL370.85122.5043.6010.573.401.831.311.121.04MDRL86.0035.009.003.001.001.001.001.001.00SDRL401.74123.0737.527.792.441.120.610.350.21LaplaceARL370.8593.1932.588.252.871.701.301.141.06MDRL247.0068.0027.007.002.001.001.001.001.00SDRL401.7490.8326.506.122.031.010.600.380.25CNARL370.25128.8246.6311.223.581.871.301.091.02MDRL248.0091.0037.0010.003.001.001.001.001.00SDRL400.04129.3940.588.262.611.170.600.310.15(0.25, 2.448)N(0, 1)ARL370.18180.5268.4013.214.442.451.691.331.14MDRL254.00126.0050.0011.004.002.002.001.001.00SDRL375.22180.1563.649.232.531.280.780.530.36t4ARL369.69143.2947.669.553.412.041.511.261.14MDRL255.00101.0035.009.003.002.001.001.001.00SDRL375.73142.4642.576.151.951.060.690.490.36t8ARL369.78168.2360.5211.794.052.301.631.311.15MDRL254.00117.0044.0010.004.002.001.001.001.00SDRL374.36166.3655.567.862.301.190.760.530.37LogisticARL370.66168.7060.2111.694.012.281.621.311.15MDRL255.00116.0043.0010.004.002.001.001.001.00SDRL376.79168.9855.647.932.291.180.750.530.37LaplaceARL370.66131.8243.169.073.432.101.571.311.16MDRL255.0092.0032.008.003.002.001.001.001.00SDRL376.79130.9938.695.861.971.100.740.530.39CNARL371.72175.2465.0512.484.222.351.631.291.12MDRL256.00123.0047.0011.004.002.001.001.001.00SDRL376.29173.6760.768.602.421.220.740.500.33Furthermore, atn=5andλ=0.05 with 10 percent shift in process location under the Logistic distribution, the TEWMA-SN control chart reduces the ARL1 by 79.95 percent and the proposed TEWMA−SRRSS control chart reduces the ARL1 by 91.79 percent (see Table 8). The proposed TEWMA−SRRSS control chart efficiency over the TEWMA-SN control chart can be seen in Figure 6. In short, the proposed TEWMA−SRRSS 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.


## 4.3. Proposed versusTEWMA−X¯ Control Chart

TheTEWMA−SRRSS 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−SRRSS control chart shows the OOC signal after 105.76 observations (see Table 8). The superiority of the proposed control chart can also be seen in the case of outliers under CN distribution. For example, the proposed TEWMA−SRRSS 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). The performance of the proposed control chart over the TEWMA−X¯ control chart is depicted in Figures 7 and 8. In brief, the proposed TEWMA−SRRSS control chart is quite sensitive for detecting process location shifts relative to the TEWMA−X¯ control chart.Table 10
RL characteristics of the TEWMA-SN control chart whenARL0=370.
δλL0.000.050.100.250.500.751.001.251.50n = 50.051.755ARL370.16300.16127.3665.3923.6312.297.373.341.76MDRL226.00189.0090.0052.0019.0010.006.003.001.00SDRL433.64343.98132.8261.4320.0710.436.222.641.150.102.057ARL370.66317.07150.5975.5125.5813.518.323.902.07MDRL244.00209.00104.0056.0022.0012.007.003.001.00SDRL403.07343.76154.9671.4420.3310.126.202.871.430.252.444ARL370.71341.92201.57108.1632.9015.619.404.712.86MDRL255.00236.00140.0077.0025.0013.008.004.002.00SDRL376.22344.07198.79103.5127.8711.135.812.491.160.502.744ARL370.00354.05254.57160.7352.5621.9411.655.153.10MDRL256.00244.00176.00112.0037.0016.009.005.003.00SDRL373.11356.39252.52158.1949.2918.878.652.691.100.752.807ARL369.67360.66294.33215.1088.4338.7919.316.743.34MDRL260.00249.00205.00149.0063.0027.0014.005.003.00SDRL367.82367.82358.70291.34213.8586.5036.7217.464.80n=100.051.774ARL370.65254.2080.0238.6313.346.563.771.681.09MDRL224.00157.0061.0031.0010.005.003.001.001.00SDRL441.63297.4780.6335.4311.915.933.371.240.360.102.047ARL370.66280.3697.5145.0315.257.844.722.191.30MDRL247.00189.0072.0036.0013.007.004.002.001.00SDRL395.10293.4893.1338.3511.105.683.381.370.550.252.454ARL370.39316.88137.4461.4417.268.435.162.441.35MDRL252.00219.0096.0044.0014.008.005.002.001.00SDRL379.72320.57135.9757.8913.235.503.251.520.660.502.761ARL370.52337.38189.4197.1624.8310.235.772.841.77MDRL254.00233.00133.0069.0018.008.005.003.002.00SDRL375.53337.71186.4994.3622.077.673.561.300.660.752.891ARL370.35351.34244.79148.5142.5115.957.763.111.79MDRL255.00243.00169.00103.0030.0012.006.003.002.00SDRL369.68351.31243.05148.0640.8414.166.061.700.70Figure 7
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under t8 distribution when n=5, λ=0.05, and ARL0=370.Figure 8
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under logistic distribution when n=5, λ=0.05, and ARL0=370.


## 4.4. Proposed versusDEWMA−SRRSS Control Chart

The proposedTEWMA−SRRSS control chart provides better performance than the DEWMA−SRRSS control chart. For example, under a normal environment, at n=10 and λ=0.50 for δ=2.5 percent, the proposed TEWMA−SRRSS control chart reduces the ARL1 by 44.92 percent, while the DEWMA−SRRSS control chart reduces the ARL1 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−SRRSS control chart decreases the ARL1 by 34.59 percent and the DEWMA−SRRSS control chart decreases the ARL1 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−SRRSS control chart gives OOC signal after 45.28 observations, whereas the TEWMA−SRRSS control chart does after 50.84 observations (see Tables 4 and 13). The comprehensive RL features of the DEWMA−SRRSS control chart for process location shift under selected distributions are shown in Tables 12 and 13. The TEWMA−SRRSS control chart outperforms the DEWMA−SRRSS control chart, as shown in Figures 9 and 10. Results indicate that the proposed structure works effectively in all environments relative to the DEWMA−SRRSS control chart.Table 11
RL characteristics of the TEWMA-X¯ control chart under different distributions when ARL0=370.
δλ,L0.000.050.100.250.500.751.001.251.50n = 5(0.10, 2.052)N(0, 1)ARL369.03307.06200.7265.8721.8311.547.174.793.55MDRL250.00202.00136.0049.0019.0010.006.004.003.00SDRL395.95328.82211.0760.9416.998.575.373.602.72t4ARL370.29186.9977.4518.305.802.861.751.261.10MDRL242.00124.0056.0016.005.002.001.001.001.00SDRL400.55194.0974.0413.244.362.151.180.600.35t8ARL369.13187.8475.7318.586.072.911.781.311.10MDRL241.50129.5056.0017.005.002.001.001.001.00SDRL401.65197.8671.4813.834.562.181.160.650.34LogisticARL369.25182.8276.0118.405.952.941.801.281.10MDRL251.00129.0057.0016.005.002.001.001.001.00SDRL395.31190.5571.1514.084.512.171.200.630.35LaplaceARL369.16181.9676.6318.555.952.851.781.311.10MDRL253.00130.0057.0016.005.002.001.001.001.00SDRL396.30191.0772.1314.254.472.071.170.640.33CNARL370.86184.2175.8418.455.882.841.761.301.10MDRL238.00127.5057.0016.005.002.001.001.001.00SDRL406.48187.5670.4013.784.482.111.140.650.35(0.25, 2.456)N(0, 1)ARL369.58337.05253.3192.0326.4512.908.045.494.12MDRL255.00237.00180.0063.0021.0011.007.005.004.00SDRL371.62344.14249.8990.2721.869.255.233.442.57t4ARL370.01244.15112.6822.796.563.352.091.501.20MDRL259.00163.0080.0018.006.003.002.001.001.00SDRL374.25244.05106.5718.024.022.031.240.790.47t8ARL369.15245.69113.5722.106.603.422.121.501.22MDRL252.00170.0079.0018.006.003.002.001.001.00SDRL378.00244.08113.2517.964.072.081.260.770.48LogisticARL370.81235.39109.9722.426.723.442.101.501.22MDRL259.50165.0076.0018.006.003.002.001.001.00SDRL379.06236.63106.0317.584.192.151.210.770.48LaplaceARL370.35238.73113.3522.416.523.392.111.501.21MDRL259.00165.0079.0018.006.003.002.001.001.00SDRL385.77241.88109.2917.633.992.111.240.770.49CNARL370.73240.89111.7221.686.623.292.111.521.21MDRL246.50171.0079.0017.006.003.002.001.001.00SDRL382.48237.76108.4617.444.192.041.250.790.48n = 10(0.10, 2.052)N(0, 1)ARL371.96311.91204.0264.0521.8711.436.994.933.50MDRL249.00206.00139.5048.0019.0010.006.004.003.00SDRL403.38328.93212.0260.2416.618.445.353.772.64t4ARL370.15128.1045.7710.673.171.601.151.041.01MDRL246.0091.0037.009.003.001.001.001.001.00SDRL397.13125.5039.697.952.350.990.440.220.11t8ARL369.95124.7944.4810.743.241.631.151.031.01MDRL257.0091.0036.009.003.001.001.001.001.00SDRL381.78120.8738.667.792.441.060.430.180.08LogisticARL370.40123.9644.6310.633.201.621.161.031.00MDRL242.0091.0037.009.002.001.001.001.001.00SDRL402.56121.8137.007.852.421.000.460.180.07LaplaceARL370.40124.1144.6210.643.181.611.161.031.00MDRL239.0091.0036.009.002.001.001.001.001.00SDRL399.49121.4338.267.942.421.000.450.190.06CNARL368.84123.6343.3910.443.181.641.161.031.00MDRL240.0085.5035.009.003.001.001.001.001.00SDRL397.07123.3536.987.952.321.020.440.170.06(0.25, 2.456)N(0, 1)ARL369.37330.89250.4091.1326.7312.707.735.474.09MDRL253.00226.00172.0064.0021.0011.007.005.004.00SDRL371.41339.08252.8788.3222.378.835.083.372.56t4ARL370.58179.7164.0712.033.761.911.291.071.01MDRL251.00122.0046.0010.003.002.001.001.001.00SDRL376.77186.9558.958.472.271.070.570.270.11t8ARL370.68171.9963.6011.623.761.931.261.081.01MDRL251.00123.0045.5010.003.002.001.001.001.00SDRL378.68167.4559.578.062.341.100.540.280.12LogisticARL370.68178.2462.8811.513.711.931.281.071.01MDRL244.50123.0046.0010.003.002.001.001.001.00SDRL382.00179.3159.267.902.341.100.560.270.10LaplaceARL369.07179.3163.9611.673.811.911.291.081.01MDRL241.00127.0047.0010.003.002.001.001.001.00SDRL387.13177.0659.188.242.301.090.580.280.12CNARL370.01165.6661.8811.623.721.941.281.061.01MDRL245.00115.0045.0010.003.002.001.001.001.00SDRL377.58167.7958.148.182.351.120.560.260.11Table 12
RL characteristics of theDEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=5.
δλ,L0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.742)N(0, 1)ARL370.75210.72169.5195.7750.5136.1218.8311.068.032.681.651.241.031.00MDRL242.50144.00111.5068.0039.0029.0016.009.006.002.002.001.001.001.00SDRL404.79229.03191.5096.9146.7931.0515.648.826.231.690.770.450.170.04CNARL371.09210.55162.7695.5151.9434.7017.8310.847.562.541.421.111.021.00MDRL221.00140.50110.5066.0037.5027.0014.009.006.002.001.001.001.001.00SDRL431.56229.54176.3299.1750.6531.6514.869.156.521.850.770.380.130.03LogisticARL369.58199.90156.3788.8148.9230.1615.779.916.622.381.531.231.031.00MDRL238.00134.00114.0069.0038.0024.0012.008.005.002.001.001.001.001.00SDRL412.81210.52162.9885.8543.4326.8612.727.955.271.460.710.440.190.03LaplaceARL368.58162.91122.1563.1334.1522.2311.657.405.322.041.491.221.041.01MDRL238.00114.5089.5049.0028.0019.009.006.004.002.001.001.001.001.00SDRL412.81162.22120.2958.1728.5518.039.335.694.091.180.680.450.210.10t4ARL369.11260.66221.13116.9068.7545.0824.5116.1412.073.311.941.491.151.06MDRL216.50164.00137.0077.0049.0032.0017.0011.008.003.002.001.001.001.00SDRL395.36262.45218.25119.4265.4736.6018.9011.467.962.241.070.690.380.25t8ARL370.65224.56205.5799.4060.7337.2720.8112.118.742.921.761.361.071.02MDRL250.00146.50139.0073.0046.0030.0017.0010.007.002.002.001.001.001.00SDRL423.83246.91220.3994.6857.4131.4017.189.486.981.980.900.600.270.15(0.10, 2.009)N(0, 1)ARL368.51248.23212.33122.6066.0339.8819.9612.358.793.041.711.271.031.00MDRL243.50172.50143.0086.0048.0032.0017.0011.008.003.001.001.001.001.00SDRL403.68249.20224.65119.5364.0833.6416.109.275.941.970.950.510.170.03CNARL371.13228.27192.20114.4964.7440.0521.1612.989.293.191.801.331.051.01MDRL255.00159.50140.5084.0049.0031.0017.0011.008.003.002.001.001.001.00SDRL391.82230.71186.98112.0359.5534.1216.859.906.452.040.960.550.230.07LogisticARL370.86227.50204.89106.7856.3435.2817.3811.278.292.611.641.221.031.01MDRL256.00159.00134.5077.5042.0027.0015.0010.008.002.001.001.001.001.00SDRL402.88233.86220.06106.8152.1329.4513.218.285.451.660.870.490.170.09LaplaceARL369.36189.52158.1579.3341.1525.4213.298.505.992.381.511.231.041.01MDRL263.00135.00111.5060.0033.0021.0011.007.005.002.001.001.001.001.00SDRL412.96194.09164.2874.5834.3522.2510.105.824.171.460.760.500.210.09t4ARL370.78292.00249.00149.6886.8858.7929.2217.8913.343.842.151.551.181.05MDRL223.50179.50161.0097.0058.0041.0022.0013.5010.003.002.001.001.001.00SDRL360.78286.72250.57141.8674.4848.3221.1611.958.112.491.350.800.450.22t8ARL371.53256.23227.76132.7070.1346.4222.9214.639.973.371.901.391.081.01MDRL256.50175.50156.5093.0051.0038.0020.0012.509.003.002.001.001.001.00SDRL390.44267.12235.60134.5366.1838.6317.3610.857.302.121.040.670.280.11(0.25, 2.362)N(0, 1)ARL370.58301.13273.03167.12102.3358.7030.5417.3710.933.472.041.471.081.01MDRL258.50199.50189.00110.0075.0043.0023.0014.009.003.002.001.001.001.00SDRL362.34306.15284.31169.2699.9354.4326.5113.307.401.990.990.650.270.08CNARL371.19294.66257.26177.09104.4963.7929.4417.4611.153.602.141.511.101.02MDRL274.50202.00176.50128.0072.0047.0021.0014.009.003.002.001.001.001.00SDRL353.65299.44256.86167.69105.0558.3526.8114.128.052.051.070.670.310.15LogisticARL371.17277.17257.67149.2683.1352.4523.7413.679.303.111.911.391.081.02MDRL275.00188.00176.00101.5060.0040.0018.0011.008.003.002.001.001.001.00SDRL348.81281.41258.33150.3481.1943.8120.299.886.421.750.990.620.290.14LaplaceARL371.11247.40222.46116.1159.4036.8616.149.957.102.721.681.361.111.02MDRL275.00162.50165.0085.0043.0027.0013.008.006.003.001.001.001.001.00SDRL348.81260.58212.18110.4055.1632.4912.307.194.631.480.860.620.330.15t4ARL369.52307.12297.61207.84132.8477.7537.6422.4214.384.452.561.791.281.10MDRL243.00211.00208.50150.0089.5057.0029.0017.0012.004.002.002.001.001.00SDRL376.56315.27279.56200.11139.0572.2031.4318.3410.672.661.440.890.550.34t8ARL369.32285.23271.09184.52109.2169.6131.6918.2511.773.822.251.621.151.05MDRL255.50196.50196.50130.0081.0052.0024.0015.0010.003.002.001.001.001.00SDRL354.63280.49266.28186.00102.3665.6327.8314.368.302.231.180.770.390.22(0.50, 2.601)N(0, 1)ARL371.14336.08330.95232.29171.68101.8750.1126.0216.064.192.381.791.311.11MDRL258.00240.00221.00164.50120.5073.0034.0018.0012.004.002.002.001.001.00SDRL380.72326.29339.71222.02164.6395.4549.0624.4613.482.441.020.650.470.31CNARL370.72316.65306.03239.43164.08103.6753.5328.8917.494.272.551.831.371.15MDRL251.00219.00209.00168.50115.0072.0039.0022.0013.004.002.002.001.001.00SDRL361.63317.66294.81225.10163.16101.6752.2826.5214.782.471.130.670.510.36LogisticARL369.36316.07300.44220.68133.2985.1840.4319.9513.483.672.201.721.331.12MDRL251.50217.50218.00148.0094.0056.0030.0014.0010.003.002.002.001.001.00SDRL357.96306.79298.34222.27128.5887.0736.9917.9811.632.050.960.680.470.33LaplaceARL371.17291.10261.49170.05102.2061.0026.5613.839.453.132.051.641.311.15MDRL252.50208.00190.50120.0076.0042.0019.0010.007.003.002.002.001.001.00SDRL363.24282.51261.00168.3597.9457.9524.3011.947.291.710.940.650.480.36t4ARL369.99341.67332.01268.76178.68128.8863.6436.7222.195.232.912.121.571.29MDRL256.50224.00226.00189.00125.0089.0048.0027.0016.004.003.002.002.001.00SDRL375.89347.85335.27261.40177.72130.4359.6834.0820.123.491.460.920.620.47t8ARL370.07326.33319.56233.66169.06118.7556.0531.4019.154.502.661.951.391.19MDRL234.50223.00222.00164.00123.0088.0041.0022.0014.004.002.002.001.001.00SDRL383.44327.84325.64228.44162.83113.1253.1629.2516.762.691.240.760.510.40Figure 9
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under Laplace distribution when n=5, λ=0.05, and ARL0=370.Figure 10
ARLs of the proposed TEWMA−SRRSS versus TEWMA-SR, TEWMA-SN, TEWMA−X¯, and DEWMA−SRRSS control charts under CN distribution when n=5, λ=0.05, and ARL0=370.Table 13
RL characteristics of theDEWMA−SRRSS control chart under different distributions with nominal ARL0=370 and n=10.
δλ,LDistr.0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00(0.05, 1.503)N(0, 1)ARL369.5595.0177.7834.7719.0011.585.853.562.541.131.001.001.001.00MDRL236.0069.0059.0029.0016.0010.005.003.002.001.001.001.001.001.00SDRL408.6097.3274.5330.6315.459.084.412.541.820.370.030.000.000.00CNARL370.52105.6283.6336.6819.3912.236.143.882.621.151.011.001.001.00MDRL243.5077.0062.0028.0016.0010.005.003.002.001.001.001.001.001.00SDRL413.79106.3483.5632.1315.939.824.742.801.770.380.080.000.000.00LogisticARL371.3885.9266.9931.1615.769.685.133.102.221.081.011.001.001.00MDRL236.0065.0048.0026.0013.008.004.002.002.001.001.001.001.001.00SDRL443.4282.7967.5726.8412.957.643.922.281.420.280.070.000.000.00LaplaceARL370.2068.9651.9224.0212.467.503.872.521.851.061.001.001.001.00MDRL261.0052.0039.0020.0010.006.003.002.002.001.001.001.001.001.00SDRL467.2465.3850.6119.889.925.842.871.701.100.240.060.030.000.00t4ARL369.84121.67100.8344.8022.0714.687.144.703.171.281.041.001.001.00MDRL263.5088.0072.0037.0018.0012.006.004.002.001.001.001.001.001.00SDRL427.48121.9699.1039.3718.4012.185.713.492.340.540.210.050.000.00t8ARL371.06109.2883.6340.2220.9712.626.744.072.711.171.021.001.001.00MDRL236.5073.0059.0032.0018.0010.005.003.002.001.001.001.001.001.00SDRL416.69118.7884.2436.8517.3210.105.403.101.930.410.120.030.000.00(0.10, 1.709)N(0, 1)ARL368.73118.7292.4842.7820.9713.356.564.182.831.171.021.001.001.00MDRL247.5082.0067.0032.0017.5012.006.004.002.001.001.001.001.001.00SDRL366.84118.5090.4137.8816.379.304.532.741.820.420.120.000.000.00CNARL369.38124.70100.4245.8422.5013.726.884.382.961.211.011.001.001.00MDRL251.5086.0069.5036.0019.0012.006.004.002.001.001.001.001.001.00SDRL382.14120.4799.0739.7316.959.634.833.042.000.460.120.000.000.00LogisticARL370.94106.2983.5035.0617.8911.395.973.592.531.131.001.001.001.00MDRL245.5077.0059.0028.0015.0010.005.003.002.001.001.001.001.001.00SDRL405.08104.5881.3928.2913.938.143.972.441.610.350.040.000.000.00LaplaceARL369.3084.6661.2826.8513.358.474.442.872.151.101.011.001.001.00MDRL257.0062.0046.0022.0011.508.004.002.002.001.001.001.001.001.00SDRL448.5883.6357.5321.019.815.863.201.821.290.330.080.000.000.00t4ARL370.49145.27120.5453.2926.2316.738.485.353.601.371.051.011.001.00MDRL251.00104.0082.0040.0022.0015.007.005.003.001.001.001.001.001.00SDRL396.26140.85118.9050.6920.5412.506.073.612.520.600.220.080.000.00t8ARL371.30136.63106.1447.2023.8814.287.294.683.321.251.011.001.001.00MDRL258.0093.0077.0036.0020.0012.006.004.003.001.001.001.001.001.00SDRL383.52136.48105.0142.2419.1211.385.143.152.150.510.100.030.000.00(0.25, 1.995)N(0, 1)ARL368.94163.89125.8861.0228.9715.797.674.753.411.361.031.001.001.00MDRL261.50110.0086.0045.0020.0013.007.004.003.001.001.001.001.001.00SDRL361.42161.56121.8557.6926.8011.945.132.781.900.570.160.000.000.00CNARL369.14179.78142.1460.2330.0316.788.485.103.621.391.051.001.001.00MDRL267.00121.5099.5044.0023.0013.007.005.003.001.001.001.001.001.00SDRL370.83176.44139.8455.7026.5013.805.813.132.130.560.210.040.000.00LogisticARL371.19151.80119.4952.9522.9213.926.634.082.941.261.021.001.001.00MDRL248.00105.0085.0037.0017.5011.006.004.003.001.001.001.001.001.00SDRL383.41152.05116.7151.9718.6110.784.242.371.600.470.140.000.000.00LaplaceARL370.19121.1891.2136.8416.409.615.063.382.521.181.021.001.001.00MDRL248.0088.0062.0027.0013.008.005.003.002.001.001.001.001.001.00SDRL383.41116.5791.7630.9612.756.642.991.821.270.410.130.040.000.00t4ARL369.05196.86156.3578.6736.0520.409.985.874.221.601.131.021.001.00MDRL247.50140.00108.0059.0026.0015.009.005.004.001.001.001.001.001.00SDRL376.25193.47162.1070.8630.5017.246.793.742.440.720.340.150.030.00t8ARL370.83197.61155.4076.3932.2619.259.265.473.891.501.061.001.001.00MDRL273.00134.50108.0055.0023.0015.008.005.003.001.001.001.001.001.00SDRL361.65203.48149.0573.4829.8115.246.873.442.160.630.240.050.000.00(0.50, 2.205)N(0, 1)ARL370.24222.16196.7798.1547.6826.3010.285.803.801.471.041.001.001.00MDRL261.00164.00138.5069.0033.0019.008.005.003.001.001.001.001.001.00SDRL362.63215.71194.2397.3447.3823.668.333.792.160.600.200.030.000.00CNARL370.38228.32202.24102.6150.8428.1010.926.134.101.511.081.001.001.00MDRL270.00159.00147.5072.5036.0020.008.005.004.001.001.001.001.001.00SDRL362.46225.70192.8098.2744.7525.798.484.192.370.630.260.040.000.00LogisticARL371.47226.11167.0189.5040.3622.698.904.973.401.371.051.001.001.00MDRL263.50147.00121.0067.0028.0016.507.004.003.001.001.001.001.001.00SDRL381.73255.35168.8383.1940.2720.297.193.101.830.540.220.000.000.00LaplaceARL371.27192.89133.9760.9727.5814.456.253.762.701.271.041.001.001.00MDRL263.50125.5095.0040.0019.5011.005.003.002.001.001.001.001.001.00SDRL381.73222.77130.9560.0025.7712.074.272.111.420.470.190.060.000.00t4ARL370.71267.22213.56128.2564.6135.5014.427.455.031.751.191.041.001.00MDRL255.00186.00148.5092.0046.0025.0010.006.004.002.001.001.001.001.00SDRL386.36261.54218.84123.8861.6333.5212.575.343.180.780.390.180.030.00t8ARL370.13246.85218.05118.3058.2830.5412.456.784.401.591.101.011.001.00MDRL266.00169.00152.5085.0041.0022.009.005.004.001.001.001.001.001.00SDRL353.53244.73216.05116.5155.8628.0010.304.862.720.700.300.090.000.00


## 5. Illustrative Example

This section demonstrates practical application to implement the proposedTEWMA−SRRSS control chart in practice. For this purpose, the proposed TEWMA−SRRSS with TEWMA-SR and DEWMA−SRRSS control charts are considered to monitor the process location shift. For the practical implementation of the proposed TEWMA−SRRSS control chart, we assumed the data set used by Muttlak and Al-Sabah [20]. These data were also used by Abid et al. [37] for the execution of the EWMA−SRRSS control chart. The collected data were about the Pepsi Cola Company in AL-Khobar, Saudi Arabia. The 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. They used the RSS method to hold the appropriate volume of soft drinks in bottles during the filling process.The RSS technique is used to collect 27 ranked set samples by repeating the two cycles four times, with each cycle having a size ofn=3. To compare the proposed TEWMA−SRRSS control chart with TEWMA-SR and DEWMA−SRRSS control charts, 27 samples with r=12 are collected using RSS. The design parameters L,λ of the DEWMA−SRRSS, TEWMA-SR, and TEWMA−SRRSS are (1.578, 0.05), (1.75, 0.05), and (1.305, 0.05), respectively, at ARL0=370. Comparison of the proposed TEWMA−SRRSS control chart with the TEWMA-SR and DEWMA−SRRSS control charts can be visualized in Figures 11–13. The DEWMA−SRRSS control chart shows 4 OOC signals (at sample numbers 24–27), whereas the TEWMA-SR control chart shows no OOC signals. In contrast, the proposed TEWMA−SRRSS control chart shows 7 OOC signals (at sample numbers 21–27) in the process location parameter. These findings indicate that the proposed TEWMA−SRRSS has improved performance for the monitoring of the process location as compared to DEWMA−SRRSS and TEWMA-SR control charts. Hence, the proposed TEWMA−SRRSS control chart under RSS has a better ability to detect shifts in process location.Figure 11
Application for theDEWMA−SRRSS control chart when L=1.578,λ=0.05, and ARL0=370.Figure 12
Application for the TEWMA–SR control chart whenL=1.75,λ=0.05, and ARL0=370.Figure 13
Application for the proposedTEWMA−SRRSS control chart when L=1.305,λ=0.05, and ARL0=370.


## 6. Summary, Conclusions, and Recommendations

The 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 asTEWMA−SRRSS control chart) using different continuous symmetric distributions. The Monte Carlo simulation method is used to obtain the numerical results of the proposed TEWMA−SRRSS control chart along with other existing control charts. The performance of TEWMA−SRRSS control chart is substantially better than the SRS-based NP TEWMA-SR, TEWMA sign, TEWMA−X¯, and double EWMA-SR DEWMA−SRRSS control charts. A real-life example is also provided to demonstrate how the proposed control chart can be used in practice. Therefore, to adopt a robust and efficient control chart, the proposed control chart gives an alternate choice to quality practitioners. The proposed work can be extended to the multivariate scenario.



---

*Source: 2570198-2021-10-26.xml*