# A Distribution-Free THWMA Control Chart under Ranked Set Sampling

Authors: Hongying Zhang; Zahid Rasheed; Majid Khan; Jimmy Joseph Namangale; Syed Masroor Anwar; Aamir Hamid
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The basic assumption of the parametric control charts is that the underlying process follows the specific distribution. The appropriateness of parametric control charts is questionable when different industrial applications do not support this desired assumption. Recently, nonparametric control charts have been introduced to overcome this deficiency of parametric control charts. Nonparametric control charts have the same in-control run-length characteristics for all continuous distributions and are in-control robust. On the other hand, the ranked set sampling technique is preferred over the simple random sampling technique with control charts because it reduces variability and improves the control chart’s performance. So, this study aims to propose a nonparametric triple homogenously weighted moving average control chart under Wilcoxon signed-rank test with a ranked set sampling technique (regarded asNPTHWMARSS) to further enhance the process location monitoring. Monte Carlo simulations are used for computational purposes. The proposed control chart’s run-length performance is compared with competing control charts, like TEWMA-SR, TEWMA-X¯, TEWMA-SN, TEWMA-SRRSS, and NPDHWMARSS control charts. The comparison revealed that the proposed NPTHWMARSS control chart outperformed the other competing control charts, particularly for small to moderate shifts in process location. Finally, a real-life application is presented for quality practitioners to illustrate the effectiveness of the proposed control chart.


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## Body

## 1. Introduction

The main concern in the manufacturing process is the product’s quality, and the variations are the primary source that influences the quality of the product. The variations are generally categorized into the natural cause of variations and unnatural causes of variations. Natural variations are harmless while unnatural variations seriously impact the quality of the ultimate product. The process remains in-control (IC) with natural variations, whereas the process is regarded as an out-of-control (OOC) process in the presence of unnatural variations. The existence of unnatural variations causes the shift in process parameters (location and/or dispersion). Control charts are used to monitor these shifts in the process parameters. Control charts are typically divided into memoryless and memory-type control charts based on their structural system. Shewhart and Van Nostrand [1] presented the first memoryless control chart, regarded as the Shewhart control chart, whereas Page [2] and Roberts [3] introduced the concept of memory-type control charts, like cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts, respectively.Several extensions to the main structure of classical control charts have already been made in order to improve their ability to detect earlier shifts. For example, Alevizakos et al. [4] offered a triple EWMA (TEWMA) control chart that outperformed the EWMA and DEWMA control charts. Similarly, Chatterjee et al. [5] investigated the use of a new S2 TEWMA control chart for detecting shifts in process dispersion. Likewise, Alevizakos et al. [6] and Alevizakos et al. [7] introduced the TEWMA-SR and the TEWMA-SN control charts, respectively, to enhance their shift detection ability in the process location. Also, Mahmood et al. [8] suggested a control chart for joint monitoring of process location and dispersion. Later, Alevizakos et al. [9] presented one and two-sided TEWMA control chart, observing the time between k≥1 events using the gamma distribution. After that, Rasheed et al. [10] and Rasheed et al. [11] presented two control charts for efficiently monitoring the process mean.Conventional parametric control charts are often used when an ongoing process follows a specified probability distribution. In many cases, the ongoing process may deviate from a particular distribution, or the distribution of the current process may be in question. Nonparametric (NP) control charts are a good alternative to parametric control charts. The IC run-length (RL) features of the NP control charts remain the same for all symmetric continuous distributions. The sign (SN), as well as Wilcoxon signed-rank (SR), are quite well NP approaches used in statistical process monitoring (SPM). Likewise, simple random sampling (SRS) and ranked set sampling (RSS) techniques are quite often used in SPM with both parametric and NP control charts to observe the ongoing process data. The SPM literature supports RSS because it reduces variability and enhances the efficiency of the associated control charts (Haq et al. [12] and Noor-ul-Amin and Tayyab [13]). Many researchers, including Tsai et al. [14], Abid et al. [15], Abbasi [16], Chakraborti and Graham [17], Rasheed et al. [18], and Abbasi et al. [19], also used these sampling techniques along with NP control charts.Hunter [20] pointed out that the EWMA control chart gives more weight to recent observations and less weight to former observations. Abbas [21] introduced the homogeneously weighted moving average (HWMA) control chart to monitor shifts in process location effectively. It gives a specific weight to current observations and the remaining weight to all previous observations equally. The distribution of weights in such a way enhances the HWMA control chart’s capacity to detect early shifts. Similarly, Adeoti and Koleoso [22] developed a hybrid HWMA (HHWMA) control chart to monitor the process location’s shifts. Furthermore, Anwar et al. [23] suggested a double HWMA (DHWMA) control chart for observing shifts in process location. They reported that the DHWMA control chart detects shifts better than the HWMA control chart. Also, Riaz et al. [24] designed a triple HWMA (THWMA) control chart to identify the small shifts in the process location. The THWMA control chart outperforms the HWMA and DHWMA control charts for monitoring process location.Takahasi and Wakimoto [25] suggested that an RSS-based mean estimator is more unbiased and efficient than an SRS-based estimator. Similarly, the THWMA control chart is more effective for monitoring process location parameters. A thorough literature review observed that no one had explored an NP THWMA control chart based on RSS to date. To fill this gap in this study, we are aimed to propose an NP THWMA-SR control chart under RSS (NPTHWMARSS) for monitoring shifts in process location for continuous and symmetric distribution. The proposed control chart’s RL characteristics, including average RL (ARL), median RL (MDRL), and standard deviation of RL (SDRL), are measured using various distributions like normal, student’s t, contaminated normal (CN), Laplace, and logistic distributions. These RL characteristics of the proposed NPTHWMARSS control chart is compared to the other competing control charts, like TEWMA-SR, TEWMA-X¯, TEWMA-SN, TEWMA-SRRSS, and NPDHWMARSS control charts.The remainder of the paper is ordered as follows: Section2 offers the design structure of the competing and the proposed control chart. Section 3 defines the proposed control chart’s IC and out-of-control (OOC) performance. Section 4 covers a comparative study of the proposed control chart, whereas Section 5 provides a real-life application. Section 6 concludes with closing remarks.


## 2. Competing and Proposed Control Charts

This section provides the design structure of the competing and the proposedNPTHWMARSS control chart. More details can be found in the subsequent subsections.


### 2.1.NPHWMARSS Control Chart

Various researchers like Kim and Kim [26] and Abid et al. [15] used the following RSS-based Wilcoxon signed-rank statistic to monitor the shift in process location.(1)SRRSSt=∑j=1n∑h=1msignXtjh−θ0Rtjh+,where θ0 symbolizes the process median and t,j,andh denote the number of samples, observations, and cycles used in the RSS approach, respectively. The mean and variance of SRRSSt statistic are ESRRSSt=0 and VarSRRSSt=rr+12r+1/6ϖ02., respectively, where r denotes the number of replications and can be defined as r=nm. In this study, we assume m=1 for a valid assessment of the proposed control chart to the competitors, so r=nm, which is r=n. The quantity ϖ02 is used to improve the efficiency of the control chart and can be defined as ϖ02=1−4/n∑j=1nFk0−1/22. The values of Fk0 can be obtained by solving the following mathematical expression Fk0=r!/j−1!r−j!∫−∞0Ftj−11−Ftr−jftdt. The plotting statistic of the NP HWMA under RSS (NPHWMARSS) control chart can be defined as follows:(2)NPHt=ηSRRSSt+1−ηSR¯RSSt−1,where η is a smoothing parameter and meets the condition ηϵ0,1. Also, the term SR¯RSSt−1 is the mean of SRRSS of t−1 samples. The mean and variance of the statistic NPHt are ENPHt=μ0 and(3)VarNPHt=η2rr+12r+16ϖ02,ift=1η2+1−η2t−1rr+12r+16ϖ02,ift>1,respectively. The control limits of the NPHWMARSS control chart are as follows:(4)LCLNPHWMARSSt=μ0−Lη2rr+12r+16ϖ02,ift=1μ0−Lη2+1−η2t−1rr+12r+16ϖ02,ift>1CLNPHWMARSSt=μ0UCLNPHWMARSSt=μ0+Lη2rr+12r+16ϖ02,ift=1μ0+Lη2+1−η2t−1rr+12r+16ϖ02,ift>1.IfNPHt>UCLNPHWMARSSt or NPHt<LCLNPHWMARSSt, the process is considered OOC; otherwise, it remains IC.


### 2.2.NPDHWMARSSControl Chart

The plotting statistic of theNPDHWMARSS control chart is defined as follows:(5)NPHt=ηSRRSSt+1−ηSR¯RSSt−1,NPDHt=ηNPHt+1−ηSR¯RSSt−1,.The simplified version of the plotting statistic of the proposedNPDHWMARSS control chart is expressed as follows:(6)NPDHt=η2SRRSSt+1−η2SR¯RSSt−1.TheinitialvalueofSRRSSt is 0, whereas the IC mean and variance of the plotting statistic NPDHt are(7)EDHt=μ0VarDHt=η4rr+12r+16ϖ02ift=1,and(8)EDHt=μ0VarDHt=η4+1−η21+η2t−1rr+12r+16ϖ02ift>1.The control limits of the proposedNPDHWMARSS control chart are defined as follows:(9)LCLNPDHWMARSSt=μ0−Lη4rr+12r+16ϖ02,ift=1μ0−Lη4+1−η21+η2t−1rr+12r+16ϖ02,ift>1CLNPDHWMARSSt=μ0UCLNPDHWMARSSt=μ0+Lη4rr+12r+16ϖ02,ift=1μ0+Lη4+1−η21+η2t−1rr+12r+16ϖ02,ift>1.PlotNPDHt against LCLDNPDHWMARSSt and UCLDNPDHWMARSSt. If NPDHt>UCLDNPDHWMARSSt or NPDHt<LCLDNPDHWMARSSt, the underlying process is OOC; else, it is IC.


### 2.3. ProposedNPTHWMARSS Control Chart

This subsection describes the design structure of the proposedNPTHWMARSS control chart to monitor shifts in process location. The plotting statistic of the NPTHWMARSS control chart based on RSS is defined as follows:(10)NPHt=ηSRRSSt+1−ηSR¯RSSt−1,NPDHt=ηNPHt+1−ηSR¯RSSt−1,NPTHt=ηNPDHt+1−ηSR¯RSSt−1,.The simplified version of (10) is(11)NPTHt=η3SRRSSt+1−η3SR¯RSSt−1.The mean and variance of the statisticNPTHt are ENPTHt=μ and(12)VarTHt=η6+1−η32t−1rr+12r+16ϖ02.Based on this mean and variance, the control limits for theNPTHWMARSS control chart are as follows:(13)LCLNPTHWMAt=μ−Lη6rr+12r+16ϖ02,t=1μ−Lη6+1−η32t−1rr+12r+16ϖ02,t>1CLTHWMAt=μUCLNPTHWMAt=μ+Lη6rr+12r+16ϖ02,t=1μ+Lη6+1−η32t−1rr+12r+16ϖ02,t>1.WhenLCLNPTHWMAt<NPTHt<UCLNPTHWMAt, the process is in IC state; otherwise, it is OOC.


## 2.1.NPHWMARSS Control Chart

Various researchers like Kim and Kim [26] and Abid et al. [15] used the following RSS-based Wilcoxon signed-rank statistic to monitor the shift in process location.(1)SRRSSt=∑j=1n∑h=1msignXtjh−θ0Rtjh+,where θ0 symbolizes the process median and t,j,andh denote the number of samples, observations, and cycles used in the RSS approach, respectively. The mean and variance of SRRSSt statistic are ESRRSSt=0 and VarSRRSSt=rr+12r+1/6ϖ02., respectively, where r denotes the number of replications and can be defined as r=nm. In this study, we assume m=1 for a valid assessment of the proposed control chart to the competitors, so r=nm, which is r=n. The quantity ϖ02 is used to improve the efficiency of the control chart and can be defined as ϖ02=1−4/n∑j=1nFk0−1/22. The values of Fk0 can be obtained by solving the following mathematical expression Fk0=r!/j−1!r−j!∫−∞0Ftj−11−Ftr−jftdt. The plotting statistic of the NP HWMA under RSS (NPHWMARSS) control chart can be defined as follows:(2)NPHt=ηSRRSSt+1−ηSR¯RSSt−1,where η is a smoothing parameter and meets the condition ηϵ0,1. Also, the term SR¯RSSt−1 is the mean of SRRSS of t−1 samples. The mean and variance of the statistic NPHt are ENPHt=μ0 and(3)VarNPHt=η2rr+12r+16ϖ02,ift=1η2+1−η2t−1rr+12r+16ϖ02,ift>1,respectively. The control limits of the NPHWMARSS control chart are as follows:(4)LCLNPHWMARSSt=μ0−Lη2rr+12r+16ϖ02,ift=1μ0−Lη2+1−η2t−1rr+12r+16ϖ02,ift>1CLNPHWMARSSt=μ0UCLNPHWMARSSt=μ0+Lη2rr+12r+16ϖ02,ift=1μ0+Lη2+1−η2t−1rr+12r+16ϖ02,ift>1.IfNPHt>UCLNPHWMARSSt or NPHt<LCLNPHWMARSSt, the process is considered OOC; otherwise, it remains IC.


## 2.2.NPDHWMARSSControl Chart

The plotting statistic of theNPDHWMARSS control chart is defined as follows:(5)NPHt=ηSRRSSt+1−ηSR¯RSSt−1,NPDHt=ηNPHt+1−ηSR¯RSSt−1,.The simplified version of the plotting statistic of the proposedNPDHWMARSS control chart is expressed as follows:(6)NPDHt=η2SRRSSt+1−η2SR¯RSSt−1.TheinitialvalueofSRRSSt is 0, whereas the IC mean and variance of the plotting statistic NPDHt are(7)EDHt=μ0VarDHt=η4rr+12r+16ϖ02ift=1,and(8)EDHt=μ0VarDHt=η4+1−η21+η2t−1rr+12r+16ϖ02ift>1.The control limits of the proposedNPDHWMARSS control chart are defined as follows:(9)LCLNPDHWMARSSt=μ0−Lη4rr+12r+16ϖ02,ift=1μ0−Lη4+1−η21+η2t−1rr+12r+16ϖ02,ift>1CLNPDHWMARSSt=μ0UCLNPDHWMARSSt=μ0+Lη4rr+12r+16ϖ02,ift=1μ0+Lη4+1−η21+η2t−1rr+12r+16ϖ02,ift>1.PlotNPDHt against LCLDNPDHWMARSSt and UCLDNPDHWMARSSt. If NPDHt>UCLDNPDHWMARSSt or NPDHt<LCLDNPDHWMARSSt, the underlying process is OOC; else, it is IC.


## 2.3. ProposedNPTHWMARSS Control Chart

This subsection describes the design structure of the proposedNPTHWMARSS control chart to monitor shifts in process location. The plotting statistic of the NPTHWMARSS control chart based on RSS is defined as follows:(10)NPHt=ηSRRSSt+1−ηSR¯RSSt−1,NPDHt=ηNPHt+1−ηSR¯RSSt−1,NPTHt=ηNPDHt+1−ηSR¯RSSt−1,.The simplified version of (10) is(11)NPTHt=η3SRRSSt+1−η3SR¯RSSt−1.The mean and variance of the statisticNPTHt are ENPTHt=μ and(12)VarTHt=η6+1−η32t−1rr+12r+16ϖ02.Based on this mean and variance, the control limits for theNPTHWMARSS control chart are as follows:(13)LCLNPTHWMAt=μ−Lη6rr+12r+16ϖ02,t=1μ−Lη6+1−η32t−1rr+12r+16ϖ02,t>1CLTHWMAt=μUCLNPTHWMAt=μ+Lη6rr+12r+16ϖ02,t=1μ+Lη6+1−η32t−1rr+12r+16ϖ02,t>1.WhenLCLNPTHWMAt<NPTHt<UCLNPTHWMAt, the process is in IC state; otherwise, it is OOC.


## 3. Implementation of the Proposed Control Chart

The performance metrics of the proposedNPTHWMARSS control chart for monitoring shifts in process location is given in this section. Subsection 3.1 offers the performance metrics of the proposed NPTHWMARSS control chart. Similarly, the proposed control chart’s robustness, IC, and OOC performance are presented in Subsection 3.2.


### 3.1. Performance Metrics

TheARL is commonly used to evaluate the control chart’s performance. The ARL is the expected number of sample points before the first OOC signal from the control chart. In this study, we set ARL0=370, with sample sizes n=5 and 10. Monte Carlo simulations with 50,000 simulations in the R program are used to determine the RL characteristics. To examine the performance behaviour of the proposed NPTHWMARSS control chart, various values of η∈ (0.10, 0.25, 0.50, and 0.75) and δ∈ (0.025, 0.05, 0.075, 0.10, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00, and 5.00) are used. The following steps are used to obtain the nominal (prespecified) ARL0 value:(i)
To select a random sample from the evaluated distribution.(ii)
Specify the process parameters (ηandL).(iii)
Determine theNPTHt plotting statistics using equation (8).(iv)
FindLCLNPTHWMARSSt and UCLNPTHWMARSSt from equation (9).(v)
Plot the plotting statisticNPTHt against LCLNPTHWMARSSt and UCLNPTHWMARSSt over t.(vi)
IfNPTHt>UCLNPTHWMARSSt or NPTHt<LCLNPTHWMARSSt, note this sample of NPTHt statistic as an RL. For instance, at t=195, if NPTHt>UCLNPTHWMARSSt or NPTHt<LCLNPTHWMARSSt, record 195 as a first RL.(vii)
Repeat steps (ii) to (vi) 50,000 times and record RLs.(viii)
Using 50,000 simulations, calculateARL0 and record RLs.(ix)
IfARL0=300; otherwise, adjust the constant in step (ii) as needed, and then, repeat steps (ii) to (ix) to achieve ARL0=300.(x)
To acquireARL1 values, draw a shifted sample from the considered distribution again and repeat steps (ii) to (ix).


### 3.2. Robustness, IC, and OOC Performances of theNPTHWMARSS Control Chart

This subsection highlights the proposedNPTHWMARSS control chart’s robustness, IC, and OOC behaviour when a process location is shifted. Normal and nonnormal continuous and symmetrical distributions are used to evaluate such properties. For this study, the distributions used were the standard normal distribution, i.e., N0,1, Laplace or double exponential, i.e., D.E0.1/2, student’s t distribution, i.e., tυ, logistic distribution i.e., 0,3/π, contaminated normal (CN) distribution, which is the mixture of N0,σ02 and N0,σ12 (see Table 1). Tables 2 and 3 demonstrate the RL characteristics of the proposed NPTHWMARSS control chart for location shift. For comparison purposes, all these distributions were reparametrized with zero mean/median and unit variance. For all symmetric continuous distributions, the IC RL characteristics of the NP control chart remain constant.Table 1
PDFs of the continuous distributions used for this study.
DistributionsPDF(i)Standard normalfX=e−X2/2/2π, where M0=0 and σ2=1(ii)Student’stvfX=Γv+1/2/Γv/2vπ1+X2/vv+1/2, where M0=0, σ2=v/v−2 and v=4and8 are taken(iii)LogisticfX=e−πX/3/3/π1+e−πX/32 where M0=0 and σ2=3/π2(iv)LaplacefX=1/2e−X, where M0=0 and σ2=1/2(v)Contaminated normal (CN)fX=0.95e−X2/2/2π+0.05e−X2/2σ02/σ02π, where M0=0 and σ2=0.95+0.05σ02Table 2
RL characteristics of the proposedNPTHWMARSS control chart under different distributions with nominal ARL0=370 and n=5.
δ(η, L)DistributionsMetrics00.0250.050.0750.10.250.50.7511.522.5350.10, 1.26NormalARL371.4650.9423.1515.1311.273.981.931.261.061.001.001.001.001.00MDRL8.009.007.007.005.003.001.001.001.001.001.001.001.001.00SDRL2361.91132.3041.9522.7714.613.231.310.710.350.060.000.000.000.00t(4)ARL371.5159.5528.2317.9313.264.712.291.561.261.071.031.021.011.00MDRL9.008.008.007.006.004.001.001.001.001.001.001.001.001.00SDRL2348.39164.0455.1728.1718.294.081.571.010.690.380.240.180.130.05t(8)ARL372.0152.2125.4315.3311.034.101.961.321.091.011.001.001.001.00MDRL9.237.056.736.225.873.411.001.001.001.001.001.001.001.00SDRL2375.02139.3150.0724.1715.002.791.110.810.490.150.090.020.000.00CNARL372.1752.3824.7415.9511.184.152.011.341.091.011.001.001.001.00MDRL9.008.008.007.005.004.001.001.001.001.001.001.001.001.00SDRL2337.53137.5045.1124.1914.633.461.370.790.430.150.070.040.000.00LaplaceARL371.0938.0517.9311.148.173.121.671.251.101.021.011.001.001.00MDRL9.009.007.005.005.003.001.001.001.001.001.001.001.001.00SDRL2433.9582.5228.7514.668.912.311.090.680.440.210.100.050.030.00LogisticARL371.5846.2121.1313.069.203.231.411.091.021.001.001.001.001.00MDRL8.777.816.795.904.722.750.830.530.210.120.010.000.000.00SDRL2464.11113.0935.8018.5111.472.161.010.530.280.080.010.000.000.000.25, 1.615NormalARL370.10161.6966.1535.8922.955.832.401.501.151.011.001.001.001.00MDRL126.0070.5035.0022.0015.005.003.001.001.001.001.001.001.001.00SDRL481.04210.9280.7339.8524.584.451.500.920.530.130.000.000.000.00t(4)ARL371.23177.1978.2643.0128.026.402.211.301.111.011.001.001.001.00MDRL137.0376.1140.3123.1917.535.492.331.311.091.011.001.001.001.00SDRL494.57242.8996.6150.3930.175.081.201.080.820.310.190.070.010.00t(8)ARL371.60169.2270.2938.9025.035.112.101.191.091.031.011.001.001.00MDRL134.0273.9235.0822.7116.014.292.011.201.061.011.001.001.001.00SDRL493.61227.4987.5044.1326.054.631.391.020.620.210.140.050.010.00CNARL371.25160.1869.0537.6024.396.012.191.471.101.011.001.001.001.00MDRL138.0373.0535.1022.0114.514.312.101.031.011.001.001.001.001.00SDRL496.32210.2184.1342.0125.124.101.191.010.700.190.080.050.010.00LaplaceARL371.83109.3139.8121.4714.083.511.851.461.071.011.001.001.001.00MDRL134.6352.3123.5714.019.023.652.011.371.021.011.001.001.001.00SDRL496.02141.3246.5923.1113.692.031.110.820.590.200.090.010.000.00LogisticARL370.99139.2057.1131.0119.704.892.051.101.071.011.001.001.001.00MDRL134.0164.1129.3018.0412.113.612.331.031.011.001.001.001.001.00SDRL490.10184.1767.8933.6820.413.591.180.690.440.180.090.010.000.000.50, 2.67NormalARL370.77248.78133.0878.5051.2212.294.452.822.111.391.121.031.011.00MDRL307.00201.00108.0066.0044.0011.004.003.002.001.001.001.001.001.00SDRL284.49197.20101.5155.6134.876.871.951.130.930.620.350.170.080.00t(4)ARL370.85269.93155.2094.0663.1015.375.453.352.551.741.381.211.121.02MDRL310.00221.00125.0078.0053.0014.005.003.003.001.001.001.001.001.00SDRL282.03211.07119.1468.6943.769.022.651.431.130.870.670.500.380.17t(8)ARL369.03264.34143.1886.1157.4813.864.903.072.321.561.221.101.041.00MDRL302.00214.00117.0072.0049.0013.005.003.003.001.001.001.001.001.00SDRL285.75207.19108.6762.9139.217.962.261.281.020.760.490.320.220.06CNARL368.81258.94138.4982.5454.4112.994.652.952.211.471.161.061.021.00MDRL304.00210.00113.0070.0047.0012.004.003.002.001.001.001.001.001.00SDRL280.42206.50104.5458.4936.777.392.101.180.970.690.430.260.160.04LaplaceARL369.34195.5590.9851.0433.048.453.572.451.911.391.171.091.041.00MDRL307.00161.0076.0044.0029.008.003.003.002.001.001.001.001.001.00SDRL281.59152.8565.4434.6121.154.571.611.130.940.650.440.300.200.04LogisticARL370.13237.57121.4970.0345.8510.964.092.662.031.391.141.061.021.00MDRL307.00192.00100.0059.0039.0010.004.003.002.001.001.001.001.001.00SDRL281.59186.5590.1249.6430.866.101.811.130.940.640.390.250.150.020.75, 2.859NormalARL371.01353.40272.81191.02110.8520.745.012.661.891.341.121.031.011.00MDRL286.00256.00188.00133.0083.5016.004.003.002.001.001.001.001.001.00SDRL327.29341.61281.16205.18114.8316.922.941.230.770.490.330.160.090.00t(4)ARL372.03355.04291.69213.52144.0326.496.513.392.341.611.321.181.111.02MDRL286.00261.00207.00147.00100.0021.006.003.002.001.001.001.001.001.00SDRL327.18337.98286.71223.50152.1121.834.191.781.130.720.530.420.330.14t(8)ARL369.36355.52288.97207.82130.4223.255.723.002.101.451.211.091.041.00MDRL275.02263.10201.00138.0091.0018.005.003.002.001.001.001.001.001.00SDRL329.72335.93286.60213.56130.8318.733.551.500.940.580.420.300.200.06CNARL372.01358.82287.96198.80125.9022.205.362.831.991.391.151.061.021.00MDRL285.00256.00201.00129.5087.0017.005.003.002.001.001.001.001.001.00SDRL329.53342.61281.79207.94125.3618.013.231.340.850.540.380.240.150.04LaplaceARL368.77307.89196.55103.5263.2911.613.652.291.731.321.161.081.041.00MDRL281.00221.00133.0075.0048.009.003.002.002.001.001.001.001.001.00SDRL323.25294.27199.7596.2955.288.512.051.140.810.540.380.270.200.04LogisticARL369.51343.92259.29172.7898.2217.084.452.501.821.321.131.051.021.00MDRL283.00245.00179.00112.0073.0013.004.002.002.001.001.001.001.001.00SDRL325.98329.09256.64181.8690.1413.262.501.170.790.500.340.220.150.02Table 3
RL characteristics of the proposedNPTHWMARSS control chart under different distributions at nominal ARL0=370 and n=10.
δ(η, L)DistributionsMetrics00.0250.050.0750.10.250.50.7511.522.5350.25, 1.362NormalARL369.9566.0523.7512.598.402.521.141.001.001.001.001.001.001.00MDRL124.0035.0015.009.006.003.001.001.001.001.001.001.001.001.00SDRL479.1879.8724.7711.637.061.560.520.080.000.000.000.000.000.00t(4)ARL370.8984.1529.7916.0110.263.061.341.031.001.001.001.001.001.00MDRL128.0044.0018.0011.008.003.001.001.001.001.001.001.001.001.00SDRL478.00101.2632.0915.399.151.910.770.250.080.010.000.000.000.00t(8)ARL369.6675.0326.3214.159.242.771.231.011.001.001.001.001.001.00MDRL129.0038.0016.0010.007.003.001.001.001.001.001.001.001.001.00SDRL485.3792.1828.3913.597.931.740.640.150.020.000.000.000.000.00CNARL371.4270.3524.9413.418.772.611.171.011.001.001.001.001.001.00MDRL121.0036.0016.009.007.003.001.001.001.001.001.001.001.001.00SDRL479.2686.8826.8112.667.491.630.570.100.000.000.000.000.000.00LaplaceARL370.0745.6916.188.755.811.951.071.001.001.001.001.001.001.00MDRL126.0027.0011.007.005.001.001.001.001.001.001.001.001.001.00SDRL478.4251.9415.737.494.421.230.360.060.000.000.000.000.000.00LogisticARL371.4261.3221.0911.437.482.311.101.001.001.001.001.001.001.00MDRL129.0034.0013.008.006.003.001.001.001.001.001.001.001.001.00SDRL480.3771.4322.0210.166.161.440.440.050.000.000.000.000.000.000.50, 2.22NormalARL371.17127.1549.5426.4816.844.151.661.051.001.001.001.001.001.00MDRL298.00102.0042.0023.0015.004.001.001.001.001.001.001.001.001.00SDRL306.2798.0034.1516.6710.211.930.870.260.030.000.000.000.000.00t(4)ARL368.97152.0560.9632.9821.355.102.091.271.051.001.001.001.001.00MDRL292.00123.0052.0029.0019.005.002.001.001.001.001.001.001.001.00SDRL306.34118.7243.5521.8613.292.511.040.590.240.040.010.000.000.00t(8)ARL370.66139.5055.3429.7419.044.601.871.131.011.001.001.001.001.00MDRL291.00112.0047.0026.0017.004.002.001.001.001.001.001.001.001.00SDRL309.72110.1938.8619.1411.812.240.950.420.120.000.000.000.000.00CNARL369.50134.0751.7128.0717.884.351.751.091.001.001.001.001.001.00MDRL292.00107.0044.0025.0016.004.001.001.001.001.001.001.001.001.00SDRL307.98106.0036.2618.2310.872.050.910.350.060.010.000.000.000.00LaplaceARL369.3690.8733.1817.7311.213.121.381.041.001.001.001.001.001.00MDRL287.0076.0029.0016.0010.003.001.001.001.001.001.001.001.001.00SDRL308.2967.5421.6310.796.411.450.700.240.070.000.000.000.000.00LogisticARL370.26117.0543.9524.0214.913.801.541.051.001.001.001.001.001.00MDRL290.0095.0037.0021.0013.004.001.001.001.001.001.001.001.001.00SDRL309.2990.5629.5414.988.971.750.800.250.070.000.000.000.000.000.75, 2.319NormalARL371.13228.89100.4347.2226.674.141.501.041.001.001.001.001.001.00MDRL262.00160.0072.0034.0020.004.001.001.001.001.001.001.001.001.00SDRL361.86228.8795.9242.7822.452.380.640.200.040.000.000.000.000.00t(4)ARL370.19255.05120.9662.3835.395.461.861.201.031.001.001.001.001.00MDRL260.00175.0087.0046.0027.005.002.001.001.001.001.001.001.001.00SDRL362.84257.07114.8757.1630.693.420.880.420.180.040.000.000.000.00t(8)ARL371.31246.14109.4254.5630.504.761.651.111.011.001.001.001.001.00MDRL259.00169.0078.0040.0023.004.002.001.001.001.001.001.001.001.00SDRL368.61250.95104.5749.4026.312.900.750.320.100.010.000.000.000.00CNARL369.71237.43104.0550.2128.554.401.551.071.001.001.001.001.001.00MDRL258.00164.0075.0037.0021.004.001.001.001.001.001.001.001.001.00SDRL359.39236.6799.1945.2524.932.560.680.250.060.000.000.000.000.00LaplaceARL371.00174.1362.1427.4715.272.931.281.041.001.001.001.001.001.00MDRL260.00123.0046.0021.0012.003.001.001.001.001.001.001.001.001.00SDRL364.84167.9856.3423.2711.821.560.500.190.060.010.000.000.000.00LogisticARL369.71217.7988.6640.9522.373.671.401.041.001.001.001.001.001.00MDRL259.00153.0063.0031.0017.003.001.001.001.001.001.001.001.001.00SDRL362.01217.2785.1035.7418.452.060.570.190.050.000.000.000.000.00The same parameters are used for comparison purposes as reported in numerous relevant articles. TheARL measures are used to compare the proposed and competing control charts. Table 2 and Figures 1 and 2 provide the following outcomes:(i)
The proposed control chart’s ICRL distribution looks remarkably similar to all distributions examined in this study. For example, at η=0.10,0.25,0.50,0.75, n=5and10 and the ARL0=370 for all investigated distributions (see Tables 2 and 3).(ii)
As the smoothing parameter reduces, the proposed control chart becomes more effective in detecting shifts. This illustrates that the proposedNPTHWMARSS control chart is more sensitive to small smoothing parameters (see Figure 1).(iii)
As the sample size increases, the proposedNPTHWMARSS control chart’s performance increases.(iv)
The Laplace distribution outperforms the other distributions regarding OOC RL performance (see Figure2).(v)
The proposedNPTHWMARSS control chart’s ARL1 values increase as η increases at a certain size of the shift. For instance, under normal distribution at η=0.25,n=5,andδ=0.025, the ARL1=161.69, whereas when η=0.50,n=5,andδ=0.025, the ARL1=248.78 (see Tables 2 and 3).(vi)
The distribution of RL values is positively skewed, i.e.,ARL>MRDL (see Tables 2 and 3).(vii)
TheARL1 values of the proposed NPTHWMARSS control chart are smaller than competing control charts with different shift sizes in process location (see Figure 3).Figure 1
ARL characteristic of the proposedNPTHWMARSS control chart under various distributions for different values of η when n=5 and ARL0=370.Figure 2
ARL characteristic of the proposedNPTHWMARSS control chart under various distributions when n=5and10 and ARL0=370.Figure 3
ARL characteristic of the proposedNPTHWMARSS control chart under various distributions when η=0.10and0.25,n=5, and ARL0=370.


## 3.1. Performance Metrics

TheARL is commonly used to evaluate the control chart’s performance. The ARL is the expected number of sample points before the first OOC signal from the control chart. In this study, we set ARL0=370, with sample sizes n=5 and 10. Monte Carlo simulations with 50,000 simulations in the R program are used to determine the RL characteristics. To examine the performance behaviour of the proposed NPTHWMARSS control chart, various values of η∈ (0.10, 0.25, 0.50, and 0.75) and δ∈ (0.025, 0.05, 0.075, 0.10, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00, and 5.00) are used. The following steps are used to obtain the nominal (prespecified) ARL0 value:(i)
To select a random sample from the evaluated distribution.(ii)
Specify the process parameters (ηandL).(iii)
Determine theNPTHt plotting statistics using equation (8).(iv)
FindLCLNPTHWMARSSt and UCLNPTHWMARSSt from equation (9).(v)
Plot the plotting statisticNPTHt against LCLNPTHWMARSSt and UCLNPTHWMARSSt over t.(vi)
IfNPTHt>UCLNPTHWMARSSt or NPTHt<LCLNPTHWMARSSt, note this sample of NPTHt statistic as an RL. For instance, at t=195, if NPTHt>UCLNPTHWMARSSt or NPTHt<LCLNPTHWMARSSt, record 195 as a first RL.(vii)
Repeat steps (ii) to (vi) 50,000 times and record RLs.(viii)
Using 50,000 simulations, calculateARL0 and record RLs.(ix)
IfARL0=300; otherwise, adjust the constant in step (ii) as needed, and then, repeat steps (ii) to (ix) to achieve ARL0=300.(x)
To acquireARL1 values, draw a shifted sample from the considered distribution again and repeat steps (ii) to (ix).


## 3.2. Robustness, IC, and OOC Performances of theNPTHWMARSS Control Chart

This subsection highlights the proposedNPTHWMARSS control chart’s robustness, IC, and OOC behaviour when a process location is shifted. Normal and nonnormal continuous and symmetrical distributions are used to evaluate such properties. For this study, the distributions used were the standard normal distribution, i.e., N0,1, Laplace or double exponential, i.e., D.E0.1/2, student’s t distribution, i.e., tυ, logistic distribution i.e., 0,3/π, contaminated normal (CN) distribution, which is the mixture of N0,σ02 and N0,σ12 (see Table 1). Tables 2 and 3 demonstrate the RL characteristics of the proposed NPTHWMARSS control chart for location shift. For comparison purposes, all these distributions were reparametrized with zero mean/median and unit variance. For all symmetric continuous distributions, the IC RL characteristics of the NP control chart remain constant.Table 1
PDFs of the continuous distributions used for this study.
DistributionsPDF(i)Standard normalfX=e−X2/2/2π, where M0=0 and σ2=1(ii)Student’stvfX=Γv+1/2/Γv/2vπ1+X2/vv+1/2, where M0=0, σ2=v/v−2 and v=4and8 are taken(iii)LogisticfX=e−πX/3/3/π1+e−πX/32 where M0=0 and σ2=3/π2(iv)LaplacefX=1/2e−X, where M0=0 and σ2=1/2(v)Contaminated normal (CN)fX=0.95e−X2/2/2π+0.05e−X2/2σ02/σ02π, where M0=0 and σ2=0.95+0.05σ02Table 2
RL characteristics of the proposedNPTHWMARSS control chart under different distributions with nominal ARL0=370 and n=5.
δ(η, L)DistributionsMetrics00.0250.050.0750.10.250.50.7511.522.5350.10, 1.26NormalARL371.4650.9423.1515.1311.273.981.931.261.061.001.001.001.001.00MDRL8.009.007.007.005.003.001.001.001.001.001.001.001.001.00SDRL2361.91132.3041.9522.7714.613.231.310.710.350.060.000.000.000.00t(4)ARL371.5159.5528.2317.9313.264.712.291.561.261.071.031.021.011.00MDRL9.008.008.007.006.004.001.001.001.001.001.001.001.001.00SDRL2348.39164.0455.1728.1718.294.081.571.010.690.380.240.180.130.05t(8)ARL372.0152.2125.4315.3311.034.101.961.321.091.011.001.001.001.00MDRL9.237.056.736.225.873.411.001.001.001.001.001.001.001.00SDRL2375.02139.3150.0724.1715.002.791.110.810.490.150.090.020.000.00CNARL372.1752.3824.7415.9511.184.152.011.341.091.011.001.001.001.00MDRL9.008.008.007.005.004.001.001.001.001.001.001.001.001.00SDRL2337.53137.5045.1124.1914.633.461.370.790.430.150.070.040.000.00LaplaceARL371.0938.0517.9311.148.173.121.671.251.101.021.011.001.001.00MDRL9.009.007.005.005.003.001.001.001.001.001.001.001.001.00SDRL2433.9582.5228.7514.668.912.311.090.680.440.210.100.050.030.00LogisticARL371.5846.2121.1313.069.203.231.411.091.021.001.001.001.001.00MDRL8.777.816.795.904.722.750.830.530.210.120.010.000.000.00SDRL2464.11113.0935.8018.5111.472.161.010.530.280.080.010.000.000.000.25, 1.615NormalARL370.10161.6966.1535.8922.955.832.401.501.151.011.001.001.001.00MDRL126.0070.5035.0022.0015.005.003.001.001.001.001.001.001.001.00SDRL481.04210.9280.7339.8524.584.451.500.920.530.130.000.000.000.00t(4)ARL371.23177.1978.2643.0128.026.402.211.301.111.011.001.001.001.00MDRL137.0376.1140.3123.1917.535.492.331.311.091.011.001.001.001.00SDRL494.57242.8996.6150.3930.175.081.201.080.820.310.190.070.010.00t(8)ARL371.60169.2270.2938.9025.035.112.101.191.091.031.011.001.001.00MDRL134.0273.9235.0822.7116.014.292.011.201.061.011.001.001.001.00SDRL493.61227.4987.5044.1326.054.631.391.020.620.210.140.050.010.00CNARL371.25160.1869.0537.6024.396.012.191.471.101.011.001.001.001.00MDRL138.0373.0535.1022.0114.514.312.101.031.011.001.001.001.001.00SDRL496.32210.2184.1342.0125.124.101.191.010.700.190.080.050.010.00LaplaceARL371.83109.3139.8121.4714.083.511.851.461.071.011.001.001.001.00MDRL134.6352.3123.5714.019.023.652.011.371.021.011.001.001.001.00SDRL496.02141.3246.5923.1113.692.031.110.820.590.200.090.010.000.00LogisticARL370.99139.2057.1131.0119.704.892.051.101.071.011.001.001.001.00MDRL134.0164.1129.3018.0412.113.612.331.031.011.001.001.001.001.00SDRL490.10184.1767.8933.6820.413.591.180.690.440.180.090.010.000.000.50, 2.67NormalARL370.77248.78133.0878.5051.2212.294.452.822.111.391.121.031.011.00MDRL307.00201.00108.0066.0044.0011.004.003.002.001.001.001.001.001.00SDRL284.49197.20101.5155.6134.876.871.951.130.930.620.350.170.080.00t(4)ARL370.85269.93155.2094.0663.1015.375.453.352.551.741.381.211.121.02MDRL310.00221.00125.0078.0053.0014.005.003.003.001.001.001.001.001.00SDRL282.03211.07119.1468.6943.769.022.651.431.130.870.670.500.380.17t(8)ARL369.03264.34143.1886.1157.4813.864.903.072.321.561.221.101.041.00MDRL302.00214.00117.0072.0049.0013.005.003.003.001.001.001.001.001.00SDRL285.75207.19108.6762.9139.217.962.261.281.020.760.490.320.220.06CNARL368.81258.94138.4982.5454.4112.994.652.952.211.471.161.061.021.00MDRL304.00210.00113.0070.0047.0012.004.003.002.001.001.001.001.001.00SDRL280.42206.50104.5458.4936.777.392.101.180.970.690.430.260.160.04LaplaceARL369.34195.5590.9851.0433.048.453.572.451.911.391.171.091.041.00MDRL307.00161.0076.0044.0029.008.003.003.002.001.001.001.001.001.00SDRL281.59152.8565.4434.6121.154.571.611.130.940.650.440.300.200.04LogisticARL370.13237.57121.4970.0345.8510.964.092.662.031.391.141.061.021.00MDRL307.00192.00100.0059.0039.0010.004.003.002.001.001.001.001.001.00SDRL281.59186.5590.1249.6430.866.101.811.130.940.640.390.250.150.020.75, 2.859NormalARL371.01353.40272.81191.02110.8520.745.012.661.891.341.121.031.011.00MDRL286.00256.00188.00133.0083.5016.004.003.002.001.001.001.001.001.00SDRL327.29341.61281.16205.18114.8316.922.941.230.770.490.330.160.090.00t(4)ARL372.03355.04291.69213.52144.0326.496.513.392.341.611.321.181.111.02MDRL286.00261.00207.00147.00100.0021.006.003.002.001.001.001.001.001.00SDRL327.18337.98286.71223.50152.1121.834.191.781.130.720.530.420.330.14t(8)ARL369.36355.52288.97207.82130.4223.255.723.002.101.451.211.091.041.00MDRL275.02263.10201.00138.0091.0018.005.003.002.001.001.001.001.001.00SDRL329.72335.93286.60213.56130.8318.733.551.500.940.580.420.300.200.06CNARL372.01358.82287.96198.80125.9022.205.362.831.991.391.151.061.021.00MDRL285.00256.00201.00129.5087.0017.005.003.002.001.001.001.001.001.00SDRL329.53342.61281.79207.94125.3618.013.231.340.850.540.380.240.150.04LaplaceARL368.77307.89196.55103.5263.2911.613.652.291.731.321.161.081.041.00MDRL281.00221.00133.0075.0048.009.003.002.002.001.001.001.001.001.00SDRL323.25294.27199.7596.2955.288.512.051.140.810.540.380.270.200.04LogisticARL369.51343.92259.29172.7898.2217.084.452.501.821.321.131.051.021.00MDRL283.00245.00179.00112.0073.0013.004.002.002.001.001.001.001.001.00SDRL325.98329.09256.64181.8690.1413.262.501.170.790.500.340.220.150.02Table 3
RL characteristics of the proposedNPTHWMARSS control chart under different distributions at nominal ARL0=370 and n=10.
δ(η, L)DistributionsMetrics00.0250.050.0750.10.250.50.7511.522.5350.25, 1.362NormalARL369.9566.0523.7512.598.402.521.141.001.001.001.001.001.001.00MDRL124.0035.0015.009.006.003.001.001.001.001.001.001.001.001.00SDRL479.1879.8724.7711.637.061.560.520.080.000.000.000.000.000.00t(4)ARL370.8984.1529.7916.0110.263.061.341.031.001.001.001.001.001.00MDRL128.0044.0018.0011.008.003.001.001.001.001.001.001.001.001.00SDRL478.00101.2632.0915.399.151.910.770.250.080.010.000.000.000.00t(8)ARL369.6675.0326.3214.159.242.771.231.011.001.001.001.001.001.00MDRL129.0038.0016.0010.007.003.001.001.001.001.001.001.001.001.00SDRL485.3792.1828.3913.597.931.740.640.150.020.000.000.000.000.00CNARL371.4270.3524.9413.418.772.611.171.011.001.001.001.001.001.00MDRL121.0036.0016.009.007.003.001.001.001.001.001.001.001.001.00SDRL479.2686.8826.8112.667.491.630.570.100.000.000.000.000.000.00LaplaceARL370.0745.6916.188.755.811.951.071.001.001.001.001.001.001.00MDRL126.0027.0011.007.005.001.001.001.001.001.001.001.001.001.00SDRL478.4251.9415.737.494.421.230.360.060.000.000.000.000.000.00LogisticARL371.4261.3221.0911.437.482.311.101.001.001.001.001.001.001.00MDRL129.0034.0013.008.006.003.001.001.001.001.001.001.001.001.00SDRL480.3771.4322.0210.166.161.440.440.050.000.000.000.000.000.000.50, 2.22NormalARL371.17127.1549.5426.4816.844.151.661.051.001.001.001.001.001.00MDRL298.00102.0042.0023.0015.004.001.001.001.001.001.001.001.001.00SDRL306.2798.0034.1516.6710.211.930.870.260.030.000.000.000.000.00t(4)ARL368.97152.0560.9632.9821.355.102.091.271.051.001.001.001.001.00MDRL292.00123.0052.0029.0019.005.002.001.001.001.001.001.001.001.00SDRL306.34118.7243.5521.8613.292.511.040.590.240.040.010.000.000.00t(8)ARL370.66139.5055.3429.7419.044.601.871.131.011.001.001.001.001.00MDRL291.00112.0047.0026.0017.004.002.001.001.001.001.001.001.001.00SDRL309.72110.1938.8619.1411.812.240.950.420.120.000.000.000.000.00CNARL369.50134.0751.7128.0717.884.351.751.091.001.001.001.001.001.00MDRL292.00107.0044.0025.0016.004.001.001.001.001.001.001.001.001.00SDRL307.98106.0036.2618.2310.872.050.910.350.060.010.000.000.000.00LaplaceARL369.3690.8733.1817.7311.213.121.381.041.001.001.001.001.001.00MDRL287.0076.0029.0016.0010.003.001.001.001.001.001.001.001.001.00SDRL308.2967.5421.6310.796.411.450.700.240.070.000.000.000.000.00LogisticARL370.26117.0543.9524.0214.913.801.541.051.001.001.001.001.001.00MDRL290.0095.0037.0021.0013.004.001.001.001.001.001.001.001.001.00SDRL309.2990.5629.5414.988.971.750.800.250.070.000.000.000.000.000.75, 2.319NormalARL371.13228.89100.4347.2226.674.141.501.041.001.001.001.001.001.00MDRL262.00160.0072.0034.0020.004.001.001.001.001.001.001.001.001.00SDRL361.86228.8795.9242.7822.452.380.640.200.040.000.000.000.000.00t(4)ARL370.19255.05120.9662.3835.395.461.861.201.031.001.001.001.001.00MDRL260.00175.0087.0046.0027.005.002.001.001.001.001.001.001.001.00SDRL362.84257.07114.8757.1630.693.420.880.420.180.040.000.000.000.00t(8)ARL371.31246.14109.4254.5630.504.761.651.111.011.001.001.001.001.00MDRL259.00169.0078.0040.0023.004.002.001.001.001.001.001.001.001.00SDRL368.61250.95104.5749.4026.312.900.750.320.100.010.000.000.000.00CNARL369.71237.43104.0550.2128.554.401.551.071.001.001.001.001.001.00MDRL258.00164.0075.0037.0021.004.001.001.001.001.001.001.001.001.00SDRL359.39236.6799.1945.2524.932.560.680.250.060.000.000.000.000.00LaplaceARL371.00174.1362.1427.4715.272.931.281.041.001.001.001.001.001.00MDRL260.00123.0046.0021.0012.003.001.001.001.001.001.001.001.001.00SDRL364.84167.9856.3423.2711.821.560.500.190.060.010.000.000.000.00LogisticARL369.71217.7988.6640.9522.373.671.401.041.001.001.001.001.001.00MDRL259.00153.0063.0031.0017.003.001.001.001.001.001.001.001.001.00SDRL362.01217.2785.1035.7418.452.060.570.190.050.000.000.000.000.00The same parameters are used for comparison purposes as reported in numerous relevant articles. TheARL measures are used to compare the proposed and competing control charts. Table 2 and Figures 1 and 2 provide the following outcomes:(i)
The proposed control chart’s ICRL distribution looks remarkably similar to all distributions examined in this study. For example, at η=0.10,0.25,0.50,0.75, n=5and10 and the ARL0=370 for all investigated distributions (see Tables 2 and 3).(ii)
As the smoothing parameter reduces, the proposed control chart becomes more effective in detecting shifts. This illustrates that the proposedNPTHWMARSS control chart is more sensitive to small smoothing parameters (see Figure 1).(iii)
As the sample size increases, the proposedNPTHWMARSS control chart’s performance increases.(iv)
The Laplace distribution outperforms the other distributions regarding OOC RL performance (see Figure2).(v)
The proposedNPTHWMARSS control chart’s ARL1 values increase as η increases at a certain size of the shift. For instance, under normal distribution at η=0.25,n=5,andδ=0.025, the ARL1=161.69, whereas when η=0.50,n=5,andδ=0.025, the ARL1=248.78 (see Tables 2 and 3).(vi)
The distribution of RL values is positively skewed, i.e.,ARL>MRDL (see Tables 2 and 3).(vii)
TheARL1 values of the proposed NPTHWMARSS control chart are smaller than competing control charts with different shift sizes in process location (see Figure 3).Figure 1
ARL characteristic of the proposedNPTHWMARSS control chart under various distributions for different values of η when n=5 and ARL0=370.Figure 2
ARL characteristic of the proposedNPTHWMARSS control chart under various distributions when n=5and10 and ARL0=370.Figure 3
ARL characteristic of the proposedNPTHWMARSS control chart under various distributions when η=0.10and0.25,n=5, and ARL0=370.


## 4. Comparative Study

This section compares the proposedNPTHWMARSS control chart with other competing control charts, including TEWMA-SR, TEWMA-X¯, TEWMA-SN, TEWMA-SRRSS, and NPDHWMARSS.


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

TheARL profile demonstrates that the proposed NPTHWMARSS control chart outperforms the TEWMA-SR control chart. For example, under normal distribution, at η=0.10,n=5, and δ=0.05,0.10,0.25,0.50,and0.75, the ARL1 values of the proposed NPTHWMARSS control chart are 23.15,11.27,3.98,1.93,and1.26, whereas the ARL1 values of the TEWMA-SR control charts are 201.99,87.06,21.39,7.37,and4.12 (see Tables 2 and 4). Similarly, when we consider Laplace distribution for comparison, we observed the same behaviour of the proposed NPTHWMARSS control chart. For instance, at η=0.25,n=5, and δ=0.25and0.50 the ARL1 values of the proposed NPTHWMARSS control chart are 4.34and2.10 while the ARL1 values of the TEWMA-SR control chart are 16.59and6.25 (see Tables 2 and 4). Figure 4 depicts the efficacy of the proposed control chart over the TEWMA-SR control chart. These findings demonstrate that the proposed NPTHWMARSS control chart performs better than the TEWMA-SR control chart for detecting shifts in process location.Table 4
RL characteristics of the TEWMA-SR control chart under different distribution atARL0=370 and n=5.
DistriNormalCNLaplacet(4)t(8)LogisticδARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRL(0.10, 2.04)0.00370.38251.00386.98370.01251.00388.09369.82251.00390.04371.41254.00390.11370.69249.00398.13370.92251.00390.270.05201.99140.00206.95196.07136.00201.10143.21101.00142.05161.05113.00163.04186.01129.00190.51183.86126.00186.310.1087.0665.0082.0584.0563.0078.1356.1445.0049.2562.5151.0055.6177.0259.0071.9277.0158.0071.630.2521.3920.0016.0220.3819.0015.0214.6114.0010.9815.7215.0010.9919.0519.0014.8719.1317.0013.800.507.377.005.027.077.004.815.576.003.795.616.003.726.728.004.996.657.004.550.754.124.002.514.014.001.963.413.001.953.343.001.863.763.002.253.754.002.241.002.813.001.292.893.001.232.643.001.192.562.001.052.703.001.232.713.001.221.252.292.000.822.273.000.812.312.000.732.272.000.662.312.000.742.302.000.711.502.072.000.422.052.000.432.052.000.482.042.000.392.091.000.532.082.000.40(0.25, 2.435)0.00370.41256.00372.81371.12256.00371.25370.89255.00375.36368.89257.00369.99371.25256.00370.99372.04256.00375.420.05250.98174.00250.11245.11169.00245.36192.03136.00189.21209.02146.00208.11235.56167.00235.23233.89163.00231.590.10124.0389.00121.23118.5186.00113.3278.1157.0073.8989.0566.00106.35111.2080.00106.21110.2576.00105.360.2526.2121.0021.0925.0121.0020.6316.5915.0012.0317.9216.0013.5122.8419.0018.0223.0218.0018.520.508.238.004.707.968.004.716.258.003.396.317.003.417.538.004.237.518.004.170.754.725.002.304.555.002.164.015.001.893.924.001.814.414.002.144.393.002.091.003.394.001.363.274.001.323.143.001.263.013.001.153.223.001.293.233.001.311.252.752.000.902.683.000.922.672.000.902.592.000.832.693.000.882.702.000.881.502.411.000.632.362.000.602.412.000.702.322.000.652.412.000.652.412.000.63Figure 4
ARL characteristic of the proposedNPTHWMARSS and competing control charts under various distributions when n=5 and ARL0=370.


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

The proposedNPTHWMARSS control chart performs better than the TEWMA-SN control chart. The RL profiles of NPTHWMARSS and TEWMA-SN control charts are shown in Tables 2 and 5 for a nominal ARL0 of 370. It is preferable to use the control chart with the smaller ARL1 value in a given shift because it detects the shift more quickly. The proposed NPTHWMARSS control charts’ efficiency can be observed under the normal distribution. For example, at n=5,η=0.25,andδ=0.05,0.10,and0.5, the ARL1 values for the proposed NPTHWMARSS control charts are 66.15,22.95,and2.40, whereas the ARL1 values for the TEWMA-SN control charts are 238.00,141.00,and26.00 (see Tables 2 and 5). Figure 4 shows the supremacy of the proposed control chart over the TEWMA-SN control chart.Table 5
RL characteristics of the TEWMA-SN control chart under normal distribution atARL0=370 and n=5.
δ(η,L)(η,L)(η,L)(η,L)(η,L)0.05, 1.7550.10, 2.0570.25, 2.4440.50, 2.7440.75, 2.807ARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRL0.00370.19228.00434.01370.91245.00404.01371.21256.00377.14370.90257.00374.52370.25262.00368.250.05300.18190.00344.05317.22211.00344.23342.63238.00345.36354.63245.00357.31362.71250.00359.810.10127.3991.00133.11151.00105.00155.12202.21141.00199.58255.01178.00253.69295.36207.00392.110.2565.4153.0062.0076.0357.0071.51108.2578.00104.23161.54114.00159.01215.14150.00214.560.5023.6520.0021.0126.1123.0020.3933.0126.0028.1453.2638.0050.4188.4864.0086.610.7512.3011.0010.5213.5713.0011.2115.6714.0011.1822.0410.0019.0339.3128.0037.011.007.397.006.248.358.006.279.459.005.8511.6810.008.9019.4715.0017.621.253.363.002.653.924.002.894.725.002.535.174.003.006.786.005.011.501.772.001.212.092.001.512.882.001.203.123.002.003.373.452.01


### 4.3. Proposed Versus TEWMA-X¯ Control Chart

The proposedNPTHWMARSS control chart’s ARL values are more sensitive at all combinations of δ and η than the TEWMA-X¯ control chart. For instance, in case of logistic distribution, when n=5,η=0.10, and δ= 0.05, 0.10, 0.25, 0.50, and 0.75, the ARL1 values of the proposed NPTHWMARSS and TEWMA-X¯ control charts are 22.05, 10.14, 3.69, 1.81, and 1.26 and 182.82, 76.01, 18.40, 5.95, and 2.94, respectively (see Tables 2 and 6). The supremacy of the proposed NPTHWMARSS control chart to the TEWMA-X¯ can also be seen in Figure 4. When compared to the normal distribution, the NPTHWMARSS and TEWMA-X¯ control charts reduce ARL 82.12% and 8.80%, respectively, when η=25% and δ=5%, (see Tables 2 and 6). In all investigated distributions, the proposed NPTHWMARSS control chart is more responsive to identifying shifts than the TEWMA-X¯ control chart.Table 6
RL characteristics of the TEWMA-X¯ control chart under different distributions at ARL0=370 and n=5.
ShiftNormalCNLaplacet(4)t(8)LogisticARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRL(0.10, 2.052)0.00369.03250.00395.95370.86238.00406.48369.16253.00396.30369.29242.00400.55369.13241.50401.65369.25251.00395.310.05307.06202.00328.82184.21127.50187.56181.96130.00191.07186.99124.00194.09187.84129.50197.86182.82129.00190.550.10200.72136.00211.0775.8457.0070.4076.6357.0072.1377.4556.0074.0475.7356.0071.4876.0157.0071.150.2565.8749.0060.9418.4516.0013.7818.5516.0014.2518.3016.0013.2418.5817.0013.8318.4016.0014.080.5021.8319.0016.995.885.004.485.955.004.475.805.004.366.075.004.565.955.004.510.7511.5410.008.572.842.002.112.852.002.072.862.002.152.912.002.182.942.002.171.007.176.005.371.761.001.141.781.001.171.751.001.181.781.001.161.801.001.201.254.794.003.601.301.000.651.311.000.641.261.000.601.311.000.651.281.000.631.503.553.002.721.101.000.351.101.000.331.101.000.351.101.000.341.101.000.35(0.25, 2.456)0.00369.58255.00371.62370.73246.50382.48370.35259.00385.77369.93259.00374.25369.15252.00378.00370.81259.50379.060.05337.05237.00344.14240.89171.00237.76238.73165.00241.88244.15163.00244.05245.69170.00244.08235.39165.00236.630.10253.31180.00249.89111.7279.00108.46113.3579.00109.29112.6880.00106.57113.5779.00113.25109.9776.00106.030.2592.0363.0090.2721.6817.0017.4422.4118.0017.6322.7918.0018.0222.1018.0017.9622.4218.0017.580.5026.4521.0021.866.626.004.196.526.003.996.566.004.026.606.004.076.726.004.190.7512.9011.009.253.293.002.043.393.002.113.353.002.033.423.002.083.443.002.151.008.047.005.232.112.001.252.112.001.242.092.001.242.122.001.262.102.001.211.255.495.003.441.521.000.791.501.000.771.501.000.791.501.000.771.501.000.771.504.124.002.571.211.000.481.211.000.491.201.000.471.221.000.481.221.000.48


### 4.4. Proposed versus TEWMA-SRRSS Control Chart

TheARL study reveals that the proposed NPTHWMARSS control chart outperforms the TEWMA-SRRSS control chart for all possible values of ηandδ (see Tables 2 and 7). As an illustration, for CN distribution, at η=0.10andδ=0.05,0.075,0.10,0.5,and0.75, the ARL1 values of the NPTHWMARSS and TEWMA-SRRSS control charts are 24.74,15.95,11.18,2.01,and1.34 and 110.71,64.19,40.58,3.28,and1.79, respectively (see Tables 2 and 7). Under normal distribution, at η=25%, a 2.5% increase in process location parameter reduces the ARL by 23.31% for the TEWMA-SRRSS control chart while the NPTHWMARSS reduces ARL by 56.31% (see Tables 2 and 7). Figure 4 also shows the superiority of the NPTHWMARSS control chart over the TEWMA-SRRSS control chart. The results show that the NPTHWMARSS control chart is superior to the TEWMA-SRRSS control chart in process location’s shift detection.Table 7
RL characteristics of the TEWMA-SRRSS control chart under different distributions at n=5 and ARL0=370.
δ(0.1, 1.837)DistributionsMetrics0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00N(0, 1)ARL371.18240.21201.32108.0858.0139.0022.1013.349.223.111.811.301.041.01(0.25, 2.204)MDRL242.01163.90142.9976.1244.1331.8920.0211.937.992.011.021.101.011.00SDRL401.88254.19201.14107.0153.3932.8717.0110.007.052.200.980.570.190.04CNARL369.87223.16190.11111.6764.2840.6520.9614.0210.003.301.811.361.081.00MDRL241.10154.82135.9876.0147.9633.9017.9911.957.972.971.981.021.011.01SDRL403.28234.26190.11110.0159.8831.9917.2110.147.052.411.050.580.300.10LogisticARL368.99219.80180.2198.0551.9236.0117.8911.197.932.751.711.321.101.04MDRL260.00151.01125.1173.0741.2130.1015.809.956.872.021.011.011.011.00SDRL381.31231.52178.0391.2549.3932.6714.137.996.081.880.930.610.300.09LaplaceARL370.01180.11146.4971.9838.9326.0413.999.006.382.521.621.311.111.09MDRL260.10120.00104.0056.2032.8221.9612.907.914.932.101.081.071.051.01SDRL380.32193.11143.6367.7135.2920.4711.535.984.891.730.910.610.290.03t(4)ARL371.01258.90230.50143.1080.9549.0227.0017.0311.984.012.321.631.181.07MDRL240.38168.61156.10103.0061.5339.0223.6114.3112.013.132.091.211.081.01SDRL384.00291.02242.26136.3576.4143.5220.0013.219.032.911.631.000.630.30t(8)ARL369.92241.83204.59126.8368.0545.0123.0016.2010.613.531.991.401.091.02MDRL251.23170.25140.0190.3053.2137.6120.1114.359.223.052.021.021.0011.00SDRL418.75261.49203.81130.2160.3636.8917.2811.327.632.581.250.680.300.09N(0, 1)ARL370.74284.31253.51151.8488.7854.5724.9215.5710.383.592.091.441.071.01MDRL267.50199.50167.00108.5064.0038.5019.0013.009.003.002.001.001.001.00SDRL369.16291.85267.11151.3588.6650.8020.1711.016.992.131.060.630.260.08CNARL371.10293.39250.56154.4888.7957.9427.4515.7411.223.782.121.581.131.02MDRL273.00214.50172.50109.5064.0042.0021.0013.0010.003.502.001.001.001.00SDRL366.10279.85252.76141.0183.9053.5423.5911.447.552.101.080.750.370.15LogisticARL370.38271.57242.25137.5877.8747.2022.3413.599.283.231.951.441.081.02MDRL265.50180.50169.50103.0058.0036.0018.5012.008.003.002.001.001.001.00SDRL360.62282.24242.01135.3869.5640.4416.299.515.851.871.020.650.280.13LaplaceARL370.38230.11199.04105.7652.4430.7215.0810.527.072.821.831.351.101.02MDRL265.50156.50144.0075.0038.5023.0013.009.006.003.002.001.001.001.00SDRL360.62223.94196.34100.8147.6726.9711.197.144.411.680.970.650.330.17t(4)ARL371.04304.19282.63193.67113.3872.9632.7819.3713.044.522.521.851.281.11MDRL251.00208.50200.00136.5080.0051.0025.0015.0011.004.002.002.001.001.00SDRL367.97313.54280.19186.75112.5370.9227.1616.718.952.581.480.980.550.36t(8)ARL370.07116.22108.7681.7354.0536.0819.7012.128.853.081.891.351.071.01MDRL261.5082.5074.0060.5039.0026.0016.0011.008.003.002.001.001.001.00SDRL366.37115.99114.0880.4953.3236.9016.959.096.201.921.040.590.260.11


### 4.5. Proposed versusNPDHWMARSS Control Chart

TheNPTHWMARSS control chart outperforms the NPDHWMARSS control chart in detecting shifts in process location. For example, using t(8) distribution at n=5,η=0.10,andδ=0.025,0.10,and0.25, the ARL1 values of the proposed NPTHWMARSS and NPDHWMARSS control charts are 53.91,12.06,and4.37 and 142.35,21.43,and5.93, respectively (see Tables 2 and 8). Similarly, under Laplace distribution, when n=5,η=0.25,andδ=0.05and0.50, the ARL1 values of the proposed NPTHWMARSS control chart are 41.74and2.10 while the ARL1 values for NPDHWMARSS control chart are 77.27and3.20, respectively (see Tables 2 and 8). As illustrated in Figure 4, the NPTHWMARSS control chart outperforms the NPDHWMARSS control chart.Table 8
RL characteristics of theNPDHWMARSS control chart under various distributions at nominal ARL0=370 and n=5.
δ(η, L)DistributionsMetrics00.0250.050.0750.10.250.50.7511.522.535(0.10, 1.449)NormalARL370.46130.3353.1429.5519.285.412.411.541.171.011.001.001.001.00MDRL50.0037.0022.0015.0011.004.003.001.001.001.001.001.001.001.00SDRL631.89202.9872.0636.3321.474.051.470.950.570.120.010.000.000.00CNARL371.89135.0556.1831.2920.345.662.521.621.231.031.011.001.001.00MDRL50.0036.0023.0016.0012.004.003.001.001.001.001.001.001.001.00SDRL629.45211.0676.7238.7622.854.311.541.010.650.230.110.060.030.00t(4)ARL371.56152.0565.2436.2223.726.522.891.921.471.141.051.031.011.00MDRL49.0037.5026.0018.0013.005.003.001.001.001.001.001.001.001.00SDRL628.70241.6190.2145.8027.695.271.801.200.890.520.320.230.160.08t(8)ARL372.17142.3559.1633.0621.435.932.641.731.311.051.011.001.001.00MDRL47.0038.0024.0016.0012.005.003.001.001.001.001.001.001.001.00SDRL628.03223.9681.0341.1724.614.611.621.080.740.320.150.090.050.01LaplaceARL370.5890.4134.8618.8312.754.102.101.481.231.051.011.001.001.00MDRL49.0031.0017.0011.009.004.001.001.001.001.001.001.001.001.00SDRL631.32133.7343.2821.1212.782.751.290.900.650.300.150.080.040.00LogisticARL370.48118.8747.9226.2116.974.942.261.511.181.021.001.001.001.00MDRL49.0036.0021.0014.0010.004.003.001.001.001.001.001.001.001.00SDRL631.32182.1163.7931.6518.293.581.390.920.580.220.060.030.000.00(0.25, 2.365)NormalARL368.05227.94114.9467.5144.5910.563.972.511.821.151.011.001.001.00MDRL333.00191.0098.0059.0039.009.004.003.001.001.001.001.001.001.00SDRL257.53174.1485.1147.8830.836.471.891.241.020.530.150.030.010.00CNARL367.90235.00120.5270.8046.8011.174.162.621.901.221.041.011.001.00MDRL331.00198.00102.0061.0041.0010.004.003.001.001.001.001.001.001.00SDRL255.79178.6890.0650.8732.526.872.031.291.060.620.290.140.080.00t(4)ARL369.09250.46134.3281.3354.3813.184.823.002.231.481.191.071.041.00MDRL330.00212.00113.0070.0047.0012.004.003.003.001.001.001.001.001.00SDRL257.37188.46101.0459.2138.328.382.511.511.220.870.590.380.260.08t(8)ARL368.82240.57124.2074.3849.3511.854.372.752.011.301.071.021.011.00MDRL333.00203.00105.0064.0043.0010.004.003.001.001.001.001.001.001.00SDRL256.51181.3993.0353.1934.517.392.191.361.110.720.380.190.100.00LaplaceARL368.23174.6677.2744.4329.237.303.202.151.631.171.041.011.001.00MDRL334.00146.0067.0039.0026.007.003.003.001.001.001.001.001.001.00SDRL256.71132.2755.5431.2119.794.261.631.210.970.570.290.150.060.00LogisticARL368.23212.61104.4760.2139.349.333.652.351.721.171.031.011.001.00MDRL334.00179.0089.0052.0034.008.004.003.001.001.001.001.001.001.00SDRL256.71162.2578.1442.4427.065.631.781.221.000.560.240.100.040.00


## 4.1. Proposed versus TEWMA-SR Control Chart

TheARL profile demonstrates that the proposed NPTHWMARSS control chart outperforms the TEWMA-SR control chart. For example, under normal distribution, at η=0.10,n=5, and δ=0.05,0.10,0.25,0.50,and0.75, the ARL1 values of the proposed NPTHWMARSS control chart are 23.15,11.27,3.98,1.93,and1.26, whereas the ARL1 values of the TEWMA-SR control charts are 201.99,87.06,21.39,7.37,and4.12 (see Tables 2 and 4). Similarly, when we consider Laplace distribution for comparison, we observed the same behaviour of the proposed NPTHWMARSS control chart. For instance, at η=0.25,n=5, and δ=0.25and0.50 the ARL1 values of the proposed NPTHWMARSS control chart are 4.34and2.10 while the ARL1 values of the TEWMA-SR control chart are 16.59and6.25 (see Tables 2 and 4). Figure 4 depicts the efficacy of the proposed control chart over the TEWMA-SR control chart. These findings demonstrate that the proposed NPTHWMARSS control chart performs better than the TEWMA-SR control chart for detecting shifts in process location.Table 4
RL characteristics of the TEWMA-SR control chart under different distribution atARL0=370 and n=5.
DistriNormalCNLaplacet(4)t(8)LogisticδARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRL(0.10, 2.04)0.00370.38251.00386.98370.01251.00388.09369.82251.00390.04371.41254.00390.11370.69249.00398.13370.92251.00390.270.05201.99140.00206.95196.07136.00201.10143.21101.00142.05161.05113.00163.04186.01129.00190.51183.86126.00186.310.1087.0665.0082.0584.0563.0078.1356.1445.0049.2562.5151.0055.6177.0259.0071.9277.0158.0071.630.2521.3920.0016.0220.3819.0015.0214.6114.0010.9815.7215.0010.9919.0519.0014.8719.1317.0013.800.507.377.005.027.077.004.815.576.003.795.616.003.726.728.004.996.657.004.550.754.124.002.514.014.001.963.413.001.953.343.001.863.763.002.253.754.002.241.002.813.001.292.893.001.232.643.001.192.562.001.052.703.001.232.713.001.221.252.292.000.822.273.000.812.312.000.732.272.000.662.312.000.742.302.000.711.502.072.000.422.052.000.432.052.000.482.042.000.392.091.000.532.082.000.40(0.25, 2.435)0.00370.41256.00372.81371.12256.00371.25370.89255.00375.36368.89257.00369.99371.25256.00370.99372.04256.00375.420.05250.98174.00250.11245.11169.00245.36192.03136.00189.21209.02146.00208.11235.56167.00235.23233.89163.00231.590.10124.0389.00121.23118.5186.00113.3278.1157.0073.8989.0566.00106.35111.2080.00106.21110.2576.00105.360.2526.2121.0021.0925.0121.0020.6316.5915.0012.0317.9216.0013.5122.8419.0018.0223.0218.0018.520.508.238.004.707.968.004.716.258.003.396.317.003.417.538.004.237.518.004.170.754.725.002.304.555.002.164.015.001.893.924.001.814.414.002.144.393.002.091.003.394.001.363.274.001.323.143.001.263.013.001.153.223.001.293.233.001.311.252.752.000.902.683.000.922.672.000.902.592.000.832.693.000.882.702.000.881.502.411.000.632.362.000.602.412.000.702.322.000.652.412.000.652.412.000.63Figure 4
ARL characteristic of the proposedNPTHWMARSS and competing control charts under various distributions when n=5 and ARL0=370.


## 4.2. Proposed versus TEWMA-SN Control Chart

The proposedNPTHWMARSS control chart performs better than the TEWMA-SN control chart. The RL profiles of NPTHWMARSS and TEWMA-SN control charts are shown in Tables 2 and 5 for a nominal ARL0 of 370. It is preferable to use the control chart with the smaller ARL1 value in a given shift because it detects the shift more quickly. The proposed NPTHWMARSS control charts’ efficiency can be observed under the normal distribution. For example, at n=5,η=0.25,andδ=0.05,0.10,and0.5, the ARL1 values for the proposed NPTHWMARSS control charts are 66.15,22.95,and2.40, whereas the ARL1 values for the TEWMA-SN control charts are 238.00,141.00,and26.00 (see Tables 2 and 5). Figure 4 shows the supremacy of the proposed control chart over the TEWMA-SN control chart.Table 5
RL characteristics of the TEWMA-SN control chart under normal distribution atARL0=370 and n=5.
δ(η,L)(η,L)(η,L)(η,L)(η,L)0.05, 1.7550.10, 2.0570.25, 2.4440.50, 2.7440.75, 2.807ARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRL0.00370.19228.00434.01370.91245.00404.01371.21256.00377.14370.90257.00374.52370.25262.00368.250.05300.18190.00344.05317.22211.00344.23342.63238.00345.36354.63245.00357.31362.71250.00359.810.10127.3991.00133.11151.00105.00155.12202.21141.00199.58255.01178.00253.69295.36207.00392.110.2565.4153.0062.0076.0357.0071.51108.2578.00104.23161.54114.00159.01215.14150.00214.560.5023.6520.0021.0126.1123.0020.3933.0126.0028.1453.2638.0050.4188.4864.0086.610.7512.3011.0010.5213.5713.0011.2115.6714.0011.1822.0410.0019.0339.3128.0037.011.007.397.006.248.358.006.279.459.005.8511.6810.008.9019.4715.0017.621.253.363.002.653.924.002.894.725.002.535.174.003.006.786.005.011.501.772.001.212.092.001.512.882.001.203.123.002.003.373.452.01


## 4.3. Proposed Versus TEWMA-X¯ Control Chart

The proposedNPTHWMARSS control chart’s ARL values are more sensitive at all combinations of δ and η than the TEWMA-X¯ control chart. For instance, in case of logistic distribution, when n=5,η=0.10, and δ= 0.05, 0.10, 0.25, 0.50, and 0.75, the ARL1 values of the proposed NPTHWMARSS and TEWMA-X¯ control charts are 22.05, 10.14, 3.69, 1.81, and 1.26 and 182.82, 76.01, 18.40, 5.95, and 2.94, respectively (see Tables 2 and 6). The supremacy of the proposed NPTHWMARSS control chart to the TEWMA-X¯ can also be seen in Figure 4. When compared to the normal distribution, the NPTHWMARSS and TEWMA-X¯ control charts reduce ARL 82.12% and 8.80%, respectively, when η=25% and δ=5%, (see Tables 2 and 6). In all investigated distributions, the proposed NPTHWMARSS control chart is more responsive to identifying shifts than the TEWMA-X¯ control chart.Table 6
RL characteristics of the TEWMA-X¯ control chart under different distributions at ARL0=370 and n=5.
ShiftNormalCNLaplacet(4)t(8)LogisticARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRLARLMDRLSDRL(0.10, 2.052)0.00369.03250.00395.95370.86238.00406.48369.16253.00396.30369.29242.00400.55369.13241.50401.65369.25251.00395.310.05307.06202.00328.82184.21127.50187.56181.96130.00191.07186.99124.00194.09187.84129.50197.86182.82129.00190.550.10200.72136.00211.0775.8457.0070.4076.6357.0072.1377.4556.0074.0475.7356.0071.4876.0157.0071.150.2565.8749.0060.9418.4516.0013.7818.5516.0014.2518.3016.0013.2418.5817.0013.8318.4016.0014.080.5021.8319.0016.995.885.004.485.955.004.475.805.004.366.075.004.565.955.004.510.7511.5410.008.572.842.002.112.852.002.072.862.002.152.912.002.182.942.002.171.007.176.005.371.761.001.141.781.001.171.751.001.181.781.001.161.801.001.201.254.794.003.601.301.000.651.311.000.641.261.000.601.311.000.651.281.000.631.503.553.002.721.101.000.351.101.000.331.101.000.351.101.000.341.101.000.35(0.25, 2.456)0.00369.58255.00371.62370.73246.50382.48370.35259.00385.77369.93259.00374.25369.15252.00378.00370.81259.50379.060.05337.05237.00344.14240.89171.00237.76238.73165.00241.88244.15163.00244.05245.69170.00244.08235.39165.00236.630.10253.31180.00249.89111.7279.00108.46113.3579.00109.29112.6880.00106.57113.5779.00113.25109.9776.00106.030.2592.0363.0090.2721.6817.0017.4422.4118.0017.6322.7918.0018.0222.1018.0017.9622.4218.0017.580.5026.4521.0021.866.626.004.196.526.003.996.566.004.026.606.004.076.726.004.190.7512.9011.009.253.293.002.043.393.002.113.353.002.033.423.002.083.443.002.151.008.047.005.232.112.001.252.112.001.242.092.001.242.122.001.262.102.001.211.255.495.003.441.521.000.791.501.000.771.501.000.791.501.000.771.501.000.771.504.124.002.571.211.000.481.211.000.491.201.000.471.221.000.481.221.000.48


## 4.4. Proposed versus TEWMA-SRRSS Control Chart

TheARL study reveals that the proposed NPTHWMARSS control chart outperforms the TEWMA-SRRSS control chart for all possible values of ηandδ (see Tables 2 and 7). As an illustration, for CN distribution, at η=0.10andδ=0.05,0.075,0.10,0.5,and0.75, the ARL1 values of the NPTHWMARSS and TEWMA-SRRSS control charts are 24.74,15.95,11.18,2.01,and1.34 and 110.71,64.19,40.58,3.28,and1.79, respectively (see Tables 2 and 7). Under normal distribution, at η=25%, a 2.5% increase in process location parameter reduces the ARL by 23.31% for the TEWMA-SRRSS control chart while the NPTHWMARSS reduces ARL by 56.31% (see Tables 2 and 7). Figure 4 also shows the superiority of the NPTHWMARSS control chart over the TEWMA-SRRSS control chart. The results show that the NPTHWMARSS control chart is superior to the TEWMA-SRRSS control chart in process location’s shift detection.Table 7
RL characteristics of the TEWMA-SRRSS control chart under different distributions at n=5 and ARL0=370.
δ(0.1, 1.837)DistributionsMetrics0.000.0250.030.050.0750.100.150.200.250.500.751.001.502.00N(0, 1)ARL371.18240.21201.32108.0858.0139.0022.1013.349.223.111.811.301.041.01(0.25, 2.204)MDRL242.01163.90142.9976.1244.1331.8920.0211.937.992.011.021.101.011.00SDRL401.88254.19201.14107.0153.3932.8717.0110.007.052.200.980.570.190.04CNARL369.87223.16190.11111.6764.2840.6520.9614.0210.003.301.811.361.081.00MDRL241.10154.82135.9876.0147.9633.9017.9911.957.972.971.981.021.011.01SDRL403.28234.26190.11110.0159.8831.9917.2110.147.052.411.050.580.300.10LogisticARL368.99219.80180.2198.0551.9236.0117.8911.197.932.751.711.321.101.04MDRL260.00151.01125.1173.0741.2130.1015.809.956.872.021.011.011.011.00SDRL381.31231.52178.0391.2549.3932.6714.137.996.081.880.930.610.300.09LaplaceARL370.01180.11146.4971.9838.9326.0413.999.006.382.521.621.311.111.09MDRL260.10120.00104.0056.2032.8221.9612.907.914.932.101.081.071.051.01SDRL380.32193.11143.6367.7135.2920.4711.535.984.891.730.910.610.290.03t(4)ARL371.01258.90230.50143.1080.9549.0227.0017.0311.984.012.321.631.181.07MDRL240.38168.61156.10103.0061.5339.0223.6114.3112.013.132.091.211.081.01SDRL384.00291.02242.26136.3576.4143.5220.0013.219.032.911.631.000.630.30t(8)ARL369.92241.83204.59126.8368.0545.0123.0016.2010.613.531.991.401.091.02MDRL251.23170.25140.0190.3053.2137.6120.1114.359.223.052.021.021.0011.00SDRL418.75261.49203.81130.2160.3636.8917.2811.327.632.581.250.680.300.09N(0, 1)ARL370.74284.31253.51151.8488.7854.5724.9215.5710.383.592.091.441.071.01MDRL267.50199.50167.00108.5064.0038.5019.0013.009.003.002.001.001.001.00SDRL369.16291.85267.11151.3588.6650.8020.1711.016.992.131.060.630.260.08CNARL371.10293.39250.56154.4888.7957.9427.4515.7411.223.782.121.581.131.02MDRL273.00214.50172.50109.5064.0042.0021.0013.0010.003.502.001.001.001.00SDRL366.10279.85252.76141.0183.9053.5423.5911.447.552.101.080.750.370.15LogisticARL370.38271.57242.25137.5877.8747.2022.3413.599.283.231.951.441.081.02MDRL265.50180.50169.50103.0058.0036.0018.5012.008.003.002.001.001.001.00SDRL360.62282.24242.01135.3869.5640.4416.299.515.851.871.020.650.280.13LaplaceARL370.38230.11199.04105.7652.4430.7215.0810.527.072.821.831.351.101.02MDRL265.50156.50144.0075.0038.5023.0013.009.006.003.002.001.001.001.00SDRL360.62223.94196.34100.8147.6726.9711.197.144.411.680.970.650.330.17t(4)ARL371.04304.19282.63193.67113.3872.9632.7819.3713.044.522.521.851.281.11MDRL251.00208.50200.00136.5080.0051.0025.0015.0011.004.002.002.001.001.00SDRL367.97313.54280.19186.75112.5370.9227.1616.718.952.581.480.980.550.36t(8)ARL370.07116.22108.7681.7354.0536.0819.7012.128.853.081.891.351.071.01MDRL261.5082.5074.0060.5039.0026.0016.0011.008.003.002.001.001.001.00SDRL366.37115.99114.0880.4953.3236.9016.959.096.201.921.040.590.260.11


## 4.5. Proposed versusNPDHWMARSS Control Chart

TheNPTHWMARSS control chart outperforms the NPDHWMARSS control chart in detecting shifts in process location. For example, using t(8) distribution at n=5,η=0.10,andδ=0.025,0.10,and0.25, the ARL1 values of the proposed NPTHWMARSS and NPDHWMARSS control charts are 53.91,12.06,and4.37 and 142.35,21.43,and5.93, respectively (see Tables 2 and 8). Similarly, under Laplace distribution, when n=5,η=0.25,andδ=0.05and0.50, the ARL1 values of the proposed NPTHWMARSS control chart are 41.74and2.10 while the ARL1 values for NPDHWMARSS control chart are 77.27and3.20, respectively (see Tables 2 and 8). As illustrated in Figure 4, the NPTHWMARSS control chart outperforms the NPDHWMARSS control chart.Table 8
RL characteristics of theNPDHWMARSS control chart under various distributions at nominal ARL0=370 and n=5.
δ(η, L)DistributionsMetrics00.0250.050.0750.10.250.50.7511.522.535(0.10, 1.449)NormalARL370.46130.3353.1429.5519.285.412.411.541.171.011.001.001.001.00MDRL50.0037.0022.0015.0011.004.003.001.001.001.001.001.001.001.00SDRL631.89202.9872.0636.3321.474.051.470.950.570.120.010.000.000.00CNARL371.89135.0556.1831.2920.345.662.521.621.231.031.011.001.001.00MDRL50.0036.0023.0016.0012.004.003.001.001.001.001.001.001.001.00SDRL629.45211.0676.7238.7622.854.311.541.010.650.230.110.060.030.00t(4)ARL371.56152.0565.2436.2223.726.522.891.921.471.141.051.031.011.00MDRL49.0037.5026.0018.0013.005.003.001.001.001.001.001.001.001.00SDRL628.70241.6190.2145.8027.695.271.801.200.890.520.320.230.160.08t(8)ARL372.17142.3559.1633.0621.435.932.641.731.311.051.011.001.001.00MDRL47.0038.0024.0016.0012.005.003.001.001.001.001.001.001.001.00SDRL628.03223.9681.0341.1724.614.611.621.080.740.320.150.090.050.01LaplaceARL370.5890.4134.8618.8312.754.102.101.481.231.051.011.001.001.00MDRL49.0031.0017.0011.009.004.001.001.001.001.001.001.001.001.00SDRL631.32133.7343.2821.1212.782.751.290.900.650.300.150.080.040.00LogisticARL370.48118.8747.9226.2116.974.942.261.511.181.021.001.001.001.00MDRL49.0036.0021.0014.0010.004.003.001.001.001.001.001.001.001.00SDRL631.32182.1163.7931.6518.293.581.390.920.580.220.060.030.000.00(0.25, 2.365)NormalARL368.05227.94114.9467.5144.5910.563.972.511.821.151.011.001.001.00MDRL333.00191.0098.0059.0039.009.004.003.001.001.001.001.001.001.00SDRL257.53174.1485.1147.8830.836.471.891.241.020.530.150.030.010.00CNARL367.90235.00120.5270.8046.8011.174.162.621.901.221.041.011.001.00MDRL331.00198.00102.0061.0041.0010.004.003.001.001.001.001.001.001.00SDRL255.79178.6890.0650.8732.526.872.031.291.060.620.290.140.080.00t(4)ARL369.09250.46134.3281.3354.3813.184.823.002.231.481.191.071.041.00MDRL330.00212.00113.0070.0047.0012.004.003.003.001.001.001.001.001.00SDRL257.37188.46101.0459.2138.328.382.511.511.220.870.590.380.260.08t(8)ARL368.82240.57124.2074.3849.3511.854.372.752.011.301.071.021.011.00MDRL333.00203.00105.0064.0043.0010.004.003.001.001.001.001.001.001.00SDRL256.51181.3993.0353.1934.517.392.191.361.110.720.380.190.100.00LaplaceARL368.23174.6677.2744.4329.237.303.202.151.631.171.041.011.001.00MDRL334.00146.0067.0039.0026.007.003.003.001.001.001.001.001.001.00SDRL256.71132.2755.5431.2119.794.261.631.210.970.570.290.150.060.00LogisticARL368.23212.61104.4760.2139.349.333.652.351.721.171.031.011.001.00MDRL334.00179.0089.0052.0034.008.004.003.001.001.001.001.001.001.00SDRL256.71162.2578.1442.4427.065.631.781.221.000.560.240.100.040.00


## 5. Real-Life Application

This section illustrates a real-life application of the nonisothermal continuous stirred tank reactor (CSTR) process to demonstrate the applicability of the proposedNPTHWMARSS control chart. Adegoke et al. [27] recently considered this real-life data. The CSTR process has nine different variables, one of which we choose as the variable of interest (X), which represents the output temperature, and this variable is used in real-life application with parameters μX=369.88 and σX2=0.32. It started with 1000 observations, 600 of which were made while the process was in an IC state. After the 24th observation, a shift in the process location is introduced following Anwar et al. [28] and Anwar et al. [29]. The estimation is carried out with the help of the mentioned parameters, and the control limits are obtained. In order to use the proposed control chart and the existing (TEWMA-SRRSS) control chart in practise, the variable of interest (X) is used. We used the RSS approach to generate 40 paired observations of size n = 5 and m=1 from a normal distribution. The parameters of the proposed NPTHWMARSS and TEWMA-SRRSS control charts used for real-life analysis are L=1.26,η=0.10,ARL0=370 and L=1.837,η=0.10,ARL0=370, respectively. Figure 5 indicates that the proposed NPTHWMARSS control chart triggers the first OOC signal at sample number 29 while the TEWMA-SRRSS control chart detects the first OOC point at sample number 35. Similarly, the proposed NPTHWMARSS control chart detects overall 12 OOC points, whereas the TEWMA-SRRSS control chart detects 06 OOC points (see Figure 5).Figure 5
Real-life application of the proposedNPTHWMARSS control chart against the TEWMA-SRRSS control chart.


## 6. Conclusions and Recommendations

The basic assumption of the parametric control charts is that the underlying process follows the specific distribution. When the features do not follow a particular distribution, using nonparametric (NP) control charts to monitor the quality characteristics is recommended. The triple homogeneously weighted moving average (THWMA) is the advanced version of the triple exponentially weighted moving average (TEWMA) control chart for enhanced process location monitoring. Similarly, the ranked set sampling (RSS) technique is preferred over simple random sampling (SRS) when the sampling procedure is costly and complicated. To this end, this study uses the NP THWMA control chart and the RSS technique, which presents an NP THWMA Wilcoxon signed-rank control chart (denoted byNPTHWMARSS) for enhanced monitoring of process location shifts. The proposed control chart’s performance is evaluated in ARL, MDRL, and SDRL. The findings demonstrated that the proposed control chart outperforms competitor control charts such as TEWMA-SR, TEWMA-X¯, TEWMA-SN, TEWMA-SRRSS, and NPDHWMARSS. Furthermore, a real-life application is presented to illustrate the proposed control chart’s application in practice. The proposed study can be extended to the more ranked set sampling techniques and multivariate scenarios.where Xϵℜ.



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*Source: 3823013-2022-08-22.xml*