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

Authors: Zahid Rasheed; Majid Khan; Nafiu Lukman Abiodun; Syed Masroor Anwar; G. Khalaf; Saddam Akber Abbasi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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


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## Body

## 1. Introduction

A quality assurance system offers a variety of management approaches that save time and money while producing a high-quality final product. These approaches are commonly utilized in the manufacturing process to detect irregularities and improve product quality. Statistical process control (SPC) is a key aspect in detecting abnormalities of ultimate products. SPC techniques are widely used in industrial applications, biological sciences, environmental studies, and healthcare departments to monitor the ongoing processes. The quality of the products is influenced by unnatural variation. The existence of unnatural variations causes the shift in process parameters (location and/or dispersion). Control charts are popular tools in SPC that helps in identifying the shifts in process parameters. Usually, control charts are generally classified as memoryless and memory-type charts based on their design structure. Shewhart [1] introduced the first memoryless chart known as the Shewhart chart, whereas Page [2] and Roberts [3] introduced the concept of memory charts, known as a cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts, respectively.Classical parametric control charts are usually used when an ongoing process follows a predefined probability distribution. The ongoing process may not follow a specific distribution, or the distribution of the ongoing process may be in doubt. Nonparametric (NP) control charts are a reliable substitute for parametric control charts to handle such scenarios. NP control charts are convenient because their in-control (IC) run-length (RL) distribution is the same for all continuous distributions. The sign (SN) and Wilcoxon signed-rank (SR) are two well-known NP techniques practiced in statistical process monitoring (SPM) with control charts. Likewise, simple random sampling (SRS) and ranked set sampling (RSS) techniques are quite often used in SPM with both parametric and NP control charts to observe the data of the ongoing processes [4]. RSS is recommended in the SPM literature as it decreases variability and improves the efficiency of the associated control charts [5, 6]. For instance, hazardous waste sites with varying levels of contamination can be classified visually based on soil staining, whereas actual statistics of toxic chemicals and quantification of their ecological impact are prohibitively expensive.Numerous researchers introduced a wide range of NP control charts using different NP statistics. Amin and Searcy [7] introduced an efficient NP EWMA-SR control chart to monitor the process location shift efficiently. Similarly, Bakir [8] presented NP Shewhart-SR control chart for process location monitoring. Subsequently, Yang, et al. [9] offered an EWMA-SN control chart for process location. Moreover, Graham, et al. [10] and Graham et al. [11] designed an NP EWMA-SN control chart based on a single observation and NP EWMA-SR control chart, respectively, for monitoring shifts in process location. Likewise, Lu [12] and Tsai et al. [13] presented enhanced NP EWMA control charts based on SN statistics and used RSS techniques in conjunction with NP control charts. Eventually, Chakraborty et al. [14] and Lu [15] developed a generally weighted moving average-SR (GWMA-SR) and SN statistic-based NP double GWMA control charts for process proportion, respectively, to improve the shift detection ability. Furthermore, Chakraborti and Graham [16] reviewed some latest development in both univariate and multivariate NP control charts. Also, Rasheed et al. [17] and Rasheed et al. [18] advocated RSS-based parametric and NP control charts for efficiently monitoring the process mean, respectively.The recently introduced homogeneously weighted moving average (HWMA) control chart by Abbas [19] is efficient for monitoring process location. Adegoke et al. [20] provided an auxiliary information-based (AIB) HWMA control chart for process location more efficient than the HWMA control chart. Later, Anwar et al. [21] extended the AIB-HWMA and suggested the AIB-DHWMA control chart for improved process location monitoring. Based on a thorough literature review, it is observed that no one has developed an NP DHWMA control chart under SR along with RSS methodology to date. This is a research gap that needs to be explored. So this study proposes an NP DHWMA-SR control chart under RSS (NPDHWMARSS) for monitoring shifts in process location for continuous and symmetric distribution. The study’s main goal is to propose a control chart for detecting small and moderate shifts in process location more quickly because small changes in process parameters can have a significant financial impact on an organization’s operations. The run-length (RL) characteristics of the proposed control chart, including average run length (ARL), median run length (MDRL), and standard deviation of run-length (SDRL), are obtained using various distributions like normal, student’s t, contaminated normal (CN), Laplace, and logistic distributions. The performance of proposed NPDHWMARSS control chart is decided by providing a valid comparison with other existing control charts such as DEWMA-X¯, NPDEWMA-SR, NPRDEWMA-SR, DHWMA, and NPHWMARSS control charts.The rest of the paper is organized as follows: Section2 presents the design structure of the competing and the proposed control chart. Similarly, Section 3 describes the proposed control chart’s IC and out-of-control (OOC) performance. Likewise, Section 4 contains a comparative study of the proposed control chart, whereas Section 5 provides a real-life application. Finally, concluding remarks are given in Section 6.


## 2. Competing and Proposed Control Charts

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


### 2.1. NPRDEWMA-SR Control Chart

Abbas et al. [4] presented an NP double EWMA SR under the RSS (NPRDEWMA-SR) control chart that outperforms the NPREWMA-SR control chart in terms of shift detection in process location. The plotting statistics of the NPRDEWMA-SR control chart are given as follows:(1)ESRRSSt=λSRRSSt+1−λESRRSSt−1DESRRSSt=λESRRSSt+1−λDESRRSSt−1,where λ∈0,1 is a smoothing constant. The control limits of the NPRDEWMA-SR control chart can be designed as follows:(2)LCLNPRDEWMA−SRt=−Lrr+12r+16ϖ02λ41+λ2−t2+2t+1λ2t+2t2+2t−1λ2t+2−t21−λ23UCLNPRDEWMA−SRt=+Lrr+12r+16ϖ02λ41+λ2−t2+2t+1λ2t+2t2+2t−1λ2t+2−t21−λ23.The process goes OOC whenDESRRSSt>UCLNPRDEWMA−SRtorDESRRSSt<LCLNPRDEWMA−SRt; otherwise, it will remain an IC state.


### 2.2. DHWMA Control Chart

Abid et al. [22] developed a DHWMA control chart that detects shifts more efficiently than the HWMA control chart. The DHWMA control chart plotting statistic is given as follows:(3)DHt=λ2X¯t+1−λ2X¯¯t−1,where X¯t and X¯¯t−1 are the mean of tth and mean of the previous t−1 samples, respectively. The control limits of the DHWMA control chart based on this EDHt and VarDHt are defined as follows:(4)LCLDHWMAt=μ0−Lλ4σ02/n,ift=1μ0−Lλ4σ02/n+1−λ21+λ2σ02/nt−1,ift>1CLDHWMAt=μ0UCLDHWMAt=μ0+Lλ4σ02/n,ift=1μ0+Lλ4σ02/n+1−λ21+λ2σ02/nt−1,ift>1.The process remain is IC ifLCLDHWMAt<DHt<UCLDHWMAt; otherwise, it goes OOC.


### 2.3. ProposedNPDHWMARSS Control Chart

Various researchers like Kim and Kim [23], Abid et al. [24], and Abbas et al. [4] used the following RSS-based Wilcoxon signed-rank statistic to monitor shifts in process location:(5)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. 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:(6)Fk0=r!j−1!r−j!∫−∞0Ftj−11−Ftr−jftdt.Abid et al. [24] have more information on the RSS approach and related terms. The methodology of the proposed NPDHWMARSS control chart is defined as follows:(7)NPHt=λSRRSSt+1−λSR¯RSSt−1NPDHt=λNPHt+1−λSR¯RSSt−1,where SR¯RSSt−1 is the mean of SRRSS of t−1 samples. The simplified form of the plotting statistic NPDHt is(8)NPDHt=λ2SRRSSt+1−λ2SR¯RSSt−1,NPDH0=0.The control limits of the proposedNPDHWMARSS control chart are(9)LCLNPDHWMARSSt=μ0−Lλ4rr+12r+1/6ϖ02,ift=1μ0−Lλ4+1−λ21+λ2/t−1rr+12r+1/6ϖ02,ift>1CLNPDHWMARSSt=μ0UCLNPDHWMARSSt=μ0+Lλ4rr+12r+1/6ϖ02,ift=1μ0+Lλ4+1−λ21+λ2/t−1rr+12r+1/6ϖ02,ift>1.IfNPDHt>UCLDNPDHWMARSSt or NPDHt<LCLDNPDHWMARSSt, the underlying process is OOC; else, it is IC.


## 2.1. NPRDEWMA-SR Control Chart

Abbas et al. [4] presented an NP double EWMA SR under the RSS (NPRDEWMA-SR) control chart that outperforms the NPREWMA-SR control chart in terms of shift detection in process location. The plotting statistics of the NPRDEWMA-SR control chart are given as follows:(1)ESRRSSt=λSRRSSt+1−λESRRSSt−1DESRRSSt=λESRRSSt+1−λDESRRSSt−1,where λ∈0,1 is a smoothing constant. The control limits of the NPRDEWMA-SR control chart can be designed as follows:(2)LCLNPRDEWMA−SRt=−Lrr+12r+16ϖ02λ41+λ2−t2+2t+1λ2t+2t2+2t−1λ2t+2−t21−λ23UCLNPRDEWMA−SRt=+Lrr+12r+16ϖ02λ41+λ2−t2+2t+1λ2t+2t2+2t−1λ2t+2−t21−λ23.The process goes OOC whenDESRRSSt>UCLNPRDEWMA−SRtorDESRRSSt<LCLNPRDEWMA−SRt; otherwise, it will remain an IC state.


## 2.2. DHWMA Control Chart

Abid et al. [22] developed a DHWMA control chart that detects shifts more efficiently than the HWMA control chart. The DHWMA control chart plotting statistic is given as follows:(3)DHt=λ2X¯t+1−λ2X¯¯t−1,where X¯t and X¯¯t−1 are the mean of tth and mean of the previous t−1 samples, respectively. The control limits of the DHWMA control chart based on this EDHt and VarDHt are defined as follows:(4)LCLDHWMAt=μ0−Lλ4σ02/n,ift=1μ0−Lλ4σ02/n+1−λ21+λ2σ02/nt−1,ift>1CLDHWMAt=μ0UCLDHWMAt=μ0+Lλ4σ02/n,ift=1μ0+Lλ4σ02/n+1−λ21+λ2σ02/nt−1,ift>1.The process remain is IC ifLCLDHWMAt<DHt<UCLDHWMAt; otherwise, it goes OOC.


## 2.3. ProposedNPDHWMARSS Control Chart

Various researchers like Kim and Kim [23], Abid et al. [24], and Abbas et al. [4] used the following RSS-based Wilcoxon signed-rank statistic to monitor shifts in process location:(5)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. 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:(6)Fk0=r!j−1!r−j!∫−∞0Ftj−11−Ftr−jftdt.Abid et al. [24] have more information on the RSS approach and related terms. The methodology of the proposed NPDHWMARSS control chart is defined as follows:(7)NPHt=λSRRSSt+1−λSR¯RSSt−1NPDHt=λNPHt+1−λSR¯RSSt−1,where SR¯RSSt−1 is the mean of SRRSS of t−1 samples. The simplified form of the plotting statistic NPDHt is(8)NPDHt=λ2SRRSSt+1−λ2SR¯RSSt−1,NPDH0=0.The control limits of the proposedNPDHWMARSS control chart are(9)LCLNPDHWMARSSt=μ0−Lλ4rr+12r+1/6ϖ02,ift=1μ0−Lλ4+1−λ21+λ2/t−1rr+12r+1/6ϖ02,ift>1CLNPDHWMARSSt=μ0UCLNPDHWMARSSt=μ0+Lλ4rr+12r+1/6ϖ02,ift=1μ0+Lλ4+1−λ21+λ2/t−1rr+12r+1/6ϖ02,ift>1.IfNPDHt>UCLDNPDHWMARSSt or NPDHt<LCLDNPDHWMARSSt, the underlying process is OOC; else, it is IC.


## 3. Implementation of the Proposed Control Chart

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


### 3.1. Performance Metrics

ARL is widely used to evaluate the performance of the control chart. The ARL is the expected number of sample points before the first OOC signal from the control chart. The ARL is categorized as IC ARL (ARL0) and out-of-control ARL (ARL1). If a process is functioning in an in-control state, the ARL0 needed to be large enough to avoid frequent false alarms. However, the ARL1 should be small enough; it quickly detects the shift. It is necessary for better performance of a control chart; it should have a smaller ARL1 as compared to other control charts at the fixed value of ARL0. In this study, we set ARL0 to 370 and 500, with sample sizes (n) of 5 and 10. Monte Carlo simulations with 50,000 simulations in the R program are used to determine the RL characteristics. To examine the performance behavior of the proposed NPDHWMARSS control chart, various values of λ∈ (0.05, 0.10, 0.25, 0.50) and δ∈ (0.025, 0.05, 0.075, 0.10, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00, 5.00) are used. The following algorithm is used for simulations:(i)
To create samples from considered distributions, a finite loop is used.(ii)
Specify the process parameters (λandL).(iii)
Draw a sample from a distribution used in this study.(iv)
Determine theNPDHt plotting statistics using equation (7).(v)
FindLCLNPDHWMARSSt and UCLNPDHWMARSSt from equation (8).(vi)
Plot the plotting statisticNPDHt against LCLNPDHWMARSSt and UCLNPDHWMARSSt over t.(vii)
IfNPDHt>UCLNPDHWMARSSt or NPDHt<LCLNPDHWMARSSt, note this sample of NPDHt statistic as an RL. For instance, at t=105, if NPDHt>UCLNPDHWMARSSt or NPDHt<LCLNPDHWMARSSt, record 105 as a first RL.(viii)
Repeat steps (ii) through (vi) for 50,000 times and record RLs.(ix)
CalculateARL0 from 50,000 and recorded RLs.(x)
ARL0=100; if not, adjust constant accordingly in step (ii) and repeat from (ii) to (ix) steps to obtain ARL0=100.(xi)
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 theNPDHWMARSS Control Chart

This subsection highlights the proposedNPDHWMARSS control chart’s robustness, IC, and OOC behavior when a process location is shifted. Table 1 shows the RL characteristics of the proposed NPDHWMARSS control chart for location shift. These characteristics are assessed using normal and non-normal continuous symmetric distributions. The distributions used for this study are standard normal distribution, that is, N0,1; double exponential or Laplace distribution, that is, DE0,1/2; heavy tail student’s t distribution, that is, tυ; logistic distribution, that is, 0,3/π; and contaminated normal (CN) distribution, which is the mixture of N0,σ02 and N0,σ12. All these distributions were reparametrized with zero mean/median and unit variance for comparison purposes. For all symmetric continuous distributions, the IC RL characteristics of the NP control chart remain constant [4].Table 1
RL characteristics of the proposedNPDHWMARSS control chart under different distributions with nominal ARL0=500 and n=10.
(λ, L)Distr.Metricsδ00.0250.050.0750.10.250.50.7511.522.535(0.05, 1.064)NormalARL499.4931.7413.257.965.651.941.051.001.001.001.001.001.001.00MDRL12.009.007.005.004.001.001.001.001.001.001.001.001.001.00SDRL1,851.3055.8417.048.315.231.330.310.030.000.000.000.000.000.00CNARL503.0834.3613.868.415.842.021.071.001.001.001.001.001.001.00MDRL12.009.007.005.004.001.001.001.001.001.001.001.001.001.00SDRL1,890.2861.3218.389.165.561.400.370.050.000.000.000.000.000.00t(4)ARL498.7040.6216.399.836.642.301.141.011.001.001.001.001.001.00MDRL11.0010.007.006.005.001.001.001.001.001.001.001.001.001.00SDRL1,850.0774.6122.4211.146.561.630.530.130.030.000.000.000.000.00t(8)ARL499.2236.6014.818.756.192.111.091.001.001.001.001.001.001.00MDRL12.0010.007.005.004.001.001.001.001.001.001.001.001.001.00SDRL1,852.4168.1019.919.735.931.480.430.060.000.000.000.000.000.00LaplaceARL498.4024.069.465.834.231.551.021.001.001.001.001.001.001.00MDRL13.009.006.004.003.001.001.001.001.001.001.001.001.001.00SDRL1,856.3837.1910.665.453.501.010.190.020.000.000.000.000.000.00LogisticARL498.5530.3412.037.215.211.821.041.001.001.001.001.001.001.00MDRL12.009.006.005.004.001.001.001.001.001.001.001.001.001.00SDRL1,869.4750.4914.947.384.641.230.270.030.000.000.000.000.000.00(0.10, 1.306)NormalARL499.4861.7621.7211.827.792.461.121.001.001.001.001.001.001.00MDRL81.0028.0013.008.006.003.001.001.001.001.001.001.001.001.00SDRL781.0980.8523.9511.146.591.520.450.050.000.000.000.000.000.00CNARL499.1665.7223.0712.448.172.551.151.001.001.001.001.001.001.00MDRL78.0029.0014.008.006.003.001.001.001.001.001.001.001.001.00SDRL772.9786.0425.5811.966.971.580.510.080.010.000.000.000.000.00t(4)ARL498.2879.2427.2014.729.642.921.311.031.001.001.001.001.001.00MDRL83.0034.0016.0010.007.003.001.001.001.001.001.001.001.001.00SDRL779.49105.1130.9514.688.631.830.720.220.060.000.000.000.000.00t(8)ARL499.0770.3024.3112.998.662.671.201.011.001.001.001.001.001.00MDRL87.0030.0014.009.006.003.001.001.001.001.001.001.001.001.00SDRL781.0294.0627.7112.617.601.650.580.120.020.000.000.000.000.00LaplaceARL499.2142.4914.758.185.541.901.051.001.001.001.001.001.001.00MDRL81.0022.0010.006.004.001.001.001.001.001.001.001.001.001.00SDRL780.7951.8314.916.944.261.180.300.040.000.000.000.000.000.00LogisticARL498.8857.0119.4410.567.022.261.091.001.001.001.001.001.001.00MDRL82.0027.5012.007.005.002.001.001.001.001.001.001.001.001.00SDRL771.6772.0621.019.775.651.400.380.050.000.000.000.000.000.00(0.25, 2.113)NormalARL502.19132.3249.7426.3316.574.071.581.041.001.001.001.001.001.00MDRL435.00112.0044.0023.0015.004.001.001.001.001.001.001.001.001.00SDRL369.4396.2933.3016.9910.301.980.910.250.030.000.000.000.000.00CNARL500.19137.9352.6527.9317.654.271.681.061.001.001.001.001.001.00MDRL432.00118.0046.0025.0016.004.001.001.001.001.001.001.001.001.00SDRL368.45100.1535.1418.1510.962.080.960.330.060.000.000.000.000.00t(4)ARL499.91160.2061.9933.1521.174.962.021.221.031.001.001.001.001.00MDRL434.00136.0054.0030.0019.005.001.001.001.001.001.001.001.001.00SDRL365.58115.7142.6021.6713.362.511.120.600.240.040.010.000.000.00t(8)ARL498.30145.7555.9129.7618.824.521.821.101.011.001.001.001.001.00MDRL428.00123.0049.0026.0017.004.001.001.001.001.001.001.001.001.00SDRL366.98107.8937.8719.2711.842.271.020.410.090.000.000.000.000.00LaplaceARL501.0592.8533.3017.5210.993.071.331.031.001.001.001.001.001.00MDRL435.0080.0030.0016.0010.003.001.001.001.001.001.001.001.001.00SDRL368.4164.6821.8710.986.611.530.720.240.090.020.000.000.000.00LogisticARL502.04119.7744.6223.7014.703.751.471.041.001.001.001.001.001.00MDRL439.00103.0039.0021.0013.004.001.001.001.001.001.001.001.001.00SDRL364.6885.8229.8414.949.231.780.830.250.050.000.000.000.000.00(0.50, 2.376)NormalARL498.71201.0170.6934.3420.174.261.671.081.001.001.001.001.001.00MDRL355.00147.0055.0028.0017.004.002.001.001.001.001.001.001.001.00SDRL482.39184.9558.2425.7613.712.050.760.280.050.000.000.000.000.00CNARL497.95210.0875.0536.3521.564.491.761.121.011.001.001.001.001.00MDRL349.00155.0058.0030.0018.004.002.001.001.001.001.001.001.001.00SDRL486.93191.4862.2427.3314.782.200.800.330.090.000.000.000.000.00t(4)ARL499.23234.1289.7844.6826.265.372.081.291.061.001.001.001.001.00MDRL353.00172.0068.0036.0022.005.002.001.001.001.001.001.001.001.00SDRL483.45216.3076.4534.4718.702.800.930.530.250.050.010.010.000.00t(8)ARL501.84220.4279.5639.2023.384.771.881.161.021.001.001.001.001.00MDRL355.50160.5060.0031.0020.004.002.001.001.001.001.001.001.001.00SDRL490.89206.0867.9729.9816.202.360.850.400.130.020.000.000.000.00LaplaceARL498.36142.1944.4721.3012.973.141.391.061.011.001.001.001.001.00MDRL351.00106.0036.0018.0011.003.001.001.001.001.001.001.001.001.00SDRL489.74126.5833.9014.608.051.410.620.240.070.010.000.000.000.00LogisticARL499.28186.3562.6129.7917.683.841.561.071.001.001.001.001.001.00MDRL358.00138.0049.0025.0015.004.001.001.001.001.001.001.001.001.00SDRL479.40170.7050.1421.4012.021.760.710.260.070.000.000.000.000.00For comparison purposes, the same parameters are used as reported in numerous relevant articles. TheARL measures are used to compare the proposed and competing control charts. Based on the research findings and sensitivity analysis, the following observations have been made.(i)
The proposed control chart’s ICRL distribution looks remarkably similar for all distributions examined in this study. For example, at λ= (0.05, 0.10, 0.25, 0.50) and n=5,10, the ARL0=370,500 for all investigated distributions (see Table 1).(ii)
As the smoothing parameter reduces, the proposed control chart becomes more effective in detecting shifts. This illustrates that the proposedNPDHWMARSS control chart is more sensitive to small smoothing parameters (see Figure 1).(iii)
As the sample size increases, the proposedNPDHWMARSS control chart’s shift detection ability for process location improves (see Figure 2).(iv)
The Laplace distribution outperforms the other distributions in terms of OOC RL performance (see Figures3).(v)
The proposedNPDHWMARSS control chart’s ARL1 values increase as λ increases at a certain size of the shift. For instance, under normal distribution at λ=0.25,n=10,andδ=0.025, the ARL1=132.32, whereas when λ=0.50,n=10,andδ=0.025, the ARL1=201.01 (see Table 1).(vi)
TheARL1 of the proposed NPDHWMARSS control chart are smaller than those in the competing control charts with different shift sizes in process location (see Figure 4).(vii)
The distribution of RL values is positively skewed, that is,ARL>MRDL (see Table 1).Figure 1
ARL characteristics of the proposedNPDHWMARSS control chart for different values of λ when n=10 and ARL0=500.Figure 2
ARL characteristics of the proposedNPDHWMARSS control chart when n=5,10 and ARL0=500.Figure 3
ARL characteristics of the proposedNPDHWMARSS control chart under various distributions when n=10 and ARL0=500.Figure 4
ARL characteristics of the proposedNPDHWMARSS and competing control charts under various distributions when n=10 and ARL0=500.


## 3.1. Performance Metrics

ARL is widely used to evaluate the performance of the control chart. The ARL is the expected number of sample points before the first OOC signal from the control chart. The ARL is categorized as IC ARL (ARL0) and out-of-control ARL (ARL1). If a process is functioning in an in-control state, the ARL0 needed to be large enough to avoid frequent false alarms. However, the ARL1 should be small enough; it quickly detects the shift. It is necessary for better performance of a control chart; it should have a smaller ARL1 as compared to other control charts at the fixed value of ARL0. In this study, we set ARL0 to 370 and 500, with sample sizes (n) of 5 and 10. Monte Carlo simulations with 50,000 simulations in the R program are used to determine the RL characteristics. To examine the performance behavior of the proposed NPDHWMARSS control chart, various values of λ∈ (0.05, 0.10, 0.25, 0.50) and δ∈ (0.025, 0.05, 0.075, 0.10, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00, 5.00) are used. The following algorithm is used for simulations:(i)
To create samples from considered distributions, a finite loop is used.(ii)
Specify the process parameters (λandL).(iii)
Draw a sample from a distribution used in this study.(iv)
Determine theNPDHt plotting statistics using equation (7).(v)
FindLCLNPDHWMARSSt and UCLNPDHWMARSSt from equation (8).(vi)
Plot the plotting statisticNPDHt against LCLNPDHWMARSSt and UCLNPDHWMARSSt over t.(vii)
IfNPDHt>UCLNPDHWMARSSt or NPDHt<LCLNPDHWMARSSt, note this sample of NPDHt statistic as an RL. For instance, at t=105, if NPDHt>UCLNPDHWMARSSt or NPDHt<LCLNPDHWMARSSt, record 105 as a first RL.(viii)
Repeat steps (ii) through (vi) for 50,000 times and record RLs.(ix)
CalculateARL0 from 50,000 and recorded RLs.(x)
ARL0=100; if not, adjust constant accordingly in step (ii) and repeat from (ii) to (ix) steps to obtain ARL0=100.(xi)
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 theNPDHWMARSS Control Chart

This subsection highlights the proposedNPDHWMARSS control chart’s robustness, IC, and OOC behavior when a process location is shifted. Table 1 shows the RL characteristics of the proposed NPDHWMARSS control chart for location shift. These characteristics are assessed using normal and non-normal continuous symmetric distributions. The distributions used for this study are standard normal distribution, that is, N0,1; double exponential or Laplace distribution, that is, DE0,1/2; heavy tail student’s t distribution, that is, tυ; logistic distribution, that is, 0,3/π; and contaminated normal (CN) distribution, which is the mixture of N0,σ02 and N0,σ12. All these distributions were reparametrized with zero mean/median and unit variance for comparison purposes. For all symmetric continuous distributions, the IC RL characteristics of the NP control chart remain constant [4].Table 1
RL characteristics of the proposedNPDHWMARSS control chart under different distributions with nominal ARL0=500 and n=10.
(λ, L)Distr.Metricsδ00.0250.050.0750.10.250.50.7511.522.535(0.05, 1.064)NormalARL499.4931.7413.257.965.651.941.051.001.001.001.001.001.001.00MDRL12.009.007.005.004.001.001.001.001.001.001.001.001.001.00SDRL1,851.3055.8417.048.315.231.330.310.030.000.000.000.000.000.00CNARL503.0834.3613.868.415.842.021.071.001.001.001.001.001.001.00MDRL12.009.007.005.004.001.001.001.001.001.001.001.001.001.00SDRL1,890.2861.3218.389.165.561.400.370.050.000.000.000.000.000.00t(4)ARL498.7040.6216.399.836.642.301.141.011.001.001.001.001.001.00MDRL11.0010.007.006.005.001.001.001.001.001.001.001.001.001.00SDRL1,850.0774.6122.4211.146.561.630.530.130.030.000.000.000.000.00t(8)ARL499.2236.6014.818.756.192.111.091.001.001.001.001.001.001.00MDRL12.0010.007.005.004.001.001.001.001.001.001.001.001.001.00SDRL1,852.4168.1019.919.735.931.480.430.060.000.000.000.000.000.00LaplaceARL498.4024.069.465.834.231.551.021.001.001.001.001.001.001.00MDRL13.009.006.004.003.001.001.001.001.001.001.001.001.001.00SDRL1,856.3837.1910.665.453.501.010.190.020.000.000.000.000.000.00LogisticARL498.5530.3412.037.215.211.821.041.001.001.001.001.001.001.00MDRL12.009.006.005.004.001.001.001.001.001.001.001.001.001.00SDRL1,869.4750.4914.947.384.641.230.270.030.000.000.000.000.000.00(0.10, 1.306)NormalARL499.4861.7621.7211.827.792.461.121.001.001.001.001.001.001.00MDRL81.0028.0013.008.006.003.001.001.001.001.001.001.001.001.00SDRL781.0980.8523.9511.146.591.520.450.050.000.000.000.000.000.00CNARL499.1665.7223.0712.448.172.551.151.001.001.001.001.001.001.00MDRL78.0029.0014.008.006.003.001.001.001.001.001.001.001.001.00SDRL772.9786.0425.5811.966.971.580.510.080.010.000.000.000.000.00t(4)ARL498.2879.2427.2014.729.642.921.311.031.001.001.001.001.001.00MDRL83.0034.0016.0010.007.003.001.001.001.001.001.001.001.001.00SDRL779.49105.1130.9514.688.631.830.720.220.060.000.000.000.000.00t(8)ARL499.0770.3024.3112.998.662.671.201.011.001.001.001.001.001.00MDRL87.0030.0014.009.006.003.001.001.001.001.001.001.001.001.00SDRL781.0294.0627.7112.617.601.650.580.120.020.000.000.000.000.00LaplaceARL499.2142.4914.758.185.541.901.051.001.001.001.001.001.001.00MDRL81.0022.0010.006.004.001.001.001.001.001.001.001.001.001.00SDRL780.7951.8314.916.944.261.180.300.040.000.000.000.000.000.00LogisticARL498.8857.0119.4410.567.022.261.091.001.001.001.001.001.001.00MDRL82.0027.5012.007.005.002.001.001.001.001.001.001.001.001.00SDRL771.6772.0621.019.775.651.400.380.050.000.000.000.000.000.00(0.25, 2.113)NormalARL502.19132.3249.7426.3316.574.071.581.041.001.001.001.001.001.00MDRL435.00112.0044.0023.0015.004.001.001.001.001.001.001.001.001.00SDRL369.4396.2933.3016.9910.301.980.910.250.030.000.000.000.000.00CNARL500.19137.9352.6527.9317.654.271.681.061.001.001.001.001.001.00MDRL432.00118.0046.0025.0016.004.001.001.001.001.001.001.001.001.00SDRL368.45100.1535.1418.1510.962.080.960.330.060.000.000.000.000.00t(4)ARL499.91160.2061.9933.1521.174.962.021.221.031.001.001.001.001.00MDRL434.00136.0054.0030.0019.005.001.001.001.001.001.001.001.001.00SDRL365.58115.7142.6021.6713.362.511.120.600.240.040.010.000.000.00t(8)ARL498.30145.7555.9129.7618.824.521.821.101.011.001.001.001.001.00MDRL428.00123.0049.0026.0017.004.001.001.001.001.001.001.001.001.00SDRL366.98107.8937.8719.2711.842.271.020.410.090.000.000.000.000.00LaplaceARL501.0592.8533.3017.5210.993.071.331.031.001.001.001.001.001.00MDRL435.0080.0030.0016.0010.003.001.001.001.001.001.001.001.001.00SDRL368.4164.6821.8710.986.611.530.720.240.090.020.000.000.000.00LogisticARL502.04119.7744.6223.7014.703.751.471.041.001.001.001.001.001.00MDRL439.00103.0039.0021.0013.004.001.001.001.001.001.001.001.001.00SDRL364.6885.8229.8414.949.231.780.830.250.050.000.000.000.000.00(0.50, 2.376)NormalARL498.71201.0170.6934.3420.174.261.671.081.001.001.001.001.001.00MDRL355.00147.0055.0028.0017.004.002.001.001.001.001.001.001.001.00SDRL482.39184.9558.2425.7613.712.050.760.280.050.000.000.000.000.00CNARL497.95210.0875.0536.3521.564.491.761.121.011.001.001.001.001.00MDRL349.00155.0058.0030.0018.004.002.001.001.001.001.001.001.001.00SDRL486.93191.4862.2427.3314.782.200.800.330.090.000.000.000.000.00t(4)ARL499.23234.1289.7844.6826.265.372.081.291.061.001.001.001.001.00MDRL353.00172.0068.0036.0022.005.002.001.001.001.001.001.001.001.00SDRL483.45216.3076.4534.4718.702.800.930.530.250.050.010.010.000.00t(8)ARL501.84220.4279.5639.2023.384.771.881.161.021.001.001.001.001.00MDRL355.50160.5060.0031.0020.004.002.001.001.001.001.001.001.001.00SDRL490.89206.0867.9729.9816.202.360.850.400.130.020.000.000.000.00LaplaceARL498.36142.1944.4721.3012.973.141.391.061.011.001.001.001.001.00MDRL351.00106.0036.0018.0011.003.001.001.001.001.001.001.001.001.00SDRL489.74126.5833.9014.608.051.410.620.240.070.010.000.000.000.00LogisticARL499.28186.3562.6129.7917.683.841.561.071.001.001.001.001.001.00MDRL358.00138.0049.0025.0015.004.001.001.001.001.001.001.001.001.00SDRL479.40170.7050.1421.4012.021.760.710.260.070.000.000.000.000.00For comparison purposes, the same parameters are used as reported in numerous relevant articles. TheARL measures are used to compare the proposed and competing control charts. Based on the research findings and sensitivity analysis, the following observations have been made.(i)
The proposed control chart’s ICRL distribution looks remarkably similar for all distributions examined in this study. For example, at λ= (0.05, 0.10, 0.25, 0.50) and n=5,10, the ARL0=370,500 for all investigated distributions (see Table 1).(ii)
As the smoothing parameter reduces, the proposed control chart becomes more effective in detecting shifts. This illustrates that the proposedNPDHWMARSS control chart is more sensitive to small smoothing parameters (see Figure 1).(iii)
As the sample size increases, the proposedNPDHWMARSS control chart’s shift detection ability for process location improves (see Figure 2).(iv)
The Laplace distribution outperforms the other distributions in terms of OOC RL performance (see Figures3).(v)
The proposedNPDHWMARSS control chart’s ARL1 values increase as λ increases at a certain size of the shift. For instance, under normal distribution at λ=0.25,n=10,andδ=0.025, the ARL1=132.32, whereas when λ=0.50,n=10,andδ=0.025, the ARL1=201.01 (see Table 1).(vi)
TheARL1 of the proposed NPDHWMARSS control chart are smaller than those in the competing control charts with different shift sizes in process location (see Figure 4).(vii)
The distribution of RL values is positively skewed, that is,ARL>MRDL (see Table 1).Figure 1
ARL characteristics of the proposedNPDHWMARSS control chart for different values of λ when n=10 and ARL0=500.Figure 2
ARL characteristics of the proposedNPDHWMARSS control chart when n=5,10 and ARL0=500.Figure 3
ARL characteristics of the proposedNPDHWMARSS control chart under various distributions when n=10 and ARL0=500.Figure 4
ARL characteristics of the proposedNPDHWMARSS and competing control charts under various distributions when n=10 and ARL0=500.


## 4. Comparative Study

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


### 4.1. Proposed versus DEWMA-X¯ Control Chart

TheARL profile demonstrates that the proposed NPDHWMARSS control chart outperforms the DEWMA-X¯ control chart. For example, under normal distribution, at =0.05,n=10, and δ=0.025,0.05,0.075,0.10,0.25,0.50,0.75, the ARL1 values of the proposed NPDHWMARSS control chart are 31.74,13.25,7.96,5.65,1.94,1.05,1.00, whereas the ARL1 values of the DEWMA-X¯ control charts are 284.02,127.00,69.34,43.71,9.68,2.93,1.57 (see Tables 1 and 2). Similarly, when we consider Laplace distribution for comparison, we observed the same behavior in the proposed NPDHWMARSS control chart. For instance, at λ=0.05,n=10, and δ=0.025,0.05,0.075,0.10,0.25,0.50,0.75, the ARL1 values of the proposed NPDHWMARSS control chart are 24.06,9.46,5.83,4.23,1.55,1.02,1.00, while the ARL1 values of DEWMA-X¯ control chart are 287.40,129.08,68.32,44.01,9.73,2.91,1.57 (see Figure 4 and Tables 1 and 2).Table 2
RL characteristic of theDEWMA−X¯ control chart under different distributions with ARL0=500 and n=10.
Distr.λ,LMetricδ00.0250.030.050.0750.10.150.20.250.50.7511.52Normal0.05, 2.081ARL500.04284.02239.41127.0069.3443.7123.0214.109.682.931.571.161.001.00t(4)0.05, 2.08ARL498.01287.13241.49127.4069.3543.2622.6214.139.712.911.551.141.021.00CN0.05, 3.204ARL498.11390.49359.60240.51146.5596.5451.0332.0722.336.903.392.091.221.01Laplace0.05, 2.083ARL498.67287.40245.01129.0868.3244.0122.4814.109.732.911.571.161.011.00Logistic0.05, 2.083ARL503.92287.07241.11129.3069.4544.1422.6114.019.592.931.561.141.011.00


### 4.2. Proposed versus NPDEWMA-SR Control Chart

The proposedNPDHWMARSS control chart is more sensitive than the NPDEWMA-SR control chart for all combinations of δ and λ. For instance, in case of logistic distribution, when n=10,λ=0.05, and δ= 0.025, 0.05, 0.075, 0.10, 0.25, 0.50, 0.75, the ARL1 values of the proposed NPDHWMARSS and NPDEWMA-SR control charts are 30.34, 12.03, 7.21, 5.21, 1.82, 1.04, and 1.00 and 280.87, 123.83, 66.59, 42.46, 9.40, 3.08, and 1.734, respectively (see Tables 1 and 3). The supremacy of the proposed NPDHWMARSS control chart to the NPDEWMA-SR can also be seen in Figure 4. Likewise, in the scenario of t8 distribution comparison, we noted the same behavior of the proposed NPDHWMARSS control chart. As an illustration, at λ=0.05,n=10, and δ=0.05,0.10,0.50, the ARL1 values of the proposed NPDHWMARSS control chart are 14.81,6.19,1.09, respectively, while the ARL1 values of the NPDEWMA-SR control chart are 123.93,42.69,3.05 (see Tables 1 and 3).Table 3
RL characteristics of the SRS-based NP double EWMA SR (NPDEWMA-SR) control chart under different distributions with nominalARL0=500 and n=10.
(λ, L)Distr.Metricsδ00.0250.030.050.0750.10.150.20.250.50.7511.52(0.05, 6.57)NormalARL499.98299.203248.03138.0375.7448.0025.0115.5610.613.341.831.321.041.00SDRL541.40328.41263.05139.8270.7641.3420.3412.278.2122.331.060.620.180.04t(4)ARL501.45247.01203.44101.3853.6034.7218.0011.227.652.591.591.261.051.01SDRL546.82262.02212.4598.2747.5728.8314.128.695.841.710.810.550.250.19t(8)ARL500.89276.24233.07123.9367.0442.6922.1113.809.633.051.751.311.031.00SDRL538.51294.92249.16125.1962.1037.0017.4510.717.342.060.950.610.250.11CNARL498.95287.54241.30131.1070.6745.1323.5214.5810.103.211.791.281.051.00SDRL545.49310.49257.85131.2065.1239.2718.5811.427.712.221.000.580.230.09LaplaceARL499.36231.49188.3692.0949.4531.5016.3810.377.192.581.641.291.081.01SDRL535.19247.33197.4490.2042.6526.1113.018. 005.461.710.890.590.290.15LogisticARL499.45280.87236.702123.8366.5942.4621.7913.529.403.081.7341.301.031.00SDRL549.41300.01250.82124.0161.0336.5517.4710.577.322.121.000.560.320.05


### 4.3. Proposed versus NPRDEWMA-SR Control Chart

The proposedNPDHWMARSS control chart performs better than the NPRDEWMA-SR control chart. For instance, using the t4 distribution with n=10,λ=0.05,andδ=0.05,0.50, the ARL1 values of the proposed NPDHWMARSS and the NPRDEWMA-SR control charts are 16.39,1.14, and 49.00,1.28, respectively (see Tables 1 and 4). The proposed NPDHWMARSS charts’ efficiency may also be observed in the case of CN distribution. For example, at n=10,λ=0.50andδ=0.025,0.05,0.075,0.10, the ARL1 values for the proposed NPDHWMARSS control chart are 34.36,13.86,8.41,5.84, whereas the ARL1 values for the NPRDEWMA-SR control chart are 121.25,40.58,21.37,and13.11 (see Figure 4 and Tables 1 and 4).Table 4
RL characteristics of the RSS-based NP double EWMA SR (NPRDEWMA-SR) control chart under different distributions with nominalARL0=500 and n=10.
(λ, L)Distr.Metricsδ00.0250.030.050.0750.10.150.20.250.50.7511.52(0.05, 2.975)NormalARL501.58117.1387.7838.3420.0112.546.293.822.651.131.001.001.001.00SDRL562.31119.8185.9032.9016.029.694.802.881.800.400.070.000.000.00t(4)ARL499.96141.02111.2149.0025.8115.878.234.863.401.281.031.001.001.00SDRL557.27138.50109.7644.5820.9313.206.393.702.490.580.200.060.000.00t(8)ARL500.13129.0197.7044.0022.6414.157.044.322.951.191.011.001.001.00SDRL555.79131.1896.1239.7318.2812.036.253.232.130.490.110.010.000.00CNARL500.89121.2594.8940.5821.3713.116.804.042.811.161.001.001.001.00SDRL553.38122.5192.4438.1017.3011.365.202.981.960.430.080.010.010.00LaplaceARL498.9479.0057.7025.4013.638.204.182.601.921.061.001.001.001.00SDRL565.2973.1852.3520.6511.007.023.071.791.190.250.060.000.010.00LogisticARL499.71103.1278.0234.3618.0111.075.483.322.361.101.001.001.001.00SDRL552.61100.7072.6029.7715.779.114.172.461.620.350.030.000.000.00


### 4.4. Proposed versus DHWMA Control Chart

TheARL study reveals that the proposed NPDHWMARSS control chart outperforms the DHWMA control chart for all combinations of ηandδ (see Tables 1 and 5). As an illustration, for normal distribution, at λ=0.05,n=10,andδ=0.075,0.10,0.25, the ARL1 values of the NPDHWMARSS and DHWMA control charts are 7.96,5.65,1.94 and 21.24,14.81,4.63, respectively (see Tables 1 and 5). Figure 4 also shows the superiority of the NPDHWMARSS control chart over the DHWMA control chart. The results show that the NPDHWMARSS control chart is superior to the DHWMA control chart for monitoring process location shifts.Table 5
RL characteristics of the DHWMA control chart under normal distribution with nominalARL0=500 and n=10.
λ, Lδ00.0250.050.0750.10.250.50.7511.522.5350.05, 1.39ARL501.1487.0834.9721.2414.814.632.101.351.081.001.001.001.001.00SDRL1,930.36198.0363.1732.3319.693.911.460.810.390.040.000.000.000.000.10, 1.7105ARL502.29167.3867.1737.1524.276.192.561.541.151.001.001.001.001.00SDRL805.27248.7988.8745.5427.404.951.650.970.530.060.000.000.000.000.25, 2.7422ARL498.17294.65141.7183.1054.3412.594.292.481.641.031.001.001.001.00SDRL373.80227.63105.0257.9537.247.672.131.280.930.220.000.000.000.000.50, 3.075ARL502.05371.48210.92121.2876.5014.664.542.491.651.051.001.001.001.00SDRL489.97355.66195.21107.1364.309.402.321.140.790.220.020.000.000.00


### 4.5. Proposed versusNPHWMARSS Control Chart

The findings show that the proposedNPDHWMARSS control chart is better than the NPHWMARSS control chart in terms of OOC performance. For example, when we examine the t8 distribution at n=10,λ=0.10andδ=0.025,0.10,0.25, the ARL1 values of the proposed NPDHWMARSS and NPHWMARSS control charts are 70.30,8.66,2.67, and 153.55,20.27,4.77, respectively (see Figure 4 and Tables 1 and 6). Likewise, under Laplace distribution, when n=10,λ=0.05andδ=0.05,0.50, the ARL1 values of the proposed NPDHWMARSS control chart are 9.46,1.02, while the ARL1 values for NPHWMARSS control chart are 31.42,1.31, respectively (see Tables 1 and 6). The statistics show that the proposed structure works effectively for all distributions when compared to the NPHWMARSS control chartTable 6
RL characteristics of theNPHWMARSS control chart under various distributions with nominal ARL0=500 and n=10.
(λ, L)Distr.Metricsδ00.0250.050.0750.10.250.50.7511.522.535(0.05, 2.011)NormalARL502.18126.4747.4424.8315.553.861.561.041.001.001.001.001.001.00MDRL450.00108.0041.0022.0014.004.001.001.001.001.001.001.001.001.00SDRL369.2093.5732.8616.6610.061.910.870.240.030.000.000.000.000.00CNARL498.95131.6749.9226.0116.504.061.651.061.001.001.001.001.001.00MDRL447.00114.0044.0023.0015.004.001.001.001.001.001.001.001.001.00SDRL354.9896.6235.1917.6510.852.000.930.310.040.000.000.000.000.00LaplaceARL503.0289.9731.4216.3410.242.931.311.031.001.001.001.001.001.00MDRL447.0078.0028.0014.009.003.001.001.001.001.001.001.001.001.00SDRL357.8466.1021.2810.816.451.460.690.200.070.010.000.000.000.00LogisticARL502.13115.2542.1322.1713.633.571.451.031.001.001.001.001.001.00MDRL447.0099.0036.0019.0012.003.001.001.001.001.001.001.001.001.00SDRL357.8483.4929.5614.548.811.750.810.230.030.000.000.000.000.00t(4)ARL503.11153.2059.6531.3419.664.771.971.221.031.001.001.001.001.00MDRL449.00130.0052.0027.0017.004.001.001.001.001.001.001.001.001.00SDRL361.52115.0442.0121.9313.152.471.070.590.230.020.030.000.000.00t(8)ARL500.36138.2053.1828.3517.564.241.741.101.001.001.001.001.001.00MDRL437.00115.5046.0025.0016.004.001.001.001.001.001.001.001.001.00SDRL369.84105.1436.7719.0511.452.150.980.400.080.000.000.000.000.00(0.10, 2.268)NormalARL501.08141.1853.7828.3617.914.331.691.061.001.001.001.001.001.00MDRL416.00116.0047.0025.0016.004.001.001.001.001.001.001.001.001.00SDRL409.43104.6135.4817.4510.472.030.920.300.040.000.000.000.000.00CNARL498.89188.3984.6947.5131.267.052.631.351.101.021.001.001.001.00MDRL401.00123.0049.0027.0017.004.001.001.001.001.001.001.001.001.00SDRL392.85106.5837.0518.4511.472.120.970.360.060.000.000.000.000.00LaplaceARL499.36100.2535.7018.9311.973.221.421.051.001.001.001.001.001.00MDRL409.0084.0032.0017.0011.003.001.001.001.001.001.001.001.001.00SDRL401.0972.3322.5911.486.811.500.760.270.080.000.000.000.000.00LogisticARL500.14129.1247.6625.2315.933.921.581.051.001.001.001.001.001.00MDRL409.00108.0041.0022.0014.004.001.001.001.001.001.001.001.001.00SDRL401.0994.7731.6415.469.241.780.860.250.060.000.000.000.000.00t(4)ARL498.01169.3566.2736.0922.575.342.191.281.051.001.001.001.001.00MDRL398.50141.0056.0032.0020.005.002.001.001.001.001.001.001.001.00SDRL395.19124.8545.2222.7713.462.651.120.630.270.030.010.000.000.00t(8)ARL499.21153.5560.1632.0220.274.771.921.151.011.001.001.001.001.00MDRL400.50125.0052.0029.0018.004.001.001.001.001.001.001.001.001.00SDRL400.94119.0239.9119.7111.992.231.020.460.140.000.000.000.000.00


## 4.1. Proposed versus DEWMA-X¯ Control Chart

TheARL profile demonstrates that the proposed NPDHWMARSS control chart outperforms the DEWMA-X¯ control chart. For example, under normal distribution, at =0.05,n=10, and δ=0.025,0.05,0.075,0.10,0.25,0.50,0.75, the ARL1 values of the proposed NPDHWMARSS control chart are 31.74,13.25,7.96,5.65,1.94,1.05,1.00, whereas the ARL1 values of the DEWMA-X¯ control charts are 284.02,127.00,69.34,43.71,9.68,2.93,1.57 (see Tables 1 and 2). Similarly, when we consider Laplace distribution for comparison, we observed the same behavior in the proposed NPDHWMARSS control chart. For instance, at λ=0.05,n=10, and δ=0.025,0.05,0.075,0.10,0.25,0.50,0.75, the ARL1 values of the proposed NPDHWMARSS control chart are 24.06,9.46,5.83,4.23,1.55,1.02,1.00, while the ARL1 values of DEWMA-X¯ control chart are 287.40,129.08,68.32,44.01,9.73,2.91,1.57 (see Figure 4 and Tables 1 and 2).Table 2
RL characteristic of theDEWMA−X¯ control chart under different distributions with ARL0=500 and n=10.
Distr.λ,LMetricδ00.0250.030.050.0750.10.150.20.250.50.7511.52Normal0.05, 2.081ARL500.04284.02239.41127.0069.3443.7123.0214.109.682.931.571.161.001.00t(4)0.05, 2.08ARL498.01287.13241.49127.4069.3543.2622.6214.139.712.911.551.141.021.00CN0.05, 3.204ARL498.11390.49359.60240.51146.5596.5451.0332.0722.336.903.392.091.221.01Laplace0.05, 2.083ARL498.67287.40245.01129.0868.3244.0122.4814.109.732.911.571.161.011.00Logistic0.05, 2.083ARL503.92287.07241.11129.3069.4544.1422.6114.019.592.931.561.141.011.00


## 4.2. Proposed versus NPDEWMA-SR Control Chart

The proposedNPDHWMARSS control chart is more sensitive than the NPDEWMA-SR control chart for all combinations of δ and λ. For instance, in case of logistic distribution, when n=10,λ=0.05, and δ= 0.025, 0.05, 0.075, 0.10, 0.25, 0.50, 0.75, the ARL1 values of the proposed NPDHWMARSS and NPDEWMA-SR control charts are 30.34, 12.03, 7.21, 5.21, 1.82, 1.04, and 1.00 and 280.87, 123.83, 66.59, 42.46, 9.40, 3.08, and 1.734, respectively (see Tables 1 and 3). The supremacy of the proposed NPDHWMARSS control chart to the NPDEWMA-SR can also be seen in Figure 4. Likewise, in the scenario of t8 distribution comparison, we noted the same behavior of the proposed NPDHWMARSS control chart. As an illustration, at λ=0.05,n=10, and δ=0.05,0.10,0.50, the ARL1 values of the proposed NPDHWMARSS control chart are 14.81,6.19,1.09, respectively, while the ARL1 values of the NPDEWMA-SR control chart are 123.93,42.69,3.05 (see Tables 1 and 3).Table 3
RL characteristics of the SRS-based NP double EWMA SR (NPDEWMA-SR) control chart under different distributions with nominalARL0=500 and n=10.
(λ, L)Distr.Metricsδ00.0250.030.050.0750.10.150.20.250.50.7511.52(0.05, 6.57)NormalARL499.98299.203248.03138.0375.7448.0025.0115.5610.613.341.831.321.041.00SDRL541.40328.41263.05139.8270.7641.3420.3412.278.2122.331.060.620.180.04t(4)ARL501.45247.01203.44101.3853.6034.7218.0011.227.652.591.591.261.051.01SDRL546.82262.02212.4598.2747.5728.8314.128.695.841.710.810.550.250.19t(8)ARL500.89276.24233.07123.9367.0442.6922.1113.809.633.051.751.311.031.00SDRL538.51294.92249.16125.1962.1037.0017.4510.717.342.060.950.610.250.11CNARL498.95287.54241.30131.1070.6745.1323.5214.5810.103.211.791.281.051.00SDRL545.49310.49257.85131.2065.1239.2718.5811.427.712.221.000.580.230.09LaplaceARL499.36231.49188.3692.0949.4531.5016.3810.377.192.581.641.291.081.01SDRL535.19247.33197.4490.2042.6526.1113.018. 005.461.710.890.590.290.15LogisticARL499.45280.87236.702123.8366.5942.4621.7913.529.403.081.7341.301.031.00SDRL549.41300.01250.82124.0161.0336.5517.4710.577.322.121.000.560.320.05


## 4.3. Proposed versus NPRDEWMA-SR Control Chart

The proposedNPDHWMARSS control chart performs better than the NPRDEWMA-SR control chart. For instance, using the t4 distribution with n=10,λ=0.05,andδ=0.05,0.50, the ARL1 values of the proposed NPDHWMARSS and the NPRDEWMA-SR control charts are 16.39,1.14, and 49.00,1.28, respectively (see Tables 1 and 4). The proposed NPDHWMARSS charts’ efficiency may also be observed in the case of CN distribution. For example, at n=10,λ=0.50andδ=0.025,0.05,0.075,0.10, the ARL1 values for the proposed NPDHWMARSS control chart are 34.36,13.86,8.41,5.84, whereas the ARL1 values for the NPRDEWMA-SR control chart are 121.25,40.58,21.37,and13.11 (see Figure 4 and Tables 1 and 4).Table 4
RL characteristics of the RSS-based NP double EWMA SR (NPRDEWMA-SR) control chart under different distributions with nominalARL0=500 and n=10.
(λ, L)Distr.Metricsδ00.0250.030.050.0750.10.150.20.250.50.7511.52(0.05, 2.975)NormalARL501.58117.1387.7838.3420.0112.546.293.822.651.131.001.001.001.00SDRL562.31119.8185.9032.9016.029.694.802.881.800.400.070.000.000.00t(4)ARL499.96141.02111.2149.0025.8115.878.234.863.401.281.031.001.001.00SDRL557.27138.50109.7644.5820.9313.206.393.702.490.580.200.060.000.00t(8)ARL500.13129.0197.7044.0022.6414.157.044.322.951.191.011.001.001.00SDRL555.79131.1896.1239.7318.2812.036.253.232.130.490.110.010.000.00CNARL500.89121.2594.8940.5821.3713.116.804.042.811.161.001.001.001.00SDRL553.38122.5192.4438.1017.3011.365.202.981.960.430.080.010.010.00LaplaceARL498.9479.0057.7025.4013.638.204.182.601.921.061.001.001.001.00SDRL565.2973.1852.3520.6511.007.023.071.791.190.250.060.000.010.00LogisticARL499.71103.1278.0234.3618.0111.075.483.322.361.101.001.001.001.00SDRL552.61100.7072.6029.7715.779.114.172.461.620.350.030.000.000.00


## 4.4. Proposed versus DHWMA Control Chart

TheARL study reveals that the proposed NPDHWMARSS control chart outperforms the DHWMA control chart for all combinations of ηandδ (see Tables 1 and 5). As an illustration, for normal distribution, at λ=0.05,n=10,andδ=0.075,0.10,0.25, the ARL1 values of the NPDHWMARSS and DHWMA control charts are 7.96,5.65,1.94 and 21.24,14.81,4.63, respectively (see Tables 1 and 5). Figure 4 also shows the superiority of the NPDHWMARSS control chart over the DHWMA control chart. The results show that the NPDHWMARSS control chart is superior to the DHWMA control chart for monitoring process location shifts.Table 5
RL characteristics of the DHWMA control chart under normal distribution with nominalARL0=500 and n=10.
λ, Lδ00.0250.050.0750.10.250.50.7511.522.5350.05, 1.39ARL501.1487.0834.9721.2414.814.632.101.351.081.001.001.001.001.00SDRL1,930.36198.0363.1732.3319.693.911.460.810.390.040.000.000.000.000.10, 1.7105ARL502.29167.3867.1737.1524.276.192.561.541.151.001.001.001.001.00SDRL805.27248.7988.8745.5427.404.951.650.970.530.060.000.000.000.000.25, 2.7422ARL498.17294.65141.7183.1054.3412.594.292.481.641.031.001.001.001.00SDRL373.80227.63105.0257.9537.247.672.131.280.930.220.000.000.000.000.50, 3.075ARL502.05371.48210.92121.2876.5014.664.542.491.651.051.001.001.001.00SDRL489.97355.66195.21107.1364.309.402.321.140.790.220.020.000.000.00


## 4.5. Proposed versusNPHWMARSS Control Chart

The findings show that the proposedNPDHWMARSS control chart is better than the NPHWMARSS control chart in terms of OOC performance. For example, when we examine the t8 distribution at n=10,λ=0.10andδ=0.025,0.10,0.25, the ARL1 values of the proposed NPDHWMARSS and NPHWMARSS control charts are 70.30,8.66,2.67, and 153.55,20.27,4.77, respectively (see Figure 4 and Tables 1 and 6). Likewise, under Laplace distribution, when n=10,λ=0.05andδ=0.05,0.50, the ARL1 values of the proposed NPDHWMARSS control chart are 9.46,1.02, while the ARL1 values for NPHWMARSS control chart are 31.42,1.31, respectively (see Tables 1 and 6). The statistics show that the proposed structure works effectively for all distributions when compared to the NPHWMARSS control chartTable 6
RL characteristics of theNPHWMARSS control chart under various distributions with nominal ARL0=500 and n=10.
(λ, L)Distr.Metricsδ00.0250.050.0750.10.250.50.7511.522.535(0.05, 2.011)NormalARL502.18126.4747.4424.8315.553.861.561.041.001.001.001.001.001.00MDRL450.00108.0041.0022.0014.004.001.001.001.001.001.001.001.001.00SDRL369.2093.5732.8616.6610.061.910.870.240.030.000.000.000.000.00CNARL498.95131.6749.9226.0116.504.061.651.061.001.001.001.001.001.00MDRL447.00114.0044.0023.0015.004.001.001.001.001.001.001.001.001.00SDRL354.9896.6235.1917.6510.852.000.930.310.040.000.000.000.000.00LaplaceARL503.0289.9731.4216.3410.242.931.311.031.001.001.001.001.001.00MDRL447.0078.0028.0014.009.003.001.001.001.001.001.001.001.001.00SDRL357.8466.1021.2810.816.451.460.690.200.070.010.000.000.000.00LogisticARL502.13115.2542.1322.1713.633.571.451.031.001.001.001.001.001.00MDRL447.0099.0036.0019.0012.003.001.001.001.001.001.001.001.001.00SDRL357.8483.4929.5614.548.811.750.810.230.030.000.000.000.000.00t(4)ARL503.11153.2059.6531.3419.664.771.971.221.031.001.001.001.001.00MDRL449.00130.0052.0027.0017.004.001.001.001.001.001.001.001.001.00SDRL361.52115.0442.0121.9313.152.471.070.590.230.020.030.000.000.00t(8)ARL500.36138.2053.1828.3517.564.241.741.101.001.001.001.001.001.00MDRL437.00115.5046.0025.0016.004.001.001.001.001.001.001.001.001.00SDRL369.84105.1436.7719.0511.452.150.980.400.080.000.000.000.000.00(0.10, 2.268)NormalARL501.08141.1853.7828.3617.914.331.691.061.001.001.001.001.001.00MDRL416.00116.0047.0025.0016.004.001.001.001.001.001.001.001.001.00SDRL409.43104.6135.4817.4510.472.030.920.300.040.000.000.000.000.00CNARL498.89188.3984.6947.5131.267.052.631.351.101.021.001.001.001.00MDRL401.00123.0049.0027.0017.004.001.001.001.001.001.001.001.001.00SDRL392.85106.5837.0518.4511.472.120.970.360.060.000.000.000.000.00LaplaceARL499.36100.2535.7018.9311.973.221.421.051.001.001.001.001.001.00MDRL409.0084.0032.0017.0011.003.001.001.001.001.001.001.001.001.00SDRL401.0972.3322.5911.486.811.500.760.270.080.000.000.000.000.00LogisticARL500.14129.1247.6625.2315.933.921.581.051.001.001.001.001.001.00MDRL409.00108.0041.0022.0014.004.001.001.001.001.001.001.001.001.00SDRL401.0994.7731.6415.469.241.780.860.250.060.000.000.000.000.00t(4)ARL498.01169.3566.2736.0922.575.342.191.281.051.001.001.001.001.00MDRL398.50141.0056.0032.0020.005.002.001.001.001.001.001.001.001.00SDRL395.19124.8545.2222.7713.462.651.120.630.270.030.010.000.000.00t(8)ARL499.21153.5560.1632.0220.274.771.921.151.011.001.001.001.001.00MDRL400.50125.0052.0029.0018.004.001.001.001.001.001.001.001.001.00SDRL400.94119.0239.9119.7111.992.231.020.460.140.000.000.000.000.00


## 5. Illustrative Example

The proposed charts’ implementation is generally associated with industrial processes and finished products, and it can be adapted to different of many other fields such as medicine, planning, financial reporting, neutrosophic statistics, and so on. This section provides a real-life application of the non-isothermal continuous stirred tank reactor (CSTR) process to demonstrate the applicability of the proposedNPDHWMARSS control chart. This real-life data was originally proposed by Marlin and Marlin [25], and has since been widely used as a standard in fault diagnosis, for instance, Xiangrong et al. [26], Ridwan et al. [27], Adegoke et al. [20], and many more. The CSTR process has nine different variables, one of which we choose as the variable of interest (X) represents the output temperature.The data initially consists of 1,000 observations, with the first 600 occurring when the process was in an IC condition. The phase I sample’s parameters are as follows:μY=368.2328,μX=369.8789,σY2=0.2185915,σX2=0.3180327, and ρYX=0.08974039. We used the RSS approach to generate 40 paired observations of size n = 5 from a normal distribution. After the 24th sample, a shift in the process location is introduced following Anwar et al. [28]. The parameters of the proposed and NPRDHWMA-SR control charts used for real-life analysis are L=1.535,λ=0.10,ARL0=500, and L=2.117,λ=0.10,ARL0=500, respectively. Figure 5 indicates that the proposed NPDHWMARSS control chart triggers the first OOC signal at sample number 25, while the NPRDEWMA-SR control chart detects the first OOC point at sample number 29. Similarly, the proposed NPDHWMARSS control chart detects overall 16 OOC points, whereas the NPRDEWMA-SR control chart detects 12 OOC points (see Table 7 and Figure 5).Figure 5
Real-life application of the proposedNPDHWMARSS and the NPRDEWMA-SR control charts.Table 7
Application of the proposed versus NPRDEWMA-SR control charts.
Sample #ProposedNPRDEWMA-SRStatisticUCLLCLStatisticUCLLCL10.070.08−0.080.070.11−0.1126.867.89−7.890.060.23−0.2330.035.58−5.580.070.35−0.3541.064.56−4.560.160.47−0.4752.433.95−3.950.170.60−0.6060.983.53−3.530.180.71−0.7170.673.22−3.220.190.82−0.8280.722.98−2.980.200.92−0.9290.772.79−2.790.241.02−1.02100.942.63−2.630.221.10−1.10110.392.50−2.500.191.18−1.18120.302.38−2.380.201.25−1.25130.512.28−2.280.211.31−1.31140.562.19−2.190.251.37−1.37150.762.11−2.110.331.42−1.42161.042.04−2.040.441.47−1.47171.311.97−1.970.591.51−1.51181.561.92−1.920.701.54−1.54191.401.86−1.860.751.57−1.57201.191.81−1.810.781.60−1.60211.081.77−1.770.781.63−1.63220.961.72−1.720.741.65−1.65230.821.68−1.680.721.67−1.67240.811.65−1.650.681.68−1.68250.891.61−1.610.791.70−1.70261.461.58−1.581.021.71−1.71271.961.55−1.551.331.72−1.72282.391.52−1.521.721.73−1.73292.821.49−1.492.141.74−1.74303.191.47−1.472.621.74−1.74313.561.44−1.443.101.75−1.75323.891.42−1.423.621.76−1.76334.211.40−1.404.131.76−1.76344.501.38−1.384.651.76−1.76354.811.36−1.365.181.77−1.77365.041.34−1.345.641.77−1.77375.211.32−1.326.111.77−1.77385.471.30−1.306.581.77−1.77395.731.28−1.287.041.78−1.78405.901.27−1.277.441.78−1.78


## 6. Summary, Conclusions, and Future Recommendations

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



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*Source: 7350204-2022-06-13.xml*