Abstract
The nonparametric (NP) control charts are famous for detecting a shift in the process parameters (location and/or dispersion) when the underlying process characteristic does not follow the distributional assumptions. Similarly, when the cost of estimations is very high and the ranking of observational is relatively simple, the ranked set sampling (RSS) technique is preferred over the simple random sampling (SRS) technique. On the other hand, the NP triple exponentially weighted moving average (EWMA) control chart based on SRS is superior to the NP EWMA and NP double EWMA (NP DEWMA) based on the SRS technique to detect a shift in the process location. This study designed an advanced form of NP TEWMA Wilcoxon signed-rank based on RSS, denoted as control chart to identify a shift in the process location parameter. The Monte Carlo simulation method is used to assess the performance of the proposed control chart along with SRS-based NP TEWMA (TEWMA-SR), SRS-based NP TEWMA sign (TEWMA-SN), SRS-based , and RSS-based NP DEWMA-SR control charts. The study shows that the proposed control chart is more efficient in identifying shifts (especially in small shifts) in the process location than the existing control charts. Finally, a real-life application is also provided for the practical implementation of the proposed control chart.
1. Introduction
Variations are an essential part of every manufacturing and service process, and these variations can be categorized as common and special cause variations. The common cause variations are harmless, and the mechanism or process that operates under these variations is called in-control (IC). However, the special cause variations affect the product quality, and the process that runs in these variations is referred to as out-of-control (OOC). Statistical process control (SPC) tool (e.g., cause-and-effect diagram, check sheet, control charts, histogram, Pareto chart, scatter diagram, and stratification) kit is famous for monitoring the shift (i.e., special cause variations) in the process parameters (location and/or dispersion). Compared to other SPC tools, the control charts got special attention because they are competent and easy to implement to detect the shift in the process parameters. Furthermore, the control charts can be classified into memoryless and memory-type depending on their design structures. Shewhart [1] introduced control charts known as the Shewhart control charts; these control charts are also named memoryless-type control charts. Shewhart control charts only use current information to detect a large shift quickly in the process parameters. Later on, Page [2] and Roberts [3] proposed cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts, respectively; these control charts are also known as memory-type. Memory-type control charts are famous for identifying a small-to-moderate shift in the process parameters.
Generally, for classical parametric control charts (i.e., Shewhart, CUSUM, and EWMA), the underlying process characteristics follow a normal distribution or any other specified probability distribution to identify a shift in the process parameters. Occasionally, there may be a situation when the underlying process characteristics do not follow any specific distribution or distribution of the underlying process characteristics is in doubt. In this case, nonparametric (NP) control charts are the robust substitute for parametric control charts. These control charts are convenient because their IC run length (RL) distribution is similar for all continuous distributions. The sign (SN) and Wilcoxon signed-rank (SR) are well-known NP techniques. The SN technique only required the assumption of continuity, whereas the SR method required symmetry assumption as well [4].
Bakir and Reynolds [5] introduced an NP CUSUM control chart (NPCUSUM-SR) based on the signed-rank statistic for the process location. Similarly, Amin and Searcy [6] developed an NP EWMA-SR control chart by combining the SR technique with the EWMA control chart to monitor the process location shift effectively. In the same lines, Bakir [7] presented the SR-based Shewhart control chart. After that, Balakrishnan et al. [8] suggested an NP Shewhart control chart using runs rules to make it statistically sensitive. Correspondingly, Yang and Cheng [9] and Yang et al. [10] introduced CUSUM-SN and EWMA-SN control charts, respectively, for effective process location monitoring. In the same way, Graham et al. [11] presented the design structure of the single observation-based NP EWMA control chart. Furthermore, Chakraborty et al. [12] developed the GWMA-SR control chart to improve the EWMA-SR control chart’s ability to detect small shifts. Later on, Raza et al. [13] introduced the double NP EWMA-SR based (DEWMA-SR) control chart to monitor shift and showed that it is more sensitive than the EWMA-SR control chart. Likewise, Malela-Majika [14] developed new distribution-free control charts based on the Wilcoxon rank-sum test for efficiently monitoring the process location. Also, He et al. [15] designed an NP multivariate control chart for the time between events and amplitude data. This control chart is used to track the time intervals’ location shifts and the amplitudes of an event.
In SPC, sampling techniques such as simple random sampling (SRS) and perfect ranked set sampling (RSS) are well known and are commonly used with SN and SR techniques to observe the underlying process data. Furthermore, both SRS and RSS sampling techniques are widely used with parametric and NP control charts. RSS is a necessary and valuable statistical technique commonly used in statistical quality control when the precise measurement of a selected unit is either difficult or prohibitively expensive and time-consuming [16]. It is self-evident that RSS efficiency is dependent on the precision with which the randomly selected units are defined. Errors in the ranking have a negative impact on the estimator’s efficiency and result in imprecise estimates. Dell and Clutter [17] investigated the effect of error and imperfect ranking on mean estimator performance. They presented that even when imperfect rankings are used, the RSS mean estimator remains unbiased and outperforms the SRS mean estimator. However, the performance of the RSS remains superior to that of the imperfect RSS. Visual inspection of the study variable or based on the auxiliary variable may easily lead to ranking a small set of the selected units [18]. For example, hazardous waste sites with different contamination levels can be ranked by a visual inspection of soil discoloration, when the actual measurements of toxic chemicals and quantifying their environmental impact are very costly. In this regard, Salazar and Sinha [19] introduced RSS with control charts to monitor the process location shift. They demonstrated that the RSS-based control charts outperformed against SRS-based control charts. Similarly, Muttlak and Al-Sabah [20] and Abujiya and Muttlak [21] presented an RSS-based (median ranked set sampling and extreme ranked set sampling) and double RSS (DRSS) based control charts, respectively, to monitor the shift in the process location. Later, Al-Omari and Haq [22] proposed a Shewhart-type control chart based on the DRSS technique.
Haq et al. [23] suggested a maximum EWMA (MaxEWMA) control chart based on order DRSS (ODRSS) and order imperfect DRSS (OIDRSS) sampling techniques. In addition, they [24] also developed a new synthetic control chart for process location and dispersion using different RSS techniques. Furthermore, based on RSS, Haq and Khoo [25] developed synthetic EWMA and synthetic CUSUM control charts to monitor the process location. Recently, Abbas et al. [26] introduced an NP DEWMA-SR control chart using the RSS technique for process location monitoring, labelled as control chart.
Similarly, in the SPC, modifications and enhancements are continually practiced to enhance the performance of the memory-type control charts. For example, Shamma and Shamma [27] extended the classical EWMA control chart and suggested a double EWMA (DEWMA) control chart for the process location. The DEWMA control chart is more responsive than the classical EWMA control chart. Anwar et al. [28] and Aslam and Anwar [29] introduced modified-EWMA control charts, respectively, in the presence of auxiliary information and Bayesian methodology. Similarly, Anwar et al. [30] introduced auxiliary information-based (AIB) combined MEC for the simultaneous monitoring of process location and dispersion.
Recently, Chatterjee et al. [31] proposed a TEWMA control chart to monitor process dispersion shift. Later on, Alevizakos et al. [32] and Alevizakos et al. [4] suggested an NP TEWMA-SN (TEWMA-SN) and NP TEWMA-SR (TEWMA-SR) control charts, respectively. Furthermore, Alevizakos et al. [33] extended EWMA and DEWMA control charts and proposed a triple EWMA (TEWMA) control chart further to enhance the efficiency of the EWMA control chart. The TEWMA control chart is more responsive than the EWMA and DEWMA control charts to monitor small-to-moderate shifts in process location.
As discussed earlier, the SRS-based TEWMA-SN and TEWMA-SR control charts are well known for timely monitoring of the quality characteristics easily available. Still, when selecting quality characteristics is either difficult or expensive and time-consuming, these SRS-based control charts fail to monitor the process efficiently. To monitor such quality characteristics, this study suggested an NP triple EWMA-SR control chart with the RSS technique for effectively monitoring the process location of a continuous and symmetric distribution. Various processing environments like normal, Contaminated normal (CN), Laplace, Student’s , and Logistic are used to measure the characteristics of the proposed control chart. The average run length , the median of the run length (MDRL), and the standard deviation of the run length are assessed to determine the efficiency of the proposed control chart against other control charts. The sample points before the control chart signal are referred to as , and the RL’s expected values are referred to as . is the of the IC process while is the of the OOC process. A control chart is considered to be efficient when it has smaller values as compared to its contestant at different shifts.
The remainder of the article is as follows: Section 2 presents the design structures of existing and the proposed control charts. Similarly, Section 3 offers the implementation of the proposed control chart. Also, the comparative study is given in Section 4, while Section 5 provides an illustrative example related to the proposed control chart. Finally, Section 6 offers the summary, conclusions, and recommendations of the study.
2. Existing and Proposed Methods
This section explains the design structures of some existing and the proposed control charts. For example, Section 2.1 presents the RSS-based Wilcoxon signed-rank statistic. Similarly, methodologies of the NP EWMA SR with RSS and control charts are given in Sections 2.2 and 2.3, respectively. Finally, the design structure of the proposed control chart is provided in Section 2.4.
2.1. RSS-Based Wilcoxon Signed-Rank Statistic
Mclntyre [16] gives the idea of the RSS technique for data collection. Suppose RSS of size n is taken from a quality variable , where , and . denotes each sample, denotes the number of observations, and denotes the number of cycles used in the RSS technique. is the number of iterations used in the RSS technique with a sample of size and number of cycles. Let , where denote the ranks of absolute deviations of RSS values from median . According to Kim and Kim [34], the statistic of RSS under SR can be written aswhere
The mean and variance of the statistic are and , respectively [34]. The can be defined as follows:where the values of can be obtained by solving the following expression for :whereas is used to enhance the performance of the control chart. The values of at different sample sizes taken from [35] are given as (2, 0.750), (3, 0.625), (4, 0.547), (5, 0.490), (6, 0.451), (7, 0.416), (8, 0.393), (9, 0.371), and (10, 0.352).
2.2. Control Chart
The control chart is more sensitive for detecting shifts in the process location/median than the NPEWMA-SR control chart. The plotting statistic of the control chart based on equation (1) is defined aswhere is a smoothing constant and the initial value of the is equal to 0. The mean and variance of the statistic are and , respectively. The variance of is reduced to , when the term approaches to unity. The control limits of the control chart (see Table 1) are defined below:where is the width of the control limits, which helps to determine . The process will be OOC if any or ; otherwise, it will remain in an IC state.
2.3. Control Chart
Abbas et al. [26] introduced the following plotting statistics of the control chart:
The mean and variance of ( control chart) for IC process are and , respectively. The control limits of the control chart can be designed aswhere is the control chart width and the initial value of the . The process will be OOC when or ; otherwise, it will remain in an IC state.
2.4. Proposed Control Chart
This section offers a proposed control chart structure. In more detail, the proposed control chart is designed for monitoring the process location shift. The plotting statistics of the proposed control chart are defined as
The initial values of are all equal to 0 (zero). The mean and variance of the statistic under IC state are andrespectively, where . The time-varying control limits of the proposed control chart are specified aswhere is the width of the control limits. For large values of q, we have and the asymptotic control limits become
The process goes OOC if any . However, if , the process will be in IC state.
3. Implementation of the Proposed Control Chart
This section provides the implementation of the proposed control chart. Section 3.1 describes the choice of design parameters. Similarly, Section 3.2 presents the evaluation of the proposed control chart.
3.1. Choice of Design Parameters
and are the design parameters of the proposed control chart. The control chart width is chosen with various values of , such that the prespecified is obtained. The best subgroup size is between 5 and 10, depending on and shift size [5]. Different properties are assessed to study the performance behavior of the proposed control chart with and . is assumed to be 370 for this study.
3.2. Evaluation
This section provides the characteristics of the different distributions (normal and non-normal distributions) for IC robustness and IC performance of the proposed control chart. The proposed control chart’s features are evaluated using normal and non-normal distributions.
3.2.1. Evaluation of the Run-Length Distribution
The performance of the proposed control chart can be assessed in both normal and non-normal symmetric distributions. The distributions used in this analysis are as follows:(i)Standard normal distribution, i.e., (ii)Double exponential (Laplace) distribution, i.e., (iii)Heavy tail Student’s distribution , with the degree of freedom (iv)The logistic distribution, i.e., (v)Contaminated normal (CN) distribution, which is a mixture of and , i.e., , with such that , where and . For comparison purpose, all the considered distributions in this analysis were reparametrized with mean zero and unit standard deviation. The probability density functions of the distributions used in this study are shown in Table 2.
3.2.2. Monte Carlo Simulation
The Monte Carlo simulation is used to obtain the characteristics of the proposed control chart. The simulation algorithm is developed in R software to compute the characteristics. random samples of the size from any distribution covered by the study for a shift were generated. The simulation algorithm to determine the nominal values of under various distributions can be explained in the following steps [36]:(a)An finite loop is practiced to generate samples from different distributions(b)Select a particular value for (c)Determine the plotting statistic offered in equation (9)(d)Let us assume (e)Calculate the from equation (11)(f)Plot the statistic against control limits in step (e) over q(g)If the , record the sample number of statistic as an . For example, at , if record 255 as a first .(h)Repeat steps from (a) to () for times and record RLs.(i)Compute the average of times noted RLs, which is called .(j)If ; otherwise, adjust (i.e., ) constant accordingly in step (d) and repeat steps from (a) to (i) to obtain .(k)To compute the values, again generate samples from relevant distribution and repeat steps from (b) to (i).
3.2.3. In-Control Robustness of the Proposed Control Chart
The EWMA and control charts have the limitation that their IC RL features do not remain the same for all continuous distributions. The IC characteristics of the NP control chart’s characteristics remain the same for all symmetric continuous distributions [4]. The simulated results of the proposed control chart with different combinations of to obtain the nominal are shown in Tables 3 and 4. The IC distribution of the proposed control chart seems to be similar for all distributions used in this study. The distribution is positively skewed when .
3.2.4. Out-of-Control Performance
OOC performance of the proposed control chart can be discussed under perfect RSS and imperfect RSS. The OOC control chart’s efficiency is primarily compared, and it demonstrates how sensitive the control chart is to detect shifts in process parameters. Tables 3 and 4 show the OOC performance of the proposed control chart for the various distributions evaluated in this study. For small and moderate shifts, the OOC performance of the proposed control chart increases as increases; i.e., values decrease by increasing . The proposed scheme’s values reduce with the rise in n and m values. The OOC RL performance of the Laplace distribution is higher as compared to the other distributions used in this study (see Figures 1 and 2). The proposed control chart’s ability to detect shifts deteriorates when it occurs later for small values of ( or ), whereas this trend is noticed for large values of ( or ) for large shifts (see Figures 3 and 4). The proposed control chart’s OOC performance under imperfect RSS is also determined to illustrate the superiority of perfect RSS or generally RSS over imperfect RSS (see Tables 5–7). For example, under normal distribution, at and the ARL values of the proposed control chart under RSS are , whereas the ARL values of the proposed control chart under imperfect RSS are (see Tables 3 and 5). In general, the proposed control chart under RSS is very sensitive to detect small and moderate shifts in the process location.
In terms of OOC performance at a specific shift , the proposed control chart outperforms all other control charts. For instance, at , and , the values of the proposed , TEWMA-SR, TEWMA-SN, , and control charts are 151.84, 251.08, 341.92, 337.05, and 167.12, respectively. The is used as a metric to compare the proposed and existing control charts. Some findings are presented in the percentage decreasing in . Under normal setup, the control chart displaying a 3 percent improvement in the process median minimizes the by 54.56 percent for , whereas at the same shift , the proposed control chart reduced the by 80.55 percent for (see Tables 3 and 4).
4. Comparative Study
This section provides the proposed control chart’s comparisons to several existing control charts, which includes [26], [33], TEWMA-SN [32], and TEWMA-SR [4] control charts. More detail is provided in the following sections.
4.1. Proposed versus TEWMA-SR Control Chart
The proposed control chart outperforms the TEWMA-SR control chart. For instance, at and with percent shift in the Logistic distribution, the proposed control chart reduces the value by 78.28 percent, whereas the TEWMA-SR control chart reduces the value by 58.74 percent (see Table 8). Similarly, under distribution, when and along with percent, the values of the proposed control chart are reduced by 86.46 percent, while the TEWMA-SR control chart reduces the values by 72.30 percent (see Tables 4 and 9). Figures 5 and 6 highlight the proposed control chart superiority over the TEWMA-SR control chart. It is worth mentioning that, for small shifts, the proposed control chart outperforms the TEWMA-SR control chart for different combinations of .
4.2. Proposed versus TEWMA-SN Control Chart
The features of the TEWMA-SN control chart for the process location are listed in Table 10. The proposed control chart is more efficient than the TEWMA-SN control chart. For example, under Laplace distribution, the proposed control chart with with percent shows OOC situation almost after 7.75 observations, whereas the TEWMA-SN control chart gives OOC state nearly after 25.67 observations (see Tables 4 and 10). Similarly, in the distribution scenario, at with a shift of 5 percent, the TEWMA-SN control chart decreases the by 67.83 percent, while the proposed control chart decreases the by 89.13 percent (see Table 8).
Furthermore, at with 10 percent shift in process location under the Logistic distribution, the TEWMA-SN control chart reduces the by 79.95 percent and the proposed control chart reduces the by 91.79 percent (see Table 8). The proposed control chart efficiency over the TEWMA-SN control chart can be seen in Figure 6. In short, the proposed control chart is quite sensitive under normal and non-normal distributions for detecting small to moderate shifts in process location against the TEWMA-SN control chart.
4.3. Proposed versus Control Chart
The control chart outperforms the control chart in terms of early shift detection ability. For instance, using the Laplace distribution, the control chart with and for percent gives OOC signal after 238.73 observations, whereas the proposed control chart shows the OOC signal after 105.76 observations (see Table 8). The superiority of the proposed control chart can also be seen in the case of outliers under CN distribution. For example, the proposed control chart with and for percent produces OOC signal after 14.61 observations, while the control chart shows OOC signal after 43.39 observations (see Tables 4 and 11). The performance of the proposed control chart over the control chart is depicted in Figures 7 and 8. In brief, the proposed control chart is quite sensitive for detecting process location shifts relative to the control chart.
4.4. Proposed versus Control Chart
The proposed control chart provides better performance than the control chart. For example, under a normal environment, at and for percent, the proposed control chart reduces the by 44.92 percent, while the control chart reduces the by 39.99 percent (see Table 8). Similarly, when we examine the Logistic distribution at and for percent, the proposed control chart decreases the by 34.59 percent and the control chart decreases the by 30.57 percent (see Tables 3 and 12). Furthermore, under a CN environment, at and with percent, the proposed control chart gives OOC signal after 45.28 observations, whereas the control chart does after 50.84 observations (see Tables 4 and 13). The comprehensive RL features of the control chart for process location shift under selected distributions are shown in Tables 12 and 13. The control chart outperforms the control chart, as shown in Figures 9 and 10. Results indicate that the proposed structure works effectively in all environments relative to the control chart.
5. Illustrative Example
This section demonstrates practical application to implement the proposed control chart in practice. For this purpose, the proposed with TEWMA-SR and control charts are considered to monitor the process location shift. For the practical implementation of the proposed control chart, we assumed the data set used by Muttlak and Al-Sabah [20]. These data were also used by Abid et al. [37] for the execution of the control chart. The collected data were about the Pepsi Cola Company in AL-Khobar, Saudi Arabia. The company uses different production lines, like a line used for filling soft drinks in bottles is referred to as an interest production and the volume of drink in the bottle is referred to as a variable of interest. They used the RSS method to hold the appropriate volume of soft drinks in bottles during the filling process.
The RSS technique is used to collect 27 ranked set samples by repeating the two cycles four times, with each cycle having a size of . To compare the proposed control chart with TEWMA-SR and control charts, 27 samples with are collected using RSS. The design parameters of the , TEWMA-SR, and are (1.578, 0.05), (1.75, 0.05), and (1.305, 0.05), respectively, at . Comparison of the proposed control chart with the TEWMA-SR and control charts can be visualized in Figures 11–13. The control chart shows 4 OOC signals (at sample numbers 24–27), whereas the TEWMA-SR control chart shows no OOC signals. In contrast, the proposed control chart shows 7 OOC signals (at sample numbers 21–27) in the process location parameter. These findings indicate that the proposed has improved performance for the monitoring of the process location as compared to and TEWMA-SR control charts. Hence, the proposed control chart under RSS has a better ability to detect shifts in process location.
6. Summary, Conclusions, and Recommendations
The ranked set sampling (RSS) technique is preferred over the simple random sampling (SRS) for the processes when the estimations are destructive or costly, and the ranking of observational data is comparatively simple. Similarly, the nonparametric (NP) control charts are very useful to monitor shifts in the process parameters when the distribution of the underlying process is questionable or unknown. So, this study proposes an NP triple exponentially weighted moving average (TEWMA) Wilcoxon signed-rank (SR) control chart under the RSS technique (represented as control chart) using different continuous symmetric distributions. The Monte Carlo simulation method is used to obtain the numerical results of the proposed control chart along with other existing control charts. The performance of control chart is substantially better than the SRS-based NP TEWMA-SR, TEWMA sign, , and double EWMA-SR control charts. A real-life example is also provided to demonstrate how the proposed control chart can be used in practice. Therefore, to adopt a robust and efficient control chart, the proposed control chart gives an alternate choice to quality practitioners. The proposed work can be extended to the multivariate scenario.
Abbreviations
| : | Average run length |
| : | In-control average run length |
| : | Out-of-control average run length |
| CUSUM: | Cumulative sum |
| CUSUM-SR: | CUSUM Wilcoxon signed-rank |
| CUSUM-SN: | CUSUM sign |
| CN: | Contaminated normal |
| DEWMA-SR: | Double EWMA-SR |
| DEWMA: | Double EWMA |
| : | Double EWMA-SR with RSS |
| : | One-sided DEWMA for time between events |
| DEWMA-SR: | Double NP EWMA-SR |
| : | Double EWMA mean |
| DRSS: | Double RSS |
| EWMA: | Exponentially weighted moving average |
| : | NP EWMA-SR with RSS |
| : | One-sided EWMA for time between events |
| EWMA-SR: | EWMA Wilcoxon signed-rank |
| EWMA-SN: | EWMA sign |
| GEWMA-SR: | Generalised EWMA-SR |
| IC: | In-control |
| LOG: | Logistic distribution |
| : | Lower control limit of |
| : | Lower control limit of |
| : | Lower control limit of |
| MEC: | Mixed EWMA-CUSUM |
| MCE: | Mixed CUSUM-EWMA |
| MxMEC: | Auxiliary information based MEC |
| MxMCE: | Auxiliary information based MCE |
| NP: | Nonparametric |
| OOC: | Out-of-control |
| ODRSS: | Order DRSS |
| OIDRSS: | Order imperfect DRSS |
| RSS: | Ranked set sampling |
| RL: | Run length |
| SDRL: | Standard deviation of the run length |
| SRS: | Simple random sampling |
| SPC: | Statistical process control |
| SR: | Wilcoxon signed-rank |
| TEWMA: | Triple TEWMA |
| : | Triple EWMA with SR under RSS |
| TEWMA-SR: | TEWMA with SR |
| TEWMA-SN: | TEWMA signed-rank test |
| : | One-sided TEWMA for time between events |
| : | TEWMA mean |
| : | Upper control limit of |
| : | Upper control limit of |
| : | Upper control limit of |
| : | Amount of shift in process location |
| : | Standard deviation |
| : | Sample size |
| : | Smoothing constant |
| : | Belong to |
| : | Student’s t distribution with degree of freedom |
| : | Variance |
| : | Initial value of |
| : | Control limit coefficient |
| : | Plotting statistic of |
| : | Plotting statistic of |
| : | Plotting statistic of |
| : | Median of ranked values |
| : | Rank of absolute deviations |
| : | Initial value of |
| : | Initial value of |
| : | Number of cycles |
| : | Number of iterations. |
Data Availability
The real-life data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The author Saddam Akber Abbasi acknowledges Qatar University for providing excellent research facilities.