Efficient monitoring of coefficient of variation with an application to chemical reactor process

Control chart is a useful tool to monitor the performance of the industrial or production processes. Control charts are mostly adopted to detect unfavorable variations in process location (mean) and dispersion (standard deviation) parameters. In the literature, many control charts are designed for the monitoring of process variability under the assumption that the process mean is constant over time and the standard deviation is independent of the mean. However, for many real‐life processes, the standard deviation may be proportional to mean, and hence it is more appropriate to monitor the process coefficient of variation (CV). In this study, we are proposing a design structure of the Shewhart type CV control chart under neoteric ranked set sampling with an aim to improve the detection ability of the usual CV chart. A comprehensive simulation study is conducted to evaluate the performance of the proposed CV[NRSS] chart in terms of ARL , MDRL, and SDRL measures. Moreover, the comparison of CV[NRSS] chart is made with the existing competitive charts, based on simple random sampling, ranked set sampling (RSS), median RSS, and extreme RSS schemes. The results revealed that the proposed chart has better detection ability as compared to all existing competitive charts. Finally, a real‐life example is presented to illustrate the working of the newly proposed CV chart.

control limit (UCL). When the measurement of a quality characteristic falls outside of the decision lines, the production process is alarmed with an indication of the production of nonconforming items.
In the literature, most control chart structures are designed to monitor the location or dispersion parameter of the process. The Shewhart̄chart and the median chart are popular charts for monitoring process location Montgomery, 1 and other location-based Shewhart charts are discussed in Schoonhoven et al 2 and Nazir et al. 3 The Shewhart charts based on the dispersion parameters are presented in Abbasi and Miller, 4 Ali et al, 5 Abbas et al, 6 and references therein. The location-based charts were designed under the assumption of equal variances. However, such assumption is not realistic. Therefore, joint or simultaneous structures of these charts were proposed, in which two charts were embedded together in such a way that one monitors the location parameter and other monitors dispersion parameter. Thē− and̄− charts are the basic simultaneous structures, while other simultaneous structures are discussed in Mahmood et al, 7 Zafar et al, 8 Sanusi et al, 9 and references therein.
Mostly in control chart settings, it is assumed that the process mean is stable over time, and the standard deviation is independent of the mean. However, such assumptions may not be valid for many real-life processes. Under these circumstances, it is recommended to monitor the coefficient of variation (CV). The CV-based chart was initially proposed by Kang et al 10 that involves plotting the sample CV (i.e., = ∕̄), where and̄represent the sample standard deviation and sample mean, respectively.
In the last few years, many CV control charts were proposed in the SPC literature. For instance, Connett and Lee 11 utilized a CV chart in the clinical trials to monitor split specimens of urine. Hong et al 12 provided an exponentially weighted moving average (EWMA) chart for the monitoring of process CV. The variance and CV charts based on their predictive distributions were designed by Menzefricke. 13 Castagliola et al 14 proposed two one-sided EWMA charts for monitoring the square of CV statistics, while this proposal is modified by Zhang et al 15 to enhance the sensitivity of EWMA charts. For efficient detection of the small shifts in the process, the double EWMA-based CV chart was proposed by Hong et al. 16 Further, the synthetic version of the CV chart was designed by Calzada and Scariano, 17 and run-rules with Shewhart CV chart were implemented by Castagliola et al. 18 The one-sided Shewhart chart for the monitoring of process CV under short production runs was investigated by Castagliola et al, 19 and the economic design of the CV chart was discussed by Yeong et al. 20 Similar to the proposal of Castagliola et al, 14 the cumulative sum (CUSUM) structure was proposed by Tran and Tran. 21 Further, a side sensitive group runs chart for the monitoring of CV was proposed by You et al. 22 The CV control chart with variable sampling interval (VSI) strategy, variable sample size (VSS) strategy, and the combination of these two strategies was studied by Castagliola et al, 23 Castagliola et al, 24 Amdouni et al, 25 Amdouni et al, 26 Khaw et al, 27 Yeong et al, 28 and Muhammad et al. 29 Furthermore, the one-sided CV control chart under the measurement errors was proposed by Yeong et al, 30 and a Bayesian CV control chart was discussed by Van Zyl and Van der Merwe. 31 All of the studies mentioned above have considered zero-state performance, while for the steady-state performance, Teoh et al 32 designed a chart, namely, the run-sum chart for the monitoring of process CV. The multivariate extension of the CV chart under zero-state structure was proposed by Yeong et al, 33 while under the steady-state structure was designed by Lim et al. 34 The one-sided synthetic version of CV chart with measurement errors was discussed by Tran et al, 35 and the ARL-unbiased Shewhart CV chart was proposed by Guo and Wang. 36 The adaptive version of multivariate CV chart with VSS and VSI strategies was proposed by Khaw et al, 37 while the phase I properties of univariate and multivariate CV chart were studied by Dawod et al, 38 and Abbasi and Adegoke. 39 The CV control chart under repetitive group sampling plan was discussed by Yan et al, 40 under multiple repetitive group sampling plan was designed by Yan et al, 41 and under the double sampling plan was proposed by Ng et al. 42 The optimal design of EWMA-based CV chart was proposed by Teoh et al, 43 and a new EWMA chart for the monitoring of CV was proposed by Zhang et al. 44 This new EWMA-based CV chart was designed such that negative normalized observations of traditional CV-EWMA statistic were set at zero. By this modification, the new chart has better detection ability as compared to the conventional CV-based EWMA chart. Recently, Du Nguyen et al 45 studied Shewhart CV control chart with VSI strategy under measurement errors. The variable parameter chart for the monitoring of the univariate CV was discussed by Yeong et al, 46 and multivariate CV chart was proposed by Chew et al. 47 The one-sided CV control chart for the multivariate CV under short production runs was proposed by Khatun et al. 48 Most of the studies mentioned above were designed under the simple random sampling (SRS) structure. However, ranked set sampling (RSS) structure proposed by McIntyre 49 is a more efficient sampling structure as compared to the SRS. Abbasi et al 50 proposed Shewhart CV chart under RSS and its modified strategies, and the EWMA-based CV chart under RSS schemes was designed by Noor-ul-Amin and Riaz. 51 Recently, Zamanzade and Al-Omari 52 proposed a new ranked set scheme, namely, neoteric ranked set sampling (NRSS). The NRSS is a more efficient sampling structure as compared to the SRS and RSS schemes. Recent control charting structures under NRSS scheme may be found in Nawaz et al, 53 Nawaz and Han, 54 Abbas et al, 55 Hussain et al, 56 and references therein. In the literature, the Shewhart CV chart is not designed under the NRSS scheme, in the best of our knowledge. Hence, this study is intended to propose a Shewhart CV chart under the NRSS structure to obtain an efficient method for the monitoring of process CV. Further, the proposed structure is compared with the Shewhart CV chart under the SRS scheme Kang et al 10 and RSS strategies Abbasi et al. 50 Rest of the article is organized as follows: the design structure of the proposed study is given in Section 2. Section 3 presents the performance evaluation of the proposed charts, while simulation study results and discussions are provided in Section 4. Section 5 consists of a real-life example to illustrate the working of new control chart, and finally, the conclusions and recommendations are reported in Section 6.

THE DESIGN STRUCTURE OF THE PROPOSED STUDY
This section is designed to present different sampling schemes and the structure of the proposed CV chart.

Sampling strategies
The choice of sampling scheme plays a significant role in decision making. In the SPC literature, control charts are designed under several sampling schemes, such as SRS, RSS, median RSS, extreme RSS, and so on. The SRS is the most straightforward sampling mechanism, in which all units in the population have an equal probability of being selected in a sample. Further, the description of RSS and NRSS methods is given in the following subsections.

Ranked set sampling schemes
The structure of RSS was initiated by McIntyre. 49 The method of RSS is outlined as follows: draw 2 ( , ) samples from a bivariate normal distribution using specified correlation and divide them into sets having observations in each set.
In each set, actual observations ( ) are ranked (in ascending order) according to the concomitant variable ( ). For actual RSS observations of size , choose the f irst smallest actual observation from f irst set, second smallest actual observation from second set, and continue this procedure until the largest actual observation is selected from the ℎ set. The median RSS (MRSS) was introduced by Muttlak, 57 and its structure can be defined as follows: draw 2 ( , ) samples from a bivariate normal distribution using specified correlation and divide them into sets having observations in each set. In each set, actual observations ( ) are ranked (in ascending order) according to the concomitant variable ( ). For actual MRSS observations of size , when the size of the set is even, choose half observations from the first ( ∕2) sets and other from the last (( + 2)∕2) sets. However, when the size of the set is odd, select (( + 1)∕2) observations from each set.
Samawi et al 58 introduced another RSS scheme, namely, extreme RSS (ERSS). The mechanism of ERSS is defined as follows: draw 2 ( , ) samples from a bivariate normal distribution using specified correlation and divide them into sets having observations in each set. In each set, actual observations ( ) are ranked (in ascending order) according to the concomitant variable ( ). For actual ERSS observations of size , when the size of the set is even, choose the smallest ranked observations of first ( ∕2) sets and remaining were selected as the largest ranked observations of last ( ∕2) sets. However, when the size of the set is odd, select the smallest sample from the first (( − 1)∕2) sets, largest sample from last (( − 1)∕2) sets, and the median of the rest of the sets.

2.2
The Let represents a quality characteristic of interest having mean and standard deviation . The process CV ( ) can be defined as: = . (1) The CV given in (1) is a standardized measure of the dispersion of and has several benefits over the other measures of the dispersion, such as , which is always understood in the context of . Usually, parameters and are unknown and are estimated from the sample observations, where and̄represent the sample standard deviation and sample mean, respectively.
Assume [ℵ] is the observed variable where sample number is indexed with ; = 1, 2, … , and time is indexed with ; = 1, 2, … , for the sampling scheme ℵ. Hence, the ℎ estimated CV statistic can be defined as follows: .
The probability limits are considered as the decision lines for [ℵ] chart, which is defined on the base of the prespecified probability of a type I error as follows: , with Hence, the values of plotting statistiĉ[ ℵ] are plotted against [ℵ] and [ℵ] , and if any value falls outside of these limits, then the process is declared as out-of-control (OOC); otherwise, it is assumed in-control (IC) point. For a given sampling scheme, the limits, as mentioned above, depend on the choice of . Therefore, to obtain the TA B L E 1 Control chart constants and probability limits for the [NRSS] ∕2) ), the following procedure is adopted.
i. Generate 100,000 random samples of size 2 from the bivariate normal distribution having mean vector and covariance matrix Σ, where ii. Obtain 100,000 ranked samples of size using above-mentioned ranked set schemes.

PERFORMANCE EVALUATIONS
This section consists of a discussion on performance measures, which are used to evaluate the stated proposal and comparative study. Moreover, the design of the simulated study is also reported in this section.

Performance measures
The average run-length ( ) is the most important measure, used to describe the performance of a control chart. The is defined as the average number of plotting points until a point falls outside the limits (cf. Mahmood and Xie 59 ).
Generally, the measure is classified into two categories: IC ( 0 ) and OUC ( 1 ). When the process is assumed under stable conditions or based on IC parameters, then the measure is known by 0 otherwise, the measure is termed as 1 . A control chart is declared as the best chart among the competing charts, if for a fixed 0 , it has a minimum 1 for detecting shifts in process parameters (CV in this case). Furthermore, due to the skewed nature of measure, few researchers prefer to evaluate the performance using the median run-length ( ) and standard deviation of the run-length ( ). The provides a robust estimate of the run-length of a chart, while is used to show the variation in run-length values. All three measures ( , , and ) are used for performance evaluation of the proposed [NRSS] chart. Further, its performance is compared with the CV charts (i.e., [SRS] , [RSS] , [MRSS] , and [ERSS] ) based on SRS, RSS, MRSS, and ERSS schemes.

Design of simulation study
To evaluate the performance of the proposed [NRSS] chart and for the comparative analysis, a comprehensive simulation study is designed, as described below: i. Generate a random sample of size 2 from the bivariate normal distribution with = ( , ) and Σ = ⎛ ⎜ ⎜ ⎝ , where 1 is equals to (1 + ). The process is considered as IC when = 0 and when > 0, the process is assumed to be OOC. ii. Obtain ranked set sample of size using above-mentioned ranked set schemes.
iii. Based on the ranked set sample of size , compute the 1[ℵ] using (2) and plot it against the control limits based on the choice of provided in Table 1. iv. If 1[ℵ] falls in the limits, then repeat steps (i-iii) until [ℵ] falls outside of the limits. Note the as the first run-length value. v. Repeat steps i-iv, a large number of times say 100,000 times and record the run-length values.
Further, the performance measures, such as , , and , were computed as the mean, median, and standard deviation of these 100,000 run-length values, respectively.
The performance of the [NRSS] control chart is evaluated based on the different choices of sample size ( = 3, 5, 7, and 10) and several choices of

RESULTS AND DISCUSSIONS
In this section, first, we will evaluate the performance of the proposed [NRSS] chart in terms of , , and with respect to different choices of 0 , such as 100, 200, 370, and 500 in Tables 2-5. Moreover, the effect of the perfect and imperfect ranking situations on the performance of the [NRSS] chart is reported in • The correlation factor often selected to determine the ranking strategy used for the generation of NRSS samples. When < 1, a ranking strategy is known by imperfect ranking, and it is assumed perfect ranking when = 1. The shift detection ability of the  (Table 6). Hence, it is concluded that the [NRSS] chart showed superior performance under perfect ranking.
Abbasi et al 50 proposed Shewhart-type CV control charts based on RSS, MRSS, and ERSS strategies. Their respective charts are referred as [RSS] , [MRSS] , and [ERSS] charts. They have compared these charts with the conventional [SRS] chart and concluded that the [ERSS] chart has better detection ability as compared to all other charts. In this study, we are comparing the proposed [NRSS] chart with the [SRS] , [RSS] , [MRSS] , and [ERSS] charts. The ( ) for each chart, with respect to several levels of shifts in process CV, is plotted in Figures 1 and 2. In this comparative study, perfect sampling mechanism (i.e., = 1) is considered with the fixed 0 = 370. • From Figures 1 and 2, it is clearly observed that the [NRSS] chart has the lowest ARL curves as compared to the [SRS] , [RSS] , [MRSS] , and [ERSS] charts, considering samples sizes n = 5 and 10, respectively. • Hence, it is concluded that the [NRSS] chart is the best performing chart among all other competing CV charts, to detect a change in the process CV.

AN ILLUSTRATIVE EXAMPLE
For the illustration of [ℵ] charts, we have considered nonisothermal Continuous Stirred Tank Chemical Reactor (CSTR) dataset obtained from Yoon and MacGregor. 60 Usually, a nonisothermal CSTR is used to illustrate the fault isolation. The process plotted in Figure 3 consists of a feed stream, a product stream, and a cooling water flow to the coils. The feed stream is a merged stream of the reactant A and the solvent. The first-order reaction ( → ) was considered and reactor system involves heat transfer and heat of reaction. Further, it was also assumed that the tank is well mixed, and the physical TA B L E 6 Run length characteristics of [NRSS] chart with respect to perfect and imperfect ranking when = 5 and 0 = 370  Among all fault detection indicators, outlet temperature is an important variable that shows the variation in the temperature of the whole system. Therefore, we have considered outlet temperature as a study variable and obtained 40 sets each of size 5 (i.e., = 5) using the different sampling schemes, such as RSS, MRSS, ERSS, and NRSS. Further, for checking the consistency of [ℵ] statistics,̄[ ℵ] ; = 1, 2, … , 40, are plotted against thê2 [ℵ] in Figure 4, following Kang et al 10 and Abbasi et al. 50 The scatter plots of̄[ ℵ] and̂2 [ℵ] showed that the [ℵ]   Under the null hypothesis, it is assumed that the [ℵ] statistics are constant against̄[ ℵ] , while nonconstant behavior is considered under the alternative hypothesis. The regression results with respect to sampling schemes are reported in Table 7, where all P-values are higher than the level of significance (i.e., = 0.05), which supports the null hypothesis that the [ℵ] statistics are constant against̄ [ ℵ] . From Figure 4, it is also revealed that there is no apparent correlation structure between̄[ ℵ] and̂2 [ℵ] . Hence, based on these two findings, it is justified to use [ℵ] charts to detect a change in outlet temperature.
Based on all sets, the estimated IĈ[ ℵ] were obtained, which are reported in Table 8. Further, at fixed 0 = 200 and by usinĝ[ ℵ] , the control limits (i.e., [ℵ] and [ℵ] ) were obtained, which are also given in Table 8. Furthermore, the [ℵ] statistics for each set are presented in Table 8 Table 9 and are plotted against their respected limits (i.e., [ℵ] and [ℵ] ) in Figure 5.

CONCLUSIONS AND RECOMMENDATIONS
The CV chart is often used to monitor the relative variability in the process. In this study, we have proposed a Shewhartbased CV chart, considering a newly introduced NRSS, which is referred as the [NRSS] chart. An extensive simulation study is designed to evaluate the performance of [NRSS] chart in terms of , , and measures. The findings depict that the performance of the [NRSS] chart improves with an increase of sample size . The [NRSS] has highest detection ability under the perfect RSS mechanism. Moreover, the comparison of [NRSS] chart with the existing competitive charts (such as [SRS] , [RSS] , [MRSS] , and charts) revealed the superiority of the [NRSS] chart, in terms of detecting shifts in process CV. The implementation of the CV charts on the CSTR dataset also indicated the superiority of [NRSS] chart, compared to other charts. The study can be extended for efficient detection of shifts in multivariate CV or by considering the auxiliary-based estimates of process CV, mentioned in Abbasi. 61 Moreover, to obtain further improvements for the detection of small or moderate shifts, EWMA and CUSUM-type structures can be the possible extensions to the [NRSS] chart.