A New Hybrid Exponentially Weighted Moving Average Control Chart with Repetitive Sampling for Monitoring the Coefficient of Variation

Abstract

The implementation of Statistical Quality Control (SQC) has been tracked in various areas, such as agriculture, environment, industry, and health services. The employment of SQC methodologies is frequently employed for monitoring and identification of process irregularities across various fields. This research proposes and implements a novel SQC methodology in agricultural areas. A control chart is one of the SQC tools that facilitates real-time monitoring of multiple activities, including agricultural yield, industrial yield, and hospital outcomes. Advanced control charts with symmetrical data are being subjected to the new SQC method, which is suitable for this purpose. This research aims to develop a novel hybrid exponentially weighted moving average control chart for detecting the coefficient of variation (CV) using a repetitive sampling method called the HEWMARS-CV control chart. It is an effective tool for monitoring the mean and variance of a process simultaneously. The HEWMARS-CV control chart used the repetitive sampling scheme to generate two pairs of control limits to enhance the performance of the control chart. The proposed control chart is compared with the classical HEWMA and Shewhart control charts regarding the average run length (ARL) when the data has a normal distribution. The Monte Carlo simulation method is utilized to approximate the ARL values of the proposed control charts to determine their performance. The proposed control chart detects small shifts in CV values more effectively than the existing control chart. An illustrative application related to monitor the wheat yield at Rothamsted Experimental Station in Great Britain is also incorporated to demonstrate the efficiency of the proposed control chart. The efficiency of the proposed HEWMARS-CV control chart on the real data shows that the proposed control chart can detect a shift in the CV of the process, and it is superior to the existing control chart in terms of the average run length.

1. Introduction

Generally, we use one control chart to monitor the process mean and another to monitor the variance. The Shewhart $\overline{X}$ and S control charts are used for quality control, where their nominal values must control the mean and standard deviation of the process. The basic assumption is that the nominal values are fixed constants, and there are many applications for which this assumption is reasonable. In various processes, the mean and standard deviation can vary simultaneously in unusual circumstances. Detecting a process out-of-control by controlling the mean and standard deviation based on a single control chart prevents the inflated rate of false alarms caused by using two control charts. In real-life situations, not all processes have constant means or standard deviations that allow control charts for the mean or the standard deviation to be utilized for process monitoring. Thus, it is reasonable to monitor the process mean and standard deviation simultaneously with a coefficient of the variation (CV) control chart.

The motivation for this paper comes from the importance of detecting variation in data using a CV. The CV is defined as the ratio of the population standard deviation to the mean, which can be regarded as a measure of stability or uncertainty and can also indicate the relative dispersion of data to the population mean. Therefore, developing a control chart that can quickly detect changes in the CV would be the most beneficial. Many researchers designed the control chart for CV monitoring, such as Kang et al. [1], who first proposed the CV problem in a control chart. They developed a Shewhart control chart for monitoring the CV using rational subgroups. The dominant part of this control chart is sensitive for large shifts but is not sensitive for moderate and small shifts. Later, Hong et al. [2] developed an exponentially weighted moving average (EWMA) control chart for monitoring moderate and small shifts in CV. Later, Castagliola et al. [3] proposed a two-sided EWMA for CV-squared charts. This control chart consists of upward and downward one-sided EWMA CV square charts to monitor increases and decreases in a CV, respectively. Furthermore, they found the proposed control chart was more effective for detecting changes in the CV than a classical EWMA CV.

Haq [4] proposed a new control chart using two EWMA statistics and called it a hybrid EWMA control chart (HEWMA). The HEWMA statistics consist of two EWMA statistics and two smoothing constants. He also shows that the HEWMA control chart is more sensitive to detect small changes in the mean than the classical EWMA control chart. Traditionally, the control charts usually incorporate single sampling techniques, while double, sequential, and repetitive sampling techniques are more efficient in the area of the sampling plan. The repetitive sampling technique was first introduced by Sherman [5] for the attribute acceptance sampling plan. A repetitive sampling scheme is employed when there needs to be more information from the initial sample to decide. Sherman [5] discovered this sampling method to be more efficient than the single sampling method in terms of the average run length (ARL). Balamurali and Jun [6] show that the variables of a repetitive acceptance sampling plan perform better than single and double acceptance sampling plans in terms of the average sample number. The repetitive sampling technique has become popular in the area of control charts. Ahmad et al. [7] first introduced repetitive sampling in the area of the control chart. They studied the $\overline{X}$ control chart using the process capability index based on the repetitive sampling technique. They found that the newly suggested sampling technique is better than the existing single sampling technique for detecting the mean shift of the process. Azam et al. [8] proposed the hybrid exponentially weighted moving average (HEWMA) control chart using repetitive sampling for monitoring the process mean. It is more effective to detect very small shifts than the HEWMA control chart. Recently, Aslam et al. [9] presented the HEWMACV control chart for monitoring the CV. The simulation study shows that the HEWMACV control chart has the ability to detect a shift in the manufacturing process. Muhammad [10] provides a repetitive sampling technique to create control charts utilizing EWMA and double exponentially weighted moving averages (DEWMA) control charts to detect process shifts based on a non-normal distribution. Later, Phanyaem [11] discussed the efficiency of this sampling technique with EWMA-sign and GWMA-sign control charts for monitoring the mean change in the process. Peh et al. [12] propose the double sampling control chart to monitor the CV of the process and compared it with the standard CV control chart.

Aslam et al. [13] discussed the effectiveness of repetitive sampling with the EWMA control chart for monitoring blood glucose in type-II diabetic patients. Other researchers apply the control chart, utilizing a repetitive sampling method to increase its performance. The Shewhart control chart was not as sensitive as this method. In order to monitor what the changes in the process mean, Azam et al. [8] utilized the repeating sampling method based on the HEWMA control chart to data from the industry. Industry adoption of the HEWMA control chart could lower the number of non-conforming products. Repetitive sampling for in silico data was used by Huang et al. [14] to illustrate the applicability of the generally weighted moving average control charts. Naveed et al. [15] employed the extended EWMA control chart with a repeating sampling method to monitor the process mean of the industrial data.

The research mentioned above discovered that by utilizing the coefficient of variation in measurements, it was possible to track the variation of the process by simultaneously monitoring its mean and standard deviation. Additionally, several studies use repeated sampling techniques instead of single sampling due to their higher efficiency than single sampling. Previous studies have demonstrated no work on designing the HEWMA control chart using repetitive sampling techniques for monitoring CV. Using repetitive sampling combined with the HEWMA control chart will improve the efficiency of detecting the process shift. For this reason, we present the design of the hybrid exponentially weighted moving average control chart based on a repetitive sampling scheme called the HEWMARS-CV control chart for detecting a shift in the CV. The proposed control chart utilizes current and previous observations to make decisions about the state of the control chart. In addition, we will compare the efficacy of this proposed HEWMARS-CV control chart to the classical HEWMA and Shewhart control charts. The design of the proposed control chart on real data shows that it can apply in the agricultural field.

The remainder of this paper is organized into the following sections: Section 2 presents the materials and methods. Section 3 lays out the design structure of the HEWMARS-CV control chart, and the performance evaluation measures are included in Section 4. Furthermore, Section 5 consists of the comparison and performance analysis of the proposed HEWMARS-CV control chart against some existing control charts. Section 6 offers a real-life data application to enhance the performance of the proposed HEWMARS-CV control chart. The final section addresses the concluding remarks.

2. Materials and Methods

This section presents a HEWMARS-CV control chart for monitoring the CV using repetitive sampling. The CV is commonly used to compare numerical distributions obtained on different scales and as a measure of precision for the data set dispersion. There are several practical applications, such as estimating product variability generated by manufacturing process quality control. This study’s proposed control chart is formulated based on the normality assumption. If the normality assumption is violated, the data used to create control charts should be subgroups. The central limit theorem is used to determine the robustness of the control chart when the data violates a normal distribution. This paper assumes that $X_{t1},X_{t2},\dots ,X_{tn}$ are the subgroups of size $n$ at time $t=1,2,\dots ,$ which follows the normal distribution with the population mean $\mu$ and variance ${\sigma }^2.$ There is independence within and between these subgroups.

The population CV is defined as:

$$\gamma =\frac{\sigma }{\mu }.$$

(1)

The sample mean and standard deviation of these subgroups can be calculated by:

$${\overline{x}}_t={\displaystyle \sum }_{j=1}^n{x}_{tj}/n.$$

(2)

$$s_t=\sqrt{\frac{{{\displaystyle \sum }}_{j=1}^n{(x_{tj}-{\overline{x}}_t)}^2}{n-1}.}$$

(3)

Hence, sample CV is denoted by $W_t.$

$$W_t={\frac{s_t}{\overline{x}}}_t.$$

(4)

Here, the mean and variance of $W_t$ are given by Hong et al. [2], as follows:

$$\left.E(W_t)=\gamma \left[1+\frac{1}{n}\right.\left({\gamma }^2-\frac{1}{4}\right)+\frac{1}{n^2}\left(3{\gamma }^4-\frac{{\gamma }^2}{4}-\frac{7}{32}\right)+\frac{1}{n^3}\left(15{\gamma }^6-\frac{3{\gamma }^4}{4}-\frac{7{\gamma }^2}{32}-\frac{19}{128}\right) \right],$$

(5)

$$\left. V(W_t)={\gamma }^2\left[\frac{1}{n}\right.\left({\gamma }^2+\frac{1}{2}\right)+\frac{1}{n^2}\left(8{\gamma }^4+{\gamma }^2+\frac{3}{8}\right)+\frac{1}{n^3}\left(69{\gamma }^6+\frac{7{\gamma }^4}{2}+\frac{3{\gamma }^2}{4}+\frac{3}{16}\right) \right],$$

(6)

where $\gamma$ is the population of CV.

Hong et al. [2] developed the CV control chart using the EWMA technique for detecting a small shift in the CV more than the CV Shewhart control chart. It is usually used to monitor and detect a small change in a process mean. The EWMA control chart has an advantage over the Shewhart chart since it allows for a more appropriate accumulation of prior information for the purpose of process monitoring.

The recursive equation of EWMA-CV statistics is denoted by $Z_t$, which can be calculated as follows:

$$Z_t=\left(1-\lambda \right)Z_{t-1}+\lambda W_t$$

(7)

where $W_t$ is the sequence of sample CV; $\lambda$ is an exponential smoothing parameter; $0\le \lambda \le 1$.

In the case of a process under in-control, the approximations for $E(Z_t)$ and $V(Z_t)$ are provided by Hong et al. [2], as follows:

$$E(Z_t)=E(W_t),$$

(8)

$$V(Z_t)=V(W_t)\left(\frac{\lambda }{2-\lambda }\right)\left[1-{\left(1-\lambda \right)}^{2t}\right].$$

(9)

The upper and lower control limits of the EWMA-CV control chart are given as follows:

$$UCL=E(W_t)+L\sqrt{V(W_t)\left(\frac{\lambda }{2-\lambda }\right)\left[1-{\left(1-\lambda \right)}^{2t}\right],}$$

(10)

$$LCL=E(W_t)-L\sqrt{V(W_t)\left(\frac{\lambda }{2-\lambda }\right)\left[1-{\left(1-\lambda \right)}^{2t}\right].}$$

(11)

Later, Haq [4] proposed a hybrid exponentially weighted moving average (HEWMA) control chart for monitoring the process mean. The advantage of the HEWMA control chart is that it can identify small shifts much more quickly than the traditional CUSUM and EWMA control charts. The superiority of the HEWMA control chart over the classical EWMA control chart is due to the weighting of the past data accumulated through the two exponential smoothing parameters. Suppose $X_1,X_2,\dots$ are independent random variables with a normal distribution, mean $\mu$ and variance ${\sigma }^2$. Here, ${\lambda }_1$ and ${\lambda }_2$ are the exponential smoothing parameters $(0<{\lambda }_1\le 1)$ and $\left(0<{\lambda }_2\le 1\right).$The EWMA statistics, represented by $E_t$.

$$E_t=\left(1-{\lambda }_2\right)E_{t-1}+{\lambda }_2{X}_t; t=1,2,\dots$$

(12)

The HEWMA statistics denoted by $H{E}_t$ are based on the statistics $E_t$ and ${\lambda }_1.$ Therefore, the HEWMA statistics can be written in this form

$$H{E}_t=\left(1-{\lambda }_1\right)H{E}_{t-1}+{\lambda }_1{E}_t; t=1,2,\dots$$

(13)

Suppose the mean of the HEWMA statistics is set to $\mu ={\mu }_0$ when the process is in-control; the mean of $H{E}_t$ statistics is given by:

$$E\left(H{E}_t\right)={\mu }_0.$$

(14)

The variance of HEWMA statistics $H{E}_t$ is given by: see Haq [4]

$$\begin{array}{cc}\hfill V\left(H{E}_t\right)& =Var\left[{\lambda }_1\left[\left(1-{\lambda }_2\right)E_{t-1}+{\lambda }_2{X}_t\right]+\left(1-{\lambda }_1\right)\left[\left(1-{\lambda }_1\right)H{E}_{t-2}+{\lambda }_1{E}_{t-1}\right]\right] \hfill \\ \hfill & =Var\left[{\lambda }_1{\lambda }_2{X}_t+{\lambda }_1^{2}H{E}_{t-2}-2{\lambda }_{1}H{E}_{t-2}+H{E}_{t-2}-{\lambda }_1^2{E}_{t-1}-{\lambda }_1{\lambda }_2{E}_{t-1}+2{\lambda }_1{E}_{t-1}\right] \hfill \\ \hfill & =\frac{{\lambda }_1^2{\lambda }_2{\sigma }^2}{\left(2-{\lambda }_2\right)}\left[\frac{1-{\left(1-{\lambda }_1\right)}^{2t}}{{\lambda }_1\left(2-{\lambda }_1\right)}-\frac{{\left(1-{\lambda }_1\right)}^2\left\{{\left(1-{\lambda }_2\right)}^{2t}-{\left(1-{\lambda }_1\right)}^{2t}\right\}}{{\left(1-{\lambda }_2\right)}^2-{\left(1-{\lambda }_1\right)}^2}\right].\hfill \end{array}$$

(15)

In the situation where $t$ is sufficiently large, the variance reduces to

$$V\left(H{E}_t\right)=\frac{{\lambda }_1{\lambda }_2{\sigma }^2}{\left(2-{\lambda }_2\right)\left(2-{\lambda }_1\right)}.$$

(16)

The upper and lower control limits of the HEWMA control chart are given by:

$$UC{L}_{\mathrm{HEWMA}}={\mu }_0+L\sigma \sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_2\right)\left(2-{\lambda }_1\right)}},$$

(17)

$$LC{L}_{\mathrm{HEWMA}}={\mu }_0-L\sigma \sqrt[]{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_2\right)\left(2-{\lambda }_1\right)},}$$

(18)

where L is the control limit coefficient to be determined for the HEWMA control chart.

Recently, Aslam et al. [13] presented the exponentially weighted moving average (EWMA) control chart using repetitive sampling for monitoring the process mean based on the normal distribution. Additionally, the EWMA control chart with repetitive sampling efficiently detects a relatively small process mean shift more than the classical EWMA control chart.

The EWMA statistics at each time t, are given by Aslam et al. [11]

$$EWM{A}_t=\left(1-\lambda \right)EWM{A}_{t-1}+\lambda X_t;t=1,2,\dots$$

(19)

According to Aslam et al. [13], the EWMA repetitive sampling control chart has two pairs of control limits. The outer and inner control limits coefficient are determined by the value $L_1$ and $L_2$, respectively. Establishing these two pairs of control limits is used for repetitive sampling to calculate statistics to detect process changes. The formula for calculating the two pairs of control limits based on the $EWM{A}_t$ statistics can also be calculated. The two outer control limits denoted by $LC{L}_{\mathrm{EWMA}1}$ and $UC{L}_{\mathrm{EWMA}1}$ are given as follows:

$$LC{L}_{\mathrm{EWMA}1}={\mu }_0-L_1\sigma \sqrt{\frac{\lambda }{\left(2-\lambda \right)},}$$

(20)

$$UC{L}_{\mathrm{EWMA}1}={\mu }_0+L_1\sigma \sqrt[]{\frac{\lambda }{\left(2-\lambda \right)}}.$$

(21)

The two inner control limits denoted by $LC{L}_{\mathrm{EWMA}2}$ and $UC{L}_{\mathrm{EWMA}2}$ are given as follows:

$$LC{L}_{\mathrm{EWMA}2}={\mu }_0-L_2\sigma \sqrt{\frac{\lambda }{\left(2-\lambda \right)}},$$

(22)

$$UC{L}_{\mathrm{EWMA}2}={\mu }_0+L_2\sigma \sqrt[]{\frac{\lambda }{\left(2-\lambda \right)},}$$

(23)

where $L_1$ and $L_2$ are the control limit coefficients; $L_1>L_2>0$ is to be determined for the EWMA control chart using repetitive sampling, which are customarily taken to correspond to the desired value of the in-control average run length (${\mathrm{ARL}}_0$).

3. The Design Structure of the New Hybrid EWMA Control Chart

This section introduces a new hybrid exponentially weighted moving average chart with a repetitive sampling technique for monitoring the CV, which is called HEWMARS-CV.

3.1. Repetitive Sampling Technique

Sherman [5] introduced the repetitive sampling technique in the field of the acceptance sampling plan; it is more efficient than the single sampling technique. The repetitive sampling scheme is used to construct the HEWMA control charts. The HEWMA control chart with repetitive sampling has two pairs of control limits. The two pairs of control limits include the inner control limits ($UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}$ and $LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}$) and the outer control limits ($UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}$ and $LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}$). The operational procedure of the repetitive sampling technique is working such that if the HEWMARS-CV statistic value falls within the inner control limit, we say that process is working in-control conditions. If the HEWMARS-CV statistic value falls outside the outer control limits, the process works in out-of-control conditions. If the HEWMARS-CV statistic value falls between the inner and outer control limit, the procedure is repeated until the out-of-control or in-control decision is reached. It will be clear from the proposed control chart’s working approach whether the process should be classified as an in-control state, a repeated sampling situation, or an out-of-control state. If the running process is established as being in-control, repeat the computation of the HEWMARS-CV statistics and the calculation of the HEWMARS-CV control limits by selecting the appropriate control limit coefficients ($L_1$ and $L_2$) using an iterative method such that the in-control $\mathrm{ARL}$ of the HEWMARS-CV control chart achieves the required value of ${\mathrm{ARL}}_0$. If the present process is a repetitive situation, count the number of repetitions. Otherwise, the number of subgroups and repetitions should be used to define the run length of the control chart.

3.2. The Proposed HEWMARS-CV Control Chart

The proposed control chart based on the HEMWARS-CV control chart with a repetitive sampling scheme has the following steps:

Step 1: Let $X_t;t=1,2,\dots$ be a quality characteristic of a process whose distribution is normal with mean $\mu$ and variance ${\sigma }^2$.

Step 2: Select a random sample of size$n$ from the current subgroup at time t and compute the sample CV; $W_t=s_t/{\overline{x}}_t$, where ${\overline{x}}_t$ is the sample mean and $s_t$ is the standard deviation.

Step 3: Calculate the HEWMARSCV statistics based on sample CV and the exponential smoothing parameters $(0<{\lambda }_1\le 1)$ and $\left(0<{\lambda }_2\le 1\right)$. Here, EWMARS-CV statistics are represented by $EWMARSC{V}_t$.

$$EWMARSC{V}_t=\left(1-{\lambda }_2\right)EWMARSC{V}_{t-1}+{\lambda }_2{W}_t;t=1,2,\dots$$

(24)

Therefore, the HEWMARS-CV statistics are represented by $HEWMARSC{V}_t$.

$$HEWMARSC{V}_t=\left(1-{\lambda }_1\right)HEWMARSC{V}_{t-1}+{\lambda }_{1}EWMARSC{V}_t;t=1,2,\dots$$

(25)

Step 4: Construct the two pairs of control limits (the outer and inner control limits) using the repetitive sampling technique. In sequential sampling, the process continues to select a sample until a final decision about the state of the process is made. In repetitive sampling, we repeat the process when uncertain of the initial sampling decision. For the indecision case, the process is repeated, and a new sample is selected to decide the process’s state.

The two outer control limits of the HEWMARS-CV control chart consist of the outer upper control limit and the outer lower control limit, denoted by $UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}$ and $LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}$, respectively,

$$UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}={\mu }_0+L_1\sigma \sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_2\right)\left(2-{\lambda }_1\right)},}$$

(26)

$$LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}={\mu }_0-L_1\sigma \sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_2\right)\left(2-{\lambda }_1\right)}.}$$

(27)

The two inner control limits of the HEWMARS-CV control chart consist of the inner upper control limit and the inner lower control limit, denoted by $UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}$ and $LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}$, respectively,

$$UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}={\mu }_0+L_2\sigma \sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_2\right)\left(2-{\lambda }_1\right)},}$$

(28)

$$LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}={\mu }_0-L_2\sigma \sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_2\right)\left(2-{\lambda }_1\right)},}$$

(29)

where $L_1$ and $L_2$ are the control limit coefficients; $L_1>L_2>0$ is to be determined for the HEWMARS-CV control chart. Monte Carlo simulations are conducted for a specified shift size; $\delta$ = 1.00 for a normal distribution to calculate ARL₀ by choosing different pairs of values setting ${\lambda }_1$ and${\lambda }_2$. The values of $L_1$were constant and the values of $L_2$ were decreased until ARL₀ was almost equal to the desired value.

Step 5: Declare the process out-of-control if the HEWMARS-CV statistics are located outside of the outer control limits, while in-control is declared if the HEWMARS-CV statistics are located inside the inner control limits. When the HEWMARS-CV statistics are located between the inner and outer control limits, the decision is postponed, and resampling is performed.

(i)

If $LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}\le HEWMARSC{V}_t\le UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}$, the process is in-control.

If $HEWMARSC{V}_t<LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}$ or $HEWMARSC{V}_t>UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1},$ the process is out-of-control.

(ii)

If $LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}\le HEWMARSC{V}_t\le LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}$ or $UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}\le HEWMARSC{V}_t\le UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}$, go to step 1.

3.3. Algorithmic Steps for HEWMARS-CV Control Chart

The algorithmic steps for the HEWMARS-CV control chart under a repetitive sampling scheme are given below:

• 3.3.1. Generate the random samples from a normal distribution with the parameter ${\gamma }_0$ = 0.08, 0.10, 0.15 and 0.20, respectively.
• 3.3.2. Calculate HEWMARS-CV statistics; $HEWMARSC{V}_{\mathrm{t}}$.
• 3.3.3. Calculate the mean and variance of the HEWMARS-CV statistics.
• 3.3.4. Calculate the pair’s control limits, which consist of the two outer control limits $(LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1},$ $UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}$) and the two inner control limits $(LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}, UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}$), where the values of $L_1$ were constant and the values of $L_2$ were decreased until ARL₀ was almost equal to the desired value.
• 3.3.5. The $HEWMARSC{V}_t$ statistics calculated in step 2 are compared with the pairs of control limits in step 4 until the first out-of-control value is found. The in-control run length is recorded.
• 3.3.6. In each situation, the in-control run length (RL) is recorded after repeating steps 1 through 10,000 times.
• 3.3.7. For the values of RL obtained in step 6, the ARL is calculated.
• 3.3.8. Steps 1 through 7 are repeated for various values of $L_1$ and $L_{2 }.$ The values of $L_{1 }$and $L_{2 }$are recorded when ${\mathrm{ARL}}_0$ = 370.
• 3.3.9. Repeat steps 1–7, using different values of shift size $\delta$ = 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09, 1.10, 1.20, 1.30, 1.40 and 1.50 in step 1 and the control limit coefficients $L_1$ and $L_{2 }$ values from step 8; thus, the ${\mathrm{ARL}}_1$ is calculated for the shifted process.

4. Average Run Length

The performance of the in-control process is denoted by ${\mathrm{ARL}}_0$, whereas the performance of the out-of-control process is designated by${\mathrm{ARL}}_1$. The process is defined as out-of-control if $HEWMARSC{V}_t\ge UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}$ or $HEWMARSC{V}_t\ge LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}$, so the probability that a process is considered out-of-control based on a single subgroup when it is actually in control can be determined as follows:

$$\begin{gathered} P_{out}^0=P\left\{HEWMARSC{V}_t<LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}\left|{\gamma }_0\right.\right\} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+P\left\{HEWMARSC{V}_t>UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}\left|{\gamma }_0\right.\right\}. \end{gathered}$$

(30)

The probability of repetition when the process is in-control is defined as follows:

$$\begin{gathered} P_{rep}^0=P\left\{UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}<HEWMARSC{V}_t<UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}\left|{\gamma }_0\right.\right\} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +P\left\{LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}<HEWMARSC{V}_t<LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}\left|{\gamma }_0\right.\right\}. \end{gathered}$$

(31)

Equation (31) can be rewritten as follows:

$$\begin{gathered} P_{rep}^0=\mathsf{\Phi }\left(L_1\right)-\mathsf{\Phi }\left(L_2\right)+\mathsf{\Phi }\left(-L_2\right)-\mathsf{\Phi }\left(-L_1\right) \\ =2\left(\Phi \left(L_1\right)-\Phi \left(L_2\right)\right). \end{gathered}$$

(32)

Consequently, the ${\mathrm{ARL}}_0$ and ${\mathrm{ARL}}_1$ of the control chart can be calculated, and the error probabilities are as follows:

The probability of a Type I error ($P_{out}^0$) is the probability of producing an alarm signal when there is no real change (false alarm),

$$P_{out}^0=\frac{2\left(1-\Phi \left(L_1\right)\right)}{1-2\left(\Phi \left(L_1\right)-\Phi \left(L_2\right)\right)}=\frac{P_{out}^0}{1-P_{rep}^0}.$$

(33)

The performance of the control chart is customarily taken to correspond to the desired value of the in-control average run length ${\mathrm{ARL}}_0$ and the out-of-control average run length ${\mathrm{ARL}}_1$. The in-control $\mathrm{ARL}$ is the number of subgroups that should be plotted until the process is considered out-of-control. When the process is in-control, the ${\mathrm{ARL}}_0$ is calculated as follows:

$${\mathrm{ARL}}_0=\frac{1}{P_{out}^0}.$$

(34)

Suppose the observations have a normal distribution with a mean $\mu$ and standard deviation $\sigma$, and $3\sigma$ limits are used. In that case, $P_{out}^0$ = 0.0027 is the probability that any observation exceeds the control limits of an in-control process, and ${\mathrm{ARL}}_0$ = (1/0.0027) = 370.

As noted previously, a process mean may change due to a number of uncontrollable circumstances. Assume now that the process has been shifted from ${\gamma }_0$ to ${\gamma }_1=\delta {\gamma }_0$, where $\delta$ is the shift size in the CV. We assume that ${\gamma }_0=$ 0.08, 0.10, 0.15 and 0.20, respectively. The probability that perhaps a process is judged out-of-control based on a single sample for a shifted process, denoted by$P_{out}^1$, is determined as follows:

$$\begin{gathered} P_{out}^1=P\left\{HEWMARSC{V}_t<LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}|{\gamma }_1\right\} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +P\left\{HEWMARSC{V}_t>UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}|{\gamma }_1\right\}. \end{gathered}$$

(35)

Equation (35) is rewritten as follows:

$$P_{out}^1=\mathsf{\Phi }\left(-L_1+\frac{\delta }{\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}}\right)+1-\mathsf{\Phi }\left(-L_1+\frac{\delta }{\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}}\right).$$

(36)

Consequently, the probability of repeats for a shifted process is given by

$$\begin{gathered} P_{out}^1=P\left\{UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}|{\gamma }_1<HEWMARSC{V}_t\left.〈UC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}\right|{\gamma }_1\right\} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left\{LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}1}|{\gamma }_1<HEWMARSC{V}_t<LC{L}_{\mathrm{HEWMARS}\text{-}\mathrm{CV}2}|{\gamma }_1\right\}. \end{gathered}$$

(37)

Then, Equation (37) can be written as follows:

$$\begin{gathered} P_{rep}^1=\mathsf{\Phi }\left(\frac{L_1\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}+\delta }{\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}}\right)-\mathsf{\Phi }\left(\frac{L_2\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}+\delta }{\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} + \mathsf{\Phi }\left(\frac{-L_2\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}+\delta }{\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}}\right)-\mathsf{\Phi }\left(\frac{L_1\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}+\delta }{\sqrt{\frac{{\lambda }_1{\lambda }_2}{\left(2-{\lambda }_1\right)\left(2-{\lambda }_2\right)}}}\right). \end{gathered}$$

(38)

The probability of type II error ($P_{out}^1$) is the probability of having no signal when there is a real change (false negative).

$$P_{out}^1=\frac{P_{out}^1}{1-P_{rep}^1}.$$

(39)

Therefore, the $\mathrm{ARL}$ for the shifted process that is out of control, represented by ${\mathrm{ARL}}_1$, is given by

$${\mathrm{ARL}}_1=\frac{1}{P_{out}^1}.$$

(40)

Based on Equations (34) and (40) present the in-control $\mathrm{ARL}$ and out-of-control $\mathrm{ARL}$performance characteristics of the control charts.

The selection of parameters ${\lambda }_1, {\lambda }_2$ and $L_1, L_2$ are the two design characteristics of the proposed chart that have a direct impact on the performance of the control chart. Typically, the selection of ${\lambda }_1, {\lambda }_2$ and $L_1, L_2$ depends on two steps: Using a search technique, we identify the combinations of ${\lambda }_1, {\lambda }_2$ and $L_1, L_2$ that provide the nominal ${\mathrm{ARL}}_0$. In the second stage, the (${\lambda }_1, {\lambda }_2$ and $L_1, L_2$) combinations with the smallest ${\mathrm{ARL}}_1$ for the shift ($\delta$) to be detected were chosen. We typically picked ${\lambda }_1, {\lambda }_2$ and then chose $L_1, L_2$ from these two-stage design parameters.

5. Research Results

In this section, we compare the $\mathrm{ARL}$ obtained from the HEWMARS-CV, HEWMA-CV, and Shewhart-CV control charts. The comparative performance of the proposed control charts was conducted, assuming the data were of normal distribution. The data in the study were simulated using the Monte Carlo simulation technique. The Monte Carlo simulation technique calculates the ARL₀ and ARL₁ of the HEWMARS-CV, HEWMA-CV, and Shewhart-CV control charts via the R programming language. There are 10,000 iterations in the trial cycle. We design the exponential smoothing parameter: ${\lambda }_1$ = 0.10, 0.30 and ${\lambda }_2$ = 0.25, 0.50, 0.75, respectively. The assigned $\gamma$, representing the CV equals 0.08, 0.10, 0.15, and 0.20. Set the sample size $n$ to equal 10 and 20. Let $\delta$ be the shift size in the CV; $\delta$ = 1.00 determines the in-control parameter, while $\delta$ = 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09, 1.10, 1.20, 1.30, 1.40, and 1.50 identify the out-of-control parameters. The criterion for comparing the performance of these control charts is the ARL values. In a situation where the process is in-control, determine ${\mathrm{ARL}}_0$ = 370. When the process is out-of-control, we consider the control chart with the lowest ${\mathrm{ARL}}_1$ value, meaning this control chart is the most effective at detecting changes in CV values.

From

Table 1. Comparison of ARL between HEWMARS-CV, HEWMA-CV, and Shewhart Control Charts given $\gamma$ = 0.08 and ${\lambda }_1$ = 0.10.

Sample Size $\mathit{n}$	Shift Size $\mathsf{\delta }$		HEWMARS-CV				HEWMA-CV				Shewhart-CV
		${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75	${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75
		L1, L2	3.720, 3.610	2.480, 2.380	1.864, 1.856	L	3.710	2.475	1.863	L	2.250
10	1.00		370.994	369.019	370.302		370.991	370.177	370.556		370.312
	1.01		349.203	348.099	349.962		351.132	352.090	352.563		363.059
	1.02		319.118	319.004	321.177		320.636	323.997	325.174		357.059
	1.03		279.586	280.171	284.137		284.292	285.769	286.929		344.108
	1.04		236.516	236.193	241.173		238.384	244.117	246.208		334.108
	1.05		191.323	191.974	198.990		196.195	202.736	203.088		322.072
	1.06		151.740	153.921	162.192		157.733	163.493	165.504		308.733
	1.07		111.450	120.433	130.226		125.891	131.115	131.841		293.250
	1.08		95.342	97.053	105.904		101.038	106.680	107.446		277.034
	1.09		78.854	79.613	87.013		85.861	88.841	89.597		258.553
	1.10		66.110	66.427	73.602		72.958	75.735	75.626		240.411
	1.20		28.157	26.241	29.950		32.373	31.361	31.213		82.531
	1.30		20.193	18.233	19.963		23.069	21.534	21.248		27.921
	1.40		16.718	14.499	15.470		16.732	17.067	16.648		12.193
	1.50		14.524	12.210	12.716		14.560	14.349	13.868		6.175
		${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75	${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75
		L1, L2	4.540, 4.525	3.013, 2.980	2.261, 2.252	L	4.538	3.010	2.260	L	3.010
20	1.00		369.917	370.134	369.228		370.380	370.416	370.183		370.559
	1.01		335.234	335.390	337.395		337.579	337.221	341.432		361.402
	1.02		281.225	286.748	287.063		284.771	289.990	291.113		350.244
	1.03		218.747	224.371	229.707		223.970	227.949	230.909		334.545
	1.04		159.266	166.920	169.223		164.240	169.834	174.310		318.723
	1.05		115.123	121.949	124.246		119.990	125.432	126.821		300.025
	1.06		89.220	90.840	94.002		91.570	94.550	96.280		279.018
	1.07		71.246	71.650	74.330		72.296	75.853	76.404		253.826
	1.08		59.205	59.207	61.097		61.066	62.245	62.514		229.839
	1.09		50.533	50.764	51.850		53.201	53.337	54.210		201.499
	1.10		45.592	44.450	45.736		47.346	47.359	47.461		178.254
	1.20		24.800	22.699	23.015		26.211	24.673	24.413		36.586
	1.30		19.104	16.765	16.674		20.309	18.289	17.908		10.604
	1.40		16.024	13.646	13.348		17.161	14.991	14.480		4.097
	1.50		13.999	11.591	11.198		15.056	12.871	12.309		1.952

,

Table 2. Comparison of ARL between HEWMARS-CV, HEWMA-CV, and Shewhart Control Charts given $\gamma$ = 0.08 and ${\lambda }_1$ = 0.30.

Sample Size $\mathit{n}$	Shift Size $\mathsf{\delta }$		HEWMARS-CV				HEWMA-CV				Shewhart-CV
		${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75	${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75
		L1, L2	2.182, 2.176	1.520, 1.380	1.162, 1.145	L	2.181	1.510	1.161	L	2.250
10	1.00		370.363	370.231	369.983		370.317	370.220	370.164		370.312
	1.01		352.514	354.319	355.298		354.914	357.861	357.116		363.059
	1.02		330.883	335.231	337.604		332.620	338.231	339.202		357.059
	1.03		300.883	310.835	314.751		301.988	314.669	317.548		344.108
	1.04		263.935	279.657	287.332		269.491	285.713	288.795		334.108
	1.05		227.352	245.563	254.964		229.351	252.176	258.841		322.072
	1.06		189.591	213.415	222.439		193.256	217.097	226.381		308.733
	1.07		155.338	174.930	188.732		157.704	182.044	191.976		293.250
	1.08		122.831	143.953	152.690		126.573	151.933	160.047		277.034
	1.09		99.167	115.200	130.988		102.607	124.754	133.145		258.553
	1.10		81.126	94.765	107.595		82.685	101.189	109.093		240.411
	1.20		22.126	19.356	23.395		23.371	23.752	25.029		82.531
	1.30		12.802	9.356	10.688		13.850	12.123	12.120		27.921
	1.40		9.408	6.250	6.782		10.509	8.385	8.057		12.193
	1.50		7.675	4.816	4.889		8.753	6.601	6.093		6.175
		${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75	${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75
		L1, L2	2.614, 2.599	1.780, 1.735	1.359, 1.328	L	2.615	1.779	1.357	L	3.010
20	1.00		370.273	370.037	370.537		370.280	370.220	370.182		370.559
	1.01		343.600	346.823	350.590		344.693	350.905	351.129		361.402
	1.02		305.053	315.028	322.309		307.042	319.436	323.073		350.244
	1.03		256.331	274.119	281.928		258.696	275.761	283.931		334.545
	1.04		200.059	222.608	235.847		203.737	231.874	239.278		318.723
	1.05		148.417	173.169	189.600		151.400	180.052	191.684		300.025
	1.06		108.814	130.294	146.280		111.570	153.955	148.480		279.018
	1.07		79.853	97.215	111.121		82.554	100.780	109.926		253.826
	1.08		61.347	71.228	81.451		62.870	76.306	83.446		229.839
	1.09		48.376	55.093	62.865		50.000	59.191	64.716		201.499
	1.10		39.601	43.340	48.965		41.197	47.034	50.886		178.254
	1.20		13.392	11.057	11.473		15.265	13.376	13.330		36.586
	1.30		9.483	6.539	6.265		10.686	8.237	7.799		10.604
	1.40		7.631	4.932	4.347		8.765	6.300	5.799		4.097
	1.50		6.501	3.984	3.349		7.583	5.300	4.600		1.952

,

Table 3. Comparison of ARL between HEWMARS-CV, HEWMA-CV and Shewhart Control Charts given $\gamma$ = 0.10 and ${\lambda }_1$ = 0.10.

Sample Size $\mathit{n}$	Shift Size $\mathsf{\delta }$		HEWMARS-CV				HEWMA-CV				Shewhart-CV
		${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75	${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75
		L1, L2	4.135, 4.128	2.761, 2.755	2.083, 2.005	L	4.135	2.761	2.083	L	2.235
10	1.00		370.278	370.464	370.165		370.063	370.200	370.447		370.723
	1.01		348.824	347.351	350.407		350.246	350.083	353.997		364.989
	1.02		310.543	320.099	323.986		319.047	319.591	325.724		350.007
	1.03		283.373	285.794	286.788		283.706	283.867	289.888		349.744
	1.04		236.498	240.937	245.799		242.375	243.282	252.391		337.979
	1.05		193.459	200.707	200.566		197.920	199.265	210.185		329.297
	1.06		153.477	159.972	161.105		156.970	162.178	169.872		301.211
	1.07		125.808	128.045	127.575		125.848	131.082	137.465		293.936
	1.08		101.101	103.692	102.882		103.256	105.889	111.346		271.031
	1.09		82.626	86.627	83.911		85.805	87.224	91.431		268.908
	1.10		70.929	73.016	69.760		72.285	74.998	78.038		241.279
	1.20		30.747	29.994	27.191		31.821	31.426	31.289		84.285
	1.30		21.732	20.239	18.047		22.837	21.371	21.146		28.330
	1.40		17.608	15.819	14.150		18.668	16.916	16.529		11.728
	1.50		15.103	13.163	11.714		16.132	14.188	13.764		6.407
		${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75	${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75
		L1, L2	5.071, 5.054	3.368, 3.355	2.527, 2.515	L	5.071	3.368	2.527	L	2.990
20	1.00		370.197	370.845	370.167		370.560	370.698	370.441		370.870
	1.01		336.136	343.096	337.367		340.285	341.867	338.217		362.121
	1.02		289.520	293.128	289.735		289.818	292.878	289.679		347.938
	1.03		221.018	229.114	230.158		226.589	234.234	232.322		335.171
	1.04		159.216	172.399	172.955		172.190	175.273	172.235		317.302
	1.05		116.324	126.402	125.512		116.823	128.533	127.572		300.914
	1.06		89.724	95.107	95.082		90.220	96.813	96.332		278.986
	1.07		71.078	75.467	74.596		71.780	76.400	76.341		254.704
	1.08		60.759	61.493	61.460		61.224	63.665	63.073		230.906
	1.09		50.761	52.491	52.725		53.138	53.852	54.085		202.941
	1.10		45.432	45.940	45.886		47.095	47.272	47.333		179.432
	1.20		24.724	23.234	22.853		25.988	24.612	24.318		38.258
	1.30		18.963	17.019	16.513		20.115	18.182	17.683		10.524
	1.40		15.882	13.763	13.194		16.986	14.863	14.337		4.194
	1.50		13.847	11.634	11.037		14.929	12.715	12.148		1.994

,

Table 4. Comparison of ARL between HEWMARS-CV, HEWMA-CV, and Shewhart Control Charts given $\gamma$ = 0.10 and ${\lambda }_1$ = 0.30.

Sample Size n	Shift Size $\mathsf{\delta }$		HEWMARS-CV				HEWMA-CV				Shewhart-CV
		${\mathsf{\lambda }}_1, {\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75	${\mathsf{\lambda }}_1, {\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75
		L1, L2	2.439, 2.430	1.685,1.675	1.299, 1.290	L	2.439	1.685	1.299	L	2.235
10	1.00		369.648	369.256	370.715		370.593	370.623	370.731		370.723
	1.01		353.636	352.458	355.372		355.554	354.739	355.449		364.989
	1.02		331.934	332.597	333.755		331.797	333.286	335.516		350.007
	1.03		305.946	306.991	312.821		305.010	310.372	313.069		349.744
	1.04		270.623	278.384	284.826		271.365	277.306	285.781		337.979
	1.05		233.991	244.647	254.175		232.533	247.734	253.182		329.297
	1.06		192.880	211.190	219.484		195.327	210.317	219.095		301.211
	1.07		157.989	173.904	186.266		159.377	177.044	187.690		293.936
	1.08		127.449	146.665	157.316		127.221	145.067	156.370		271.031
	1.09		100.216	116.932	129.884		103.301	117.278	130.089		268.908
	1.10		83.675	96.604	105.434		85.294	96.568	106.599		241.279
	1.20		22.427	22.049	23.369		23.414	23.101	24.445		84.285
	1.30		12.753	10.864	10.835		13.852	12.088	11.962		28.330
	1.40		9.370	7.146	6.910		10.426	8.336	7.964		11.728
	1.50		7.634	5.419	4.949		8.693	6.536	6.027		6.407
		${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75	${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75
		L1, L2	2.925, 2.915	1.995, 1.990	1.523, 1.518	L	2.925	1.995	1.523	L	2.990
20	1.00		370.812	370.917	370.209		370.983	370.984	370.755		370.870
	1.01		347.666	353.638	351.636		347.386	353.316	352.912		362.121
	1.02		308.833	320.910	323.733		309.324	321.929	326.688		347.938
	1.03		259.886	281.868	289.519		261.934	282.110	286.469		335.171
	1.04		205.754	231.844	242.173		206.590	237.014	244.398		317.302
	1.05		153.682	183.443	196.377		153.401	184.270	195.778		300.914
	1.06		111.950	138.644	150.663		112.136	140.725	153.579		278.986
	1.07		82.363	104.081	115.662		82.712	105.578	114.768		254.704
	1.08		62.171	77.229	86.938		63.450	78.581	87.542		230.906
	1.09		49.560	60.524	65.853		50.492	60.911	67.833		202.941
	1.10		40.607	47.215	51.674		41.446	48.584	53.402		179.432
	1.20		14.001	12.248	12.355		15.206	13.331	13.467		38.258
	1.30		9.551	7.137	6.710		10.656	8.215	7.779		10.524
	1.40		7.630	5.248	4.625		8.692	6.281	5.713		4.194
	1.50		6.504	4.208	3.537		7.534	5.242	4.611		1.994

,

Table 5. Comparison of ARL between HEWMARS-CV, HEWMA-CV, and Shewhart Control Charts given $\gamma$ = 0.15 and ${\lambda }_1=0.10$.

Sample Size $\mathit{n}$	Shift Size $\mathsf{\delta }$		HEWMARS-CV				HEWMA-CV				Shewhart-CV
		${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75	${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75
		L1, L2	4.836, 4.815	3.232, 3.100	2.436, 2.414	L	4.836	3.232	2.436	L	2.205
10	1.00		370.046	370.545	370.153		370.162	370.350	370.847		370.713
	1.01		348.798	343.668	350.031		349.685	353.057	348.925		363.408
	1.02		319.560	320.606	323.024		319.839	320.916	321.470		357.367
	1.03		280.270	267.917	284.861		284.445	282.813	286.855		46.168
	1.04		236.534	226.970	246.092		241.236	224.561	245.293		332.012
	1.05		195.943	191.432	204.007		192.118	197.464	202.769		322.639
	1.06		156.444	149.176	164.483		157.466	162.834	165.453		309.205
	1.07		123.021	113.573	131.744		126.626	136.697	132.266		294.718
	1.08		101.456	99.414	107.056		102.908	103.085	108.633		277.690
	1.09		84.886	77.493	88.320		85.108	89.470	89.930		262.326
	1.10		71.666	74.213	74.349		72.746	75.308	76.490		243.924
	1.20		30.281	29.923	29.396		31.786	30.912	30.824		85.366
	1.30		21.340	20.049	19.314		22.496	21.082	20.782		29.431
	1.40		17.176	15.540	14.783		18.346	16.542	16.153		13.003
	1.50		14.687	12.858	12.148		15.792	13.858	13.366		6.750
		${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75	${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75
		L1, L2	5.970, 5.869	3.961, 3.960	2.975, 2.965	L	5.970	3.961	2.975	L	2.975
20	1.00		370.570	370.450	370.662		370.267	370.479	370.083		370.898
	1.01		337.676	331.952	339.109		343.264	337.476	339.767		359.399
	1.02		286.948	290.236	288.891		291.885	290.350	290.265		351.302
	1.03		223.539	228.588	231.633		232.813	238.003	231.110		337.361
	1.04		164.149	175.956	173.814		172.635	175.987	174.802		321.599
	1.05		118.005	126.497	126.648		125.258	131.618	129.807		303.489
	1.06		88.724	93.198	95.945		94.793	95.837	97.743		280.582
	1.07		69.235	76.151	75.598		75.616	76.229	77.291		260.188
	1.08		57.071	60.423	61.975		63.098	61.554	64.155		232.593
	1.09		48.505	52.166	52.375		53.741	53.127	53.822		207.865
	1.10		42.911	46.203	46.017		47.827	47.201	47.617		182.083
	1.20		23.282	23.201	22.701		25.740	24.176	23.921		39.074
	1.30		18.098	16.818	16.231		19.807	17.823	17.386		11.291
	1.40		15.134	13.507	12.925		16.654	14.511	13.990		4.485
	1.50		13.203	11.397	10.745		14.617	12.395	11.845		2.110

,

Table 6. Comparison of ARL between HEWMARS-CV, HEWMA-CV, and Shewhart Control Charts given $\gamma$ = 0.15 and ${\lambda }_1=0.30$.

Sample Size $\mathit{n}$	Shift Size $\mathsf{\delta }$		HEWMARS-CV				HEWMA-CV				Shewhart-CV
		${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75	${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75
		L1, L2	2.865, 2.850	1.990, 1.985	1.540, 1.535	L	2.865	1.990	1.540	L	2.205
10	1.00		370.386	370.348	370.739		369.830	369.791	370.002		370.713
	1.01		355.792	356.090	357.528		354.528	356.020	360.141		363.408
	1.02		332.336	338.257	341.108		334.573	337.650	342.545		357.367
	1.03		303.942	313.520	319.727		305.077	312.645	320.574		346.168
	1.04		269.631	286.346	294.800		270.487	286.861	294.578		332.012
	1.05		232.259	253.710	265.217		231.954	252.959	266.166		322.639
	1.06		195.681	217.215	232.456		197.013	219.712	231.389		309.205
	1.07		160.039	184.599	199.570		161.697	184.066	202.138		294.718
	1.08		129.527	150.981	166.101		129.012	151.575	169.833		277.690
	1.09		103.810	126.228	139.785		103.810	127.694	138.965		262.326
	1.10		84.504	101.139	115.614		86.463	104.953	115.540		243.924
	1.20		22.402	23.074	25.120		23.723	24.101	26.159		85.366
	1.30		12.593	11.040	11.308		13.660	12.238	12.308		29.431
	1.40		9.246	7.282	7.073		10.290	8.309	8.058		13.003
	1.50		7.469	5.459	5.039		8.562	6.528	6.074		6.750
		${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75	${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75
		L1, L2	3.450, 3.445	2.359, 2.350	1.805, 1.795	L	3.450	2.359	1.805	L	2.975
20	1.00		370.161	370.487	369.534		370.814	370.903	370.870		370.898
	1.01		345.069	352.521	351.638		346.010	351.545	350.742		359.399
	1.02		310.309	322.531	326.772		305.664	321.163	325.401		351.302
	1.03		259.106	278.847	289.351		255.873	279.602	291.561		337.361
	1.04		205.310	230.498	244.575		204.371	234.185	245.392		321.599
	1.05		153.714	185.322	199.188		153.916	184.319	199.518		303.489
	1.06		113.636	140.684	153.304		114.174	140.584	153.948		280.582
	1.07		84.657	106.091	116.334		83.846	105.556	119.239		260.188
	1.08		63.672	78.576	87.823		65.148	79.851	88.689		232.593
	1.09		50.176	60.390	67.639		51.483	62.115	69.197		207.865
	1.10		40.930	46.677	53.278		42.073	49.226	53.518		182.083
	1.20		14.104	12.305	12.469		15.181	13.558	13.677		39.074
	1.30		9.514	7.054	6.647		10.508	8.190	7.729		11.291
	1.40		7.548	5.117	4.538		8.558	6.200	5.637		4.485
	1.50		6.387	4.105	3.448		7.398	5.137	4.525		2.110

,

Table 7. Comparison of ARL between HEWMARS-CV, HEWMA-CV, and Shewhart Control Charts given $\gamma$ = 0.20 and ${\lambda }_1=0.10$.

Sample Size $\mathit{n}$	Shift Size $\mathsf{\delta }$		HEWMARS-CV				HEWMA-CV				Shewhart-CV
		${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75	${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75
		L1, L2	5.331, 5.315	3.572, 3.549	2.695, 2.685	L	5.313	3.549	2.695	L	2.193
10	1.00		370.545	370.237	370.191		370.907	370.839	370.136		370.364
	1.01		351.120	353.490	351.682		351.155	352.527	352.970		364.558
	1.02		320.144	323.914	324.892		320.813	326.774	327.014		354.944
	1.03		281.777	289.968	290.934		284.239	291.548	292.628		345.996
	1.04		241.023	249.953	252.208		243.290	250.577	251.541		336.386
	1.05		196.591	209.215	208.220		198.497	208.637	209.267		324.123
	1.06		158.713	166.244	171.646		160.275	168.367	172.977		309.150
	1.07		126.770	134.381	137.548		127.222	135.553	137.828		294.483
	1.08		103.745	111.001	112.472		104.772	112.216	112.306		279.421
	1.09		86.291	90.121	91.052		86.988	92.007	93.851		265.427
	1.10		72.282	76.427	77.627		74.670	77.955	78.601		242.006
	1.20		30.399	29.533	29.837		31.614	31.092	30.768		90.228
	1.30		21.045	19.543	19.451		22.283	20.864	20.567		30.948
	1.40		16.883	15.058	14.720		17.980	16.321	15.912		13.989
	1.50		14.396	12.399	12.009		15.466	13.554	13.108		7.147
		${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75	${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.10, 0.25	0.10, 0.50	0.10, 0.75
		L1, L2	6.631, 6.610	4.412, 4.409	3.315, 3.310	L	6.631	4.412	3.315	L	2.993
20	1.00		370.450	370.988	370.149		370.445	370.838	370.518		370.320
	1.01		338.016	340.797	341.538		338.446	342.438	341.757		361.470
	1.02		287.926	293.949	295.018		289.641	294.954	295.341		349.509
	1.03		225.019	237.311	237.201		225.536	237.820	237.415		335.222
	1.04		169.383	178.566	176.123		169.387	179.550	179.566		319.512
	1.05		121.888	129.676	131.417		122.824	131.112	132.322		297.868
	1.06		92.827	98.708	99.941		93.875	99.761	100.188		281.957
	1.07		17.375	77.999	77.828		74.489	78.051	78.584		259.506
	1.08		60.975	63.536	63.924		62.288	63.979	64.934		234.765
	1.09		52.260	53.559	54.267		53.412	55.077	55.047		209.703
	1.10		45.546	46.926	46.648		47.112	47.687	47.664		185.877
	1.20		24.117	22.911	22.643		25.409	23.995	23.787		40.611
	1.30		18.285	16.493	16.064		19.362	17.505	17.133		12.044
	1.40		15.188	13.193	12.616		16.280	14.148	13.658		4.828
	1.50		13.152	11.063	10.496		14.248	12.078	11.533		2.310

and

Table 8. Comparison of ARL between HEWMARS-CV, HEWMA-CV, and Shewhart Control Charts given $\gamma$ = 0.20 and ${\lambda }_1=0.30$.

Sample Size $\mathit{n}$	Shift Size $\mathsf{\delta }$		HEWMARS-CV				HEWMA-CV				Shewhart-CV
		${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75	${\mathsf{\lambda }}_1$$, {\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75
		L1, L2	3.173, 3.150	2.216, 2.178	1.717, 1.651	L	3.174	2.217	1.718	L	2.193
10	1.00		369.888	370.106	369.844		370.081	370.328	370.542		370.364
	1.01		354.024	357.120	354.613		354.065	357.693	357.696		364.558
	1.02		331.060	338.109	338.460		332.974	338.192	340.402		354.944
	1.03		301.690	314.014	317.075		302.577	315.615	318.377		345.996
	1.04		269.392	285.313	291.268		271.081	286.036	293.641		336.386
	1.05		232.577	254.243	262.540		232.854	256.705	264.752		324.123
	1.06		193.637	220.564	229.468		195.599	223.257	230.882		309.150
	1.07		160.236	187.136	195.996		163.451	188.461	200.638		294.483
	1.08		128.366	156.156	164.734		131.679	158.972	170.754		279.421
	1.09		104.742	128.579	137.372		105.330	130.209	141.365		265.427
	1.10		85.590	104.907	114.052		86.690	107.103	117.122		242.006
	1.20		22.652	23.313	23.826		23.793	25.010	26.577		90.228
	1.30		12.572	10.925	10.683		13.822	12.430	12.471		30.948
	1.40		9.116	7.051	6.449		10.267	8.375	8.105		13.989
	1.50		7.294	5.257	4.564		8.467	6.491	6.032		7.147
		${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75	${\mathsf{\lambda }}_1$, ${\mathsf{\lambda }}_2$	0.30, 0.25	0.30, 0.50	0.30, 0.75
		L1, L2	3.853, 3.760	2.643, 2.560	2.024, 1.998	L	3.854	2.642	2.025	L	2.993
20	1.00		369.120	370.126	370.463		370.667	370.652	370.998		370.320
	1.01		342.961	349.165	349.617		347.327	351.445	351.120		361.470
	1.02		308.384	320.179	322.054		312.222	321.054	324.679		349.509
	1.03		258.112	282.322	286.754		263.606	284.962	288.170		335.222
	1.04		205.205	231.788	243.416		211.288	234.969	244.768		319.512
	1.05		153.314	186.083	197.054		160.725	188.138	200.732		297.868
	1.06		111.566	140.980	152.938		118.708	144.740	157.965		281.957
	1.07		81.791	104.430	116.809		87.839	109.048	119.888		259.506
	1.08		62.282	78.920	88.256		66.596	82.266	91.243		234.765
	1.09		48.416	60.693	69.160		53.240	63.168	70.891		209.703
	1.10		38.866	47.050	53.894		43.036	50.512	55.547		185.877
	1.20		12.776	11.408	12.338		15.296	13.608	13.889		40.611
	1.30		8.724	6.246	6.503		10.424	8.153	7.780		12.044
	1.40		6.988	4.638	4.419		8.433	6.139	5.602		4.828
	1.50		5.941	3.775	3.323		7.260	5.094	4.444		2.310

, the process is in the in-control state ($\gamma ={\gamma }_0$); the process is in the out-of-control state $\left(\gamma ={\gamma }_1=\delta {\gamma }_0\right),$ where $\delta$ is the shift size, and $\delta$ = 1.00, 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09, 1.10, 1.20, 1.30, 1.40, and 1.50, respectively. Therefore, the process is in-control, meaning $\delta$ = 1.00. For this research, the shift sizes parameters were categorized by a small shift size ($1.01\le \delta \le 1.20$) and a large shift size $(1.30\le \delta \le 1.50$). The in-control $\mathrm{ARL}$ was set as ${\mathrm{ARL}}_0$ = 370 with the parameter ${\gamma }_0$ set as 0.08, 0.10, 0.15 and 0.20, respectively. The sample sizes $\left(n=10 \mathrm{and} 20\right),$the exponential smoothing parameters $\left({\lambda }_1=0.10, 0.30\right)$ and $\left({\lambda }_2=0.25, 0.30, 0.75\right)$, and the control limit coefficients $L_1$ were constant, and the values of $L_2$ were decreased until ARL₀ is almost equal to the desired value.

Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 present the behavior of results from the HEWMARS-CV, HEWMA-CV, and Shewhart-CV control charts, which can be summarized as follows:

• In the case of a process that is in-control, the $\mathrm{ARL}$value obtained is very close to the desired ${\mathrm{ARL}}_0$ values.
• In cases where the magnitude of the shift is small$\left(1.01\le \delta \le 1.20\right)$, the HEWMARS-CV and HEWMA-CV control charts provide a smaller ${\mathrm{ARL}}_1$ than the Shewhart control chart. In addition, it was found that the ${\mathrm{ARL}}_1$ of the proposed HEWMARS-CV control chart is slightly less than the HEWMA-CV control chart. This result shows that the HEWMARS-CV control chart can be an efficient alternative to the HEWMA-CV control chart. In cases where the magnitude of the shift was large $\left(1.30\le \delta \le 1.50\right),$ the Shewhart control chart provides an ${\mathrm{ARL}}_1$ that is smaller than the HEWMARS-CV and the HEWMA-CV control charts. These results demonstrate that the HEWMARS-CV and HEWMA-CV control charts agree better for detecting a small shift in the CV than the Shewhart control chart. On the other hand, the Shewhart control chart was more effective at detecting large shifts in CV values faster than the HEWMARS-CV and HEWMA-CV control charts.
• In the out-of-control cases where the shift sizes $\delta >1.00,$ the values of ${\mathrm{ARL}}_1$ from their control chart decrease rapidly as the shift size increases.
• The control chart was more sensitive to detecting changes with a large sample size $\left(n=20\right)$ than with a small sample size $\left(n=10\right)$.
• The selection of the exponential smoothing parameter for the HEWMARS-CV control chart, in the case of a small shift in CV value, defining pairs of exponential smoothing parameters (${\lambda }_1$, ${\lambda }_2$) as (0.10, 0.25), (0.10, 0.50), (0.30, 0.25), and (0.30, 0.50), were more sensitive to detecting changes than (0.10, 0.75) and (0.30, 0.75). Whereas, in the case of a large shift size in CV, a pair of exponential smoothing parameters (${\lambda }_1$, ${\lambda }_2$) with values of (0.10, 0.75) and (0.30, 0.75) have better sensitivity in detecting changes.

6. Applications

This section presents a real data set representing the wheat yield data from the experiment at Rothamsted Experimental Station in Great Britain, which is part of the Institute of Arable Crops Research, Mercer, and Hall [16]. We used the Anderson–Darling (AD) test to test model fitting for the wheat yield data set. The result of this analysis is shown in Figure 9, which shows the AD statistics on a normal probability plot (PP). The p-value for the AD statistical test is 0.178, according to the findings. As can be observed in Figure 9, the p-value is higher than 0.05, indicating that the null hypothesis is not rejected and that the wheat yield data came from a population with a normally distributed population. It indicates that the normal distribution reasonably fits the wheat yield data set. Using this data set to construct the Shewhart, HEWMA-CV, and the proposed HEWMARS-CV control charts, we consider the ${\mathrm{ARL}}_0$ to be equal to 370. We have estimated the parameters of the normal distribution. The estimators of the mean and standard deviation are 3.949 and 0.3034, respectively. Thus, the in-control CV is equal to 0.08.

This section shows the sensitivity of the Shewhart, HEWMA-CV, and HEWMARS-CV control charts in detecting changes in CV values. The efficiency of the Shewhart control chart in detecting changes in CV values is presented in Figure 10. The HEWMA-CV control chart with a pair of exponential smoothing parameters of ${\lambda }_1=$0.10, ${\lambda }_2=$0.25, and $L$ = 3.710 is presented in Figure 11, and the proposed HEWMARS-CV control chart with ${\lambda }_1=$0.10, ${\lambda }_2=$0.50, $L_1$ = 3.720, and $L_2$ = 3.610 is displayed in Figure 12. The graphic shows that the proposed HEWMARS-CV control chart can detect the change in CV on the 28th statistics, whereas the HEWMA-CV control chart can detect the shift in CV on the 29th statistics. Consequently, it was reasonable to conclude that the HEWMARS-CV and HEWMA-CV control charts were superior to the Shewhart control chart for detecting the change in CV.

7. Conclusions

This research aims to develop a novel HEWMARS-CV control chart for monitoring the CV using a repetitive sampling technique. The efficiency of HEWMARS-CV, HEWMA-CV, and Shewhart control charts in terms of ${\mathrm{ARL}}_1$ values found that the HEWMARS-CV and HEWMA-CV control charts are the best in the sense that they have minimized ${\mathrm{ARL}}_1$ values when the processes are a slight shift $\left(1.01\le \delta \le 1.20\right)$, whereas the Shewhart control chart has minimized values when the processes are large shift $\left(1.30\le \delta \le 1.50\right)$. The efficiency of the proposed HEWMARS-CV control chart on the real data shows that the proposed control chart can detect a shift in the CV of the process, and it is superior to the existing control chart in terms of the average run length.

The restriction of this research is that it investigates the efficiency of the CV control chart under the normality assumption. As a result, future research can extend to detect shifts in the multivariate CV control chart or by considering the other distributions, i.e., the Weibull and Gamma distributions. Furthermore, extending the CV further to improve the detection of the CUSUM control chart may be possible.