Analysis of Average Run Length of Extended and New Extended Exponentially Weighted Moving Average Control Charts Using Markov Chain Approach Under Symmetric Distribution

Abstract

Statistical Process Control (SPC) plays a crucial role in monitoring and improving manufacturing processes to ensure product quality. Control charts using exponentially weighted moving averages (EWMA) and their extensions, including Extended EWMA (EEWMA) and New Extended EWMA (NEEWMA), have been developed to increase the sensitivity for detecting small to medium process changes. This research proposes a method for calculating the Average Run Length (ARL) and Standard Deviation of Run Length (SDRL) of control charts under a symmetric distribution using the Markov Chain Approach (MCA). This method is based on the probability of state transitions between controlled and uncontrolled states. The MCA method is more efficient than the Monte Carlo Simulation Approach (MC) in terms of accuracy and significantly reduces processing time. This research also demonstrates the application of ARL and SDRL calculations using the MCA method in various studies. Firstly, the performance of control charts is compared using the Mean Percentage Error (MPE) and Mean Absolute Percentage Error (MAPE). Secondly, the impact of symmetrically distributed process parameters on the performance of control charts is examined. Thirdly, a practical application of the control charts is presented. This research applies the proposed method to detect changes in unemployment insurance claims (UI) using seasonally adjusted initial claims assessment (ICSA) and continuing claims assessment (CCSA) rates from 2021 to 2025. The results show that the MCA method is more efficient than the MC method in terms of accuracy and significantly reduces processing time.

1. Introduction

Statistical Process Control (SPC) is a fundamental methodology for monitoring, controlling, and improving manufacturing processes to ensure compliance with specified quality standards. Among the most powerful and extensively utilized SPC tools are control charts, which are generally classified into two categories based on the type of data they monitor:

• Quantitative control charts are used for continuous data or data that follows a continuous distribution, such as water weight, rainfall amount, or stock prices.
• Qualitative control charts are used for categorical or discrete data, such as the number of defective items or the number of surface defects on a product.

A control chart provides a graphical representation of process data over time, with control limits defining the expected range of variation for a stable process. When data points remain within these limits, the process is said to be in control; otherwise, it indicates a potential out-of-control condition requiring corrective action.

The concept of control charts was initially introduced by Shewhart [1,2,3], whose chart proved particularly effective in detecting large and sudden shifts in the process mean. However, the traditional Shewhart chart generally exhibits lower sensitivity to small and moderate process changes. To overcome this limitation, Roberts [4] developed the Exponentially Weighted Moving Average (EWMA) control chart, which integrates historical data into the monitoring statistic, thereby enhancing sensitivity to persistent small shifts.

Among various statistical process control (SPC) techniques—including Shewhart, CUSUM, and EWMA charts—the EWMA chart has garnered significant attention due to its computational simplicity, strong detection capability for small to moderate shifts, and wide applicability in continuous process monitoring. Compared to CUSUM charts, EWMA charts offer a smoother monitoring structure and are often easier to implement and interpret in practical settings. Moreover, EWMA-based frameworks can be flexibly extended to accommodate different distributional assumptions and performance evaluation methods.

Lucas and Saccucci [5] conducted an extensive investigation into the statistical properties and optimization of EWMA charts for detecting small to moderate shifts. To further enhance detection performance, Naveed et al. [6] proposed the Extended EWMA (EEWMA) chart, which introduces an additional weighting factor to provide greater flexibility and improved sensitivity while maintaining robustness. More recently, Javed et al. [7] introduced the New Extended EWMA (NEEWMA) chart by incorporating another exponential weighting parameter. This additional smoothing factor enables the NEEWMA chart to achieve a better balance between responsiveness to subtle process changes and noise filtering, resulting in superior detection performance compared to traditional EWMA and EEWMA charts, particularly for small to moderate shifts.

The Average Run Length (ARL) is widely recognized as a fundamental performance metric for evaluating and comparing control charts in statistical process control. ARL is defined as the expected number of samples, known as the Run Length (RL), collected before a control chart signals an out-of-control condition. A smaller out-of-control ARL corresponds to quicker detection of process shifts, while a larger in-control ARL indicates greater process stability and a lower false alarm rate during monitoring. Despite its widespread use due to its interpretability and practical significance, ARL has several well-documented limitations. As ARL represents only the expected value of the run length, it may not fully capture the variability and distributional properties of the run length distribution. As a result, different control charts may exhibit similar ARL values yet demonstrate markedly different detection behaviors in practice.

Various methods have been developed to estimate ARL, with Monte Carlo (MC) simulation being among the most commonly employed techniques. The MC method estimates ARL by generating numerous simulated sequences of process observations and recording their respective run lengths. Due to its flexibility and straightforward implementation, MC simulation is widely used to assess control chart performance under diverse distributional scenarios.

However, the accuracy of MC simulation heavily depends on the number of simulation replications performed. Obtaining highly reliable ARL estimates, especially for large in-control ARL values, often necessitates a substantial number of simulations, which results in considerable computational expense and longer execution times. Additionally, simulation-based estimates are inherently subject to sampling variability, which can influence the stability and precision of the results. These limitations have motivated the development and adoption of analytical approaches, such as the Markov chain approach (MCA), which offer more stable and computationally efficient ARL estimation for control chart performance evaluation.

Alternatively, analytical methods provide more computationally efficient approaches for ARL evaluation. Among these, the Markov Chain Approach (MCA), initially proposed by Brook and Evans [1] and later compared with other analytical techniques by Champ and Rigdon [8], models the evolution of the monitoring statistic as a discrete-state Markov process by Sericola [9]. This framework facilitates the computation of the ARL by solving a system of linear equations. The applicability of the MCA has been widely demonstrated in the performance assessment of various weighted moving average control charts. For example, Sukparungsee [10] applied the MCA to approximate the ARL of the generally weighted moving average (GWMA) control chart for defect count monitoring, while Phengsalae et al. [10] extended this methodology to analyze the ARL of the Poisson GWMA control chart, and Zhao et al. [11] applies MCA to approximate the run length distributions of Poisson EWMA control charts under linear drift conditions, providing an effective method for monitoring process changes over time. Furthermore, Kraichok et al. [12] employed the MCA to evaluate the ARL performance of a modified exponentially weighted moving average (MEWMA) control chart. The theoretical foundations and applications of ARL-based performance evaluation in statistical process control have also been comprehensively documented in the literature [13]. Compared to the Monte Carlo simulation method, the MCA provides deterministic ARL estimates free from simulation variability and significantly reduces computational effort, all while maintaining high numerical accuracy. Consequently, the MCA has become one of the most widely adopted analytical tools for evaluating the performance of EWMA-type control charts.

This research proposes a method for calculating the ARL and SDRL values of EWMA, EEWMA, and NEEWMA control charts under symmetric distributions using the MCA. Although MCA has been widely used for ARL approximation in traditional control charts, limited studies have investigated analytical ARL and SDRL estimation for EEWMA and NEEWMA control charts under multiple symmetric distributions within a unified framework. Therefore, this study extends the existing literature by developing analytical approximations for EWMA, EEWMA, and NEEWMA control charts under normal, logistic, and Laplace distributions using MCA. The accuracy of the proposed method is compared with MC using Mean Percentage Error (MPE) and Mean Absolute Percentage Error (MAPE). In addition, the study investigates the effects of process and smoothing parameters on control chart performance and demonstrates the practical applicability of the proposed framework through the detection of changes in unemployment insurance claims (UI) using Initial Claims (ICSA) and Continuing Claims (CCSA) rate data from 2021 to 2025.

2. Symmetric Distribution

Symmetric distributions are a class of probability distributions in which the left and right sides of the distribution are mirror images of each other around a central point, typically the mean ($\mu$). Formally, a distribution is symmetric if its probability density function (PDF) satisfies:

$$f(\mu -x)=f(\mu +x)$$

This property implies that the mean, median, and mode coincide at the center of the distribution. Symmetric distributions are commonly used in statistical modeling due to their mathematical simplicity, ease of analysis, and clear interpretability. However, it is important to acknowledge that many real-world industrial and service processes often display skewness, heavy tails, or discrete characteristics because of operational variability and rare events. Thus, assuming symmetry should be considered mainly a simplifying assumption that aids theoretical development rather than an accurate reflection of all practical data scenarios.

In this section, we focus on three important symmetric distributions: the normal, logistic, and Laplace distributions.

2.1. Normal Distribution

The normal distribution, commonly referred to as the Gaussian distribution, is one of the most widely used symmetric distributions in statistics and process monitoring. Its bell-shaped curve distinguishes it and is widely adopted due to its robust theoretical properties and analytical tractability. PDF is defined as follows:

$$f\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi {\beta }^2}}}exp\{-{\displaystyle \frac{{(x-\mu )}^2}{2{\beta }^2}}\}$$

where $\mu$ is the location parameter and $\beta >0$ is the scale parameter.

The mean and variance of the distribution are $E\left(X\right)=\mu$ and $V\left(X\right)={\beta }^2$. The moment generating function (MGF) is

$$M_X\left(t\right)=exp\{\mu t+{\displaystyle \frac{{\beta }^2{t}^2}{2}}\}$$

Theorem 1.

Let $X\sim N(\mu ,{\beta }^2)$, then $Y=aX+b\sim N(a\mu +b,a^2{\beta }^2)$.

Proof of Theorem 1.

$$\begin{array}{c}\hfill M_Y\left(t\right)=e^{bt}{M}_X\left(at\right)=exp\{(a\mu +b)t+{\displaystyle \frac{{\left(a\beta \right)}^2{t}^2}{2}}\}.\end{array}$$

So that $M_Y\left(t\right)=exp\{(a\mu +b)t+{\displaystyle \frac{{\left(a\beta \right)}^2{t}^2}{2}}\}$, hence $Y\sim \mathrm{N}(a\mu +b,a^2{\beta }^2)$.    □

Remark 1.

If $a={\displaystyle \frac{1}{\beta }}$ and $b=-{\displaystyle \frac{\mu }{\beta }}$ then $Y={\displaystyle \frac{X-\mu }{\beta }}\sim \mathrm{N}(0,1)$.

2.2. Logistic Distribution

The logistic distribution is another symmetric distribution that closely resembles the normal distribution in shape but features heavier tails, thereby providing increased robustness to moderate deviations and extreme observations. Owing to its analytical simplicity and heavy-tailed characteristics, it is frequently employed in probabilistic modeling and robust statistical analysis. PDF is expressed as follows:

$$f\left(x\right)={(1+exp(-{\displaystyle \frac{x-\mu }{\beta }}))}^{-2}(exp(-{\displaystyle \frac{x-\mu }{\beta }}))$$

where $\mu$ the location parameter and $\beta >0$ the scale parameter.

The mean and variance of the distribution are $E\left(X\right)=\mu$ and $V\left(X\right)={\displaystyle \frac{{\pi }^2{\beta }^2}{3}}$. The moment-generating function (MGF) is

$$M_X\left(t\right)={\displaystyle \frac{1}{1-{\beta }^2{t}^2}}exp\left\{\mu t\right\},\left|t\right|<{\displaystyle \frac{1}{\beta }}$$

Theorem 2.

Let $X\sim \mathrm{LG}(\mu ,\beta )$, then $Y=aX+b\sim \mathrm{LG}(a\mu +b,a\beta )$.

Proof of Theorem 2.

$$\begin{array}{c}\hfill M_Y\left(t\right)=e^{bt}{M}_X\left(at\right)={\displaystyle \frac{1}{1-{\left(a\beta \right)}^2{t}^2}}exp\left\{(a\mu +b)t\right\}\end{array}$$

so that $M_Y\left(t\right)=exp\left\{(a\mu +b)t\right\}B(1-a\beta t,1+a\beta t)$; hence, $Y\sim \mathrm{LG}(a\mu +b,a\beta )$.    □

Remark 2.

If $a={\displaystyle \frac{1}{\beta }}$ and $b=-{\displaystyle \frac{\mu }{\beta }}$ then $Y={\displaystyle \frac{X-\mu }{\beta }}\sim \mathrm{LG}(0,1)$.

2.3. Laplace Distribution

The Laplace distribution, also referred to as the double exponential distribution, is characterized by a sharper peak at the center and heavier tails compared to the normal distribution. These features render it particularly suitable for modeling data exhibiting abrupt changes or greater variability around the central value. PDF is expressed as follows:

$$f\left(x\right)={\displaystyle \frac{1}{2\beta }}exp(-{\displaystyle \frac{|x-\mu |}{\beta }})$$

where $\mu$ is the location parameter and $\beta >0$ is the scale parameter.

The mean and variance of the distribution are $E\left(X\right)=\mu$ and $V\left(X\right)=2{\beta }^2$. The moment generating function (MGF) is

$$M_X\left(t\right)=exp\left\{\mu t\right\}B(1-\beta t,1+\beta t),t\in (-{\displaystyle \frac{1}{\beta }},{\displaystyle \frac{1}{\beta }})$$

where $\mathrm{B}(a,b)$ the beta function is $\mathrm{B}(x,y)={\int }_0^1{t}^{x-1}{(1-t)}^{y-1}dt$.

Theorem 3.

Let $X\sim \mathrm{LP}(\mu ,\beta )$, then $Y=aX+b\sim \mathrm{LP}(a\mu +b,a\beta )$.

Proof of Theorem 3.

$$\begin{array}{c}\hfill M_Y\left(t\right)=e^{bt}{M}_X\left(at\right)=exp\left\{(a\mu +b)t\right\}B(1-a\beta t,1+a\beta t)\end{array}$$

so that $M_Y\left(t\right)={\displaystyle \frac{1}{1-{\left(a\beta \right)}^2{t}^2}}exp\left\{(a\mu +b)t\right\}$; hence, $Y\sim \mathrm{LP}(a\mu +b,a\beta )$.    □

Remark 3.

If $a={\displaystyle \frac{1}{\beta }}$ and $b=-{\displaystyle \frac{\mu }{\beta }}$ then $Y={\displaystyle \frac{X-\mu }{\beta }}\sim \mathrm{LP}(0,1)$.

3. Control Charts

In this section, we present the control charts employed in this study, including the EWMA, EEWMA, and NEEWMA control charts. Control charts are statistical monitoring tools used to assess whether a process operates under stable conditions by comparing calculated monitoring statistics against predefined control limits. The process observations are assumed to be sequentially collected over time and are represented as follows:

Let $X_t$ be a random variable from a symmetric distribution with a mean ${\mu }_0$ and variance ${\sigma }_0^2$ with time $t=1,2,\dots$.

3.1. Exponentially Weighted Moving Average Control Chart (EWMA)

The EWMA control chart was originally proposed by Roberts [4]. It is designed to enhance the detection of small to moderate shifts in the process mean by assigning greater weight to recent observations, while still incorporating historical data through a recursive structure. Due to its computational simplicity, smoothing ability, and effectiveness in continuous monitoring, the EWMA chart is widely adopted in statistical quality control. However, its performance relies on assumptions such as independent observations and appropriate parameter selection, which may affect effectiveness in practical scenarios involving autocorrelation or non-standard process behaviors. The EWMA statistic is expressed as follows:

$$Z_t={\lambda }_1{X}_t+(1-{\lambda }_1)Z_{t-1}$$

(1)

where $Z_t$ is the EWMA statistic at time t, and ${\lambda }_1$ is the weighted parameter; $0<{\lambda }_1\le 1$.

The mean and variance of the EWMA statistic are $E\left(Z_t\right)={\mu }_0$ and $V\left(Z_t\right)=Q^Z·{\sigma }_0^2$, where:

$$Q^Z={\displaystyle \frac{{\lambda }_1}{2-{\lambda }_1}}.$$

(2)

The upper control limit ($UC{L}^Z$) and lower control limit ($LC{L}^Z$) for the monitoring process are determined according to the EWMA structure as follows:

$$UC{L}^Z,LC{L}^Z={\mu }_0\pm L_Z{\sigma }_0\sqrt{Q^Z},$$

(3)

where $L_Z$ is the limit parameter of EWMA, which represents the width of the upper control limit and lower control limit of the EWMA control chart. The EWMA stopping time (${\tau }_Z$) can then be written as follows:

$${\tau }_Z=inf\{t>0:Z_t>UC{L}^Z\mathrm{or}{Z}_t<LC{L}^Z\}.$$

3.2. Extended Exponentially Weighted Moving Average Control Chart (EEWMA)

The EEWMA control chart was introduced by Naveed et al. [6] as an extension of the traditional EWMA. By incorporating an additional weighting mechanism, the EEWMA chart offers greater flexibility and improved sensitivity for detecting small to moderate shifts while maintaining robustness against random variability. This additional smoothing enhances adaptability to various monitoring environments; however, it also increases the chart’s dependence on parameter configuration and model specification. Careful parameter calibration is therefore necessary to balance shift detection sensitivity with false alarm control. The EEWMA statistic is defined as follows:

$$E_t={\lambda }_1{X}_t-{\lambda }_2{X}_{t-1}+(1-{\lambda }_1+{\lambda }_2)E_{t-1}$$

(4)

where $E_t$ is the EEWMA statistic at time t; ${\lambda }_1$ is the weighted parameter; $0<{\lambda }_1\le 1$, ${\lambda }_2$ is the lag-1 weighted parameter; $0\le {\lambda }_2<{\lambda }_1$.

The mean and variance of the EEWMA statistic are $E\left(E_t\right)={\mu }_0$ and $V\left(E_t\right)=Q^E·{\sigma }_0^2$, where:

$$Q^E={\displaystyle \frac{{\lambda }_1^2+{\lambda }_2^2-2{\lambda }_1{\lambda }_2(1-{\lambda }_1+{\lambda }_2)}{2({\lambda }_1-{\lambda }_2)-{({\lambda }_1-{\lambda }_2)}^2}}$$

It can be expanded to the simplified formula, that is,

$$Q^E={\displaystyle \frac{{\lambda }_1-{\lambda }_2+2{\lambda }_1{\lambda }_2}{2-{\lambda }_1+{\lambda }_2}}.$$

(5)

The upper control limit ($UC{L}^E$) and lower control limit ($LC{L}^E$) for the monitoring process are determined according to the EEWMA structure as follows:

$$UC{L}^E,LC{L}^E={\mu }_0\pm L_E{\sigma }_0\sqrt{Q^E},$$

(6)

where $L_E$ is the limit parameter of EEWMA, which represents the width of the upper control limit and lower control limit of the EEWMA control chart. The EEWMA stopping time (${\tau }_E$) can then be written as follows:

$${\tau }_E=inf\{t>0:E_t>UC{L}^E\mathrm{or}{E}_t<LC{L}^E\}.$$

Furthermore, the EEWMA control chart becomes the EWMA control chart when ${\lambda }_2=0$.

3.3. New Extended Exponentially Weighted Moving Average Control Chart (NEWWMA)

To further enhance monitoring performance, Javed et al. [7] proposed the NEEWMA control chart by adding another exponential weighting parameter to the EEWMA framework. This additional smoothing parameter allows the NEEWMA chart to better balance responsiveness to subtle process shifts with noise reduction, improving detection performance for small to moderate changes. Nevertheless, the presence of multiple smoothing parameters increases model complexity and sensitivity to parameter selection, potentially complicating practical implementation and interpretation. Improper parameter choices may negatively impact monitoring effectiveness or robustness under varying process conditions. The NEEWMA statistic is expressed as follows:

$$N_t={\lambda }_1{X}_t-{\lambda }_2{X}_{t-1}-{\lambda }_3{X}_{t-2}+(1-{\lambda }_1+{\lambda }_2+{\lambda }_3)N_{t-1}$$

(7)

where $N_t$ is the NEEWMA statistic at time t; ${\lambda }_1$ is the weighted parameter; ${\lambda }_2+{\lambda }_3<{\lambda }_1\le 1$; ${\lambda }_2$ is the lag-1 weighted parameter; ${\lambda }_3<{\lambda }_2<{\lambda }_1$, ${\lambda }_3$ is the lag-2 weighted parameter; $0\le {\lambda }_3<{\lambda }_2$.

The mean and variance of the NEEWMA statistic are $E\left(N_t\right)={\mu }_0$ and $V\left(N_t\right)=Q^N·{\sigma }_0^2$, where:

$$Q^N={\displaystyle \frac{({\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2)-2{\lambda }_2({\lambda }_1-{\lambda }_3)(1-b)-2{\lambda }_1{\lambda }_3{(1-b)}^2}{1-{(1-b)}^2}},$$

where $b={\lambda }_1-{\lambda }_2-{\lambda }_3$, and it can be expanded to the simplified formula, that is,

$$Q^N={\displaystyle \frac{{\lambda }_1-{\lambda }_2-{\lambda }_3+2{\lambda }_2({\lambda }_1-{\lambda }_3)}{2-{\lambda }_1+{\lambda }_2+{\lambda }_3}}+2{\lambda }_1{\lambda }_3.$$

(8)

The upper control limit ($UC{L}^N$) and lower control limit ($LC{L}^N$) for the monitoring process are determined according to the NEEWMA structure as follows:

$$UC{L}^N,LC{L}^N={\mu }_0\pm L_N{\sigma }_0\sqrt{Q^N/n}$$

(9)

where $L_N$ is the limit parameter of NEEWMA, which represents the width of the upper control limit and lower control limit of the NEEWMA control chart. The NEEWMA stopping time (${\tau }_N$) can then be written as follows:

$${\tau }_N=inf\{t>0:N_t>UC{L}^N\mathrm{or}{N}_t<LC{L}^N\}.$$

Furthermore, the NEEWMA control chart becomes the EEWMA control chart when ${\lambda }_3=0$, and the EWMA control chart when both ${\lambda }_2=0$ and ${\lambda }_3=0$

4. Average Run Length (ARL)

Run Length (RL) is the number of samples obtained before a change in process is detected. RL is related to the stopping time given by $RL=\tau -1$. Average Run Length (ARL) is the expected value of RL; it is given by:

$$ARL={\displaystyle \sum _{k=0}^{\infty }}k·Pr(RL=k).$$

(10)

The ARLs have two values, as follows. First, the ARL before an out-of-control is detected when the process is in control, defined as $AR{L}_0$. Second, ARL before an out-of-control is detected after the process mean changed, defined as $AR{L}_1$.

The standard deviations of RL are given by:

$$SDRL=\sqrt{E\left(R{L}^2\right)-AR{L}^2}$$

(11)

where $E\left(R{L}^2\right)$ is the second moment of RL given by:

$$E\left(R{L}^2\right)={\displaystyle \sum _{k=0}^{\infty }}{k}^2·Pr(RL=k).$$

(12)

However, calculating these probabilities is challenging because the exact distribution of the control chart statistics is often unknown, making analytical evaluation difficult. As a result, ARL and SDRL are typically estimated using numerical or simulation-based methods. Among these, Monte Carlo Simulation (MC) is widely used due to its flexibility and ease of implementation across various distributional settings. Nevertheless, reliable MC estimation often requires a large number of simulation runs, especially for high in-control ARL values, which increases computational cost, extends execution time, and introduces sampling variability in the results.

To address these limitations, analytical approaches such as the Markov Chain Approach (MCA) have been developed. MCA provides a more stable and computationally efficient framework for estimating ARL and SDRL by modeling the transition behavior of the monitoring statistic through transition probability matrices. Therefore, this study employs both the MCA and MC to evaluate and compare control chart performance.

4.1. Monte Carlo Simulation Approach (MC)

MC simulation is a traditional method used to estimate ARL and SDRL of control charts when these measures cannot be obtained analytically. Furthermore, MC results can be used to validate the accuracy of alternative methods, as the approach relies on statistical data generation and applies control charts to detect shifts in simulated data, thereby evaluating the run length performance under specified conditions. The approximations of ARL and SDRL are given by

$$AR{L}_{MC}={\displaystyle \frac{1}{M}}{\displaystyle \sum _{t=1}^M}R{L}_t.$$

(13)

and

$$SAR{L}_{MC}=\sqrt{{\displaystyle \frac{1}{M-1}}\sum _{t=1}^M{(R{L}_t-AR{L}_{MC})}^2}.$$

(14)

where M is the number of simulation loops to approximate ARL and SDRL in this research, assuming $M=100,000$.

4.2. Markov Chain Approach (MCA)

MCA is one of the most effective methods to study the behavior of a control chart. This approach has been discussed by many authors (e.g., Champ and Rigdon [13]; Lucas and Saccucci [5]). They introduced the MCA to approximate an ARL state in an in-control process, assuming the observation is an in-control state and an out-of-control state. The transition probability $P_{ij}$ is the probability of moving from state to state in one step and is given by

$$P_{ij}=Pr(X_{t+1}=x_j\mid X_t=x_i),$$

the transition probability matrix $\mathbf{P}$ and the elements of matrix $P_{ij}$ can be rewritten as

$$\mathbf{P}={\left[P_{ij}\right]}_{(n+1)\times (n+1)}=\left[\begin{array}{cc}\mathbf{R}& (\mathbf{I}-\mathbf{R})\mathbf{1}\\ \mathbf{0}& 1\end{array}\right]$$

where:

$\mathbf{P}$ is the one-step transition probability matrix.

$\mathbf{R}$ is the matrix of probabilities of statistics changing states from in to in, as follows:

$$\mathbf{R}={\left[P_{ij}\right]}_{n\times n};i,j=1,\dots ,n.$$

$(\mathbf{I}-\mathbf{R})\mathbf{1}$ is the vector of probabilities of statistics changing states from in to out, as follows:

$$\mathbf{I}={\left[\begin{array}{ccc}1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & 1\end{array}\right]}_{n\times n},\mathbf{1}=1^T={[1\cdots 1]}_{1\times n},\mathbf{0}={[0\cdots 0]}_{1\times n}.$$

The k-stage transition probability matrix is useful for evaluating ARL because it contains the probability that the chain goes from one state to another state in k steps. This matrix is

$${\mathbf{P}}_k=\left[\begin{array}{cc}{\mathbf{R}}^k& (\mathbf{I}-{\mathbf{R}}^k)\mathbf{1}\\ \mathbf{0}& 1\end{array}\right].$$

The vector $(\mathbf{I}-{\mathbf{R}}^k)\mathbf{1}$ is the vector of transition probabilities from state $i\le n$ to state $n+1$ in k steps. Hence,

$$Pr({\tau }_r\le k\mid X_0=x_i)={\mathbf{p}}_i^{\left(0\right)}(\mathbf{I}-{\mathbf{R}}^k)\mathbf{1}$$

(15)

where ${\mathbf{p}}_i^{\left(0\right)}$ is the initial probability vector with 1 at position ith and 0 otherwise, which represents the probability that the stopping time is given by

$$\begin{array}{cc}\hfill Pr(\tau =k\mid X_0=x_i)& =Pr(\tau \le k\mid X_0=x_i)-Pr(\tau \le k-1\mid X_0=x_i)={\mathbf{p}}_i^{\left(0\right)}({\mathbf{R}}^{k-1}-{\mathbf{R}}^k)\mathbf{1}.\hfill \end{array}$$

So, the transition probability of RL is given by

$$Pr(RL=k)=Pr({\tau }_r=k+1\mid X_0=x_i)={\mathbf{p}}_i^{\left(0\right)}{\mathbf{R}}^k(\mathbf{I}-\mathbf{R})\mathbf{1},$$

(16)

where $k=0,1,2,\dots$

In calculating ARL and SDRL values, the key mathematical equation is the sum of the transition matrices, with the following equation:

• ${\sum }_{k=0}^{\infty }{R}^k=R{(1-R)}^{-1}$,
• ${\sum }_{k=0}^{\infty }k{R}^k=R{(1-R)}^{-2}$,
• ${\sum }_{k=0}^{\infty }{k}^2{R}^k=R(1+R){(1-R)}^{-3}$

Thus, the first moment of RL (ARL) and the second moment of RL can approximately be written as

$$\begin{array}{c}\hfill E\left(RL\right)={\displaystyle \sum _{k=0}^{\infty }}kPr(RL=k)={\mathbf{p}}_i^{\left(0\right)}\mathbf{R}{(\mathbf{I}-\mathbf{R})}^{-1}\mathbf{1},\end{array}$$

and

$$\begin{array}{c}\hfill E\left(R{L}^2\right)={\displaystyle \sum _{k=0}^{\infty }}{k}^{2}Pr(RL=k)={\mathbf{p}}_i^{\left(0\right)}\mathbf{R}(\mathbf{I}+\mathbf{R}){(\mathbf{I}-\mathbf{R})}^{-\mathbf{2}}\mathbf{1}.\end{array}$$

so that the ARL and SDRL can approximately follow:

$$AR{L}_{MCA}={\mathbf{p}}_i^{\left(0\right)}\mathbf{R}{(\mathbf{I}-\mathbf{R})}^{-1}\mathbf{1},$$

(17)

and

$$SDR{L}_{MCA}=\sqrt{{\mathbf{p}}_i^{\left(0\right)}\mathbf{R}(\mathbf{I}+\mathbf{R}){(\mathbf{I}-\mathbf{R})}^{-\mathbf{2}}\mathbf{1}-AR{L}_{MCA}^2}.$$

(18)

where ${\mathbf{p}}_i^{\left(0\right)}$ is the initial probability vector with 1 at position ith and 0 otherwise, $\mathbf{I}$ is the identity matrix, $\mathbf{1}$ is a vector of 1, $\mathbf{R}$ is the matrix of probabilities of statistics changing states from in to in, as follows:

$$\mathbf{I}={\left[\begin{array}{ccc}1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & 1\end{array}\right]}_{n\times n},\mathbf{1}=1^T={[1\cdots 1]}_{1\times n},\mathbf{R}={\left[P_{ij}\right]}_{n\times n};i,j=1,\dots ,n.$$

A key component in approximating ARL and SDRL is the transition probability, which depends on the control chart structure. In the proposed MCA, n denotes the number of partitions in the state space. Following previous studies [10,14], this study uses $n=1000$ to achieve stable and accurate ARL approximations.

5. The Transition Probability

The transition probability $P_{ij}$ is the probability of moving from state to state in one step and is given by

$$P_{ij}=Pr(X_{t+1}=x_j\mid X_t=x_i).$$

(19)

and $\mathbf{R}$ is the transition probability of the in-control state matrix, which is given by

$$\mathbf{R}={\left[P_{ij}\right]}_{\left(n\right)\times \left(n\right)}$$

(20)

It is a key component in approximate ARL and SDRL, where each control chart has its own probability equation format. The basis of the transition probability of the in-control state is as follows:

Let $X_t$ be a random variable from a symmetric distribution with a mean $\mu ={\mu }_0+\delta$ and variance ${\sigma }_0^2$ with time $t=1,2,\dots$, and let $S_t$ denote the control chart statistic for $X_t$. The control limits, namely the lower control limit ($LC{L}^S$) and upper control limit ($UC{L}^S$), are defined as

$$UC{L}^S,LC{L}^S={\mu }_0\pm L_S{\sigma }_0\sqrt{Q^S}$$

(21)

where $L_S$ is a control limit coefficient and $Q^S$ is a parameter depending on the type of control chart. Then, the transition probability of the control chart is as follows:

$$P_{ij}^S=Pr(L_j^S\le S_t\le U_j^S\mid S_{t-1}=z_i^s),$$

(22)

where $S_t$ is statistics of control chart at time t.

$z_i^S$ is the middle of the control chart at time $t-1$ in i position as follows:

$$z_i^S=LC{L}^S+{\displaystyle \frac{(2i-1)(UC{L}^S-LC{L}^S)}{2n}}={\mu }_0-L_S{\sigma }_0\sqrt{Q^S}(1-{\displaystyle \frac{2i-1}{n}})$$

(23)

$L_j^S$ is the lower limit of the control chart in the partition j as follows:

$$L_j^S=LC{L}^S+{\displaystyle \frac{(j-1)(UC{L}^S-LC{L}^S)}{n}}={\mu }_0-L_S{\sigma }_0\sqrt{Q^S}(1-{\displaystyle \frac{2(j-1)}{n}})$$

(24)

$U_j^S$ is the upper limit of the control chart in the partition j as follows:

$$U_j^S=LC{L}^S+{\displaystyle \frac{j(UC{L}^S-LC{L}^S)}{n}}={\mu }_0-L_S{\sigma }_0\sqrt{Q^S}(1-{\displaystyle \frac{2j}{n}})$$

(25)

For EWMA control charts, the transition probability equation $P_{ij}$ can be rewritten in the EWMA control charts by Equations (1) and (3) as follows:

$$\begin{array}{cc}\hfill P_{ij}^Z& =Pr(L_j^Z\le Z_t\le U_j^Z\mid Z_{t-1}=z_i^Z)\hfill \\ & =Pr(L_j^Z\le {\lambda }_1{X}_t+(1-{\lambda }_1)z_i^Z\le U_j^Z)\hfill \\ & =Pr(X_t\le {\displaystyle \frac{U_j^Z-(1-{\lambda }_1)z_i^Z}{{\lambda }_1}})-Pr(X_t\le {\displaystyle \frac{L_j^Z-(1-{\lambda }_1)z_i^Z}{{\lambda }_1}})\hfill \end{array}$$

Thus, the transition probability of EWMA is given by

$$P_{ij}^Z=Pr(X\le {\mu }_0-A_{{\sigma }_0}^Z·B_{ij}^Z)-Pr(X\le {\mu }_0-A_{{\sigma }_0}^Z(B_{ij}^Z+{\displaystyle \frac{2}{n}}))$$

(26)

where $A_{{\sigma }_0}^Z={\displaystyle \frac{L{\sigma }_0\sqrt{Q^Z}}{{\lambda }_1}}$ and $B_{ij}^Z=(1-{\displaystyle \frac{2j}{n}})+(1-{\lambda }_1)(1-{\displaystyle \frac{2i-1}{n}})$. By Remarks 1–3 of symmetric distributions, the transition probability of EWMA can be evaluated as:

$$P_{ij}^Z=F_Y({\displaystyle \frac{A_{{\sigma }_0}^Z(B_{ij}^Z+\frac{2}{n})+\delta }{\beta }})-F_Y({\displaystyle \frac{A_{{\sigma }_0}^Z·B_{ij}^Z+\delta }{\beta }})$$

(27)

where Y is a symmetric distribution with location and scale parameters of 0 and 1.

Consequently, in the EEWMA and NEEWMA control charts, the transition probability equation $P_{ij}$ can be rewritten in the EEWMA control charts by Equations (4) and (6), and the NEEWMA control charts by Equations (7) and (9) as follows:

For EEWMA:

$$\begin{array}{cc}\hfill P_{ij}^E& =Pr(L_j^E\le E_t\le U_j^E\mid E_{t-1}=z_i^E)\hfill \\ & =Pr(L_j^E\le {\lambda }_1{X}_t-{\lambda }_2{X}_{t-1}+(1-{\lambda }_1+{\lambda }_2)z_i^E\le U_j^E),\hfill \end{array}$$

For NEEWMA:

$$\begin{array}{cc}\hfill P_{ij}^N& =Pr(L_j^N\le N_t\le U_j^N\mid N_{t-1}=z_i^N)\hfill \\ & =Pr(L_j^N\le {\lambda }_1{X}_t-{\lambda }_2{X}_{t-1}-{\lambda }_3{X}_{t-2}+(1-{\lambda }_1+{\lambda }_2+{\lambda }_3)z_i^N\le U_j^N).\hfill \end{array}$$

This study approximates $X_{t-1}$ and $X_{t-2}$ by their expected values under the in-control process condition. This simplification is used to make the analytical derivation of the transition probability matrix in MCA more manageable. According to statistical theory, the unique minimum variance unbiased estimator (UMVUE) of the process mean $\mu$ is $\overline{X}$, which naturally serves as a reference for the in-control process mean ${\mu }_0$.

In practical process monitoring, the exact process shift is typically unknown, whereas the in-control mean can be estimated from historical data using $\overline{X}$. Thus, the expected-value approximation simplifies the transition structure and makes the analytical estimation of ARL and SDRL more feasible.

Therefore, this research presents an estimation for $X_{t-1}$ and $X_{t-2}$ using the expectation of $X_{t-1}$ and $X_{t-2}$ is ${\mu }_0$; thus, we obtain the following:

For EEWMA,

$$\begin{array}{cc}\hfill P_{ij}^E& =Pr(L_j^E\le {\lambda }_1{X}_t-{\lambda }_2{\mu }_0+(1-{\lambda }_1+{\lambda }_2)z_i^E\le U_j^E)\hfill \\ & =Pr(X_t\le (U_j^E+{\lambda }_2{\mu }_0-(1-{\lambda }_1+{\lambda }_2)z_i^E)/{\lambda }_1)\hfill \\ & -Pr(X_t\le (L_j^E+{\lambda }_2{\mu }_0-(1-{\lambda }_1+{\lambda }_2)z_i^E)/{\lambda }_1)\hfill \end{array}$$

For NEEWMA,

$$\begin{array}{cc}\hfill P_{ij}^N& =Pr(L_j^N\le {\lambda }_1{X}_t-{\lambda }_2{\mu }_0-{\lambda }_3{\mu }_0+(1-{\lambda }_1+{\lambda }_2+{\lambda }_3)z_i^N\le U_j^N)\hfill \\ & =Pr(X_t\le (U_j^N+({\lambda }_2+{\lambda }_3){\mu }_0-(1-{\lambda }_1+{\lambda }_2+{\lambda }_3)z_i^N)/{\lambda }_1)\hfill \\ & -Pr(X_t\le (L_j^N+({\lambda }_2+{\lambda }_3){\mu }_0-(1-{\lambda }_1+{\lambda }_2+{\lambda }_3)z_i^N)/{\lambda }_1)\hfill \end{array}$$

Similarly, EWMA can be structured, and a remark of a symmetric probability distribution can rewrite the transition probability of EEWMA and NEEWMA as follows:

For EEWMA,

$$P_{ij}^E=F_Y({\displaystyle \frac{A_{{\sigma }_0}^E(B_{ij}^E+\frac{2}{n})+\delta }{\beta }})-F_Y({\displaystyle \frac{A_{{\sigma }_0}^E·B_{ij}^E+\delta }{\beta }})$$

(28)

where $A_{{\sigma }_0}^E={\displaystyle \frac{L{\sigma }_0\sqrt{Q^E}}{{\lambda }_1}}$ and $B_{ij}^E=(1-{\displaystyle \frac{2j}{n}})+(1-{\lambda }_1+{\lambda }_2)(1-{\displaystyle \frac{2i-1}{n}})$.

Furthermore, the transition probability of the EEWMA control chart becomes the transition probability of the EWMA control chart when ${\lambda }_2=0$.

For NEEWMA,

$$P_{ij}^N=F_Y({\displaystyle \frac{A_{{\sigma }_0}^N(B_{ij}^N+\frac{2}{n})+\delta }{\beta }})-F_Y({\displaystyle \frac{A_{{\sigma }_0}^N·B_{ij}^N+\delta }{\beta }})$$

(29)

where $A_{{\sigma }_0}^N={\displaystyle \frac{L{\sigma }_0\sqrt{Q^N}}{{\lambda }_1}}$ and $B_{ij}^N=(1-{\displaystyle \frac{2j}{n}})+(1-{\lambda }_1+{\lambda }_2+{\lambda }_3)(1-{\displaystyle \frac{2i-1}{n}})$.

Furthermore, the transition probability of the NEEWMA control chart becomes the transition probability of the EEWMA control chart when ${\lambda }_3=0$, and the transition probability of the EWMA control chart when both ${\lambda }_2=0$ and ${\lambda }_3=0$

However, this assumption diminishes the temporal dependence on lagged observations and may not fully represent the stochastic behavior of the EEWMA and NEEWMA statistics. As a result, the approximation might underestimate variability in certain cases, so the ARL and SDRL results should be considered within this limitation.

6. Numerical Result

In this section, we will present the results of the accuracy, behavior, and performance of EWMA, EEWMA, and NEEWMA, evaluated by MCA. The scope of this research is as follows:

• Control charts are EWMA, EEWMA, and NEEWMA with ${\lambda }_1=0.1,0.3,0.7$, ${\lambda }_2=0.01,0.03,0.05$, and ${\lambda }_3={\lambda }_2/4$;
• Process distribution is normal, logistic, and Laplace distributions with location, ${\mu }_0$ = 0, 5, 10, and scale, $\beta$ = 1, 5, 10;
• Process shifts are $\delta$ = 0, $\pm 0.01$, $\pm 0.03$, $\pm 0.05$, $\pm 0.07$, $\pm 0.09$, $\pm 0.1$, $\pm 0.3$, $\pm 0.5$, $\pm 0.7$, $\pm 0.9$, $\pm 1.0$, $\pm 2.0$, $\pm 3.0$;
• The in-control ARLs are $AR{L}_0$ = 200, 370, 500.

First, the control limits of the control chart were determined using the bisection method, as shown in Algorithm 1, and the results of the control limits are shown in

Table 1. Limit parameters (L) of the EWMA, EEWMA, and NEEWMA control charts for symmetric distribution.

${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	${\mathit{A}\mathit{R}\mathit{L}}_0$
			Normal Distribution			Logistic Distribution			Laplace Distribution
			200	370	500	200	370	500	200	370	500
0.1	-	-	2.4561	2.7021	2.8151	2.4877	2.7555	2.8801	2.5385	2.8350	2.9745
0.3	-	-	2.7144	2.9256	3.0237	2.8933	3.1720	3.3056	3.1114	3.4539	3.6192
0.7	-	-	2.8010	2.9952	3.0864	3.2301	3.5559	3.7151	3.6381	4.0543	4.2580
0.1	0.010	-	2.6633	2.9379	3.0639	2.6911	2.9870	3.1245	2.7370	3.0618	3.2142
	0.025	-	3.0517	3.3833	3.5353	3.0728	3.4248	3.5879	3.1099	3.4905	3.6685
	0.050	-	4.0495	4.5477	4.7759	4.0557	4.5719	4.8101	4.0715	4.6146	4.8672
0.3	0.010	-	2.7737	2.9908	3.0916	2.9491	3.2332	3.3692	3.1650	3.5131	3.6811
	0.025	-	2.8680	3.0948	3.2000	3.0378	3.3306	3.4706	3.2501	3.6073	3.7794
	0.050	-	3.0416	3.2867	3.4003	3.2012	3.5106	3.6579	3.4061	3.7803	3.9601
0.7	0.010	-	2.8126	3.0078	3.0995	3.2392	3.5656	3.7250	3.6463	4.0632	4.2671
	0.025	-	2.8306	3.0274	3.1198	3.2533	3.5806	3.7404	3.6590	4.0768	4.2812
	0.050	-	2.8625	3.0622	3.1559	3.2782	3.6070	3.7676	3.6810	4.1006	4.3058
0.1	0.01	0.0025	2.7147	2.9968	3.1263	2.7414	3.0447	3.1855	2.7859	3.1180	3.2737
	0.025	0.00625	3.2322	3.5925	3.7577	3.2500	3.6302	3.8061	3.2824	3.6908	3.8815
	0.05	0.0125	4.8039	5.4530	5.7510	4.7999	5.4649	5.7718	4.8004	5.4900	5.8102
0.3	0.01	0.0025	2.7841	3.0024	3.1038	2.9583	3.2433	3.3797	3.1733	3.5223	3.6906
	0.025	0.00625	2.8971	3.1272	3.2340	3.0638	3.3593	3.5004	3.2735	3.6332	3.8064
	0.05	0.0125	3.1122	3.3655	3.4829	3.2650	3.5811	3.7315	3.4638	3.8444	4.0272
0.7	0.01	0.0025	2.8134	3.0088	3.1005	3.2391	3.5654	3.7248	3.6457	4.0624	4.2663
	0.025	0.00625	2.8332	3.0304	3.1229	3.2534	3.5805	3.7403	3.6577	4.0752	4.2795
	0.05	0.0125	2.8693	3.0698	3.1639	3.2798	3.6083	3.7686	3.6797	4.0987	4.3036

.

Algorithm 1 Bisection-based procedure for determining the control limit L

Input:Distribution, $\mu$, $\sigma$, ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, $\delta =0$ and $AR{L}_0$. Output:$L_{AR{L}_0}$.   1:  Initialize $L=1$.   2:  Compute $AR{L}_L=ARL\left(L\right)$ using the Monte Carlo approximation (MCA) method.   3:  while  $AR{L}_L>AR{L}_0$  do   4:        $L=L+1$   5:        Compute $AR{L}_L=ARL\left(L\right)$ using MCA.   6:  Set $L_{\mathrm{lower}}=L-1$ and $L_{\mathrm{upper}}=L$.   7:  repeat   8:        $L={\displaystyle \frac{L_{\mathrm{upper}}+L_{\mathrm{lower}}}{2}}$   9:        Compute $AR{L}_L=ARL\left(L\right)$ using MCA. 10:        if $|AR{L}_L-AR{L}_0|<{10}^{-7}$ then 11:        $L_{AR{L}_0}=L$ 12:        break 13:        else if $AR{L}_L-AR{L}_0>{10}^{-7}$ then 14:        $L_{\mathrm{upper}}=L$ 15:        else 16:        $L_{\mathrm{lower}}=L$ 17:  until convergence 18:  return  $L_{AR{L}_0}=L$

6.1. Accuracy of ARL Evaluation Method

This section presents a comparison of the calculation methods for ARL and SDRL values of EWMA, EEWMA, and NEEWMA control charts between the MCA and MC methods, with control chart parameters set at ${\lambda }_1=0.1$, ${\lambda }_2=0.05$, and ${\lambda }_3={\lambda }_2/4$ under a symmetrically distributed process with position 0 and scale 1. There are three symmetrical distributions: normal, logistic, and Laplace. The comparison of the MCA and MC methods measures performance using the Mean Percentage Error (MPE) and Mean Absolute Percentage Error (MAPE), which are defined as follows:

$$MPE={\displaystyle \frac{100\%}{n\left(\delta \right)}}\sum _{\forall \delta }{\displaystyle \frac{AR{L}_{MC}-AR{L}_{MCA}}{AR{L}_{MC}}}$$

and

$$MAPE={\displaystyle \frac{100\%}{n\left(\delta \right)}}\sum _{\forall \delta }\left|{\displaystyle \frac{AR{L}_{MC}-AR{L}_{MCA}}{AR{L}_{MC}}}\right|.$$

The results indicate that the MCA method outperforms the MC method in terms of both accuracy and computational efficiency. Specifically, the MCA method yields MPE and MAPE close to 0, demonstrating that its estimates are very close to those obtained by the MC method. The detailed results are presented in

Table 2. Performance of MCA to evaluate ARL and SDRL for the EWMA, EEWMA, and NEEWMA control charts for symmetric distribution with location $\mu =0$ and scale $\beta =1$.

Distribution	${\mathit{ARL}}_0$	Measures	EWMA		EEWMA		NEEWMA
			MPE	MAPE	MPE	MAPE	MPE	MAPE
Normal	200	ARL	0.0627%	0.3660%	0.1018%	0.3325%	$-0.0338$%	0.2763%
		SDRL	0.2074%	0.5515%	$-0.2152$%	0.5901%	0.1036%	0.4665%
	370	ARL	0.0546%	0.3927%	0.0344%	0.2164%	$-0.0147$%	0.2329%
		SDRL	0.2714%	0.4893%	0.2369%	0.5808%	$-2.1889$%	2.7809%
	500	ARL	0.4326%	0.5913%	$-0.1696$%	0.3108%	0.1887%	0.5975%
		SDRL	$-0.8499$%	1.0996%	$-0.6674$%	0.8900%	0.2511%	1.5391%
Logistic	200	ARL	0.1693%	0.4125%	$-0.0239$%	0.2939%	$-0.1651$%	0.4050%
		SDRL	$-0.2308$%	0.3968%	$-0.3383$%	1.0066%	0.3102%	0.4071%
	370	ARL	$-0.0584$%	0.2498%	$-0.0563$%	0.2250%	0.0937%	0.2454%
		SDRL	0.2668%	0.4778%	$-0.2570$%	0.4454%	$-0.1140$%	0.3158%
	500	ARL	0.0907%	0.2245%	$-0.0085$%	0.1953%	$-0.1060$%	0.1923%
		SDRL	0.0274%	0.3328%	0.0186%	0.3220%	$-0.0774$%	0.4086%
Laplace	200	ARL	0.0767%	0.2828%	0.0393%	0.2662%	$-0.0128$%	0.2349%
		SDRL	0.0846%	0.4659%	0.4768%	0.9670%	$-0.0510$%	0.4522%
	370	ARL	0.0210%	0.3387%	$-0.0232$%	0.2656%	0.0718%	0.2504%
		SDRL	$-0.1851$%	0.5618%	0.2473%	0.5500%	0.4572%	0.7841%
	500	ARL	$-0.1334$%	1.4219%	0.0419%	0.4090%	$-0.0088$%	0.1647%
		SDRL	$-0.1068$%	0.3971%	0.1408%	0.6606%	0.1450%	1.1668%

and

Table 3. Accuracy of ARL and SDRL for the EWMA, EEWMA, and NEEWMA control charts using MCA and MC for Laplace distribution with location $\mu =0$ and scale $\beta =1$ with $AR{L}_0=370$.

$\mathit{\delta }$	Measure	EWMA		EEWMA		NEEWMA
		MCA	MC	MCA	MC	MCA	MC
0.00	ARL	370.00	370.64	370.00	370.03	370.00	370.59
	SDRL	364.92	364.97	364.24	364.11	364.05	364.23
	CPU time	3.12	291.46	3.12	289.25	3.08	290.37
0.01	ARL	369.36	369.10	369.29	369.50	369.27	369.47
	SDRL	364.27	364.94	363.52	363.46	363.31	363.58
	CPU time	3.17	287.62	3.10	291.67	3.17	286.78
0.03	ARL	364.30	364.61	363.70	363.48	363.54	363.15
	SDRL	359.13	359.20	357.84	357.37	357.48	357.83
	CPU time	3.19	280.38	3.09	285.08	3.20	285.73
0.05	ARL	354.56	354.73	352.99	352.81	352.57	352.58
	SDRL	349.25	349.95	346.96	346.56	346.33	346.95
	CPU time	3.06	277.66	3.15	279.74	3.15	279.12
0.07	ARL	340.85	340.44	338.02	338.59	337.25	337.69
	SDRL	335.33	335.07	331.75	331.84	330.77	330.85
	CPU time	3.14	265.66	3.16	261.84	3.07	262.55
0.09	ARL	324.06	324.79	319.85	319.57	318.71	318.88
	SDRL	318.29	318.20	313.29	313.46	311.93	311.65
	CPU time	3.25	255.13	3.11	251.52	3.19	251.15
0.10	ARL	314.81	314.05	309.92	309.55	308.60	308.68
	SDRL	308.91	308.18	303.20	303.52	301.65	301.70
	CPU time	3.11	247.06	3.15	244.77	3.08	243.67
0.30	ARL	138.47	138.83	130.40	130.02	128.38	128.94
	SDRL	130.15	130.62	121.15	121.53	118.88	118.59
	CPU time	3.15	114.48	3.17	108.02	3.11	108.62
0.50	ARL	62.01	62.65	58.06	58.45	57.12	57.80
	SDRL	53.31	53.67	48.66	48.38	47.54	47.57
	CPU time	3.14	53.86	3.14	49.54	3.09	49.80
0.70	ARL	33.33	33.40	31.62	31.72	31.23	31.97
	SDRL	25.32	25.87	23.15	23.29	22.64	22.54
	CPU time	3.15	29.88	3.14	28.65	2.91	27.64
0.90	ARL	20.89	20.74	20.15	20.33	19.99	19.89
	SDRL	13.88	13.65	12.84	12.86	12.60	12.51
	CPU time	2.70	18.93	3.13	18.48	3.12	18.00
1.00	ARL	17.26	17.14	16.78	16.75	16.67	16.68
	SDRL	10.75	10.51	10.02	10.19	9.85	9.88
	CPU time	3.00	15.26	2.78	15.11	3.00	15.00
2.00	ARL	5.59	5.57	5.65	5.65	5.68	5.67
	SDRL	2.31	2.29	2.27	2.25	2.26	2.24
	CPU time	3.02	6.06	3.12	5.54	3.09	5.75
3.00	ARL	3.10	3.07	3.19	3.17	3.21	3.21
	SDRL	1.11	1.09	1.10	1.10	1.10	1.12
	CPU time	2.71	3.92	2.72	4.34	2.81	3.52
total of CPU time		42.91	2147.36	43.08	2133.55	43.07	2127.70

and Figure 1, Figure 2 and Figure 3.

For the normal distribution, comparing the MCA and MC methods for calculating the ARL and SDRL at $AR{L}_0=370$, the MCA method provides a closer approximation and uses less processing time than the MC method, with MPE and MAPE values less than $1\%$. The analysis results are shown in

Table 4. Accuracy of ARL and SDRL for the EWMA, EEWMA, and NEEWMA control charts using MCA and MC for normal distribution with location $\mu =0$ and scale $\beta =1$ with $AR{L}_0=370$.

$\mathit{\delta }$	Measure	EWMA		EEWMA		NEEWMA
		MCA	MC	MCA	MC	MCA	MC
0.00	ARL	370.00	370.71	370.00	370.58	370.00	370.56
	SDRL	363.25	363.29	362.56	362.51	362.37	362.79
	CPU time	3.36	253.12	3.25	257.80	3.23	257.26
0.01	ARL	368.19	368.99	368.07	368.42	368.04	368.44
	SDRL	361.42	361.88	360.61	360.61	360.39	360.41
	CPU time	3.30	255.40	3.21	253.09	3.22	256.80
0.03	ARL	354.32	354.44	353.34	353.69	353.08	353.71
	SDRL	347.43	347.83	345.74	345.47	345.27	345.20
	CPU time	3.39	243.01	3.30	243.04	3.22	246.36
0.05	ARL	329.45	329.71	327.12	327.74	326.49	326.63
	SDRL	322.32	322.01	319.25	319.13	318.42	318.90
	CPU time	3.33	229.68	3.22	224.60	3.19	227.71
0.07	ARL	297.95	297.08	294.26	294.42	293.28	293.37
	SDRL	290.54	290.18	286.07	286.78	284.87	284.42
	CPU time	3.34	208.91	3.25	209.85	3.20	204.91
0.09	ARL	264.12	264.47	259.39	259.99	258.14	258.26
	SDRL	256.41	256.61	250.88	250.38	249.40	249.24
	CPU time	3.33	183.85	3.20	183.31	3.31	181.15
0.10	ARL	247.37	247.76	242.29	242.58	240.96	240.29
	SDRL	239.52	239.51	233.62	233.03	232.05	232.88
	CPU time	3.31	175.33	3.28	172.01	3.25	168.50
0.30	ARL	65.89	65.12	63.51	63.54	62.92	62.18
	SDRL	57.39	57.42	54.30	54.92	53.52	53.42
	CPU time	3.36	50.18	3.18	48.01	3.28	47.20
0.50	ARL	27.23	27.20	26.68	26.66	26.55	26.71
	SDRL	20.05	20.52	19.04	19.12	18.79	18.12
	CPU time	3.25	21.46	3.24	20.72	3.11	21.36
0.70	ARL	15.37	15.33	15.31	15.35	15.30	15.22
	SDRL	9.58	9.52	9.21	9.20	9.11	9.95
	CPU time	3.25	12.50	3.12	12.79	3.25	12.83
0.90	ARL	10.28	10.32	10.35	10.46	10.38	10.40
	SDRL	5.59	5.63	5.43	5.48	5.39	5.37
	CPU time	3.11	9.17	2.91	9.02	3.19	8.99
1.00	ARL	8.74	8.83	8.84	8.81	8.87	8.88
	SDRL	4.49	4.56	4.38	4.34	4.35	4.34
	CPU time	3.14	8.10	2.94	8.02	3.00	7.81
2.00	ARL	3.18	3.19	3.29	3.29	3.32	3.34
	SDRL	1.22	1.21	1.21	1.22	1.21	1.21
	CPU time	2.96	3.68	2.93	4.11	2.94	3.26
3.00	ARL	1.76	1.75	1.84	1.84	1.87	1.86
	SDRL	0.66	0.66	0.66	0.66	0.66	0.67
	CPU time	2.94	2.53	2.92	2.79	2.94	2.36
total of CPU time		45.37	1656.92	43.95	1649.16	44.33	1646.50

. For $AR{L}_0=200,500$, the graph shows that the MCA method provides a close approximation to the MC method, as evidenced by the similar trends between the MCA and MC methods. The MPE and MAPE values in Table 2 are also less than $1\%$, demonstrating that the MCA method outperforms the MC method in both computational and time aspects across all $AR{L}_0$ levels under a normal distribution with location 0 and scale 1.

For the logistic distribution, comparing the MCA and MC methods for calculating the ARL and SDRL at $AR{L}_0=370$, the MCA method provides a closer approximation and uses less processing time than the MC method, with MPE and MAPE values less than $1\%$. The analysis results are shown in

Table 5. Accuracy of ARL and SDRL for the EWMA, EEWMA, and NEEWMA control charts using MCA and MC for logistic distribution with location $\mu =0$ and scale $\beta =1$ with $AR{L}_0=370$.

$\mathit{\delta }$	Measure	EWMA		EEWMA		NEEWMA
		MCA	MC	MCA	MC	MCA	MC
0.00	ARL	370.00	370.83	370.00	370.90	370.00	370.98
	SDRL	364.07	364.80	363.37	363.51	363.17	363.47
	CPU time	3.20	260.29	3.27	260.41	3.12	255.20
0.01	ARL	369.53	369.20	369.49	369.70	369.47	369.52
	SDRL	363.59	363.16	362.85	362.71	362.64	362.38
	CPU time	3.28	257.41	3.17	257.97	3.19	257.82
0.03	ARL	365.78	365.98	365.42	365.36	365.33	365.15
	SDRL	359.80	359.36	358.73	358.26	358.44	358.86
	CPU time	3.16	255.93	3.17	251.94	3.18	252.31
0.05	ARL	358.51	358.97	357.55	357.12	357.30	357.15
	SDRL	352.44	352.49	350.76	350.71	350.30	350.29
	CPU time	3.20	249.12	3.22	250.02	3.20	249.40
0.07	ARL	348.11	348.37	346.35	346.01	345.87	345.69
	SDRL	341.91	341.31	339.41	339.77	338.73	338.05
	CPU time	3.22	237.84	3.19	243.59	3.15	242.16
0.09	ARL	335.11	335.98	332.43	332.29	331.71	331.73
	SDRL	328.76	328.23	325.31	325.81	324.38	324.91
	CPU time	3.25	233.02	3.20	235.69	3.16	231.21
0.10	ARL	327.83	327.85	324.67	324.97	323.82	323.32
	SDRL	321.39	321.52	317.45	317.45	316.39	316.62
	CPU time	3.14	227.90	3.16	223.06	3.19	228.54
0.30	ARL	168.64	168.40	161.61	161.61	159.82	159.19
	SDRL	160.43	160.38	152.51	152.28	150.47	150.57
	CPU time	3.23	122.03	3.16	119.59	3.16	115.35
0.50	ARL	83.19	83.30	79.00	79.64	77.98	78.00
	SDRL	74.51	74.82	69.55	69.24	68.32	68.41
	CPU time	3.16	62.78	3.17	60.00	3.16	60.09
0.70	ARL	46.71	46.88	44.64	44.17	44.15	44.50
	SDRL	38.43	38.62	35.76	35.07	35.11	35.28
	CPU time	3.00	36.40	3.15	35.22	3.19	34.93
0.90	ARL	29.70	29.68	28.71	28.78	28.48	28.54
	SDRL	22.16	22.94	20.71	20.89	20.36	20.33
	CPU time	3.17	23.52	3.12	23.65	3.11	22.87
1.00	ARL	24.59	24.48	23.91	23.84	23.75	23.45
	SDRL	17.45	17.31	16.37	16.17	16.11	16.81
	CPU time	3.11	19.73	3.10	19.44	2.95	19.21
2.00	ARL	7.79	7.78	7.87	7.85	7.89	7.91
	SDRL	3.75	3.74	3.66	3.61	3.63	3.65
	CPU time	2.70	7.11	3.02	7.46	3.11	6.76
3.00	ARL	4.28	4.31	4.38	4.41	4.41	4.43
	SDRL	1.70	1.74	1.69	1.69	1.69	1.70
	CPU time	2.72	4.16	3.09	4.42	3.04	4.48
total of CPU time		43.54	1997.24	44.19	1992.46	43.91	1980.33

. For $AR{L}_0=200,500$, the graph shows that the MCA method provides a close approximation to the MC method, as evidenced by the similar trends between the MCA and MC methods. The MPE and MAPE values in Table 2 are also less than $1\%$, demonstrating that the MCA method outperforms the MC method in both computational and time aspects across all $AR{L}_0$ levels under a logistic distribution with location 0 and scale 1.

For the Laplace distribution, comparing the MCA and MC methods for calculating the ARL and SDRL at $AR{L}_0=370$, the MCA method provides a closer approximation and uses less processing time than the MC method, with MPE and MAPE values less than $1\%$. The analysis results are shown in Table 3. For $AR{L}_0=200,500$, the graph shows that the MCA method provides a close approximation to the MC method, as evidenced by the similar trends between the MCA and MC methods. The MPE and MAPE values shown in Table 2 are also less than $1\%$, demonstrating that the MCA method outperforms the MC method in both computational and time aspects across all $AR{L}_0$ levels under a Laplace distribution with location 0 and scale 1.

6.2. Efficiency of Control Charts

This section presents the use of the MCA method to calculate ARL and SDRL values and compares the performance of EWMA, EEWMA, and NEEWMA control charts to show that the MCA method can be used to measure performance. By defining the processes used to measure performance with a symmetrical distribution of location 0 and scale 1, namely normal, logistic, and Laplace distributions, the research results are shown in

Table 6. Detection performance of the EWMA, EEWMA, and NEEWMA control charts using MCA for normal distribution with location $\mu =0$ and scale $\beta =1$ with $AR{L}_0=370$.

Chart		EWMA			EEWMA			NEEWMA
${\lambda }_1$		0.1	0.3	0.7	0.1	0.1	0.1	0.1	0.1	0.1
${\lambda }_2$		-	-	-	0.01	0.025	0.05	0.01	0.025	0.05
${\lambda }_3$		-	-	-	-	-	-	0.0025	0.00625	0.0125
Large $\delta$	$-3.00$	1.763	1.094	0.782	1.847	1.998	2.351	1.870	2.073	2.615
		(0.66)	(0.644)	(0.89)	(0.664)	(0.665)	(0.672)	(0.664)	(0.665)	(0.694)
	$-2.00$	3.185	2.392	2.921	3.294	3.494	3.984	3.324	3.594	4.363
		(1.215)	(1.41)	(2.759)	(1.215)	(1.218)	(1.237)	(1.215)	(1.221)	(1.258)
	$-1.00$	8.746	9.905	21.913	8.849	9.074	9.747	8.880	9.199	10.337
		(4.487)	(7.748)	(21.55)	(4.38)	(4.234)	(4.039)	(4.354)	(4.179)	(3.975)
Medium $\delta$	$-0.90$	10.289	12.512	28.813	10.364	10.557	11.209	10.389	10.673	11.815
		(5.586)	(10.199)	(28.453)	(5.427)	(5.207)	(4.9)	(5.389)	(5.122)	(4.787)
	$-0.70$	15.389	21.995	52.285	15.324	15.332	15.774	15.316	15.382	16.349
		(9.583)	(19.393)	(51.954)	(9.207)	(8.672)	(7.877)	(9.116)	(8.461)	(7.544)
	$-0.50$	27.254	45.594	100.849	26.703	26.023	25.487	26.576	25.806	25.701
		(20.05)	(42.809)	(100.591)	(19.042)	(17.567)	(15.247)	(18.793)	(16.969)	(14.184)
	$-0.30$	65.920	114.813	199.444	63.536	60.085	54.970	62.949	58.711	53.020
		(57.39)	(112.118)	(199.321)	(54.301)	(49.589)	(41.536)	(53.522)	(47.598)	(37.447)
	$-0.10$	247.398	302.473	339.732	242.328	233.856	216.818	240.993	229.971	206.603
		(239.522)	(300.377)	(339.785)	(233.622)	(223.576)	(202.408)	(232.054)	(218.878)	(188.759)
Small $\delta$	$-0.09$	264.147	313.469	345.132	259.428	251.464	235.118	258.179	247.778	225.070
		(256.414)	(311.411)	(345.192)	(250.877)	(241.36)	(220.92)	(249.399)	(236.871)	(207.441)
	$-0.07$	297.977	333.758	354.586	294.293	287.951	274.386	293.309	284.960	265.615
		(290.544)	(331.768)	(354.658)	(286.075)	(278.238)	(260.7)	(284.871)	(274.471)	(248.558)
	$-0.05$	329.467	350.643	361.988	327.143	323.068	314.007	326.516	321.112	307.858
		(322.324)	(348.71)	(362.069)	(319.251)	(313.753)	(300.896)	(318.416)	(311.057)	(291.502)
	$-0.03$	354.334	362.808	367.080	353.359	351.624	347.643	353.094	350.779	344.834
		(347.427)	(360.917)	(367.167)	(345.738)	(342.646)	(335.054)	(345.274)	(341.098)	(329.153)
	$-0.01$	368.194	369.189	369.674	368.077	367.867	367.376	368.045	367.764	367.022
		(361.422)	(367.32)	(369.765)	(360.612)	(359.088)	(355.108)	(360.387)	(358.304)	(351.771)
Initial $\delta$	0.00	370.000	370.001	370.001	370.000	370.000	370.000	370.000	370.000	370.000
		(363.247)	(368.136)	(370.092)	(362.558)	(361.25)	(357.779)	(362.365)	(360.573)	(354.814)
Small $\delta$	0.01	368.190	369.188	369.674	368.072	367.860	367.365	368.040	367.757	367.008
		(361.422)	(367.32)	(369.765)	(360.612)	(359.088)	(355.108)	(360.387)	(358.304)	(351.771)
	0.03	354.321	362.804	367.079	353.343	351.605	347.614	353.078	350.759	344.795
		(347.427)	(360.917)	(367.167)	(345.737)	(342.646)	(335.053)	(345.273)	(341.097)	(329.151)
	0.05	329.446	350.638	361.987	327.119	323.039	313.963	326.492	321.080	307.801
		(322.323)	(348.71)	(362.069)	(319.251)	(313.752)	(300.894)	(318.416)	(311.057)	(291.5)
	0.07	297.950	333.752	354.585	294.263	287.915	274.333	293.278	284.920	265.546
		(290.544)	(331.768)	(354.658)	(286.074)	(278.237)	(260.699)	(284.87)	(274.47)	(248.555)
	0.09	264.116	313.461	345.131	259.393	251.423	235.059	258.143	247.733	224.996
		(256.413)	(311.411)	(345.192)	(250.876)	(241.359)	(220.918)	(249.399)	(236.87)	(207.438)
Medium $\delta$	0.10	247.366	302.464	339.730	242.292	233.813	216.757	240.956	229.925	206.528
		(239.522)	(300.377)	(339.785)	(233.621)	(223.575)	(202.406)	(232.053)	(218.877)	(188.756)
	0.30	65.892	114.801	199.441	63.506	60.051	54.927	62.918	58.676	52.970
		(57.388)	(112.118)	(199.321)	(54.3)	(49.587)	(41.533)	(53.521)	(47.596)	(37.442)
	0.50	27.234	45.584	100.845	26.681	26.000	25.458	26.554	25.781	25.668
		(20.049)	(42.809)	(100.591)	(19.04)	(17.565)	(15.244)	(18.791)	(16.967)	(14.18)
	0.70	15.373	21.988	52.282	15.308	15.314	15.753	15.299	15.364	16.325
		(9.582)	(19.393)	(51.954)	(9.206)	(8.671)	(7.875)	(9.114)	(8.459)	(7.541)
	0.90	10.276	12.505	28.810	10.351	10.542	11.192	10.376	10.658	11.796
		(5.585)	(10.199)	(28.453)	(5.426)	(5.206)	(4.898)	(5.388)	(5.121)	(4.785)
Large $\delta$	1.00	8.735	9.899	21.910	8.837	9.061	9.732	8.868	9.186	10.320
		(4.485)	(7.748)	(21.55)	(4.379)	(4.232)	(4.037)	(4.353)	(4.177)	(3.973)
	2.00	3.179	2.389	2.919	3.288	3.488	3.976	3.318	3.588	4.355
		(1.215)	(1.41)	(2.759)	(1.214)	(1.217)	(1.237)	(1.214)	(1.22)	(1.258)
	3.00	1.759	1.091	0.781	1.843	1.994	2.346	1.866	2.068	2.609
		(0.66)	(0.644)	(0.889)	(0.664)	(0.665)	(0.671)	(0.664)	(0.665)	(0.694)

Note: Bold values indicate the smallest ARL among EWMA, EEWMA, and NEEWMA in each row.

,

Table 7. Detection performance of the EWMA, EEWMA, and NEEWMA control charts using MCA for logistic distribution with location $\mu =0$ and scale $\beta =1$ with $AR{L}_0=370$.

Chart		EWMA			EEWMA			NEEWMA
${\lambda }_1$		0.1	0.3	0.7	0.1	0.1	0.1	0.1	0.1	0.1
${\lambda }_2$		-	-	-	0.01	0.025	0.05	0.01	0.025	0.05
${\lambda }_3$		-	-	-	-	-	-	0.0025	0.00625	0.0125
Large $\delta$	$-3.00$	4.287	4.157	13.351	4.391	4.590	5.111	4.420	4.694	5.533
		(1.704)	(2.547)	(12.881)	(1.691)	(1.677)	(1.676)	(1.688)	(1.673)	(1.692)
	$-2.00$	7.805	10.620	46.667	7.877	8.055	8.640	7.900	8.161	9.170
		(3.753)	(8.198)	(46.307)	(3.658)	(3.533)	(3.381)	(3.635)	(3.488)	(3.341)
	$-1.00$	24.610	56.357	171.960	23.927	23.114	22.484	23.773	22.860	22.685
		(17.447)	(53.528)	(171.925)	(16.371)	(14.881)	(12.741)	(16.112)	(14.305)	(11.846)
Medium $\delta$	$-0.90$	29.724	70.093	194.034	28.730	27.489	26.263	28.501	27.071	26.236
		(22.156)	(67.321)	(194.045)	(20.712)	(18.697)	(15.756)	(20.363)	(17.911)	(14.501)
	$-0.70$	46.738	110.711	242.976	44.667	41.896	38.408	44.176	40.875	37.437
		(38.431)	(108.152)	(243.084)	(35.763)	(31.956)	(26.169)	(35.112)	(30.442)	(23.574)
	$-0.50$	83.221	176.536	294.326	79.034	73.072	64.365	78.011	70.728	60.932
		(74.51)	(174.35)	(294.53)	(69.551)	(62.215)	(50.336)	(68.319)	(59.2)	(44.614)
	$-0.30$	168.669	270.058	339.357	161.642	150.795	132.184	159.859	146.183	122.976
		(160.432)	(268.397)	(339.644)	(152.506)	(139.996)	(117.26)	(150.474)	(134.546)	(104.846)
	$-0.10$	327.849	355.859	366.370	324.691	319.199	307.248	323.843	316.586	299.378
		(321.389)	(354.674)	(366.705)	(317.452)	(310.476)	(294.574)	(316.387)	(307.093)	(283.341)
Small $\delta$	$-0.09$	335.130	358.471	367.055	332.452	327.768	317.461	331.731	325.529	310.577
		(328.759)	(357.301)	(367.392)	(325.314)	(319.17)	(304.97)	(324.379)	(316.172)	(294.767)
	$-0.07$	348.122	362.949	368.214	346.367	343.269	336.309	345.893	341.774	331.541
		(341.911)	(361.803)	(368.552)	(339.413)	(334.899)	(324.165)	(338.73)	(332.668)	(316.17)
	$-0.05$	358.519	366.373	369.087	357.566	355.872	352.003	357.308	355.049	349.296
		(352.437)	(365.246)	(369.427)	(350.761)	(347.69)	(340.153)	(350.301)	(346.152)	(334.31)
	$-0.03$	365.788	368.688	369.671	365.430	364.791	363.314	365.333	364.479	362.265
		(359.797)	(367.574)	(370.012)	(358.731)	(356.744)	(351.681)	(358.435)	(355.732)	(347.568)
	$-0.01$	369.528	369.854	369.964	369.488	369.415	369.247	369.477	369.380	369.126
		(363.586)	(368.748)	(370.305)	(362.845)	(361.44)	(357.732)	(362.638)	(360.714)	(354.588)
Initial $\delta$	0.00	370.000	370.001	370.000	370.000	370.000	370.000	370.000	370.000	370.000
		(364.065)	(368.895)	(370.342)	(363.365)	(362.035)	(358.502)	(363.17)	(361.346)	(355.485)
Small $\delta$	0.01	369.526	369.854	369.964	369.485	369.412	369.241	369.474	369.376	369.119
		(363.586)	(368.748)	(370.305)	(362.845)	(361.44)	(357.732)	(362.638)	(360.714)	(354.588)
	0.03	365.781	368.686	369.671	365.422	364.781	363.298	365.325	364.468	362.243
		(359.797)	(367.574)	(370.012)	(358.731)	(356.744)	(351.681)	(358.435)	(355.732)	(347.567)
	0.05	358.508	366.371	369.087	357.554	355.856	351.977	357.295	355.031	349.261
		(352.437)	(365.246)	(369.427)	(350.761)	(347.69)	(340.152)	(350.3)	(346.151)	(334.308)
	0.07	348.107	362.946	368.214	346.350	343.247	336.275	345.874	341.750	331.495
		(341.911)	(361.803)	(368.552)	(339.412)	(334.899)	(324.164)	(338.73)	(332.668)	(316.169)
	0.09	335.111	358.468	367.055	332.430	327.741	317.419	331.708	325.499	310.522
		(328.759)	(357.301)	(367.392)	(325.314)	(319.17)	(304.969)	(324.378)	(316.171)	(294.765)
Medium $\delta$	0.10	327.828	355.855	366.370	324.668	319.170	307.204	323.819	316.554	299.319
		(321.389)	(354.674)	(366.705)	(317.452)	(310.476)	(294.573)	(316.386)	(307.092)	(283.338)
	0.30	168.635	270.048	339.356	161.605	150.752	132.126	159.821	146.138	122.907
		(160.431)	(268.397)	(339.644)	(152.505)	(139.994)	(117.257)	(150.473)	(134.545)	(104.842)
	0.50	83.191	176.525	294.323	79.002	73.037	64.319	77.979	70.690	60.880
		(74.509)	(174.35)	(294.53)	(69.549)	(62.213)	(50.333)	(68.318)	(59.199)	(44.61)
	0.70	46.714	110.700	242.973	44.641	41.867	38.372	44.150	40.845	37.396
		(38.43)	(108.152)	(243.084)	(35.762)	(31.955)	(26.167)	(35.111)	(30.44)	(23.57)
	0.90	29.703	70.083	194.031	28.708	27.465	26.234	28.479	27.046	26.203
		(22.155)	(67.321)	(194.045)	(20.71)	(18.695)	(15.753)	(20.362)	(17.91)	(14.497)
Large $\delta$	1.00	24.591	56.347	171.957	23.907	23.092	22.458	23.752	22.837	22.655
		(17.446)	(53.528)	(171.925)	(16.369)	(14.879)	(12.738)	(16.111)	(14.303)	(11.843)
	2.00	7.794	10.614	46.664	7.866	8.043	8.626	7.889	8.149	9.155
		(3.752)	(8.198)	(46.307)	(3.656)	(3.531)	(3.38)	(3.634)	(3.486)	(3.339)
	3.00	4.280	4.153	13.349	4.383	4.582	5.102	4.413	4.686	5.523
		(1.704)	(2.547)	(12.881)	(1.69)	(1.676)	(1.675)	(1.687)	(1.673)	(1.691)

Note: Bold values indicate the smallest ARL among EWMA, EEWMA, and NEEWMA in each row.

and

Table 8. Detection performance of the EWMA, EEWMA, and NEEWMA control charts using MCA for Laplace distribution with location $\mu =0$ and scale $\beta =1$ with $AR{L}_0=370$.

Chart		EWMA			EEWMA			NEEWMA
${\lambda }_1$		0.1	0.3	0.7	0.1	0.1	0.1	0.1	0.1	0.1
${\lambda }_2$		-	-	-	0.01	0.025	0.05	0.01	0.025	0.05
${\lambda }_3$		-	-	-	-	-	-	0.0025	0.00625	0.0125
Large $\delta$	$-3.00$	3.110	2.930	11.896	3.191	3.345	3.749	3.213	3.425	4.077
		(1.108)	(1.516)	(11.383)	(1.104)	(1.103)	(1.117)	(1.104)	(1.104)	(1.135)
	$-2.00$	5.595	7.797	44.950	5.663	5.821	6.309	5.685	5.912	6.742
		(2.313)	(5.388)	(44.591)	(2.27)	(2.218)	(2.17)	(2.26)	(2.201)	(2.171)
	$-1.00$	17.277	48.131	170.508	16.795	16.275	16.058	16.691	16.137	16.387
		(10.754)	(45.242)	(170.508)	(10.023)	(9.076)	(7.863)	(9.853)	(8.731)	(7.421)
Medium $\delta$	$-0.90$	20.904	61.016	192.694	20.167	19.315	18.687	20.003	19.057	18.886
		(13.876)	(58.205)	(192.743)	(12.839)	(11.482)	(9.719)	(12.596)	(10.984)	(9.061)
	$-0.70$	33.351	100.124	241.931	31.646	29.498	27.164	31.253	28.759	26.752
		(25.318)	(97.589)	(242.083)	(23.153)	(20.245)	(16.298)	(22.641)	(19.153)	(14.749)
	$-0.50$	62.035	165.725	293.656	58.088	52.754	45.783	57.148	50.766	43.481
		(53.308)	(163.661)	(293.912)	(48.662)	(42.14)	(32.629)	(47.537)	(39.594)	(28.596)
	$-0.30$	138.500	262.496	339.070	130.432	118.430	99.520	128.419	113.525	91.215
		(130.15)	(261.097)	(339.413)	(121.151)	(107.468)	(84.609)	(118.883)	(101.736)	(73.424)
	$-0.10$	314.832	354.538	366.335	309.940	301.391	282.934	308.623	297.325	271.144
		(308.905)	(353.752)	(366.729)	(303.199)	(293.085)	(270.424)	(301.654)	(288.201)	(255.067)
Small $\delta$	$-0.09$	324.079	357.388	367.027	319.871	312.458	296.168	318.735	308.903	285.536
		(318.291)	(356.621)	(367.422)	(313.289)	(304.347)	(283.937)	(311.929)	(299.993)	(269.789)
	$-0.07$	340.866	362.280	368.197	338.039	332.984	321.508	337.271	330.525	313.709
		(335.333)	(361.545)	(368.594)	(331.75)	(325.239)	(309.824)	(330.769)	(322.018)	(298.635)
	$-0.05$	354.573	366.026	369.078	353.008	350.174	343.563	352.580	348.779	338.912
		(349.249)	(365.316)	(369.477)	(346.962)	(342.74)	(332.369)	(346.332)	(340.618)	(324.47)
	$-0.03$	364.304	368.561	369.668	363.708	362.619	360.030	363.545	362.079	358.161
		(359.13)	(367.868)	(370.068)	(357.838)	(355.413)	(349.209)	(357.48)	(354.174)	(344.217)
	$-0.01$	369.360	369.840	369.963	369.292	369.167	368.867	369.274	369.105	368.648
		(364.265)	(369.155)	(370.364)	(363.515)	(362.083)	(358.253)	(363.306)	(361.337)	(354.984)
Initial $\delta$	0.00	370.001	370.000	370.000	370.000	370.000	370.000	370.001	370.000	370.000
		(364.917)	(369.317)	(370.401)	(364.237)	(362.933)	(359.414)	(364.047)	(362.252)	(356.376)
Small $\delta$	0.01	369.358	369.839	369.963	369.289	369.163	368.861	369.270	369.101	368.639
		(364.265)	(369.155)	(370.364)	(363.515)	(362.083)	(358.252)	(363.306)	(361.337)	(354.984)
	0.03	364.297	368.560	369.668	363.699	362.608	360.011	363.536	362.067	358.136
		(359.13)	(367.868)	(370.068)	(357.838)	(355.413)	(349.209)	(357.479)	(354.173)	(344.216)
	0.05	354.561	366.024	369.078	352.994	350.156	343.533	352.565	348.759	338.872
		(349.249)	(365.316)	(369.477)	(346.962)	(342.74)	(332.368)	(346.331)	(340.617)	(324.468)
	0.07	340.849	362.277	368.196	338.020	332.960	321.470	337.251	330.498	313.657
		(335.332)	(361.545)	(368.594)	(331.75)	(325.239)	(309.823)	(330.769)	(322.017)	(298.633)
	0.09	324.058	357.385	367.026	319.848	312.428	296.122	318.710	308.871	285.476
		(318.291)	(356.621)	(367.422)	(313.289)	(304.346)	(283.935)	(311.929)	(299.992)	(269.786)
Medium $\delta$	0.10	314.810	354.534	366.334	309.915	301.359	282.885	308.597	297.290	271.080
		(308.905)	(353.752)	(366.729)	(303.199)	(293.084)	(270.422)	(301.653)	(288.2)	(255.064)
	0.30	138.468	262.487	339.068	130.398	118.391	99.469	128.383	113.483	91.154
		(130.149)	(261.097)	(339.413)	(121.15)	(107.467)	(84.606)	(118.882)	(101.734)	(73.42)
	0.50	62.009	165.715	293.654	58.060	52.724	45.745	57.120	50.734	43.437
		(53.307)	(163.661)	(293.912)	(48.661)	(42.138)	(32.626)	(47.536)	(39.593)	(28.592)
	0.70	33.329	100.114	241.928	31.623	29.474	27.135	31.230	28.733	26.719
		(25.317)	(97.589)	(242.083)	(23.152)	(20.243)	(16.295)	(22.639)	(19.151)	(14.745)
	0.90	20.886	61.007	192.691	20.149	19.295	18.664	19.985	19.037	18.860
		(13.875)	(58.205)	(192.743)	(12.838)	(11.48)	(9.716)	(12.595)	(10.982)	(9.058)
Large $\delta$	1.00	17.261	48.122	170.505	16.778	16.257	16.037	16.674	16.118	16.363
		(10.753)	(45.242)	(170.508)	(10.022)	(9.074)	(7.861)	(9.852)	(8.729)	(7.418)
	2.00	5.586	7.792	44.947	5.654	5.812	6.298	5.676	5.902	6.730
		(2.312)	(5.388)	(44.591)	(2.269)	(2.217)	(2.169)	(2.259)	(2.2)	(2.17)
	3.00	3.104	2.926	11.894	3.185	3.339	3.741	3.207	3.418	4.069
		(1.107)	(1.516)	(11.383)	(1.104)	(1.102)	(1.116)	(1.103)	(1.103)	(1.135)

Note: Bold values indicate the smallest ARL among EWMA, EEWMA, and NEEWMA in each row.

and Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, which are detailed as follows.

For the Normal distribution, with $AR{L}_0=370$, it was found that as ${\lambda }_1$ increases, the performance in detecting small and medium changes decreases, but conversely, the performance in detecting large changes increases. When ${\lambda }_1$ is constant, it was found that as ${\lambda }_2$ increases, the performance in detecting small and medium changes increases, but conversely, the performance in detecting large changes decreases. When ${\lambda }_1$ is constant and ${\lambda }_3={\lambda }_2/4$, it was found that increasing ${\lambda }_3$ increases the performance in detecting small and medium changes, but conversely, the performance in detecting large changes decreases. Furthermore, it was found that the increase in performance in detecting changes does not differ significantly when that change decreases, as shown in Table 6. For $AR{L}_0=200$ and 500, the analysis results are similar to $AR{L}_0=370$. The control chart is more effective at detecting small and medium changes when ${\lambda }_1$ is low, and ${\lambda }_2$ and ${\lambda }_3$ are high. Conversely, setting ${\lambda }_1$ high, ${\lambda }_2$ and ${\lambda }_3$ low results in higher efficiency in detecting large changes, as shown in Figure 4 and Figure 7.

For the logistic distribution, with $AR{L}_0=370$, it was found that as ${\lambda }_1$ increases, the efficiency in detecting small and medium changes decreases. Conversely, when ${\lambda }_1$ is low, the control chart’s efficiency in detecting large changes increases. When ${\lambda }_1$ is constant, it was found that when ${\lambda }_2$ increases, the efficiency in detecting small and medium changes increases, but conversely, the efficiency in detecting large changes decreases. When ${\lambda }_1$ is constant and ${\lambda }_3={\lambda }_2/4$, it was found that increasing ${\lambda }_3$ increases the efficiency in detecting small and medium changes, but conversely, the efficiency in detecting large changes decreases. Furthermore, it was found that the increase in efficiency in detecting changes does not differ significantly when that change decreases. Equal values are shown in Table 7. For $AR{L}_0=200$ and 500, the analysis results are similar to $AR{L}_0=370$. The control chart is more effective at detecting small and medium changes when ${\lambda }_1$ is low and ${\lambda }_2$ and ${\lambda }_3$ are high. Conversely, setting ${\lambda }_1$ high, ${\lambda }_2$ and ${\lambda }_3$ low results in higher efficiency in detecting large changes, as shown in Figure 6 and Figure 9.

For the Laplace distribution, for $AR{L}_0=370$, it is found that as ${\lambda }_1$ increases, the efficiency of detecting small and medium changes decreases. Conversely, when ${\lambda }_1$ is low, the control chart is less efficient at detecting large changes. The trend shows that when the change is large enough, a higher value of ${\lambda }_1$ indicates better performance. When ${\lambda }_1$ is constant, it is found that as ${\lambda }_2$ increases, the performance in detecting small and medium changes improves, but conversely, the performance in detecting large changes decreases. When ${\lambda }_1$ is constant and ${\lambda }_3={\lambda }_2/4$ is set, it is found that increasing ${\lambda }_3$ increases the performance in detecting small and medium changes, but conversely, the performance in detecting large changes decreases. Furthermore, it is observed that the increase in performance in detecting changes is not significantly different when the change decreases. Equal values are shown in Table 8. For $AR{L}_0=200$ and 500, the analysis results are similar to $AR{L}_0=370$. The control chart is more effective at detecting small and medium changes when ${\lambda }_1$ is low and ${\lambda }_2$ and ${\lambda }_3$ are high. Conversely, setting ${\lambda }_1$ high and ${\lambda }_2$ and ${\lambda }_3$ low results in higher efficiency in detecting large changes, as shown in Figure 5 and Figure 8.

6.3. Behavior of Detecting Under Symmetric Processes

This section presents the use of the MCA method to calculate ARL and SDRL to study the impact on the performance of EWMA, EEWMA, and NEEWMA control charts from the Location and Scale parameters of symmetrically distributed processes: Normal, Logistic, and Laplace. The control chart parameters were set to ${\lambda }_1=0.1$, ${\lambda }_2=0.05$, and ${\lambda }_3={\lambda }_2/4$, and the change level was greater than or equal to 0. The research results are shown in

Table 9. Detection behavior of the EWMA control charts using MCA for symmetric distribution with scale $\beta =1$ with $AR{L}_0=370$.

$\mathit{\delta }$	Normal Distribution			Logistic Distribution			Laplace Distribution
	$\mathit{\mu }=\mathbf{0}$	$\mathit{\mu }=\mathbf{5}$	$\mathit{\mu }=\mathbf{10}$	$\mathit{\mu }=\mathbf{0}$	$\mathit{\mu }=\mathbf{5}$	$\mathit{\mu }=\mathbf{10}$	$\mathit{\mu }=\mathbf{0}$	$\mathit{\mu }=\mathbf{5}$	$\mathit{\mu }=\mathbf{10}$
0	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00
	(363.247)	(363.247)	(363.247)	(364.065)	(364.065)	(364.065)	(364.917)	(364.917)	(364.917)
0.01	368.19	368.19	368.19	369.53	369.53	369.53	369.36	369.36	369.36
	(361.422)	(361.422)	(361.422)	(363.586)	(363.586)	(363.586)	(364.265)	(364.265)	(364.265)
0.03	354.32	354.32	354.32	365.78	365.78	365.78	364.30	364.30	364.30
	(347.427)	(347.427)	(347.427)	(359.797)	(359.797)	(359.797)	(359.13)	(359.13)	(359.13)
0.05	329.45	329.45	329.45	358.51	358.51	358.51	354.56	354.56	354.56
	(322.323)	(322.323)	(322.323)	(352.437)	(352.437)	(352.437)	(349.249)	(349.249)	(349.249)
0.07	297.95	297.95	297.95	348.11	348.11	348.11	340.85	340.85	340.85
	(290.544)	(290.544)	(290.544)	(341.911)	(341.911)	(341.911)	(335.332)	(335.332)	(335.332)
0.09	264.12	264.12	264.12	335.11	335.11	335.11	324.06	324.06	324.06
	(256.413)	(256.413)	(256.413)	(328.759)	(328.759)	(328.759)	(318.291)	(318.291)	(318.291)
0.1	247.37	247.37	247.37	327.83	327.83	327.83	314.81	314.81	314.81
	(239.522)	(239.522)	(239.522)	(321.389)	(321.389)	(321.389)	(308.905)	(308.905)	(308.905)
0.3	65.89	65.89	65.89	168.64	168.64	168.64	138.47	138.47	138.47
	(57.388)	(57.388)	(57.388)	(160.431)	(160.431)	(160.431)	(130.149)	(130.149)	(130.149)
0.5	27.23	27.23	27.23	83.19	83.19	83.19	62.01	62.01	62.01
	(20.049)	(20.049)	(20.049)	(74.509)	(74.509)	(74.509)	(53.307)	(53.307)	(53.307)
0.7	15.37	15.37	15.37	46.71	46.71	46.71	33.33	33.33	33.33
	(9.582)	(9.582)	(9.582)	(38.43)	(38.43)	(38.43)	(25.317)	(25.317)	(25.317)
0.9	10.28	10.28	10.28	29.70	29.70	29.70	20.89	20.89	20.89
	(5.585)	(5.585)	(5.585)	(22.155)	(22.155)	(22.155)	(13.875)	(13.875)	(13.875)
1	8.74	8.74	8.74	24.59	24.59	24.59	17.26	17.26	17.26
	(4.485)	(4.485)	(4.485)	(17.446)	(17.446)	(17.446)	(10.753)	(10.753)	(10.753)

,

Table 10. Detection behavior of the EWMA control charts using MCA under symmetric distribution with location $\mu =0$ with $AR{L}_0=370$.

$\mathit{\delta }$	Normal Distribution			Logistic Distribution			Laplace Distribution
	$\mathit{\beta }=\mathbf{1}$	$\mathit{\beta }=\mathbf{5}$	$\mathit{\beta }=\mathbf{10}$	$\mathit{\beta }=\mathbf{1}$	$\mathit{\beta }=\mathbf{5}$	$\mathit{\beta }=\mathbf{10}$	$\mathit{\beta }=\mathbf{1}$	$\mathit{\beta }=\mathbf{5}$	$\mathit{\beta }=\mathbf{10}$
0	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00
	(363.247)	(363.247)	(363.247)	(364.065)	(364.065)	(364.065)	(364.917)	(364.917)	(364.917)
0.01	368.19	369.93	369.98	369.53	369.98	370.00	369.36	369.98	369.99
	(361.422)	(363.174)	(363.229)	(363.586)	(364.046)	(364.061)	(364.265)	(364.891)	(364.91)
0.03	354.32	369.35	369.84	365.78	369.83	369.96	364.30	369.77	369.94
	(347.427)	(362.588)	(363.082)	(359.797)	(363.893)	(364.022)	(359.13)	(364.682)	(364.858)
0.05	329.45	368.19	369.55	358.51	369.53	369.88	354.56	369.36	369.84
	(322.323)	(361.422)	(362.789)	(352.437)	(363.586)	(363.945)	(349.249)	(364.265)	(364.754)
0.07	297.95	366.47	369.11	348.11	369.07	369.77	340.85	368.74	369.69
	(290.544)	(359.686)	(362.351)	(341.911)	(363.128)	(363.831)	(335.332)	(363.642)	(364.597)
0.09	264.12	364.20	368.53	335.11	368.47	369.62	324.06	367.93	369.48
	(256.413)	(357.396)	(361.767)	(328.759)	(362.518)	(363.677)	(318.291)	(362.814)	(364.389)
0.1	247.37	362.87	368.19	327.83	368.11	369.53	314.81	367.44	369.36
	(239.522)	(356.05)	(361.422)	(321.389)	(362.157)	(363.586)	(308.905)	(362.324)	(364.265)
0.3	65.89	314.24	354.32	168.64	353.67	365.78	138.47	348.15	364.30
	(57.388)	(306.979)	(347.427)	(160.431)	(347.538)	(359.797)	(130.149)	(342.741)	(359.13)
0.5	27.23	247.37	329.45	83.19	327.83	358.51	62.01	314.81	354.56
	(20.049)	(239.522)	(322.323)	(74.509)	(321.389)	(352.437)	(53.307)	(308.905)	(349.249)
0.7	15.37	186.96	297.95	46.71	295.25	348.11	33.33	274.80	340.85
	(9.582)	(178.647)	(290.544)	(38.43)	(288.418)	(341.911)	(25.317)	(268.307)	(335.332)
0.9	10.28	140.59	264.12	29.70	260.41	335.11	20.89	234.38	324.06
	(5.585)	(132.007)	(256.413)	(22.155)	(253.181)	(328.759)	(13.875)	(227.307)	(318.291)
1	8.74	122.41	247.37	24.59	243.24	327.83	17.26	215.30	314.81
	(4.485)	(113.762)	(239.522)	(17.446)	(235.812)	(321.389)	(10.753)	(207.958)	(308.905)

,

Table 11. Detection behavior of the EEWMA control charts using MCA under symmetric distribution with scale $\beta =1$ with $AR{L}_0=370$.

$\mathit{\delta }$	Normal Distribution			Logistic Distribution			Laplace Distribution
	$\mathit{\mu }=\mathbf{0}$	$\mathit{\mu }=\mathbf{5}$	$\mathit{\mu }=\mathbf{10}$	$\mathit{\mu }=\mathbf{0}$	$\mathit{\mu }=\mathbf{5}$	$\mathit{\mu }=\mathbf{10}$	$\mathit{\mu }=\mathbf{0}$	$\mathit{\mu }=\mathbf{5}$	$\mathit{\mu }=\mathbf{10}$
0	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00
	(357.779)	(357.779)	(357.779)	(358.502)	(358.502)	(358.502)	(359.414)	(359.414)	(359.414)
0.01	367.37	367.37	367.37	369.24	369.24	369.24	368.86	368.86	368.86
	(355.108)	(355.108)	(355.108)	(357.732)	(357.732)	(357.732)	(358.252)	(358.252)	(358.252)
0.03	347.61	347.61	347.61	363.30	363.30	363.30	360.01	360.01	360.01
	(335.053)	(335.053)	(335.053)	(351.681)	(351.681)	(351.681)	(349.209)	(349.209)	(349.209)
0.05	313.96	313.96	313.96	351.98	351.98	351.98	343.53	343.53	343.53
	(300.894)	(300.894)	(300.894)	(340.152)	(340.152)	(340.152)	(332.368)	(332.368)	(332.368)
0.07	274.33	274.33	274.33	336.28	336.28	336.28	321.47	321.47	321.47
	(260.699)	(260.699)	(260.699)	(324.164)	(324.164)	(324.164)	(309.823)	(309.823)	(309.823)
0.09	235.06	235.06	235.06	317.42	317.42	317.42	296.12	296.12	296.12
	(220.918)	(220.918)	(220.918)	(304.969)	(304.969)	(304.969)	(283.935)	(283.935)	(283.935)
0.1	216.76	216.76	216.76	307.20	307.20	307.20	282.89	282.89	282.89
	(202.406)	(202.406)	(202.406)	(294.573)	(294.573)	(294.573)	(270.422)	(270.422)	(270.422)
0.3	54.93	54.93	54.93	132.13	132.13	132.13	99.47	99.47	99.47
	(41.533)	(41.533)	(41.533)	(117.257)	(117.257)	(117.257)	(84.606)	(84.606)	(84.606)
0.5	25.46	25.46	25.46	64.32	64.32	64.32	45.75	45.75	45.75
	(15.244)	(15.244)	(15.244)	(50.333)	(50.333)	(50.333)	(32.626)	(32.626)	(32.626)
0.7	15.75	15.75	15.75	38.37	38.37	38.37	27.14	27.14	27.14
	(7.875)	(7.875)	(7.875)	(26.167)	(26.167)	(26.167)	(16.295)	(16.295)	(16.295)
0.9	11.19	11.19	11.19	26.23	26.23	26.23	18.66	18.66	18.66
	(4.898)	(4.898)	(4.898)	(15.753)	(15.753)	(15.753)	(9.716)	(9.716)	(9.716)
1	9.73	9.73	9.73	22.46	22.46	22.46	16.04	16.04	16.04
	(4.037)	(4.037)	(4.037)	(12.738)	(12.738)	(12.738)	(7.861)	(7.861)	(7.861)

,

Table 12. Detection behavior of the EEWMA control charts using MCA under symmetric distribution with location $\mu =0$ with $AR{L}_0=370$.

$\mathit{\delta }$	Normal Distribution			Logistic Distribution			Laplace Distribution
	$\mathit{\beta }=\mathbf{1}$	$\mathit{\beta }=\mathbf{5}$	$\mathit{\beta }=\mathbf{10}$	$\mathit{\beta }=\mathbf{1}$	$\mathit{\beta }=\mathbf{5}$	$\mathit{\beta }=\mathbf{10}$	$\mathit{\beta }=\mathbf{1}$	$\mathit{\beta }=\mathbf{5}$	$\mathit{\beta }=\mathbf{10}$
0	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00
	(357.779)	(357.779)	(357.779)	(358.502)	(358.502)	(358.502)	(359.414)	(359.414)	(359.414)
0.01	367.37	369.89	369.97	369.24	369.97	369.99	368.86	369.95	369.99
	(355.108)	(357.672)	(357.752)	(357.732)	(358.471)	(358.494)	(358.252)	(359.368)	(359.403)
0.03	347.61	369.05	369.76	363.30	369.73	369.93	360.01	369.59	369.90
	(335.053)	(356.813)	(357.537)	(351.681)	(358.224)	(358.433)	(349.209)	(358.995)	(359.309)
0.05	313.96	367.37	369.34	351.98	369.24	369.81	343.53	368.86	369.71
	(300.894)	(355.108)	(357.108)	(340.152)	(357.732)	(358.309)	(332.368)	(358.252)	(359.123)
0.07	274.33	364.88	368.70	336.28	368.52	369.63	321.47	367.78	369.44
	(260.699)	(352.579)	(356.465)	(324.164)	(356.995)	(358.124)	(309.823)	(357.144)	(358.844)
0.09	235.06	361.61	367.86	317.42	367.56	369.39	296.12	366.34	369.08
	(220.918)	(349.262)	(355.612)	(304.969)	(356.017)	(357.878)	(283.935)	(355.675)	(358.473)
0.1	216.76	359.70	367.37	307.20	366.99	369.24	282.89	365.49	368.86
	(202.406)	(347.32)	(355.108)	(294.573)	(355.439)	(357.732)	(270.422)	(354.809)	(358.252)
0.3	54.93	294.44	347.61	132.13	344.60	363.30	99.47	333.06	360.01
	(41.533)	(281.083)	(335.053)	(117.257)	(332.64)	(351.681)	(84.606)	(321.66)	(349.209)
0.5	25.46	216.76	313.96	64.32	307.20	351.98	45.75	282.89	343.53
	(15.244)	(202.406)	(300.894)	(50.333)	(294.573)	(340.152)	(32.626)	(270.422)	(332.368)
0.7	15.75	156.23	274.33	38.37	264.34	336.28	27.14	230.83	321.47
	(7.875)	(141.384)	(260.699)	(26.167)	(250.981)	(324.164)	(16.295)	(217.344)	(309.823)
0.9	11.19	114.78	235.06	26.23	223.07	317.42	18.66	185.50	296.12
	(4.898)	(99.918)	(220.918)	(15.753)	(209.084)	(304.969)	(9.716)	(171.253)	(283.935)
1	9.73	99.53	216.76	22.46	204.25	307.20	16.04	166.21	282.89
	(4.037)	(84.795)	(202.406)	(12.738)	(190.002)	(294.573)	(7.861)	(151.702)	(270.422)

,

Table 13. Detection behavior of the NEEWMA control charts using MCA under symmetric distribution with scale $\beta =1$ with $AR{L}_0=370$.

$\mathit{\delta }$	Normal Distribution			Logistic Distribution			Laplace Distribution
	$\mathit{\mu }=\mathbf{0}$	$\mathit{\mu }=\mathbf{5}$	$\mathit{\mu }=\mathbf{10}$	$\mathit{\mu }=\mathbf{0}$	$\mathit{\mu }=\mathbf{5}$	$\mathit{\mu }=\mathbf{10}$	$\mathit{\mu }=\mathbf{0}$	$\mathit{\mu }=\mathbf{5}$	$\mathit{\mu }=\mathbf{10}$
0	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00
	(354.814)	(354.814)	(354.814)	(355.485)	(355.485)	(355.485)	(356.376)	(356.376)	(356.376)
0.01	367.01	367.01	367.01	369.12	369.12	369.12	368.64	368.64	368.64
	(351.771)	(351.771)	(351.771)	(354.588)	(354.588)	(354.588)	(354.984)	(354.984)	(354.984)
0.03	344.80	344.80	344.80	362.24	362.24	362.24	358.14	358.14	358.14
	(329.151)	(329.151)	(329.151)	(347.567)	(347.567)	(347.567)	(344.216)	(344.216)	(344.216)
0.05	307.80	307.80	307.80	349.26	349.26	349.26	338.87	338.87	338.87
	(291.5)	(291.5)	(291.5)	(334.308)	(334.308)	(334.308)	(324.468)	(324.468)	(324.468)
0.07	265.55	265.55	265.55	331.50	331.50	331.50	313.66	313.66	313.66
	(248.555)	(248.555)	(248.555)	(316.169)	(316.169)	(316.169)	(298.633)	(298.633)	(298.633)
0.09	225.00	225.00	225.00	310.52	310.52	310.52	285.48	285.48	285.48
	(207.438)	(207.438)	(207.438)	(294.765)	(294.765)	(294.765)	(269.786)	(269.786)	(269.786)
0.1	206.53	206.53	206.53	299.32	299.32	299.32	271.08	271.08	271.08
	(188.756)	(188.756)	(188.756)	(283.338)	(283.338)	(283.338)	(255.064)	(255.064)	(255.064)
0.3	52.97	52.97	52.97	122.91	122.91	122.91	91.15	91.15	91.15
	(37.442)	(37.442)	(37.442)	(104.842)	(104.842)	(104.842)	(73.42)	(73.42)	(73.42)
0.5	25.67	25.67	25.67	60.88	60.88	60.88	43.44	43.44	43.44
	(14.18)	(14.18)	(14.18)	(44.61)	(44.61)	(44.61)	(28.592)	(28.592)	(28.592)
0.7	16.33	16.33	16.33	37.40	37.40	37.40	26.72	26.72	26.72
	(7.541)	(7.541)	(7.541)	(23.57)	(23.57)	(23.57)	(14.745)	(14.745)	(14.745)
0.9	11.80	11.80	11.80	26.20	26.20	26.20	18.86	18.86	18.86
	(4.785)	(4.785)	(4.785)	(14.497)	(14.497)	(14.497)	(9.058)	(9.058)	(9.058)
1	10.32	10.32	10.32	22.66	22.66	22.66	16.36	16.36	16.36
	(3.973)	(3.973)	(3.973)	(11.843)	(11.843)	(11.843)	(7.418)	(7.418)	(7.418)

and

Table 14. Detection behavior of the NEEWMA control charts using MCA for symmetric distribution with location $\mu =0$ with $AR{L}_0=370$.

$\mathit{\delta }$	Normal Distribution			Logistic Distribution			Laplace Distribution
	$\mathit{\beta }=\mathbf{1}$	$\mathit{\beta }=\mathbf{5}$	$\mathit{\beta }=\mathbf{10}$	$\mathit{\beta }=\mathbf{1}$	$\mathit{\beta }=\mathbf{5}$	$\mathit{\beta }=\mathbf{10}$	$\mathit{\beta }=\mathbf{1}$	$\mathit{\beta }=\mathbf{5}$	$\mathit{\beta }=\mathbf{10}$
0	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00	370.00
	(354.814)	(354.814)	(354.814)	(355.485)	(355.485)	(355.485)	(356.376)	(356.376)	(356.376)
0.01	367.01	369.88	369.97	369.12	369.96	369.99	368.64	369.95	369.99
	(351.771)	(354.691)	(354.783)	(354.588)	(355.449)	(355.476)	(354.984)	(356.32)	(356.362)
0.03	344.80	368.92	369.73	362.24	369.68	369.92	358.14	369.51	369.88
	(329.151)	(353.712)	(354.537)	(347.567)	(355.161)	(355.404)	(344.216)	(355.873)	(356.25)
0.05	307.80	367.01	369.25	349.26	369.12	369.78	338.87	368.64	369.66
	(291.5)	(351.771)	(354.048)	(334.308)	(354.588)	(355.26)	(324.468)	(354.984)	(356.027)
0.07	265.55	364.19	368.53	331.50	368.28	369.57	313.66	367.35	369.33
	(248.555)	(348.897)	(353.316)	(316.169)	(353.731)	(355.044)	(298.633)	(353.657)	(355.692)
0.09	225.00	360.49	367.57	310.52	367.17	369.29	285.48	365.63	368.90
	(207.438)	(345.137)	(352.345)	(294.765)	(352.594)	(354.758)	(269.786)	(351.904)	(355.247)
0.1	206.53	358.34	367.01	299.32	366.51	369.12	271.08	364.63	368.64
	(188.756)	(342.941)	(351.771)	(283.338)	(351.923)	(354.588)	(255.064)	(350.87)	(354.984)
0.3	52.97	286.81	344.80	122.91	340.88	362.24	91.15	326.82	358.14
	(37.442)	(270.154)	(329.151)	(104.842)	(325.75)	(347.567)	(73.42)	(312.114)	(344.216)
0.5	25.67	206.53	307.80	60.88	299.32	349.26	43.44	271.08	338.87
	(14.18)	(188.756)	(291.5)	(44.61)	(283.338)	(334.308)	(28.592)	(255.064)	(324.468)
0.7	16.33	147.25	265.55	37.40	253.50	331.50	26.72	216.49	313.66
	(7.541)	(129.11)	(248.555)	(23.57)	(236.657)	(316.169)	(14.745)	(199.357)	(298.633)
0.9	11.80	108.08	225.00	26.20	211.09	310.52	18.86	171.40	285.48
	(4.785)	(90.2)	(207.438)	(14.497)	(193.582)	(294.765)	(9.058)	(153.57)	(269.786)
1	10.32	93.92	206.53	22.66	192.26	299.32	16.36	152.84	271.08
	(3.973)	(76.32)	(188.756)	(11.843)	(174.517)	(283.338)	(7.418)	(134.831)	(255.064)

and Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15, which are detailed as follows.

For the EWMA control chart, when $AR{L}_0=370$ is set in Table 9, under a process with a constant $\sigma$ of 1, it was found that the ARL values at each level $\mu$ did not differ significantly across all symmetric distributions and all $\delta$ levels, demonstrating that the location parameter of a symmetric distribution does not affect the performance of the EWMA control chart. In Table 10, under a process with a constant $\mu$ of 0, the EWMA control chart performance decreased as the process had a higher value, indicating that the scale parameter of a symmetric distribution has an inverse effect on the EWMA control chart performance. For $AR{L}_0=200$ and 500 in Figure 10 and Figure 13, the Location parameter did not affect the EWMA control chart performance; conversely, the Scale parameter resulted in decreased EWMA control chart performance.

For the EEWMA control chart, when $AR{L}_0=370$ in Table 11, under processes with a constant $\mu$ scale of 1, the ARL values at each $\mu$ level were not significantly different across all symmetric distributions and all $\delta$ levels. This demonstrates that the Location parameter of a symmetric distribution does not affect the performance of the EEWMA control chart. In Table 12, under processes with a constant $\mu$ scale of 0, the performance of the EEWMA control chart decreased as the process had a higher $\sigma$, indicating that the scale parameter of a symmetric distribution has an inverse effect on the performance of the EEWMA control chart. For $AR{L}_0=200$ and 500 in Figure 11 and Figure 14, the Location parameter did not affect the performance of the EEWMA control chart; conversely, the Scale parameter resulted in decreased performance.

For the NEEWMA control chart, when $AR{L}_0=370$ in Table 13, under processes with a constant $\mu$ scale of 1, the ARL values at each $\mu$ level were not significantly different across all symmetric distributions and all $\delta$ levels. This demonstrates that the Location parameter of a symmetric distribution does not affect the performance of the NEEWMA control chart. In Table 14, under processes with a constant $\mu$ scale of 0, the performance of the NEEWMA control chart decreased as the process had a higher $\sigma$, indicating that the scale parameter of a symmetric distribution has an inverse effect on the performance of the NEEWMA control chart. For $AR{L}_0=200$ and 500 in Figure 12 and Figure 15, the Location parameter did not affect the performance of the NEEWMA control chart; conversely, the Scale parameter resulted in decreased performance.

Furthermore, a comparison of the performance of control charts across different distributions revealed that control charts were best at detecting changes occurring in processes with normal distributions, followed by the Laplace distribution and the logistic distribution, respectively.

7. Application

This section discusses the application of control charts to real-world data to demonstrate how they can be used in various fields. The application involves setting the control chart parameters to ${\lambda }_1=0.1$, ${\lambda }_2=0.05$ and ${\lambda }_3={\lambda }_2/4$ using Unemployment Insurance Claims (UI).

Unemployment insurance claims (UI) refer to the number of individuals applying for unemployment benefits, measured by the Initial Claims (ICSA) and Continued Claims (CCSA) indicators derived from Federal Reserve Economic Data (FRED). These indicators are important because they provide timely and high-frequency insights into labor market conditions, capturing both the inflow of newly unemployed individuals and the persistence of unemployment. In this study, the variables are constructed as rates of change in ICSA and CCSA to better capture short-term fluctuations and dynamic movements in labor market conditions. The dataset covers the period from January 2021 to December 2025, as illustrated by the time-series plots and ACF/PACF of the difference of rate of change in Figure 16 and Figure 17, and is divided into a training set (2021–2024) and a test set (2025). Before this division, an ARIMA(3,1,0) model was fitted to the data, and the residuals were extracted for structural change detection using the proposed control charts. The model was identified based on the time series behavior and the ACF and PACF plots shown in Figure 16 and Figure 17. After first-order differencing (d = 1), the series became approximately stationary. The ACF gradually decayed, while the PACF showed significant spikes up to lag 3 followed by a cutoff, indicating an AR(3) structure. Consequently, the ARIMA(3,1,0) model was selected for analysis.

First, an ARIMA(3,1,0) model is fitted to extract residuals for change detection. Because control charts require independent observations, the ARIMA model helps remove autocorrelation and isolate the independent components of the series. Residuals that meet the assumptions of independence and normality are then used for further analysis. The results show that residuals from the ARIMA(3,1,0) model exhibit an approximately symmetrical distribution, as shown in

Table 15. Distributional tests of ARIMA residuals for ICSA and CCSA.

UI Index	Mean	SD	Kolmogorov-Smirnov Test (p-Value)
			Normal	Logistic	Laplace
ICSA	0.0005	0.0635	0.7098 *	0.0207	0.4756 *
CCSA	0.0005	0.0259	0.3387 *	0.0054	0.2212 *

* $p-\mathrm{value}\ge 0.05$.

, and demonstrate approximate independence, supported by ACF and PACF plots where most autocorrelation values fall within confidence bounds in Figure 18 and Figure 19. The data are then split into a training set (2021–2024) for constructing control charts and a test set (2025) for detecting changes.

For the training set, the construction of EWMA, EEWMA, and NEEWMA control charts is based on the calculation of the mean and variance of the residual (error) values. These statistics are then used to determine the control limit parameters (L), including the upper control limit (UCL) and lower control limit (LCL). The results of the analysis are presented in the table below.

For the test dataset, the change detection results indicate that, for Initial Claims (ICSA), the EWMA, EEWMA, and NEEWMA control charts detected the first signal in months 5, 3, and 1, respectively, as illustrated in Figure 20. For Continued Claims (CCSA), the EWMA, EEWMA, and NEEWMA control charts detected the first signal in months 6, 5, and 1, respectively, as shown in Figure 21. These results demonstrate that the NEEWMA control chart exhibits higher sensitivity and is capable of detecting shifts more promptly than the EWMA and EEWMA approaches.

8. Discussion

This research shows that MCA provides stable and computationally efficient ARL estimation compared to MC under the symmetric distribution conditions considered. Unlike MC simulation, which requires repeated data generation and may suffer from sampling variability, the MCA uses analytical transition probability structures to estimate ARL and SDRL values more systematically. These findings align with the pioneering work of Brook and Evans [1], who first introduced the MCA for run length analysis, and with the comparative study by Champ and Rigdon [13], which demonstrated MCA’s effectiveness for control chart performance evaluation. Similar applications of MCA for weighted moving average control charts have also been reported in previous studies [10,12,14,15]. However, the conclusions of this study should be interpreted within the scope of its underlying assumptions, including the assumption of symmetric distributions and simplified transition structures. The results can be explained using Equation (29). By setting (i), (j), and (n) equal to 1, the transition probability can be expressed as

$$P^N=F_N\left(A\right)-F_N\left(B\right),$$

(30)

where $A={\displaystyle \frac{\delta }{\beta }}+L·{\displaystyle \frac{{\sigma }_0}{\beta }}·{\displaystyle \frac{\sqrt{Q^N}}{{\lambda }_1}}$ and $B={\displaystyle \frac{\delta }{\beta }}-L·{\displaystyle \frac{{\sigma }_0}{\beta }}·{\displaystyle \frac{\sqrt{Q^N}}{{\lambda }_1}}$. Accordingly, the ARL can be rewritten as $AR{L}^N$, as follows:

$$AR{L}^N=P^N{(1-P^N)}^{-1}$$

(31)

First, concerning the control limit parameter, the results indicate it remains unchanged across different location and scale parameters under symmetric distributions, as shown in Table 1. From Equation (30), when $\delta =0$,

$$P^N=F_N(L·{\displaystyle \frac{{\sigma }_0}{\beta }}·{\displaystyle \frac{\sqrt{Q^N}}{{\lambda }_1}})-F_N(-L·{\displaystyle \frac{{\sigma }_0}{\beta }}·{\displaystyle \frac{\sqrt{Q^N}}{{\lambda }_1}})=2F_N(L·{\displaystyle \frac{{\sigma }_0}{\beta }}·{\displaystyle \frac{\sqrt{Q^N}}{{\lambda }_1}})-1$$

This shows that the transition probability does not depend on the location parameter. Additionally, since ${\sigma }_0/\beta$ remains constant for the symmetric distributions considered, neither the location nor scale parameters directly affect the determination of the control limit. This property aligns with the theoretical behavior of symmetric distributions discussed in SPC literature [8]. However, this may not hold for asymmetric or discrete distributions, where skewness and distribution shape could influence transition probabilities differently.

Second, regarding control chart performance, the results show that monitoring efficiency improves as ${\lambda }_1$ decreases and ${\lambda }_2$ and ${\lambda }_3$ increase. From Equation (30), the term $(\sqrt{Q^N}/{\lambda }_1)$ increases when ${\lambda }_1$ decreases, leading to larger transition probabilities and faster shift detection. Conversely, increasing ${\lambda }_2$ and ${\lambda }_3$ decreases $\sqrt{Q^N}/{\lambda }_1$ through the relationship $Q^N\le Q^E\le Q^Z$ when ${\lambda }_1$ is fixed. These findings are consistent with the smoothing and sensitivity properties of EWMA-type control charts reported by Lucas and Saccucci [5], Naveed et al. [6], and Javed et al. [7]. In particular, the additional weighting structures in the EEWMA and NEEWMA charts improve flexibility and responsiveness to small and moderate process shifts. However, while added smoothing parameters enhance detection capability, they also increase parameter sensitivity and model complexity. Thus, inappropriate parameter choices may affect robustness and practical performance.

Third, under symmetric distribution settings, the location parameter does not significantly affect monitoring performance, whereas larger-scale parameters reduce control chart efficiency. This can be explained using Equation (30). Since the probability expression does not explicitly involve the location parameter, changes in location do not affect transition probabilities or ARL values. In contrast, the scale parameter appears in the denominator of the probability expression. As the scale increases, the cumulative probability decreases, resulting in larger ARL values and slower shift detection. Therefore, greater process variability reduces monitoring efficiency. Practically, this suggests that EWMA-type control charts may be less sensitive in highly variable process environments, even under symmetric distribution conditions.

Overall, the findings suggest that the proposed MCA framework offers an effective analytical alternative for evaluating EWMA, EEWMA, and NEEWMA control chart performance under symmetric distributions. Compared with MC simulation, MCA reduces computational burden and provides deterministic estimates without simulation variability, consistent with prior analytical SPC studies [1,12,13]. However, this study is limited to symmetric distributions and simplified transition assumptions. The method’s performance may vary when applied to asymmetric, discrete, autocorrelated, or more complex stochastic processes. Additionally, although the MCA showed favorable estimation performance here, comparisons with other analytical approximation methods, such as integral equation approaches discussed by Champ and Rigdon [13], were beyond this work’s scope. Future research could extend this framework to broader distributional settings and alternative analytical methods.

9. Conclusions

This research introduced a method for estimating ARL and SDRL values of EWMA, EEWMA, and NEEWMA control charts under symmetric distributions using MCA. The framework relies on transition probabilities between in-control and out-of-control states, and its performance was compared to MC. Results showed that MCA provides stable and highly accurate ARL and SDRL estimates, with MPE and MAPE values close to zero compared to MC, while significantly reducing computational time. These findings suggest that MCA is an effective analytical alternative for evaluating EWMA-type control charts under the symmetric distribution conditions studied.

The comparison of EWMA, EEWMA, and NEEWMA charts revealed that smaller ${\lambda }_1$ values improve sensitivity to larger process shifts, while larger ${\lambda }_2$ and ${\lambda }_3$ values enhance detection of small to moderate changes. This highlights the importance of carefully selecting smoothing parameters to balance sensitivity and monitoring stability. In practical settings with highly sensitive processes—such as in medical, food, or chemical industries—choosing appropriate smoothing parameters can improve the detection of subtle process changes.

The study also examined the effects of process parameters across symmetric distributions, including normal, logistic, and Laplace. It found that the location parameter has little impact on chart performance, whereas increasing the scale parameter reduces monitoring efficiency by increasing ARL values. Practically, this suggests that processes with high variability may benefit from standardization or rescaling before monitoring.

Additionally, the proposed method was applied to unemployment insurance claims data, using ICSA and CCSA from 2021 to 2025. The results showed that the NEEWMA chart detected structural changes faster than the EEWMA and EWMA charts, demonstrating the advantage of additional smoothing for identifying subtle shifts in real-world data.

However, these conclusions are based on assumptions of symmetric distributions and simplified transition structures. The method’s performance may differ with asymmetric, discrete, autocorrelated, or more complex stochastic processes. Also, while MCA outperformed MC simulation here, comparisons with other analytical methods like integral equations or quasi-Markov approximations were beyond this study’s scope.

Future research could explore ARL and SDRL estimation for asymmetric and discrete distributions such as gamma, Weibull, and Poisson, and for processes with temporal dependence. Further studies may also adapt this framework to other control charts, including nonparametric, multivariate, and variance-based monitoring schemes.