Enhancing Sensitivity of Nonparametric Tukey Extended EWMA-MA Charts for Effective Process Mean Monitoring

Abstract

A control chart is a crucial statistical process control (SPC) instrument for identifying method variances that may undermine product efficacy. The combined control chart has been utilized to enhance recognition capability. When testing a methodology, nonparametric statistics make a strong and compelling case when the distribution of a quality feature is uncertain. The primary focus of monitoring this work is to offer a novel control chart to support the surveillance of mean activities. This chart will incorporate a Tukey method, an extended exponentially weighted moving average control chart, and a moving average control chart called the Nonparametric EEWMA-MA chart. The Monte Carlo simulation facilitates assessments for evaluating system performance using average run lengths (ARL) based on zero-state. The comparison analysis demonstrates that the sensitivity of the suggested chart surpasses that of the conventional control chart (including the moving average (MA) chart, the extended exponentially weighted moving average (EEWMA) chart, and the mixed extended exponentially weighted moving average-moving average (EEWMA-MA) chart) in rapidly detecting changes that fluctuate with varying parameter settings by examining the minimal ARL. A simplified monitoring scenario using data on vinyl chloride can be employed to demonstrate the feasibility of the proposed technique.

1. Introduction

The Statistical Process Control (SPC) happens in real-time throughout the manufacturing process, gathering input data from the operations that produce the products. Statistical approaches are employed to determine when the procedure is under control. Statistically generated data processing provides a graphical representation of variation, enhancing comprehension of the process. The control chart is an effective SPC instrument that ensures replacements are executed before producing defective items and that the process is restored to control. The critical assumption in SPC is that the process adheres to a normal distribution, also known as symmetry. Control charts commonly utilized in tracking processes include the Shewhart [1] chart; the CUSUM chart, frequently called cumulative sum, suggested in Page [2]; the EWMA chart, or exponentially weighted moving average control chart, developed through Robert [3]; and the MA chart, often known as the moving average, presented by Khoo [4]. In 2018, to improve process monitoring, Naveed et al. [5] devised an EEWMA chart, or extended exponentially weighted moving average control chart, that builds on the EWMA framework; this chart outperforms both the Shewhart and EWMA charts in terms of detecting changes quickly. The MA and EEWMA control charts promptly identify minor variations in operations. Moreover, numerous control charts were constructed for use in the process. Patel & Divecha [6] introduced a modified exponentially weighted moving average control chart (MEWMA), which was later reapproved by Khan et al. [7]. Shamma & Shamma [8] introduced a double exponentially weighted moving average control chart (DEWMA). A TEWMA chart, meaning triple exponentially weighted moving average control chart, was delineated by Alevizakos et al. [9]. Abbas promoted an HWMA or homogeneously weighted moving average control chart [10].

A combined control chart approach uses different control charts to keep track of other parts of a process, like the mean and variability, and it also uses different statistical methods to better find shifts. This could involve using charts to keep track of the mean and standard deviation together, or using a mix of methods like CUSUM and EWMA [11]. Some benefits of using integrated control charts are as follows: First, they have better sensitivity, which makes it easier to find changes in process parameters, even if they are small or happen slowly over time. Consequently, this ensures comprehensive monitoring. Third, they have the ability to adapt by changing the combinations to meet the needs of both the process and management. The MA control chart having span w is derived from the mean of the w newest data. The structure of this chart is less complex compared to the similarities of EWMA and EEWMA charts; nonetheless, it is less proficient in identifying minor process alterations [12]. Even so, the EEWMA control chart offers an enhanced structure derived from EWMA and is excellent at identifying minor process shifts. Consequently, the integrated EEWMA-MA can enhance the monitoring of the process mean.

Currently, the mixed control chart is the preferred method for enhancing the efficacy of control charts in process monitoring. Numerous authors developed the mathematics and evaluated the success of mixed control charts. In order to keep track of changes in procedures, Aslam et al. [13] created a combined DMA-EWMA method to monitor quality that follows an exponential distribution, and their results show that this new chart works better than the existing control charts. Abid et al. [14] developed the HWMA-CUSUM chart, which monitors the process mean, proving that conventional control charts are inferior to the suggested chart, particularly in detecting minor changes. Raza et al. [15] developed an EWMA mixed MA control chart using statistics with covariance terms, and the findings show that this graph is better than the ones that came before it. Naveed et al. [16] developed the EEWMA-MA control chart utilizing auxiliary information to assess performance based on average run length. Zubair et al. [17] evaluated the efficacy of the mixed HWMA-CUSUM control chart using auxiliary information, focusing on their run length characteristics. A mixed MEWMA and MCUSUM control chart was constructed by Devianto et al. [18] to monitor the process. The investigation indicates that the proposed chart surpasses all alternative methods for identifying shifts in the process location parameter. Furthermore, Raza et al. [19] propose that the mixed control chart with MA emphasizes the most recent w samples, with the influence of prior observations decreasing exponentially over time, in contrast to DEWMA and TEWMA, which blend EWMA statistics with additional EWMA charting metrics, thereby continuously using information from the most recent to the earliest observations.

Traditional and mixed control charts were developed based on normal distribution or recall parametric control charts. In reality, specific processes may not conform to standard assumptions; thus, nonparametric or distribution-free control charts are being investigated as alternate methods for monitoring procedures based on target values. In cases where the workflow distribution is unclear or when non-normal occurrences occur, Alemi [20] proposed the Tukey control chart as a simple and effective tool for tracking the mean operation. It has become an effective alternative to parametric control charts for monitoring processes. Utilizing the Tukey control chart offers several significant advantages. First, it exhibits strong robustness against outliers and non-normal distributions, as it avoids using mean and standard deviation, which are influenced by extreme values. Employing the median and interquartile range ensures the chart remains effective even in the event of unusual data [21]. Second, the chart is effective with small sample sizes, when parametric estimators may lack precision. This renders it beneficial for early-stage monitoring or in scenarios where data collection is costly or challenging [22]. Third, the Tukey chart is responsive to alterations in the process median, making it particularly effective for detecting shifts in central tendency when the mean is unreliable or uninformative due to data skewness. The chart is easily comprehensible, as it is typically presented as a boxplot with upper and lower bounds. This renders it a valuable instrument for quality professionals within industrial and non-industrial environments [23]. There is also literature on creating better or hybrid TCC layouts. Riaz et al. [24] introduced the MEC-TCC. The Tukey (TCC) and individual/moving range control charts are shown by Khaliq et al. [25]. Adsiz et al. [26] offered the Tukey-EWMA control chart with a variable sample period. The Tukey MEWMA-MA was delineated by Talordphop et al. [27]. Instead of utilizing the Tukey control chart, Abu-Shawiesh et al. [28] suggested using other robust estimator-based control charts.

Nevertheless, Khaliq et al. [29] developed the Tukey-EWMA chart, which was established by Khaliq et al. [30] with appropriate subgrouping. Mahmood et al. [31] introduced the TEWMA-Tukey control charts, utilizing both repeat and single-sample techniques for both normal and abnormal processes. Talordphop et al. [32] proposed the EEWMA-Tukey. All review results indicate that control charts can swiftly respond to modifications and be utilized in diverse contexts with few limitations. However, References [33,34] relate to applied improvements in work.

None of the control charts is universally superior for detecting changes. This means numerous researchers have developed novel control charts to identify changes and restore the process to normalcy rapidly. So, this research employs the concept of merging two effective charts to recognize modifications quickly. Embracing the benefits of mixed control charts and their user-friendliness in non-standard process situations, the Nonparametric EEWMA-MA was proposed to enhance the detection capabilities for small changes in the procedure mean parameter. Using the Monte Carlo simulation to get zero-state average run length (ARL) values allowed us to assess the control charts’ efficacy. Zero-state ARL property is effective in preventing the production of defective products since it promptly detects the initial out-of-control signal when a process is initially assumed to be in control. It is highly effective for beginning monitoring or when a process is anticipated to remain stable from the outset. Meanwhile, zero-state metrics are better at rapidly detecting significant shifts. Ultimately, we apply the proposed chart in a real-world context and contrast it with current control charts to illustrate its practical relevance.

2. The General Model of Control Charts

This section delineates the comprehensive model of control charts, encompassing (i) the moving average (MA) chart, (ii) the extended exponentially weighted moving average (EEWMA) chart, (iii) the mixed extended exponentially weighted moving average—moving average (EEWMA-MA) chart, and (iv) the proposed Nonparametric EEWMA-MA chart. Let $T_{ij}$ be the i sample $i=1,2,3,\dots$ and the j observation $j=1,2,3,\dots ,n$ considering that the variables in concern are generally distributed with a mean of $\mu$ and a variance of ${\sigma }^2$ and are entirely independent.

Furthermore, the concept for the combination of two control charts utilizes the statistics of the first chart and employs the control limits of the second chart. A control chart typically looks like the following:

2.1. The MA Control Chart

To obtain the MA statistic, we use the following formula to calculate the moving average of period i for each span ($w$):

$$M{A}_i=\left\{\begin{array}{l}{\displaystyle \frac{T_i+T_{i-1}+T_{i-2}+\dots }{i}}, i<w\\ {\displaystyle \frac{T_i+T_{i-1}+\dots +T_{i-w+1}}{w}}, i\ge w\end{array}\right.$$

(1)

The control boundary coefficients provide the MA chart’s control limits $C_1,$ as follows:

$$\begin{array}{l}UCL=\mu +C_1{\displaystyle \frac{\sigma }{\sqrt{i}}}, i<w\\ LCL=\mu -C_1{\displaystyle \frac{\sigma }{\sqrt{w}}}, i\ge w\end{array}$$

(2)

2.2. The EEWMA Control Chart

A description of the EEWMA statistic is provided below:

$$E_i={\beta }_1{T}_i-{\beta }_2{T}_{i-1}+(1-{\beta }_1+{\beta }_2)E_{i-1},$$

(3)

where the smoothing parameters ${\beta }_1$, ${\beta }_2$ range from 0 to 1, which $0<{\beta }_1\le 1$ and $0\le {\beta }_2<{\beta }_1$. The value of $E_0$ and $X_0$ is taken as the target mean. A comprehensive derivation of the mean, variance, and control limits is presented in Naveed et al. [5]. The EEWMA chart’s control boundaries are provided by

$$\begin{array}{l}UCL=\mu +C_2\sigma \sqrt{{\displaystyle \frac{{\beta }_1^2+{\beta }_2^2-2{\beta }_1{\beta }_2\left(1-{\beta }_1+{\beta }_2\right)}{2\left({\beta }_1-{\beta }_2\right)-{\left({\beta }_1-{\beta }_2\right)}^2}}}\\ LCL=\mu -C_2\sigma \sqrt{{\displaystyle \frac{{\beta }_1^2+{\beta }_2^2-2{\beta }_1{\beta }_2\left(1-{\beta }_1+{\beta }_2\right)}{2\left({\beta }_1-{\beta }_2\right)-{\left({\beta }_1-{\beta }_2\right)}^2}}}\end{array}$$

(4)

where $C_2$, an indication of the appropriate limit, is provided on the EEWMA control chart.

2.3. The EEWMA-MA Control Chart

With the incorporation of a double-smoothing structure, the suggested EEWMA-MA framework adds methodological uniqueness by applying two different layers of memory-based filtering in succession: Extended Exponentially Weighted Moving Average (EEWMA) and Moving Average (MA). The first layer, called EEWMA, extends the conventional EWMA structure by using dual weighting parameters (${\beta }_1$ and ${\beta }_2$) to provide the current and past data varying degrees of relevance. This improves the chart’s ability to capture both current and lag changes, allowing it to adjust its sensitivity to a wider variety of shift patterns. To further stabilize the control statistic and lessen the impact of short-term process noise or fluctuation, the second layer, MA, then applies a smoothing window to the EEWMA outputs.

The EEWMA-MA mixed control chart combines the strengths of both EEWMA and MA charts. Its key strength points are

(i)

Enhanced sensitivity to small and moderate shifts: EEWMA is particularly sensitive to slight changes in a process because it gives more weight to newer data. MA smooths down short-term changes, making them less noisy and easier to find when they are modest. When combined, the EEWMA-MA chart can detect a wider variety of shift sizes, which improves its ability to find shifts in general.

(ii)

Less noise and smoother signals: MA smooths out random noise, which reduces variability. EEWMA already smooths data by giving more weight to more recent data. They work together to make a control chart that is less likely to give false alerts but still responds to important changes.

(iii)

Customization flexibility: The weights in the EEWMA and the span size in the MA can be changed to fit the needs of the process. The EEWMA-MA chart can be used in a wide range of industrial and non-industrial settings since it is so flexible.

(iv)

A better Average Run Length (ARL) profile: Mixed charts usually have superior ARL performance, namely longer latency to false alarm when in control and higher ARL₀. Lower ARL₁ means that things may be found more quickly when they are out of hand. Due to this double benefit, EEWMA-MA is more effective than either EEWMA or MA alone.

Here is the definition of the EEWMA-MA statistic:

$$E{M}_i={\beta }_{1}M{A}_i-{\beta }_{2}M{A}_{i-1}+(1-{\beta }_1+{\beta }_2)E{M}_{i-1}.$$

(5)

The expected value and asymptotic variance of the EEWMA-MA statistic are given as

$$E\left(E{M}_i\right)=\mu$$

(6)

$$V\left(E{M}_i\right)={\displaystyle \frac{{\sigma }^2}{w}}\left[{\displaystyle \frac{{\beta }_1^2+{\beta }_2^2-2{\beta }_1{\beta }_2\left(1-{\beta }_1+{\beta }_2\right)}{2\left({\beta }_1-{\beta }_2\right)-{\left({\beta }_1-{\beta }_2\right)}^2}}\right].$$

(7)

Specification of certain operations in algebra is also characterized as

$$E{M}_i={\beta }_{1}M{A}_i+bM{A}_{i-1}+abM{A}_{i-2}+a^{2}bM{A}_{i-3}+\dots +a^{i-2}bM{A}_1-a^{i-1}{\beta }_{2}M{A}_0+a^{i}E{M}_0$$

$$E(E{M}_i)=\mu \left[({\beta }_1-{\beta }_2)\{1+a+a^2+\dots +a^{i-2}+a^{i-1}\}+a^i\right]$$

where

$$a=(1-{\beta }_1+{\beta }_2), b=(a{\beta }_1-{\beta }_2).$$

Then,

$$E(E{M}_i)=\mu \left[\left({\beta }_1-{\beta }_2\right)\left\{{\displaystyle \frac{1-a^i}{1-a}}\right\}+a^i\right]$$

$$V\left(E{M}_i\right)={\displaystyle \frac{{\sigma }^2}{w}}\left[\left({\beta }_1^2+{\beta }_2^2\right)\left\{{\displaystyle \frac{1-a^{2i}}{1-a^2}}\right\}-2a{\beta }_1{\beta }_2\left\{{\displaystyle \frac{1-{(a^2)}^{i-1}}{1-a^2}}\right\}\right].$$

Therefore, the EEWMA-MA charts control boundaries by

$$\begin{array}{l}UCL=\mu +C_3\sigma \sqrt{\left({\displaystyle \frac{1}{w}}\right){\displaystyle \frac{{\beta }_1^2+{\beta }_2^2-2{\beta }_1{\beta }_2\left(1-{\beta }_1+{\beta }_2\right)}{2\left({\beta }_1-{\beta }_2\right)-{\left({\beta }_1-{\beta }_2\right)}^2}}}\\ LCL=\mu -C_3\sigma \sqrt{\left({\displaystyle \frac{1}{w}}\right){\displaystyle \frac{{\beta }_1^2+{\beta }_2^2-2{\beta }_1{\beta }_2\left(1-{\beta }_1+{\beta }_2\right)}{2\left({\beta }_1-{\beta }_2\right)-{\left({\beta }_1-{\beta }_2\right)}^2}}}\end{array}$$

(8)

where $C_3$ is an indication of the appropriate limit is provided on the EEWMA-MA control chart.

2.4. The Tukey Control Chart

Tukey is a nonparametric control chart that employs the following control limits:

$$\begin{array}{l}UCL=Q_3+C (IQR)\\ LCL=Q_1-C (IQR)\end{array}$$

(9)

where IQR is the quartile range $(Q_3-Q_1)$, $Q_1$ and $Q_3$ are the first and the third quartiles, and $C$ is the control limits coefficient for the Tukey.

2.5. The Proposed Nonparametric EEWMA-MA Control Chart

In modern quality control applications, it is increasingly common to encounter process data that violate classical assumptions—particularly the assumption of normality. Real-world data often exhibit skewness, heavy tails, or outliers due to measurement errors, process drifts, or environmental disturbances. Under such conditions, traditional parametric control charts like Shewhart, EWMA, or even the EEWMA-MA may experience inflated false alarm rates or degraded shift detection performance. The EEWMA-MA framework retains memory of past data and smooths short-term variation, and when combined with the Tukey function, it becomes both sensitive and robust. The suggested chart blends the benefits of the mixed EEWMA-MA and Tukey control charts, including the former being responsive to slight shifts. At the same time, the other one is independent of distribution parameters. The proposed chart concept is analogous to EEWMA-MA, but substitutes the mean with the first and third quartiles. The upper and lower control limits do not utilize the standard deviation parameter; instead, their are derived using the interquartile range (IQR). The statistics can be conveyed by the EEWMA-MA statistic, and the optimal control limit from Equation (8) can be written as follows:

$$\begin{array}{l}UCL=Q_3+C_4\left(IQR\right)\sqrt{\left({\displaystyle \frac{1}{w}}\right){\displaystyle \frac{{\beta }_1^2+{\beta }_2^2-2{\beta }_1{\beta }_2\left(1-{\beta }_1+{\beta }_2\right)}{2\left({\beta }_1-{\beta }_2\right)-{\left({\beta }_1-{\beta }_2\right)}^2}}}\\ LCL=Q_1-C_4\left(IQR\right)\sqrt{\left({\displaystyle \frac{1}{w}}\right){\displaystyle \frac{{\beta }_1^2+{\beta }_2^2-2{\beta }_1{\beta }_2\left(1-{\beta }_1+{\beta }_2\right)}{2\left({\beta }_1-{\beta }_2\right)-{\left({\beta }_1-{\beta }_2\right)}^2}}}\end{array}$$

(10)

where $C_4$ represents the control limit coefficient for the process considered in control. Moreover, these $Q_1$ and $Q_3$ are the first and third quartiles, respectively, and IQR is the interquartile range $(Q_3-Q_1)$. The structure diagram of the proposed model is presented in Figure 1.

3. The Run Length Method of the Control Chart

The run length method and its related aspects are usually used to assess the efficacy of a control chart. The number of statistics provided on a chart before it signals an out-of-control state is called its run length. The average run length is an important performance metric. The mean quantity of samples recorded on a control chart is the average run length (ARL) before the first out-of-control alarm occurs. Scholarly works have provided a variety of approaches to ARL analysis [35,36]. Two types of ARL are ARL₀, which is the average run length when the equipment is under control, and ARL₁, which is the average run length when the mechanism is not under control. Under typical circumstances, when operations are managed effectively, ARL₀ is generally configured sufficiently high enough to avert false alerts. Conversely, the ARL₁ must be conservative to detect any process shift promptly. However, some publications look at the ARL’s steady state and zero states, for example, Alevizakos et al. [37]. The ARL is divided as follows:

$$ARL={\displaystyle \frac{{\displaystyle \sum _{i=1}^{N}R{L}_i}}{N}}.$$

(11)

where $R{L}_i$ is the quantity of samples required before the method becomes unmanageable for the initial instance.

This study employs a Monte Carlo simulation, utilizing an analysis technique, to produce computational results that evaluate the precision of control charts throughout 100,000 iterations. An increased quantity of repetitions leads to a reduced variance [38]. Here is the description of the simulation algorithm:

Step 1: Construct 10,000 random variables with a specified distribution.

Step 2: Establish the pattern parameters $\left(\beta \right)$ and control limit coefficient $\left(C\right)$ for an already-determined ARL₀ = 370 value.

Step 3: Calculate the test statistics, establish control limits, and record the number of data points while the tracking statistic attains the control limit, commonly called the run length.

Step 4: Execute steps 1–3 100,000 times to attain the ARL for the specified process mean shifts.

Step 5: Evaluate the efficacy of control charts exhibiting the minimal ARL₁.

Additionally, to determine the ideal parameters $\omega ,{\beta }_1,{\beta }_2$ for a suggested chart, extensive simulations will be conducted to ensure that the standard deviation of the ARL estimation error is below one percent of the real ARL [39].

Roles and interpretation of parameters: The parameters ${\beta }_1$ and ${\beta }_2$ define the weighting of current and previous observations in the extended EWMA component. When ${\beta }_1$ and ${\beta }_2$ decrease, the memory duration is extended, making it more responsive to invisible, enduring alterations. When ${\beta }_1$ and ${\beta }_2$ increase, they exhibit a more rapid response to abrupt alterations, but with increased variability [40,41]. The parameter $\omega$ indicates the size of the moving average (MA) window utilized for post-smoothing. Increasing $\omega$ enhances noise absorption but slows detection, whereas dropping $\omega$ simplifies reaction time but may lead to a rise in false alarms [3,42]. Researchers typically select initial numbers based on prior research and studies. For instance, ${\beta }_1$ and ${\beta }_2$ often range from 0.05 to 0.2, whereas $\omega$ generally falls between 3 and 7. Subsequently, practitioners enhance these values using process-specific simulation or retrospective analysis to determine the optimal ARLs (ARL₀ and ARL₁) for maintaining control and identifying out-of-control conditions [41,43]. Suggestions have been made to employ optimization techniques such as response surface methodology [44] or evolutionary algorithms to assist in parameter adjustment in more complex monitoring scenarios. To obtain solutions, follow the method described in Figure 2.

4. Analysis Study

This study examines the efficacy of the proposed Nonparametric EEWMA-MA, MA, EEWMA, and EEWMA-MA charts when using the normal (0,1), logistic (6,2), and Student’s t (10) distributions. These distributions are commonly utilized because of their mathematical characteristics and their widespread application in modeling diverse events. The normal distribution is frequently selected because of the Central Limit Theorem, the logistic distribution is advantageous for modeling symmetrical data, and the Student’s t distribution is suitable for small sample sizes or if the population standard deviation is unknown. We ran Monte Carlo simulations with the control chart’s smoothing parameters set to 0.1 and 0.25 to see how they performed, with the moving average chart spans (w) set at 3, 5, and 10, constant shifts of 0.05, 0.1, 0.25, 0.5, 0.75, 1, 1.5, and 2. ARL₀ = 370 was utilized due to its common application in process management. We employed the smoothing parameter within [5,9] in the simulation, as it is widely used in manufacturing. The beneficial effect for all control charts was assessed using the average run length based on zero-state, quantified by the minimum average run length (ARL₁). The indicated values in the table below demonstrate that the chart minimized ARL₁.

Table 1. Analyzing the ARL values for control charts with a normal distribution when w = 3.

Shift	MA ${\mathit{C}}_1=2.98$	${\mathit{\beta }}_1=0.25$$, {\mathit{\beta }}_2=0.1$			${\mathit{\beta }}_1=0.10$$, {\mathit{\beta }}_2=0.03$
		EEWMA ${\mathit{C}}_2=2.97$	EEWMA-MA ${\mathit{C}}_3=4.04$	Nonparametric EEWMA-MA ${\mathit{C}}_4=8.056$	EEWMA ${\mathit{C}}_2=2.7$	EEWMA-MA ${\mathit{C}}_3=4.11$	Nonparametric EEWMA-MA ${\mathit{C}}_4=11.96$
0	370.14	370.62	370.44	370.08	370.95	370.50	370.18
0.05	360.60	345.27	335.46	227.14	321.40	314.25	220.55
0.1	338.53	307.31	263.43	201.44	233.16	228.17	198.31
0.2	233.25	153.20	141.18	89.55	118.70	106.13	87.26
0.3	112.63	96.69	76.14	54.86	61.18	56.93	52.21
0.5	97.62	41.66	29.14	25.84	26.23	24.54	22.66
1	11.14	13.16	7.52	7.48	9.18	8.35	7.46
1.5	4.51	5.23	4.64	4.59	5.18	4.52	4.57
2	2.52	2.66	2.61	2.60	3.43	2.60	2.59
3	1.15	1.71	1.66	1.80	1.69	1.49	1.38

Note: The lowest score symbolizes strength.

,

Table 2. Analyzing the ARL values for control charts with a normal distribution when w = 5.

Shift	MA ${\mathit{C}}_1=2.88$	${\mathit{\beta }}_1=0.25$$, {\mathit{\beta }}_2=0.1$			${\mathit{\beta }}_1=0.10$$, {\mathit{\beta }}_2=0.03$
		EEWMA ${\mathit{C}}_2=2.97$	EEWMA-MA ${\mathit{C}}_3=4.67$	Nonparametric EEWMA-MA ${\mathit{C}}_4=5.87$	EEWMA ${\mathit{C}}_2=2.7$	EEWMA-MA ${\mathit{C}}_3=4.11$	Nonparametric EEWMA-MA ${\mathit{C}}_4=14.66$
0	370.94	370.62	370.56	370.45	370.95	370.60	370.42
0.05	348.61	345.27	334.99	226.94	321.40	313.78	210.65
0.1	307.91	307.31	260.57	200.73	233.16	224.82	200.66
0.2	163.76	153.20	134.33	88.08	118.70	104.47	82.35
0.3	99.87	96.69	71.63	49.21	61.18	55.13	42.79
0.5	50.99	41.66	26.91	24.88	26.23	23.81	21.67
1	10.06	13.16	6.66	6.37	9.18	7.73	6.33
1.5	3.76	5.23	3.95	3.77	5.18	4.17	3.78
2	1.98	2.66	2.05	1.99	3.43	2.03	1.99
3	1.10	1.71	1.47	1.45	1.69	1.32	1.30

Note: The lowest score symbolizes strength.

,

Table 3. Analyzing the ARL values for control charts with a normal distribution when w = 10.

Shift	MA ${\mathit{C}}_1=2.75$	${\mathit{\beta }}_1=0.25$$, {\mathit{\beta }}_2=0.1$			${\mathit{\beta }}_1=0.10$$, {\mathit{\beta }}_2=0.03$
		EEWMA ${\mathit{C}}_2=2.97$	EEWMA-MA ${\mathit{C}}_3=5.495$	Nonparametric EEWMA-MA ${\mathit{C}}_4=12.94$	EEWMA ${\mathit{C}}_2=2.7$	EEWMA-MA ${\mathit{C}}_3=6.275$	Nonparametric EEWMA-MA ${\mathit{C}}_4=19.24$
0	370.46	370.62	370.65	370.52	370.95	370.18	370.51
0.05	250.54	345.27	331.76	223.42	321.40	321.16	220.11
0.1	204.39	307.31	246.41	189.55	233.16	222.08	187.48
0.2	160.16	153.20	119.56	87.23	118.70	98.83	85.67
0.3	93.33	96.69	61.02	47.73	61.18	51.06	46.64
0.5	21.13	41.66	22.17	20.95	26.23	21.96	19.32
1	13.62	13.16	5.26	5.11	9.18	6.98	4.92
1.5	1.46	5.23	2.16	2.09	5.18	3.71	2.03
2	1.08	2.66	1.17	1.16	3.43	1.31	1.13
3	1.00	1.71	1.05	1.04	1.69	1.07	1.02

Note: The lowest score symbolizes strength.

,

Table 4. Analyzing the ARL values for control charts with a Logistic distribution when w = 3.

Shift	MA ${\mathit{C}}_1=5.72$	${\mathit{\beta }}_1=0.25,$${\mathit{\beta }}_2=0.1$			${\mathit{\beta }}_1=0.10,$${\mathit{\beta }}_2=0.03$
		EEWMA ${\mathit{C}}_2=6.693$	EEWMA-MA ${\mathit{C}}_3=13.71$	Nonparametric EEWMA-MA ${\mathit{C}}_4=27.32$	EEWMA ${\mathit{C}}_2=12.481$	EEWMA-MA ${\mathit{C}}_3=31.534$	Nonparametric EEWMA-MA ${\mathit{C}}_4=80.56$
0	370.45	370.32	370.36	370.55	370.45	370.53	370.54
0.05	369.73	359.45	359.36	327.43	356.95	346.43	326.12
0.1	366.51	341.72	339.16	284.44	339.95	303.43	282.33
0.2	360.20	331.95	316.44	245.95	303.46	155.96	242.46
0.3	355.09	318.95	289.94	157.97	271.60	128.97	155.44
0.5	318.45	286.46	230.45	111.98	196.38	112.97	90.06
1	224.40	190.41	180.47	127.99	170.82	164.98	116.54
1.5	145.23	139.83	109.48	82.49	123.01	93.64	62.23
2	92.48	77.87	55.99	50.00	77.61	53.97	40.03
3	36.39	15.15	15.09	10.10	15.14	10.08	9.67

Note: The lowest score symbolizes strength.

,

Table 5. Analyzing the ARL values for control charts with a Logistic distribution when w = 5.

Shift	MA ${\mathit{C}}_1=5.414$	${\mathit{\beta }}_1=0.25,$${\mathit{\beta }}_2=0.1$			${\mathit{\beta }}_1=0.10,$${\mathit{\beta }}_2=0.03$
		EEWMA ${\mathit{C}}_2=6.693$	EEWMA-MA ${\mathit{C}}_3=18.614$	Nonparametric EEWMA-MA ${\mathit{C}}_4=38.12$	EEWMA ${\mathit{C}}_2=12.481$	EEWMA-MA ${\mathit{C}}_3=40.746$	Nonparametric EEWMA-MA ${\mathit{C}}_4=107.05$
0	370.19	370.32	370.42	370.82	370.45	370.92	370.80
0.05	369.43	359.45	357.92	320.43	356.95	345.42	318.83
0.1	366.20	341.72	336.43	281.94	339.95	304.43	280.05
0.2	348.82	331.95	315.43	242.45	303.46	305.43	240.23
0.3	335.06	318.95	257.44	154.46	271.60	252.93	152.84
0.5	290.34	286.46	229.45	184.98	196.38	228.45	182.25
1	170.06	190.41	152.96	121.99	170.82	124.47	115.33
1.5	93.46	139.83	91.13	82.39	123.01	68.98	62.20
2	52.75	67.87	51.03	42.19	67.61	33.99	42.03
3	19.75	15.15	11.58	10.21	15.14	8.99	8.80

Note: The lowest score symbolizes strength.

,

Table 6. Analyzing the ARL values for control charts with a Logistic distribution when w = 10.

Shift	MA ${\mathit{C}}_1=5.042$	${\mathit{\beta }}_1=0.25,$${\mathit{\beta }}_2=0.1$			${\mathit{\beta }}_1=0.10,$${\mathit{\beta }}_2=0.03$
		EEWMA ${\mathit{C}}_2=6.693$	EEWMA-MA ${\mathit{C}}_3=26.286$	Nonparametric EEWMA-MA ${\mathit{C}}_4=61.19$	EEWMA ${\mathit{C}}_2=12.481$	EEWMA-MA ${\mathit{C}}_3=57.56$	Nonparametric EEWMA-MA ${\mathit{C}}_4=160.05$
0	370.38	370.32	370.43	370.42	370.45	370.44	370.45
0.05	365.33	359.45	353.42	317.43	356.95	345.43	315.22
0.1	362.27	341.72	325.43	261.44	339.95	303.93	260.16
0.2	338.36	331.95	303.93	197.46	303.46	282.44	196.67
0.3	310.82	318.95	256.44	155.46	271.60	246.94	152.19
0.5	242.92	286.46	222.95	196.48	196.38	205.95	181.78
1	111.09	139.41	132.97	118.49	130.82	121.97	114.43
1.5	53.63	39.83	30.98	23.99	23.01	30.68	22.18
2	29.74	27.87	24.29	20.99	27.61	21.99	20.98
3	11.86	15.15	11.02	10.00	15.14	8.49	8.32

Note: The lowest score symbolizes strength.

,

Table 7. Analyzing the ARL values for control charts with a Student’s t distribution when w = 3.

Shift	MA ${\mathit{C}}_1=3.48$	${\mathit{\beta }}_1=0.25,$${\mathit{\beta }}_2=0.1$			${\mathit{\beta }}_1=0.10,$${\mathit{\beta }}_2=0.03$
		EEWMA ${\mathit{C}}_2=3.54$	EEWMA-MA ${\mathit{C}}_3=4.66$	Nonparametric EEWMA-MA ${\mathit{C}}_4=8.56$	EEWMA ${\mathit{C}}_2=3.10$	EEWMA-MA ${\mathit{C}}_3=4.65$	Nonparametric EEWMA-MA ${\mathit{C}}_4=12.25$
0	370.65	370.04	370.65	370.58	370.72	370.44	370.62
0.05	369.06	357.90	345.57	342.25	336.50	329.20	325.87
0.1	341.59	322.36	299.42	295.96	264.04	252.93	250.64
0.2	293.12	219.37	180.22	171.84	139.39	128.64	124.27
0.3	222.76	143.31	104.86	101.61	78.59	71.72	68.80
0.5	123.75	59.71	41.44	39.55	34.16	30.66	25.54
1	27.35	14.18	10.02	8.44	11.29	10.00	8.36
1.5	8.28	6.56	4.53	4.43	6.33	5.43	4.41
2	3.47	3.87	2.58	2.33	4.20	3.54	2.30
3	1.32	1.78	1.00	1.00	2.28	1.85	1.00

Note: The lowest score symbolizes strength.

,

Table 8. Analyzing the ARL values for control charts with a Student’s t distribution when w = 5.

Shift	MA ${\mathit{C}}_1=3.32$	${\mathit{\beta }}_1=0.25,$${\mathit{\beta }}_2=0.1$			${\mathit{\beta }}_1=0.10,$${\mathit{\beta }}_2=0.03$
		EEWMA ${\mathit{C}}_2=3.54$	EEWMA-MA ${\mathit{C}}_3=5.323$	Nonparametric EEWMA-MA ${\mathit{C}}_4=10.28$	EEWMA ${\mathit{C}}_2=3.10$	EEWMA-MA ${\mathit{C}}_3=5.62$	Nonparametric EEWMA-MA ${\mathit{C}}_4=14.92$
0	370.87	370.04	370.62	370.61	370.72	371.02	370.53
0.05	358.69	357.90	341.30	340.51	336.50	340.50	323.32
0.1	333.61	322.36	283.74	280.61	264.04	250.77	241.84
0.2	253.55	219.37	168.13	166.68	139.39	129.39	123.88
0.3	171.63	143.31	94.22	91.95	78.59	68.76	65.51
0.5	76.95	59.71	35.93	33.39	34.16	29.24	24.43
1	15.38	14.18	8.66	8.25	11.29	9.35	8.11
1.5	5.31	6.56	3.81	3.52	6.33	5.04	3.42
2	2.64	3.87	2.15	2.10	4.20	3.29	2.08
3	1.24	1.78	0.73	0.70	2.28	1.67	1.00

Note: The lowest score symbolizes strength.

and

Table 9. Analyzing the ARL values for control charts with a Student’s t distribution when w = 10.

Shift	MA ${\mathit{C}}_1=3.10$	${\mathit{\beta }}_1=0.25,$${\mathit{\beta }}_2=0.1$			${\mathit{\beta }}_1=0.10,$${\mathit{\beta }}_2=0.03$
		EEWMA ${\mathit{C}}_2=3.54$	EEWMA-MA ${\mathit{C}}_3=6.20$	Nonparametric EEWMA-MA ${\mathit{C}}_4=12.18$	EEWMA ${\mathit{C}}_2=3.10$	EEWMA-MA ${\mathit{C}}_3=7.04$	Nonparametric EEWMA-MA ${\mathit{C}}_4=16.82$
0	370.80	370.04	370.82	370.71	370.72	370.88	370.58
0.05	352.95	357.90	339.63	338.85	336.50	323.59	320.04
0.1	303.51	322.36	274.00	272.27	264.04	237.71	235.81
0.2	194.77	219.37	143.22	141.34	139.39	117.38	116.21
0.3	114.94	143.31	78.65	76.22	78.59	63.32	61.19
0.5	44.65	59.71	29.16	27.78	34.16	26.25	24.22
1	9.86	14.18	6.66	6.41	11.29	8.33	7.65
1.5	4.08	6.56	2.91	2.83	6.33	4.44	3.41
2	2.29	3.87	1.43	1.21	4.20	2.78	2.05
3	1.17	1.78	0.42	0.41	2.28	1.35	1.00

Note: The lowest score symbolizes strength.

exhibit the ARL for different smoothing constants (${\beta }_1=0.1, {\beta }_2=0.03$ and ${\beta }_1=0.25, {\beta }_2=0.10$) values, varied spans (w = 3, 5, and 10), and different distributions. The suggested Tukey EEWMA-MA chart is better than the MA, EEWMA, and EEWMA-MA charts for all shifts because it finds process changes faster, as shown by ARL values. Nonetheless, significant changes in the normal distribution indicate that the MA chart is doing better, as demonstrated in Table 1, Table 2 and Table 3.

The remark from the simulation findings with mixed control chart statistics indicated that as the smoothing parameter score increased, the control limits’ width decreased. Nevertheless, as the span size expanded, the length of the control limits grew, resulting in a decrease in ARL₁.

The comparative analysis of the suggested chart against existing control charts (MA, EEWMA, and EEWMA-MA) was presented in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. In the context of normal distribution, it has been observed that the ARL₁ values for the Tukey EEWMA-MA control chart are inferior to those of the MA, EEWMA, and EEWMA-MA charts for all shift values, except for the significant shift, where the MA chart exceeds them, as indicated in Table 1, Table 2 and Table 3.

Table 4, Table 5 and Table 6 illustrate the Logistic distribution; our investigation demonstrated that the ARL₁ values for the proposed chart remain consistently below those of the MA, EEWMA, and EEWMA-MA charts across all shift values. The examination of Student’s t distribution yields results consistent with the Logistic distribution, indicating that the suggested chart surpasses the MA, EEWMA, and EEWMA-MA charts across all shifts, as displayed in Table 7, Table 8 and Table 9.

According to the comparison review, we assess the ARL performance of the suggested chart in comparison to the TEWMA-Tukey chart [31], the Tukey CUSUM-MA chart [45], and the Tukey MA-DEWMA control chart [46]. Analogous to a Tukey-based control chart’s ability to swiftly detect a shift in the mean process, the modeling findings indicated that this event is likewise feasible.

5. Application

For the practical application of the MA, EEWMA, EEWMA-MA, and the proposed Tukey EEWMA-MA control chart, the dataset contains vinyl chloride measurements collected from clean-up gradient monitoring water sources (unit: mg/L) [47]. Applying control charts to monitor vinyl chloride concentrations in water sources involves monitoring these levels over time to identify potential pollution and ensure the water remains drinkable. It was employed to monitor fluctuations in the average concentration of vinyl chloride. This enables the identification of quantities that deviate from the usual range, potentially indicating an issue that requires investigation. The whole data set is confirmed to exhibit a non-normal distribution with a p-value of 0.000029. Figure 3 presents the outcomes of the data utilized in control charts. Figure 3a demonstrates that the MA control chart fails to recognize shifts, whereas the EEWMA control chart similarly does not detect the shift, as depicted in Figure 3b. This suggests that the EEWMA-MA and the proposed Tukey EEWMA-MA control charts exhibit enhanced capability to swiftly identify shifts, as depicted in Figure 3c,d. The mentioned detections clearly indicate that the proposed layout offers greater sensitivity for the identification of unique causes. This implies that the process may suggest a potential problem demanding more investigation. These results could be applicable to several quality control contexts, including hazard recognition, risk evaluation, process regulation and monitoring, waste management, and conservation of the environment.

6. Conclusions

Nonparametric control charts provide a reliable and flexible tool for evaluating a process when the true dispersion in an indicator metric is unclear. This article presented the Nonparametric EEWMA-MA control chart to identify shifts in the process mean without distribution. Monte Carlo simulations with symmetric distributions aim to test control charts with the lowest possible ARL₁ and see how well they work. As a nonparametric plug-in, we added the Tukey function to the EEWMA-MA structure to increase robustness even more. Through the use of weighting data according to their proximity to the median, this innovation reduces the impact of extreme or non-normal observations. A nonparametric EEWMA-MA chart is the end product, demonstrating enhanced resilience in the presence of noisy or non-normal process circumstances while retaining sensitivity to slight changes. This progress is consistent with recent studies that support hybrid and reliable SPC technologies in challenging settings. According to the findings, the suggested chart works better for spotting different changes. Additionally, a concrete case study illustrates the proposed chart’s viability and efficacy in identifying procedural modifications compared to alternative control charts. Nonetheless, the constraints of the suggested chart may be addressed by disregarding the covariance element, the Average Run Length (ARL) derived from simulation may require an extended duration, the computational complexity, and non-study performance under autocorrelated data. Finally, a thorough analysis might be conducted to identify the ideal values in the smoothing factor and span based on various shifts of concern—the efficacy of the proposed plotting scheme warrants further investigation to monitor procedural dispersion. This concept can be expanded to include autocorrelated processes, applied to differing sample sizes, or construct the corresponding single-filter weighting function and directly optimize its form. A systematic parameter selection framework, including empirical guidelines and optimization algorithms, along with an assessment of the proposed charts in terms of their realistic steady-state performance and the conditional expected delay, will be developed in future work to enhance the practical applicability of the EEWMA-MA control chart.