Variable sample size based EWMA control chart with an exponential scaling mechanism for production process monitoring

Abstract

Statistical Process Control is essential for ensuring process stability and detecting variations in a production environment. This study introduces a control chart based on the Exponentially Weighted Moving Average (EWMA) that uses an adaptive sample size. The proposed approach enhances shift detection by dynamically adjusting the sample size in response to changes in process variation. Extensive Monte Carlo simulations were performed to assess the performance of the proposed control chart, focusing on metrics such as the Average Run Length (ARL) and the Standard Deviation of Run Length (SDRL). The findings show that the new chart surpasses both the Fixed Sample Size EWMA (FEWMA) and the Variable Sample Size EWMA charts, particularly in detecting small to moderate shifts in the process. This approach strikes a balance between detection sensitivity and computational efficiency, enabling prompt identification of process changes while maintaining robustness during in-control conditions. To illustrate its practical applicability, a real-world dataset was analyzed, demonstrating the effectiveness of the proposed method in actual process monitoring scenarios.

Introduction

Statistical process control (SPC) involves the use of statistical methods for the monitoring, control, and improvement of processes within a given manufacturing industry. Control charts (CC) are statistical tools utilized in SPC that aim at analyzing both the normal and abnormal deviations in a process. CCs are frequently categorized into memory-less and memory-type types. Page²⁷ introduced CUSUM charts and Robert²⁸ proposed the EWMA chart. Because these Shewhart control charts are memoryless, they are perfect for the detection of large shifts occurring during a process. Opposite to this, both CUSUM and EWMA charts are conditional on stored past information and prove more admirable in spotting small to moderate shifts. This offers a lot more since process changes can vary greatly in magnitude and thus monitoring techniques become as critical as possible to such shifts¹. The idea of joining Shewhart-style control limits to CUSUM and EWMA charts was first introduced by Lucas² and later on by Lucas & Saccucci³. It helps in detecting large shifts while simultaneously being sensitive to small shifts. These hybrid control schemes are expected to perform well across a fairly wide range of shift magnitudes. From this perspective, the Adapted Exponentially Weighted Moving Average (AEWMA) chart was developed to advance further compared to the standard EWMA chart. The AEWMA chart is designed to dynamically adjust its parameters so that it achieves greater flexibility and robustness in detecting large and small process changes. Different methods of adaptive control charting were also developed using M-estimators and score functions for different types of shifts in processes such as assignable causes, anomalies, and outliers.

Examples of such functions include the Huber function and the Hampel function. These functions are commonly used because they reduce the influence of extreme values and are more responsive to moderate changes. For instance, Capizzi and Masarotto⁴ proposed using an adaptive EWMA chart incorporating the Huber and Bi-square functions to detect shifts of varying magnitudes more effectively. Jiang et al.⁵ adopted the Huber function in their ACUSUM chart to identify both small and large process shifts. Recent developments have further advanced control chart designs based on score functions. Zaman et al.^6,7 introduced several adaptive EWMA and CUSUM charts using Huber and Hampel functions to improve the detection of both deteriorations and improvements in process quality. These efforts highlight the adaptability and improved sensitivity achieved by integrating robust score functions into adaptive statistical process control methods. Additional enhancements to adaptive control charts have been proposed in various studies, including those by Zhao et al.⁸, Abubakar et al.^9,10, Arshad et al.¹¹, Sarwar and Noor-ul-Amin¹², and Rasheed et al.¹³.

A lot of research regarding adaptive control charts has been conducted in terms of variable sampling intervals (VSI) and variable sample sizes (VSS), using which one can manipulate the timing or size of samples collected according to the process state. In this article, we discuss the VSS approach, which adjusts the sample size based on what happens in the current state as a means of capturing even the slightest of changes. The early works in this regard were: Prabhu et al.¹⁴ and Costa¹⁵, who were the first to apply VSS to control charts. De Magalhães et al.¹⁶ provided a detailed review of the various designs of adaptive charts and VSS strategies. Zhou and Lian¹⁷ went further to present a VSS-NP chart with adaptive sampling whereby the sample size varied according to whether or not the process mean remained within a safe range or indicated potential problems. The next step has been to replicate these findings in the context of EWMA charts. Ji et al.^18,19 developed an EWMA chart with variable sampling, while Chou and Chen²⁰ proposed a cost-efficient EWMA scheme with variable sampling intervals. Recently, Ayesha et al.²¹ proposed a refined version of the EWMA control chart for responding more adequately to process shifts by tracking changes in the variability of the process. Li, Qin, and Wu²² contributed to designing an adaptive EWMA control chart for applications where the distribution of the process is unknown. The charts are best applied in environments where the process mean as well as the sample size changes very quickly. Their flexible construction guarantees trustworthy monitoring while reducing the possibility of spurious alarms. This EWMA chart is effective in applications demanding accurate tracking of small and large shifts, particularly when obtaining the samples is easy. This chart readjusts its control limits with respect to the sample size and a dynamic weighting factor, which improves its sensitivity to detect process instability. This highlights the significant role that adaptive control charts play in modern quality management. In this study, the process data is assumed to follow a normal distribution, which is a standard and widely accepted approach in the design and application of control charts. This assumption simplifies both analysis and implementation. However, comparative evaluations show that real-world processes often deviate from normality, involving both symmetric and skewed non-normal distributions. It is important to note that the current analysis focuses solely on the robustness of the charts under normal conditions and does not account for other influencing factors. All calculations have been carried out using data that follows a normal distribution. If readers want to conduct research in non-normal situations, they are advised to consult the literature that deals with those cases. Graham et al.²³, Abbas et al.^24,25 have discussed the nonnormal cases in control charting.

In contrast to previous designs like the VSS EWMA (VEWMA) chart, the new method presented by Ji et al.^18,19. and subsequently improved upon by Riaz and Abbas (2021) utilizes distinct, adaptive processes to make changes in real-time. This model surpasses previous models by utilizing a more continuous and dynamic form. It provides for real-time addition or subtraction of sample size, often in subsequent process stages, depending on the extent to which sample values drift away from the process mean. Smooth adjustment guarantees that there are no sudden changes, and it facilitates methods that are efficient in detecting small changes. In the AEWMA method, the smoothing constant is not static it adjusts automatically. The process has a constant λ value but variable sample sizes. It is based on comparatively stable statistics and does not need highly accurate calibration. Through exponential scaling, the proposed approach is superior to previous variable sample size control charts. This creates more stable monitoring and less emphasis on small fluctuations, such as witnessed with EWMA and VEWMA charts. Also, since exponential functions are easy to calculate, this approach steers clear of complicated decisions that are typical in most conventional charting methods. Further, the results in Sects. 4 and 5 indicate that the approach performs better than VEWMA for small to medium-sized shifts in the process. Because the proposed control chart can detect changes, runs easily on computers, and its estimate improves as more data is used, it is very useful for ongoing industrial monitoring. Section 2 of the paper will introduce the basic methods of the EWMA control chart, the conventional EWMA, and VEWMA. In Sect. 3, a new exponential scaling-based EWMA control chart with adaptive sample size is developed. Section 4 is an evaluation of the ARL and SDRL of these charts and provides a comprehensive comparison of the CCs and an outline of the main findings, whereas practical applications are thoroughly analyzed in Sect. 5. Finally, Sect. 6 presents a summary of the major discoveries and conclusions of the paper.

Existing control charts

This section describes the present methodologies and procedures. Section 2.1 describes the EWMA CC with a fixed sample size. The variable sample size EWMA is discussed in Sect. 2.2.

Fixed sample size EWMA (FEWMA) control chart

In this subsection, we describe the classical EWMA control chart denoted as FEWMA when the sample size is fixed in process monitoring. Then the EWMA statistic for the tth sample can be expressed as follows.

1

where = smoothing constant. The initial value of the statistic is mostly chosen as the target value of the process. We have the control limits as follows:

2

here stands for the number of samples within each subgroup, and represents the coefficient determining the control limits to achieve a desirable Average Run Length (ARL₀) under control.

VSS EWMA control chart

Control charts that can swiftly and precisely identify shifts are essential for efficient process monitoring. Variable Sample Size charts adjust sample sizes according to the current process state, enabling greater sensitivity to slight changes, whereas Fixed Sample Size charts employ a constant sample size for all observations. On the other hand, a fixed sample size selects the same size groups for every sample. The variable sample size control chart keeps adjusting the right sample size according to what is happening in the process. It helps to identify small to moderate changes in the mean of the process. In the case of variable sample size EWMA (VEWMA), the unbiased estimates can be obtained with , given by

3

where λ is the amount that determines the importance of the most recent observations. This combining past and current observations gives statistic value in prediction to be able to spot small and steady changes in the meaning of the process. Amirhossein et al.²⁹ presented a VEWMA control chart in which they used the same statistic and control limits as mentioned in Sect. 2.1, as well as upper and lower warning limits (UWL and LWL). These warning limits are defined as follows:

4

where is the warning coefficient of VEWMA.

The space between LWL and UWL and between LCL and LWL is called a warning zone; the space between UWL and UCL and between LWL and LWL is safe zone. Let and stand for the minimum and maximum sample sizes, respectively. Thus, when the sample falls inside the safe zone, the is used in the tth sample; when it falls in the warning zone, the is used such that:

5

here sample size is a function of and can be written in this form

6

Given that the value of the sample size must be an integer and that the relation between and is linear, note that here is a positive integer value between and as sample size is variable hence control limits will also vary for each sample. The upper and lower control limit and the plotting statistic for this chart are:

7

and the control limits are as follows for the VSS case

8

For , and the limiting form of the control limits can be determined as follows:

9

for every sample, it’s necessary to compute the sample size , upper and lower control limits ( and ), and the statistic value . The Variable Sample Size EWMA triggers an alert when exceeds the or falls below the . A side-by-side comparison of the main distinctions and benefits of each strategy is shown in Table 1. Table 1 below highlights the comparison between fixed sample size and variable sample size control charts.Table 1

Control chart strategies.

Criterion	Fix sample size (FSS)	Variable sample size (VSS)
EWMA statistic
Control limits
Sample size function	Fixed sample size n = N (constant for all samples)
Interpretation of	Not applicable (sample size remains constant)	depends on
Sensitivity to shifts	Lower sensitivity to small shifts, as n is fixed	Higher sensitivity to small shifts, as increases with ( )
ARL performance	Higher ARL (slower detection) for small shifts	Lower ARL (faster detection) for small shifts, as adjusts adaptively
Computational complexity	A simple, consistent sample size makes calculations straightforward	More complex, due to the dynamic calculation of based on

Design of the proposed adaptive EWMA control chart

This section porposed the new adaptive sample size based EWMA control chart named as ASEWMA control chart. The technique relies on the fact that process data would follow a normal distribution. But in real life, non-normal departures like skewed distribution or heavy tails can affect the reliability of the chart, and robust or non-parametric alternative techniques may be required. In addition, effective application in real-time monitoring relies on having access to a correct estimate of the process mean (μ), which is not always immediately available and may necessitate an initial Phase I analysis for calibration.Through dynamic sample size adjustment, this improved ASEWMA methodology seeks to produce more reliable and responsive outcomes. Let represent a randomly and normaly distributed variable at time }. For a certain amount of time, , the process is believed to be in an in-control (IC) state. This is and after δ, the system moves to an out-of-control (OOC) state because of a change in process parameter. Meanwhile, the term μ shows the value of mean during the IC state and μ₁ is used for the mean in OOC state, i.e. X_t ~ (μ, σ₂) in the situation of the IC state and for X_t ~ N(μ₁, σ₂) to the OOC state. The UWL and LWL are used in VSS control chart to divide the data. zones that are safe and those that are not safe to enter. In different zones, the size of the samples used is not the same; Small samples are part of the safety zone, whereas big samples are included in the warning zones. Through this type of skill, a process can be more accurately tracked and reviewed improving how quickly and well the VSS control chart can tell if the process is in OCC state. We suggest that here, we should look at the use of an adaptable method to manage the number of products needed for monitoring production by using a VSS control chart. The approach depends on real-time data and sample size for each iteration follows the shape of an exponential function based on how far results are from the target value. Small changes become visible on the chart due to this way of plotting data that adjust the sample size when deviations are bigger and ensure the method remains effective and efficient during normal times. Detecting changes in a process is mainly possible with the help of control charts in SPC. Making use of such as FEWMA and VEWMA, traditional control charts have long been common in monitoring any uncertainties in a process. VEWMA control charts change the sample size stepwise when the warning limits are reached, which does not always provide the most effective way to catch defects. Thus, we develop a dynamic adjustment system that lets the sample size adjust easily along with actual changes in the overall process. The function being suggested includes an exponential scaling approach that allows the sample size to grow gradually as the deviations grow. Unlike what we saw in traditional VEWMA charts, exponential functions let us easily adjust sample size on the go and still be computationally efficient. The formulation for the adaptive sample size function is shown in this section as below.

the sample size at the t^th iteration,

: the minimum allowable sample size (when the process is stable),

: the maximum allowable sample size (when deviation is high),

: the EWMA statistic at iteration t,

μ: the target process mean,

θ: the scaling constant controlling the rate of exponential growth to deviation,

1. The sample size at each iteration is bounded between a minimum and maximum value.

2. The adjustment is driven by the deviation | −μ| of the EWMA statistic from the process mean μ.

3. The , or exponential scaling fraction, is used to calculate the intermediate sample size.

4. The resulting sample size is then constrained within bounds using:

   10

This function ensures responsiveness to process deviations while avoiding excessive computational demands associated with unbounded sample sizes.

The control chart monitors the process through the following steps:

The initial sample size is set to . The EWMA statistic is initialized at the process mean ( =μ). At each iteration ( ) the mean of a random sample x_t of size _t is computed from a normal distribution with a mean equal to the shifted process mean and standard deviation. The EWMA statistic is updated recursively using:

11

The dynamic control limits are computed based on the adaptive sample size as

12

and the limiting form of the control limits can be determined as follows:

13

Decision rule

In the proposed one-sided ASEWMA control chart, an out-of-control (OOC) signal is triggered if the proposed statistic exceeds a threshold . For the two-sided ASEWMA, the OOC signal is indicated if or . The threshold is always supposed to be positive and determined so that IC ARL is fixed at a specified level (say ARL0). It is determined separately for each parameter combination, ensuring adaptability across scenarios.

Performance evaluation

The control chart is often assessed by its RL characteristics, which include mean, standard deviation, and percentiles. These RL properties can be computed using a variety of methods, including the probability technique, Markov chain, integral equations, and Monte Carlo (MC) simulation. In this study, we employed the MC approach, which is widely known for its versatility, to calculate the RL profiles of the proposed ASEWMA control chart.

Algorithm

The following procedures should be followed to use the MC simulation approach to determine the RL profiles of the proposed ASEWMA CC.

• Step 1:The algorithm starts by setting initial and maximum sample sizes n₁ and n₂ and then within the loop, adjusts the sample size n_t based on how much the data deviates from the target and the control threshold h, then setting the smoothing constant λ, control limit multiplier L3, and target mean μ.

• Step 2: The sample size for each iteration is adjusted based on an exponential scaling function of the deviation from the target mean. If the deviation increases, the sample size is increased toward otherwise, it remains close to .

• Step 3: Generate a random item based on Normal distribution i.e., . The ASEWMA statistic is updated recursively as described in Sect. 3.

• Step 4: The control limits are computed dynamically, considering the estimated standard deviation of the ASEWMA statistic.

• Step 5:

  • Regarding the workflow is IC

    1. Choose the desired IC ARL₀ value, let’s say 370 or 500.

    2. Choose a threshold value of h so that, in 100,000 replications, the IC ARL reaches a predetermined level denoted as ARL₀.

    3. A similar exercise is completed with other parametric choices before the application of the chart for the shifted OOC condition at a certain predetermined shift .

    4. Regarding the workflow is OCC

       1. When assign RL to the iteration number. If not, proceed with steps two and three.

       2. Continue choosing the sampling components until 100,000 repetitions are completed.

       3. Determine the mean and standard deviation of the RLs.

       The results presented in Tables 2, 3, 4 and 5 provide a detailed evaluation of the ARL and SDRL for different configurations of the ASEWMA control chart. They are required to see if the chart can spot changes in the process. A quick review of the main findings is offered to offer an overview. It is confirmed that the ASEWMA chart performs well because its actual ARL values are similar to the expected 370 when there is no shift. Because of this stability, the chart does not report an out-of-control situation when the process continues as normal. As the parameter δ increases in the OOC scenario, the ARL values go down a lot, which means the control chart can show deviations more rapidly. As an example, if n₂ = 6 and n₁ = 3 and δ = 0.2 at λ = 0.15, the ARL becomes 143.67, which reflects better detection abilities. Likewise, when λ = 0.25, the ARL is 176.01, so a larger smoothing constant postpones the time of detection. So, if λ is very low, it lets us detect slight fluctuations; however, the biggest shifts will remain undetected. The results show that different sample sizes affected the study. Extending the sample size range from n₁ = 3, n₂ = 6 to n₁ = 5, n₂ = 10 mean that process shifts are detected faster. As an illustration, when δ = 0.3, the ARL goes from 77.19 (with smaller sample sizes) to 44.91 (with larger sample sizes). So, setting a higher sample size can make the control chart better to detect process changes and react faster. As the δ value in the ASEWMA chart exceeds 0.5, ARL values fall quickly so that the chart catches any significant increase in the mean almost right away. When δ takes a value of 1.0, the ARL drops to 8.22 for the combination of λ = 0.15, n₁ = 3, and n₂ = 6, suggesting that the chart points out the out-of-control state rapidly. Also, when the shift is larger, the SDRL values decrease a lot, meaning it is easier to detect the fault with greater consistency. From the results of various control charts, it is clear that detecting small changes happens most quickly when the sample sizes are higher (n₁ = 5,n₂ = 10). On the other hand, models where n₁ = 3 and n₂ = 6 are acceptable for resource-constrained settings as they are both sensitive and do not slow down. Keeping in mind, a low λ is speedier when reacting to even small changes. As revealed in Tables 2 and 3, having a low smoothing constant (such as λ = 0.15) helps detect even small changes in the process, making ARL go down for δ values like 0.2 or 0.3, however, this could lead to a slightly longer SDRL. On the other hand, greater values of λ mean that changes are detected with greater reliability, but there could be a delay in noticing small movements. Both n₁ and n₂ should be increased (for example, from 3–6 to 5–10) to make detecting moderate and large changes faster and by lowering the detection delay for most δ values. Yet, it involves more computational work and may take more effort and funds to sample the data in reality. In other words, the result confirms a balance among sensitivity, running time, and how easy it is to implement, which makes careful setting of parameters necessary depending on the monitored process. Tables 2 and 3 clearly show that different control chart performances happen according to the chosen parameters. To be specific, the chart with λ = 0.15 and (n₁ = 5, n₂ = 10) performs better than others in picking up small to moderate shifts (δ = 0.2 to 0.5) since it achieves the shortest Average Run Length. For this same configuration and δ = 0.3, the ARL is 44.91, and surpasses both EWMA and VEWMA. If λ = 0.25 and n₁ = 3, n₂ = 6, then ARLs are higher and there is less variability. Thus, this configuration is more appropriate when there are computing or sampling restrictions. All in all, the best-performing configuration for processes that need to spot early changes in small amounts is λ = 0.15 and (n₁, n₂) = (5, 10). The similary pattern of results can be observed from Tables 4 and 5 when the ARL₀ is set at 500. It indicates that it is important to adjust the standards based on the different areas of their importance among speed, stability, and how much must be sampled.

       Table 2. Run length profile of control charts for λ = 0.15 and ARL₀ = 370.

       δ	n1 = 3, n2 = 6						n1 = 5, n2 = 10
       	FEWMA		VEWMA		ASEWMA		FEWMA		VEWMA		ASEWMA
       	L1 = 2.8109		L2 = 3.1114		L3 = 2.8100		L1 = 2.8091		L2 = 3.1001		L3 = 2.8100
       	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL
       0	370.92	370.14	370.58	370.54	370.33	370.30	370.38	370.36	370.90	370.88	370.46	370.44
       0.1	268.62	267.48	264.32	264.18	232.96	232.19	269.46	267.13	216.82	215.22	184.68	182.68
       0.2	143.67	139.90	136.43	131.70	107.33	106.08	141.07	139.06	94.07	89.87	69.27	65.93
       0.3	77.19	73.00	73.19	68.71	52.48	49.55	77.75	73.81	44.91	40.91	33.19	29.73
       0.4	46.06	41.43	42.64	37.73	30.69	26.65	46.26	41.72	25.88	21.54	19.0	15.1
       0.6	21.19	17.17	19.43	15.18	14.19	10.85	21.16	16.94	11.83	8.27	8.89	6.05
       0.8	12.30	8.89	11.14	7.64	8.45	5.74	12.28	8.94	7.21	4.35	5.54	3.39
       1.0	8.22	5.51	7.52	4.61	5.81	3.58	8.21	5.46	5.02	2.72	3.95	2.16
       2.0	2.65	1.39	2.77	1.16	2.28	0.9	2.63	1.37	2.08	0.75	1.81	0.6
       3.0	1.50	0.64	1.81	0.62	1.61	0.54	1.50	0.65	1.41	0.50	1.3	0.46
       4.0	1.11	0.32	1.36	0.49	1.26	0.44	1.12	0.33	1.08	0.27	1.05	0.21
       5.0	1.01	0.11	1.11	0.31	1.06	0.25	1.01	0.12	1.00	0.07	1	0.04

       Table 3. Run length profile of control charts for λ = 0.25 for ARL₀ = 370.

       δ	n1 = 3, n2 = 6						n1 = 5, n2 = 10
       	FEWMA		VEWMA		ASEWMA		FEWMA		VEWMA		ASEWMA
       	L1 = 2.8996		L2 = 3.1540		L3 = 2.9060		L1 = 2.8995		L2 = 3.1351		L3 = 2.9060
       	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL
       0	371.27	371.23	371.94	371.92	370.82	370.80	370.26	370.23	370.33	366.50	370.08	370.05
       0.1	292.64	292.62	294.82	292.30	262.17	260.48	296.63	296.60	249.02	249.00	225.42	225.39
       0.2	176.01	173.14	173.12	170.98	137.62	136.67	179.31	177.55	124.00	121.67	93.41	90.10
       0.3	102.32	99.11	100.16	98.23	72.17	69.98	101.92	99.46	61.46	58.60	44.60	41.48
       0.4	62.26	58.50	58.47	55.50	41.06	38.45	61.72	59.17	33.73	30.48	24.26	21.55
       0.6	27.06	24.01	24.86	21.47	17.62	14.77	27.26	24.22	14.35	11.23	10.49	7.77
       0.8	14.93	12.00	13.33	10.20	9.94	7.36	14.95	11.98	7.89	5.26	6.09	3.90
       1.0	9.44	6.91	8.56	5.79	6.47	4.23	9.47	6.91	5.29	3.03	4.23	2.38
       2.0	2.78	1.46	2.83	1.20	2.36	0.94	2.79	1.46	2.10	0.76	1.87	0.60
       3.0	1.55	0.68	1.83	0.62	1.65	0.54	1.54	0.67	1.41	0.50	1.31	0.46
       4.0	1.13	0.35	1.38	0.49	1.29	0.45	1.14	0.35	1.09	0.28	1.05	0.23
       5.0	1.01	0.13	1.11	0.32	1.08	0.26	1.01	0.12	1.00	0.07	1	0.07

       Table 4. Run length profile of control charts for λ = 0.15 for ARL₀ = 500.

       δ	n1 = 3, n2 = 6, λ = 0.15						n1 = 5, n2 = 10, λ = 0.15
       	FEWMA		VEWMA		ASEWMA		FEWMA		VEWMA		ASEWMA
       	L = 2.9100		L = 3.2300		L = 2.9200		L = 2.9100		L = 3.2103		L = 2.9200
       	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL
       0	500.49	499.39	500.20	499.12	500.84	498.80	500.87	499.90	500.52	198.96	500.79	499.35
       0.1	348.66	343.86	345.63	344.97	300.14	298.14	342.41	340.81	284.71	282.05	241.84	239.72
       0.2	174.92	172.84	168.75	163.67	130.42	128.04	174.93	170.29	113.50	109.26	84.69	81.16
       0.3	92.55	88.30	88.00	83.15	62.84	59.35	92.94	88.47	52.31	47.10	37.36	32.98
       0.4	53.39	48.75	50.14	44.82	35.60	31.18	54.27	49.28	29.35	24.65	21.04	17.15
       0.6	23.81	19.14	21.59	17.10	15.78	12.01	23.64	19.07	12.80	8.90	9.70	6.61
       0.8	13.37	9.66	12.08	8.04	9.28	6.25	13.53	9.76	7.66	4.61	5.95	3.60
       1.0	8.87	5.88	8.14	4.97	6.25	3.87	8.83	5.75	5.30	2.89	4.16	2.25
       2.0	2.77	1.42	2.90	1.23	2.34	0.91	2.77	1.42	2.15	0.77	1.85	0.60
       3.0	1.55	0.67	1.85	0.63	1.65	0.53	1.56	0.67	1.44	0.52	1.33	0.47
       4.0	1.14	0.35	1.42	0.50	1.29	0.45	1.14	0.35	1.10	0.30	1.05	0.23
       5.0	1.01	0.13	1.13	0.34	1.08	0.27	1.02	0.14	1.00	0.09	1.00	0.06

       Table 5. Run length profile of control charts for λ = 0.25 for ARL₀ = 500.

       δ	n1 = 3, n2 = 6, λ = 0.25						n1 = 5, n2 = 10, λ = 0.25
       	FEWMA		VEWMA		ASEWMA		FEWMA		VEWMA		ASEWMA
       	L = 3.0		L = 3.2600		L = 3.002		L = 3.0		L = 3.2501		L = 3.0100
       	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL
       0	500.96	499.12	500.34	497.26	500.00	495.64	500.67	498.76	500.84	498.77	500.74	498.86
       0.1	389.24	385.33	385.15	383.77	352.28	350.03	384.68	382.57	341.02	339.54	295.88	293.53
       0.2	226.66	224.07	221.95	220.06	175.75	173.52	225.71	223.33	158.93	156.26	119.60	118.51
       0.3	126.48	124.21	120.45	117.93	87.29	84.63	127.77	125.27	74.52	70.74	53.46	51.29
       0.4	74.95	71.45	69.75	66.09	48.74	45.83	75.26	72.12	40.16	36.38	28.13	24.77
       0.6	31.66	28.36	28.69	25.67	20.07	17.19	31.74	28.35	15.89	12.54	11.60	8.85
       0.8	16.64	13.44	14.78	11.48	10.74	7.99	16.81	13.69	8.59	5.81	6.58	4.30
       1.0	10.46	7.62	9.28	6.32	6.91	4.52	10.41	7.66	5.67	3.26	4.45	2.50
       2.0	2.94	1.53	2.95	1.25	2.43	0.97	2.92	1.53	2.17	0.76	1.89	0.60
       3.0	1.60	0.70	1.88	0.63	1.69	0.52	1.62	0.71	1.46	0.51	1.36	0.48
       4.0	1.16	0.37	1.41	0.50	1.32	0.46	1.16	0.37	1.11	0.31	1.07	0.25
       5.0	1.02	0.15	1.14	0.34	1.09	0.28	1.02	0.14	1.00	0.09	1.00	0.06

       Real life application

       The researchers frequently assess the efficiency and dependability of Control Charts by examining both real-world and simulated datasets. We demonstrate the application of the proposed chart using an actual dataset. The dataset, originally presented by Montgomery²⁶, consists of 25 samples, each comprising five wafers. Within semiconductor manufacturing, this dataset corresponds to the hard-bake stage in the photolithography process. We classify the first 25 samples as an IC process, forming the phase-I dataset. To introduce an upward shift of 0.5 in the process mean. This results in 20 additional samples representing an OOC process, constituting the phase-II dataset. Figures 1 and 2 illustrate the behavior of ASEWMA control chart under different parameter settings. These charts depict the process monitoring results, highlighting how the control chart responds to process shifts. In Fig. 1, the control chart was constructed using a smoothing parameter λ = 0.1, which gives more weight to past observations. The sample sizes range from n₁ = 5 to n₂ = 10, indicating an adaptive approach where the sample size increases when deviations from the process mean are detected. The control limit multiplier L = 31.9301 was selected to maintain the desired in-control ARL. In Fig. 2, a similar control chart is used, but with a lower sample size range of n₁ = 3 to n₂ = 6 and a slightly lower control limit multiplier L = 27.9301. Figure 1 (higher sample size range) detects process shifts more quickly due to its larger sample sizes, leading to lower ARL and improved detection of small process shifts. Figure 2 (lower sample size range) provides a more balanced detection approach, reducing computational complexity while still being effective in identifying shifts.

       Fig. 1 [Images not available. See PDF.]

       Proposed control chart for λ = 0.1, , .

       Fig. 2 [Images not available. See PDF.]

       Proposed control chart for λ = 0.1, , .

       Conclusion

       This study presents an enhanced VAEWMA control chart designed to improve process monitoring efficiency. By incorporating variable sample sizes and adaptive control limits, the proposed method significantly improves shift detection speed compared to conventional FEWMA and VEWMA charts. The Monte Carlo simulation results demonstrate that larger sample sizes (e.g., n₁ = 5, n₂ = 10) enhance detection capabilities, particularly for small process shifts, while lower smoothing parameters (λ = 0.15) result in quicker responses to deviations. The findings suggest that the VAEWMA control chart is particularly beneficial in environments where both small and large shifts must be identified efficiently. The capability of adjusting sample sizes according to each process variation enables faster action response with greater flexibility while reducing false alarm frequencies. In addition, the analysis comparison validates that the feasible methodology improves the performance of process detection when out of control while maintaining the process stability within in-control limits.

       Further work could be done by implementing the VAEWMA chart in real-time industrial processes when it can be expanded for non-normal and auto-correlated processes. Furthermore, its performance can be enhanced with additional machine learning-based static adaptive mechanisms. This control chart will be beneficial in high-quality control modern applications, providing maximum reliability, accuracy, and efficiency in the processes of various industries.

       Acknowledgements

       The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/70/46 ", and this study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446).

       Author contributions

       I.A.N. and M.M.A.A. conceptualized the study and developed the theoretical framework. A.Y.A. conducted the simulations and performed the data analysis. S.H. wrote the initial draft of the manuscript and organized the literature review. I.A.N. and A.Y.A. contributed to result interpretation and technical validation. M.M.A.A. and S.H. revised the manuscript critically for important intellectual content. All authors reviewed and approved the final version of the manuscript.

       Data availability

       The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.

       Competing interests

       The authors declare no competing interests.

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References

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