Monitoring of manufacturing process using bayesian EWMA control chart under ranked based sampling designs

Control charts, including exponentially moving average (EWMA) , are valuable for efficiently detecting small to moderate shifts. This study introduces a Bayesian EWMA control chart that employs ranked set sampling (RSS) with known prior information and two distinct loss functions (LFs), the Square Error Loss function (SELF) and the Linex Loss function (LLF), for posterior and posterior predictive distributions. The chart's performance is assessed using average run length (ARL) and standard deviation of run length (SDRL) profiles, and it is compared to the Bayesian EWMA control chart based on simple random sampling (SRS). The results indicate that the proposed control chart detects small to moderate shifts more effectively. The application in semiconductor manufacturing provides concrete evidence that the Bayesian EWMA control chart, when implemented with RSS schemes, demonstrates a higher degree of sensitivity in detecting deviations from normal process behavior. Comparison to the Bayesian EWMA control chart using SRS, it exhibits a superior ability to identify and flag instances where the manufacturing process is going out of control. This heightened sensitivity is critical for promptly addressing and rectifying issues, which ultimately contributes to improved quality control in semiconductor production.


Bayesian approach
This section presents a concise discussion of the Bayesian approach, which incorporates both sample knowledge and prior information through the P and PP distribution.The prior distribution is a pivotal element in Bayesian estimation and can be divided into two primary categories: informative and non-informative priors.Informative priors come into play when there exists prior knowledge or information regarding the population parameter, enabling the integration of this prior information into the Bayesian analysis.Conversely, non-informative priors, sometimes referred to as vague or uninformative priors, are selected when limited to no prior knowledge is accessible.This choice ensures that the prior distribution does not introduce any undue influence or bias into the Bayesian inference process.Conjugate prior distributions are employed when both the informative prior and the sampling distribution belong to the same family of probability distributions.This choice streamlines Bayesian analysis by simplifying the calculation of the P distribution.The advantage lies in the P distribution maintaining the same mathematical form as the prior distribution, making computations more straightforward.In essence, selecting a conjugate prior ensures mathematical compatibility between the prior and P distributions, facilitating efficient Bayesian estimation and belief updating when new data is introduced.In the present study, the variable under consideration X is used which follows a normal distribution with unknown mean θ and known variance δ 2 .We considered normal prior (conjugate prior) with hyperparameters θ 0 and δ 2 0 given as: when no information is available about the prior distribution θ , it has minimum effect on the P distribution.Thus, in cases where no prior knowledge is available, it is often preferred to utilize a uniform distribution as the prior.This choice of using a non-informative prior, often referred to as a vague or uninformative prior, ensures that in the absence of prior knowledge or information, all conceivable values of the unknown population parameter are treated with equal probability or weight.In other words, the prior distribution does not favor any specific value or range of values for the parameter, effectively representing a state of maximum uncertainty.This approach is particularly useful when a researcher or analyst wants to maintain objectivity and avoid introducing any bias into the Bayesian analysis, allowing the data to exert the most significant influence on the final inference about the parameter of interest.It is a way to approach Bayesian analysis when one wants to be as neutral as possible regarding the parameter's possible values.The probability function of the uniform prior can be represented by p(θ) , and its definition is as follows: where c represents the constant of proportionality and n is the sample size.The uniform prior is not known for exhibiting the invariance property in Bayesian analysis.However, Jeffrey 24 introduced an alternative prior distribution, called Jeffrey's prior.This prior is proportional to the Fisher information matrix, a fundamental concept in statistics that quantifies the amount of information contained in a sample from a probability distribution.Jeffrey's prior is particularly valuable because it provides a prior distribution that remains invariant under certain transformations of parameters, making it a useful choice in cases where maintaining invariance is important for robust Bayesian inference.This property ensures that the prior distribution does not change when expressing information in a different parameterization, enhancing the flexibility and applicability of Bayesian analysis.The Jeffrey's prior for the parameter θ is given by p(θ) ∝ √ I(θ) , where

The P distribution is denoted by
The The PP distribution in Bayesian analysis combines updated beliefs from the P distribution with the prior distribution to predict future data Y probabilistically, considering Bayesian process uncertainty.This aids informed, probabilistic forecasting across various fields.The PP distribution is mathematizied as In Bayesian analysis, the risk associated with the estimator is mitigated through the use of a LF.In this study, we employed both symmetric and asymmetric LFs to address this aspect.

Square error loss function
In Bayesian analysis, the symmetric type LF SELF suggested by Gauss 25 .It is commonly employed to measure the disparity between a point estimate (e.g., posterior mean or median) and the true but typically unknown parameter value.It is favored in Bayesian analysis when the objective is to minimize the expected value of the squared difference between the estimate and the true parameter value, exhibiting sensitivity to larger errors.The choice (4) p y|x = p y|θ p(θ|x)dθ.

Linex loss function
Varian 26 was the first to introduce the asymmetric type LF Linex LF (LLF), defined as: When the estimator θ is employed to estimate the unknown population parameter, it is mathematically represented as:

Ranked set sampling
In the RSS procedure, we select m 2 units randomly from the underlying population.The m 2 units are distributed into m sets, with m similar set size randomly proposed by McIntyre 27 .The study variable Z is considered without taking into account the actual measurement and the ordering (ranking) of the m units in each set visually.The The process unfolds systematically by arranging all m sets and sequentially selecting the smallest unit from the first set for measurement.This sequential selection continues as the second smallest unit is chosen from the second set, and so on, until it culminates with the selection of the largest unit from the mth set.This constitutes a single cycle of RSS with a size of m.This cycle is then replicated r times iteratively until the desired sample size, denoted as n, is achieved.The procedure for RSS can be expressed as Zi(j), r, where i and j signify the sample set and order statistic, respectively, with values ranging from 1 to m, while r denotes the cycle number.In the case where c = 1, the mean and variance of the ranked set sample estimator can be described as follows: The RSS mean estimator is given as and the variance of the estimator is given by where µ is the overall mean.

Median ranked set sampling
Muttlak 28 introduced the median ranked set sampling (MRSS) method as an estimation technique for the population mean.MRSS exhibits improved performance by minimizing ranking errors compared to the mean estimator used in RSS.Like RSS, MRSS divides the sampling units into m sets and arranges them within each set based on the study variable.If m is odd, then (m + 1/2)th units are picked as a sample from all sets.When m is an even, the selection process entails choosing the smallest order units from the two middle sampling units of the first (m/2)th set and selecting the highest order units from the two middle sampling units of the remaining (m/2)th sets.This method involves completing a single cycle of MRSS with a size of m.If needed, the cycle can be repeated r times to reach the desired sample size n = mr.For a single cycle, the mean estimator for MRSS in case of odd sample size is given by and the variance of the estimator is given by In the case of an even sample size, the population mean estimator of MRSS with one cycle is

Extreme ranked set sampling
ERSS scheme is the modified form of the RSS suggested by Samawi et al. 29 and is useful in situations where the collection of ith ordered units is more difficult than extreme units.In the ERSS method, a random selection of m 2 units is made from the underlying population, and they are then randomly distributed into m sets of equal size.The units within each set are arranged based on the study variable.When m is an odd, the procedure involves selecting the smallest units from the first m−1 2 th ordered set and the highest units from the following m−1 2 th ordered sets, ultimately concluding by choosing the median unit from the last set.When m is even, the process entails selecting the smallest units from the first m 2 th ordered units and the highest units from the remaining m 2 th ordered sets.This constitutes a single iteration, and if necessary, the process is repeated r times to achieve the desired sample size, denoted as n = mr.
Using ERSS for a single cycle, the mean estimator for an odd sample size is mathematically expressed as and the variance of the estimator is given by The mean estimator for ERSS, in the scenario of an odd sample size and a single cycle, is defined as follows: and the variance of the estimator is given by

EWMA control-chart
Consider a quality characteristic X that follows a normal distribution with an unknown location parameter θ and a known scale parameter σ 2 .In this scenario, we can express the probability density function and likelihood function of the random variable X as follows: Let sample observation x 1, x 2, x 3, ...x n be independently and normally distributed with mean θ and variance δ 2 .The statistic of EWMA CC is given by In this context, it represents a smoothing constant that adheres to the 0 ≤ ≤ 1 constraint.The values of assign weights to both current and past sample observations.When α is set to 1, all weights are exclusively attributed to the current sample.The initial value is initialized to the mean value of the process so that z 0 = θ 0 .
The mean and variance of the EWMA statistic is given as The control limits UCL, CL and LCL of EWMA CC are as follow: Vol:.( 1234567890 In this context, the abbreviations UCL, CL, and LCL correspond to the upper control limit, center line, and lower control limit, respectively.The control constant L is employed to modify the predefined average run length ARL 0 .The asymptotic expression for the control limits of the EWMA statistic is presented as follows:

Proposed Bayesian-EWMA applying differnt RSS designs
The plotting statistic for the recomended Bayesian EWMA CC under different RSS schemes (RSS, MRSS, and ERSS) by using LFs and informative prior for P and PP distribution is defined as: where RSS stands for ranked set sampling and LF is used for the loss function, the above structure is work under prior distribution given in "Utilizing normal prior the posterior distribution".

Utilizing normal prior the posterior distribution
The P distribution is the resulting probability distribution achieved by combining the likelihood function with a normal prior distribution and mathematically expressed as The P(θ|x) is normally distributed with mean and variance are θ n and δ 2 n respectively and given as θ/X ∼ N θ n , δ 2 n , where . The construction of the P and PP distribution under normal prior is derived in Suppl.Appendix B.
The control limits for the Bayesian EWMA CC, established using both P and PP distributions and factoring in various LFs within the RSS schemes, are presented below.

Under RSS, control limits utilizing SELF
The SELF serves as a LF that facilitates the establishment of control limits for Bayesian-EWMA CCs through the utilization of RSS schemes.The estimator θ based on SELF is mathematizied as: The properties of the estimator θSELF are mathematically expressed as: and In Suppl.Appendix C, we explained the process of deriving the Bayes estimator using SELF under RSS strategies for both the P and PP distributions.The asymptotic control limits of the EWMA CC under the Bayesian approach with RSS and SELF are formulated as follows:

Under LLF, control limits utilizing RSS schemes
The LLF serves as an instance of an asymmetric LFs applied in establishing control limits for the EWMA CC within the Bayesian framework when employing RSS schemes.The estimator of θLLF is defined as The properties of the estimator θLLF are given below: and The asymptotic control limits of Bayesian-EWMA CC with RSS schemes and LLF loss function are defined as

PP distribution using normal distribution
In this section, the EWMA CC utilizing Bayesian theory has been constructed using PP distribution.Let the future observation of size k are y 1 , y 2 , ...., y k , then the PP distribution Y |X is derived as: where , the Bayesian-EWMA control limits for PP distribution using various loss functions by using RSS schemes are given in subsection "Under LLF, control limits utilizing RSS schemes".

Under LLF, control limits utilizing RSS schemes
The Bayesian-EWMA CC under asymmetric LLF and with different RSS designs, the estimator θLLF is math- ematized as where δ2 . The properties of θLLF described as: and (34)

Simulation study
The The assessment of the Bayesian EWMA CCs performance in this study adheres to standard quality control methodologies, relying on ARL and SDRL as primary performance indicators.These metrics gauge the control charts efficiency across diverse ranked-based sampling designs while considering two distinct LFs and the inclusion of informative priors.Specifically, ARL 0 and SDRL 0 quantify the ARL and SDRL under normal, in-control conditions, while ARL 1 and SDRL 1 reflect scenarios where the process is out of control.The effectiveness of a CC is typically judged by its capacity to achieve a smaller ARL 1 for a specific shift.To compute efficiency metrics, the Monte Carlo simulation method is employed, encompassing 20,000 iterations.The control constant is adjusted to achieve a predetermined ARL 0 under varying LFs and different values of , such as = 0.10 and 0.25.
The simulation study encompasses the use of both the P and PP distributions to anlyze Bayesian EWMA CC using ranked-based sampling designs with sample size n = 5 and assumed standard normal distribution as prior distribution.Detailed steps for simulating ARL and SDRL are provided below.
Step 1: for in-control process i.We choice a specific value of and L smoothing constants for fixed ARL 0 .ii. Draw a sample with a RSS stratigies for an in-control process from normal distribution such that X ∼ N E θ , δ .iii.Computing the mean and standard deviation for the P distribution under different LFs, while assuming standard normal distributions for the sampling and prior distributions.iv.Based on Bayesian theory, the proposed Bayesian EWMA statistics are computed to evaluate the process according to the projected strategy.v. Considered Implemented the CC methodology, and in case of out-of-control process signals, documented the number of subgroups as the in-control run length.length.vi.Repeating the steps 100,000 times to estimate in-control ARL 0 .
Step 2: evaluation of the out-of-control process i. Draw a ranked set sample of size n from the shifted process by utilizing normal distribution, such that , where ∂ represents the shift in the process mean.ii.Compute the recommended Bayesian-EWMA statistic and assess the process based on the proposed study design.iii.If out-of-control signals are detected during the process, implement the CC method and record the number of run lengths for the out-of-control process.iv.Repeating the steps 100,000 times to estimate out-of-control ARL.

Results, discussion and findings
The study results, as presented in Tables 1, 2, 3, 4, 5 and 6, provide a comprehensive comparison of the Bayesian EWMA CC utilizing informative prior distributions and two distinct LFs across various ranked-based sampling designs.These findings emphasize the charts performance in comparison to the traditional Bayesian-EWMA CC designed for SRS.The analysis illuminates the strengths of the proposed Bayesian EWMA CC under diverse conditions, underscoring the importance of ranked-based sampling methods and the choice of LFs.In summary, this study demonstrates the considerable potential of the proposed method to significantly enhance process monitoring and improve quality control practices.The findings highlight a significant advantage of the suggested Bayesian EWMA CC for RSS, MRSS, and ERSS exhibits superior performance compared to the Bayesian EWMA CC for SRS at every shift.For example, Table 1  www.nature.com/scientificreports/Suppl.Appendix D. The results show that the variances under the RSS schemes are consistently lower than those of SRS.This comparison reinforces the efficiency of the suggested CC based on RSS schemes.In the following section, we will delve into the key findings and implications of the proposed CC.
• The The appraisal of the offered CC involved analyzing its performance across various smoothing constant (λ) values.Tables 1 and 2 provided comprehensive results on the ARL and SDRL for the both P and PP distributions, with a normal prior and the SELF.These tables illuminated how the Bayesian-EWMA CCs efficiency varied with different λ values.The key finding was that the Bayesian EWMA CC achieved its optimal performance at the smallest λ value.In essence, when the smoothing constant was set to its minimum, the CC exhibited the highest sensitivity in detecting process deviations.This observation underscores the critical role of selecting an appropriate λ value, as a smaller λ enhances the ability to detect process changes promptly-an essential aspect of quality control and process monitoring.For example, at ARL 0 = 370 , δ = 0.30, and = 0.1,  • With an increase in the magnitude of the shift, the ARL values of the suggested Bayesian EWMA CC for RSS schemes decrease faster than the considered Bayesian-EWMA CC.For example, from Tables 1 and 2, we can observe that at ARL 0 = 370 , and with = 0.1, the ARL value for δ = 0.20 is 55.66 and for δ = 0.50 is 10.36 using RSS, 43.99 and 8.53 utilizing MRSS and under the similar condition, the ARL result using ERSS is 60.05 and 11.80, which shows that the suggested design is unbiased.• The outcomes presented in Tables 3 and 4 1, 2, 3, 4, 5 and 6, which detail the per- formance of the proposed CC utilizing both P and PP distributions, and considering both SELF and LLF, it becomes apparent that the Bayesian EWMA CC incorporating MRSS excels in terms of efficiency when it comes to detecting shifts in the monitored process.When compared to other sampling design strategies, the offered CC utilizing MRSS consistently demonstrates a higher level of effectiveness in promptly and accurately identifying and responding to shifts in the process.This superior performance suggests that MRSS is a particularly suitable approach when aiming to enhance the efficiency of process monitoring and quality control through the Bayesian EWMA CC framework.

Real data application
In the field of SPC, it is a customary approach for researchers to evaluate the effectiveness of CC by employing a combination of simulated and real datasets.For this study, we utilized a dataset obtained from Montgomery 30 , consisting of flow width during the hard-bake process.The primary focus of this study is to assess the performance of the newly proposed Bayesian CCs within the context of ranked-based sampling strategies.Our primary objective revolves around the development of a robust statistical CC tailored to effectively monitor variations in the flow width of the resist.The flow width measurements were taken in microns at hourly intervals.The dataset comprises 45 samples, each consisting of measurements from five wafers, totaling 225 observations.It's important to note that we designate the first 30 samples as representing the in-control process, constituting the phase-I dataset, which includes 150 observations.Control limits are established based on this phase-I dataset, assuming a standard normal distribution.The remaining 15 samples are categorized as part of the out-of-control process, involving an ascending shift of 0.017 added to the central process mean, thus forming the phase-II dataset.The subsequent steps delineate the procedure for implementing the proposed Bayesian EWMA CC under various ranked-based sampling schemes.
Step 1: Select twenty-five observations by SRS from 150 control observations and divide them into five groups of equal size randomly at the time t.Note that the sample from the in-control dataset is generated for t = 1, 2, …., 30.Step 2: Rank the observations within each group and select the five observations for measurement purposes by following the sampling schemes such as RSS, MRSS, and ERSS.For instance, the diagonal elements will be selected for the RSS scheme.
Step Step 5: Calculate the Bayesian-EWMA statistic by following the design of the CC and plotting the graph with the control limits.
The Bayes estimator using SELF for P and PP distribution utilizing informative prior is 1.5056 for the incontrol dataset.The control limits such as LCL and UCL are 1.46 and 1.54 respectively using SRS.The LCL and UCL for RSS are 1.47 and 1.53, LCL and UCL for MRSS are 1.48 and 1.51 respectively.The respective control limits for the Bayesian-EWMA for ERSS are 1.47 and 1.53.Figure 5 presents the Bayesian-EWMACC under SRS for P and PP distribution in which all the points are within the control limits.Figures 6, 7, and 8 present visual representations of the Bayesian EWMA CC as applied to various RSS strategies, including RSS, MRSS, and ERSS schemes.These charts serve as visual indicators of instances where the process exhibits out-of-control behavior, with such deviations occurring at the 37th, 34th, and 36th samples, respectively.Upon a comprehensive analysis of all Figs. 1, 2, 3, 4, 5, 6, 7 and 8, it becomes evident that the Bayesian EWMA CC, particularly when employed within the RSS frameworks displays superior efficiency in promptly identifying out-of-control signals in comparison to the pre-existing Bayesian CC.These findings underscore the enhanced capabilities of the proposed CC in the context of process monitoring and quality control practices.

Conclusion
The present study uses the ranked-based sampling designs to improve Bayesian CC.The suggested CC applying P and PP distribution is constructed to evaluate the variations in the production process.The run length findings were adapted to asses the effectiveness of recomended CC using RSS schemes.The simulation study was conducted to assess the comparative effectiveness of the Bayesian CC using RSS schemes in contrast to the   established Bayesian EWMA CC using SRS.The results presented in Tables 1, 2, 3, 4, 5 and 6 consistently indicate that the proposed Bayesian CC, when employed in conjunction with RSS, MRSS, and ERSS and utilizing both LFs for both P and PP distributions, outperforms the traditional Bayesian EWMA CC based on SRS in terms of its ability to promptly and accurately detect out-of-control signals.This underscores the excellence of the suggested Bayesian CC in the realm of quality control and process monitoring.Furthermore, the superiority of the proposed CC becomes evident when examining the ARL plots (Fig. 1, 2, and 3).These plots provide a visual representation of how quickly the CC can detect out-of-control signals.Additionally, apart from the ARL plots, we demonstrate the effectiveness of the recommended CC through a practical example involving the hard bake process in semiconductor manufacturing.The results of this real-world application affirm that the Bayesian EWMA CC, when deployed under RSS schemes, displays a heightened sensitivity in promptly identifying out-of-control signals in comparison to its counterpart operating under the SRS framework.This heightened sensitivity underscores the practical advantages of implementing the proposed Bayesian EWMA CC in the context of quality control and process monitoring within semiconductor manufacturing.

Limitations of the study
In scenarios involving large sample sizes, the implementation of Bayesian CCs under RSS designs may encounter computational challenges.This complexity stems from the necessity of conducting Bayesian updates for both the process mean and variance at each individual sample point.As a result, this procedure can become computationally intensive and time-consuming due to the substantial computational workload it entails, rendering it resource-intensive.Another limitation is that the Bayesian approach specifies prior distributions for the process mean and variance.If the prior distributions are not chosen carefully, the performance of the CC may be affected.Moreover, the selection of prior distributions can be subjective and may require expert knowledge, potentially introducing bias into the analysis.
Finally, using RSS in the Bayesian-EWMA CC assumes that the data is exchangeable.If the data is not exchangeable, the CC may not perform as expected, and the results may be unreliable.Therefore, it is important to assess the exchangeability of the data before using RSS in the Bayesian-EWMA CC.

Future recommendations
The proposed Bayesian EWMA CCs under RSS schemes can be applied to other memory-type CCs.This approach is also versatile enough to handle distributions beyond the normal distribution, including Poisson or binomial distributions, with the requirement of appropriate adjustments to the likelihood function in Bayesian updating.This extension of the approach to cover diverse types of CCs and non-normal distributions has the potential to enhance the effectiveness and efficiency of quality control processes across various industries, such as healthcare, finance, and manufacturing.

Figure 1 .
Figure 1.Flow chart for the suggested CC applying RSS schems.

Figure 2 .
Figure 2. Employing SELF, ARL plots for the recommended CC applying the RSS, MRSS, and ERSS schemes.
show the run length results under LLF for Pdistribution for ARL 0 = 370 , = 0.10 and for δ = 0.50 the ARL value is 10.43 and 13.91 for = 0.25 , 8.47 and 11.25 for MRSS and in the same case, the run length outcomes using ERSS is 11.81 and 15.71, which clearly shows that as the value of increases the performance of the recommended CC distribution decreases.• The results for PP distribution under LLF are presented in Tables 5 and 6, which show that the ARL values at ARL 0 = 370 , δ = 0.20 and 54.68 and the ARL values for = 0.25 is 85.76 using RSS, In the identical situation the ARL results are 44.30and 71.03.Under ERSS the ARL outcomes are 62.82 and 97.99.The results presented in Tables 3, 4, 5 and 6 indicate that the proposed Bayesian-EWMA CC for the Pdistribution utilizing LLF yields a similar performance to the PP distribution under LLF.• After a comprehensive analysis of the outcomes provided in Tables

Figure 3 .
Figure 3. Utlizing LLF, ARL plots of the suggested CC under RSS, MRSS, and ERSS for P distribution.

3 :
Compute the sample means such as Z SRS , Z RSS , Z O MRSS and Z O ERSS for t = 1, 2, ..., 45 .The shifted dataset is used for t = 31, 32, ..., 45.Step 4: Compute the Bayes estimator and its standard deviation using different loss functions such as (SELF and LLF) from the sample means ( Z SRS , Z RSS ,

Figure 4 .
Figure 4. Using LLF, ARL plots of the suggested CC under different RSS strategies for PP distribution.
The control limits for Bayesian-EWMA CC under LLF and utilizing various RSS schemes are given below.

Table 1 .
gives the values of ARLs and SDRLs of proposed Bayesian EWMA for RSS schemes (i.e.RSS, MRSS, ERSS) and Bayesian-EWMA CC by using SRS for P and PP distribution utilizing SELF as 371, 55.66, 10.36, 4.69, 1.29, 1 for RSS, 369.42, 43.99, 8.53, 3.95, 1.16, 1 for MRSS, 371, 60.05, 11.80, 5.29, 1.39, 1 for ERSS and Bayesian EWMA CC for SRS values for the same shift are 371.62,125.58,28.35, 13.41, 4.16, 2.12 at = 0.10.The results suggest that, for every shift value, the proposed method consistently exhibits significantly lower ARL values compared to the aviable Bayesian CC.This observation underscores the superior performance of the proposed Bayesian EWMA CC when applied to RSS strategies in comparison to the Under SELF, AEL and SDRL outcomes of Bayesian CC with = 0.10.
existing Bayesian EWMA CC designed for SRS.For LLF and at = 0.25, we considered Table 4 which show that the ARL results for the suggested method are 371.79,85.76, 13.91, 5.59, 1 for RSS, 370.89, 66.78, 11.25, 4.59, 1 for MRSS, and 369.55, 100.52, 15.71, 6.33, 1 for ERSS, the values of ARL for Bayesian-EWMA SRS are 371.05,179.81, 28.05, 15.89, 1.66.The run length profiles, which represent the number of consecutive samples required to detect an out-of-control condition, consistently exhibit smaller values for the suggested CC when compared

Table 2 .
ARL and SDRL outcomes of Bayesian CC with = 0.25 based on SELF.

Table 3 .
Using LLF, run length profiles of Bayesian CC for P distribution with = 0.10.

Bayes-EWMA SRS Bayes-EWMA RSS Bayes-EWMA MRSS Bayes-EWMA ERSS ARL SDRL ARL SDRL ARL SDRL ARL SDRL L = 2.7047 L = 2.7331 L = 2.7245 L = 2.7142
Vol:.(1234567890) Scientific Reports | (2023) 13:18240 | https://doi.org/10.1038/s41598-023-45553-xwww.nature.com/scientificreports/ to the considered CC.This trend suggests that the proposed CC outperforms the considered CC in terms of its ability to detect process deviations more quickly and efficiently.To further illustrate this efficiency, Figs. 1, 2 and 3 in the ARL demonstrate the effectiveness of the proposed CC when utilizing various RSS schemes.These figures provide visual evidence of the proposed CC's effectiveness in different scenarios.The comparison of the variances obtained through simulations under the considered sampling schemes is presented in Table D1 of

Table 4 .
Using LLF, run length outcomes of Bayesian CC for P distribution with = 0.25.

Table 5 .
Using LLF, the run length values of Bayesian CC for PP distribution with = 0.10.

Table 6 .
Using LLF, the run length outcomes of Bayesian-EWMA CC for PP distribution with = 0.25.