Memory type Max-EWMA control chart for the Weibull process under the Bayesian theory

The simultaneous monitoring of both process mean and dispersion, particularly in normal processes, has garnered significant attention within the field. In this article, we present a new Bayesian Max-EWMA control chart that is intended to track a non-normal process mean and dispersion simultaneously. This is accomplished through the utilization of the inverse response function, especially in cases where the procedure follows a Weibull distribution. We used the average run length (ARL) and the standard deviation of run length (SDRL) to assess the efficacy of our suggested control chart. Next, we contrast our suggested control chart's performance with an already-existing Max-EWMA control chart. Our results show that compared to the control chart under consideration, the proposed control chart exhibits a higher degree of sensitivity. Finally, we present a useful case study centered around the hard-bake process in the semiconductor manufacturing sector to demonstrate the performance of our Bayesian Max-EWMA control chart under different Loss Functions (LFs) for a Weibull process. The case study highlights how flexible the chart is to various situations. Our results offer strong proof of the outstanding ability of the Bayesian Max-EWMA control chart to quickly identify out-of-control signals during the hard-bake procedure. This in turn significantly contributes to the enhancement of process monitoring and quality control.


Squared error loss function
To assess the discrepancy between expected and actual values, the squared error loss function-also called mean square error or MSE-is essential in Bayesian methodology.It is essential to both Bayesian inference and decision theory.Predictions or parameter estimates are represented as probability distributions in Bayesian modeling.The squared error loss measures the cost of differences by squaring them mathematically, which penalizes larger discrepancies more severely.The primary goal of Bayesian practice is to minimize the expected squared error loss by averaging it under the posterior distribution, resulting in point estimates or predictive distributions.This loss function is widely used in Bayesian applications, especially for continuous variables, and is closely related to the mean squared error.Ultimately, it helps assess the quality of predictions and parameter estimates by considering both estimate uncertainty and their proximity to actual values.The SELF is endorsed by Gauss 23 , incorporating both the variable of interest, denoted as X, and the estimator θ used for estimating an unknown population parameter θ , denoted as theta.Its mathematical expression is as follows: And the Bayes estimator using SELF is mathematized.

Linex loss function
The Linex loss function employed in Bayesian analysis measures the cost of prediction errors by balancing linear and exponential components.It assesses the difference between predicted and true values and is valuable in various Bayesian applications, allowing flexibility in quantifying asymmetric losses and adapting to scenarios with varying consequences of overestimation and underestimation errors.Varian 24 proposed LLF to mitigate risks in Bayes estimation.The LLF is mathematically described Under LLF, the Bayesian estimator θ is mathematizied as

Proposed Bayesian Max-EWMA CC for joint monitoring of the Weibull distribution
We have a series of random samples, denoted as V i1 , V i2 , …, V in , drawn from a Weibull distribution denoted as W(α, ) at different time points, i.e., i = 1, 2, 3. Typically, the Weibull distribution's parameters ( α and ) are not known in advance and need to be estimated using available historical data.To estimate these parameters, an appropriate method is employed, usually with the consideration that the process is under control.Letus denote the estimated scale and shape parameters as α 0 and 0 , respectively.These parameter estimates are derived by leveraging a relationship between the Weibull and standard normal distributions, as given by Faraz et al. 25 in Eq. (7) as follows: where the mean and variance are given by And Equations (8) and (9) provides insights into how shifts in the parameters of a Weibull distribution influence the mean and variance of a random variable following a standard normal distribution.Essentially, they quantify the impact of changing Weibull distribution parameters on the characteristics of the standard normal distribution. (1) (3) L θ, θ = θ − θ 2 (4) θ(SELF) = E θ/x (θ).where H(n, ν) is a chi-square distribution characterized by ν degrees of freedom, and φ −1 denotes the inverse of the standard normal distribution function.The computations for EWMA EWMA statistics regarding both the process mean and variance are outlined as follows: In this context, P 0 and Q 0 represent the initial values for the EWMA sequences P t and Q t , respectively, with (a constant within the range [0, 1]) denoting the smoothing constant.P t and Q t are also mutually independent because of the independence of P t and Q t .When considering an in-control process, both P t and Q t follow normal distributions, each with a mean of zero and variances of δ 2 P t and δ 2 Q t , respectively.This is defined as follows The plotting statistics, Bayesian Max-EWMA for jointly monitoring using P t(LF) and Q t(LF) is mathematically defined For t = 1, 2, .. As the Bayesian Max-EWMA statistic is a positive value, we required to plot only the upper control limit for jointly monitoring the process mean and variance.If the plotting statistic A t within the UCL, then the process is in control and if the A t cross the UCL, the process is out of control.

Results and discussion
In this analysis, Tables 1, 2, 3 and 4 serve as a central platform for presenting the outcomes derived from the application of the Bayesian Max-EWMA CC for the Weibull process.This study undertakes a rigorous examination, specifically focusing on the influence of two distinct LFs designed to emphasize the significance of the posterior distribution.Importantly, these assessments are conducted within the framework of informative priors, introducing prior knowledge and beliefs into the analytical process.To ensure the reliability and robustness of our statistical conclusions, a substantial replication of 50, 000 replicates is employed for the calculation of both the ARL and SDRL.Furthermore, we exercise precision by carefully selecting smoothing constants, namely λ values of 0.10 and 0.25, which fine-tune our analysis and enable us to evaluate the performance of the Bayesian Max-EWMA CC method under diverse conditions.Furthermore, this study expands its scope and examines an extensive range of combinations with variance shift values (b) covering values from 0.25 to 3.00 and mean shift values (a) ranging from 0.00 to 3.00.This comprehensive analysis allows us to evaluate the performance of the Bayesian Max-EWMA CC approach, which specifically aims to comprehensively monitor process variance and mean simultaneously.The results of our study clearly show how sensitively the method can detect deviations from the standard within production processes.This demonstrates its enormous potential as a useful and trustworthy tool for continuous quality control and monitoring in a variety of industrial environments.It is important to ensure that the calculated plot statistic Si remains below the UCLi at all times.Each trial ends when the plot statistic exceeds the UCLi, indicating a significant change in the process mean and standard deviation.Changes in the parameters of the Weibull distribution are related to these variations.In particular, we analyze shifts in scale parameters ranging from 0.0 to 5.00 and shifts in shape parameters ranging from 0.25 to 4.00The process initially follows a normal distribution N(0, 1) and remains within the control limits when it comes to the Weibull distribution with parameters W(1, 1.5).The ARL under control conditions (ARL0) for the two different cases of the smoothing constant λ = 0.10 and 0.25 was found to be 370.The results shown in Tables 1 and 2 provide strong evidence for the effectiveness of the Bayesian Max-EWMA CC, particularly when used in conjunction with the SELF for the posterior distribution.Maintaining process stability and product quality depends on the combined approach's exceptional ability to simultaneously detect shifts in both process mean and variance.The results show a remarkable trend: the ARLs continuously decrease as the magnitude of the mean shift increases.Similarly, ARLs decrease when variance shifts occur.These recurring trends strongly suggest that the Bayesian Max-EWMA CC has the important ability to detect process changes in a timely manner and indicate what is needed for process control and early intervention.Due to these properties, it is an extremely valuable tool for thorough monitoring of production processes and ensures timely detection and correction of deviations from established standards.Ultimately, the use of these CCs improves process effectiveness and product quality, making them a valuable asset in a variety of industries.For example, if you look at the ARL results.The resulting ARL values for these shifts are as follows: 369.15, 80.19, 24.05, 13.71, 10.14, 8.16, 7.12, 4.88, and 4.13.Interestingly, the corresponding ARL values noticeably decrease with increasing displacement magnitude.This result shows the extent to which the proposed Bayesian Max-EWMA CC can be used to quickly identify changes in the shape parameter.The ability of the CC to quickly detect even small deviations from the process mean suggests that it is very sensitive.This means that these changes can be responded to quickly, which is critical to maintaining the consistency and quality of the process.The effects of changing the value of the shape parameter from a = 1.50 to 4.00 while maintaining the scaling parameter values are similar.The resulting ARL values are as follows: 369.15, 24.91, 16.92, 7.06, 5.80, 3.60, 2.60, 2.11, and 1.50.These ARL values show a clear trend: the ARL values sharply decline as the shape parameter deviates from the baseline value of 1.This pattern highlights how well the suggested Bayesian Max-EWMA CC performs in quickly identifying changes in process variance.Moreover, it is noteworthy that when examining the performance of the proposed Bayesian Max-EWMA CC in Table 2, we find that the CC becomes less effective as the smoothing constant increases.This observation suggests that in specific scenarios, opting for a lower smoothing constant might be more advantageous in achieving optimal performance.Similarly, Tables 3  and 4 present the ARL outcomes of the Bayesian Max-EWMA CC using the LLF with a consistent λ = 0.25 and n = 5.Across various trials involving shifts in the shape parameter ranging from 1.50 to 5.00 and corresponding shifts in the scale parameter fixed at 1.0, the resulting ARL values were 370.09, 20.34, 6.34, 2.81, 1.99, 1.57 and 1.17.These findings highlight a clear trend: as the magnitude of process shifts increases, the ARL values exhibit a rapid decrease, underscoring the exceptional accuracy of the proposed Max-EWMA CC in swiftly detecting shifts in both process mean and variance.Moreover, it is essential to note that the efficiency of the proposed CC for the simultaneous monitoring of the process mean and variance is influenced by the sample size.Across all the tables, a consistent pattern emerges: as the sample size increases, the corresponding ARL values decrease, indicating the enhanced effectiveness of the suggested CC in promptly identifying deviations from the expected process parameters.The following simulation steps have been considered for the calculations of ARLs and SDRLs.
Step 1: Establishing the control limits i.To commence, establish the initial control limits by computing the values for UCL and λ. ii.Generate a random sample of size n to depict the in-control process, utilizing normal distributions.iii.Calculate the statistic required for the suggested control chart.
Verify whether the plotted statistic lies within the UCL; if so, proceed to steps (iii-iv) once more.
Step 2: Assessing the out-of-control average run length (ARL)  samples.Both charts are employed to monitor variations in the process mean, and the resulting computations are presented in Table 6.
Figures 1 and 2 provide a visual representation of the implementation of the provided Bayesian Max-EWMA CC, designed for the simultaneous monitoring of both process mean and dispersion.This monitoring employs both the SELF and LLF approaches.A thorough examination of these charts reveals clear signals indicating that the process has gone out of control in the 23rd and 21st samples, especially when considering a smoothing constant value of 0.10.The identified departure from the normal process state, resulting in an out-of-control scenario, can be ascribed to two main factors: alterations in either the process mean or variance.These alterations stem from modifications in the shape and scale parameters of the Weibull distribution, thereby influencing the distribution's properties and giving rise to the observed deviations.

Conclusion
In this study, we introduce an innovative Bayesian Max-EWMA CC using the Weibull process designed for the simultaneous monitoring of the process mean and variance.This CC incorporates informative prior distributions and integrates two distinct LFs within the context of posterior distribution.The performance of this novel approach was rigorously evaluated through a comprehensive analysis, with results presented in Tables 1, 2, 3 and 4.These assessments employ crucial metrics such as ARL and SDRL.We conduct a practical case study focused on the hard bake process in semiconductor manufacturing.Interestingly, the proposed Bayesian Max-EWMA CC shows excellent performance in identifying out-of-control signals in the process when applied to posterior distributions.Crucially, the knowledge acquired from this research could be applied to the creation of other memory-type CCs, improving their efficacy in a variety of industrial applications.Extending this novel method to different kinds of CC instead of just nonnormal distributions can lead to a more thorough comprehension of the underlying data patterns.This broader application enables the early detection of potential quality issues in different domains and allows for swift corrective actions, thereby reducing the risk of costly errors and defects.This method is essential for quickly spotting irregularities in patient data, enabling prompt interventions, and enhancing the quality of patient care in real-world situations like healthcare.In manufacturing, extending this approach to non-normal distributions and diverse CC types aids in identifying variations in the production process, ultimately leading to improved product quality and a reduction in waste.

Figure 1 .
Figure 1.Using SELF, the Bayesian Max-EWMA CC using Weibull process for jointly monitoring with = 0.10.

Table 1 .
Run length profile of Bayesian Max-EWMA control chart as a result of a shift in Weibull parameters W(1, 1.5) with γ = 0.10 and considering subgroup size as 3, 5, and 7.

Table 4 .
The ARL results of Bayesian Max-EWMA control chart as a result of a shift in Weibull parameters W(1, 1.5) with γ = 0.25 and considering subgroup size as 3, 5, and 7 under LLF.α n

Table 5 .
Data set related to breaking strengths of carbon fibers.

Table 6 .
The values and out of control status proposed Bayesian Max-EWMA under SELF and LLF, with = 0.10.