All processes, whether in manufacturing or non-manufacturing settings, inherently feature variations, which can be classified as either natural or abnormal. Natural variations are inescapable and do not pose any harm. However, abnormal variations can have detrimental effects on the process and compromise the quality of the final outcome. To ensure the efficiency of the process and promptly deal with these variations, it is vital to implement corrective measures. The term “shift” denotes the magnitude of irregular fluctuations in parameters, such as location and dispersion. In essence, the term “shift” denotes a systematic alteration of the location parameter of a probability distribution, such as a mean or median shift. The uneven fluctuation of parameters is not frequently used to characterize it. In statistical process control (SPC), quality control charts (CCs) are a crucial instrument for tracking and managing the caliber of a manufacturing process. Plotting process data across time and setting control limits based on past process performance is how quality CCs operate. The process is considered as “control” if the data lies within the control limits, meaning that it is operating within its expected range of variation. If the data deviates from the predetermined range, it may indicate a process issue that requires further detailed investigation. There are several kinds of quality CCs, each one intended to track a particular step in the process. Operators may immediately recognize changes and take appropriate action by using these charts to analyses trends, patterns, and shifts in the data produced by the process. Walter Shewhart [
1] is the inventor of memoryless-type CCs, which are less susceptible to small or moderate shifts and effectively monitor significant shifts in the manufacturing process by using only the most recent sample information. In contrast, traditional memory-type control charts, such as the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts, as suggested by [
2] and [
3], respectively, are commonly employed to effectively monitor small-to-moderate changes. The fundamental designs of traditional memory type CCs are continually being modified and enhanced (see for example, [
4,
5,
6,
7,
8]). Haq and Woodall [
9] highlight the close relationship between the modified EWMA CC and a moving average-based EWMA CC. Both exhibit unfavorable weighting functions that impact run length properties, leading to the suggestion that an ordinary EWMA CC may offer better overall performance. Yeganeh et al. [
10] present adaptive GLR CCs for monitoring linear profiles using variable sampling intervals and sequential sampling, particularly focusing on scenarios where explanatory variables are uncontrollable. The SS approach exhibits superior performance with lower ATS values in simulations and real-life applications. Riaz et al. [
11] introduces two adaptive CCs for process mean vector shift monitoring, using PCA dimensionality reduction and adaptive techniques like Huber and Bi-square functions. The PCA-based multivariate CUSUM CC outperforms classical EWMA CCs in simulations and real-life wind turbine manufacturing. Woodall et al. [
12] assess CCs using RSS techniques, emphasizing their improved average run length performance due to reduced parameter estimation error. Nevertheless, they underscore the importance of caution when evaluating RSS benefits over time, especially in the context of monitoring process means. Haq et al. [
13] has examined the capability of the EWMA CC in the existence of measurement inaccuracy. When shift size is already available or when it is needed to create a CC for a specific shift, CCs may be a beneficial tool. Before employing CCs, the amount of the shift is often already decided. To facilitate the identification of changes of various magnitudes, a quality investigator may focus on improving double and adaptive control charting techniques. The Shewhart and EWMA CC elements are combined in a user-friendly manner to create the adaptive exponentially weighted moving average (AEWMA) CC. Sabahno et al. [
14] introduces adaptive schemes for simultaneous monitoring of the mean and variability in a multivariate normal quality characteristic, extending existing bivariate non-adaptive CCs and demonstrating the applicability of the adaptive scheme through a numerical example. Santorea et al. [
15] introduce adaptive CCs utilizing RSS, featuring variable sample sizes and multiple dependent state sampling, demonstrating improved performance in comparison to non-adaptive RSS and adaptive SRS approaches, supported by comprehensive simulations and practical applications. For monitoring location parameters, the authors of [
16] have established a comparison technique and have suggested using the AEWMA CC. The AEWMA CC was created with the Huber score function, which includes elements of both the EWMA and Shewhart CCs. The study has shown that while monitoring shifts of various magnitudes, the AEWMA CC outperformed the conventional Shewhart, optimal EWMA, Shewhart EWMA, and optimal CUSUM CCs. In the literature, various studies are conducted to explore the ACUSUM and AEWMA CCs for monitoring the process mean [
17,
18,
19,
20,
21]. Abbas et al. [
22] presents nonparametric DEWMA charts using the Wilcoxon signed-rank test for efficient process location monitoring, demonstrating superior performance compared to classical and nonparametric alternatives. A practical application involving piston ring diameter is also provided. Abbas et al. [
23] introduces a PAEWMA chart for monitoring nonconformities per unit in industrial processes, demonstrating superior performance compared to existing schemes in detecting unknown shifts, with real-life applications from various datasets. The AEWMA CC is examined by Zaman et al. [
24] utilizing the Tukey Bi-square function, which effectively tracks the location parameter of process. All of the aforementioned research has been conducted using the traditional methodology, which simply relies on sample data and leaves out previous knowledge. The Bayesian approach is an estimating technique that incorporates sample data and prior knowledge, which is updated to provide a posterior distribution. Girshick and Rubin [
25] discuss SPC and continuous inspection methods optimized for a specific income function and a production model with four states, including known transition probabilities. The study utilizes Markov processes and integral equations to derive optimal procedures approaching a limiting distribution. Riaz et al. [
26] analyze the Bayesian EWMA CC under three LFs with various informative and non-informative priors. Performance is assessed using ARL and SDRL via Monte Carlo simulations across different smoothing constant values, accompanied by a practical illustrative example. A Bayesian modified EWMA CC that includes posterior (P) and posterior predictive (PP) distribution is introduced in Asalam et al.’s [
27] study, which has revealed the CC’s greater capacity to detect out-of-control signals compared to other CCs by evaluating its performance using ARL and SDRL metrics. Noor et al. [
28] have suggested the Bayesian AEWMA CC for monitoring process mean under various loss functions. Du et al. [
29] propose a Bayesian-based lubricating oil replacement scheme using a hidden Markov chain model, enhancing machine health, lowering costs, and improving availability, surpassing age-based and failure-based methods in fault detection and average availability. Ali [
30] presents Bayesian predictive monitoring using CUSUM and EWMA CCs, eliminating the need for large Phase-I datasets, enabling online monitoring, comparing Bayesian memory-type charts with frequentist ones, and assessing performance under practitioner-to-practitioner variability using AARL and SDARL. A brand-new Bayesian EWMA CC is developed by Lin et al. [
31] for detecting changes in process variance without depending on presumptions on its underlying distribution and investigated the sampling properties of the proposed method to make its implementation simple for processes with time-varying distributions. They have also carried out a simulation study to show the value of the CC. In their unique Bayesian Hybrid EWMA CC, Imad et al. [
32] have proposed a variety of RSS techniques and an instructive prior to monitor the process mean. The efficiency of the proposed method is assessed by using run length profiles, and it is then contrasted with that of other Bayesian CCs under SRS, such as Bayesian HEWMA and Bayesian AEWMA. Wang et al. [
33] explore how measurement error affects the Bayesian EWMA CC, incorporating different RSS sampling designs and LFs, and find that the median ranked set sampling scheme performs best under these conditions.
PRSS is a technique that aims to reduce data collection costs and time while maintaining or improving accuracy. It involves collecting paired samples based on the ranked order of units, showing promise in various applications. Integrating Bayesian CCs with PRSS enhances its efficiency and effectiveness. Bayesian CCs enable the incorporation of prior information, aiding informed decisions and improved estimates. They offer more precise and reliable control limits compared to traditional charts by accommodating various data distributions. This integration also addresses the small sample size issue by utilizing paired sample information effectively. The Bayesian framework allows flexible modeling of complex relationships, useful in situations with nonlinear or non-monotonic dependencies. Overall, PRSS with Bayesian control charts improves sampling by incorporating prior knowledge, providing robust control limits, addressing sample size concerns, and allowing flexible modeling for better decision making and process improvement. The focus in the current study is to develop a novel Bayesian AEWMA CC that utilizes various paired RSS schemes, such as paired ranked set sampling (PRSS), quartiles paired ranked set sampling (QPRSS), and extreme paired ranked set sampling (EPRSS), including an informative prior for the P and PP distributions based on distinct LFs such as SELF and LLF. The study evaluates the capability of the suggested Bayesian CC using ARL and SDRL. The remaining work is organized as follows:
Section 2 describes the Bayesian theory,
Section 3 discusses several PRSS schemes, and
Section 4 describes how to build a Bayesian AEWMA CC.
Section 5 includes the simulation study. Results, debates, and findings are found in
Section 6, and real-world applications are found in
Section 7.
Section 8 contains the conclusion.