Standard Deviation Of Run Length Research Articles

Robustness of Moving Average-Exponential Weighted Moving Average Control Chart with the Light-Tailed Distribution

Control charts are the most significant statistical process control tools for ensuring industrial process reliability and efficiency. This research uses moving average – exponentially weighted moving average (MA-EWMA) control charts to investigate the dispersion characteristics. The control chart performance compares the arithmetic mean of run length (AMRL) and standard deviation of run length (SDRL) profiles of MA-EWMA and synthetic charts. Student's t-distributions are compared for dispersion processes. Finally, we present a case to show how essential control charts are in practice.

Sensitivity and reliability comparisons of EWMA mean control chart based on robust scale estimators under non‐normal process: COVID data application

AbstractRobust control charts are getting vital importance in statistical process control theory as they are insensitive to the departure from normality. Therefore, the main objective of current work is to determine the effects of non‐normal process on the Exponentially Weighted Moving Average (EWMA) control chart. To achieve this goal, the sensitivity and reliability comparisons are made under the non‐normal process by comparing five robust M‐scale estimators, suggested in literature to modify the EWMA control limits for monitoring process mean and on the basis of percentile bootstrap estimator. The paper addresses the run length (RL) distribution of a robust EWMA control chart under the non‐normal process for which the exponential distribution is used as non‐normal process. The standard deviation of RL, out‐of‐control average run length (ARL), and shift detection probabilities are examined to assess the sensitivity and reliability of robust EWMA control charts for mean of monitoring process. The results of this research indicate that the classical EWMA control chart's performance is substantially impacted by the non‐normal distribution and the proposed EWMA control charts show higher sensitivity than classical one in terms of having smaller values of out‐of‐control ARLs. A real‐life example from the medical sciences field is provided the practical usage of the proposed control charts. The simulation analysis and practical example have shown that the suggested control charts are effective in quickly monitoring the out‐of‐control process.

Exponentially weighted moving average chart using zero-inflated negative binomial distribution

Zero-inflated models are frequently used to deal with data having many zeros. A commonly used model for over-dispersed data containing zeros is known as the zero-inflated Poisson model. However, to account for the heterogeneity of counts that leads to excess variance besides inflation of zeros in the data using a more flexible model than the zero-inflated Poisson model, a zero-inflated negative binomial (ZINB) is suggested. In the present study, Shewhart and exponentially weighted moving average (EWMA) control charts are suggested to monitor the ZINB data. The charts are compared using the average run length and standard deviation of run length by using extensive Monte Carlo simulations. Besides a comprehensive simulation study assuming different settings of parameters of ZINB, a real data set is used to show the practicality of the proposed charts. The results indicate that the EWMA chart is better than the Shewhart chart.

Shewhart Type Control Chart Based on Median for Location Parameter

Control charts are prominent tools in monitoring various variables of manufacturing processes. In this paper, the control chart based on sample median is proposed for process location parameter using Shewhart?s approach. The proposed control chart is developed under some distributions including normal distribution. Its performance is studied in terms of power, average run length (ARL) and standard deviation of run length (SDRL). Also, the effect of non normality on this control chart is discussed.. KEYWORDS :ARL, Control limits, Location parameter, Median, Non-normality, Power.

Designing Bayesian paradigm-based CUSUM scheme for monitoring shape parameter of the Inverse Gaussian distribution

Bayesian Control charts have gained prominence as highly effective statistical instruments in overseeing manufacturing processes and ensuring adept management of process variability. The Bayesian methodology proves especially adept in managing uncertainties related to parameters within the manufacturing sector. In this research endeavor, our focus lies in establishing the threshold for monitoring the shape parameter of the Inverse Gaussian distribution (IGD) in phase-II based on cumulative sum (CUSUM) chart in classical and Bayesian frameworks using a variety of loss functions for enhancing their utility and applicability and have compared them with the Shewhart and EWMA control charts. We have also compared the proposed IGD-based CUSUM charts with the Lomax-based CUSUM charts. The evaluation of the proposed approach hinges on performance measures such as average run length (ARL), standard deviation of run length (SDRL), and median of run length (MDRL). Through simulation, the study showcases the efficiency of the Bayesian approach-based CUSUM charts, highlighting their superiority over the conventional classical setup-oriented CUSUM charts. Additionally, the proposed approach is employed in practical scenarios using real-data case studies from the aerospace manufacturing sector. The outcomes derived from real-data analysis substantiate and reinforce the conclusions drawn from the simulated results.

Developing Bayesian EWMA chart for change detection in the shape parameter of Inverse Gaussian process.

Bayesian Control charts are emerging as the most efficient statistical tools for monitoring manufacturing processes and providing effective control over process variability. The Bayesian approach is particularly suitable for addressing parametric uncertainty in the manufacturing industry. In this study, we determine the monitoring threshold for the shape parameter of the Inverse Gaussian distribution (IGD) and design different exponentially-weighted-moving-average (EWMA) control charts based on different loss functions (LFs). The impact of hyperparameters is investigated on Bayes estimates (BEs) and posterior risks (PRs). The performance measures such as average run length (ARL), standard deviation of run length (SDRL), and median of run length (MRL) are employed to evaluate the suggested approach. The designed Bayesian charts are evaluated for different settings of smoothing constant of the EWMA chart, different sample sizes, and pre-specified false alarm rates. The simulative study demonstrates the effectiveness of the suggested Bayesian method-based EWMA charts as compared to the conventional classical setup-based EWMA charts. The proposed techniques of EWMA charts are highly efficient in detecting shifts in the shape parameter and outperform their classical counterpart in detecting faults quickly. The proposed technique is also applied to real-data case studies from the aerospace manufacturing industry. The quality characteristic of interest was selected as the monthly industrial production index of aircraft from January 1980 to December 2022. The real-data-based findings also validate the conclusions based on the simulative results.

Evaluation of different DEWMA schemes for monitoring the ratio of two normal random variables

In many fields such as baking industry and cosmetic factory, the ratio of two normal random variables (RZ) is one of the key quality characteristics that needs to be monitored. In this paper, we study the properties of three double exponentially weighted moving average (DEWMA) schemes for monitoring the RZ. The run length (RL) properties, average run length (ARL) and standard deviation of run length (SDRL), and charting parameters of the proposed DEWMA RZ schemes are simulated for different correlation parameter between the two normal variables. These numerous simulation results are presented in tables and figures to make comparisons with the existing exponentially weighted moving average (EWMA) RZ scheme. The comparisons show that the proposed DEWMA RZ schemes are more effective than the EWMA RZ in detecting small changes. Additionally, the proposed three one-sided DEWMA RZ schemes’ properties are compared in detail. A food industry example details the implementation of the proposed RZ schemes.

Two Bayesian approaches of monitoring mean of Gaussian process using Bayes factor

AbstractThis paper develops two novel process monitoring schemes for the mean of a Gaussian process: the Bayes factor (BF) and the improved Bayes factor (IBF) schemes. Conjugate priors are used to construct the plotting statistics. The performance of the proposed schemes is evaluated in terms of average run length (ARL), standard deviation of run length (SDRL), and several percentiles, and these performance metrics across different hyper‐parameters and various sample sizes are evaluated via Monte Carlo simulations. Both zero‐state and steady‐state out‐of‐control (OOC) performances are investigated comprehensively. The simulation results show that the IBF scheme outperforms the existing Bayesian exponentially weighted moving average (EWMA) schemes under different loss functions in zero‐state. In steady‐state conditions, the IBF scheme outperforms for small shifts. Finally, we present two examples to illustrate the practical application of the proposed schemes.

Memory type Bayesian adaptive max-EWMA control chart for weibull processes

The simultaneous monitoring of both the process mean and dispersion has gained considerable attention in statistical process control, especially when the process follows the normal distribution. This paper introduces a novel Bayesian adaptive maximum exponentially weighted moving average (Max-EWMA) control chart, designed to jointly monitor the mean and dispersion of a non-normal process. This is achieved through the utilization of the inverse response function, particularly suitable for processes conforming to a Weibull distribution. To assess the effectiveness of the proposed control chart, we employed the average run length (ARL) and the standard deviation of run length (SDRL). Subsequently, we compared the performance of our proposed control chart with that of an existing Max-EWMA control chart. Our findings suggest that the proposed control chart demonstrates a higher level of sensitivity in detecting out-of-control signals. Finally, to illustrate the effectiveness of our Bayesian Max-EWMA control chart under various Loss Functions (LFs) for a Weibull process, we present a practical case study focusing on the hard-bake process in the semiconductor manufacturing industry. This case study highlights the adaptability of the chart to different scenarios. Our results provide compelling evidence of the exceptional performance of the suggested control chart in rapidly detecting out-of-control signals during the hard-bake process, thereby significantly contributing to the improvement of process monitoring and quality control.

The Development and Evaluation of Homogenously Weighted Moving Average Control Chart based on an Autoregressive Process

This research aims to investigate a Homogenously Weighted Moving Average (HWMA) control chart for detecting minor and moderate shifts in the process mean. A mathematical model for the explicit formulae of the average run length (ARL) of the HWMA control chart based on the autoregressive (AR) process is presented. The efficacy of the HWMA control chart is evaluated based on the average run length, the standard deviation of run length (SDRL), and the median run length (MRL). As illustrations of the design and implementation of the HWMA control chart, numerical examples are provided. In numerous instances, a comparative analysis of the HWMA control chart relative to the Extended Exponentially Weighted Moving Average (Extended EWMA) and cumulative sum (CUSUM) control charts with mean process shifts is performed in detail. Additionally, the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI) are utilized to evaluate the performance of control charts. For various shift sizes, the HWMA control chart is superior to the Extended EWMA and CUSUM control charts. This study applies empirical data from the area of economics to validate the explicit formula of ARL values for the HWMA control chart. Doi: 10.28991/HIJ-2024-05-01-02 Full Text: PDF

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.

Performance Evaluation of HWMA Control Chart based on AR(p) with Trend Model to Detect Shift Process Mean

The main goal of this study is to establish explicit solutions for the average run length (ARL) of the Homogenously Weighted Moving Average control chart when subjected to autoregressive with trend process. The accuracy of the explicit formula for the ARL is evaluated in comparison to the numerical integral equation method. To evaluate the two approaches, the accuracy percentage was employed. A determination is carried out of the HWMA control chart’s effectiveness using the median run length (MRL), the standard deviation of run length (SDRL), and the average run length (ARL). A comprehensive comparison is performed between the HWMA control chart, the Extended Exponentially Weighted Moving Average (EEWMA), and the cumulative sum (CUSUM) control charts with mean process shifts to illustrate the design and implementation of the HWMA control chart. As criteria for various values of design parameters, the performance of these control charts can also be evaluated using the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI). To evaluate the effectiveness of our explicit formula approach, we employ this formula on copper price data.

A nonparametric mixed exponentially weighted moving average-moving average control chart with an application to gas turbines.

This study aims to develop a nonparametric mixed exponentially weighted moving average-moving average (NPEWMA-MA) sign control chart for monitoring shifts in process location, particularly when the distribution of a critical quality characteristic is either unknown or non-normal. In literature, the variance expression of the mixed exponentially weighted moving average-moving average (EWMA-MA) statistic is calculated by allowing sequential moving averages to be independent, and thus the exclusion of covariance terms results in an inaccurate variance expression. Furthermore, the effectiveness of the EWMA-MA control chart deteriorates when the distribution of a critical quality characteristic deviates from normality. The proposed NPEWMA-MA sign control chart addresses these by utilizing the corrected variance of the EWMA-MA statistic and incorporating the nonparametric sign test into the EWMA-MA charting structure. The chart integrates the moving average (MA) statistic into the exponentially weighted moving average (EWMA) statistic. The EWMA-MA charting statistic assigns more weight to recent w samples, with weights for previous observations decling exponentially. Monte Carlo simulations assess the chart's performance using various run length (RL) characteristics such as average run length (ARL), standard deviation of run length (SDRL), and median run length (MRL). Additional measures for overall performance include the average extra quadratic loss (AEQL) and relative mean index (RMI). The proposed NPEWMA-MA sign control chart demonstrates superior performance compared to existing nonparametric control charts across different symmetrical and asymmetric distributions. It efficiently detects process shifts, as validated through both a simulated study and a real-life example from a combined cycle power plant.

Monitoring of semiconductor manufacturing process on Bayesian AEWMA control chart under paired ranked set sampling schemes

Quality control often employs memory-type control charts, including the exponentially weighted moving average (EWMA) and Shewhart control charts, to identify shifts in the location parameter of a process. This article pioneers a new Bayesian Adaptive EWMA (AEWMA) control chart, built on diverse loss functions (LFs) such as the square error loss function (SELF) and the Linex loss function (LLF). The proposed chart aims to enhance the process of identifying small to moderate as well as significant shifts in the mean, signifying a notable advancement in the field of quality control. These are implemented utilizing an informative prior for both posterior and posterior predictive distributions, employing various paired ranked set sampling (PRSS) schemes. The effectiveness of the suggested chart is appraised using average run length (ARL) and the standard deviation of run length (SDRL). Monte Carlo simulations are employed to contrast the recommended approach against other control charts. The outcomes demonstrate the dignitary performance of the recommended chart in identifying out-of-control signals, especially applying PRSS designs, in comparison to simple random sampling (SRS). Finally, a practical application was conducted in the semiconductor manufacturing context to appraise the efficacy of the offered chart using various paired ranked set sampling strategies. The results reveal that the suggested control chart performed well in capturing the out-of-control signals far better than the already in use control charts. Overall, this study interposes a new technique with diverse LFs and PRSS designs, improving the precision and effectiveness in detecting process mean shifts, thereby contributing to advancements in quality control and process monitoring.

Reassessment of performance evaluation of EWMA control chart for exponential process

AbstractMemory control charts or charts, such as cumulative sums and exponentially weighted moving average (EWMA), are widely used for tracking minor to moderate changes in process parameters. The literature presented an EWMA chart for monitoring exponential processes, assuming probability limits to determine both in‐ and out‐of‐control average run length (ARL). The use of probability limits for memory charts is an incorrect method. So, we are aiming to correct the EWMA chart by Aslam et al. and the proposed EWMA chart (represented by ) for the exponential process using the run‐length (RL) method based on the simulation method instead of the probability‐based method. The ARL, standard deviation of RL, and median of RL are used to evaluate the proposed chart's performance. For practical consideration of the proposed chart, a real‐world example of a urinary tract infection is also presented.

Profile monitoring using synthetic T2 control chart

In this paper, we propose synthetic T2 chart for monitoring different parameters including Slope, Intercept and Error variance for simple and multiple linear regression profiles in Phase II. The construction and optimal design of the proposed synthetic T2 chart are discussed in detail. The steady-state performance of the proposed chart is evaluated by the Average Run Length (ARL) and Standard Deviation of Run Length (SDRL) and is compared with the existing charts in the literature for monitoring simple and multiple linear profiles. It turns out that the proposed synthetic T2 chart performs uniformly better than the traditional T2 chart in detecting different kinds of shifts in the parameters of the simple and multiple linear profiles under steady-state run length performance. An example with a real data set is also presented to explain the implementation of the proposed chart.

Improving Sensitivity of the DEWMA Chart with Exact ARL Solution under the Trend AR(p) Model and Its Applications

The double exponentially weighted moving average (DEWMA) chart is a control chart that is a vital analytical tool for keeping track of the quality of a process, and the sensitivity of the control chart to the process is evaluated using the average run length (ARL). Herein, the aim of this study is to derive the explicit formula of the ARL on the DEWMA chart with the autoregressive with trend model and its residual, which is exponential white noise. This study shows that this proposed method was compared to the ARL derived using the numerical integral equation (NIE) approach, and the explicit ARL formula decreased the computing time. By changing exponential parameters that were relevant to evaluating in various circumstances, the sensitivity of AR(p) with the trend model with the DEWMA chart was investigated. These were compared with the EWMA and CUSUM charts in terms of the ARL, standard deviation run length (SDRL), and median run length (MRL). The results indicate that the DEWMA chart has the highest performance, and when it was small, the DEWMA chart had high sensitivity for detecting processes. Digital currencies are utilized to demonstrate the efficacy of the proposed method; the results are consistent with the simulated data. Doi: 10.28991/ESJ-2023-07-06-03 Full Text: PDF

Process Monitoring Using Truncated Gamma Distribution

The time-between-events idea is commonly used for monitoring high-quality processes. This study aims to monitor the increase and/or decrease in the process mean rapidly using a one-sided exponentially weighted moving average (EWMA) chart for the detection of upward or downward mean shifts using a truncated gamma distribution. The use of the truncation method helps to enhance and improve the sensitivity of the proposed chart. The performance of the proposed chart with known and estimated parameters is analyzed by using the run length properties, including the average run length (ARL) and standard deviation run length (SDRL), through extensive Monte Carlo simulation. The numerical results show that the proposed scheme is more sensitive than the existing ones. Finally, the chart is implemented in real-world situations to highlight the significance of the proposed chart.

Bayesian AEWMA control chart under ranked set sampling with application to reliability engineering

The article introduces a novel Bayesian AEWMA Control Chart that integrates different loss functions (LFs) like the square error loss function and Linex loss function under an informative prior for posterior and posterior predictive distributions, implemented across diverse ranked set sampling (RSS) designs. The main objective is to detect small to moderate shifts in the process mean, with the average run length and standard deviation of run length serving as performance measures. The study employs a hard bake process in semiconductor production to demonstrate the effectiveness of the proposed chart, comparing it with existing control charts through Monte Carlo simulations. The results underscore the superiority of the proposed approach, particularly under RSS designs compared to simple random sampling (SRS), in identifying out-of-control signals. Overall, this study contributes a comprehensive method integrating various LFs and RSS schemes, offering a more precise and efficient approach for detecting shifts in the process mean. Real-world applications highlight the heightened sensitivity of the suggested chart in identifying out-of-control signals compared to existing Bayesian charts using SRS.

A Weibull process monitoring with AEWMA control chart: an application to breaking strength of the fibrous composite

In recent times, there has been a growing focus among researchers on memory-based control charts. The Exponentially Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) charts and the adaptive control charting approaches got the attention. Control charts are commonly employed to oversee processes, assuming the monitored variable follows a normal distribution. However, it's worth noting that this assumption does not hold true in many real-world situations. The use of the algebraic expression for normalization, which can be used for all kinds of skewed distributions with a closed-form distribution function, using the proposed continuous function to adapt a smoothing constant, motivates this study. In the present manuscript, we design an EWMA statistic-based adaptive control chart to monitor the irregular variations in the mean of two parametric Weibull distribution and use Hasting approximation for normalization. The adaptive control charts are used to update the smoothing constant according to the estimated shift. Here we use the proposed continuous function to adapt the smoothing constant. The average run length and standard deviation of run length are calculated under different parameter settings. The effectiveness of the proposed chart is argued in terms of ARLs over the considered EWMA chart through Monte-Carlo (MC) simulation method. The proposed chart is examined, followed by a real data set to demonstrate the design and application procedures.