Standard Deviation Of The Run Length Research Articles

Optimal triple sampling [formula omitted] charts for autocorrelated processes

Autocorrelation is a phenomenon in which the characteristics of interest being monitored are dependent. In the presence of autocorrelation, the detection ability of classical control charts deteriorates significantly, and this poses great concerns to practitioners as some processes in real-life settings produce autocorrelated observations. This paper deals with the problem of autocorrelation in Shewhart charts by incorporating the AR(1) and MA(1) models into the classical TS X¯ chart, which is an extension of the Shewhart chart. The performance of the introduced TS X¯ charts for AR(1) and MA(1) is improved by optimizing the parameters of the charts when the in-control average run length (ARL) of the charts is 370. The TS X¯chart for AR(1) is found to outperform the DS X¯chart for AR(1) in the presence of autocorrelation. The performance of the proposed charts is assessed using ARL, standard deviation of the run length (SDRL), average number of observations to signal (ANOS), and standard deviation of the number of observations to signal (SDNOS) criteria. Based on these criteria, the TS X¯chart for MA(1) is found to surpass the TS X¯chart for AR(1) as the autocorrelation level increases.

A new EWMA chart for simultaneously monitoring the parameters of a shifted exponential distribution

In various scenarios where products and services are accompanied by warranties to ensure their reliability over a specified time, the two-parameter (shifted) exponential distribution serves as a fundamental model for time-to-event data. In modern production process, the products often come with warranties, and their quality can be manifested by the changes in the scale and origin parameters of a shifted exponential (SE) distribution. This paper introduces the Max-EWMA chart, employing maximum likelihood estimators and exponentially weighted moving average (EWMA) statistics, to jointly monitor SE distribution parameters. Additionally, we extend two additional charts, namely the Max-DEWMA and Max-TEWMA charts to enhance early-stage shift detection. Performance evaluations under zero-state and steady-state conditions compare these charts with the existing Max-CUSUM chart in terms of expected value and standard deviation of the run length (RL) distribution. Our findings reveal that among the Max-EWMA schemes, the Max-EWMA SE chart outperforms the others in terms of steady-state performance, while the Max-TEWMA chart surpasses the Max-EWMA and Max-DEWMA SE charts in respect to zero-state performance. Moreover, the proposed Max-EWMA schemes demonstrate advantages over Max-CUSUM, especially for small to moderate smoothing constants. We also provide an illustrative example to demonstrate the implementation of the proposed schemes.

A Robust Design for the Omnibus SPRT Control Chart Under Skewed Data Distributions

Control charts are widely used in manufacturing industries to ensure that production levels are stable and satisfactory. Recently, the omnibus sequential probability ratio test (OSPRT) control chart was developed for the purpose of monitoring the mean and variability of a process simultaneously. As the OSPRT chart was proposed for the first time in literature, its development relied entirely on the assumption that data follow the Normal distribution. Nonetheless, researchers are frequently reminded that the quality characteristics of manufacturing processes do not necessarily follow the Normal distribution, e.g., strengths of glass fibres, and lifetimes of products. In this paper, we investigate the extent to which the performances of the OSPRT chart designed for the Normal model deteriorate, in situations where the data distributions are Gamma and Lognormal. Results show that the in-control average run length (ARL) and standard deviation of the run length of the OSPRT chart designed for the Normal distribution deteriorate rapidly as skewness increases. To address this issue, we propose a robust design for the OSPRT chart by adjusting its control limits, known as the skewness correction method. It is shown that the skewness-corrected OSPRT chart enjoys a guaranteed in-control ARL, with a justifiable degradation in its out-of-control performances. Besides, we also show some insights into selecting the charting parameters for the skewness-corrected OSPRT chart in order to achieve an optimum out-of-control ARL performance over various shift sizes. The paper wraps up with an illustrative example of the skewness-corrected OSPRT chart for monitoring the weights of radial tyres.

A double sampling ridge penalized likelihood ratio control charting method with variable sample size for Phase II monitoring of high-dimensional covariance matrix

Today, manufacturers face two challenges: the financial limitation of sampling and the large number of study quality characteristics. In such situations, the problem dimension is usually larger than the sample size, which is known in the literature of statistical quality control as a high dimensional process. The use of conventional multivariate charts to monitor the covariance matrix of such processes is not possible for two reasons: (1) the non-reversibility of the sample covariance matrix, (2) the possibility of the occurrence of sparse shifts in which only a few elements of the covariance matrix are affected. This paper focuses on Phase II monitoring of the covariance matrix of high-dimensional processes based on integrating the ridge penalized likelihood ratio (RPLR) statistic and the combined DS-VSS sampling method, in which DS and VSS features are used concurrently. The performance of the proposed DS-VSS RPLR control chart is compared with the RPLR one in terms of three metrics of the average run length, standard deviation of the run length, and expected value of sample size under three types of joint diagonal/off-diagonal, diagonal and off-diagonal shift patterns. The results show that the DS-VSS scheme is more sensitive than the RPLR in detecting both sparse and non- sparse changes.

A robust cusum control chart for median absolute deviation based on trimming and winsorization.

Statistical quality control is concerned with the analysis of production and manufacturing processes. Control charts are process control techniques, commonly applied to observe and control deviations. Shewhart control charts are very sensitive and used for large shifts based on the basic assumption of normality. Cumulative Sum (CUSUM) control charts are effective for identifying that may have special causes, such as outliers or excessive variability in subgroup means. This study uses a CUSUM control chart problems structure to evaluate the performance of robust dispersion parameters. We investigated the design structure features of various control charts, based on currently defined estimators and some new robust scale estimators using trimming and winsorization in different scenarios. The Median Absolute Deviation based on trimming and winsorization is introduced. The effectiveness of CUSUM control charts based on these estimators is evaluated in terms of average run length (ARL) and Standard Deviation of the Run Length (SDRL) using a simulation study. The results show the robustness of the CUSUM chart in observing small changes in magnitude for both normal and contaminated data. In general, robust estimators MADTM and MADWM based on CUSUM charts outperform in all environments.

Analytical Explicit Formulas of Average Run Length of Homogenously Weighted Moving Average Control Chart Based on a MAX Process

Statistical process control (SPC) is used for monitoring and detecting anomalies in processes in the areas of manufacturing, environmental studies, economics, and healthcare, among others. Herein, we introduce an innovative SPC approach via mathematical modeling and report on its application via simulation studies to examine its suitability for monitoring processes involving correlated data running on advanced control charts. Specifically, an approach for detecting small to moderate shifts in the mean of a process running on a homogenously weighted moving average (HWMA) control chart, which is symmetric about the center line with upper and lower control limits, is of particular interest. A mathematical model for the average run length (ARL) of a moving average process with exogenous variables (MAX) focused only on the zero-state performance of the HWMA control chart is derived based on explicit formulas. The performance of our approach was investigated in terms of the ARL, the standard deviation of the run length (SDRL), and the median run length (MRL). Numerical examples are given to illustrate the efficacy of the proposed method. A detailed comparative analysis of our method for processes on HWMA and cumulative sum (CUSUM) control charts was conducted for process mean shifts in many situations. For several values of the design parameters, the performances of these two control charts are also compared in terms of the expected ARL (EARL), expected SDRL (ESDRL), and expected MRL (EMRL). It was found that the performance of the HWMA control chart was superior to that of the CUSUM control chart for several process mean shift sizes. Finally, the applicability of our method on a HWMA control chart is provided based on a real-world economic process.

Performance Analysis of Interval Type-2 Fuzzy X¯ and R Control Charts

Statistical process control (SPC) is one of the most powerful techniques for improving quality, as it is able to detect special causes of problems in processes, products and services with a remarkable degree of accuracy. Among SPC tools, X¯ and R control charts are widely employed in process monitoring. However, scenarios involving vague, imprecise and even subjective data require a type-2 fuzzy set approach. Thus, X¯ and R control charts should be coupled with interval type-2 triangular fuzzy numbers (IT2TFN) in order to add further information to traditional control charts. This paper proposes a performance analysis of IT2TFN and X¯ and R control charts by means of average run length (ARL), standard deviation of the run length (SDRL) and RL percentile. Computer simulations were carried out considering 10,000 runs to obtain ARL, SDRL and the 5th, 25th, 50th, 75th and 95th RL percentiles. Simulation results reveal that the proposed control charts increased fault detection capability (speed of response) and slightly reduced the number of false alarms in processes under control. Moreover, it was observed that, in addition to superior performance, IT2TFN X¯-R control charts proved to be more robust and flexible when compared to traditional control charts.

EWMA monitoring schemes for MCV in short production runs with an application to the steel sleeve manufacturing process

In industries, multi-variety and small batch processes are extremely common due to the fact that the production process is moving towards flexible manufacturing to meet the increasing demand of consumers for individualized products. In recent years, monitoring techniques based on the multivariate coefficients of variation (MCV) have been widely developed owing to their applicability in assessing the relative variability of multivariate processes. It is worth noting that few of these studies have targeted monitoring the MCV in a short production run context. In this paper, two new one-sided MCV monitoring schemes are proposed by adopting the exponentially weighted moving average (EWMA) scheme in a finite horizon production. Based on the Markov chain approach, the performance measures, including the truncated average run length (TARL) and truncated standard deviation of the run length (TSDRL), of the EWMA MCV schemes are derived. Moreover, the superiority of the proposed monitoring schemes is illustrated by comparing their performance with that of the conventional Shewhart MCV monitoring schemes in a short production run context. The results show that increasing the batches enhances the advantages of the EWMA monitoring schemes over the Shewhart schemes. Finally, the superiority of the proposed EWMA monitoring scheme is illustrated by a real steel sleeve manufacturing case study.

Optimal design of S2‐EWMA control chart for short production runs

AbstractIn recent fast pace smart manufacturing environment, the short production runs (SPR) control charts offer a higher degree of flexibility compared to the traditional control charts. This is due to Phase I data collection not needed for SPR control charts and this eases practitioners in estimating the parameters and setting up the control charts. This paper proposes the methods to optimally design the SPR S‐chart and SPR S2‐EWMA control chart for process variance monitoring in terms of truncated average run length (TARL), truncated standard deviation of the run length (TSDRL), and expected truncated average run length (ETARL) performance measures. The optimal parameters in minimizing the TARL and ETARL values for SPR S2‐EWMA control chart are presented. The control chart performances are compared between both the SPR S‐chart and SPR S2‐EWMA control charts. Results show that SPR S2‐EWMA control chart has a higher chart sensitivity and it triggers the out‐of‐control signal faster than the SPR S‐chart. An illustration using the real electronic manufacturing industry data is given in this paper to explain the implementation of the SPR S‐chart and SPR S2‐EWMA control chart.

Monitoring the Weibull Scale Parameter Based on Type I Censored Data Using a Modified EWMA Control Chart

In industrial production, the exponentially weighted moving average scheme is widely used to monitor shifts in product quality, especially small-to-moderate shifts. In this paper, we propose a modified one-sided EWMA scheme for Type I right-censored Weibull lifetime data for detecting shifts in the scale parameter with the shape parameter fixed. A comparative analysis with existing cumulative sum and exponentially weighted moving average results from the literature is provided. The zero-state and steady-state behaviour of the new scheme are considered with regard to the average run length, the standard deviation of the run length, and other performance measures. Our simulation shows stronger power in detecting changes in the censored lifetime data using the modified scheme than that using the traditional exponentially weighted moving average scheme, and the new scheme is superior to the cumulative sum scheme in most situations. A real-data example further demonstrates the effectiveness of the proposed method.

Memory‐type control charts for censored reliability data

AbstractControl charts are commonly used to monitor a process to detect undesirable changes. The main goal of this work is to propose exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts to track a process by utilizing type‐I censored generalized exponential (GE) distributed data. In particular, the censored data are replaced with the conditional expected value (CEV). A comparison between CUSUM and CUSUM ignoring unobserved covariates (CUSUM‐IUC) charts is also a part of this study. The GE distribution is considered due to its application in reliability analysis. The performance of the charts is evaluated by using the average run length along with the standard deviation of the run length. Furthermore, the study also examines the effect of smoothing parameters, censoring rate, and shifts on the proposed methodologies. The results indicate that the EWMA chart performs better than other schemes (CUSUM and CUSUM‐IUC) for shifts in the shape parameter of the GE distribution. Finally, the proposed schemes are also applied to a real‐life data set.

Monitoring of Location Parameters with a Measurement Error under the Bayesian Approach Using Ranked-Based Sampling Designs with Applications in Industrial Engineering

To detect sustainable changes in the production processes, memory-type control charts are frequently utilized. This study is conducted to assess the performance of the Bayesian adaptive exponentially weighted moving average (AEWMA) control chart using ranked set sampling schemes following two different loss functions in the presence of a measurement error for posterior and posterior predictive distributions using conjugate priors. This study is based on the covariate model and multiple measurement methods in the presence of a measurement error (ME). The performance of the proposed Bayesian-AEWMA control chart with ME has been evaluated through the average run length and the standard deviation of the run length. Finally, a real-life application in semiconductor manufacturing was conducted to evaluate the effectiveness of the proposed Bayesian-AEWMA control chart with a measurement error based on different ranked set sampling schemes. The results demonstrate that the proposed control chart, in the presence of a measurement error, performed well in detecting out-of-control signals compared to the existing control chart. However, the median ranked set sampling scheme (MRSS) proved to be better than the other two schemes in the presence of a measurement error.

Hybrid EWMA Control Chart under Bayesian Approach Using Ranked Set Sampling Schemes with Applications to Hard-Bake Process

A memory-type control chart is an important tool of statistical process control for monitoring small to moderate shifts in the manufacturing process. Using the prior information by the Bayesian approach is helpful in control charts. In this paper, a new hybrid exponentially weighted moving average (HEWMA) control chart is suggested under the Bayesian theory using ranked set sampling (RSS) schemes for posterior and posterior predictive distribution with informative prior and different loss functions (LFs). The extensive Monto Carlo simulation is conducted to evaluate the overall performance of the proposed Bayesian HEWMA control chart through average-run-length (ARL) and standard-deviation of the run-length (SDRL). Finally, a numerical example of the hard-bake process in semiconductor manufacturing is used to check the working and execution of the proposed Bayesian HEWMA control-chart under different RSS schemes. The results reveal that the suggested Bayesian HEWMA control-chart under RSS schemes is more sensitive in detecting out-of-control signals than the Bayesian HEWMA and Bayesian AEWMA control-charts under simple random sampling.

Efficient control chart-based monitoring of scale parameter for a process with heavy-tailed non-normal distribution

<abstract> <p>Statistical process control is a procedure of quality control that is widely used in industrial processes to enable monitoring by using statistical techniques. All production processes are faced with natural and unnatural variations. To maintain the stability of the production process and reduce variation, different tools are used. Control charts are significant tools to monitor a production process. In this article, we design an extended exponentially weighted moving average (EEWMA) chart under the assumption of inverse Maxwell (IM) distribution, an IM EEWMA (IMEEWMA) control chart. We have estimated the performance of the proposed chart in terms of various run-length (RL) properties, including the average RL, standard deviation of the RL and median RL. We have also carried out a comparative analysis of the proposed chart with the existing Shewhart-type chart for IM distribution (VIM chart) and IM exponential weighted moving average (IMEWMA) chart. We observed that the proposed IMEEWMA chart performed better than the VIM chart and IMEWMA chart in terms of the ability to detect small and moderate shifts. To demonstrate its practical application, we have applied the IMEEWMA chart, along with existing control charts, to monitor the lifetime of car brake pad data. This real-world example illustrates the superiority of the IMEEWMA chart over its counterparts in industrial scenarios.</p> </abstract>

Time and magnitude monitoring using Weibull based Max-EWMA chart

To monitor time and magnitude, different types of control charts have been proposed in the literature. This article proposes a new maximum exponentially weighted moving average (Max-EWMA) control chart by assuming Weibull distribution. The test-statistic of the Max-EWMA chart consists of two EWMA statistics, where one statistic is used for monitoring magnitude while the other to monitor time interval between events. The performances of the proposed chart is assessed on the basis of run length characteristics like, average run length (ARL), standard deviation of the run length (SDRL), and quantiles of the run length distribution. Besides extensive simulation studies, real data sets are also considered to show the applications of the proposed chart. The results show that the proposed chart is efficient in detecting different size of small shifts.

New EWMA t charts for process mean

AbstractIn practice, the process standard deviation is usually unknown or is estimated inaccurately. The ‐type charts, such as the exponentially weighted moving average (EWMA) chart, which works under the assumption that the process standard deviation is either known or it is estimated accurately, may not perform well when the process standard deviation is actually unknown or estimated inaccurately. To address the aforesaid setback of the EWMA‐ chart, the EWMA‐t chart was developed for monitoring the process mean when errors in estimating the process standard deviation occur or when the standard deviation is unknown. In this paper, new EWMA‐t charts, which are also robust against estimation errors in the process standard deviation and are applicable when the standard deviation is unknown, are developed for detecting mean shifts of a normal process. The new EWMA‐t charts are developed using the newly derived t‐statistics, based on the fixed sampling interval (FSI) and variable sampling interval (VSI) features. The new FSI/VSI EWMA‐t charts are found to outperform the existing FSI/VSI EWMA‐t chart, in terms of the zero‐state (ZS) and steady‐state (SS) average, median and standard deviation of the run‐length (SDRL)/time‐to‐signal criteria, for various shift sizes in the process mean, values of smoothing parameter and shift sizes in the process standard deviation. The application of the existing and new FSI/VSI‐EWMA‐t charts for detecting increases and decreases in the process mean is demonstrated with the automobile piston rings dataset.

Comparisons of some memory‐type control chart for monitoring Weibull‐distributed time between events and some new results

AbstractMonitoring Weibull time between event (TBE) processes is essential to avoid deterioration of quality characteristics in various reliability analysis problems. Statistical process monitoring (SPM) charts are widely used to monitor the possible decreasing mean shift. For monitoring such data, many control charts have been proposed in the literature, such as the Weibull exponentially weighted moving average (EWMA) chart, the Weibull cumulative sum (CUSUM) chart, and Mixed EWMA–CUSUM (MEC) chart and Mixed Hybrid EWMA–CUSUM (MHEC) chart. The current investigation revisits their efficacy and compares them with a CUSUM chart based on the generally weighted moving average (GWMA) statistic, labelled as the Mixed GWMA–CUSUM (MGC) chart, to monitor the decreasing mean shift in Weibull TBEs processes. We offer some numerical results based on the Monte‐Carlo simulation, such as average run length (ARL), the standard deviation of the run length (SDRL) and the relative mean index (RMI). Our findings show that the MGC chart often outperforms the existing control charts in detecting small downward shifts. Finally, an illustrative example is given to display the application of the MGC chart for Weibulll data.

On designing TEWMA-Tukey control charts for normal and non-normal processes using single and repetitive sampling schemes

Technology advancement and the need to produce quality products have necessitated the development of more efficient process monitoring schemes. In the form of a combined triple EWMA (TEWMA) and Tukey control chart (TCC), this study is an analogous attempt to develop a better charting framework for normal and non-normal data using single and repetitive schemes. The proposed and competing charts are compared based on average run length, standard deviation of the run length and median run length criteria under both zero-state and steady-state conditions. The proposed charts display dominance in detecting mean shifts in both directions not only for symmetric distributions but they are also robust for skewed distributions in that they are devoid of the average run length biased problem. The proposed charts beat the Tukey and Shewhart charts under the repetitive scheme. The TEWMA-TCC under the repetitive scheme has emerged as the best chart under all distributional setups. The steady-state findings of the TEWMA-TCC and its repetitive variant are as beneficial as those of the zero-state results, if not more. A practical application for the proposed designs in the surveillance of industrial production index data in the aerospace manufacturing is illustrated. The proposed charts issue more and earlier signals in the Phase-II process compared with their existing counterparts, most notably the two Tukey charts, i.e. the TCC and EWMA-TCC. Hence, quality and industrial engineers can rely on the proposed charts in process monitoring.

On the performances of the X ¯ and EWMA charts with auxiliary information for monitoring the process mean in short production runs

This article investigates the performances of auxiliary information based X ¯ (called X ¯ − AI ) and exponentially weighted moving average (EWMA) (called EWMA-AI) charts for detecting shifts in the process mean in short production runs using the truncated average run length (TARL), truncated standard deviation of the run length (TSDRL) and expected truncated average run length (ETARL) criteria. The findings reveal that the EWMA-AI chart is better in detecting small and moderate shifts, while the two charts are comparable in detecting large shifts. An example is given to demonstrate the applications of the X ¯ − AI and EWMA-AI charts in short production runs. The significance of this article is based on the fact that it is the first article that studies the performances of auxiliary information (AI) charts in short production runs.

A new EWMA median chart with measurement errors in short production runs–An application in yogurt cup filling process

AbstractThis article presents an exponentially weighted moving average (EWMA) median chart for short production runs (SPR) (called EWMA SPR median chart). This chart improves the sensitivity of the Shewhart median chart for SPR (called Shewhart SPR median chart) proposed by Khoo et al15 in the detection of mean shifts. The proposed chart's performance is evaluated by means of the truncated average run length (TARL), truncated standard deviation of the run length (TSDRL), and expected TARL (ETARL) criteria. For all sizes of mean shifts, regardless of the sample size, the findings show that the EWMA SPR median chart outperforms the Shewhart SPR median chart. The EWMA SPR median chart is also investigated when measurement errors are present, as these errors affect the chart's performance. An example is given to illustrate the working of the chart when measurement errors are present.