Joint Monitoring of Process Location and Dispersion Based on CUSUM Procedure and Generalized Likelihood Ratio in the Presence of Measurement Errors

In this paper, we propose a new methodology based on the combination of cumulative sum procedure and generalized likelihood ratio statistic for joint monitoring of the process location and dispersion. Then, we explore the effect of measurement errors on detecting ability of the proposed control chart when (i) the variance of measurement error is constant (ii) the variance of measurement error increases linearly as the level of the process mean increases. We also utilize multiple measurements on each sample point in order to decrease the adverse effects of measurement errors on the performance of the proposed control charts. Two numerical examples based on simulation studies are given to evaluate the ability of the proposed methods in terms of average run length, median run length, standard deviation of run length, and the first and third quantile points of the run length distribution (Q₁ and Q₃). Finally, a real life example is given to illustrate the application of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

EWMA Control Chart Performance with Estimated Parameters under Non‐normality

Exponentially weighted moving average (EWMA) control charts can be designed to detect shifts in the underlying process parameters quickly while enjoying robustness to non‐normality. Past studies have shown that performance of various EWMA control charts can be adversely affected when parameters are estimated or observations do not follow a normal distribution. To the best of our knowledge, simultaneous effect of parameter estimation and non‐normality has not been studied so far. In this paper, a Markov chain approach is used to model and evaluate performance of EWMA control charts when parameter estimation is subject to non‐normality using skewed and heavy‐tailed symmetric distributions. Using standard deviation of the run length (SDRL), average run length (ARL), and percentiles of run lengths for various phase I sample sizes, we show that larger phase I sample sizes do not necessarily lead to a better performance for non‐normal observations. Copyright © 2015 John Wiley & Sons, Ltd.     

Monitoring multivariate coefficient of variation with upward Shewhart and EWMA charts in the presence of measurement errors using the linear covariate error model

In practice, measurement errors exist and ignoring their presence may lead to erroneous conclusions in the actual performance of control charts. The implementation of the existing multivariate coefficient of variation (MCV) charts ignores the presence of measurement errors. To address this concern, the performances of the upward Shewhart-MCV and exponentially weighted moving average MCV charts for detecting increasing MCV shifts, using a linear covariate error model, are investigated. Explicit mathematical expressions are derived to compute the limits and average run lengths of the charts in the presence of measurement errors. Finally, an illustrative example using a real-life dataset is presented to demonstrate the charts’ implementation.     

A double exponentially weighted moving average control chart for monitoring COM‐Poisson attributes

The Conway‐Maxwell‐Poisson (COM‐Poisson) distribution is a two‐parameter generalization of the Poisson distribution, which can be used for overdispersed or underdispersed count data and also contains the geometric and Bernoulli distributions as special cases. This article presents a double exponentially weighted moving average control chart with steady‐state control limits to monitor COM‐Poisson attributes (regarded as CMP‐DEWMA chart). The performance of the proposed control chart has been evaluated in terms of the average, the median, and the standard deviation of the run‐length distribution. The CMP‐DEWMA control chart is studied not only to detect shifts in each parameter individually but also in both parameters simultaneously. The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the CMP‐DEWMA chart is more effective than the EWMA chart at detecting downward shifts of the process mean. Finally, a real data set is presented to demonstrate the application of the proposed chart.     

A double exponentially weighted moving average chart based on likelihood ratio for monitoring an inflated Pareto process

In this paper, we present a new chart called a likelihood ratio based double exponentially weighted moving average (LR_DEWMA) chart to monitor the shape parameter of the inflated Pareto process. Three other control charts such as the Shewhart type, the classical cumulative sum (CUSUM), and the likelihood ratio based EWMA (LR_EWMA) charts are also investigated. The performance of the control charts is evaluated by the average run length (ARL) and standard deviation of run lengths (SDRL) computed through the Monte Carlo simulation approach. Moreover, the median run length (MRL) and some other run length (RL) percentiles are also considered in some cases. Different charts have shown the best performance in different cases. In detecting smaller shifts, while the LR_DEWMA chart outperformed the other charts in terms of ARL and MRL, the CUSUM chart has shown the best performance in terms of SDRL and IQR of RLs. The application of the proposed control charts is illustrated using a chromatography analyses data from the food industry.     

Phase II monitoring of general linear profiles in the presence of between‐profile autocorrelation

In this paper, an approach based on the U statistic is first proposed to eliminate the effect of between‐profile autocorrelation of error terms in Phase‐II monitoring of general linear profiles. Then, a control chart based on the adjusted parameter estimates is designed to monitor the parameters of the model. The performance of the proposed method is compared with the ones of some existing methods in terms of average run length for weak, moderate, and strong autocorrelation coefficients under different shift scenarios. The results show that the proposed method provides significantly better results than the competing methods to detect shifts in the regression parameters, while the competing methods perform better in detecting shifts in the standard deviation. At the end, the applicability of the proposed method is illustrated by an example. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian EWMA control chart with measurement error under different loss functions

This paper presents the Bayesian EWMA control chart under two different loss functions (i) squared error loss function (SELF) and (ii) linex loss function (LLF) in the presence of measurement error (ME). We take posterior and posterior predictive distribution under the conjugate prior. We used a linear covariate model in the existence of ME to evaluate the control chart. We also studied the effects of multiple measurements and linear increasing variance methods in the existence of the ME. The average run length and standard deviation of run length are used to measure the performance of the Bayesian EWMA control chart with ME. We conducted the Monte Carlo simulation study to evaluate the performance of Bayesian EWMA control chart with ME. A real-life data example is also presented to demonstrate the application of the control chart.     

Applying Bootstrap method to the types I–II errors in the measurement system

This paper attempts to evaluate different methods of calculating the type‐I and type‐II errors in the measurement system. Furthermore, we apply the Bootstrap method to construct the confidence intervals for the type‐I and type‐II errors. Also, the proposed method is compared with the generalized inference method. Several factors such as the sample size, the measurement error, the process mean, and the process variation are simulated to validate the performance. The simulation results show that both methods almost have the same performance. In addition, we develop a computer program that can evaluate the error of measurement system without changing information or data. Two case studies of the nano measurement data are used to demonstrate the application. The simulation results indicate that the sample size has an influence for all cases. The type‐I and type‐II errors are decreased when the measurement error is increased. The type‐I and type‐II errors are affected by the measurement error, the process mean, and the process deviation. The case studies show that the development of nano technology requires the immediate attention of the measurement capability. Copyright © 2007 John Wiley & Sons, Ltd.     

An exponentially weighted moving average chart based on likelihood‐ratio test for monitoring Weibull mean and variance with subgroups

Monitoring changes in the Weibull mean and variance simultaneously is of interest in quality control. The mean and variance of a Weibull process are determined by its shape and scale parameters. Most studies are focused on monitoring the Weibull scale parameter with fixed shape parameter or the Weibull shape parameter with fixed scale parameter. In this paper, we propose an exponentially weighted moving average chart based on the likelihood‐ratio test and an inverse error function called ELR chart to monitor changes in the Weibull mean and variance simultaneously. The simulation approach is used to derive the average run length. We compare our proposed chart with other existing control charts for 3 cases, including scale parameter changes with fixed shape parameter, shape parameter changes with fixed scale parameter, and both parameters changes. The results show that the ELR chart outperforms the other control charts in terms of average run length in most cases. Two numerical examples are used to illustrate the applications of the proposed control chart.     

Shewhart and EWMA t control charts for short production runs

Short‐run productions are common in manufacturing environments like job shops, which are characterized by a high degree of flexibility and production variety. Owing to the limited number of possible inspections during a short run, often the Phase I control chart cannot be performed and correct estimates for the population mean and standard deviation are not available. Thus, the hypothesis of known in‐control population parameters cannot be assumed and the usual control chart statistics to monitor the sample mean are not applicable. t‐charts have been recently proposed in the literature to protect against errors in population standard deviation estimation due to the limitation of available sampling measures. In this paper the t‐charts are tested for implementation in short production runs to monitor the process mean and their statistical properties are evaluated. Statistical performance measures properly designed to test the chart sensitivity during short runs have been considered to compare the performance of Shewhart and EWMA t‐charts. Two initial setup conditions for the short run fixing the population mean exactly equal to the process target or, alternatively, introducing an initial setup error influencing the statistic distribution have been modelled. The numerical study considers several out‐of‐control process operating conditions including one‐step shifts for the population mean and/or standard deviation. The obtained results show that the t‐charts can be successfully implemented to monitor a short run. Finally, an illustrative example is presented to show the use of the investigated t charts. Copyright © 2010 John Wiley & Sons, Ltd.     

EWMA and DEWMA Control Charts for Poisson‐Exponential Distribution: Conditional Median Approach for Censored Data

Exponentially weighted moving average (EWMA) control charts are consistently used for the detection of small shifts contrary to Shewhart charts, which are commonly used for the detection of large shifts in the process. There are many interesting features of EWMA charts that have been studied for complete data in the literature. The aim of present study is to introduce and compare the double exponentially weighted moving average (DEWMA) and EWMA control charts under type‐I censoring for Poisson‐exponential distribution. The monitoring of mean level shifts using censored data is of a great interest in many applied problems. Moreover, a new idea of conditional median is introduced and further compared with the existing conditional expected values approach for monitoring the small mean level shifts. The performance of the DEWMA and EWMA charts is evaluated using the average run length, expected quadratic loss, and performance comparison index measures. The optimum sample size comparisons for the specified and unspecified parameters are also part of this study. Two applications for practical considerations are also discussed. It is observed that different censoring rates and the size of shifts significantly affect the performance of the EWMA and DEWMA charts. Copyright © 2016 John Wiley & Sons, Ltd.     

The performance of EWMA median and CUSUM median control charts for a normal process with measurement errors

Measurement error is often occurred in statistical process control. The effect of a linearly covariate error model on the exponentially weighted moving average (EWMA) median and cumulative sum (CUSUM) median charts is investigated. The results indicate that the EWMA median and CUSUM median charts are significantly affected in the presence of measurement errors. We compared the performance of the EWMA median and CUSUM median charts by using Markov chain method in the average run length and the standard deviation of the run length. We concluded that the CUSUM median chart for small shifts and the EWMA median chart for larger shifts are recommended. Two examples are provided to illustrate the application of the EWMA and CUSUM median charts with measurement errors.     

Distribution‐free mixed cumulative sum‐exponentially weighted moving average control charts for detecting mean shifts

In this paper, we propose distribution‐free mixed cumulative sum‐exponentially weighted moving average (CUSUM‐EWMA) and exponentially weighted moving average‐cumulative sum (EWMA‐CUSUM) control charts based on the Wilcoxon rank‐sum test for detecting process mean shifts without any distributional assumption of the underlying quality process. The performances of the proposed charts are measured through the average run‐length, relative mean index, average extra quadratic loss, and average ratio of the average run‐length and performance comparison index. It is found that the proposed charts perform better than its counterparts considered in this paper under non‐normal distributions and outperform the classical mixed CUSUM‐EWMA and EWMA‐CUSUM charts in many cases under the normal distribution. The effect of the phase I sample size is also investigated on the phase II performance of the proposed charts. A numerical illustration is given to demonstrate the implementation and simplicity of the proposed charts.     

Side-sensitive modified group runs charts with and without measurement errors for monitoring the coefficient of variation

The coefficient of variation (CV) is used in process monitoring when the process mean and standard deviation are proportional to each other. In this work, a side-sensitive modified group runs CV (SSMGR CV) chart is proposed for monitoring the process CV. The run length performance of the SSMGR CV chart is compared to those of the existing CV charts in terms of the average and standard deviation of the run length criteria. The SSMGR CV chart is found to outperform the existing CV charts. In addition, the run length performance of the SSMGR CV chart is also evaluated in the presence of measurement errors, as these errors are not only unavoidable in practice but they also affect the sensitivity of a control chart in detecting an out-of-control situation. The results obtained show that the accuracy and precision errors affect the performance of the SSMGR CV chart in detecting an out-of-control situation.     

New Exponentially Weighted Moving Average Control Charts for Monitoring Process Mean and Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely accepted because of their excellent performance in detecting small to moderate shifts in the process parameters. In this paper, we propose new EWMA control charts for monitoring the process mean and the process dispersion. These EWMA control charts are based on the best linear unbiased estimators obtained under ordered double ranked set sampling (ODRSS) and ordered imperfect double ranked set sampling (OIDRSS) schemes, named EWMA‐ODRSS and EWMA‐OIDRSS charts, respectively. We use Monte Carlo simulations to estimate the average run length, median run length, and standard deviation of run length of the proposed EWMA charts. We compare the performances of the proposed EWMA charts with the existing EWMA charts when detecting shifts in the process mean and in the process variability. It turns out that the EWMA‐ODRSS mean chart performs uniformly better than the classical EWMA, fast initial response‐based EWMA, Shewhart‐EWMA, and hybrid EWMA mean charts. The EWMA‐ODRSS mean chart also outperforms the Shewhart‐EWMA mean charts based on ranked set sampling (RSS) and median RSS schemes and the EWMA mean chart based on ordered RSS scheme. Moreover, the graphical comparisons of the EWMA dispersion charts reveal that the proposed EWMA‐ODRSS and EWMA‐OIDRSS charts are more sensitive than their counterparts. We also provide illuminating examples to illustrate the implementation of the proposed EWMA mean and dispersion charts. Copyright © 2014 John Wiley & Sons, Ltd.     

CRPS chart: Simultaneously monitoring location and scale under data‐rich environment

The detection performance of a conventional control chart is usually degraded by a large sample size as in Wang and Tsung. This paper proposes a new control chart under data‐rich environment. The proposed chart is based on the continuous ranked probability score and aims to simultaneously monitor the location and the scale parameters of any continuous process. We simulate different monitoring schemes with various shift patterns to examine the chart performance. Both in‐control and out‐of‐control performances are studied through simulation studies in terms of the mean, the standard deviation, the median, and some percentiles of the average run length distribution. Simulation results show that the proposed chart keeps a high sensitivity to shifts in location and/or scale without any distributional assumptions, and the outperformance improves, as the sample size becomes larger. Examples are given for illustration.     

Mixed Exponentially Weighted Moving Average–Cumulative Sum Charts for Process Monitoring

The control chart is a very popular tool of statistical process control. It is used to determine the existence of special cause variation to remove it so that the process may be brought in statistical control. Shewhart‐type control charts are sensitive for large disturbances in the process, whereas cumulative sum (CUSUM)–type and exponentially weighted moving average (EWMA)–type control charts are intended to spot small and moderate disturbances. In this article, we proposed a mixed EWMA–CUSUM control chart for detecting a shift in the process mean and evaluated its average run lengths. Comparisons of the proposed control chart were made with some representative control charts including the classical CUSUM, classical EWMA, fast initial response CUSUM, fast initial response EWMA, adaptive CUSUM with EWMA‐based shift estimator, weighted CUSUM and runs rules–based CUSUM and EWMA. The comparisons revealed that mixing the two charts makes the proposed scheme even more sensitive to the small shifts in the process mean than the other schemes designed for detecting small shifts. Copyright © 2012 John Wiley & Sons, Ltd.     

CCC‐r charts' performance with estimated parameter for high‐quality process

CCC‐r charts are effective in detecting process shifts in the nonconforming rate especially for a high‐quality process. The implementation of the CCC‐r charts is usually under the assumption that the in‐control nonconforming rate is known. However, the nonconforming rate is never known, and accurate estimation is difficult. We investigate the effect of estimation error on the CCC‐r charts' performances through the expected value of the average number of observations to signal (EANOS) as well as the standard deviation of the average number of observations to signal (SDANOS). By comparing the in‐control performance of the CCC‐r charts, the CCC‐r chart with a larger value of r is more susceptible to the effects of parameter estimation. Meanwhile, the performance of the CCC‐r charts can converge when detecting upward shifts in p of out‐of‐control processes. We recommend the use of the CCC‐4 chart when considering its effectiveness in detecting shifts as well as its easier construction in practice. Furthermore, it is investigated that the CCC‐4 chart is less sensitive to parameter estimation while being more effective in detecting different process shifts when compared with Geometric CUSUM chart and synthetic chart.     

Change‐Point Detection for Shifts in Control Charts Using EM Change‐Point Algorithms

Knowing the time of changes in mean and variance in a process is crucial for engineers to identify the special cause quickly and correctly. Because assignable causes may give rise to changes in mean and variance at the same time, monitoring the mean and variance simultaneously is required. In this paper, a mixture likelihood approach is proposed to detect shifts in mean and variance simultaneously in a normal process. We first transfer the change point model formulation into a mixture model and then employ the expectation and maximization algorithm to estimate the time of shifts in mean and variance simultaneously. The proposed method called EMCP (expectation and maximization change point) can be used in both phase I and II applications without the knowledge of in‐control process parameters. Moreover, EMCP can detect the time of multiple shifts and simultaneously produce the estimates of shifts in each individual segment. Numerical data and real datasets are employed to compare EMCP with the direct statistical maximum likelihood method without the use of mixture models. The experimental results show the superiority and effectiveness of the proposed EMCP. The outperformance of EMCP in detecting the time of small shifts is particularly important and beneficial for engineers to identify assignable causes rapidly and accurately in phase II applications in which small shifts occur more often and hence lead to a large average run length. Copyright © 2015 John Wiley & Sons, Ltd.