An EWMA chart for monitoring the covariance matrix of a multivariate process based on dissimilarity index

In this article, we propose an exponentially weighted moving average (EWMA) control chart for monitoring the covariance matrix of a multivariate process based on the dissimilarity index of 2 matrices. The proposed control chart essentially monitors the covariance matrix by comparing the individual eigenvalues of the estimated EWMA covariance matrix with those of the estimated covariance matrix from the in‐control (IC) phase I data. It is different from the conventional EWMA charts for monitoring the covariance matrix, which are either based on comparing the sum or product or both of the eigenvalues of the estimated EWMA covariance matrix with those of the IC covariance matrix. We compare the performance of the proposed chart with that of the best existing chart under the multivariate normal process. Furthermore, to prevent the control limit of the proposed EWMA chart developed using the limited IC phase I data from having extensively excessive false alarms, we use a bootstrap resampling method to adjust the control limit to guarantee that the proposed chart has the actual IC ARL(average run length) not less than the nominal level with a certain probability. Finally, we use an example to demonstrate the applicability and implementation of the proposed EWMA chart.     

Enhanced EWMA charts for monitoring the process coefficient of variation

The coefficient of variation (CV) is an important quality characteristic when the process variance is a function of the process mean for a production process. In this paper, we develop an auxiliary information–based (AIB) estimator for estimating the squared CV, along with its approximated mean and variance. This estimator is then used to devise new one-sided EWMA charts for monitoring the increases or decreases in the squared CV of a normal process, named the AIB-EWMA CV charts. In addition, the sensitivities of these control charts are also enhanced with the fast initial response feature. The Monte Carlo simulation method is used to compute the run length characteristics of the proposed CV charts. Based on detailed run length comparisons, it is found that the proposed AIB-EWMA CV charts are uniformly and substantially better than the existing EWMA CV charts when detecting different kinds of upward/downward shifts in the squared CV. The proposed charts are also applied to a real dataset to support the proposed theory.     

An EWMA mean chart with auxiliary information

In this paper, we show that a recently proposed auxiliary information-based (AIB) adaptive EWMA (AE) chart is sensitive (not robust) to the changes in the mean of an auxiliary variable when monitoring the changes in the mean of a quality variable, called the AIB-AE chart. To circumvent the weakness of the AIB-AE chart, we develop a new AIB estimator for the mean of a quality variable that is slightly robust to the changes in the mean of an auxiliary variable. Based on this newly developed estimator, a new AIB EWMA (AIB-E) chart is proposed for monitoring the mean of a quality variable. The zero-state and steady-state average run-length profiles of the AIB-AE and AIB-E charts are estimated with Monte Carlo simulations. It is found that the AIB-E chart is not only slightly robust to the changes in the mean of an auxiliary variable, but it also outperforms the AIB-AE chart when detecting small shifts in the mean of a quality variable. Illustrative examples are also included in this study to demonstrate the implementation of the existing and proposed AIB charts.     

A sensitive non-parametric EWMA control chart

In practice, we may not always have normally distributed quality characteristics of interest. This leads to the need for non-parametric techniques which are not dependent on the assumptions about the parent distribution. This study develops a non-parametric exponentially weighted moving average (EWMA) chart (namely the NPSS_EWMA chart) for an improved monitoring of process location. The proposal is based on the use of sign statistics on a moving pattern in an EWMA setup. The design structure of the proposed chart is developed and its performance is evaluated in terms of different properties including average run length (ARL), standard deviation run length (SDRL), percentiles, relative ARL (RARL), extra quadratic loss (EQL), and performance comparison index (PCI). The proposal is compared with recently developed non-parametric counterparts namely NPS_EWMA, NPAS_EWMA, and NPS_CUSUM charts. It is observed that the design structure of the proposed NPSS_EWMA chart outshines the existing counterparts. An application example is also included in the study for practical demonstration.     

A proposed variable parameter control chart for monitoring the multivariate coefficient of variation

An efficient process monitoring system is important for achieving sustainable manufacturing. The control charting technique is one of the most effective techniques to monitor process quality. In certain processes where the process mean and variance are not independent of one another, the coefficient of variation (CV), which measures the ratio of the standard deviation to the mean, should be monitored. In line with industrial settings, where at least two or more variables are monitored simultaneously in most processes, this paper proposes a variable parameter (VP) chart to monitor the multivariate CV (MCV). Formulae and algorithms to optimize the various performance measures are discussed. The proposed VP MCV chart is designed based on a Markov chain approach. The performance comparison shows that the proposed VP MCV chart prevails over the existing MCV charts, in terms of the average time to signal (ATS), standard deviation of the time of signal (SDTS), and expected average time to signal (EATS) criteria. An example is presented to illustrate the implementation of the proposed VP MCV chart.     

A new adaptive CUSUM control chart for detecting the multivariate process mean

We propose a new multivariate CUSUM control chart, which is based on self adaption of its reference value according to the information from current process readings, to quickly detect the multivariate process mean shifts. By specifying the minimum magnitude of the process mean shift in terms of its non‐centrality parameter, our proposed control chart can achieve an overall performance for detecting a particular range of shifts. This adaptive feature of our method is based on two EWMA operators to estimate the current process mean level and make the detection at each step be approximately optimal. Moreover, we compare our chart with the conventional multivariate CUSUM chart. The advantages of our control chart detection for range shifts over the existing charts are greatly improved. The Markovian chain method, through which the average run length can be computed, is also presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Robustness properties of multivariate EWMA control charts

In this paper, the robustness of the multivariate exponentially weighted moving average (MEWMA) control chart to non‐normal data is examined. Two non‐normal distributions of interest are the multivariate distribution and the multivariate gamma distribution. Recommendations for constructing MEWMA control charts when the normality assumption may be violated are provided. Copyright © 2002 John Wiley & Sons, Ltd.     

A side-sensitive synthetic coefficient of variation chart

Control charts for monitoring the coefficient of variation (γ) are useful for processes with an inconsistent mean (μ) and a standard deviation (σ) which changes with μ, by monitoring the consistency in the ratio σ over μ. The synthetic-γ chart is one of the charts proposed to monitor γ, and its attractiveness lie in waiting until a second point to fall outside the control limits before a decision is made. However, existing synthetic-γ charts do not differentiate between the points falling outside the upper control limit (UCL) and lower control limit (LCL). Hence, this paper proposes a side-sensitive synthetic-γ chart, where successive nonconforming samples must either fall above the UCL or below the LCL. Formulae to compute the average run length (ARL), the standard deviation of the run length (SDRL) and expected average run length (EARL) are derived using the Markov chain approach, and the algorithms to obtain the optimal charting parameters are proposed. Subsequently, the optimal charting parameters, ARL, SDRL and EARL values for various numerical examples are shown. Comparisons show that the side-sensitive synthetic-γ chart consistently outperforms the existing synthetic-γ chart, especially for small shifts. The proposed chart also consistently outperforms the Shewhart-γ chart, while showing comparable or better performance than the Exponentially Weighted Moving Average (EWMA) chart for most shift sizes, except for very small shifts. Finally, this paper shows the implementation of the proposed chart on an industrial example.     

An efficient adaptive EWMA control chart for monitoring the process mean

In recent years, the memory‐type control charts—exponentially weighted moving average (EWMA) and cumulative sum (CUSUM)—along with the adaptive and dual control‐charting structures have received considerable attention because of their excellent ability in providing an overall good detection over a range of mean‐shift sizes. These adaptive memory‐type control charts include the adaptive exponentially weighted moving average (AEWMA), dual CUSUM, and adaptive CUSUM charts. In this paper, we propose a new AEWMA chart for efficiently monitoring the process mean. The idea is to first design an unbiased estimator of the mean shift using the EWMA statistic and then adaptively update the smoothing constant of the EWMA chart. The run length profiles of the proposed AEWMA chart are computed using extensive Monte Carlo simulations. Based on a comprehensive comparative study, it turns out that the proposed AEWMA chart performs better than the existing AEWMA, adaptive CUSUM, dual CUSUM, and Shewhart‐CUSUM charts, in terms of offering more balanced protection against mean shifts of different sizes. An example is also used to explain the working of the existing and proposed control charts.     

A new adaptive EWMA control chart for monitoring the process dispersion

The exponentially weighted moving average (EWMA), cumulative sum (CUSUM), and adaptive EWMA (AEWMA) control charts have had wide popularity because of their excellent speed in tracking infrequent process shifts, which are expected to lie within certain ranges. In this paper, we propose a new AEWMA dispersion chart that may achieve better performance over a range of dispersion shifts. The idea is to first consider an unbiased estimator of the dispersion shift using the EWMA statistic, and then based on the magnitude of this shift, select an appropriate value of the smoothing parameter to design an EWMA chart, named the AEWMA chart. The run length characteristics of the AEWMA chart are computed with the help of extensive Monte Carlo simulations. The AEWMA chart is compared with some of the existing powerful competitor control charts. It turns out that the AEWMA chart performs substantially and uniformly better than the EWMA‐S², CUSUM‐S², existing AEWMA, and HHW‐EWMA charts when detecting different kinds of shifts in the process dispersion. Moreover, an example is also used to explain the working and implementation of the proposed AEWMA chart.     

An adaptive approach to EWMA dispersion chart using Huber and Tukey functions

Random causes are vital part of every process in manufacturing and nonmanufacturing environments, and these do not affect the product features. Special causes, on the other hand, come because of some burden(s) in a process and requires special attention; otherwise, it ruins the products excellence. Special causes are categorized into small, moderate, and large shifts and are handled by statistical quality control charts. The Shewhart control chart is well known for large shifts, while the cumulative sum and exponentially weighted moving average are more effective in detecting small to moderate shifts. However, in practice, many processes require the simultaneous monitoring of both the small to the large shifts. In this study, we have designed an adaptive EWMA for dispersion parameter in connection with Huber and Tukey's bisquare functions. The performance measures used in this study include average run length, extra quadratic loss, relative average run length, and performance‐comparison index. We have observed that the study proposals are good competitors to the other counter parts for an efficient monitoring of shifts of varying amounts. An illustrative example using real data is given to demonstrate the implementation of the study proposal.     

A progressive approach for the detection of the coefficient of variation

A progressive average chart usually triggers initial out-of-control (OC) signals more simply and quickly than other memory-type charts . In this paper, two progressive average control procedures are proposed for monitoring the coefficient of variation (CV) of a normally distributed process variable, namely, the progressive CV (PCV) and progressive resetting CV (PRCV) control charts , respectively. The implementation of the proposed charts is presented, and the necessary design parameters are provided. Through extensive numerical simulations, it is shown that the proposed PCV and PRCV charts outperform several existing control charts to detect the initial OC signals, especially for the small and moderate CV shifts, under each combination of the shift size, the sample size, and the in-control target value of the CV. In addition, the application of the proposed control charts is illustrated by a detection example for a spinning process.     

Weighted adaptive multivariate CUSUM control charts

An adaptive multivariate cumulative sum (AMCUSUM) control chart has received considerable attention because of its ability to dynamically adjust the reference parameter whereby achieving a better performance over a range of mean shifts than the conventional multivariate cumulative sum (CUSUM) charts. In this paper, we introduce a progressive mean–based estimator of the process mean shift and then use it to devise new weighted AMCUSUM control charts for efficiently monitoring the process mean. These control charts are easy to design and implement in a computerized environment compared with their existing counterparts. Monte Carlo simulations are used to estimate the run‐length characteristics of the proposed control charts. The run‐length comparison results show that the weighted AMCUSUM charts perform substantially and uniformly better than the classical multivariate CUSUM and AMCUSUM charts in detecting a range of mean shifts. An example is used to illustrate the working of existing and proposed multivariate CUSUM control charts.     

A EWMA Control Chart for Exponential Distributed Quality Based on Moving Average Statistics

We propose an exponentially weighted moving average (EWMA) control chart for monitoring exponential distributed quality characteristics. The proposed control chart first transforms the sample data to approximate normal variables, then calculates the moving average (MA) statistic for each subgroup, and finally constructs the EWMA statistic based on the current and the previous MA statistics. The upper and the lower control limits are derived using the mean and the variance of EWMA statistics. The in‐control and the out‐of‐control average run lengths are derived and tabularized according to process shift parameters and smoothing constants. It is shown that the proposed control chart outperforms the MA control chart for all shift parameters. 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.     

Design of a Control Chart Using a Modified EWMA Statistic

In this paper, the design of a control chart is given using a modified exponentially weighted moving average statistic under the assumption that the quality characteristic of interest follows the normal distribution. The structure of the proposed control chart is developed, and the necessary measures are derived to find the average run length for in‐control and out‐of‐control processes. The efficiency of the proposed chart is compared with two existing control charts in terms of the average run length. The results are explained with the help of industrial example. Copyright © 2016 John Wiley & Sons, Ltd.     

A generally weighted moving average control chart for zero-inflated Poisson processes

Zero-inflated Poisson (ZIP) model is very useful in high-yield processes where an excessive number of zero observations exist. This model can be viewed as an extension of the standard Poisson distribution. In this paper, a one-sided generally weighted moving average (GWMA) control chart is proposed for monitoring upward shifts in the two parameters of a ZIP process (regarded as ZIP-GWMA chart). The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the ZIP-GWMA performs better than the existing control charts under shifts in both parameters. Moreover, an illustrative example is presented to display the application of the proposed chart on practitioners.     

Optimum variable-dimension EWMA chart for multivariate statistical process control

The variable-dimension T² control chart (VDT² chart) was recently proposed for monitoring the mean of multivariate processes in which some of the quality variables are easy and inexpensive to measure while other variables require substantially more effort or expense. The chart requires most of the times that only the inexpensive variables be sampled, switching to sampling all the variables only when a warning is triggered. It has good ARL performance compared with the standard T² chart, while significantly reducing the sampling cost. However, like the T² chart, it has limited sensitivity to small and moderate mean shifts. To detect such shifts faster, we developed an exponentially weighted moving average (EWMA) version of the VDT² chart, along with Markov chain models for ARL calculation, and software (made available) for optimizing the chart design. The optimization software, which is based on genetic algorithms, runs in Windows^© and has a friendly user interface. The performance analysis shows the great gain in performance achieved by the incorporation of the EWMA procedure.     

The effects of violations of the multivariate normality assumption in multivariate Shewhart and MEWMA control charts

A multivariate Shewhart and a multivariate exponentially weighted moving average control charts are types of multivariate control charts for monitoring the mean vector. For those control charts, a multivariate normal distribution is an important assumption that is used to describe a behavior of a set of quality characteristics of interest. This research explores the sensitivity of average run lengths and standard deviation of run lengths for the multivariate Shewhart and the multivariate exponentially weighted moving average control charts when the normality assumption is incorrect.