Process monitoring research with various estimator-based MEWMA control charts

Multivariate exponentially weighted moving average (MEWMA) control chart with five different estimators as population covariance matrix is rarely applied to monitor small fluctuations in the statistical process control. In this article, mathematical models of the five estimators (S₁, S₂, S₃, S₄, S₅) are established, with which the relevant MEWMA control charts are obtained, respectively. Thereafter, the process monitoring performance of the five control charts is simulated. And the simulation results show that the S₄ estimator-based MEWMA control chart is of the best performance both in step offset failure mode and ramp offset failure mode. Since the inline process monitoring of photovoltaic manufacturing is intended to be a problem of multivariate statistics process analysis, the feasibility and effectiveness of the proposed model are elaborated in the case study during the cell testing and sorting process control for the fabrication of multicrystalline silicon solar cells.     

Deviance residual-based control charts for monitoring the beta-distributed processes

Beta-distributed process outputs are common in manufacturing industry because they range from 0 to 1 based on inputs like yield. Under the normality assumption, Shewarts control charts and Hotelling's control charts based on the deviance residual have been applied to monitor the process mean of the beta-distributed process outputs. The normality assumption can be violated according to the shape of the beta distribution. Therefore, without the normality assumption, we propose antirank control charts, exponentially weighted moving average (EWMA) control charts and cumulative sum (CUSUM) control charts. The proposed control charts outperform the existing control charts in the experimental results. The previous research has been focused on monitoring the process mean only. For the first time, in order to monitor the process variance of the beta-distributed process outputs, we propose the multivariate exponentially weighted mean squared deviation (MEWMS) chart, the first norm distance of the MEWMS deviation from its expected value (MEWMSL₁) chart, the chart based on MEWMS deviation with the approximated distribution of trace (MEWMS_AT), the multivariate trace sum squared deviation (MTSSD) chart and the multivariate matrix sum squared deviation (MMSSD) chart based on the deviance residual. The proposed control charts are compared and recommended in terms of the experimental results. This research can be a guideline for practitioners who monitor the deviance residual.     

A note on GLR charts for monitoring count processes

Various generalized likelihood ratio (GLR) charts have been proposed to monitor count processes such as binomial, Bernoulli, Poisson, and multinomial processes. The advantages of GLR charts are that designing the chart is relatively easy, estimates of the process change‐point and shift size are available for post‐signal diagnosis, and they are effective in detecting a wide range of shifts in the process parameter. However, for some special cases of the observations, such as observing all defective items or all non‐defective items, the GLR chart statistic for monitoring a count process has been said to be undefined. We show that the GLR chart statistic is always well defined.     

Run Rules median control charts for monitoring process mean in manufacturing

Recent studies show that Shewhart median ( ) chart is simpler than the Shewhart chart and it is robust against outliers, but it is often rather inefficient in detecting small or moderate process shifts. The statistical sensitivity of a Shewhart control chart can be improved by using supplementary Run Rules. In this paper, we propose the Phase II median Run Rules type control charts. A Markov chain methodology is used to evaluate the statistical performance of these charts. Moreover, the performance of proposed charts is investigated in the presence of a measurement errors and modelled by a linear covariate error model. An extensive numerical analysis with several tables and figures to show the statistical performance of the investigated charts is provided for both cases of measurement errors and no measurement errors. An example illustrates the use of these charts.     

The variable sampling rate X̄ control charts for monitoring autocorrelated processes

Recent studies demonstrated that the adaptive X? control charts are more efficient than fixed parmeters (FP) X? control chart from statistical and economic criteria. The usual assumption for designing a control chart is that the observations from the process are independent. However, for many processes, such as chemical processes, consecutive measurements are often highly correlated, especially when the interval between samples is small. In the present paper, the observations are modeled as an AR(1) process plus a random error, and the properties of the variable sampling rate (VSR) X? charts are evaluated and studied under this model. Based on the study, the VSR X? chart is faster than the FP, variable sampling interval and variable sample size X? control charts in detecting mean shifts. However, when the level of autocorrelation is high or the process mean shift is large, the advantage of the VSR X? chart is reduced. Copyright © 2008 John Wiley & Sons, Ltd.     

Phase I control charts for times between events

A count of the number of defects is often used to monitor the quality of a production process. When defects rarely occur in a process, it is often desirable to monitor the time between the occurrence of each defect rather than a count of the number of defects. An exponential distribution often provides a useful model of the time between defects. Phase I control charts for exponentially distributed processes are discussed. Methods for computing the control limits are given and the overall Type I error rates of these charts are evaluated. Copyright © 2002 John Wiley & Sons, Ltd.     

Effect of the measurement errors on two one-sided Shewhart control charts for monitoring the ratio of two normal variables

Monitoring the ratio between two random normal variables plays an important role in many industrial manufacturing processes. In this paper, we suggest designing two one-sided Shewhart control charts monitoring this ratio. The numerical results show that the one-sided charts have more advantages compared with the two-sided Shewhart chart proposed previously in the literature. Moreover, we investigate the effect of measurement error on the performance of these control charts where the measurement error is supposed to follow a linear covariate error model. The change of model parameters from an in-control condition to an out-of-control is presented without using a strict assumption about the independence of the shift size from measurement errors. A valuable finding from this study is that taking multiple measurements per item is not an effective way to reduce the negative effect of measurement error on the Shewhart charts' performance.     

Control charts for the shape parameter of reflected power function distribution under classical estimators

The reflected power function distribution (RPFD) has applications in the fields of reliability engineering and survival analysis. To identify and remove the variation in different reliability processes and also to monitor the reliability of machines where the number of errors follows RPFD, we develop control charts to keep the process in control. A memory less control chart like a Shewhart control chart, and two memory-based control charts like an exponentially weighted moving average (EWMA) control chart and a hybrid exponentially weighted moving average (HEWMA) control chart are discussed and compared with each other. Proposal of these control charts is based on two different estimators, the percentile estimator (PE) and the modified maximum likelihood estimator (MMLE). This study shows that an HEWMA control chart based on PE performs better than PE-based Shewhart and EWMA control charts, as well as MMLE-based Shewhart, EWMA, and HEWMA control charts.     

A CUSUM scheme with variable sample sizes and sampling intervals for monitoring the process mean and variance

The adaptive control feature and CUSUM chart are two monitoring schemes that are much more effective than the traditional static Shewhart chart in detecting process shifts in mean and variance. However, the designs and analyses of the adaptive CUSUM chart are mathematically intractable and the operation is very laborious. This article proposes a VSSI WLC scheme, which is a weighted‐loss‐function‐based CUSUM (WLC) scheme using variable sample sizes and sampling intervals (VSSI). This scheme detects the two‐sided mean shift and increasing standard deviation shift based on a single statistic WL (the weighted loss function). Most importantly, the VSSI WLC scheme is much easier to operate and design than a VSSI CCC scheme which comprises three individual CUSUM charts (two of them monitoring the increasing and decreasing mean shifts and one monitoring the increasing variance shift). Overall, the VSSI WLC scheme is much more effective than the static &S charts (by 72.36%), the VSSI &S charts (by 30.97%) and the static WLC scheme (by 50.94%) for detection. It is even more effective than the complicated VSSI CCC scheme for most cases. Copyright © 2006 John Wiley & Sons, Ltd.     

Shewhart-type Phase II control charts for monitoring times to an event with a guaranteed in-control and good out-of-control performance

Monitoring time to event (failure) data is important in many applications. Proper monitoring and control can make the production process more efficient and provide economic advantages. In this paper, we consider the efficacy of a class of Shewhart-type control charts for monitoring time to event data following an exponential distribution with an unknown mean, which is estimated from a class of estimators. An estimator is chosen within this class, so that the in-control performance is maximized with respect to a number of popular criteria in the recent literature, and the proposed optimal charts are compared on the basis of their in-control and out-of-control performance. The comparisons include the traditional Phase II exponential Shewhart chart using the maximum likelihood estimator. Improved in-control and out-of-control performances of these charts can enhance the quality and productivity of manufacturing processes. Since no chart is best under all the criteria, a ranking system is used to choose a chart to use in practice with a good overall performance. Two illustrative examples using real data are given; summary and conclusions are offered.     

CUSUM control charts with variable sampling interval for monitoring the ratio of two normal variables

In many industrial manufacturing processes, the ratio between two normal random variables plays a key role in ensuring quality of products. Thus,  monitoring this ratio is an important task that is well worth considering. In this paper, we combine a variable sampling interval (VSI) strategy with a cumulative sum (CUSUM) scheme to create a new type of control chart for purpose of tracking the ratio between two normal variables. The average time to signal (ATS) and the expected average time to signal (EATS) criteria are used to evaluate the performance of the new VSI CUSUM RZ control chart. The  numerical results show that the proposed control chart has much more attractive performance in comparison with the standard CUSUM-RZ control chart and the VSI EWMA-RZ control chart.     

On t and EWMA t charts for monitoring changes in the process mean

The performance of an X‐bar chart is usually studied under the assumption that the process standard deviation is well estimated and does not change. This is, of course, not always the case in practice. We find that X‐bar charts are not robust against errors in estimating the process standard deviation or changing standard deviation. In this paper we discuss the use of a t chart and an exponentially weighted moving average (EWMA) t chart to monitor the process mean. We determine the optimal control limits for the EWMA t chart and show that this chart has the desired robustness property. Copyright © 2009 John Wiley & Sons, Ltd.     

The modified control charts for monitoring the shape parameter of weighted power function distribution under classical estimator

Zaka et al provided a new distribution called the Weighted Power function distribution (WPFD), which has application in reliability engineering and survival analysis. They used different estimation methods to estimate the unknown parameters of WPFD and proved that modified maximum likelihood estimator (MMLE) is best to consider for the estimation of parameters. We have constructed the memoryless and memory-based control charts based on the assumption that the distribution of the underlying process does not follow the normal distribution. In this paper, we provide modified control charts using MMLE of the shape parameter for WPFD. We develop control charts to keep the process in control when the distribution of errors of underlying process follows WPFD. We propose the modified memoryless control chart, that is, Shewhart control chart and modified memory-based control chart, that is, Exponentially weighted moving average (EWMA) and Hybrid exponentially weighted moving average (HEWMA) control charts. We have made the comparison of the proposed control charts using Monte Carlo simulation and the real-life application for both and the memoryless control charts and memory-based control charts. We see that HEWMA based on MMLE performs better as compared to other proposed control charts.     

Boxplot‐based phase I control charts for time between events

To monitor the quality/reliability of a (production) process, it is sometimes advisable to monitor the time between certain events (say occurrence of defects) instead of the number of events, particularly when the events occur rarely. In this case it is common to assume that the times between the events follow an exponential distribution. In this paper, we propose a one‐ and a two‐sided control chart for phase I data from an exponential distribution. The control charts are derived from a modified boxplot procedure. The charting constants are obtained by controlling the overall Type I error rate and are tabulated for some configurations. A numerical example is provided for illustration. The in‐control robustness and the out‐of‐control performance of the proposed charts are examined and compared with those of some existing charts in a simulation study. It is seen that the proposed charts are considerably more in‐control robust and have out‐control properties comparable to the competing charts. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive CUSUM and EWMA charts with auxiliary information and variable sampling intervals for monitoring the process mean

The variable sampling interval (VSI) feature enhances the sensitivity of a control chart that is based on fixed sampling interval (FSI). In this paper, we enhance the sensitivities of the auxiliary information-based (AIB) adaptive Crosier cumulative sum (CUSUM) (AIB-ACC) and adaptive exponentially weighted moving average (EWMA) (AIB-AE) charts using the VSI feature when monitoring a mean shift which is expected to lie within a given interval. The Monte Carlo simulations are used to compute zero-state and steady-state run length properties of these control charts. It is found that the AIB-ACC and AIB-AE charts with VSI feature are uniformly more sensitive than those based on FSI feature. Real datasets are also considered to demonstrate the implementation of these control charts.     

Bivariate semiparametric control charts for simultaneous monitoring of process mean and variance

In the present article, two semiparametric bivariate control charts are presented, which use order statistics and are effective in jointly monitoring of possible shifts in the process mean and/or variance. To achieve that both the median location (or more generally the location of a specific order statistic) and the number of specific observations of the test sample lying between the control limits are taken into account. The false alarm rate and the in-control average run length are not affected by the marginal distributions, while the effect of the dependence structure on them is negligible; therefore, they can be used as fully nonparametric charts. A performance-comparison study is carried out, and an illustrative example is provided using a real-world data set.     

Construction of double sampling s‐control charts for agile manufacturing

Double sampling (DS) ‐control charts are designed to allow quick detection of a small shift of process mean and provides a quick response in an agile manufacturing environment. However, the DS ‐control charts assume that the process standard deviation remains unchanged throughout the entire course of the statistical process control. Therefore, a complementary DS chart that can be used to monitor the process variation caused by changes in process standard deviation should be developed. In this paper, the development of the DS s‐charts for quickly detecting small shift in process standard deviation for agile manufacturing is presented. The construction of the DS s‐charts is based on the same concepts in constructing the DS ‐charts and is formulated as an optimization problem and solved with a genetic algorithm. The efficiency of the DS s‐control chart is compared with that of the traditional s‐control chart. The results show that the DS s‐control charts can be a more economically preferable alternative in detecting small shifts than traditional s‐control charts. Copyright © 2002 John Wiley & Sons, Ltd.