Multivariate EWMA Control Chart with Adaptive Sample Sizes

Standard multivariate control charts usually employ fixed sample sizes at equal sampling intervals to monitor a process. In this study, a multivariate exponential weighted moving average (MEWMA) chart with adaptive sample sizes is investigated. Performance measure of the adaptive-sample-size MEWMA chart is obtained through a Markov chain approach. The performance of the adaptive-sample-size MEWMA chart is compared with the fixed-sample-size control chart in terms of steady-state average run length for different magnitude of shifts in the process mean. It is shown that the adaptive-sample-size chart is more efficient than the fixed-sample-size MEWMA control chart in detecting shifts in the process mean.     

A Sequential Rank-Based Nonparametric Adaptive EWMA Control Chart

Nonparametric control chart is useful when the underlying distribution is unknown, or is not likely to be normal. In this article, we provide a sequential rank-based nonparametric adaptive EWMA (NAE) control chart for detecting the persistent shift in the location parameter. This NAE chart is a self-starting scheme and thus can be used to monitor processes at the start-up stages rather than waiting for the accumulation of sufficiently large calibration samples. Moreover, we do not require any prior knowledge of the underlying distribution, and to prespecify any tuning parameter either. A Markov chain model is suggested to calibrate the run-length distribution of NAE, which is shown to have approximate tail probability as a geometric distribution. A simulation study demonstrates that the proposed control chart not only performs robustly for different distributions, but also is efficient in detecting various magnitude of shifts. A real-data example from manufacturing shows that it performs quite well in practical applications.     

A new adaptive EWMA control chart using auxiliary information for monitoring the process mean

The adaptive memory-type control charts, including the adaptive exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts, have gained considerable attention because of their excellent speed in providing overall good detection over a range of mean shift sizes. In this paper, we propose a new adaptive EWMA (AEWMA) chart using the auxiliary information for efficiently monitoring the infrequent changes in the process mean. The idea is to first estimate the unknown process mean shift using an auxiliary information based mean estimator, and then adaptively update the smoothing constant of the EWMA chart. Using extensive Monte Carlo simulations, the run length profiles of the AEWMA chart are computed and explored. The AEWMA chart is compared with the existing control charts, including the classical EWMA, CUSUM, synthetic EWMA and synthetic CUSUM charts, in terms of the run length characteristics. It turns out that the AEWMA chart performs uniformly better than these control charts when detecting a range of mean shift sizes. An illustrative example is also presented to demonstrate the working and implementation of the proposed and existing control charts.     

A new nonparametric synthetic EWMA control chart for monitoring process mean

The exponentially weighted moving average (EWMA) control charts are widely used in chemical and process industries because of their excellent speed in catching small to moderate shifts in the process target. In usual practice, many data come from a process where the monitoring statistic is non-normally distributed or it follows an unknown probability distribution. This necessitates the use of distribution-free/nonparametric control charts for monitoring the deviations from the process target. In this paper, we integrate the existing EWMA sign chart with the conforming run length chart to propose a new synthetic EWMA (SynEWMA) sign chart for monitoring the process mean. The SynEWMA sign chart encompasses the synthetic sign and EWMA sign charts. Monte Carlo simulations are used to compute the run length profiles of the SynEWMA sign chart. Based on a comprehensive comparison, it turns out that the SynEWMA sign chart is able to perform substantially better than the existing EWMA sign chart. Both real and simulated data sets are used to explain the working and implementation of existing and proposed control charts.     

A Nonparametric Shewhart-Type Synthetic Control Chart

In this article, we provide a nonparametric Shewhart-type synthetic control chart based on the signed-rank statistic to monitor shifts in the known in-control process median. The synthetic control chart is a combination of a signed-rank chart due to Bakir (2004 Bakir , S. T. ( 2004 ). A distribution-free Shewhart quality control chart based on signed-ranks . Quality Engineering 16 : 613 – 623 .[Taylor & Francis Online] , [Google Scholar]) and a conforming run length chart due to Bourke (1991 Bourke , P. D. ( 1991 ). Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection . Journal of Quality Technology 23 : 225 – 238 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The operation and design of the chart are discussed and the performance of the chart has been studied. The chart has an attractive average run length behavior as compared to the parametric control chart for a class of symmetric continuous process distributions. The proposed chart performs better than the nonparametric signed-rank chart given by Bakir (2004 Bakir , S. T. ( 2004 ). A distribution-free Shewhart quality control chart based on signed-ranks . Quality Engineering 16 : 613 – 623 .[Taylor & Francis Online] , [Google Scholar]) and Chakraborti and Eryilmaz (2007 Chakraborti , S. , Eryilmaz , S. (2007). A nonparametric Shewhart-type signed-rank control chart based on runs. Communications in Statistics—Simulation and Computation 36:335–356.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

Multivariate Synthetic |S| Control Chart with Variable Sampling Interval

A variable sampling interval (VSI) feature is introduced to the multivariate synthetic generalized sample variance |S| control chart. This multivariate synthetic control chart is a combination of the |S| sub-chart and the conforming run length sub-chart. The VSI feature enhances the performance of the multivariate synthetic control chart. The comparative results show that the VSI multivariate synthetic control chart performs better than other types of multivariate control charts for detecting shifts in the covariance matrix of a multivariate normally distributed process. An example is given to illustrate the operation of the VSI multivariate synthetic chart.     

EWMA Control Charts for Monitoring High Yield Processes

The main objective of this article is to scrutinize the efficiency and verify the performance superiority of the one-sided EWMA control chart on high-yield processes. The proposed control chart is designed to detect both upward and downward shifts of the fraction of non conforming products and is developed based on non transformed geometric counts. Its algorithmic function is theoretically established and numerous performance measures are extracted using analytical methods based on the Markov modeling of the chart. Comparisons with traditional high yield control charts are conducted. Optimality tables and nomograms are included to help graphical determination of the optimal chart parameters.     

On the Run Length of a State-Space Control Chart for Multivariate Autocorrelated Data

The literature on statistical process control (SPC) describes the negative effects of autocorrelation in terms of the increase in false alarms. This has been treated by the individual modeling of each series or the application of VAR models. In the former case, the analysis of the cross correlation structure between the variables is altered. In the latter, if the cross correlation is not strong, the filtering process may modify the weakest relations. In order to improve these aspects, state-space models have been introduced in multivariate statistical process control (MSPC). This article presents a proposal for building a control chart for innovations, estimating its average run length to highlight its advantages over the VAR approach mentioned above.     

A Multivariate Adaptive Exponentially Weighted Moving Average Control Chart

A multivariate extension of the adaptive exponentially weighted moving average (AEWMA) control chart is proposed. The new multivariate scheme can detect small and large shifts in the process mean vector effectively. The proposed scheme can be viewed as a smooth combination of a multivariate exponentially weighted moving average (MEWMA) chart and a Shewhart χ²-chart. The optimal design of the proposed chart is given according to a pre-specified in-control average run length and two shift sizes; a small and large shift each measured in terms of the non centrality parameter. The signal resistance of the newly proposed multivariate chart is also given. Comparisons among the new chart, the MEWMA chart, and the combined Shewhart-MEWMA (S-MEWMA) chart in terms of the standard and worst-case average run length profiles are presented. In addition, the three charts are compared with respect to their worst-case signal resistance values. The proposed chart gives somewhat better worst-case ARL and signal resistance values than the competing charts. It also gives better standard ARL performance especially for moderate and large shifts. The effectiveness of our proposed chart is illustrated through an example with simulated data set.     

A Double Moving Average Control Chart

The double exponentially weighted moving average (DEWMA) technique has been investigated in recent years for detecting shifts in the process mean and has been shown to be more efficient than the corresponding exponentially weighted moving average (EWMA) technique. In this article, we extend the DEWMA technique of performing exponential smoothing twice to the double moving average (DMA) technique by computing the moving average twice. Using simulation, we show that our proposed DMA chart improves upon the ARL performance of the moving average (MA) chart in detecting mean shifts of small to moderate magnitudes. It is also shown through simulation that, generally, the DMA charts with spans, w = 10 and 15 provide comparable average run length (ARL) performances to the EWMA and cumulative sum (CUSUM) charts, designed for detecting small shifts.     

The Effects of Sample Sizes in Phases I and II on p Control Chart Performance

Control charts designed for the properties of non conformities, also called p control charts, are powerful tools used for monitoring a performance of the fraction of non conforming units. Constructing a p chart is often based on the assumption that the in-control proportion of non conforming items (p ₀) is known. In practice, the value of p ₀ is rarely known and is frequently replaced by an estimate from an in-control reference sample in Phase I. This article investigates the effects of sample sizes in both Phase I and Phase II on the performance of p control charts. The conditional and marginal run length distributions are derived and the corresponding numerical studies are conducted. Moreover, the minimal sample sizes required in Phases I and II to ensure adequate statistical performance are proposed when p ₀ = 0.1 and 0.005.     

Profile Monitoring Using Nonparametric Bootstrap T 2 Control Chart

In certain statistical process control applications, performance of a product or process can be monitored effectively using a linear profile or a linear relationship between a response variable and one or more explanatory variables. In this article, we design a nonparametric bootstrap control chart for monitoring simple linear profiles based on T ² statistic. We evaluate the performance of the proposed method in phase II. The average and standard deviation of the run length under different shifts in the intercept, slope, and standard deviation are considered as the performance measures. Simulation results show that the performance of the proposed bootstrap control chart improves as the size of the available data increases.     

An Improved Distribution-free EWMA Mean Chart

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having nonnormal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used here. In this article, we propose an improved asymmetric EWMA mean chart based on a simple statistic to monitor process mean shift. We explored the sampling properties of the new monitoring statistic and calculated the average run lengths of the proposed asymmetric EWMA mean chart. We recommend the proposed improved asymmetric EWMA mean chart because the average run lengths of the modified charts are more accurate and reasonable than those of the five existed mean charts. A numerical example of service times with a right skewed distribution from a service system of a bank branch is used to illustrate the application of the improved asymmetric EWMA mean chart and to compare it with the five existing mean charts. The proposed chart showed better detection performance than those of the five existing mean charts in monitoring and detecting shifts in the process mean.     

The Synthetic Mean Square Error Control Chart

A synthetic mean square error (MSE) control chart is presented in this study for monitoring the changes in the mean and standard deviation of a normally distributed process. The synthetic MSE control chart is a combination of the standard MSE control chart and the conforming run length (CRL) control chart. From the numerical comparisons, the synthetic MSE control chart is always more efficient than the standard MSE control chart in detecting shifts in the process mean and standard deviation. The synthetic MSE chart also performs better than the exponentially weighted moving average-semicircle (EWMA-SC) chart, except for some cases where the process mean shifts are small.     

A Nonparametric Synthetic Control Chart Using Sign Statistic

This article presents a synthetic control chart for detection of shifts in the process median. The synthetic chart is a combination of sign chart and conforming run-length chart. The performance evaluation of the proposed chart indicates that the synthetic chart has a higher power of detecting shifts in process median than the Shewhart charts based on sign statistic as well as the classical Shewhart X-bar chart for various symmetric distributions. The improvement is significant for shifts of moderate to large shifts in the median. The robustness studies of the proposed synthetic control chart against outliers indicate that the proposed synthetic control chart is robust against contamination by outliers.     

An Evaluation of the Double Exponentially Weighted Moving Average Control Chart

In this article we perform a careful investigation of the double exponentially weighted moving average (DEWMA) chart performance for monitoring the process mean. We compare the performance of this chart to the usual EWMA control chart based on zero-state and worst-case average run length (ARL) measures. We also evaluate the signal resistance measure of the DEWMA chart and compare its maximum value to that of the EWMA chart. We show that the superiority of the DEWMA chart over the simpler standard EWMA chart based on zero-state ARL performance disappears when the smoothing constant of the EWMA chart is chosen to give weights to past observations closer to those given by the DEWMA chart. Moreover, our results show that the standard EWMA chart has much better performance than the DEWMA chart in terms of worst-case ARL values, especially when small smoothing constants are used. We also demonstrate using an illustrative example that the DEWMA chart can build up an exceedingly large amount of inertia when used to monitor the process mean.     

Poisson GWMA Control Chart

A generally weighted moving average (GWMA) control chart for monitoring Poisson observations is introduced. Using simulation, its average run lengths and standard deviations of run lengths are compared with those of other control charts for Poisson data. It is shown that the Poisson GWMA chart outperforms other control charts, especially when the process shift is small.     

An Examination of the Robustness to Non Normality of the EWMA Control Charts for the Dispersion

ABSTRACT

The EWMA control chart is used to detect small shifts in a process. It has been shown that, for certain values of the smoothing parameter, the EWMA chart for the mean is robust to non normality. In this article, we examine the case of non normality in the EWMA charts for the dispersion. It is shown that we can have an EWMA chart for dispersion robust to non normality when non normality is not extreme.