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.     

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.     

On moving average control charts and their conditional average run lengths

Moving average control charts have been presented in the Quality Control literature in the past 75 years; however, their conditional average run lengths have not been obtained. The objective of this article is to derive the autocorrelation function between two moving averages, and then make application of the bivariate normal distribution to compute the conditional type II error probability at the future time, t +1, given that a manufacturing process is in statistical control at the present time t. Our Tables 3 through 8 show that the values of Shewhart's average run length and the corresponding conditional first-order moving average run lengths are almost the same after one standard deviation shift from the target of a normal process mean. Our conclusion section 6 describes that the comparisons of the two average run lengths are not on a valid statistical basis.     

The impracticality of homogeneously weighted moving average and progressive mean control chart approaches

There is growing literature on new versions of “memory-type” control charts, where deceptively good zero-state average run-length (ARL) performance is misleading. Using steady-state run-length analysis in combination with the conditional expected delay (CED) metric, we show that the increasingly discussed progressive mean (PM) and homogeneously weighted moving average (HWMA) control charts should not be used in practice. Previously reported performance of methods based on these two approaches is misleading, as we found that performance is good only when a process change occurs at the very start of monitoring. Traditional alternatives, such as exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts, not only have more consistent detection behavior over a range of different change points, they can also lead to better out-of-control zero-state ARL performance when properly designed.     

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.     

Advanced multivariate cumulative sum control charts based on principal component method with application

Existing multivariate cumulative sum (MCUSUM) control charts involve entire associated variables of a process to monitor variations in the mean vector. In this study, we have offered MCUSUM control charts with principal component method (PCM). The proposed MCUSUM control charts with PCM capture the whole process variations using fewer latent variables (principal components) while preserving as much data variability as possible. To show the significance of proposed MCUSUM control charts with PCM, various performance measures are considered including average run length, extra quadratic loss, relative average run length, and performance comparison index. Furthermore, performance measures are calculated through advanced Monte Carlo simulation method to explore the behavior of proposed MCUSUM control charts and to conduct comparative analysis with existing models. Results revealed that proposed MCUSUM control charts with PCM are efficient to detect variations timely by involving smaller number of principal components instead of considering entire associated variables. Also, proposed MCUSUM control charts have the ability to accommodate the features of existing control charts, which are illustrated as the special cases. Besides, to highlight the implementation mechanism and advantages of proposed MCUSUM control charts with PCM, a real-life example from wind turbine process is included.     

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.     

A moving average control chart using a robust scale estimator for process dispersion

Control charts are one of the most powerful tools used to detect and control industrial process deviations in statistical process control. In this paper, a moving average control chart based on a robust scale estimator of standard deviation, namely, the sample median absolute deviation (MAD) statistic, for monitoring process dispersion, is proposed. A simulation study is conducted to evaluate the performance of the proposed moving average median absolute deviation (MA‐MAD) chart, in terms of average run length for various distributions. The results show that the moving average MAD chart performs well in detecting small and moderate shifts in process dispersion, especially when the normality assumption is violated. In addition, this chart is very efficient, especially when the quality characteristic follows a skewed distribution. Numerical and simulated examples are given at the end of the paper.     

On increasing the sensitivity of moving average control chart using auxiliary variable

In the statistical process control, the most useful tool to monitor the manufacturing processes in the industries is the control chart. Quality practitioners always desire the charting structure that identifies sustainable changes in the monitoring processes. The sensitivity of the control chart is improved when additional correlated auxiliary information about the study variable is introduced. The regression estimate in the form of auxiliary and supporting variables presents an unbiased and efficient statistic of the mean of the process variable. In this study, auxiliary information-based moving average (AB-MA) control chart is designed for efficient monitoring of shifts in the process location parameter. The performance of the AB-MA control chart is evaluated and compared with existing charts using average run length and other run length characteristics. The comparison reveals that the AB-MA control chart outperforms the competitors in detecting the small and medium changes in the process location parameter. The application of the proposal is also provided to implement it in real situation.     

Memory‐type multivariate control charts with auxiliary information for process mean

In statistical process control, it is a common practice to increase the sensitivity of a control chart with the help of an efficient estimator of the underlying process parameter. In this paper, we consider an efficient estimator that requires information on several study variables along with one or more auxiliary variables when estimating the mean of a multivariate normally distributed process. Using this auxiliary‐information‐based (AIB) process mean estimator, we propose new multivariate EWMA (MEWMA), double MEWMA (DMEWMA), and multivariate CUSUM (MCUSUM) charts for monitoring the process mean, denoted by the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts, respectively. The run length characteristics of the proposed multivariate charts are computed using Monte Carlo simulations. The proposed charts are compared with their existing counterparts in terms of the run length characteristics. It turns out that the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts are uniformly and substantially better than the MEWMA, DMEWMA, and MCUSUM charts, respectively, when detecting different shifts in the process mean. A real dataset is considered to explain the implementation of the proposed and existing multivariate control charts.     

Function-based adaptive exponentially weighted moving average dispersion control chart

In this study, a new process dispersion monitoring control chart is proposed based on a function-based adaptation to select the smoothing constant value, named as a function-based adaptive exponentially weighted moving average (EWMA) dispersion control chart. It is suggested to track shift ranges first expected in process dispersion by opting for smoothing constant computation with the help of a function. The shift magnitude assessment is made by an unbiased estimator that determines the smoothing constant value through the proposed function. The enhanced efficiency of the proposed chart can be assessed in terms of smaller run-length profiles, which are determined through Monte Carlo simulations. The proposed chart is compared with the existing adaptive EWMA dispersion chart, and it turned out to perform substantially efficiently in detecting all kinds of decreasing and increasing process dispersion shift magnitudes. Moreover, a real-life dataset application is explained in the example section to elaborate on the ease of implementation in the real-life scenario.     

A moving average S control chart for monitoring process variability

An efficient alternative to the S control chart for detecting shifts of small magnitude in the process variability using a moving average based on the sample standard deviation s statistic is proposed. Control limit factors are derived for the chart for different values of sample size and span w. The performance of the moving average S chart is compared to the S chart in terms of average run length. The result shows that the performance of moving average S chart for varying values of w outweigh those of the S chart for small and moderate shifts in process variability.     

Percentile‐based control chart design with an application to Shewhart X̅ and S2 control charts

We present a method to design control charts such that in‐control and out‐of‐control run lengths are guaranteed with prespecified probabilities. We call this method the percentile‐based approach to control chart design. This method is an improvement over the classical and popular statistical design approach employing constraints on in‐control and out‐of‐control average run lengths since we can ensure with prespecified probability that the actual in‐control run length exceeds a desired magnitude. Similarly, we can ensure that the out‐of‐control run length is less than a desired magnitude with prespecified probability. Some numerical examples illustrate the efficacy of this design method.     

An extended nonparametric homogeneously weighted moving average sign control chart

Parametric (or traditional) control charts are based on the assumption that the quality characteristic of interest follows a specific distribution. However, in many applications, there is a lack of knowledge about the underlying distribution. To this end, nonparametric (or distribution-free) control charts have been developed in recent years. In this article, a nonparametric double homogeneously weighted moving average (DHWMA) control chart based on the sign statistic is proposed for monitoring the location parameter of an unknown and continuous distribution. The performance of the proposed chart is measured through the run-length distribution and its associated characteristics by performing Monte Carlo simulations. The DHWMA sign chart is compared with other nonparametric sign charts, such as the homogeneously weighted moving average, generally weighted moving average (GWMA), double GWMA, and triple exponentially weighted moving average sign charts, as well as the traditional DHWMA chart. The results indicate that the proposed chart performs just as well as and in some cases better than its competitors, especially for small shifts. Finally, two examples are provided to show the application and implementation of the proposed chart.     

New memory‐type dispersion control charts

The exponentially weighted moving average (EWMA) control chart is a memory‐type process monitoring tool that is frequently used to monitor small and moderate disturbances in the process mean and/or process dispersion. In this study, we propose 2 new memory‐type control charts for monitoring changes in the process dispersion, namely, the generally weighted moving average and the hybrid EWMA charts. We use Monte Carlo simulations to compute the run length profiles of the proposed control charts. The run length comparisons of the proposed and existing charts reveal that the generally weighted moving average and hybrid EWMA charts provide better protection than the existing EWMA chart when detecting small to moderate shifts in the process dispersion. An illustrative dataset is also used to show the superiority of the proposed charts over the existing chart.     

New EWMA control charts for monitoring process dispersion using auxiliary information

The exponentially weighted moving average (EWMA) control chart is a well‐known statistical process monitoring tool because of its exceptional pace in catching infrequent variations in the process parameter(s). In this paper, we propose new EWMA charts using the auxiliary information for efficiently monitoring the process dispersion, named the auxiliary‐information–based (AIB) EWMA (AIB‐EWMA) charts. These AIB‐EWMA charts are based on the regression estimators that require information on the quality characteristic under study as well as on any related auxiliary characteristic. Extensive Monte Carlo simulation are used to compute and study the run length profiles of the AIB‐EWMA charts. The proposed charts are comprehensively compared with a recent powerful EWMA chart—which has been shown to be better than the existing EWMA charts—and an existing AIB‐Shewhart chart. It turns out that the proposed charts perform uniformly better than the existing charts. An illustrative example is also given to explain the implementation and working of the AIB‐EWMA charts.     

Monitoring the Weibull shape parameter by control charts for the sample range

In this paper, we propose control charts for monitoring changes in the Weibull shape parameter β. These charts are based on the range of a random sample from the smallest extreme value distribution. The control chart limits depend only on the sample size, the desired stable average run length (ARL), and the stable value of β. We derive control limits for both one‐ and two‐sided control charts. They are unbiased with respect to the ARL. We discuss sample size requirements if the stable value of βis estimated from past data. The proposed method is applied to data on the breaking strengths of carbon fibers. We recommend one‐sided charts for detecting specific changes in βbecause they are expected to signal out‐of‐control sooner than the two‐sided charts. Copyright © 2010 John Wiley & Sons, Ltd.     

A hybrid homogeneously weighted moving average control chart for process monitoring

Several modifications and enhancements to control charts in increasing the performance of small and moderate process shifts have been introduced in the quality control charting techniques. In this paper, a new hybrid control chart for monitoring process location is proposed by combining two homogeneously weighted moving average (HWMA) control charts. The hybrid homogeneously weighted moving average (HHWMA) statistic is derived using two smoothing constants λ₁ and λ₂ . The average run length (ARL) and the standard deviation of the run length (SDRL) values of the HHWMA control chart are obtained and compared with some existing control charts for monitoring small and moderate shifts in the process location. The results of study show that the HHWMA control chart outperforms the existing control charts in many situations. The application of the HHWMA chart is demonstrated using a simulated data.     

Directionally sensitive weighted adaptive multivariate CUSUM mean charts

In many service and manufacturing industries, process monitoring involves multivariate data, instead of univariate data. In these situations, multivariate charts are employed for process monitoring. Very often when the mean vector shifts to an out-of-control situation, the exact shift size is unknown; hence, multivariate charts for monitoring a range of the mean shift sizes in the mean vector are adopted. In this paper, directionally sensitive weighted adaptive multivariate CUSUM charts are developed for monitoring a range of the mean shift sizes. Directionally sensitive charts are useful in situations where the aim lies in monitoring either an increasing or a decreasing shift in the mean vector of the quality characteristics of interest. The Monte Carlo simulation is used to compute the run length characteristics in comparing the sensitivities of the proposed and existing multivariate CUSUM charts. In general, the directionally sensitive and weighted adaptive features enhance the sensitivities of the proposed multivariate CUSUM charts in comparison with the existing multivariate CUSUM charts without the adaptive feature or those that are directionally invariant. It is also found that the variable sampling interval feature enhances the sensitivities of the proposed and existing charts as compared to their fixed sampling interval counterparts. The implementation of the proposed charts in detecting upward and downward shifts in the in-control process mean vector is demonstrated using two different datasets.