A Multivariate Exponentially Weighted Moving Average Control Chart

A multivariate extension of the exponentially weighted moving average (EWMA) control chart is presented, and guidelines given for designing this easy-to-implement multivariate procedure. A comparison shows that the average run length (ARL) performance of this chart is similar to that of multivariate cumulative sum (CUSUM) control charts in detecting a shift in the mean vector of a multivariate normal distribution. As with the Hotelling's χ² and multivariate CUSUM charts, the ARL performance of the multivariate EWMA chart depends on the underlying mean vector and covariance matrix only through the value of the noncentrality parameter. Worst-case scenarios show that Hotelling's χ² charts should always be used in conjunction with multivariate CUSUM and EWMA charts to avoid potential inertia problems. Examples are given to illustrate the use of the proposed procedure.     

On mixed memory control charts based on auxiliary information for efficient process monitoring

Control charts are popular monitoring tools in statistical process control toolkit. These are used to identify assignable causes in the process parameters (location and/or dispersion). These assignable causes result in a shift in the process parameter(s). The shift can be categorized into three sizes (small, moderate, and large). Memory control charts such as the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts are effective for identifying small-to-moderate shift(s) in the process. Likewise, mixed memory control charts are useful for efficient process monitoring. In this study, we have proposed two new mixed memory control charts based on auxiliary information named MxMEC and MxMCE control charts to improve the efficiency of these mixed charts. The MxMEC chart is a merger of the auxiliary information based MxEWMA chart and the classical CUSUM chart. Likewise, the MxMCE chart integrates the auxiliary information based MxCUSUM with the classical EWMA chart. The proposed MxMEC and MxMCE charts are evaluated through famous performance measures including average run length, extra quadratic loss, relative average run length, and performance comparison index. The performance of the study proposals is compared with the existing counterparts such as the classical CUSUM and EWMA, MxCUSUM, MxEWMA, MEC, MCE, and runs rules-based CUSUM charts. The comparisons revealed the superiority of the proposed charts against other competing charts particularly for small-to-moderate shifts in the process location. Finally, a real-life data is used to show the implementation procedure of the proposed charts in practical situations.     

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.     

An adaptive EWMA scheme‐based CUSUM accumulation error for efficient monitoring of process location

The examination of product characteristics using a statistical tool is an important step in a manufacturing environment to ensure product quality. Several methods are employed for maintaining product quality assurance. Quality control charts, which utilize statistical methods, are normally used to detect special causes. Shewhart control charts are popular; their only limitation is that they are effective in handling only large shifts. For handling small shifts, the cumulative sum (CUSUM) and the exponential weighted moving average (EWMA) are more practical. For handling both small and large shifts, adaptive control charts are used. In this study, we proposed a new adaptive EWMA scheme. This scheme is based on CUSUM accumulation error for detection of wide range of shifts in the process location. The CUSUM features in the proposed scheme help with identification of prior shifts. The proposed scheme uses Huber and Tukey bisquare functions for an efficient shift detection. We have used average run length (ARL) as performance indicator for comparison, and our proposed scheme outperformed some of the existing schemes. An example that uses real‐life data is also provided to demonstrate the implementation of the proposed scheme.     

Efficient homogeneously weighted dispersion control charts with an application to distillation process

Monitoring disturbances in process dispersion using control chart is mostly based on the assumption that the quality characteristic follows normal distribution, which is not the case in many real-life situations. This paper proposes a set of new dispersion charts based on the homogeneously weighted moving average (HWMA) scheme, for efficient detection of shifts in process standard deviation (σ). These charts are based on a variety of σ estimators and are investigated for normal as well as heavy tailed symmetric and skewed distributions. The shift detection ability of the charts is evaluated using different run length characteristics, such as average run length (ARL), extra quadratic loss (EQL), and relative ARL measures. The performance of the proposed HWMA control charts is also compared with the existing EWMA dispersion charts, using different design parameters. Furthermore, an illustrative example is presented to monitor the vapor pressure in a distillation 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 comparison of control charting procedures for monitoring process dispersion

We consider several control charts for monitoring normal processes for changes in dispersion. We present comparisons of the average run length performances of these charts. We demonstrate that a CUSUM chart based on the likelihood ratio test for the change point problem for normal variances has an ARL performance that is superior to other procedures. Graphs are given to aid in designing this control chart.     

Computing the Percentage Points of the Run-Length Distributions of Multivariate CUSUM Control Charts

Multivariate CUSUM control charts are often used instead of the standard Hotelling's control charts in many practical problems when detection of small shifts in the process mean is important. However, design of multivariate CUSUM control charts are usually based on the average run length (ARL). In this work, we will compute the percentage points of the run-length distributions of two multivariate CUSUM control charts. It will be shown that interpretations based on ARL can be misleading since the in-control run-length distribution of a multivariate CUSUM is highly skewed. On the other hand, the percentage points of the run-length distribution provide additional information such as the median run length, early false out-of-control signals, and the skewness of the run-length distribution for a particular scheme. These extra information might provide quality control engineers further knowledge of a particular multivariate CUSUM control chart scheme.     

Dual multivariate CUSUM charts with auxiliary information for process mean

It is customary to increase the sensitivity of a control chart using an efficient estimator of the underlying process parameter which is being monitored. In this paper, using an auxiliary information-based (AIB) mean estimator, we propose dual multivariate CUSUM (DMCUSUM) and mixed DMCUSUM (MDMCUSUM) charts, called the AIB-DMCUSUM and AIB-MDMCUSUM charts, with and without fast initial response features for monitoring the mean vector of a multivariate normally distributed process. The DMCUSUM chart combines two similar-type multivariate CUSUM (MCUSUM) charts while the MDMCUSUM chart combines two different-type MCUSUM charts, into a single chart. The objective of two multivariate subcharts in the DMCUSUM/MDMCUSUM chart is to simultaneously detect small-to-moderate and moderate-to-large shifts in the process mean vector. Monte Carlo simulations are used to compute the run length characteristics, including the average run length (ARL), extra quadratic loss, and integral of the relative ARL. Based on detailed run length comparisons, it turns out that the AIB-DMCUSUM and AIB-MDMCUSUM charts uniformly and substantially outperform the DMCUSUM and MDMCUSUM charts when detecting different sizes of shift in the process mean vector. A real dataset is used to explain the implementation of proposed AIB multivariate charts.     

A new dual CUSUM mean chart

The conventional cumulative sum (CUSUM) chart is usually designed based on a known shift size. In usual practice, shift size is often unknown and can be assumed to vary within an interval. With such a range of shift size, the dual CUSUM (DCUSUM) chart provides more sensitivity than the CUSUM chart. In this paper, we propose dual Crosier CUSUM (DCCUSUM) charts with and without fast initial response features to efficiently monitor the infrequent changes in the mean of a normally distributed process. Monte Carlo simulations are used to compute the run length characteristics of one‐sided and two‐sided DCCUSUM charts. These run length characteristics are compared with those of the CUSUM, Crosier CUSUM, Shewhart‐CUSUM, and DCUSUM charts in terms of the integral relative average run length. It turns out that the proposed chart shows better performance when detecting a range of mean shift sizes. A real dataset is considered to illustrate the implementation of existing and proposed charts.     

Conditional mean- and median-based cumulative sum control charts for Weibull data

This article deals with the monitoring of censored data using the cumulative sum (CUSUM) control charts for Weibull lifetimes under type-I censoring. To develop an efficient CUSUM structure for censored data, we use the conditional expected value (CEV) and conditional median (CM) approaches. In particular, we focus on the detection of shifts in the mean of Weibull lifetimes assuming censored data. In addition to fixed/known parameter values, the effect of estimation is assessed on the detection power of control charts. The performance of the proposed charts is evaluated by the average run length (ARL). Furthermore, the ARL performance of CUSUM charts is compared with CEV- and CM-based exponentially weighted moving average (EWMA) control charts. Besides an extensive simulation study, the significance of the current work is illustrated by a data set on the response time of a thermostat experiment.     

Using FIR to Improve CUSUM Charts for Monitoring Process Dispersion

Statistical process control deals with monitoring process to detect disturbances in the process. These disturbances may be from the process mean or variance. In this study, we propose some charts that are efficient for detecting early shifts in dispersion parameter, by applying the Fast Initial Response feature. Performance measures such as average run length, standard deviation of the run length, extra quadratic loss, relative average run length, and performance comparison index are used to compare the proposed charts with their existing counterparts, including the Shewhart R chart and the Shewhart S chart with and without warning lines. Others include the CUSUM R chart, the CUSUM S chart, the EWMA of ln S², the CUSUM of ln S², the P_σ CUSUM, the χ CUSUM, and the Change Point (CP) CUSUM charts. The proposed charts do not only detect early shifts in the process dispersion faster, but also have better overall performance than their existing counterparts. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

A New Cumulative Sum Quality Control Scheme for Monitoring the Process Mean

A control chart is a powerful statistical process monitoring tool that is frequently used in many industrial and service organizations to monitor in‐control and out‐of‐control performances of the manufacturing processes. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have been recognized as potentially powerful tool in quality and management control. These control charts are sensitive to both small and moderate changes in the process. In this paper, we propose a new CUSUM (NCUSUM) quality control scheme for efficiently monitoring the process mean. It is shown that the classical CUSUM control chart is a special case of the proposed controlling scheme. The NCUSUM control chart is compared with some of the recently proposed control charts by using characteristics of the distribution of run length, i.e. average run length, median run length and standard deviation of run length. It is worth mentioning that the NCUSUM control chart detects the random shifts in the process mean substantially quicker than the classical CUSUM, fast initial response‐based CUSUM, adaptive CUSUM with EWMA‐based shift, adaptive EWMA and Shewhart–CUSUM control charts. An illustrative example is given to exemplify the implementation of the proposed quality control scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Multivariate statistical process control charts based on the approximate sequential χ2 test

Similar to the univariate CUSUM chart, a multivariate CUSUM (MCUSUM) chart can be designed to detect a particular size of the mean shift optimally based on the scheme of a sequential likelihood ratio test for the noncentrality parameter. However, in multivariate case, the probability ratio of a sequential test is intractable mathematically and the test statistic based on the ratio does not have a closed form expression which makes it impractical for real application. We drive an approximate log-likelihood ratio and propose a multivariate statistical process control chart based on a sequential χ² test to detect a change in the noncentrality parameter. The statistical properties of the proposed test statistic are explored. The average runs length (ARL) performance of the proposed charts is compared with other MCUSUM charts for process mean monitoring. The experimental results reveal that the proposed charts achieve superior, both zero-state and steady-state, ARL performance over a wide range of mean shifts, especially when the dimension of measurements is large.     

New Synthetic EWMA and Synthetic CUSUM Control Charts for Monitoring the Process Mean

Exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts are potentially powerful statistical process monitoring tools because of their excellent speed in detecting small to moderate persistent process shifts. Recently, synthetic EWMA (SynEWMA) and synthetic CUSUM (SynCUSUM) control charts have been proposed based on simple random sampling (SRS) by integrating the EWMA and CUSUM control charts with the conforming run length control chart, respectively. These synthetic control charts provide overall superior detection over a range of mean shift sizes. In this article, we propose new SynEWMA and SynCUSUM control charts based on ranked set sampling (RSS) and median RSS (MRSS) schemes, named SynEWMA‐RSS and SynEWMA‐MRSS charts, respectively, for monitoring the process mean. Extensive Monte Carlo simulations are used to estimate the run length characteristics of the proposed control charts. The run length performances of these control charts are compared with their existing powerful counterparts based on SRS, RSS and MRSS schemes. It turns out that the proposed charts perform uniformly better than the Shewhart, optimal synthetic, optimal EWMA, optimal CUSUM, near‐optimal SynEWMA, near‐optimal SynCUSUM control charts based on SRS, and combined Shewhart‐EWMA control charts based on RSS and MRSS schemes. A similar trend is observed when constructing the proposed control charts based on imperfect RSS schemes. An application to a real data is also provided to demonstrate the implementations of the proposed SynEWMA and SynCUSUM control charts. Copyright © 2014 John Wiley & Sons, Ltd.     

An Evaluation of the Crosier's CUSUM Control Chart with Estimated Parameters

We evaluate the performance of the Crosier's cumulative sum (C‐CUSUM) control chart when the probability distribution parameters of the underlying quality characteristic are estimated from Phase I data. Because the average run length (ARL) under estimated parameters is a random variable, we study the estimation effect on the chart performance in terms of the expected value of the average run length (AARL) and the standard deviation of the average run length (SDARL). Previous evaluations of this control chart were conducted while assuming known process parameters. Using the Markov chain and simulation approaches, we evaluate the in‐control performance of the chart and provide some quantiles for its in‐control ARL distribution under estimated parameters. We also compare the performance of the C‐CUSUM chart to that of the ordinary CUSUM (O‐CUSUM) chart when the process parameters are unknown. Our results show that large number of Phase I samples are required to achieve a quite reasonable performance. Additionally, the performance of the C‐CUSUM chart is found to be superior to that of the O‐CUSUM chart. Finally, we recommend the use of a recently proposed bootstrap procedure in designing the C‐CUSUM chart to guarantee, at a certain probability, that the in‐control ARL will be of at least the desired value using the available amount of Phase I data. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Mixed Cumulative Sum–Exponentially Weighted Moving Average Control Charts: An Efficient Way of Monitoring Process Location

Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) charts are famous statistical tools, to handle special causes and to bring the process back in statistical control. Shewhart charts are useful to detect large shifts, whereas EWMA and CUSUM are more sensitive for small to moderate shifts. In this study, we propose a new control chart, named mixed CUSUM‐EWMA chart, which is used to monitor the location of a process. The performance of the proposed mixed CUSUM‐EWMA control chart is measured through the average run length, extra quadratic loss, relative average run length, and a performance comparison index study. Comparisons are made with some existing charts from the literature. An example with real data is also given for practical considerations. Copyright © 2014 John Wiley & Sons, Ltd.     

A New Synthetic Exponentially Weighted Moving Average Control Chart for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely recognized as a potentially powerful process monitoring tool of the statistical process control because of their excellent speed in detecting small to moderate shifts in the process parameters. Recently, new EWMA and synthetic control charts have been proposed based on the best linear unbiased estimator of the scale parameter using ordered ranked set sampling (ORSS) scheme, named EWMA‐ORSS and synthetic‐ORSS charts, respectively. In this paper, we extend the work and propose a new synthetic EWMA (SynEWMA) control chart for monitoring the process dispersion using ORSS, named SynEWMA‐ORSS chart. The SynEWMA‐ORSS chart is an integration of the EWMA‐ORSS chart and the conforming run length chart. Extensive Monte Carlo simulations are used to estimate the run length performances of the proposed control chart. A comprehensive comparison of the run length performances of the proposed and the existing powerful control charts reveals that the SynEWMA‐ORSS chart outperforms the synthetic‐R, synthetic‐S, synthetic‐D, synthetic‐ORSS, CUSUM‐R, CUSUM‐S, CUSUM‐ln S², EWMA‐ln S² and EWMA‐ORSS charts when detecting small shifts in the process dispersion. A similar trend is observed when the proposed control chart is constructed under imperfect rankings. An application to a real data is also provided to demonstrate the implementation and application of the proposed control chart. Copyright © 2014 John Wiley & Sons, Ltd.