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.     

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.     

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.     

Monitoring the ratio of two normal variables using variable sampling interval exponentially weighted moving average control charts

We investigate in this paper a new type of control chart called VSI EWMA‐RZ by integrating the variable sampling interval feature (VSI) with the exponentially weighted moving average (EWMA) scheme to monitor the ratio of two normal random variables. Because the distribution of the ratio is skewed, we suggest designing two separated one‐sided charts instead of one two‐sided chart. A new coefficient is introduced allowing us to be free to choose a sampling interval provided that it optimizes the performance of the control chart. We also make a direct comparison between the VSI EWMA‐RZ charts and standard EWMA‐RZ control charts. The numerical simulations show that the proposed charts outperform the standard EWMA charts in detecting process shifts. An application is illustrated for the implementation of the VSI EWMA‐RZ control charts in the food industry.     

An improved variable sample size and sampling interval S control chart

The standard deviation chart (S chart) is used to monitor process variability. This paper proposes an upper‐sided improved variable sample size and sampling interval (VSSI_t) S chart by improving the existing upper‐sided variable sample size and sampling interval (VSSI) S chart through the inclusion of an additional sampling interval. The optimal designs of the VSSI_t S chart together with the competing charts under consideration, such as the VSSI S and exponentially weighted moving average (EWMA) S charts, by minimizing the out‐of‐control average time to signal (ATS₁) and expected average time to signal (EATS₁) criteria, are performed using the MATLAB programs. The performances of the standard S, VSSI S, EWMA S, and VSSI_t S charts are compared, in terms of the ATS₁ and EATS₁ criteria, where the results show that the VSSI_t S chart surpasses the other charts in detecting moderate and large shifts, while the EWMA S is the best performing chart in detecting small shifts. An illustrative example is given to explain the implementation of the VSSI_t S chart.     

Variable sampling interval exponentially weighted moving average median chart with estimated process parameters

The variable sampling interval exponentially weighted moving average median chart with estimated process parameters is proposed. The charting statistic, optimal design, performance evaluation, and implementation of the proposed chart are discussed. The average of the average time to signal (AATS) criterion is adopted to evaluate the performance of the proposed chart. The estimated process parameter‐based VSI EWMA median (VSI EWMA median‐e) chart is compared with the estimated process parameter‐based Shewhart median (SH median‐e), EWMA median (EWMA median‐e), and variable sampling interval run sum median (VSI RS median‐e) charts, in terms of the AATS criterion, where the VSI EWMA median‐e chart is shown to be superior. When process parameters are estimated, the standard deviation of the average time to signal (SDATS) criterion is used to evaluate the AATS performance of the VSI EWMA median‐e chart. Based on the SDATS criterion, the minimum number of phase‐I samples required by the VSI EWMA median‐e chart so that its performance is close to the known process parameters VSI EWMA median chart is recommended.     

EWMA control charts with variable sample sizes and variable sampling intervals

Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. Variable Sample Size (VSS) and Variable Sampling Interval (VSI) control charts vary the sampling rate from the process as a function of the data from the process. VSS and VSI control charts sample at a higher rate when there is evidence of a change in the process, and are thus able to detect process changes faster than traditional control charts. This paper considers general VSS and VSI control charts and develops integral equation and Markov chain methods for finding the statistical properties of these charts. EWMA charts with the VSS and/or the VSI features are studied in detail, and different ways of defining the EWMA control statistic are investigated. It is shown that using either the VSS or VSI feature in an EWMA control chart substantially improves the ability to detect all but very large shifts in the process mean. The VSI feature usually gives more improvement in detection ability than the VSS feature, and using both features together sometimes gives more improvement than using either one separately. Guidelines are given for choosing the possible sample sizes and the possible sampling intervals for these charts. EWMA charts with the VSS and/or VSI feature are compared to CUSUM charts and Shewhart X¯ charts with the VSS and/or VSI features.     

可变抽样区间不合格品数控制图

分析了休哈特控制图的不足,设计出具有两种抽样区间长度的可变抽样区间(VSI)np图,当点子接近控制限时,使用较短的抽样区间;当点子接近目标值时,使用较长的抽样区间.若点子超出控制限,则与固定抽样区间控制图(FSI)一样发出信号.同时还计算了在可变抽样区间下发信号前的平均时间,并与固定抽样区间np图进行比较,所设计的VSI控制图能缩短过程失控时间,从而可减少不合格品数.     

An adaptive EWMA control chart for monitoring the process mean in Bayesian theory under different loss functions

The Shewhart control chart is used for detecting the large shift and an exponentially weighted moving average (EWMA) control chart is used for detecting the small/moderate shift in the process mean. A scheme that combines both the Shewhart control chart and the EWMA control chart in a smooth way is called the adaptive EWMA (AEWMA) control chart. In this paper, we proposed a new AEWMA control chart for monitoring the process mean in Bayesian theory under different loss functions (LFs). We used informative (conjugate prior) under two different LFs: (1) squared error loss function and (2) linex loss function for posterior and posterior predictive distributions. We used the average run length and standard deviation of run length to measure the performance of the AEWMA control chart in the Bayesian theory. A comparative study is conducted for comparing the proposed AEWMA control chart in Bayesian theory with the existing Bayesian EWMA control chart. We conducted a Monte Carlo simulation study to evaluate the proposed AEWMA control chart. For the implementation purposes, we presented a real-data example.     

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.     

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.     

Design of a variable sampling interval exponentially weighted moving average median control chart in presence of measurement errors

In the literature, many control charts monitoring the median is designed under a perfect condition that there is no measurement error. This may make the practitioners confusing to apply these control charts because the measurement error is the true problem in practice. In this paper, we consider the effect of measurement error on the performance of the exponentially weighted moving average (EWMA) control chart combining with the variable sampling interval (VSI) strategy. A linear covariate error model is supposed to model the measurement error. The performance of the VSI EWMA median control chart is evaluated through the average time to signal. The numerical simulation shows that the measurement errors have a negative influence on the proposed chart.     

The Performance of the Adaptive Exponentially Weighted Moving Average Control Chart with Estimated Parameters

The adaptive exponentially weighted moving average (AEWMA) control chart has the advantage of detecting balance mixed range of mean shifts. Its performance has been studied under the assumption that the process parameters are known. Under this assumption, previous studies have shown AEWMA to provide superior statistical performance when compared with other different types of control charts. In practice, however, the process parameters are usually unknown and are required to be estimated. Using a Markov Chain approach, we show that the performance of the AEWMA control chart is affected when parameters are estimated compared with the known‐parameter case. In addition, we show the effect of different standard deviation estimators on the chart performance. Finally, a performance comparison is conducted between the exponentially weighted moving average (EWMA) chart and the AEWMA chart when the process parameters are unknown. We recommend the use of the AEWMA chart over the ordinary EWMA chart especially when a small number of Phase I samples is available to estimate the unknown parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Memory‐Type Control Charts for Monitoring the Process Dispersion

Control charts have been broadly used for monitoring the process mean and dispersion. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are memory control charts as they utilize the past information in setting up the control structure. This makes CUSUM and EWMA‐type charts good at detecting small disturbances in the process. This article proposes two new memory control charts for monitoring process dispersion, named as floating T ? S² and floating U ? S² control charts, respectively. The average run length (ARL) performance of the proposed charts is evaluated through a simulation study and is also compared with the CUSUM and EWMA charts for process dispersion. It is found that the proposed charts are better in detecting both positive as well as negative shifts. An additional comparison shows that the floating U ? S² chart has slightly smaller ARLs for larger shifts, while for smaller shifts, the floating T ? S² chart has better performance. An example is also provided which shows the application of the proposed charts on simulated datasets. Copyright © 2013 John Wiley & Sons, Ltd.     

New Synthetic Control Charts for Monitoring Process Mean and Process Dispersion

A statistical quality control chart is widely recognized as a potentially powerful tool that is frequently used in many manufacturing and service industries to monitor the quality of the product or manufacturing processes. In this paper, we propose new synthetic control charts for monitoring the process mean and the process dispersion. The proposed synthetic charts are based on ranked set sampling (RSS), median RSS (MRSS), and ordered RSS (ORSS) schemes, named synthetic‐RSS, synthetic‐MRSS, and synthetic‐ORSS charts, respectively. Average run lengths are used to evaluate the performances of the control charts. It is found that the synthetic‐RSS and synthetic‐MRSS mean charts perform uniformly better than the Shewhart mean chart based on simple random sampling (Shewhart‐SRS), synthetic‐SRS, double sampling‐SRS, Shewhart‐RSS, and Shewhart‐MRSS mean charts. The proposed synthetic charts generally outperform the exponentially weighted moving average (EWMA) chart based on SRS in the detection of large mean shifts. We also compare the performance of the synthetic‐ORSS dispersion chart with the existing powerful dispersion charts. It turns out that the synthetic‐ORSS chart also performs uniformly better than the Shewhart‐R, Shewhart‐S, synthetic‐R, synthetic‐S, synthetic‐D, cumulative sum (CUSUM) ln S², CUSUM‐R, CUSUM‐S, EWMA‐ln S², and change point CUSUM charts for detecting increases in the process dispersion. A similar trend is observed when the proposed synthetic charts are constructed under imperfect RSS schemes. Illustrative examples are used to demonstrate the implementation of the proposed synthetic charts. Copyright © 2014 John Wiley & Sons, Ltd.     

Multivariate EWMA control charts using individual observations for process mean and variance monitoring and diagnosis

Most multivariate control charts in the literature are designed to detect either mean or variation shifts rather than both. A simultaneous use of the Hotelling T ² and |S| control charts has been proposed but the Hotelling T ² reacts to mean shifts, dispersion changes, and changes of correlations among responses. The combination of two multivariate control charts into one chart sometimes loses the ability to provide detailed diagnostic information when a process is out-of-control. In this research a new multivariate control chart procedure based on exponentially weighted moving average (EWMA) statistics is proposed to monitor process mean and variance simultaneously to identify proper sources of variations. Two multivariate EWMA control charts using individual observations are proposed to achieve a quick detection of mean or variance shifts or both. Simulation studies show that the proposed charts are capable of identifying appropriate types of shifts in terms of correct detection percentages. A manufacturing example is used to demonstrate how the proposed charts can be properly set-up based on average run length values via simulations. In addition, correct detection rates of the proposed charts are explored.     

The Run Sum Hotelling's χ2 Control Chart with Variable Sampling Intervals

Control charts are widely used for process monitoring and quality control in manufacturing industries. Implementing variable sampling interval (VSI) control schemes on control charts rather than traditional fixed sampling interval procedure can significantly improve the control chart's efficiency. In this paper, the VSI run sum (RS) Hotelling's χ² chart is proposed. The optimal scores and parameters of the proposed chart are determined using an optimization technique to minimize the following: (i) out‐of‐control average time to signal (ATS); (ii) adjusted ATS (AATS), when the exact shift size can be specified; (iii) expected ATS; or (iv) expected AATS, when the exact shift size cannot be specified. The Markov chain method is used to evaluate the zero‐state ATS and expected ATS, and steady‐state AATS and expected AATS of the proposed chart. The results show that the VSI RS Hotelling's χ² chart significantly outperforms the standard RS Hotelling's χ² chart and the former also performs well compared with other competing charts. By adding more scoring regions, the efficiency of the VSI RS Hotelling's χ² chart can be further enhanced. An illustrative example using data from a manufacturing process is presented to demonstrate the application of the VSI RS Hotelling's χ² chart. The application of the proposed chart in a quality improvement program can be extended to management and service industries. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Economic statistical design of a T2 control chart with double warning lines

Recent studies have shown that enhancing the common T² control chart by using variable sample sizes (VSS) and variable sample intervals (VSI) sampling policies with a double warning line scheme (DWL) yields improvements in shift detection times over either pure VSI or VSS schemes in detecting almost all shifts in the process mean. In this paper, we look at this problem from an economical perspective, certainly at least as an important criterion as shift detection time if one considers what occurs in the industry today. Our method is to first construct a cost model to find the economic statistical design (ESD) of the DWL T² control chart using the general model of Lorenzen and Vance (Technometrics 1986; 28 :3–11). Subsequently, we find the values of the chart parameters which minimize the cost model using a genetic algorithm optimization method. Cost comparisons of Fixed ratio sampling, VSI, VSS, VSIVSS with DWL, and multivariate exponentially weighted moving average (MEWMA) charts are made, which indicate the economic efficacy of using either VSIVSS with DWL or MEWMA charts in practice if cost minimization is of interest to the control chart user. Copyright © 2010 John Wiley & Sons, Ltd.