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.     

Adaptive multivariate double sampling and variable sampling interval Hotelling's T2 charts

In this article, two adaptive multivariate charts, which combine the double sampling (DS) and variable sampling interval (VSI) features, called the adaptive multivariate double sampling variable sampling interval T² (AMDSVSI T²) and the adaptive multivariate double sampling variable sampling interval combined T² (AMDSVSIC T²) charts, are proposed. The real purpose of using the proposed charts is to provide flexibility by enabling the sampling interval length of the DS T² chart to be varied so that the chart's sensitivity can be enhanced. The fundamental difference between the two proposed charts is that when a second sample is taken, the AMDSVSI T² chart uses the information of the combined sample mean vectors while the AMDSVSIC T² chart uses the information of the combined T² statistics, in deciding about the process status. This research is motivated by existing combined DS and VSI charts in the literature, which show convincing performance improvement over the standard DS chart. Consequently, it is believed that adopting this existing approach in the multivariate case will enable superior multivariate DS charts to be proposed. Numerical results show that the proposed charts outperform the existing standard T² and other adaptive multivariate charts, in detecting shifts in the mean vector, for the zero‐state and steady‐state cases. The performances of both charts when the shift sizes in the mean vector are unknown are also measured. The application of the AMDSVSI T² chart is illustrated with an example.     

The Effect of Autocorrelation on the Hotelling T2 Control Chart

One of the basic assumptions for traditional univariate and multivariate control charts is that the data are independent in time. For the latter, in many cases, the data are serially dependent (autocorrelated) and cross‐correlated because of, for example, frequent sampling and process dynamics. It is well known that the autocorrelation affects the false alarm rate and the shift‐detection ability of the traditional univariate control charts. However, how the false alarm rate and the shift‐detection ability of the Hotelling T² control chart are affected by various autocorrelation and cross‐correlation structures for different magnitudes of shifts in the process mean is not fully explored in the literature. In this article, the performance of the Hotelling T² control chart for different shift sizes and various autocorrelation and cross‐correlation structures are compared based on the average run length using simulated data. Three different approaches in constructing the Hotelling T² chart are studied for two different estimates of the covariance matrix: (i) ignoring the autocorrelation and using the raw data with theoretical upper control limits; (ii) ignoring the autocorrelation and using the raw data with adjusted control limits calculated through Monte Carlo simulations; and (iii) constructing the control chart for the residuals from a multivariate time series model fitted to the raw data. To limit the complexity, we use a first‐order vector autoregressive process and focus mainly on bivariate data. © 2014 The Authors. Quality and Reliability Engineering International Published by 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.     

A New Combined Shewhart–Cumulative Sum S Chart for Monitoring Process Standard Deviation

The combined application of a Shewhart chart and cumulative sum (CUSUM) control chart is an effective tool for the detection of all sizes of process shifts as the scheme combines the advantages of a CUSUM at detecting small to moderate shifts and Shewhart for the quick detection of very large shifts. This article proposes new combined Shewhart–CUSUM S charts based on the extreme variations of ranked set sampling technique, for efficient monitoring of changes in the process dispersion. Using Monte Carlo simulations, the combined scheme is designed to minimize the average extra quadratic loss over the entire process shift domain. The results show that the combined Shewhart–CUSUM S charts uniformly outperform several other procedures for detecting increases and decreases in the process variability. Moreover, the proposed scheme can detect changes that are small enough to escape the Shewhart S chart or fairly large to escape detection by the CUSUM S chart. Numerical example is given to illustrate the practical application of the proposed scheme using real industrial data. Copyright © 2015 John Wiley & Sons, Ltd.     

The double sampling range chart

The R chart and the S² chart are the usual charts for monitoring process dispersion; however, the double sampling (DS) scheme has only be used with the S² chart. The difficulty in obtaining the properties of the DS R chart might explain the lack of papers dealing with this type of DS chart. The S² chart has the advantage of being more efficient than the R chart, but the same is not always observed with their DS versions. Depending on the size of the 2 samples, the DS R chart performs better. The trade‐off between operational simplicity and power of detection might lead the practitioner to choose the DS R chart, even with the DS S² chart signaling faster.     

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.     

M‐ATTRIVAR: An attribute‐variable chart to monitor multivariate process means

In this paper, an attribute‐variable control chart, namely, M‐ATTRIVAR, is introduced to monitor possible shifts in a vector of means. The monitoring starts using an attribute chart (classifying the units as approved or not using a gauge) and continues in such a way until a warning signal is given, shifting the control to a variable chart for the next sampling. If the variable chart does not confirm the warning, the monitoring returns to an attribute control. Otherwise, the monitoring remains with the variable chart. Whenever any of the charts (attribute or variable) signals an alarm, the control scheme triggers an alarm. The main advantage of this new proposal is the possibility of judging the state of the process only by the attribute chart most of the time (normally more economical and faster). The performance of the M‐ATTRIVAR control chart is compared versus the main competitor (T² control chart) in terms of performance detection (out‐of‐control average run length) but also economically (average sampling cost). The M‐ATTRIVAR is always cheaper than T², and in many scenarios, it detects quicker process shifts than the T² control chart. A numerical example illustrates a practical situation.     

Combined Application of Shewhart and Cumulative Sum R Chart for Monitoring Process Dispersion

This study analyzes the performance of combined applications of the Shewhart and cumulative sum (CUSUM) range R chart and proposes modifications based on well‐structured sampling techniques, the extreme variations of ranked set sampling, for efficient monitoring of changes in the process dispersion. In this combined scheme, the Shewhart feature enables quick detection of large shifts from the target standard deviation while the CUSUM feature takes care of small to moderate shifts from the target value. We evaluate the numerical performance of the proposed scheme in terms of the average run length, standard deviation of run length, the average ratio average run length, and average extra quadratic loss. The results show that the combined scheme can detect changes in the process that were small or large enough to escape detection by the lone Shewhart R chart or CUSUM R chart, respectively. We present a comparison of the proposed schemes with several dispersion charts for monitoring changes in process variability. The practical application of the proposed scheme is demonstrated using real industrial data. Copyright © 2014 John Wiley & Sons, Ltd.     

Hotelling's T2 control chart with adaptive sample sizes

Some quality control schemes have been developed when several related quality characteristics are to be monitored: simultaneous X¯ charts, Hotelling's T² chart, multivariate CUSUM and multivariate EWMA. Hotelling's T² control chart has the advantage of its simplicity but it is slow in detecting small process shifts. The latest developments in variable sample sizes for univariate control charts are applied in this paper to define an adaptive sample sizes T² control chart. As occurs in the univariate case the ARL improvements are very important particularly for small process shifts. An example is given to illustrate the use of the proposed scheme.     

An Efficient Nonparametric EWMA Wilcoxon Signed‐Rank Chart for Monitoring Location

The statistical performance of traditional control charts for monitoring the process shifts is doubtful if the underlying process will not follow a normal distribution. So, in this situation, the use of a nonparametric control charts is considered to be an efficient alternative. In this paper, a nonparametric exponentially weighted moving average (EWMA) control chart is developed based on Wilcoxon signed‐rank statistic using ranked set sampling. The average run length and some other associated characteristics were used as the performance evaluation of the proposed chart. A major advantage of the proposed nonparametric EWMA signed‐rank chart is the robustness of its in‐control run length distribution. Moreover, it has been observed that the proposed version of the EWMA signed‐rank chart using ranked set sampling shows better detection ability than some of the competing counterparts including EWMA sign chart, EWMA signed‐rank chart, and the usual EWMA control chart using simple random sampling scheme. An illustrative example is also provided for practical consideration. Copyright © 2016 John Wiley & Sons, Ltd.     

EWMA Control Chart for Poisson–Exponential Lifetime Distribution Under Type I Censoring

Exponentially weighted moving average (EWMA) control charts are widely used for the detection of small shifts as opposed to Shewhart charts, which are commonly used for the detection of large‐size shifts in a process. Many interesting features of EWMA charts are available in literature mainly for complete data. This study intends to investigate the EWMA control charts under Type‐I censoring for Poisson–exponential distributed lifetimes. The two commonly used sampling schemes, that is, simple random sampling and rank set sampling, are used in this study. The monitoring of mean level shifts using censored data is of a great interest in many applied problems. The idea of conditional expected values is employed in the monitoring of small mean level shifts in the current study. The performance of the EWMA charts is evaluated using the average run length extra quadratic loss and performance comparison index measures. The optimum sample‐size comparisons for the specified and unspecified parameter are also part of this study. Moreover, an illustrative example and a case study for practical considerations are also discussed. It is observed that varying censoring rates affect the performance of the chart depending upon the type of sampling scheme and the amount of shifts. Copyright © 2015 John Wiley & Sons, Ltd.     

Multivariate double sampling |S| charts for controlling process variability

A new multivariate statistical process variability control scheme, the multivariate double sampling (MDS) |S| control chart scheme for controlling shifts in a variance–covariance matrix, is proposed. An MDS |S| chart is a double-stage extension of a single-stage control chart based on the sample generalized variance. The statistical efficiency in terms of average run length of the MDS |S| charts is compared with that of traditional and adaptive sample size control charts based on the sample generalized variance for a bivariate case. The results of the comparison show that the MDS |S| charts provide a significant improvement in efficiency for detecting shifts of any range over these schemes. The performance of the MDS |S| charts in terms of detecting power is also compared with that of the modified likelihood ratio test, the sum of standardized variances for principal components charts and a decomposition procedure. The results of the comparison show that for certain out-of-control situations, the performance of the MDS |S| chart is among those of the best schemes.     

Multivariate multiple sampling charts

A new multivariate statistical process control scheme, the Multivariate Multiple Sampling (MMS) control chart scheme, is proposed in this paper. A MMS chart is a multivariate extension of a double sampling X-bar control chart with at least two sampling stages. In the paper, a statistical design optimization procedure to design the MMS chart is presented and the performance of the MMS chart is investigated. The statistical efficiency in terms of average run length of the MMS chart is compared with that of the Hotelling chart both with and without variable sampling schemes, a multivariate CUMulative SUM (CUSUM) chart, and a multivariate Exponentially Weighted Moving Average (EWMA) chart. The ability of the MMS chart to handle the worst-case scenario is also investigated and compared with that of the multivariate EWMA and CUSUM charts. The results of the investigation show that even with only two sampling stages, the MMS chart provides an improvement in efficiency in detecting small shifts over the Hotelling chart without variable sampling schemes. When the number of sampling stages is equal to two, the MMS chart is better in detecting large shifts and the multivariate EWMA and CUSUM charts are better in detecting relatively small shifts. As the number of sampling stages is increased beyond two, the improvement in sensitivity of the MMS chart in detecting the small shifts increases. When the number of sampling stages ≥3, the MMS chart begins to give a better performance than a Hotelling chart with a variable sampling scheme for small shifts and is also better than a multivariate EWMA chart for both small and large shifts. As the number of sampling stages ≥4, the MMS chart begins to give a better performance than a multivariate CUSUM chart for both small and large shifts. The results of the investigation also show that the MMS chart outperforms the multivariate EWMA and CUSUM charts in the worst-case scenario situation.     

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

Exponentially weighted moving average (EWMA) control charts have been widely recognized as an advanced statistical process monitoring tool due to their excellent performance in detecting small to moderate shifts in process parameters. In this paper, we propose a new EWMA control chart for monitoring the process dispersion based on the best linear unbiased absolute estimator (BLUAE) obtained under paired ranked set sampling (PRSS) scheme, which we name EWMA‐PRSS chart. The performance of the EWMA‐PRSS chart is evaluated in terms of the average run length and standard deviation of run length, estimated using Monte Carlo simulations. These control charts are compared with their existing counterparts for detecting both increases and decreases in the process dispersion. It is observed that the proposed EWMA‐PRSS chart performs uniformly better than the EWMA dispersion charts based on simple random sampling and ranked set sampling (RSS) schemes. We also construct an EWMA chart based on imperfect PRSS (IPRSS) scheme, named EWMA‐IPRSS chart, for detecting overall changes in the process variability. It turns out that, with reasonable assumptions, the EWMA‐IPRSS chart outperforms the existing EWMA dispersion charts. A real data set is used to explain the construction and operation of the proposed EWMA‐PRSS chart. Copyright © 2014 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.     

Progressive Mean Control Chart for Monitoring Process Location Parameter

Control charts are widely used for process monitoring. They show whether the variation is due to common causes or whether some of the variation is due to special causes. To detect large shifts in the process, Shewhart‐type control charts are preferred. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are generally used to detect small and moderate shifts. Shewhart‐type control charts (without additional tests) use only current information to detect special causes, whereas CUSUM and EWMA control charts also use past information. In this article, we proposed a control chart called progressive mean (PM) control chart, in which a PM is used as a plotting statistic. The proposed chart is designed such that it uses not only the current information but also the past information. Therefore, the proposed chart is a natural competitor for the classical CUSUM, the classical EWMA and some recent modifications of these two charts. The conclusion of this article is that the performance of the proposed PM chart is superior to the compared ones for small and moderate shifts, and its performance for large shifts is better (in terms of the average run length). Copyright © 2012 John Wiley & Sons, Ltd.     

A synthetic double sampling control chart for process mean using auxiliary information

A control chart is a simple yet powerful tool that is extensively adopted to monitor shifts in the process mean. In recent years, auxiliary‐information–based (AIB) control charts have received considerable attention as these control charts outperform their counterparts in monitoring changes in the process parameter(s). In this article, we integrate the conforming run length chart with the existing AIB double sampling (AIB DS) chart to propose an AIB synthetic DS chart for the process mean. The AIB synthetic DS chart also encompasses the existing synthetic DS chart. A detailed discussion on the construction, optimization, and evaluation of the run length profiles is provided for the proposed control chart. It is found that the optimal AIB synthetic DS chart significantly outperforms the existing AIB Shewhart, optimal AIB synthetic, and AIB DS charts in detecting various shifts in the process mean. An illustrative example is given to demonstrate the implementation of the existing and proposed AIB control charts.     

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.     

A New Exponentially Weighted Moving Average Control Chart for Monitoring the Process Mean

Exponentially weighted moving average (EWMA) quality control schemes have been recognized as a potentially powerful process monitoring tool because of their superior speed in detecting small to moderate shifts in the underlying process parameters. In quality control literature, there exist several EWMA charts that are based on simple random sampling (SRS) and ranked set sampling (RSS) schemes. Recently, a mixed RSS (MxRSS) scheme has been introduced, which encompasses both SRS and RSS schemes, and is a cost‐effective alternative to the RSS scheme. In this paper, we propose new EWMA control charts for efficiently monitoring the process mean based on MxRSS and imperfect MxRSS (IMxRSS) schemes, named EWMA–MxRSS and EWMA–IMxRSS charts, respectively. Extensive Monte Carlo simulations are used to estimate the run length characteristics of the proposed EWMA charts. The run length performances of the suggested EWMA charts are compared with the classical EWMA chart based on SRS (EWMA–SRS). It turns out that both EWMA–MxRSS and EWMA–IMxRSS charts perform uniformly better than the EWMA–SRS chart when detecting all different shifts in the process mean. An application to a real data set is provided as an illustration of the design and implementation of the proposed EWMA chart. Copyright © 2014 John Wiley & Sons, Ltd.