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.     

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.     

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.     

Directionally sensitive MCUSUM mean charts

In the context of a disease outbreak detection, a prime interest is to only detect increases in the process mean. It is thus desirable to have a directionally sensitive multivariate chart that can effectively detect either increases or decreases in the process mean vector. In this paper, with a suitable transformation that truncates multivariate observations either above or below the process mean vector, we propose one-sided and two one-sided MCUSUM charts for monitoring the mean of a multivariate normal process. Among the proposed charts, the one-sided MCUSUM charts are directionally sensitive, while the two one-sided MCUSUM charts are directionally insensitive. In addition, the fast initial response feature is also incorporated into the proposed charts to enhance their sensitivities against initial process shifts. The run length characteristics of these control charts are computed with the Monte Carlo simulation. Based on the run length comparisons, it is found that the proposed charts are more sensitive than the existing charts when detecting moderate-to-large shifts in the process mean. The proposed charts are also applied on real datasets to support the theory.     

New CUSUM and dual CUSUM mean charts

The CUSUM (C) charts are well recognized as a potentially advanced process monitoring tools because of their sensitivity against small and moderate shifts. In this paper, we first improve the sensitivity of the Brownian motion–based C (BC) chart with an appropriate transformation, named new BC (NBC) chart, for monitoring moderate and large shifts in the mean of a normal process. Then, using the control charting structure of the Crosier C (CC) chart, we propose the NBCC (NBC with CC structure) chart. In addition, for efficiently detecting a mean shift within an interval, dual version of these control charts are also proposed, named the dual NBC (DNBC) and dual NBCC (DNBCC) charts. Moreover, the fast initial response feature is also incorporated into the proposed charts. Using the Monte Carlo simulation, the run length properties of the proposed charts are computed. The run length performances of the existing and proposed charts are compared using the extra quadratic loss and integral relative average run length as performance criterion. It turns out that the NBC and NBCC (DNBC and DNBCC) charts are uniformly more sensitive than the C, CC, and NBC (dual C and dual CC) charts when detecting the mean shifts in small, moderate, and large intervals, where the DNBCC chart outperforms all considered charts. The proposed charts are also applied on real data sets to support the proposed theory.     

Memory-type control charts with multiple auxiliary information for process mean

Memory-type auxiliary-information-based (AIB) control charts are very effective in detecting small-to-moderate shifts in the process mean. In this study, we first develop a unique uniformly minimum variance unbiased estimator of the process mean that requires information on the study variable as well as on several correlated auxiliary variables. Then, based on this estimator, adaptive and nonadaptive CUSUM and EWMA charts are developed with either fixed or variable sampling interval for monitoring the process mean, namely, the multiple AIB (MAIB) charts. The proposed charts encompass existing charts with or without the auxiliary information. The run length characteristics of the proposed charts are computed with the Monte Carlo simulations when sampling from a multivariate normal distribution. Based on the run length comparisons, it is found that the MAIB charts are uniformly and substantially more sensitive than the AIB charts when monitoring the process mean. Real datasets are also considered to explain the implementation of the MAIB charts.     

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.     

Memory-type t charts with multiple auxiliary information for the process mean

Multiple auxiliary information-based (MAIB) memory-type $t$ charts are proposed with fixed and variable sampling intervals for an improved monitoring of the process mean, which include adaptive/nonadaptive cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) $t$ charts. These control charts are constructed based on a unique uniformly minimum variance unbiased estimator of the process mean that requires information on a study variable as well as on several correlated auxiliary variables. The Monte Carlo simulation technique is used to compute the run length characteristics of the proposed charts when sampling from a multivariate normal distribution. The run length comparisons show that the proposed MAIB-$t$ charts outperform their existing auxiliary information based (AIB) and non-AIB $t$ charts, where the normalizing transformation is used for all considered $t$ charts in order to have uniformity in the comparisons. A real data application is also given to support the proposed theory.     

Enhanced EWMA charts for monitoring the process coefficient of variation

The coefficient of variation (CV) is an important quality characteristic when the process variance is a function of the process mean for a production process. In this paper, we develop an auxiliary information–based (AIB) estimator for estimating the squared CV, along with its approximated mean and variance. This estimator is then used to devise new one-sided EWMA charts for monitoring the increases or decreases in the squared CV of a normal process, named the AIB-EWMA CV charts. In addition, the sensitivities of these control charts are also enhanced with the fast initial response feature. The Monte Carlo simulation method is used to compute the run length characteristics of the proposed CV charts. Based on detailed run length comparisons, it is found that the proposed AIB-EWMA CV charts are uniformly and substantially better than the existing EWMA CV charts when detecting different kinds of upward/downward shifts in the squared CV. The proposed charts are also applied to a real dataset to support the proposed theory.     

Double sampling hotelling's T2 charts

Two double sampling T² charts are discussed. They only differ in how the second sample is used to suggest to the practitioner the state of the process. An optimal method using a genetic algorithm is given for designing these charts based on the average run length (ARL). An analytical method is used to determine run length performance of the chart. Comparisons are made with various other control charting procedures. Some recommendations are given. Copyright © 2007 John Wiley & Sons, Ltd.     

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 control chart for monitoring the lognormal process variation using repetitive sampling

Control charts are developed to make the specific quality measures for a successful production process and follow normal distribution behaviors. But some real-life practices do not match such practices and exhibit some positively skewed behavior like lognormal distribution. The present study has considered this situation and proposed a monitoring control chart based on lognormal process variation using a repetitive sampling scheme. This concept proved better for detecting shifts as quickly as possible, and compared with the existing concept, results are elaborated through extensive tables. The average run lengths and standard deviations of the run lengths are being used as a performance evaluation measures and computed by using Monte Carlo simulations performed in R language. A real-life situation has been discussed in the example section to strengthen the proposed control chart concept in a real-life situation.     

Economic design of Xbar control charts with continuously variable sampling intervals

In the context of economic design, a two‐stage model is proposed that utilizes continuously variable sampling intervals. Specifically, each successive sampling interval is determined by the extremity of the latest sample. Modeling the situation as a Markov chain, the hourly cost is developed for any arbitrary set of design parameters. This proposed approach is found to be more economically desirable, possessing a smaller average out‐of‐control production time, when compared with a standard two‐stage approach where the sampling interval alternates between two fixed values. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

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.     

A CUSUM scheme with variable sample sizes and sampling intervals for monitoring the process mean and variance

The adaptive control feature and CUSUM chart are two monitoring schemes that are much more effective than the traditional static Shewhart chart in detecting process shifts in mean and variance. However, the designs and analyses of the adaptive CUSUM chart are mathematically intractable and the operation is very laborious. This article proposes a VSSI WLC scheme, which is a weighted‐loss‐function‐based CUSUM (WLC) scheme using variable sample sizes and sampling intervals (VSSI). This scheme detects the two‐sided mean shift and increasing standard deviation shift based on a single statistic WL (the weighted loss function). Most importantly, the VSSI WLC scheme is much easier to operate and design than a VSSI CCC scheme which comprises three individual CUSUM charts (two of them monitoring the increasing and decreasing mean shifts and one monitoring the increasing variance shift). Overall, the VSSI WLC scheme is much more effective than the static &S charts (by 72.36%), the VSSI &S charts (by 30.97%) and the static WLC scheme (by 50.94%) for detection. It is even more effective than the complicated VSSI CCC scheme for most cases. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

An enhanced approach for the progressive mean control charts: A discussion and comparative analysis

Recently, a new double progressive mean (DPM) control chart has been proposed in the literature of statistical process control. In the said proposal, an important term is missing in the variance expression of the DPM statistic that affects the detection ability of the proposed chart. In this study, we have derived and provided the correct version of the said variance along with its corresponding control limits. The run length profiles of the DPM chart are investigated, and the results are updated for the new version of the limits. Moreover, a sensitivity analysis between DPM and progressive mean charts based on the different choices of the design parameter is also included in this study. It is revealed that the revised version offers even more efficient outcomes than the previous ones. In addition, a real dataset application is also presented for practical considerations of the refined version.     

New EWMA charts for process variance

The EWMA chart is effective in detecting small shifts in the process mean or process variance. Numerous EWMA charts for the process variance have been suggested in the literature. In this article, new one-sided and two-sided EWMA charts are developed for monitoring the variance of a normal process. In developing these new EWMA charts, first, new unbiased estimators of the process variance are developed, followed by incorporating the developed estimators into the new EWMA charts' statistics. The Monte Carlo simulation method is adopted to evaluate the zero-state and steady-state run-length performances of the proposed EWMA variance charts, in comparison with that of three existing EWMA variance charts and the weighted adaptive CUSUM variance chart. The findings reveal that the proposed charts generally perform better than the existing charts. An example of application is given to show the implementation of the proposed and existing charts in detecting increases or decreases in the process variance.