An adaptive exponentially weighted moving average chart for the mean with variable sampling intervals

The AEWMA control chart is an adaptive EWMA (exponentially weighted moving average) type chart that combines the Shewhart and the classical EWMA schemes in a smooth way. To improve the detection performance of the FSI (fixed sampling interval) AEWMA control chart 7 in terms of the ATS(average time to signal), this paper proposes a new VSI (variable sampling interval) AEWMA control chart. A Markov chain approach is used to calculate the ATS values of the new VSI AEWMA control chart, and comparative results show that the proposed control chart performs better than the standard FSI AEWMA control chart and than other VSI control charts over a wide range of shifts.     

An economic design model for X¯ charts with random sampling policies

Recent literature contains many papers on the study of variable (but deterministic) sampling interval (VSI) policies for X¯ control charts. VSI policies perform better than their corresponding classical fixed-sampling-interval (FSI) policies. Results are also available in the literature for an adaptive VSI policy, an enhancement of the VSI policy, where the sample size also changes with the sampling interval. However, for many industrial (especially continuous) processes where X¯ charts are used, maintaining deterministic time intervals between samples is not possible owing to inherent randomness of the processes. In this paper we propose a new variable sampling policy, named random sampling policy (RSP), for X¯ charts in which the sampling intervals and the sample sizes (considered a function of the sampling intervals) are treated as random variables. An economic modeling framework for design of RSP with run rules is developed. Within the proposed modeling framework, special cases of RSP, such as VSI and FSI policies with and without run rules, can also be evaluated. An application of the economic model is demonstrated through design of RSP parameters for two numerical example problems.     

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.     

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.     

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.     

Nonparametric signed‐rank control charts with variable sampling intervals

Variable sampling interval (VSI) charts have been proposed in the literature for normal theory (parametric) control charts and are known to provide performance enhancements. In the VSI setting, the time between monitored samples is allowed to vary depending on what is observed in the current sample. Nonparametric (distribution‐free) control charts have recently come to play an important role in statistical process control and monitoring. In this paper a nonparametric Shewhart‐type VSI control chart is considered for detecting changes in a specified location parameter. The proposed chart is based on the Wilcoxon signed‐rank statistic and is called the VSI signed‐rank chart. The VSI signed‐rank chart is compared with an existing fixed sampling interval signed‐rank chart, the parametric VSI ‐chart, and the nonparametric VSI sign chart. Results show that the VSI signed‐rank chart often performs favourably and should be used.     

Monitoring autocorrelated processes using variable sampling schemes at fixed‐times

Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. A recent development in control charts is the use of adaptive control charts, i.e. the variable sampling interval (VSI) and variable sampling size charts. This paper extends this idea to the autocorrelated process. We consider a time series model which is a first‐order autoregressive process plus a random error. With variable intervals, the sampling time may be inconvenient, so using only two intervals, referred to as ‘variable sampling interval at fix times’ makes the method easier to use in practice. The sampling rate can also be adjusted by the number of samples collected, VSRFT, for ‘variable sampling rate at fixed times’. We study what we call ‘variable sampling at fixed times’, VSFT, which includes both VSIFT and VSRFT schemes, using a Markov chain model and integral equations. We show that our methods detect process shifts faster, on average, than fixed sampling X‐bar charts, and at least comparable detection ability with the less practical VSI charts. Copyright © 2007 John Wiley & Sons, Ltd.     

Monitoring the Coefficient of Variation Using a Variable Sampling Interval Control Chart

The coefficient of variation (CV) is a quality characteristic that has several applications in applied statistics and is receiving increasing attention in quality control. Few papers have proposed control charts that monitor this normalized measure of dispersion. In this paper, an adaptive Shewhart control chart implementing a variable sampling interval (VSI) strategy is proposed to monitor the CV. Tables are provided for the statistical properties of the VSI CV chart, and a comparison is performed with a Fixed Sampling Rate Shewhart chart for the CV. An example illustrates the use of these charts on real data gathered from a casting process. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Optimal Design of VSI ―X Control Charts for Monitoring Correlated Samples

This paper develops an economic design of variable sampling interval (VSI)―X control charts in which the next sample is taken sooner than usual if there is an indication that the process is off‐target. When designing VSI―X control charts, the underlying assumption is that the measurements within a sample are independent. However, there are many practical situations that violate this hypothesis. Accordingly, a cost model combining the multivariate normal distribution model given by Yang and Hancock with Bai and Lee's cost model is proposed to develop the design of VSI charts for correlated data. An evolutionary search method to find the optimal design parameters for this model is presented. Also, we compare VSI and traditional ―X charts with respect to expected cost per unit time, utilizing hypothetical cost and process parameters as well as various correlation coefficients. The results indicate that VSI control charts outperform the traditional control charts for larger mean shift when correlation is present. In addition, there is a difference between the design parameters of VSI charts when correlation is present or absent. Copyright © 2005 John Wiley & Sons, Ltd.     

An Adaptive CUSUM Chart with Single Sample Size for Monitoring Process Mean and Variance

This article proposes an adaptive absolute cumulative sum chart (called the adaptive ACUSUM chart) for statistical process control. The new development includes the variable sampling interval (VSI), variable sample size (VSS) and VSS and interval (VSSI) versions, all of which are highly effective for monitoring the mean and variance of a variable x by inspecting the absolute sample shift (where μ₀ is the in‐control mean or target value of x). While the adaptive ACUSUM chart is a straightforward extension of the ABS CUSUM chart developed by Wu, et al., it is much more effective than all other adaptive CUSUM charts. Noteworthily, the superiority of VSI ACUSUM chart over the best adaptive CUSUM chart in literature is about 35% from an overall viewpoint. Moreover, the design and implementation of the adaptive ACUSUM chart are much simpler than that of all other adaptive CUSUM schemes. All these desirable features of the adaptive ACUSUM chart may be attributable to the use of a single sample size (n = 1). Another quite interesting finding is that the simpler VSI ACUSUM chart works equally well as the more complicated VSSI ACUSUM chart. Copyright © 2012 John Wiley & Sons, Ltd.     

Guidelines for the application of adaptive control charting schemes

A comprehensive performance study and comparison of several adaptive statistical process control procedures is presented. These adaptive control chart procedures are modifications to standard Shewhart control charts that include changing the sampling interval, the sample size or both according to rules based on the value of the sample statistic. Adaptive control techniques are known to improve the performance of the standard Shewhart control charts. In this paper we develop a four-state adaptive sample size control chart and several variations of a three-state combined adaptive sample size and sampling interval control chart. We then compare these new schemes with the previously developed schemes, the two-state adaptive sampling interval, the two-state adaptive sample size and two-state combined adaptive sample size and sampling interval control chart, three-state adaptive sample size control chart and non-adaptive Shewhart control charts. These results show that the addition of the third and fourth states on the adaptive control chart schemes improve the control chart performance; however, the improvement is relatively modest.     

A survey on multivariate adaptive control charts: Recent developments and extensions

In the world of business, quality improvement is of high importance for the manufacturing industries. Statistical process control via control charts provides an online monitoring of the product's characteristic. The adaptive feature is being widely used in the design parameters of a control chart, which allows at least one of them to change during the process monitoring. Specifically, a control chart is considered adaptive if at least one of the chart's parameters (sample size, sampling interval, or control limit coefficient) is allowed to change in real time on the basis of the actual values of the sample statistics. In this paper, recent developments in the design of multivariate adaptive control schemes are presented and discussed.     

Adaptive ―X Control Charts with Sampling at Fixed Times

The idea of a variable sampling interval with sampling at fixed times (VSIFT) has been presented by Reynolds. This paper extends this idea to the other two adaptive ―X charts: the variable sampling rate with sampling at fixed times (VSRFT) ―X chart and the variable parameters with sampling at fixed times (VPFT) ―X chart. The VSIFT, VSRFT and VPFT ―X charts are inclusively called the adaptive with sampling at fixed times (AFT) ―X charts in this paper. The control scheme and the design issue are described and discussed for each of the AFT ―X charts. A comparative study shows that the AFT ―X charts have almost the same detection ability as the traditional adaptive ―X charts. However, from the practical viewpoint, the AFT ―X charts are considered to be more convenient to administer than the traditional adaptive ―X charts. Overall, this paper advances the application of ‘sampling at fixed times’ to the adaptive ―X control charts. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

VSI Q控制图应用研究

VSI Q控制图可以解决经典休哈特控制图在样本数据缺乏的情况下不能应用的问题,并且可以加快检出速度.具体方法是将样本数据转换成服从标准正态分布的统计数据,再利用可变抽样区间技术确定抽样间隔时间.应用研究证明,VSI Q控制图在缺乏样本数据时也能用于监控生产过程,快速检测出异常.     

An EWMA mean chart with auxiliary information

In this paper, we show that a recently proposed auxiliary information-based (AIB) adaptive EWMA (AE) chart is sensitive (not robust) to the changes in the mean of an auxiliary variable when monitoring the changes in the mean of a quality variable, called the AIB-AE chart. To circumvent the weakness of the AIB-AE chart, we develop a new AIB estimator for the mean of a quality variable that is slightly robust to the changes in the mean of an auxiliary variable. Based on this newly developed estimator, a new AIB EWMA (AIB-E) chart is proposed for monitoring the mean of a quality variable. The zero-state and steady-state average run-length profiles of the AIB-AE and AIB-E charts are estimated with Monte Carlo simulations. It is found that the AIB-E chart is not only slightly robust to the changes in the mean of an auxiliary variable, but it also outperforms the AIB-AE chart when detecting small shifts in the mean of a quality variable. Illustrative examples are also included in this study to demonstrate the implementation of the existing and proposed AIB charts.     

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.     

Optimal Design of the Adaptive Sample Size and Sampling Interval np Control Chart

Recent research has shown that the adaptive control charts are quicker than the traditional static charts in detecting process shifts. This paper develops the algorithm for the optimization designs of the adaptive np control charts for monitoring the process fraction non‐conforming p. It includes the variable sample size chart, the variable sampling interval chart, and the variable sample size and sampling interval chart. The performance of the adaptive np charts is measured by the average time to signal under the steady‐state mode, which allows the shift in p to occur at any time, even during the sampling inspection. By studying the performance of the adaptive np charts systematically, it is found that they do improve effectiveness significantly, especially for detecting small or moderate process shifts. Copyright © 2004 John Wiley & Sons, Ltd.     

The Design of the Variable Sampling Interval Generalized Likelihood Ratio Chart for Monitoring the Process Mean

This paper considers the problem of monitoring a normally distributed process variable in which some sustained shift in the mean may occur. A generalized likelihood ratio (GLR) control chart with a variable sampling interval (VSI) scheme is proposed to quickly detect a wide range of such shifts. The performance of the VSI GLR chart is evaluated, and the results show that using the VSI feature can greatly reduce the expected detection time. Design methodology for the VSI GLR chart is provided so that practitioners can easily use this chart in applications. An illustrative example is given to explain how to apply the VSI GLR chart. Copyright © 2013 John Wiley & Sons, Ltd.