Joint economic design of EWMA control charts for mean and variance

Control charts with exponentially weighted moving average (EWMA) statistics (mean and variance) are used to jointly monitor the mean and variance of a process. An EWMA cost minimization model is presented to design the joint control scheme based on pure economic or both economic and statistical performance criteria. The pure economic model is extended to the economic-statistical design by adding constraints associated with in-control and out-of-control average run lengths. The quality related production costs are calculated using Taguchi’s quadratic loss function. The optimal values of smoothing constants, sampling interval, sample size, and control chart limits are determined by using a numerical search method. The average run length of the control scheme is computed by using the Markov chain approach. Computational study indicates that optimal sample sizes decrease as the magnitudes of shifts in mean and/or variance increase, and higher values of quality loss coefficient lead to shorter sampling intervals. The sensitivity analysis results regarding the effects of various inputs on the chart parameters provide useful guidelines for designing an EWMA-based process control scheme when there exists an assignable cause generating concurrent changes in process mean and variance.     

Comparison of the cuscore, GLRT and cusum control charts for detecting a dynamic mean change

Although statistical process control (SPC) techniques have been focused mostly on detecting constant mean shifts, dynamic and time-varying process changes frequently occur in the monitoring of feedback-controlled and autocorrelated processes. In this research, the performances of cumulative score (Cuscore), generalized likelihood ratio test (GLRT), and cumulative sum (CUSUM) charts in detecting a dynamic mean change that finally approaches a steady-state value are compared. Theoretical results in average run length (ARL) comparison are provided. From the theretical study we find that, when the steady-state value is greater or less than a critical value,Rδ/2+δ/2, the Cuscore and CUSUM charts have a different performance in detecting the mean change. We prove also that the GLRT has the best performance among the three charts in detecting any mean change for which the steady-state value is not equal to δ or δR, when the in-control ARL is large.     

Hotelling’s T charts with variable sample size and control limit

The ideas of variable sampling interval (VSI), variable sample size (VSS), variable sample size and sampling interval (VSSI), and variable parameters (VP) in the univariate case have been successfully applied to the multivariate case to improve the efficiency of Hotelling’s T² chart with fixed sampling rate (FSR) in detecting small process shifts. However, the main disadvantage in using most of these control schemes is an increasing in the complexity due to the adaptive changes in sampling intervals. In this paper, retaining the lengths of sampling intervals constant, a variable sample size and control limit (VSSC) T² chart is proposed and described. The statistical efficiency of the VSSC T² chart in terms of the average time to signal a shift in process mean vector is compared with that of the VP, VSSI, VSS, VSI, and FSR T² charts. From the results of comparison, it shows that the VSSC T² chart for a (very) small shift in the process mean vector gives a better performance than the VSSI, VSS, VSI, and FSR T² charts; meanwhile, it presents a similar performance to the VP T² chart. Furthermore, from the viewpoint of practicability, it is more convenient for administrating the control chart than the VSI, VSSI, and VP T² chart. Thus, it may provide a good option for quick response to small shifts in a multivariate process.     

CUSUM,GLR,GEWMA以及RFCuscore控制图监测稳定过程均值漂移的效果比较(英)

本文给出了累积和控制图(CUSUM)监测稳定过程均值漂移的平均运行长度(ARL)的区间估计,并采用数字模拟的方法对CUSUM,GLR,GEWMA以及RFCuscore四种控制图监测稳定过程均值漂移的效果进行比较,结果显示CUSUM效果最好.     

Using one EWMA chart to jointly monitor the process mean and variance

Two control charts are usually used to monitor the process mean and variance separately. The mean is monitored using the [`(X)]{\bar{{X}}} chart while the variance using either the standard deviation, S chart, or the range, R chart. Recently, numerous single variable charts are proposed to jointly monitor the mean and variance. Most approaches transform the sample mean and sample variance into two statistics, each having a standard scale, and either plotting them on the same chart or combining them into a single statistic to be plotted on a chart. The R chart is more widely used than the S chart but no attempt is made to combine the [`(X)]{\bar{{X}}} and R charts in the construction of a single variable chart and to study its properties and performance. In this paper, we transform the [`(X)]{\bar{{X}}} and R statistics into two standard normal random variables, used in the computation of two corresponding exponentially weighted moving average (EWMA) statistics, which are then merged into a single plotting statistic for the proposed chart, called the EWMA [`(X)]-R{\bar{{X}}-R} chart.     

Economic design of variable parameters <Emphasis Type="Italic">X̄</Emphasis> control charts for processes with fuzzy mean shifts

Modern studies have shown that the X? control charts with variable parameters (VPs) detect process shifts faster than the traditional X? control charts. This article developed the economic design of the VP X? control chart to determine the values of the design parameters of the chart. However, this study, different from previous studies, was focused on the process that is subject to a disturbing cause, and the occurrence of the cause can result in a fuzzy mean shift (ie the magnitude of the mean shift could not be recognized exactly). The issue of economically selecting the design parameters for the chart was firstly formulated as a mathematical programming model with a fuzzy objective function that could cope with fuzzy number type of mean shift. A fuzzy-simulation-based genetic algorithm was then employed to search for the optimal values of the design parameters from the model. An industrial example was provided to illustrate the solution procedure, and was used for comparison between the VP and the traditional X? chart. Effects of model parameters on the solution of the economic design were also discussed.     

控制过程方差的CUSUMQ图及其性质

崔恒建．控制过程方差的ＣＵＳＵＭＱ图及其性质．数理统计与管理，１９９８，１７（４），３３～３８．Ｑｕｓｅｎｂｅｒｒｙ（１９９５）基于样本方差的标准化变换Φ－１［Ｈｎ－１（（ｎ－１）Ｓ２／σ２０）］提出了控制过程方差的累积和（ＣＵＳＵＭ）Ｑ控制图。本文我们描述了在控制过程方差变化中这种ＣＵＳＵＭＱ控制图的性质，并将控制图的设计方法用到单边及双边的ＣＵＳＵＭＱ图，说明它几乎是最优的。而且我们发现在控制过程方差的微小变化时，设计的ＣＵＳＵＭＱ图的性能要优于基于ｌｏｇ（Ｓ２）的ＣＵＳＵＭ和ＥＷＭＡ图     

Optimal average sample number of the SPRT chart for monitoring fraction nonconforming

The Sequential Probability Ratio Test (SPRT) control chart is a powerful tool for monitoring manufacturing processes. It is highly suitable for the applications where testing is destructive or very expensive, such as the automobile airbags test. This article studies the effect of the Average Sample Number (ASN) (i.e., the average sample size) on the chart’s performance. A design algorithm is proposed to develop the optimal SPRT chart for monitoring the fraction nonconforming p of Bernoulli processes. By optimizing the ASN and other charting parameters, the average detection speed of the SPRT chart is almost doubled. It is also found that the optimal SPRT chart significantly outperforms the optimal np and binomial CUSUM charts, in terms of Average Number of Defectives (AND), under different combinations of the design specifications. It is observed that the SPRT chart using a relatively smaller ASN and a shorter sampling interval (h) has a higher overall detection effectiveness.     

Distribution-free monitoring of univariate processes

We consider statistical process control (SPC) of univariate processes when observed data are not normally distributed. Most existing SPC procedures are based on the normality assumption. In the literature, it has been demonstrated that their performance is unreliable in cases when they are used for monitoring non-normal processes. To overcome this limitation, we propose two SPC control charts for applications when the process data are not normal, and compare them with the traditional CUSUM chart and two recent distribution-free control charts. Some empirical guidelines are provided for practitioners to choose a proper control chart for a specific application with non-normal data.     

具有可变抽样区间的非正态累积和控制图

当过程存在小波动时,累积和控制图比传统的休哈特控制图监控效果灵敏。为了提高控制图的监控效率,本文针对非正态情形下的累积和控制图进行可变抽样区间设计。首先用Burr分布近似各种非正态分布,构造可变抽样区间的非正态累积和控制图;其次利用马尓可夫链方法计算其平均报警时间;最后研究结果表明, 所设计的可变抽样区间非正态累积和控制图较固定抽样区间的非正态累积和控制图能更好地监控过程的变化。     

Max control chart with adaptive sample sizes for jointly monitoring process mean and standard deviation

This paper develops the two-state and three-state adaptive sample size control schemes based on the Max chart to simultaneously monitor the process mean and standard deviation. Since the Max chart is a single variables control chart where only one plotting statistic is needed, the design and operation of adaptive sample size schemes for this chart will be simpler than those for the joint X? and S charts. Three types of processes including on-target initial, off-target initial and steady-state conditions are considered to evaluate the chart performance. The results of this study show that both two-state and three-state schemes are more efficient than the conventional non-adaptive joint X? and S charts. The three-state procedure is only slightly better than the two-state scheme, and the most dramatic improvement occurs when the two-state scheme is compared with the non-adaptive joint X? and S charts. Moreover, with the ease of implementation, the two-state scheme is likely adequate in most practical applications.     

Variable sampling interval Shewhart control charts for monitoring the multivariate coefficient of variation

In many industrial manufacturing processes, the ratio of the variance to the mean of a quantity of interest is an important characteristic to ensure the quality of the processes. This ratio is called the coefficient of variation (CV). A lot of control charts have been designed for monitoring the CV of univariate quantity in the literature. However, the CV control charts for multivariate quantity have not received much attention yet. In this paper, we investigate a variable sampling interval (VSI) Shewhart control chart for monitoring multivariate CV. The time between two consecutive samples is allowed to vary according to the previous value of the multivariate CV, which will help the chart to detect the process shifts faster. The comparison with the fixed sampling interval Shewhart chart is implemented to highlight the advantage of the VSI method. Finally, an illustrative example is demonstrated on real data.     

Multivariate Bayesian process control for a finite production run

Most industrial products and processes are characterized by several, typically correlated measurable variables, which jointly describe the product or process quality. Various control charts such as Hotelling’s T², EWMA and CUSUM charts have been developed for multivariate quality control, where the values of the chart parameters, namely the sample size, sampling interval and the control limits are determined to satisfy given economic and/or statistical requirements. It is well known that this traditional non-Bayesian approach to a control chart design is not optimal, but very few results regarding the form of the optimal Bayesian control policy have appeared in the literature, all limited to a univariate chart design. In this paper, we consider a multivariate Bayesian process mean control problem for a finite production run under the assumption that the observations are values of independent, normally distributed vectors of random variables. The problem is formulated in the POMDP (partially observable Markov decision process) framework and the objective is to determine a control policy minimizing the total expected cost. It is proved that under standard operating and cost assumptions the control limit policy is optimal. Cost comparisons with the benchmark chi-squared chart and the MEWMA chart show that the Bayesian chart is highly cost effective, the savings are larger for smaller values of the critical Mahalanobis distance between the in-control and out-of-control process mean.     

AT&T准则下完全检验的最优设计

本文研究了完全检验的质量控制问题,将广泛用于X-控制图的AT&T准则应用于完全检验,并根据完全检验的特点,提出一种新的最优模型,数值实验结果表明AT&T准则下的完全检验优于传统的完全检验.     

�׶�II����������ֵķ��յ�����ļ��ο���ͼ

In recent years, statistical process control (SPC) has been widely used to monitor the performance of clinical practitioners, such as surgeons and general practitioners. In this paper, two risk-adjusted geometric control charts namely cumulative sum (CUSUM) and weighted likelihood ratio test (WLRT) are proposed to monitor surgery performance in phase II. The performance of the proposed control charts is evaluated and compared by simulation experiments for different shift values in the parameters of a risk-adjusted logistic regression model in terms of the average run length (ARL) criterion. The results show that all methods work well in the sense that they can effectively detect shifts in the process parameters.     

监测Lévy稳定过程均值变点的平均运行时间估计

平均运行长度(时间)(ARL)是判断一控制图监测变点效果好坏的一个重要工具.本文主要研究Lévy稳定过程的均值变点监测问题.我们给出了三个控制图,即EWMA,GEWMA和GLR的ARL估计,并通过数值模拟比较了4个控制图监测均值变点的效果和差异.     

监测L\'{e}vy稳定过程均值变点的平均运行时间估计

平均运行长度(时间) (ARL)是判断一控制图监测变点效果好坏的一个重要工具\bd 本文主要研究L\'{e}vy 稳定过程的均值变点监测问题\bd 我们给出了三个控制图, 即EWMA, GEWMA和GLR的ARL 估计, 并通过数值模拟比较了4个控制图监测均值变点的效果和差异.     

Economic design of attribute np control charts using a variable sampling policy

To use a control chart, the quality engineer should specify three decision variables, namely the sample size, the sampling interval and the critical region of the chart. A significant part of recent research relaxed the constraint of using fixed design parameters to open the way to a new type of control charts called adaptive ones where at least one of the decision variables may change in real time based on the last data information. These adaptive schemes have proven their effectiveness from economical and statistical point of views. In this paper, the economic design of an attribute np control chart using a variable sampling interval (VSI) is treated. A sensitivity analysis is conducted to search for optimal design parameters minimizing the expected total cost per hour and to reveal the impact of the process and cost parameters on the behavior of optimal solutions. An economic comparison between the classical np chart, variable sample size (VSS) np control chart and VSI chart is conducted. It is found that switching from the classical attribute chart to the VSI sampling strategy results in notable cost savings and in reduction of the average time to signal and the average number of false alarms. In most cases of the sensitivity analysis, the VSI np chart outperforms the VSS np chart based on economical and statistical considerations. Copyright © 2014 John Wiley & Sons, Ltd.     

Improving the Performance of the Chi-square Control Chart via Runs Rules

The most popular multivariate process monitoring and control procedure used in the industry is the chi-square control chart. As with most Shewhart-type control charts, the major disadvantage of the chi-square control chart, is that it only uses the information contained in the most recently inspected sample; as a consequence, it is not very efficient in detecting gradual or small shifts in the process mean vector. During the last decades, the performance improvement of the chi-square control chart has attracted continuous research interest. In this paper we introduce a simple modification of the chi-square control chart which makes use of the notion of runs to improve the sensitivity of the chart in the case of small and moderate process mean vector shifts.      

Monitoring process variability using auxiliary information

In this study a Shewhart type control chart namely V _r chart is proposed for improved monitoring of process variability (targeting large shifts) of a quality characteristic of interest Y. The proposed control chart is based on regression type estimator of variance using a single auxiliary variable X. It is assumed that (Y, X) follow a bivariate normal distribution. The design structure of V _r chart is developed and its comparison is made with the well-known Shewhart control chart namely S ² chart used for the same purpose. Using power curves as a performance measure it is observed that V _r chart outperforms the S ² chart for detecting moderate to large shifts, which is main target of Shewhart type control charts, in process variability under certain conditions on ρ _yx . These efficiency conditions on ρ _yx are also obtained for V _r chart in this study.