Robust CUSUM charts for monitoring the process mean and variance

This paper considers the problem of obtaining robust control charts for detecting changes in the mean µ and standard deviation σ of process observations that have a continuous distribution. The standard control charts for monitoring µ and σ are based on the assumption that the process distribution is normal. However, the process distribution may not be normal in many situations, and using these control charts can lead to very misleading conclusions. Although some control charts for µ can be tuned to be robust to non‐normal distributions, the most critical problem with non‐robustness is with the control chart for σ. This paper investigates the performance of two CUSUM chart combinations that can be made to be robust to non‐normality. One combination consists of the standard CUSUM chart for µ and a CUSUM chart of absolute deviations from target for σ, where these CUSUM charts are tuned to detect relatively small parameter shifts. The other combination is based on using winsorized observations in the standard CUSUM chart for µ and a CUSUM chart of squared deviations from target for σ. Guidance is given for selecting the design parameters and control limits of these robust CUSUM chart combinations. When the observations are actually normal, using one of these robust CUSUM chart combination will result in some reduction in the ability to detect moderate and large changes in µ and σ, compared with using a CUSUM chart combination that is designed specifically for the normal distribution. Copyright © 2009 John Wiley & Sons, Ltd.     

CUSUM charts for monitoring a zero‐inflated poisson process

A zero‐inflated Poisson (ZIP) process is different from a standard Poisson process in that it results in a greater number of zeros. It can be used to model defect counts in manufacturing processes with occasional occurrences of non‐conforming products. ZIP models have been developed assuming that random shocks occur independently with probability p, and the number of non‐conformities in a product subject to a random shock follows a Poisson distribution with parameter λ. In our paper, a control charting procedure using a combination of two cumulative sum (CUSUM) charts is proposed for monitoring increases in the two parameters of the ZIP process. Furthermore, we consider a single CUSUM chart for detecting simultaneous increases in the two parameters. Simulation results show that a ZIP‐Shewhart chart is insensitive to shifts in p and smaller shifts in λ in terms of the average number of observations to signal. Comparisons between the combined CUSUM method and the single CUSUM chart show that the latter's performance is worse when there are only increases in p, but better when there are only increases in λ or when both parameters increase. The combined CUSUM method, however, is much better than the single CUSUM chart when one parameter increases while the other decreases. Finally, we present a case study from the light‐emitting diode packaging industry. Copyright © 2011 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.     

The Bernoulli CUSUM Chart for Detecting Decreases in a Proportion

This article considers the Bernoulli CUSUM chart for detecting a decrease in the proportion p of nonconforming items when a continuous stream of Bernoulli observations from the process is available. The properties of the Bernoulli CUSUM chart can be obtained using a Markov chain model. However, in some cases, the number of transient states in the Markov chain may be so large that it is not feasible to work directly with the transition matrix. This article provides a solution to the equations used in defining the Markov chain so that properties of the chart can be obtained without explicitly using the transition matrix. This solution allows the practitioner to determine the control limit required to give a specified in‐control performance. A simple example is used to show that using the Bernoulli CUSUM chart is a better option than the standard practice of artificially grouping the observations into samples of size n > 1 and using a Shewhart chart. Copyright © 2012 John Wiley & Sons, Ltd.     

面向质量目标的Q控制图设计

针对大批量生产开始阶段的过程监控,提出了一种基于预定质量目标的Q控制图监控方法.其基本思路是利用面向质量目标的统计公差技术与Q控制图相结合应用,以实现大批量过程开始阶段均值和方差未知时面向质量目标的过程监控.基于质量目标建立统计公差(CP*,k*),并将该统计公差转化为基于给定置信概率的对CP和k的估计值的判定条件.通过案例分析,面向质量目标的Q控制图能够在过程保持受控状态的前提下以一定置信概率保证质量目标.     

Variable sampling interval hotelling's T2 charts with runs rules for switching between sampling interval lengths

The use of runs rules is proposed for switching between the sampling interval lengths of variable sampling interval Hotelling's T² charts. The purpose of applying these rules is to reduce the frequency of the switches which causes inconvenience in the administration of the charts. The expressions for the performance measures for the charts with these rules are derived. The effects of different runs rules on the performances are evaluated through numerical comparisons. The runs rules substantially reduce the frequency of switches during the in‐control period and during the out‐of‐control periods due to the small to moderate shifts in the process mean vector. They also fairly improve the statistical performances of the charts in detecting the small shifts and do not affect that in detecting the large shifts. However, some runs rules slightly worsen the statistical performances in detecting the moderate shifts. Copyright © 2011 John Wiley & Sons, Ltd.     

Directed control charts for detecting the shape changes from linear profiles to quadratic profiles

In profile monitoring, control charts are constructed to detect any unanticipated departures from the statistical stability of product quality over time, where product quality is characterised by a function. In many situations, due to the characteristics of a system or an operation, certain process signals can be anticipated. Thus, when a kind of departure specifically feared is identified in advance, a directed process monitoring approach can be developed. Motivated by the monitoring of cylindrical surfaces, this paper focuses on quickly detecting the shape changes from a straight line to a second-order polynomial curve. Based on the hypothesis testing on the quadratic term, two directed control charts and a combined scheme are proposed to surveillance the sampled linear shape. The performance of our proposed methods is studied and compared with the alternative charts by numerical simulations. Simulation studies show that the two proposed directed charts are almost the same, and outperform the alternative methods in some cases. Moreover, the combined scheme is robust for all the parameter combinations.     

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.     

Charts with Variable Sample Size,Sampling Interval,and Warning Limits

The notion of variable warning limits is proposed for variable sample size and sampling interval (VSSI) charts. The basic purpose is to lower down the frequency of switches between the pairs of values of the sample sizes and sampling interval lengths of VSSI charts during their implementations. Expressions for performance measures for the variable sample size, sampling interval, and warning limits (VSSIWL) charts are developed. The performances of these charts are compared numerically with that of VSSI and VSSI (1, 3) charts, where VSSI (1, 3) charts are the VSSI charts with runs rule (1, 3) for switching between the pairs of values of sample sizes and sampling interval lengths. Runs rule (1, 3) greatly reduces the frequency of the switches; however, it slightly worsens the statistical performances of the VSSI charts in detecting moderate shifts in the process mean. It is observed that the out‐of‐control statistical performance and overall switching rate of VSSIWL charts are adaptive for the same in‐control statistical performances. These charts can be set to yield exactly similar performances as that of VSSI (1, 3) charts, to yield tradeoff performances between that of VSSI (1, 3) and VSSI charts, or to yield significantly lower switching rate than even that of VSSI (1, 3) charts at the cost of slightly inferior statistical performances than that of VSSI (1, 3) charts. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimating the time of step change with Poisson CUSUM and EWMA control charts

Knowing when a process has changed would simplify the search for and identification of the special cause. Consequently, having an estimate of the process change point following a control chart signal would be useful to process engineers. Much of the literature on change point models and techniques for statistical process control applications consider processes well modelled by the normal distribution. However, the Poisson distribution is commonly used in industrial quality control applications for modelling attribute-based process quality characteristics (e.g., counts of non-conformities). Some commonly used control charts for monitoring Poisson distributed data are the Poisson cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts. In this paper, we study the effect of changes in the design of the control chart on the performances of the change point estimators offered by these procedures. In particular, we compare root mean square error performances of the change point estimators offered by the Poisson CUSUM and EWMA control charts relative to that achieved by a maximum likelihood estimator for the process change point. Results indicate that the relative performance achieved by each change point estimator is a function of the corresponding control chart design. Relative mean index plots are provided to enable users of these control charts to choose a control chart design and change point estimator combination that will yield robust change point estimation performance across a range of potential change magnitudes.     

Monitoring the Weibull shape parameter by control charts for the sample range

In this paper, we propose control charts for monitoring changes in the Weibull shape parameter β. These charts are based on the range of a random sample from the smallest extreme value distribution. The control chart limits depend only on the sample size, the desired stable average run length (ARL), and the stable value of β. We derive control limits for both one‐ and two‐sided control charts. They are unbiased with respect to the ARL. We discuss sample size requirements if the stable value of βis estimated from past data. The proposed method is applied to data on the breaking strengths of carbon fibers. We recommend one‐sided charts for detecting specific changes in βbecause they are expected to signal out‐of‐control sooner than the two‐sided charts. Copyright © 2010 John Wiley & Sons, Ltd.     

统计过程控制的进一步讨论

就统计过程控制在实施过程中遇到的若干问题进行深入研究,首先进行判异准则的小概率推算,纠正了过去小概率计算中一些错误;其次论述了过程异常的两面性及其对过程的改进作用,最后说明进行SPC不能单纯依赖控制图,还应进行能力分析(即技术稳态分析),并及时有效地采取措施对过程进行改进,才能从实施统计过程控制中获得效益.     

针对阶跃形式失控的基于门限自回归模型的SPC-EPC集成研

SPC-EPC集成是一种控制和提升产品质量的有效方法,目前在传统SPC EPC集成的研究中通常使用线性时间序列模型来描述过程的动态自相关关系,但线性模型难以对更加复杂的非线性自相关关系进行有效描述．针对这一问题,提出了使用一类非线性时间序列模型,即门限自回归模型(TAR)来描述系统的动态自相关关系,并依此建立最小均方误差控制器,并进一步建立SPC EPC集成控制体系．针对在生产过程中常见的以阶跃形式存在的过程失控,首先通过例子研究了控制器在单独使用以及集成控制方法下的控制效果并且与线性控制器相应的结果进行了对比,之后通过模拟研究进一步验证和分析了这一集成控制方法的控制效果．结果表明,基于非线性时间序列的集成SPC EPC控制方法可以针对含阶跃形式失控的复杂的非线性自相关过程进行有效的控制．     

Cuscore control charts for generalized feedback‐control systems

The cumulative score (Cuscore) statistic is devised to ‘resonate’ with deviations or signals of an expected type. When a process signal subject to feedback control occurs, it results in a fault signature in the output error. In this paper, Cuscore statistics are designed to monitor process parameters and characteristics measured by a generalized minimum variance (GMV) feedback‐control system sensitive to the fault signature of a spike, step, and bump signal. In this study, the GMV considered is a first‐order dynamic system with autoregressive moving average (ARMA) noise. We show theoretically that the performance of Cuscore charts is independent of the amount of variability transferred from the output quality characteristic to the adjustment actions in the GMV control system. Simulation is used to test the performance using the Cuscore charts. In general, the Cuscore can detect signals over a broad range of system parameter values. However, areas of low detection capability occur for certain fault signatures. In these cases, a tracking signal test is combined with the Cuscore statistics to improve detection performance. This study provides several illustrations of the underlying behavior and shows how the methodology developed can be easily applied in practice. Copyright © 2006 John Wiley & Sons, Ltd.     

Variable Sample Size and Sampling Interval Charts with Runs Rules for Switching Between Sample Sizes and Sampling Interval Lengths

Variable sample size and sampling interval (VSSI) charts are substantially more efficient than are the static charts. However, the frequent switches between sample sizes and sampling interval lengths can be a complicating factor during the implementation of these charts. In this article, runs rules are proposed for switching between the sample sizes and the sampling interval lengths of these charts to reduce the frequency of switches. The expressions for the performance measures for the charts with these runs rules are developed. The methods presented are general and can be applied to other VSSI Shewhart control charts. The effects of different runs rules on the performances of the charts are compared numerically. The runs rules substantially reduce the frequency of switches. Some runs rules do not significantly alter the statistical performances of the charts; however, some adversely affect that in detecting large shifts in the process mean. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal control limits for CCC charts in the presence of inspection errors

The control chart based on cumulative count of conforming (CCC) items between the occurrence of two non‐conforming ones, or the CCC chart, has been shown to be very useful for monitoring high‐quality processes. However, as in the implementation of other Shewhart‐type control charts, it is usually assumed that the inspection is free of error. This assumption may not be valid and this may have a significant impact on the interpretation of the control chart and the setting of control limits. This paper first investigates the effect of inspection errors and discusses the setting of control limits in such cases. Even if inspection errors are considered, the average time to alarm increases in the beginning when the process deteriorates. Since this is undesirable, the control limits in the presence of inspection errors should be set so as to maximize the average run length when the process is at the normal level. A procedure is presented for solving this problem. Copyright © 2003 John Wiley & Sons, Ltd.     

Phase II control charts for monitoring dispersion when parameters are estimated

Shewhart control charts are among the most popular control charts used to monitor process dispersion. To base these control charts on the assumption of known in-control process parameters is often unrealistic. In practice, estimates are used to construct the control charts and this has substantial consequences for the in-control and out-of-control chart performance. The effects are especially severe when the number of Phase I subgroups used to estimate the unknown process dispersion is small. Typically, recommendations are to use around 30 subgroups of size 5 each.

?We derive and tabulate new corrected charting constants that should be used to construct the estimated probability limits of the Phase II Shewhart dispersion (e.g., range and standard deviation) control charts for a given number of Phase I subgroups, subgroup size and nominal in-control average run-length (ICARL). These control limits account for the effects of parameter estimation. Two approaches are used to find the new charting constants, a numerical and an analytic approach, which give similar results. It is seen that the corrected probability limits based charts achieve the desired nominal ICARL performance, but the out-of-control average run-length performance deteriorate when both the size of the shift and the number of Phase I subgroups are small. This is the price one must pay while accounting for the effects of parameter estimation so that the in-control performance is as advertised. An illustration using real-life data is provided along with a summary and recommendations.     

A control scheme integrating the T chart and TCUSUM chart

This article proposes an integrated scheme (T&TCUSUM chart) which combines a Shewhart T chart and a TCUSUM chart (a CUSUM‐type T chart) to monitor the time interval T between the occurrences of an event or the time between events. The performance studies show that the T&TCUSUM chart can effectively improve the overall performance over the entire T shift range. On average, it is more effective than the T chart by 26.66% and the TCUSUM chart by 14.12%. Moreover, the T&TCUSUM chart performs more consistently than other charts for the detection of either small or large T shifts, because it has the strength of both the T chart (more sensitive to large shifts) and the TCUSUM chart (more sensitive to small shifts). The implementation of the new chart is almost as easy as the operation of a TCUSUM chart. Copyright © 2010 John Wiley & Sons, Ltd.     

Joint EWMA charts for multivariate process control: Markov chain and optimal design

This paper deals with the optimal design of a set of univariate exponentially weighted moving average (EWMA) quality control charts to monitor several correlated variables. The aim is to find the values for the parameters of the set of charts that minimise the average run length (ARL) for a given process shift, according to three different problem formulations, in terms of the symmetry of the directions of shift that may be expected in the particular process. The first step to achieve this objective has been the development of a multidimensional Markov chain model to compute the ARLs of the set of EWMA control charts. A genetic algorithm has been employed for the optimisation. Finally, a (non-exhaustive) performance comparison is presented between the joint EWMA charts and the equivalent multivariate EWMA (MEWMA) control chart. No scheme uniformly outperforms the other one. In some cases, the univariate charts largely outperform the MEWMA chart for the shift they are optimised but perform much worse for other shifts. Therefore, the tools described in this paper help the user to make an informed decision considering the shifts that may be expected in each particular case.