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.     

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.     

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.     

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.     

New CUSUM and Shewhart-CUSUM charts for monitoring the process mean

The CUmulative SUM (CUSUM) charts have sensitive nature against small and moderate shifts that occur in the process parameter(s). In this article, we propose the CUSUM and combined Shewhart-CUSUM charts for monitoring the process mean using the best linear unbiased estimator of the location parameter based on ordered double-ranked set sampling (RSS) scheme, where the CUSUM chart refers to the Crosier's CUSUM chart. The run-length characteristics of the proposed CUSUM charts are computed with the Monte Carlo simulations. The run-length profiles of the proposed CUSUM charts are compared with those of the CUSUM charts based on simple random sampling, RSS, and ordered RSS schemes. It is found that the proposed CUSUM charts uniformly outperform their existing counterparts when detecting all different kinds of shifts in the process mean. A real data set is also considered to explain the implementation of the proposed CUSUM charts.     

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.     

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.     

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.     

Designing a parameter-free Kullback-Leibler information control chart for monitoring process mean shift

This paper proposes a parameter-free Kullback-Leibler information control chart for monitoring sustained shifts in the process mean of a normally distributed process in phase II. Two plotted statistics are provided. One is based on our backward empirical sequential test, the other is based on the maximum log-likelihood ratio change point method. These two achieve similar performances for the control chart. The performance of the proposed chart is compared with those of the cumulative sum chart, the exponentially weighted moving average chart, and the generalized likelihood ratio (GLR) chart. The results show that our proposed chart and the GLR chart have similar performances. Both can detect a wide range of shifts in the process mean, and neither requires design parameters other than the control limits. The proposed chart outperforms GLR when the size of the shift is below 1.24 standard deviations, while GLR outperforms the proposed chart when the size of the shift is above 1.24 standard deviations.     

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.     

Dynamic probability control limits for CUSUM charts for monitoring proportions with time-varying sample sizes

We consider the problem of monitoring a proportion with time-varying sample sizes. Control charts are generally designed by assuming a fixed sample size or a priori knowledge of a sample size probability distribution. Sometimes, it is not possible to know, or accurately estimate, a sample size distribution or the distribution may change over time. An improper assumption for the sample size distribution could lead to undesirable performance of the control chart. To handle this problem, we propose the use of dynamic probability control limits (DPCLs) which are determined successively as the sample sizes become known. The method is based on keeping the conditional probability of a false alarm at a predetermined level given that there has not been any earlier false alarm. The control limits dynamically change, and the in-control performance of the chart can be controlled at the desired level for any sequence of sample sizes. The simulation results support this result showing that there is no need for any assumption of a sample size distribution with the use of this proposed approach.     

New GWMA‐CUSUM control chart for monitoring the process dispersion

The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have been widely accepted because of their fantastic speed in identifying small‐to‐moderate unusual variations in the process parameter(s). Recently, a new CUSUM chart has been proposed that uses the EWMA statistic, called the CS‐EWMA chart, for monitoring the process variability. On similar lines, in order to further improve the detection ability of the CS‐EWMA chart, we propose a CUSUM chart using the generally weighted moving average (GWMA) statistic, named the GWMA‐CUSUM chart, for monitoring the process dispersion. Monte Carlo simulations are used to compute the run length profiles of the GWMA‐CUSUM chart. On the basis of the run length comparisons, it turns out that the GWMA‐CUSUM chart outperforms the CUSUM and CS‐EWMA charts when identifying small variations in the process variability. A simulated dataset is also used to explain the working and implementation of the CS‐EWMA and GWMA‐CUSUM charts.     

CUSUM charts with controlled conditional performance under estimated parameters

We study the effect of the Phase I estimation error on the cumulative sum (CUSUM) chart. Impractically large amounts of Phase I data are needed to sufficiently reduce the variation in the in-control average run lengths (ARL) between practitioners. To reduce the effect of estimation error on the chart's performance we design the CUSUM chart such that the in-control ARL exceeds a desired value with a specified probability. This is achieved by adjusting the control limits using a bootstrap-based design technique. Such approach does affect the out-of-control performance of the chart; however, we find that this effect is relatively small.     

New EWMA control charts for monitoring process dispersion using auxiliary information

The exponentially weighted moving average (EWMA) control chart is a well‐known statistical process monitoring tool because of its exceptional pace in catching infrequent variations in the process parameter(s). In this paper, we propose new EWMA charts using the auxiliary information for efficiently monitoring the process dispersion, named the auxiliary‐information–based (AIB) EWMA (AIB‐EWMA) charts. These AIB‐EWMA charts are based on the regression estimators that require information on the quality characteristic under study as well as on any related auxiliary characteristic. Extensive Monte Carlo simulation are used to compute and study the run length profiles of the AIB‐EWMA charts. The proposed charts are comprehensively compared with a recent powerful EWMA chart—which has been shown to be better than the existing EWMA charts—and an existing AIB‐Shewhart chart. It turns out that the proposed charts perform uniformly better than the existing charts. An illustrative example is also given to explain the implementation and working of the AIB‐EWMA charts.     

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.     

Self-information-based weighted CUSUM charts for monitoring Poisson count data with varying sample sizes

In many applications, the Poisson count data with varying sample sizes are monitored using statistical process control charts. Among these applications, the weighted CUSUM charts are developed to deal with the effect of the varying sample sizes. However, some of them use limited information of the sample size or the count data while assigning the weights. To gain more information of the process, the self-information weight functions are developed based on both the sample size and the observed count data. Then, the weighted CUSUM charts are proposed with the self-information-based weight. Simulation studies show the self-information-based weighted CUSUM charts perform better than the benchmark methods in detecting small shifts. Moreover, the performance of proposed method with estimated parameters is investigated via simulation. Finally, an example is given to illustrate the application of the proposed weighted CUSUM charts.     

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.     

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.