A CCC‐r chart for high‐yield processes

The cumulative count of a conforming (CCC) chart is used to monitor high‐quality processes and is based on the number of items inspected until observing r non‐conforming ones. This charting technique is known as a CCC‐r chart. The function of the CCC‐r chart is the sensitive detection of an upward shift in the fraction defectives of the process, p. As r gets larger, the CCC‐r chart becomes more sensitive to small changes of upward shift in p. However, since many observations are required to obtain a plotting point on the chart, the cost is fairly high. For this trade‐off problem it is necessary to determine the optimal number of non‐conforming items observed before a point is plotted, the sampling (inspection) interval, and the lower control limit for the chart. In this paper a simplified optimal design method is proposed. For illustrative purposes, some numerical results for the optimal design parameter values are provided. The expected profits per cycle obtained using the proposed optimal design method are compared with those obtained using other misspecified parameter values. The effects of changing these parameters on the profit function are shown graphically. Copyright © 2001 John Wiley & Sons, Ltd.     

A comparative study of the CRL‐type control charts

The CRL (Conforming Run Length) type control charts have attracted increasing interest recently for attribute Statistical Process Control (SPC). The two most promising charts of this type are identified as the CRL‐CUSUM chart and the SCRL (Sum of CRLs) chart. This article compares the operating characteristics of these two charts in a comprehensive manner. The general findings reveal that the CRL‐CUSUM chart excels the SCRL chart in detecting downward (decreasing) fraction nonconforming (p) shifts and large‐scale upward (increasing) p shifts. However, the SCRL chart is superior to the CRL‐CUSUM chart in detecting the small and moderate scale upward p shifts, especially when the normal p value is small. The information acquired in this study will provide Quality Assurance (QA) engineers with useful guidance for selecting and applying the CRL‐type control charts. Copyright © 2000 John Wiley & Sons, Ltd.     

A Generalized Likelihood Ratio Chart for Monitoring Bernoulli Processes

This paper considers the problem of monitoring the proportion p of nonconforming items when a continuous stream of Bernoulli observations is available and the objective is to effectively detect a wide range of increases in p. The proposed control chart is based on a generalized likelihood ratio (GLR) statistic obtained from a moving window of past Bernoulli observations. The Phase II performance of this chart in detecting sustained increases in p is evaluated using the steady state average number of observations to signal. Comparisons of the Bernoulli GLR chart to the Shewhart CCC‐r chart, the Bernoulli cumulative sum chart, and the Bernoulli exponentially weighted moving average chart show that the overall performance of the Bernoulli GLR chart is better than its competitors. In addition, methods are provided for designing the Bernoulli GLR chart so that this chart can be easily applied in practice. Copyright © 2012 John Wiley & Sons, Ltd.     

Monitoring of Time‐Between‐Events with a Generalized Group Runs Control Chart

Control charting methods for time between events (TBE) is important in both manufacturing and nonmanufacturing fields. With the aim to enhance the speed for detecting shifts in the mean TBE, this paper proposes a generalized group runs TBE chart to monitor the mean TBE of a homogenous Poisson failure process. The proposed chart combines a TBE subchart and a generalized group conforming run length subchart. The zero‐state and steady‐state performances of the proposed chart were evaluated by applying a Markov chain method. Overall, it is found that the proposed chart outperforms the existing TBE charts, such as the T, T_r, EWMA‐T, Synth‐T_r, and GR‐T_r charts. Copyright © 2015 John Wiley & Sons, Ltd.     

The Effect of Parameter Estimation on Upper‐sided Bernoulli Cumulative Sum Charts

The Bernoulli cumulative sum (CUSUM) chart has been shown to be effective for monitoring the rate of nonconforming items in high‐quality processes where the in‐control proportion of nonconforming items (p₀) is low. The implementation of the Bernoulli CUSUM chart is often based on the assumption that the in‐control value p₀ is known; therefore, when p₀ is unknown, accurate estimation is necessary. We recommend using a Bayes estimator to estimate the value of p₀ to incorporate practitioner knowledge and to avoid estimation issues when no nonconforming items are observed in phase I. We also investigate the effects of parameter estimation in phase I on the upper‐sided Bernoulli CUSUM chart by using the expected value of the average number of observations to signal (ANOS) and the standard deviation of the ANOS. It is found that the effects of parameter estimation on the Bernoulli CUSUM chart are more significant than those on the Shewhart‐type geometric chart. The low p₀ values inherent to high‐quality processes imply that a very large, and often unrealistic, sample size may be needed to accurately estimate p₀. A methodology to identify a continuous variable to monitor is highly recommended when the value of p₀ is low and the required phase I sample size is impractically large. Copyright © 2012 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 Statistical High‐Quality Process Monitoring Method: The Counted Number Between Omega‐Event Control Charts

Control chart techniques for high‐quality process have attracted great attention in modern precision manufacturing. Traditional control charts are no longer applicable because of high false alarm rate. To solve this problem, in this article a new statistical process monitoring method, the counted number between omega‐event statistical process control charts, abbreviated as CBΩ charts, is proposed. The phrase omega event denotes that one observation falls into some certain interval and the CBΩ chart is to monitor the number of consecutive parts between successive r omega events. On the basis of CBΩ charts, a dual‐CBΩ monitoring scheme is developed. This scheme sets up two CBΩ charts with symmetrical omega events, (μ + ?σ, + ∞) and (? ∞, μ ? ?σ), respectively. The performance of CBΩ charts and dual‐CBΩ monitoring is investigated. Dual‐CBΩ monitoring has shown its capability in detecting both mean and variance shift and convenience in implementation compared with other traditional charts. Dual‐CBΩ monitoring can reduce false alarm rate greatly without introducing an unacceptable loss of sensitivity in detecting out‐of‐control signals in high‐quality process control. Copyright © 2011 John Wiley & Sons, Ltd.     

A new approach to setting control limits of cumulative count of conforming charts for high‐yield processes

Cumulative count of conforming (CCC‐r) charts are usually used to monitor non‐conforming fraction p in high‐yield processes. Existing approaches to setting the control limits may cause non‐maximal or biased in‐control average run length (ARL). Non‐maximal in‐control ARL implies that the chart might not quickly detect the upward shift of p from its nominal value p₀. On the other hand, biased in‐control ARL means that both the in‐control and out‐of‐control ARLs are inflated. This paper develops a new approach to setting control limits for CCC‐r charts with near‐maximal and near‐unbiased in‐control ARL. Experimental results show that the proposed approach is effective in terms of the maximization and unbiasedness of in‐control ARL. Copyright © 2009 John Wiley & Sons, Ltd.     

A Shewhart‐type Control Scheme to Monitor Weibull Data without Subgrouping

This study proposes a Shewhart control scheme to simultaneously monitor the shape parameter and the scale parameter of Weibull data without subgrouping. The proposed control scheme comprises two charts: the X chart and the moving‐ratio (MRa) chart. The X chart plots individual observations to detect the shift of the scale parameter by assuming that the shape parameter is in‐control. In contrast, the MRa chart plots moving ratios, the minimum of two consecutive Weibull data divided by the maximum of them, to detect the shift of the shape parameter. This study models the transition process of the proposed control scheme as a Markov chain to calculate two performance measures: the average number of observations to signal and the average run length. Performance analysis shows that the proposed control scheme is effective in detecting the shift of parameters, especially for the downward shift of the shape parameter. Finally, the implementation of the proposed control scheme is illustrated in two skewed data sets. Copyright © 2013 John Wiley & Sons, Ltd.     

A New Synthetic Exponentially Weighted Moving Average Control Chart for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely recognized as a potentially powerful process monitoring tool of the statistical process control because of their excellent speed in detecting small to moderate shifts in the process parameters. Recently, new EWMA and synthetic control charts have been proposed based on the best linear unbiased estimator of the scale parameter using ordered ranked set sampling (ORSS) scheme, named EWMA‐ORSS and synthetic‐ORSS charts, respectively. In this paper, we extend the work and propose a new synthetic EWMA (SynEWMA) control chart for monitoring the process dispersion using ORSS, named SynEWMA‐ORSS chart. The SynEWMA‐ORSS chart is an integration of the EWMA‐ORSS chart and the conforming run length chart. Extensive Monte Carlo simulations are used to estimate the run length performances of the proposed control chart. A comprehensive comparison of the run length performances of the proposed and the existing powerful control charts reveals that the SynEWMA‐ORSS chart outperforms the synthetic‐R, synthetic‐S, synthetic‐D, synthetic‐ORSS, CUSUM‐R, CUSUM‐S, CUSUM‐ln S², EWMA‐ln S² and EWMA‐ORSS charts when detecting small shifts in the process dispersion. A similar trend is observed when the proposed control chart is constructed under imperfect rankings. An application to a real data is also provided to demonstrate the implementation and application of the proposed control chart. Copyright © 2014 John Wiley & Sons, Ltd.     

The variable sampling rate X̄ control charts for monitoring autocorrelated processes

Recent studies demonstrated that the adaptive X? control charts are more efficient than fixed parmeters (FP) X? control chart from statistical and economic criteria. The usual assumption for designing a control chart is that the observations from the process are independent. However, for many processes, such as chemical processes, consecutive measurements are often highly correlated, especially when the interval between samples is small. In the present paper, the observations are modeled as an AR(1) process plus a random error, and the properties of the variable sampling rate (VSR) X? charts are evaluated and studied under this model. Based on the study, the VSR X? chart is faster than the FP, variable sampling interval and variable sample size X? control charts in detecting mean shifts. However, when the level of autocorrelation is high or the process mean shift is large, the advantage of the VSR X? chart is reduced. Copyright © 2008 John Wiley & Sons, Ltd.     

A GLR control chart for monitoring a multinomial process

The problem of detecting changes in the parameter p in a Bernoulli process with two possible categories for each observation has been extensively investigated in the SPC literature, but there is much less work on detecting changes in the vector parameter p in a multinomial process where there are more than two categories. A few papers have considered the case in which the direction of the change in p is known, but there is almost no work for the important case in which the direction of the change is unknown and individual observations are obtained. This paper proposes a multinomial generalized likelihood ratio (MGLR) control chart based on a likelihood ratio statistic for monitoring p when individual observations are obtained and the direction and size of the change in p are unknown. A set of 2‐sided Bernoulli cumulative sum (CUSUM) charts is proposed as a reasonable competitor of the MGLR chart. It is shown that the MGLR chart has better overall performance than the set of 2‐sided Bernoulli CUSUM charts over a wide range of unknown shifts. Equations are presented for obtaining the control limit of the MGLR chart when there are three or four components in p .     

Design and implementation of qth quantile‐unbiased tr‐chart for monitoring times between events

The times between events control charts have been proposed in literature for statistical monitoring of high‐yield processes by observing the waiting times up to r ^th (r ≥ 1  ) non‐conforming items or defects. The average run length (ARL) is the most widely used performance measure to evaluate the chart's performance, but in recent years, it has been subjected to criticisms. Because the run length distribution is highly skewed and hence, the ARL is not necessarily a typical value of the run length. Thus, evaluation of the control chart based on ARL alone could be misleading. In this paper, the quantiles of run length distribution are considered, instead of ARL, to design the t_r ‐chart. Further, we eliminate the bias in q ^th quantile function of the t_r ‐chart for both the known and unknown parameter case. In particular, the MRL‐unbiased t_r ‐chart is discussed in detail and compared with the ARL‐unbiased t_r ‐chart. It is found that the MRL‐unbiased t_r ‐chart outperforms than the corresponding ARL‐unbiased chart in unknown parameter case. It is also found that the proposed chart requires less phase I observations than that of the earlier studies has been suggested.     

Note on ‘Construction of Double Sampling s‐Control Charts for Agile Manufacturing’

He and Grigoryan (Quality and Reliability Engineering International 2002; 18 :343–355) formulated the design of a double‐sampling (DS) s control chart as an optimization problem and solved it with a genetic algorithm. They concluded that the DS s control charts can be a more economically preferable alternative in detecting small shifts than traditional s control charts. We explain that, since they only considered the average sample size when the process is in control, their conclusion is questionable. Copyright © 2006 John Wiley & Sons, Ltd.     

Properties of the Markov‐Dependent Attribute Control Chart with Estimated Parameters

The performance of attribute control charts that monitor Markov‐dependent data is usually evaluated under the assumption of known process parameters, that is, known values of a the probability an item is nonconforming given the previous item is conforming and b the probability an item is conforming given the previous item is nonconforming. In practice, these parameters are usually not known and are calculated from an in‐control Phase I‐data set. In this paper, a comparison of the in‐control ARL (average run length) properties of the attribute chart for Markov‐dependent data with known and estimated parameters is presented. The probability distribution of the estimators is developed and used to calculate the in‐control ARL and standard deviation of the run length of the chart with estimated parameters. For particular values of a and b, the in‐control ARL values of the charts with estimated parameters may be very different than those with known parameters. The size of the Phase‐I data set needed for charts with estimated parameters to exhibit the same in‐control ARL properties as those with known parameters may vary widely depending on the parameters of the process, but in general, large samples are needed to obtain accurate estimates. As the Phase‐I sample size increases, the in‐control ARL values of the charts with estimated parameters approach that of the known parameter case but not in a monotonic fashion as in the case of the X‐bar chart. Copyright © 2015 John Wiley & Sons, Ltd.     

Variable Selection‐based Multivariate Cumulative Sum Control Chart

High‐dimensional applications pose a significant challenge to the capability of conventional statistical process control techniques in detecting abnormal changes in process parameters. These techniques fail to recognize out‐of‐control signals and locate the root causes of faults especially when small shifts occur in high‐dimensional variables under the sparsity assumption of process mean changes. In this paper, we propose a variable selection‐based multivariate cumulative sum (VS‐MCUSUM) chart for enhancing sensitivity to out‐of‐control conditions in high‐dimensional processes. While other existing charts with variable selection techniques tend to show weak performances in detecting small shifts in process parameters due to the misidentification of the ‘faulty’ parameters, the proposed chart performs well for small process shifts in identifying the parameters. The performance of the VS‐MCUSUM chart under different combinations of design parameters is compared with the conventional MCUSUM and the VS‐multivariate exponentially weighted moving average control charts. Finally, a case study is presented as a real‐life example to illustrate the operational procedures of the proposed chart. Both the simulation and numerical studies show the superior performance of the proposed chart in detecting mean shift in multivariate processes. Copyright © 2016 John Wiley & Sons, Ltd.     

An assessment of the kernel‐distance‐based multivariate control chart through an industrial application

Traditional multivariate quality control charts assume that quality characteristics follow a multivariate normal distribution. However, in many industrial applications the process distribution is not known, implying the need to construct a flexible control chart appropriate for real applications. A promising approach is to use support vector machines in statistical process control. This paper focuses on the application of the ‘kernel‐distance‐based multivariate control chart’, also known as the ‘k‐chart’, to a real industrial process, and its assessment by comparing it to Hotelling's T² control chart, based on the number of out‐of‐control observations and on the Average Run Length. The industrial application showed that the k‐chart is sensitive to small shifts in mean vector and outperforms the T² control chart in terms of Average Run Length. Copyright © 2010 John Wiley & Sons, Ltd.     

Nonparametric Synthetic Control Charts for Process Variation

In this article, we propose nonparametric synthetic and side‐sensitive synthetic control charts for controlling fraction nonconforming due to increase in the process variation. Synthetic control chart is a combination of sign and conforming run length control charts. We compare performance of the proposed control charts with the Shewhart sign and S² charts. Our performance study shows that the proposed control charts have a higher power of detecting out‐of‐control signal. We also study the steady‐state behavior of a nonparametric synthetic control chart. We present a Markov chain model to evaluate the steady‐state average run length of the synthetic and side‐sensitive synthetic control charts. Copyright © 2011 John Wiley & Sons, Ltd.     

Phase II Shewhart‐type Control Charts for Monitoring Times Between Events and Effects of Parameter Estimation

Monitoring times between events (TBE) is an important aspect of process monitoring in many areas of applications. This is especially true in the context of high‐quality processes, where the defect rate is very low, and in this context, control charts to monitor the TBE have been recommended in the literature other than the attribute charts that monitor the proportion of defective items produced. The Shewhart‐type t‐chart assuming an exponential distribution is one chart available for monitoring the TBE. The t‐chart was then generalized to the t_r‐chart to improve its performance, which is based on the times between the occurrences of r (≥1) events. In these charts, the in‐control (IC) parameter of the distribution is assumed known. This is often not the case in practice, and the parameter has to be estimated before process monitoring and control can begin. We propose estimating the parameter from a phase I (reference) sample and study the effects of estimation on the design and performance of the charts. To this end, we focus on the conditional run length distribution so as to incorporate the ‘practitioner‐to‐practitioner’ variability (inherent in the estimates), which arises from different reference samples, that leads to different control limits (and hence to different IC average run length [ARL] values) and false alarm rates, which are seen to be far different from their nominal values. It is shown that the required phase I sample size needs to be considerably larger than what has been typically recommended in the literature to expect known parameter performance in phase II. We also find the minimum number of phase I observations that guarantee, with a specified high probability, that the conditional IC ARL will be at least equal to a given small percentage of a nominal IC ARL. Along the same line, a lower prediction bound on the conditional IC ARL is also obtained to ensure that for a given phase I sample, the smallest IC ARL can be attained with a certain (high) probability. Summary and recommendations are given. Copyright © 2014 John Wiley & Sons, Ltd.