Monitoring of time between events with a double generally weighted moving average control chart

Monitoring of time between events (TBE) instead of the number of events is used in high‐quality processes where the events occur rarely. This article presents a double generally weighted moving average control chart with a lower time‐varying control limit to monitor the TBE (regarded as DGWMA‐TBE chart). The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the DGWMA‐TBE chart is more effective than the DEWMA and GWMA charts in detecting moderate to large shifts. Furthermore, the DGWMA‐TBE chart is very robust for the same range of shifts when the TBE observations follow a Weibull or a lognormal distribution. Finally, examples are also presented to enhance the performance of the proposed chart.     

Distribution-free triple EWMA control chart for monitoring the process location using the Wilcoxon rank-sum statistic with fast initial response feature

The exponentially weighted moving average (EWMA) control chart is a memory-type chart known to be more efficient in detecting small and moderate shifts in the process parameter. The double EWMA (DEWMA) chart is an extension of the EWMA chart that is more effective than the latter in the detection of small-to-moderate shifts. This paper proposes a new distribution-free (or nonparametric) triple EWMA (TEWMA) control chart based on the Wilcoxon rank-sum (W) statistic to improve the detection ability in the process location parameter. Moreover, a new fast initial response (FIR) feature is added to further improve the sensitivity of the new TEWMA chart. The performance of the proposed TEWMA chart with and without FIR features is compared to those of the existing EWMA and DEWMA W charts. It is observed that the TEWMA chart with and without FIR features is superior to the competing charts in most situations. A real-life illustration is provided to show the application and implementation of the new chart.     

Monitoring of zero-inflated Poisson processes with EWMA and DEWMA control charts

The zero-inflated Poisson (ZIP) distribution is an extension of the ordinary Poisson distribution and is used to model count data with an excessive number of zeros. In ZIP models, it is assumed that random shocks occur with probability p, and upon the occurrence of random shock, the number of nonconformities in a product follows the Poisson distribution with parameter λ. In this article, we study in more detail the exponentially weighted moving average control chart based on the ZIP distribution (regarded as ZIP-EWMA) and we also propose a double EWMA chart with an upper time-varying control limit to monitor ZIP processes (regarded as ZIP-DEWMA chart). The two charts are studied to detect upward shifts not only in each parameter individually but also in both parameters simultaneously. The steady-state performance and the performance with estimated parameters are also investigated. The performance of the two charts has been evaluated in terms of the average and standard deviation of the run length, and compared with Shewhart-type and CUSUM schemes for ZIP distribution, it is shown that the proposed chart is very effective especially in detecting shifts in p when λ remains in control (IC) and in both parameters simultaneously. Finally, one real example is given to display the application of the ZIP charts on practitioners.     

The extended homogeneously weighted moving average control chart

Control charts are widely applied in many manufacturing processes to monitor the quality characteristic of interest. Recently, a homogeneously weighted moving average (HWMA) control chart was proposed as an improvement of the exponentially weighted moving average (EWMA) chart for efficiently monitoring of small shifts in the process mean. In the present article, we extend the HWMA chart by imitating exactly the double EWMA (DEWMA) technique. The proposed scheme is regarded as double HWMA (DHWMA) control chart. The run-length characteristics of the proposed chart are evaluated by performing Monte Carlo simulations. A comparison study versus the EWMA, DEWMA, HWMA, mixed EWMA cumulative sum (CUSUM), CUSUM, and GWMA charts indicates that the DHWMA chart is more effective in detecting small to moderate shifts, while it performs similarly with its competitors for large shifts. We also study the robustness of the proposed chart under several nonnormal distributions, and it is shown that the DHWMA chart is in-control robust for small values of the smoothing parameters. Finally, two examples are given to demonstrate the implementation of the proposed chart.     

The Use of Probability Limits of COM–Poisson Charts and their Applications

The conventional c and u charts are based on the Poisson distribution assumption for the monitoring of count data. In practice, this assumption is not often satisfied, which requires a generalized control chart to monitor both over‐dispersed as well as under‐dispersed count data. The Conway–Maxwell–Poisson (COM–Poisson) distribution is a general count distribution that relaxes the equi‐dispersion assumption of the Poisson distribution and in fact encompasses the special cases of the Poisson, geometric, and Bernoulli distributions. In this study, the exact k‐sigma limits and true probability limits for COM–Poisson distribution chart have been proposed. The comparison between the 3‐sigma limits, the exact k‐sigma limits, and the true probability limits has been investigated, and it was found that the probability limits are more efficient than the 3‐sigma and the k‐sigma limits in terms of (i) low probability of false alarm, (ii) existence of lower control limits, and (iii) high discriminatory power of detecting a shift in the parameter (particularly downward shift). Finally, a real data set has been presented to illustrate the application of the probability limits in practice. Copyright © 2012 John Wiley & Sons, Ltd.     

A Flexible and Generalized Exponentially Weighted Moving Average Control Chart for Count Data

The Conway–Maxwell–Poisson distribution can be used to model under‐dispersed or over‐dispersed count data. This study proposes a flexible and generalized attribute exponentially weighted moving average (EWMA), namely GEWMA, control chart for monitoring count data. The proposed EWMA chart is based on the Conway–Maxwell–Poisson distribution. The performance of the proposed chart is evaluated in terms of run length (RL) characteristics such as average RL, median RL, and standard deviation of the RL distribution. The average RL of the proposed GEWMA chart is compared with Sellers chart. The sensitivity of the standard Poisson EWMA (PEWMA) chart is also studied and compared with the proposed GEWMA chart for under‐dispersed or over‐dispersed data. It has been observed that the PEWMA chart is very sensitive for under‐dispersed or over‐dispersed data while the proposed GEWMA is very robust. Finally, the generalization of the proposed chart to the Bernoulli EWMA, PEWMA, and geometric EWMA charts is also studied using someone simulated data sets. Copyright © 2013 John Wiley & Sons, Ltd.     

EWMA and DEWMA Control Charts for Poisson‐Exponential Distribution: Conditional Median Approach for Censored Data

Exponentially weighted moving average (EWMA) control charts are consistently used for the detection of small shifts contrary to Shewhart charts, which are commonly used for the detection of large shifts in the process. There are many interesting features of EWMA charts that have been studied for complete data in the literature. The aim of present study is to introduce and compare the double exponentially weighted moving average (DEWMA) and EWMA control charts under type‐I censoring for Poisson‐exponential distribution. The monitoring of mean level shifts using censored data is of a great interest in many applied problems. Moreover, a new idea of conditional median is introduced and further compared with the existing conditional expected values approach for monitoring the small mean level shifts. The performance of the DEWMA and EWMA charts is evaluated using the average run length, expected quadratic loss, and performance comparison index measures. The optimum sample size comparisons for the specified and unspecified parameters are also part of this study. Two applications for practical considerations are also discussed. It is observed that different censoring rates and the size of shifts significantly affect the performance of the EWMA and DEWMA charts. Copyright © 2016 John Wiley & Sons, Ltd.     

A generally weighted moving average control chart for zero-inflated Poisson processes

Zero-inflated Poisson (ZIP) model is very useful in high-yield processes where an excessive number of zero observations exist. This model can be viewed as an extension of the standard Poisson distribution. In this paper, a one-sided generally weighted moving average (GWMA) control chart is proposed for monitoring upward shifts in the two parameters of a ZIP process (regarded as ZIP-GWMA chart). The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the ZIP-GWMA performs better than the existing control charts under shifts in both parameters. Moreover, an illustrative example is presented to display the application of the proposed chart on practitioners.     

A generalized statistical control chart for over‐ or under‐dispersed data

The Poisson distribution is a popular distribution used to describe count information, from which control charts involving count data have been established. Several works recognize the need for a generalized control chart to allow for data over‐dispersion; however, analogous arguments can also be made to account for potential under‐dispersion. The Conway–Maxwell–Poisson (COM‐Poisson) distribution is a general count distribution that relaxes the equi‐dispersion assumption of the Poisson distribution, and in fact encompasses the special cases of the Poisson, geometric, and Bernoulli distributions. Accordingly, a flexible control chart is developed that encompasses the classical Shewart charts based on the Poisson, Bernoulli (or binomial), and geometric (or negative binomial) distributions. Copyright © 2011 John Wiley & Sons, Ltd.     

SUBJECTS INDEX

Exponentially weighted moving average (EWMA) control charts are very widely used for the detection of small shifts. Another similar charting structure is double EWMA (DEWMA) control chart for the improved detection of the shifts. Many interesting features of EWMA and DEWMA have been described in the literature. This study intends to investigate EWMA and DEWMA control charts under Type-I censoring for gamma-distributed lifetimes. The idea of conditional expected values is used to monitor the mean level. The performance evaluations are carried out using average run length as a measure in this study. The optimum sample size comparisons for the specified and unspecified parameter are also part of the study. To assess the overall performance of the control charts, we also used extra quadratic loss and it is found DEWMA is an efficient chart for the detection of shift in scale parameter. Moreover, an illustrative example for practical considerations is included in the study. It is observed that varying censoring rates affect the performance of the chart depending upon the type of chart, the method of estimation, and the amount of shift.     

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.     

Novel design of composite generally weighted moving average and cumulative sum charts

The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts are popular statistical tools to improve the performance of the Shewhart chart in detecting small process shifts. In this study, we propose the mixed generally weighted moving average (GWMA)‐CUSUM chart and its reverse‐order CUSUM‐GWMA chart to enhance detection ability compared with existing counterparts. The simulation revealed that the mixed GWMA‐CUSUM and mixed CUSUM‐GWMA charts have the sensitivity to detect small process shifts and efficient structures compared with the mixed EWMA‐CUSUM and mixed CUSUM‐EWMA charts, respectively. Moreover, the mixed GWMA‐CUSUM chart with a large design parameter has robust performance, regardless of the high tail t distribution or right skewness gamma distribution.     

Optimal one- and two-sided adaptive EWMA scheme for monitoring Poisson count data

The Poisson distribution assumption often arises in several industrial applications for modeling defects or nonconformities. In this work, we investigate the one- and two-sided performance of a new adaptive EWMA (exponentially weighted moving average)-type chart for monitoring Poisson count data. An appropriate discrete-state Markov chain technique is provided to compute the exact ARL (average run length) properties. Moreover, comparative studies are conducted to demonstrate the higher sensitivity of the proposed chart in the detection of shifts with various magnitudes. Advices on how to select the appropriate chart parameters are provided and an illustrative numerical example is proposed.     

A Control Chart for COM–Poisson Distribution Using Resampling and Exponentially Weighted Moving Average

In this paper, we propose a control chart using exponentially weighted moving average (EWMA) statistic for count data based on the Conway–Maxwell–Poisson (called COM–Poisson) distribution. Repetitive sampling is considered by constructing two pairs of control limits for the proposed control chart. The performance of the proposed control chart is evaluated using the average run length (ARL) for various values of specified parameters. It has been observed that the proposed control chart is more efficient in terms of ARLs as compared to the existing control charts. The tables are provided and explained with the help of example. Copyright © 2015 John Wiley & Sons, Ltd.     

A new variable control chart under failure‐censored reliability tests for Weibull distribution

In this paper, we propose a new variable control chart under type II or failure‐censored reliability tests by assuming that the lifetime of a part follows the Weibull distribution with fixed and stable shape parameter. The purpose is to monitor the mean and the variance of a Weibull process. In fact, the mean and the variance are related to the scale parameter. The necessary measures are given to calculate the average run length (ARL) for in‐control and shifted processes. The tables of ARLs are presented for various shift constants and specified parameters. A simulation study is given to show the performance of the proposed control chart. The efficiency of the proposed control chart is compared with a control chart based on the conditional expected value under type II censoring. An example is also given for the illustration purpose.     

The Negative Binomial Exponentially Weighted Moving Average Chart with Estimated Control Limits

The control chart based on the compound Poisson distribution (the negative binomial exponentially weighted moving average (EWMA) chart) has been shown to be more effective than the c‐chart to monitor the wafer nonconformities in semiconductor production process. The performance of the negative binomial EWMA chart is generally evaluated with the assumption that the process parameters are known. However, in many control chart applications, the process parameters are usually unknown and are required to be estimated. For an accurate parameter estimate, a very large sample size may be required, which is seldom available in the applications. This article investigates the effect of parameter estimation on the run length properties of the negative binomial EWMA charts. Using a Markov chain approach, we show that the performance of the negative binomial EWMA chart is affected when parameters are estimated compared with the known‐parameter case. We also provide recommendations regarding phase I sample sizes, smoothing constant and clustering parameter. The sample size must be quite large for the in‐control chart performance to be close to that for the known‐parameter case. Finally, a wafer process example has been used to highlight the practical implications of estimation error and to offer advice to practitioners when constructing/analysing a phase I sample. 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.     

An enhanced approach for the progressive mean control charts

To maintain and improve the quality of the processes, control charts play an important role for reduction of variation. To detect large shifts in the process parameters, Shewhart control charts are commonly applied but for small shifts, exponentially weighted moving averages (EWMA), cumulative sum (CUSUM), double exponentially weighted moving average (DEWMA), double CUSUM, moving average (MA), double moving average (DMA), and progressive mean (PM) control charts, are used. This study proposes double progressive mean (DPM) and optimal DPM control charts to enhance the performance of the PM chart. As the proposed DPM control charts use information sequentially, hence their performance is compared with natural competitors EWMA, CUSUM, DEWMA, double CUSUM, MA, DMA, and PM control charts. Run length and its different properties are evaluated to compare the performance of the proposed charts and counterparts. Results reveal that proposed optimal DPM outperforms the other charts. An example related to voltage on fixed capacitance level is also provided to illustrate the proposed charts.     

Adaptive nonparametric CUSUM scheme for detecting unknown shifts in location

In this paper, we provide a sequential rank-based adaptive nonparametric cumulative sum control chart for detecting a range of shifts in the location parameter. This chart is a self-starting scheme, and thus can be used to monitor processes at the start-up stages rather than having to wait for the accumulation of sufficiently large calibration samples. It does not require any prior knowledge of the underlying distribution. The choice of the chart parameters is studied and a simulation study demonstrates that the proposed control chart not only performs robustly for different distributions, but also efficiently detects various magnitudes of shifts. An illustrative example is also given to introduce the implementation of this proposed control chart.     

Effects of Parameter Estimation on the Multivariate Distribution‐free Phase II Sign EWMA Chart

Multivariate nonparametric control charts can be very useful in practice and have recently drawn a lot of interest in the literature. Phase II distribution‐free (nonparametric) control charts are used when the parameters of the underlying unknown continuous distribution are unknown and can be estimated from a sufficiently large Phase I reference sample. While a number of recent studies have examined the in‐control (IC) robustness question related to the size of the reference sample for both univariate and multivariate normal theory (parametric) charts, in this paper, we study the effect of parameter estimation on the performance of the multivariate nonparametric sign exponentially weighted moving average (MSEWMA) chart. The in‐control average run‐length (ICARL) robustness and the out‐of‐control shift detection performance are both examined. It is observed that the required amount of the Phase I data can be very (perhaps impractically) high if one wants to use the control limits given for the known parameter case and maintain a nominal ICARL, which can limit the implementation of these useful charts in practice. To remedy this situation, using simulations, we obtain the “corrected for estimation” control limits that achieve a desired nominal ICARL value when parameters are estimated for a given set of Phase I data. The out‐of‐control performance of the MSEWMA chart with the correct control limits is also studied. The use of the corrected control limits with specific amounts of available reference sample is recommended. Otherwise, the performance the MSEWMA chart may be seriously affected under parameter estimation. Copyright © 2016 John Wiley & Sons, Ltd.