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 double exponentially weighted moving average control chart for monitoring COM‐Poisson attributes

The Conway‐Maxwell‐Poisson (COM‐Poisson) distribution is a two‐parameter generalization of the Poisson distribution, which can be used for overdispersed or underdispersed count data and also contains the geometric and Bernoulli distributions as special cases. This article presents a double exponentially weighted moving average control chart with steady‐state control limits to monitor COM‐Poisson attributes (regarded as CMP‐DEWMA chart). The performance of the proposed control chart has been evaluated in terms of the average, the median, and the standard deviation of the run‐length distribution. The CMP‐DEWMA control chart is studied not only to detect shifts in each parameter individually but also in both parameters simultaneously. The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the CMP‐DEWMA chart is more effective than the EWMA chart at detecting downward shifts of the process mean. Finally, a real data set is presented to demonstrate the application of the proposed chart.     

Poisson progressive mean control chart

In real life applications, many process‐monitoring problems in statistical process control are based on attribute data resulting from quality characteristics that cannot be measured on numerical or quantitative scales. For the monitoring of such data, a new attribute control chart has been proposed in this study, namely, the Poisson progressive Mean (PPM) control chart. The performance of the PPM chart is compared with the existing charts used for the monitoring of Poisson processes such as the Shewhart c‐chart, Poisson Exponentially Weighted Moving Average chart, Poisson double Exponentially Weighted Moving Average chart and the Poisson Cumulative Sum charts. The average run length comparison indicated the superior performance of the PPM chart in terms of shift detection ability. This study will help quality practitioners to choose an efficient attribute control chart.     

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.     

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.     

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.     

On designing an efficient control chart to monitor fraction nonconforming

Process control measures are mostly applied in production and manufacturing industries. The most important tool used in these disciplines is control chart. In manufacturing and production processes, when the quality characteristic of interest cannot be directly measured, it becomes essential to apply attribute control charts. To monitor fraction nonconforming of the output, quality practitioners mostly prefer p-chart. In this article, a new progressive mean (PM) control chart is being proposed for monitoring drift in proportion of nonconforming products. The design evaluations of the proposed chart are made and compared through different properties of run length distribution, such as average run length (ARL), standard deviation of run length (SDRL), and some percentile points. The performance of the proposed chart is assessed under zero-state and steady-state scenarios. The proposed PM chart is compared with p-chart, moving average (MA) chart, optimal CUSUM chart, modified exponentially weighted moving average (EWMA) chart, and runs rules p-charts for monitoring fraction nonconforming. The proposed chart spots efficiently sustained disturbances in the process as compared with their existing counterparts. Two illustrative examples are also provided; one from real-life application of nonconforming bearing and seal assemblies data and the other from simulated data for the implementation of PM chart.     

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.     

An attribute control chart for multivariate Poisson distribution using multiple dependent state repetitive sampling

In this paper, an attribute control chart for a multivariate Poisson distribution using multiple dependent state repetitive sampling (MDSRS) is presented. The evaluation of the proposed control chart is given through the average run length (ARL). The proposed control chart performs better than the existing control chart based on repetitive sampling and that using multiple dependent state sampling in terms of ARLs. A real example and a simulation study are added to explain the procedure and to demonstrate the power of the proposed control chart.     

Control Charts with Variable Dimension for Linear Combination of Poisson Variables

This article analyzes the simultaneous control of several correlated Poisson variables by using the Variable Dimension Linear Combination of Poisson Variables (VDLCP) control chart, which is a variable dimension version of the LCP chart. This control chart uses as test statistic, the linear combination of correlated Poisson variables in an adaptive way, i.e. it monitors either p₁ or p variables (p₁ < p) depending on the last statistic value. To analyze the performance of this chart, we have developed software that finds the best parameters, optimizing the out‐of‐control average run length (ARL) for a shift that the practitioner wishes to detect as quickly as possible, restricted to a fixed value for in‐control ARL. Markov chains and genetic algorithms were used in developing this software. The results show performance improvement compared to the LCP chart. Copyright © 2015 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.     

A mixed control chart to monitor the process

In this paper, we propose a mixed control chart to monitor the process quality using attribute data combined with variable data. The proposed control chart proceeds like an np control chart based on the number of nonconforming parts but requires variable data only when the decision is indeterminate. The control coefficients are determined by considering the in-control and the out-of-control average run lengths for various specified parameters. The extensive tables are provided for the industrial use. The advantages of the proposed control chart are discussed over the traditional np control 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.     

Nonparametric Progressive Mean Control Chart for Monitoring Process Target

Nonparametric control charts are widely used when the parametric distribution of the quality characteristic of interest is questionable. In this study, we proposed a nonparametric progressive mean control chart, namely the nonparametric progressive mean chart, for efficient detection of disturbances in process location or target. The proposed chart is compared with the recently proposed nonparametric exponentially weighted moving average and nonparametric cumulative sum charts using different run length characteristics such as the average run length, standard deviation of the run length, and the percentile points of the run length distribution. The comparisons revealed that the proposed chart outperformed recent nonparametric exponentially weighted moving average and nonparametric cumulative sum charts, in terms of detecting the shifts in process target. A real life example concerning the fill heights of soft drink beverage bottles is also provided to illustrate the application of the proposed nonparametric control chart. Copyright © 2012 John Wiley & Sons, Ltd.     

Bayesian Control Chart for Nonconformities

The c‐chart or the control chart for nonconformities is designed for the case where one deals with the number of defects or nonconformities observed. A control chart can be developed for the total or average number of nonconformities per unit, which is well modeled by the Poisson distribution. In this paper the c‐chart will be studied, where the usual operation of the c‐chart will be extended by introducing a Bayesian approach for the c‐chart. Control chart limits, average run lengths, and false alarm rates will be determined by using a Bayesian method. These results will be compared with the results obtained when using the classical (frequentist) method. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal EWMA of linear combination of Poisson variables for multivariate statistical process control

In this paper, we propose a new process control chart for monitoring correlated Poisson variables, the EWMA LCP chart. This chart is the exponentially weighted moving average (EWMA) version of the recently proposed LCP chart. The latter is a Shewhart-type control chart whose control statistic is a linear combination of the values of the different Poisson variables (elements of the Poisson vector) at each sampling time. As a Shewhart chart, it is effective at signalling large process shifts but is slow to signal smaller shifts. EWMA charts are known to be more sensitive to small and moderate shifts than their Shewhart-type counterparts, so the motivation of the present development is to enhance the performance of the LCP chart by the incorporation of the EWMA procedure to it. To ease the design of the EWMA LCP chart for the end user, we developed a user-friendly programme that runs on Windows© and finds the optimal design of the chart, that is, the coefficients of the linear combination as well as the EWMA smoothing constant and chart control limits that together minimise the out-of-control ARL under a constraint on the in-control ARL. The optimization is carried out by genetic algorithms where the ARLs are calculated through a Markov chain model. We used this programme to evaluate the performance of the new chart. As expected, the incorporation of the EWMA scheme greatly improves the performance of the LCP chart.     

A multivariate CUSUM control chart for monitoring Gumbel's bivariate exponential data

Exponentially distributed data are commonly encountered in high-quality processes. Control charts dedicated to the univariate exponential distribution have been extensively studied by many researchers. In this paper, we investigate a multivariate cumulative sum (MCUSUM) control chart for monitoring Gumbel's bivariate exponential (GBE) data. Some tables are provided to determine the optimal design parameters of the proposed MCUSUM GBE chart. Furthermore, both zero-state and steady-state properties of the proposed MCUSUM GBE chart for the raw and the transformed GBE data are compared with the multivariate exponentially weighted moving average (MEWMA) chart and the paired individual cumulative sum (CUSUM) chart. The results show that the proposed MCUSUM GBE chart outperforms the other two types of control charts for most shift domains. In addition, an extension to Gumbel's multivariate exponential (GME) distribution is also investigated. Finally, an illustrative example is provided in order to explain how the proposed MCUSUM GBE chart can be implemented in practice.     

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.     

Control charts for monitoring the ratio of two Poisson rates

In this paper, we are concerning in monitoring the ratio ρ of two Poisson rates by control charts. Let X and Y be two independent Poisson random variables with means λ₁ and λ₂=λ₁/ρ, respectively. The study considers that only individual observations X_i and Y_i are available at each sampling time i. The performance in detecting shifts on the ratio ρ using several statistics, some based on normalized transformations, is evaluated by an extensive simulation study. Two types of control charts, Shewhart and exponentially weighted moving average (EWMA), are considered. The one-sided control chart with upper control limit (UCL) is applied so that we are focusing on detecting when the ratio ρ shifted to higher rate in this paper. The results pointed out that EWMA control chart is a better alternative. Some guidelines indicating which statistics yield best performance are proposed for the practitioners.     

A moving average control chart using a robust scale estimator for process dispersion

Control charts are one of the most powerful tools used to detect and control industrial process deviations in statistical process control. In this paper, a moving average control chart based on a robust scale estimator of standard deviation, namely, the sample median absolute deviation (MAD) statistic, for monitoring process dispersion, is proposed. A simulation study is conducted to evaluate the performance of the proposed moving average median absolute deviation (MA‐MAD) chart, in terms of average run length for various distributions. The results show that the moving average MAD chart performs well in detecting small and moderate shifts in process dispersion, especially when the normality assumption is violated. In addition, this chart is very efficient, especially when the quality characteristic follows a skewed distribution. Numerical and simulated examples are given at the end of the paper.