Monitoring count data with Shewhart control charts based on the Touchard model

The most common control chart used to monitor count data is based on Poisson distribution, which presents a strong restriction: The mean is equal to the variance. To deal with under- or overdispersion, control charts considering other count distributions as Negative Binomial (NB) distribution, hyper-Poisson, generalized Poisson distribution (GPD), Conway–Maxwell–Poisson (COM-Poisson), Poisson–Lindley, new generalized Poisson–Lindley (NGPL) have been developed and can be found in the literature. In this paper we also present a Shewhart control chart to monitor count data developed on Touchard distribution, which is a three-parameter extension of the Poisson distribution (Poisson distribution is a particular case) and in the family of weighted Poisson models. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under- or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. Consequences in terms of speed to signal departures of stability of the parameters are obtained when incorrect control limits based on non-Touchard distribution (like Poisson, NB or COM-Poisson) are used to monitor count data generated by a Touchard distribution. Numerical examples illustrate the current proposal.     

Directionally sensitive weighted adaptive multivariate CUSUM mean charts

In many service and manufacturing industries, process monitoring involves multivariate data, instead of univariate data. In these situations, multivariate charts are employed for process monitoring. Very often when the mean vector shifts to an out-of-control situation, the exact shift size is unknown; hence, multivariate charts for monitoring a range of the mean shift sizes in the mean vector are adopted. In this paper, directionally sensitive weighted adaptive multivariate CUSUM charts are developed for monitoring a range of the mean shift sizes. Directionally sensitive charts are useful in situations where the aim lies in monitoring either an increasing or a decreasing shift in the mean vector of the quality characteristics of interest. The Monte Carlo simulation is used to compute the run length characteristics in comparing the sensitivities of the proposed and existing multivariate CUSUM charts. In general, the directionally sensitive and weighted adaptive features enhance the sensitivities of the proposed multivariate CUSUM charts in comparison with the existing multivariate CUSUM charts without the adaptive feature or those that are directionally invariant. It is also found that the variable sampling interval feature enhances the sensitivities of the proposed and existing charts as compared to their fixed sampling interval counterparts. The implementation of the proposed charts in detecting upward and downward shifts in the in-control process mean vector is demonstrated using two different datasets.     

Median control charts for monitoring asymmetric quality characteristics double bounded

In many practical situations, the quality characteristics of interest assume values in the range (0,1), like rates and proportions (but they are not results from Bernoulli experiments). Most control charts built for these quality characteristics rely on monitoring parameters of their probability distribution functions or on their averages after some reparameterization of their density probability function. However, for highly asymmetric distributions, the median is a more appropriate location parameter than the average. In this paper, we propose Shewhart-type control charts for monitoring the median of observations taken from quality characteristics double bounded after reparameterization of two probability density functions: Kumaraswamy and unit Weibull. The performance of the control charts is evaluated and compared in terms of run length (RL) analysis considering three estimators for the median. Finally, we also carry out two applications to demonstrate the applicability of these control charts.     

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.     

A control chart for monitoring the lognormal process variation using repetitive sampling

Control charts are developed to make the specific quality measures for a successful production process and follow normal distribution behaviors. But some real-life practices do not match such practices and exhibit some positively skewed behavior like lognormal distribution. The present study has considered this situation and proposed a monitoring control chart based on lognormal process variation using a repetitive sampling scheme. This concept proved better for detecting shifts as quickly as possible, and compared with the existing concept, results are elaborated through extensive tables. The average run lengths and standard deviations of the run lengths are being used as a performance evaluation measures and computed by using Monte Carlo simulations performed in R language. A real-life situation has been discussed in the example section to strengthen the proposed control chart concept in a real-life situation.     

Control charts for monitoring the median parameter of Birnbaum-Saunders distribution

In this paper, we propose control charts for monitoring the Birnbaum-Saunders (BS) median parameter (scale parameter) on the basis of three estimators. Comparison of the control charts in terms of average run length using probability control limits and those based on asymptotic distribution of three estimators for the median parameter is developed. We also present guidelines for practitioners about the minimum sample size needed to match out-of-control average run length with the asymptotic control limits in function of the median parameter after an extensive simulation study. Numerical example illustrates the applied monitoring of BS median parameter.     

A double homogeneously weighted moving average control chart for monitoring of the process mean

In the service and manufacturing industry, memory-type control charts are extensively applied for monitoring the production process. These types of charts have the ability to efficiently detect disturbances, especially of smaller amount, in the process mean and/or dispersion. Recently, a new homogeneously weighted moving average (HWMA) chart has been proposed for efficient monitoring of smaller shifts. In this study, we have proposed a new double HWMA (DHWMA) chart to monitor the changes in the process mean. The run length profile of the proposed DHWMA chart is evaluated and compared with some existing control charts. The outcomes reveal that the DHWMA chart shows better performance over its competitor charts. The effect of non-normality (in terms of robustness) and the estimation of the unknown parameters on the performance of the DHWMA chart are also investigated as a part of this study. Finally, a real-life industrial application is offered to demonstrate the proposal for practical considerations.     

Memory-type multivariate charts with fixed and variable sampling intervals for process mean when covariance matrix is unknown

Memory-type multivariate charts have been widely recognized as a potentially powerful process monitoring tool because of their excellent speed in detecting small-to-moderate shifts in the mean vector of a multivariate normally distributed process, namely, the multivariate EWMA (MEWMA), double MEWMA, Crosier multivariate CUSUM (MCUSUM), and Pignatiello and Runger MCUSUM charts. These multivariate charts are based on the assumption that the covariance matrix is known in advance; but, it may not be known in practice. It is thus not possible to use these multivariate charts unless a large Phase I dataset is available from an in-control process. In this paper, we propose multivariate charts with fixed and variable sampling intervals for the process mean vector when the covariance matrix is estimated from sample. Using the Monte Carlo simulation method, the run length characteristics of the multivariate charts are computed. It is shown that the in-control and out-of-control run length performances of the proposed multivariate charts are robust to the changes in the process covariance matrix, while the existing multivariate charts are not. A real dataset is taken to explain the implementation of the proposed multivariate charts.     

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.     

A synthetic double sampling control chart for process mean using auxiliary information

A control chart is a simple yet powerful tool that is extensively adopted to monitor shifts in the process mean. In recent years, auxiliary‐information–based (AIB) control charts have received considerable attention as these control charts outperform their counterparts in monitoring changes in the process parameter(s). In this article, we integrate the conforming run length chart with the existing AIB double sampling (AIB DS) chart to propose an AIB synthetic DS chart for the process mean. The AIB synthetic DS chart also encompasses the existing synthetic DS chart. A detailed discussion on the construction, optimization, and evaluation of the run length profiles is provided for the proposed control chart. It is found that the optimal AIB synthetic DS chart significantly outperforms the existing AIB Shewhart, optimal AIB synthetic, and AIB DS charts in detecting various shifts in the process mean. An illustrative example is given to demonstrate the implementation of the existing and proposed AIB control charts.