Shewhart-type Control Charts Research Articles

Control charts for monitoring the capability of unstable process

Stability is an important criterion to be investigated at the early stage of any industrial process and stable processes are usually preferred over unstable ones. But a process can be unstable and nevertheless capable and, therefore, there is a clear need to monitor such situations to ensure an acceptable capability. In this study we introduce, in a first step, a new Shewhart-type control chart for normal processes based on the yield index S pk to monitor the capability of potentially unstable processes and then, in a second step, we extend it using an Exponentially Weighted Moving Average (EWMA) scheme. Moreover, the optimal values of the control parameters are obtained using a nonlinear optimization problem for which the out-of-control average run length is the objective function. The introduced charts’ sensitivity is examined for deterministic shift sizes. Numerical and graphical analyses are given to display the features of the proposed charts. A simulation study shows, how in such a situation, the introduced tool is successful to recognize changes in the process capability.

Bivariate robust control charts for individual observations

This article introduces a novel robust Shewhart-type control chart designed for monitoring the location of a bivariate process using the Comedian and Sn Covariance estimators. The process of setting control limits for the proposed control charts is demonstrated. To evaluate the performance of the newly proposed control charts, simulations are conducted and compared against existing methods found in the current literature. Furthermore, the article discusses the application of the proposed method using real-life data.

Optimal design of time-between-event control charts with parameter estimation

The tr chart is a Shewhart-type control chart used to monitor the time between events, especially in high-quality processes. It has been shown to be more efficient than classical attribute control charts based on count data. In practical applications, the in-control process parameters are often unknown and need to be estimated from a Phase I reference sample. When the available Phase I data are small and the chart parameters have to be estimated, a popular approach is to adjust the control chart limits from a conditional perspective to avoid frequent false alarms using the exceedance probability criterion. However, this approach ignores the practitioner-to-practitioner (p-to-p) variation caused by the random Phase I reference samples, which results in getting different control limits and chart performance for each practitioner Large p-to-p variation makes practitioners to have a limited confidence on using their own estimated charts. Hence, in this article, we propose to optimize the tr chart via an exact method so that it has a minimum p-to-p variation. Comparisons between the optimal and conventionally adjusted charts are made in both the in- and out-of-control cases. The most important results are that the optimal chart has a far smaller p-to-p variation and its unconditional average run length values are closer to the desired ones compared to the conventional approach regardless of the in- or out-of-control cases. Finally, two real examples are presented to illustrate the implementation of the proposed chart.

Wilcoxon-Type Control Charts Based on Multiple Scans

In this article, we establish new distribution-free Shewhart-type control charts based on rank sum statistics with signaling multiple scans-type rules. More precisely, two Wilcoxon-type chart statistics are considered in order to formulate the decision rule of the proposed monitoring scheme. In order to enhance the performance of the new nonparametric control charts, multiple scans-type rules are activated, which make the proposed chart more sensitive in detecting possible shifts of the underlying distribution. The appraisal of the proposed monitoring scheme is accomplished with the aid of the corresponding run length distribution under both in- and out-of-control cases. Thereof, exact formulae for the variance of the run length distribution and the average run length (ARL) of the proposed monitoring schemes are derived. A numerical investigation is carried out and depicts that the proposed schemes acquire better performance towards their competitors.

Improved shewhart-type control charts based on weak runs in multistate trials

ABSTRACT In this work we introduce two new classes of Shewhart-type control charts with supplementary runs rules using a three or four region partition of the observed characteristic’s range. The study of the control charts’ characteristics (run length distribution, average run length) is carried over by considering an extension of the concept of r -weak runs (which was recently introduced for sequences of binary trials) under a multistate framework. Our numerical experimentation provides useful hints for the implementation of the new control charts by a practitioner and reveals that the proposed charts significantly improve the detection power (as compared to other existing charts) especially for small shifts in the process mean.

Distribution-free control charts for monitoring scale in finite horizon productions

Distribution-free (also known as nonparametric) control charts have been shown to be useful for on-line monitoring of lot production within a finite horizon production (FHP) process. Despite the partial process knowledge at the beginning of production in a FHP process, a distribution-free control chart can be started without any restrictive assumption about the underlying distribution of the quality characteristic. In this work, three nonparametric Shewhart-type control charts are investigated to monitor the unknown scale parameter in FHP processes based on some well-known nonparametric tests for scale. To this end, the violation of the assumption of equal medians of the reference and the test population distributions is investigated and a fourth control chart based on the Moses test, which does not require this assumption, is considered. An extensive numerical analysis is performed to examine chart performance based on some metrics suitable in the FHP setting. A real industrial example is presented to show a practical implementation of the investigated charts. Conclusions and future research ideas are provided.

A Combined Runs Rules Scheme for Monitoring General Inflated Poisson Processes

In this work, a control chart with multiple runs rules is proposed and studied in the case of monitoring inflated processes. Usually, Shewhart-type control charts for attributes do not have a lower control limit, especially when the in-control process mean level is very low, such as in the case of processes with a low number of defects per inspected unit. Therefore, it is not possible to detect a decrease in the process mean level. A common solution to this problem is to apply a runs rule on the lower side of the chart. Motivated by this approach, we suggest a Shewhart-type chart, supplemented with two runs rules; one is used for detecting decreases in process mean level, and the other is used for improving the chart’s sensitivity in the detection of small and moderate increasing shifts in the process mean level. Using the Markov chain method, we examine the performance of various schemes in terms of the average run length and the expected average run length. Two illustrative examples for the use of the proposed schemes in practice are also discussed. The numerical results show that the considered schemes can detect efficiently various shifts in process parameters in either direction.

Designing efficient dispersion control charts under various ranked-set sampling approaches

This study presents an in-depth evaluation of fourteen distinct ranked-set sampling methodologies employed to enhance the efficacy of Shewhart-type dispersion charts in identifying persistent shifts in underlying industrial processes. Despite the widespread use of Shewhart-type control charts under simple random sampling for monitoring process dispersion parameters, the efficiency of these dispersion charts under ranked-set sampling procedures has been largely overlooked. Through comprehensive simulation exercises, we obtained the run-length characteristics of the proposed charts, thereby laying a foundation for the comparative analysis of different sampling strategies. Our findings emphasize the superior performance of modified neoteric ranked-set sampling, which consistently outperformed the other techniques in all tested scenarios. Furthermore, we included an example to reinforce the significance and utility of the proposed methodologies in real-world industrial settings. This study provides additional results for optimal statistical control processes, highlighting the potential of ranked-set sampling approaches for enhancing process monitoring mechanisms.

On the design and performance of upper one-sided Shewhart charts for zero-inflated Poisson processes with estimated parameters

ABSTRACT In this work we study the performance of the upper one-sided Shewhart-type control charts for monitoring a zero-inflated Poisson process, in the case of estimated process parameters. In practice, the process parameters are unknown and must be estimated from a Phase I dataset. However, this affects the theoretical performance of a control chart and further investigation is needed to assess how much the performance of the chart is affected. Moreover, practical guidelines are necessary regarding the size of the Phase I sample so as the performance of the control chart in the case of estimated parameters is close to the known parameters case. In this work we use Monte Carlo simulation and evaluate various performance measures that provide useful information on chart’s performance in the case of estimated parameters. Both the in-control and the out-of-control performance are investigated. Moreover, we suggest adjustments to obtain the desired in-control performance, when the size of the Phase I sample is prespecified. The results show that very large Phase I samples are needed in order to make the chart’s performance, in the case of estimated parameters, close to the theoretical one. Using the suggested adjustments, these differences can be significantly reduced.

A control chart for monitoring images using jump location curves

Image monitoring is a relatively new research area in statistics and machine learning that has wide applications in various fields including medical diagnostics and subsequently disease monitoring, satellite imaging, security systems, and so forth. In the literature, a vast majority of the methods use image intensity changes to detect out of control images. However, this approach is often unreasonable in many real-life applications where a change in contrast between the background and foreground of an image should not indicate an out of control image as long as the boundaries of the image objects remain unchanged. In this article, we propose a Shewhart-type control chart to monitor grayscale images using detected edges of the images. The central idea is to monitor the Hausdorff distance between the point-set of detected edge pixels in each image from the corresponding point-set of the estimated true in-control image. The proposed monitoring procedure should be easy to execute many real-life applications. Numerical studies show that it performs well in various types of situations in comparison with a number of competing methods.

A new approach for designing the Shewhart-type control charts with generalized sensitizing rules

Two commonly used approaches for developing a Shewhart-type control chart with sensitizing rules are; the Markov Chain approach (MCA) and the single polynomial approach (SPA). The MCA for designing a control chart involves linear equations based on the choice of sensitizing rules. The MCA often becomes complex as the system of linear equations increases in size. The existing SPA for designing a control chart with sensitizing rules results in inconsistent in-control run length properties (RLP). To overcome the limitations of existing approaches, we present a new SPA for designing the Shewhart-type control charts with sensitizing rules. Furthermore, the new SPA has an interesting relationship with classical geometric distribution, generalized geometric distribution, and MCA. To show the significance of the new SPA, we have implemented it in designing the two-sided and one-sided mean control charts with sensitizing rules. The performance of the proposed and existing control charts is evaluated and compared using various RLP. The results reveal that the in-control RLP of the new SPA based mean control charts are sustained at their desired level. The out-of-control RLP is elaborated to highlight an optimal choice of sensitizing rules and other factors. Extensive comparative analysis shows that the new SPA based mean control charts have advantages over sustained behavior and computational reduction. Lastly, a real-life example of proposed methods is included using data on turnaround time of complete blood count (CBC) test results.

A study on the performance of the probability-limit control charts based on the geometric and negative binomial distributions

The Shewhart-type control chart based on the geometric distribution is widely used to monitor the number of normal events between two consecutive appearances of adverse events, but this control chart relies on the normality assumption and may thus result in unsatisfactory performance due to the high skewness of the geometric distribution. Thus, one can consider the probability-limit control chart based on the geometric distribution to correct the skewness, whereas the average run length or the false alarm rate of this control chart has been shown to be seriously biased despite of the skewness correction. In this paper, we first illustrate that the biased issue is mainly due to discreteness of the geometric distribution and then advocate the use of the control chart based on the negative binomial distribution. Numerical results from extensive Monte Carlo simulations clearly show that the control chart based on the negative binomial not only reduces the bias, but also outperforms the one based on the geometric control chart in a sense of smaller bias and high alarm rate.

Control charts for monitoring relative risk rate in the presence of Weibull competing risks with censored and masked data

ABSTRACT Competing risks data frequently appear in real-world operations like quality inspections, survival analysis, reliability tests, and clinical trials. From the quality point of view, relative risk rate can be considered an interesting quality indicator in analyzing the competing risks data for statistical process monitoring purposes. The relative risk rate measures the proportion of failures caused by the primary risk among a set of competing risks. This paper introduces two Shewhart-type control charts for monitoring the relative risk rate when the lifetimes of competing risks are independent Weibull random variables. The former chart is constructed based on the maximum likelihood estimation method, while the latter is developed based on the Bayesian approach. The proposed control charts can be applied in Phase II. The calculation of the Bayesian control charts and the evaluation of both process monitoring techniques have been done based on Monte Carlo simulations. The performance of the proposed control charts has been examined based on the average run length metric. The illustrative example is also discussed in detail to demonstrate the applicability of the proposed methods.

On Designing of Bayesian Shewhart-Type Control Charts for Maxwell Distributed Processes with Application of Boring Machine

The quality characteristic(s) are assumed to follow the normal distribution in many control chart constructions, although this assumption may not hold in some instances. This study proposes the Bayesian-I and Bayesian-II Shewhart-type control charts for monitoring the Maxwell scale parameter in the phase II study. The posterior and predictive distributions are used to construct the control limits for the proposed Bayesian-I and Bayesian-II Shewhart-type control charts, respectively. Various performance indicators, including average run length, quadratic loss, relative average run length, and performance comparison index, are utilized to evaluate the performance of the proposed control charts. The Bayesian-I and Bayesian-II Shewhart-type control charts are compared to their competitive CUSUMV, EWMAV and V control charts. Sensitivity analysis is also performed to study the effect of hyperparameter values on the performance behavior of the proposed control charts. Finally, real-life data is analyzed for the implementation of the proposed control charts.

A Class of Enhanced Nonparametric Control Schemes Based on Order Statistics and Runs

In this article, we establish a new class of nonparametric Shewhart-type control charts based on order statistics with signaling runs-type rules. The proposed charts offer to the practitioner the opportunity to reach, as close as possible, a pre-specified level of performance by determining appropriately their design parameters. Special monitoring schemes, already established in the literature, are ascertained to be members of the proposed class. In addition, several new nonparametric control charts that belong to the family are introduced and studied in some detail. Exact formulae for the variance of the run length distribution and the average run length (ARL) for the proposed monitoring schemes are also derived. A numerical investigation is carried out and demonstrates that the proposed schemes acquire competitive performance in detecting the shift of the underlying distribution. Although the large number of design parameters is quite hard to handle, the numerical results presented throughout the lines of the present manuscript provide practical guidance for the implementation of the proposed charts.

Simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles with a new adaptive Shewhart-type control chart

There has been a growing interest in research regarding monitoring a process through a regression model (called a profile) rather than a simple quality characteristic. This paper proposes a new monitoring scheme to simultaneously monitor the multivariate multiple linear profiles’ parameters. This scheme is based on the Shewhart control chart concept and only has one single (max-type) control chart for monitoring regression coefficients and error’s variation, which uses a new statistic to improve the variability (error’s variance-covariance matrix) shift detection in multivariate profiles. To increase the sensitivity and capability of the proposed scheme, especially in detecting small to moderate shift sizes, we add a variable parameters (VP) adaptive scheme to the developed control chart as well, considering that no adaptive monitoring schemes have so far been developed for monitoring the multivariate multiple linear profiles and neither are there any VP adaptive features for all profile monitoring schemes. Next, we develop a Markov chain model to compute the time to signal and run length performance measures. After that, we perform extensive numerical analyses to first compare the proposed control chart with the best available control charts and then evaluate its performance under different shift scenarios as well as different dimensions. The results show that the new monitoring scheme performs well compared to the best available monitoring schemes, and more importantly, it is more applicable in real practice. Finally, an illustrative example is presented to show the implementation of the proposed scheme in practice.

Phase I control chart for individual autocorrelated data: application to prescription opioid monitoring

Phase I or retrospective process monitoring plays a key part in an overall statistical process monitoring (SPM) regime and is increasingly emphasized in the recent literature. At present, a lot of the data in a variety of settings (public and private sector organizations) are collected individually and sequentially and thus are serially correlated (or autocorrelated). Though a reasonable amount of work is available in the control charting literature for prospective (Phase II) autocorrelated data monitoring, very little work exists for the retrospective phase (Phase I). In this article, we present a Shewhart-type control chart for Phase I monitoring of individual autocorrelated data, assuming normality, with estimated parameters. The methodology, while developed and presented for the first-order autoregressive (AR(1)) model for simplicity, may be adapted to more general time series models. The correct charting constants, adjusted for autocorrelation and parameter estimation, are derived, and tabulated for a nominal in-control (IC) false alarm probability (FAP). Simulation results show that the proposed chart is favorably IC FAP robust and effective for reasonably small sample sizes, moderate autocorrelation, and some model miss-specifications, compared to other approaches. An illustration using some public health data involving prescription fentanyl transactions is provided to show the potential for broader areas of applications of the proposed methodology. Along with a summary and recommendations, some future research areas are indicated. An R package is developed and made available for implementing the proposed methodology on demand.

Design of Shewhart-type control charts with estimated parameter for the Rayleigh distribution using frequentist and Bayesian approaches

ABSTRACT Studies on control charts with estimated parameters have received much attention in the recent literature. In this paper, the effect of parameter estimation on the performance of the Shewhart-type chart for the Rayleigh distribution, namely the chart, is first studied under the conditional perspective. It is found that parameter estimation has a serious effect on the performance of the frequentist chart. In order to solve this problem, the frequentist chart is adjusted by using the exceedance probability criterion to guarantee the in-control performance. Since the frequentist chart uses the sample information from Phase I, but not the process information from past experience, an alternative chart, namely the Bayesian chart, is proposed based on the predictive distribution of the plotting statistic. The performances of the Bayesian and adjusted frequentist charts are evaluated and compared in terms of the percentiles, mean, and standard deviation of the conditional average run length distribution. The results suggest that the Bayesian chart outperforms the frequentist counterpart, especially when more prior information is available.

On enhanced estimation and monitoring of population variance using robust auxiliary parameters based estimators

SummarySearching for efficient estimators of population variance as compared to sample variance has been the key area of interest of the researchers. Auxiliary variable has helped very much in finding estimators more efficient than sample variance. Auxiliary variable is greatly correlated with main study variable. Many authors including Isaki (J Am Stat Assoc 1983, 78, 117) used positively correlated auxiliary variable in the form of various conventional and non‐conventional auxiliary parameters and suggested different ratio type estimators of population variance. In this article, we have suggested a naïve class of variance estimators based on some robust auxiliary parameters, which deals with the outliers in the data. The properties including bias and Mean Square Error (MSE) of suggested family of estimators are studied up to first order of approximation. The theoretical as well as empirical comparison of introduced family of estimators is carried out with competing variance estimators. The estimator with the least MSE is recommended in different areas of applications. At last, the Shewhart type control charts based on the proposed estimators are also studies for application in quality control.

Monitoring the multivariate coefficient of variation in presence of autocorrelation with variable parameters control charts

ABSTRACT The coefficient of variation is a very important process parameter in many processes. A few control charts have been considered so far for monitoring its multivariate counterpart, i.e., the multivariate coefficient of variation (MCV). In addition, autocorrelation is very likely to occur in processes with high sampling frequency. Hence, designing suitable control charts and investigating the effect of autocorrelation on these charts is necessary. However, no control chart has been developed so far for the coefficient of variation that is capable of accounting for autocorrelation in either univariate or multivariate cases. This paper fills the gap by developing multivariate Shewhart-type control charts to monitor MCV with different autocorrelation structures for the observations: vector autoregressive, vector moving average, and vector mixed autoregressive and moving average. In addition, we add variable parameters adaptive features to the Shewhart-type scheme, in order to improve its performance. We develop a Markov chain model to get the statistical performance measures; then, we perform extensive numerical analyses to evaluate the effect of autocorrelation on adaptive and non-adaptive charts in the presence of downward and upward MCV shifts. Finally, we present an illustrative example from a healthcare process to show the implementation of this scheme in real practice.