Shewhart-type Chart Research Articles

Shewhart-EWMA chart for monitoring binomial data subject to shifts of random amounts

Attribute charts are used widely for monitoring binary events in manufacturing, service, and healthcare processes. While Shewhart type charts are efficacious in detecting large or sudden shifts in a nonconforming rate p, exponentially weighted moving average (EWMA) type charts tend to be more powerful for detecting smaller or gradual shifts. Since many processes can shift by various amounts, this paper proposes a combined optimized Shewhart-EWMA dual chart that combines the strengths of both charts to quickly detect shifts of any random magnitude within a defined range. The performance of this combined Shewhart-EWMA chart is optimized by allocating total detection power between the Shewhart chart element and EWMA chart element in a manner that yields the best overall effectiveness while maintaining the false alarm rate at a specified desired level. In numeric analysis under a range of settings, the detection speed of this combined Shewhart-EWMA chart is found to outperform individual Shewhart and EWMA charts, with up to 499% and 31% fewer average number of defective items respectively until process shifts of random sizes are detected. It also demonstrates enhanced overall efficacy when compared to synthetic, cumulative sum, combined Shewhart-synthetic, and combined Shewhart-CUSUM charts.

On the power and robustness of phase I nonparametric Shewhart-type charts using sequential normal scores

ABSTRACT Control charts play a crucial role in industry, facilitating retrospective analysis of process data during its initial phase (Phase I) and monitoring its behavior over time once a chart is set (Phase II). This study introduces four novel Phase I nonparametric control charts based on combined forms of sequential normal scores. An adequate combination of characteristics from these statistics helps address early-stage data analysis challenges, including potential contamination and lack of normality. Their performance, evaluated by empirical alarm probability, was compared, and the preferred option was later tested against parametric approaches and available nonparametric alternatives across various practical scenarios. Previous Phase I charts have demonstrated difficulties handling specific out-of-control patterns and sample size restrictions. The results of this research indicate that the SNSmax statistic exhibits superior detection power compared to existing nonparametric methods, particularly when dealing with skewed distributions. Furthermore, it demonstrates robustness against isolated and certain types of sustained changes. This positions the SNSmax statistic as a reliable and powerful tool for quality assurance practitioners dealing with the assessment of independent batches of observations.

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.

Monitoring processes with multiple dependent production lines using time between events control charts

This article develops Time Between Events (TBE) control charts to monitor processes with multiple dependent production lines. To this end, a Shewhart-type and an EWMA-type TBE chart have been proposed. The copula approach is used to describe the dependence between production lines and the homogeneous Poisson process is considered to model the number of defectives. Performance of the proposed methods is evaluated using average time to signal metric. The numerical study showed that the EWMA-TBE chart uniformly performs better than the Shewhart-type chart. Eventually, the EWMA-TBE chart is applied to monitor two real-world processes with two and four production lines.

Efficient control chart-based monitoring of scale parameter for a process with heavy-tailed non-normal distribution

<abstract> <p>Statistical process control is a procedure of quality control that is widely used in industrial processes to enable monitoring by using statistical techniques. All production processes are faced with natural and unnatural variations. To maintain the stability of the production process and reduce variation, different tools are used. Control charts are significant tools to monitor a production process. In this article, we design an extended exponentially weighted moving average (EEWMA) chart under the assumption of inverse Maxwell (IM) distribution, an IM EEWMA (IMEEWMA) control chart. We have estimated the performance of the proposed chart in terms of various run-length (RL) properties, including the average RL, standard deviation of the RL and median RL. We have also carried out a comparative analysis of the proposed chart with the existing Shewhart-type chart for IM distribution (VIM chart) and IM exponential weighted moving average (IMEWMA) chart. We observed that the proposed IMEEWMA chart performed better than the VIM chart and IMEWMA chart in terms of the ability to detect small and moderate shifts. To demonstrate its practical application, we have applied the IMEEWMA chart, along with existing control charts, to monitor the lifetime of car brake pad data. This real-world example illustrates the superiority of the IMEEWMA chart over its counterparts in industrial scenarios.</p> </abstract>

EWMA Control Chart for Rayleigh Process With Engineering Applications

Process monitoring is typically performed with quality control charts. The normality assumption is often taken into consideration for the development of control charts. However, this assumption may not hold in many real-life situations. One such circumstance is utilizing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {V}_{\mathrm {SQR}}$ </tex-math></inline-formula> chart to track the process variability of Rayleigh distributed processes. The structure of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {V}_{\mathrm {SQR}}$ </tex-math></inline-formula> chart follows the basic design of the Shewhart-type chart, which is not sensitive mainly for smaller to moderate shifts in monitoring parameter. In this paper, an enhanced approach, namely Rayleigh exponential weighted moving average (REWMA) is introduced. The suggested REWMA scheme aims to significantly improve the detection capabilities of the traditional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {V}_{\mathrm {SQR}}$ </tex-math></inline-formula> chart. Designated limits, along with charting parameters, are evaluated for different sample sizes. The effectiveness of the suggested REWMA control chart is assessed in terms of run length distributional parameters. Moreover, the Monte Carlo simulations are made to compare the run length properties of the proposed REWMA chart with the existing competitor design. The comparative assessment demonstrates that the suggested offers a considerable enhancement relative to the competing design. An application of the REWMA chart on simulated data also reveals that the proposed chart is highly sensitive to smaller and persistent shifts in the scaling parameter of Rayleigh distribution. Finally, an example from real-life has been presented to illustrate the importance of the suggested chart.

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.

Nonparametric multivariate covariance chart for monitoring individual observations

Parametric and nonparametric multivariate control charts that are proven very useful in monitoring the covariance matrix of multivariate normally or “nearly” normally distributed continuous datasets have been proposed in statistical process control (SPC) literature. However, in many recent practical applications of SPC, the underlying systems or processes are characterised by discrete or a mixture of discrete and continuous multivariate random variables. In such cases, the available multivariate control charts for monitoring the covariance matrix of continuous processes are inadequate. We propose a multivariate nonparametric Shewhart-type chart for monitoring shifts in the covariance matrix of multivariate discrete or mixture of discrete and continuous random variables. The proposed chart first projects the multivariate dataset into Euclidean space. It then uses the Alt’s likelihood ratio obtained from the least absolute shrinkage and selection operator estimator that guarantees a well-conditioned estimate of the covariance matrix as the monitoring statistic. The proposed scheme does not require any parametric model assumptions and can be based on any distance measure of choice. It has the advantage of handling multivariate datasets of any type, including discrete, continuous or a mixture of discrete and continuous random variables. It uses the relationships among the process variables to build new variables that capture the dominant structure among the original variables. A bootstrap procedure is employed to obtain the control limit of the proposed chart for a suitable distance-based model through time. Simulation results show the advantage of the proposed chart in monitoring shifts in the covariance matrix. An illustrative example involving monitoring covariance structures of the lapping process in wafer semiconductor manufacturing and diagnosis single-proton emission computed tomography are provided to show the applications of the proposed chart.

Exponentially weighed moving average charts for monitoring zero-inflated proportions with applications in health care.

In the context of public health surveillance, the aim is to monitor the occurrence of health-related events. Among them, statistical process monitoring focuses very often on the monitoring of rates and proportions (i.e. values in ) such as the proportion of patients with a specific disease. A popular control chart that is able to detect quickly small to moderate shifts in process parameters is the exponentially weighed moving average control chart. There are various models that are used to describe values in . However, especially in the case of rare health events, zero values occur very frequently which, for example, denote the absence of the disease. In this paper, we study the performance and the statistical design of exponentially weighed moving average control charts for monitoring proportions that arise in a health-related framework. The proposed chart is based on the zero-inflated Beta distribution, a mixed (discrete-continuous) distribution, suitable for modelling data in . We use a Markov chain method to study the run length distribution of the exponentially weighed moving average chart. Also, we investigate the statistical design as well as the performance of the proposed charts. Comparisons with a Shewhart-type chart are also given. Finally, we provide an example for the practical implementation of the proposed charts.

Bootstrap beta control chart for monitoring proportion data

PurposeAs an alternative to the standard p and np charts along with their various modifications, beta control charts are used in the literature for monitoring proportion data. These charts in general use average of proportions to set up the control limits assuming in-control parameters known. The purpose of the paper is to propose a control chart for detecting shift(s) in the percentiles of a beta distributed process monitoring scheme when in-control parameters are unknown. Such situations arise when specific percentile of proportion of conforming or non-conforming units is the quality parameter of interest.Design/methodology/approachParametric bootstrap method is used to develop the control chart for monitoring percentiles of a beta distributed process when in-control parameters are unknown. Extensive Monte Carlo simulations are conducted for various combinations of percentiles, false-alarm rates and sample sizes to evaluate the in-control performance of the proposed bootstrap control charts in terms of average run lengths (ARL). The out-of-control behavior and performance of the proposed bootstrap percentile chart is thoroughly investigated for several choices of shifts in the parameters of beta distribution. The proposed chart is finally applied to two skewed data sets for illustration.FindingsThe simulated values of in-control ARL are found to be closer to the theoretical results implying that the proposed chart for percentiles performs well with both positively and negatively skewed data. Also, the out-of-control ARL values for the percentiles decrease sharply with both downward and upward small, medium and large shifts in the parameters. The phenomenon indicates that the chart is effective in detecting shifts in the parameters. However, the speed of detection of shifts varies depending on the type of shift, the parameters and the percentile being considered. The proposed chart is found to be effective in comparison to the Shewhart-type chart and bootstrap-based unit gamma chart.Originality/valueIt is worthwhile to mention that the beta control charts proposed in the literature use average of proportion to set up the control limits. However, in practice, specific percentile of proportion of conforming or non-conforming items should be more useful as the quality parameter of interest than average. To the best of our knowledge, no research addresses beta control chart for percentiles of proportion in the literature. Moreover, the proposed control chart assumes in-control parameters to be unknown, and hence captures additional variability introduced into the monitoring scheme through parameter estimation. In this sense, the proposed chart is original and unique.

One-sided Shewhart control charts for monitoring the ratio of two normal variables in short production runs

Monitoring the ratio of two normal random variables plays an important role in several manufacturing environments. In addition, the traditional control charts that have been developed for infinite production horizons can not function effectively to detect anomalies in short production runs. In this paper, we tackle this problem by proposing two one-sided Shewhart-type charts to monitor the ratio of two normal random variables for a finite horizon production. The statistical performance of the proposed charts is investigated using the truncated average run length as a performance measure in short production runs. In order to help the quality practitioner to implement these control charts, we have provided ready-to-use tables of the control limit parameters. An illustrative example from the food industry is given for illustration.

Corrigendum to “Shewhart-Type Charts for Masked Data: A Strategy for Handling the Privacy Issue”

Corrigendum to “Shewhart-Type Charts for Masked Data: A Strategy for Handling the Privacy Issue”

On the Development of EWMA Control Chart for Inverse Maxwell Distribution

Abstract Variations are usually present in every manufacturing process. Control charts are implemented to detect the assignable cause variations in a process. In this article, we design an exponentially weighted moving average (EWMA) chart under the assumption of inverse Maxwell distribution, namely inverse Maxwell EWMA (IMEWMA) chart. We have evaluated the performance of the proposed chart in terms of various run length (RL) properties, including average RL, standard deviation of RL, and median RL. To examine the overall functioning ability, we have estimated extra quadratic loss, relative average RL, and performance comparison index. We have also carried out comparative analysis of the proposed chart with the existing Shewhart-type chart for Maxwell distribution, V chart. We observed that the proposed IMEWMA chart performed better than the V chart to detect small and moderate shifts. The IMEWMA and the existing charts were applied to monitor the lifetime of car brake pads and survival time for breast cancer patients. This example also depicts the superiority of the proposed chart to its existing counterparts.

Shewhart-Type Charts for Masked Data: A Strategy for Handling the Privacy Issue

Data privacy is a serious issue and therefore needs our attention. In this study, we propose masking through randomized response techniques (RRTs) to ensure the privacy and thus to avoid falsification. We assume that the process characteristic is of sensitive nature, and due to privacy issue, the actual measurements cannot be shared with the monitoring team. In such situations, the producer is very likely to falsify the measurements. Consequently, the usual control charting techniques will mislead about the process status. We discuss different data masking strategies to be used with Shewhart-type control charts. The usual Shewhart-type control chart appears to be a subchart of the proposed charts. Average run length (ARL) is used as a performance measure of the study proposals. We have evaluated the performance of the proposed charts for different shift sizes and under different intensities of masking. We have also carried out a comparative analysis for various models under varying sensitivity parameters. We have also compared the performance of the proposals with the traditional Shewhart chart. It is observed that the B-L control chart under the RRT model performs better for smaller shifts and for larger shift sizes, the G-B chart under an unrelated question model tperforms better. A real-life application of the study proposal is also considered where monitoring of thickness of paint on refrigerators is of interest.

Shewhart-type monitoring schemes with supplementary w-of-w runs-rules to monitor the mean of autocorrelated samples

The simplicity of the Shewhart charts makes it popular in practice; however, it is insensitive to detecting small shifts. In an effort to preserve its simplicity but increase its detection ability, in this paper, we propose two Shewhart-type charts supplemented with w-of-w runs-rules to monitor the mean of autocorrelated samples using a first-order autoregressive model. It is shown that the higher the level of autocorrelation, the poor the proposed schemes perform. Hence, we implement the skipping sampling strategy which involves sampling of nonconsecutive observations to form the rational subgroups to compute the corresponding sample means. The Markov chain approach is used to derive zero- and steady-state closed-form expressions of the average run-length (ARL). To supplement the specific shift performance metric, i.e. ARL, we compute the overall performance metric so that these schemes can also be evaluated from a global point of view. A real-life example is provided to illustrate the implementation of the monitoring schemes proposed here.

High-dimensional data monitoring using support machines

Dealing with high-dimensional data is a major challenge primary in statistics and consequently in statistical process control (SPC). “Shewhart-type” charts are control charts using rational subgrouping. Shewhart-type charts are effective in the detection of large shifts. In high-dimensional settings, where the number of variables is nearly as large as or larger than the number of observations, it will be very hard to use Shewhart-type charts based on rational subgroups. An alternative is to use Shewhart-type charts based on individual observations. Also, for most conventional control charts, the design of the control limits is commonly based on the assumption that the quality characteristics follow a multivariate normal distribution. However, this may not be reasonable in many real-world problems. This paper addresses these issues and suggests a monitoring methodology motivated by statistical learning theory. The proposed multivariate control chart uses tensor space model to represent a high-dimensional vector. This chart makes use of information extracted from in-control preliminary samples. Simulation studies demonstrate that the proposed control chart has very good performances.

A new auxiliary information based cumulative sum median control chart for location monitoring

Control charts are commonly used tools in statistical process control for the detection of shifts in process parameters. Shewhart-type charts are efficient for large shift values, whereas cumulative sum (CUSUM) charts are effective in detecting medium and small shifts. Control chart use commonly assumes that data are free of outliers and parameters are known or correctly estimated based on an in-control process. In practice, these assumptions are not often true because some processes occasionally have outliers. Monitoring the location parameter is usually based on mean charts, which are seriously affected by violations of these assumptions. In this paper we propose several CUSUM median control charts based on auxiliary variables, and offer comparisons with their corresponding mean control charts. To monitor the location parameter, we examined the performance of mean and median control charts in the presence and absence of outliers. Both symmetric and non-symmetric processes were studied to examine the properties of the proposed control charts to monitor the location parameter using CUSUM control charts. We used different run length measures to study in-control and out-of-control performances of CUSUM charts. Results revealed that our proposed control charts perform much better than the traditional charts in the presence of outliers. A real application of our study was provided using data on concrete compressive strength as it relates to the quality of cement manufacturing.

Comparison of joint control schemes for multivariate normal i.i.d. output

The performance of a product frequently relies on more than one quality characteristic. In such a setting, joint control schemes are used to determine whether or not we are in the presence of unfavorable disruptions in the location ( $${\varvec{\mu }}$$ ) and spread ( $${\varvec{\varSigma }}$$ ) of a vector of quality characteristics. A common joint scheme for multivariate output comprises two charts: one for $${\varvec{\mu }}$$ based on a weighted Mahalanobis distance between the vector of sample means and the target mean vector; another one for $${\varvec{\varSigma }}$$ depending on the ratio between the determinants of the sample covariance matrix and the target covariance matrix. Since we are well aware that there are plenty of quality control practitioners who are still reluctant to use sophisticated control statistics, this paper tackles Shewhart-type charts for $${\varvec{\mu }}$$ and $${\varvec{\varSigma }}$$ based on three pairs of control statistics depending on the nominal mean vector and covariance matrix, $${\varvec{\mu }}_0$$ and $${\varvec{\varSigma }}_0$$ . We either capitalize on existing results or derive the joint probability density functions of these pairs of control statistics in order to assess the ability of the associated joint schemes to detect shifts in $${\varvec{\mu }}$$ or $${\varvec{\varSigma }}$$ for various out-of-control scenarios. A comparison study relying on extensive numerical and simulation results leads to the conclusion that none of the three joints schemes for $${\varvec{\mu }}$$ and $${\varvec{\varSigma }}$$ is uniformly better than the others. However, those results also suggest that the joint scheme with the control statistics $$n \, ( \bar{\mathbf {X}}-{\varvec{\mu }}_0 )^\top \, {\varvec{\varSigma }}_0^{-1} \, ( \bar{\mathbf {X}}-{\varvec{\mu }}_0 )$$ and $$\hbox {det} \left( (n-1) \mathbf{S} \right) / \hbox {det} \left( {\varvec{\varSigma }}_0 \right) $$ has the best overall average run length performance.

Weighted EWMA charts for monitoring type I censored Weibull lifetimes

ABSTRACTIn this article, we first propose a control chart for monitoring the Weibull scale parameter in the context of Type I censored data, assuming that the shape parameter is fixed. The proposed chart is a Shewhart-type chart based on a likelihood ratio test (LRT) that utilizes an exponentially weighted moving average of the log-likelihood. In the literature, this type of chart is called a weighted exponentially weighted moving average (WEWMA) chart. The WEWMA chart is compared with a more standard EWMA chart and a CUSUM chart, which were recently studied as alternative solutions to the monitoring problem. Numerical results show that the WEWMA chart often performs better than these two alternatives. Sensitivity studies show that the WEWMA chart is more robust to variations in batch size and censoring time. A rust-resistant example is used to illustrate the proposed WEWMA chart. We extend the WEWMA and CUSUM charts to the context where the shape parameter also needs to be monitored. Joint charts that simultaneously look for change in either parameter are proposed.

Independent attributes control chart for Markov-dependent processes

ABSTRACTStatistical process monitoring (SPM) is a very efficient tool to maintain and to improve the quality of a product. In many industrial processes, end product has two or more attribute-type quality characteristics. Some of them are independent, but the observations are Markovian dependent. It is essential to develop a control chart for such situations. In this article, we develop an Independent Attributes Control Chart for Markov Dependent Processes based on error probabilities criterion under the assumption of one-step Markov dependency. Implementation of the chart is similar to that of Shewhart-type chart. Performance of the chart has been studied using probability of detecting shift criterion. A procedure to identify the attribute(s) responsible for out-of-control status of the process is given.