The density control chart: a general approach for constructing a single chart for simultaneously monitoring multiple parameters

The majority of the existing literature on simultaneous control charts, i.e. control charting mechanisms that monitor multiple population parameters such as mean and variance on a single chart, assume that the process is normally distributed. In order to adjust and maintain the overall type-I error probability, these existing charts rely largely on the property that the sample mean and sample variance are independent under the normality assumption. Furthermore, the existing charting procedures cannot be readily extended to non-normal processes. In this article, we propose and study a general charting mechanism which can be used to construct simultaneous control charts for normal and non-normal processes. The proposed control chart, which we call the density control chart, is essentially based on the premise that if a sample of observations is from an in-control process, then another sample of observations is no less likely to be also from the in-control process if the likelihood of the latter is no less than the likelihood of the former. The density control chart is developed for normal and non-normal processes where the distribution of the plotting statistic of the density control chart can be explicitly derived. Real examples are given and discussed in these cases. We also discuss how the density control chart can be constructed in cases when the distribution of the plotting statistic cannot be determined. A discussion of potential future research is also given.     

The effectiveness of Max-half-Mchart over Max-Mchart in simultaneously monitoring process mean and variability of individual observations

Control charts are widely used in industrial environments for the simultaneous or separate monitoring of the process mean and process variability. The Max-Mchart is a multivariate Shewhart-type simultaneous control chart that is used when monitoring subgroups. While this sampling design allows the computation of the generalized variance (GV) used to calculate the process variability, a GV chart cannot be plotted for individual observations. Hence, we cannot compute the single statistic in the Max-Mchart. This study aims to overcome the aforementioned issue. To this end, first, we develop a new Max-Mchart for individual observations by utilizing the statistic in the dispersion control chart. Second, instead of the standard normal distribution, we propose a new transformation using a half-normal distribution to calculate the statistic for the process mean and process variability. Thus, the proposed chart is called the Max-Half-Mchart, whose control limit is calculated using the bootstrap approach. An evaluation based on the average run length values shows the robustness of the Max-Half-Mchart for the simultaneous monitoring of the process mean and process variability. The single statistic in the Max-Half-Mchart is more consistent with the statistic in Hotelling's T² and the dispersion chart than that of the Max-Mchart.     

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.     

面向小批量生产的统计过程控制的研究

指出了在小批量生产环境下实施统计过程控制存在的问题,用概率积分变换理论,给出了控制过程均值、过程方差的标准化控制图,适用于小批量生产环境下对过程均值、过程方差的有效控制。     

Detection of process upsets—sample autocorrelation control chart and group autocorrelation control chart applications

This paper presents an application of the sample autocorrelation function to statistical process control where the process data are serially correlated. Two innovative control charts are illustrated: the sample autocorrelation control chart and the group autocorrelation control chart. The important feature is that these control charts will detect shifts in the autocorrelative structure as well as shifts in the mean of the process. The sample autocorrelation function is typically used to identify an appropriate ARIMA model for a time series. The sample autocorrelation function may also be used as the basis of control charts to detect process upsets. Two unique features distingush this application of the sample autocorrelation function to statistical process control. First, the sample autocorrelations are exponentially smoothed estimates. This allows the user to control the sensitivity of the sample autocorrelation control chart. Secondly, the sample autocorrelation control chart is applied to a continuous stream of data—rather than to a static set of data that has been used to fit an ARIMA model.     

Deviance residual-based control charts for monitoring the beta-distributed processes

Beta-distributed process outputs are common in manufacturing industry because they range from 0 to 1 based on inputs like yield. Under the normality assumption, Shewarts control charts and Hotelling's control charts based on the deviance residual have been applied to monitor the process mean of the beta-distributed process outputs. The normality assumption can be violated according to the shape of the beta distribution. Therefore, without the normality assumption, we propose antirank control charts, exponentially weighted moving average (EWMA) control charts and cumulative sum (CUSUM) control charts. The proposed control charts outperform the existing control charts in the experimental results. The previous research has been focused on monitoring the process mean only. For the first time, in order to monitor the process variance of the beta-distributed process outputs, we propose the multivariate exponentially weighted mean squared deviation (MEWMS) chart, the first norm distance of the MEWMS deviation from its expected value (MEWMSL₁) chart, the chart based on MEWMS deviation with the approximated distribution of trace (MEWMS_AT), the multivariate trace sum squared deviation (MTSSD) chart and the multivariate matrix sum squared deviation (MMSSD) chart based on the deviance residual. The proposed control charts are compared and recommended in terms of the experimental results. This research can be a guideline for practitioners who monitor the deviance residual.     

Correction on link relative based approach to CUSUM chart

Link relative-based approach was used in an article (see reference 1) to enhance the performance of the cumulative sum (CUSUM) control chart. This technique involves the use of firstly, the link relative variable to convert the process observations in a relative to the mean form and secondly, optimal constants to define a new variable which is used as the plotting statistic of the link relative CUSUM chart. In this article, it is proven through simulation study that the optimal constants with fixed values, as reported in the aforementioned article, give different results. Instead, if the regression technique is used, then the same results will be obtained.     

A sum of squares triple exponentially weighted moving average control chart

Control charts are widely known quality tools used to detect and control industrial process deviations in statistical process control. In the current paper, we propose a new single memory-type control chart, called the sum of squares triple exponentially weighted moving average control chart (referred as SS-TEWMA chart), that simultaneously detects shifts in the process mean and/or process dispersion. The run length performance of the proposed SS-TEWMA control chart is compared with that of the sum of squares EWMA, sum of squares double EWMA, sum of squares generally weighted moving average, and sum of squares double generally weighted moving average, control charts, through Monte Carlo simulations. The comparisons indicate that the proposed chart is more efficient, than the competing ones, in detecting small shifts in the process mean and/or variability for most of the considered scenarios, while it has comparable performance for some others in identifying large shifts in the process mean and small to large shifts in the process variability. Finally, two illustrative examples are provided to explain the application of the SS-TEWMA control 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.     

Regression-adjusted Poisson EWMA control chart

Exponentially weighted moving average (EWMA) control chart has a significant effect in improving product quality and is widely used in various fields. In addition to continuous data, there are many Count Data in life that need to be monitored. Poisson distribution is one of the models that study the probability distribution of discrete data, and has a wide range of applications. In previous monitoring, it was considered that the mean value of Poisson distribution in normal state was a constant value after it was determined. But in the actual situation, there are many unavoidable objective conditions that will affect the final results. We cannot monitor all situations according to the same criteria. If we ignore the conditions that affect the occurrence of the event and directly monitor the final result, on the one hand, it will increase the probability of false alarms from the control chart. On the other hand, the control chart will not be able to detect problems in time due to the untimely update of conditions. In response to this situation, this paper proposes a regression-adjusted EWMA control chart to monitor the Poisson process. The control chart can continuously adjust and update the expected values according to the actual situation. It can make the monitoring process more reasonable and the monitoring results more valuable.     

Auxiliary information-based maximum generally weighted moving average chart for simultaneously monitoring process mean and variability

An auxiliary information-based (AIB) maximum exponentially weighted moving average (MaxEWMA) chart has been proposed to simultaneously monitor both increases and decreases in the process mean and/or variability, called the AIB-MaxEWMA chart, which is superior to the existing MaxEWMA chart. In this paper, we propose the AIB maximum generally weighted moving average chart, called the AIB-MaxGWMA chart, to further enhance the sensitivity of the AIB-MaxEWMA chart. Numerical simulation studies indicate that the AIB-MaxGWMA chart is sensitive to small shifts in the process mean and/or variability. The performance of the AIB-MaxGWMA chart based on average run lengths (ARLs) also outperforms than its counterparts including AIB-MaxEWMA, MaxGWMA and MaxEWMA charts. An example is used to illustrate the efficiency of the proposed AIB-MaxGWMA chart in detecting small process shifts.     

‘Alternated Charting Statistic Modified Group Runs’ (ACS-MGR) control chart for monitoring mean vectors of bivariate and trivariate processes

In this article, we propose “Alternated Charting Statistic-Modified Group Runs” (ACS-MGR) control chart for monitoring mean vectors of bivariate and trivariate processes, as an improvement over the ACS chart. This chart is obtained by combining ACS chart with the MGR chart, and it is verified that ACS-MGR chart significantly reduces “Average Time to Signal” (ATS) as compared with ACS chart, when the process has gone out of control. Steady state performance of the chart is also studied by comparing with steady state performance of the ACS-Syn chart and ACS-GR chart.     

New EWMA charts for process variance

The EWMA chart is effective in detecting small shifts in the process mean or process variance. Numerous EWMA charts for the process variance have been suggested in the literature. In this article, new one-sided and two-sided EWMA charts are developed for monitoring the variance of a normal process. In developing these new EWMA charts, first, new unbiased estimators of the process variance are developed, followed by incorporating the developed estimators into the new EWMA charts' statistics. The Monte Carlo simulation method is adopted to evaluate the zero-state and steady-state run-length performances of the proposed EWMA variance charts, in comparison with that of three existing EWMA variance charts and the weighted adaptive CUSUM variance chart. The findings reveal that the proposed charts generally perform better than the existing charts. An example of application is given to show the implementation of the proposed and existing charts in detecting increases or decreases in the process variance.     

A New Hybrid Exponentially Weighted Moving Average Control Chart for Monitoring Process Mean: Discussion

A new hybrid exponentially weighted moving average (HEWMA) control chart has been proposed in the literature for efficiently monitoring the process mean. In that paper, the computed variance of the HEWMA statistic was, unfortunately, not correct! In this discussion, the correct variance of the HEWMA statistic is given, and the run length characteristics of the HEWMA control chart are studied and explored. It is noticed that not only the superiority of the HEWMA control chart remains over the existing (considered before) charts but also the new results based on the corrected control limits are more profound and reflective. Copyright © 2016 John Wiley & Sons, Ltd.     

Binomial CUSUM chart with curtailment

The binomial cumulative sum (CUSUM) chart has been widely used to monitor the fraction nonconforming (p) of a process. It is a powerful procedure for detecting small and moderate p shifts. This article proposes a binomial CUSUM control chart using curtailment technique (Curt_CUSUM chart in short). The new chart is able to improve the overall detection effectiveness while holding the false alarm rate at a specified level. The results of the comparative studies show that, on average, the Curt_CUSUM chart is more effective than the CUSUM chart without curtailment by 30%, in terms of Average Number of Defectives, under different circumstances. The Curt_CUSUM chart can be applied to a 100% inspection as well as a general random sampling inspection.     

A comparison study of control charts for Weibull distributed time between events

Monitoring decreases in the mean of Weibull time between events data to address process quality deteriorations is an important task in reliability analysis. Two new control charts such as Weibull exponentially weighted moving average and mixed cumulative sum‐exponentially weighted moving average by transforming the Weibull data to the exponential data are proposed and compared with 2 existing control charts such as Weibull cumulative sum and mixed exponentially weighted moving average‐cumulative sum. The performance comparison provides a way to select a specific control chart in a given situation. The average run length and the standard deviation of the run length are used as performance measures. The relative mean index is also utilized to measure the overall performance. The smaller the value of the relative mean index, the better the performance of the control chart and vice versa. Two illustrative examples are provided to show the applications of the proposed control charts.     

An Effective Double Sampling Scheme for the c Control Chart

In the present paper is developed a statistical process control inspection procedure based on a new simple‐to‐implement and effective double sampling scheme for the c control chart, aimed at the minimization of the number of inspected observation units and warranting fixed levels for the type I and II error risks. In particular, the formulations of the false alarm risk α, the power P of the chart, and the expected number of inspected observation units for the developed inspection procedure are given, whereas a macro of Microsoft Excel is adopted to solve the tackled problem. In order to illustrate the application of the developed approach and to investigate on the influence of several operating parameters, numerical examples are carried out and the related considerations are given. Finally, by comparing the performance of the developed inspection procedure with that of the related classic c chart scheme, meaningful reduction of the number of the inspected observation units can be achieved by adopting the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

The balance chart: A new SPC Concept

This paper outlines a new technique of statistical process control which goes a considerable way to resolving several existing problems. The technique described may be of particular value to automated control, small batch control and control of gauged processes. A new charting technique is described and compared with traditional control charts. The operation of the balance chart is outlined for attribute and variable processes and in precontrol mode. A graphical system for determining estimated Cp, Cpk and process mean values from limited process data is also included.     

基于小批量多品种生产环境的统计过程质量控制研究

分析了先进制造技术环境下实施统计过程质量控制所存在的问题，给出了一种基于正态过程的改进的标准化控制图，适用于小批量多品种生产环境下对过程均值、过程方差的有效控制。