Shewhart-type Chart Research Articles

Improved Shewhart-Type Charts for Monitoring Times Between Events

In high-yield processes with a low defect rate, the recommendation is to monitor the times between events (TBE) with variables control charts in order to overcome the weaknesses of conventional attribute control charts. In this paper, we incorporate supplementary runs rules into one of the existing Shewhart-type TBE charts, namely the tr -chart, in order to improve its small to moderate size-shift detection ability. Moreover, because the existing tr -chart based on probability limits is biased, we provide a procedure for designing a tr -chart, with or without runs rules, that is unbiased. The performance of several tr -charts with runs rules is investigated, whereas comparisons with EWMA and CUSUM charts are also given. It is seen that some of the runs rules enhanced tr -charts perform substantially better. A real-data example is provided for illustration and some concluding remarks are given.

Control charts for generalized exponential distribution percentiles

ABSTRACTThe generalized exponential (GE) distribution, which was introduced by Mudholkar and Srivastava in 1993, has been studied for various applications of lifetime modelings. In this article, five control charts, that comprise the Shewhart-type chart and four parametric bootstrap charts based on maximum likelihood estimation method, the moment estimation method, probability plot method, and least-square error method for the GE percentiles, are investigated. An extensive Monte Carlo simulation study is conducted to compare the performance among all five control charts in terms of average run length. Finally, an example is given for illustration.

Bayesian Monitoring of Times Between Events: The Shewhart tr-Chart

The traditional (frequentist) tr-chart is a Shewhart-type chart useful for monitoring times between events (interarrival times) following an exponential distribution. This problem often arises in high-yield processes where the defect rate is low and hence the conventional attribute charts such as the c-chart and the u-chart are often ineffective. We consider this problem under the Bayesian framework and propose a Bayesian tr-chart when the exponential rate parameter is unknown. The Bayesian tr-chart is a Shewhart-type chart that incorporates parameter uncertainty via a prior and a posterior distribution, unlike the traditional tr-chart. The control limits are constructed from the predictive distribution of a plotting statistic. The performance of the proposed chart is evaluated and comparisons are made with the traditional tr-chart. The Bayesian chart is seen to be advantageous in certain situations. An illustrative example is given and a summary and conclusions are offered.

Control charts for the coefficient of variation

The coefficient of variation (CV) has received an extensive attention in quality control. The existing control charts for monitoring the two-sided changes in the CV do not consider the average run length (ARL) unbiased criterion. In practice, the practitioners prefer the ARL-unbiased control charts. This paper proposes an ARL-unbiased Shewhart-type control chart to monitor the two-sided shifts in the CV. Then an ARL-unbiased synthetic chart in which the conforming run length sub-chart only needs a lower control limit is proposed to improve the performance of the ARL-unbiased Shewhart-type chart. The comparison results show that the proposed synthetic chart outperforms the proposed ARL-unbiased control chart in terms of the ARL. The effects of the parameter estimation on the proposed charts are also studied. Finally, a real example is used to illustrate the proposed control charts.

A Study on EWMA charts with runs rules—the Markov chain approach

ABSTRACTRuns rules are usually used with Shewhart-type charts to enhance the charts' sensitivities toward small and moderate shifts. Abbas et al. in 2011 took it a step further by proposing two runs rules schemes, applied to the exponentially weighted moving average (EWMA) chart and evaluated their average run length (ARL) performances using simulation. They showed that the proposed schemes are superior to the classical EWMA chart and other schemes being investigated. Besides pointing out some erroneous ARL and standard deviation of the run length (SDRL) computations in Abbas et al., this paper presents a Markov chain approach for computing the ARL, percentiles of the run length (RL) distribution and SDRL, for the two runs rules schemes of Abbas et al. Using Markov chain, we also propose two combined runs rules EWMA schemes to quicken the two schemes of Abbas et al. in responding to large shifts. The runs rules (basic and combined rules) EWMA schemes will be compared with some existing control charting methods, where the former charts are shown to prevail.

Comparisons of Shewhart-type rank based control charts for monitoring location parameters of univariate processes

The nonparametric (distribution-free) control charts are robust alternatives to the conventional parametric control charts when the form of underlying process distribution is unknown or complicated. In this paper, we consider two new nonparametric control charts based on the Hogg–Fisher–Randle (HFR) statistic and the Savage rank statistic. These are popular statistics for testing location shifts, especially in right-skewed densities. Nevertheless, the control charts based on these statistics are not studied in quality control literature. In the current context, we study phase-II Shewhart-type charts based on the HFR and Savage statistics. We compare these charts with the Wilcoxon rank-sum chart in terms of false alarm rate, out-of-control average run-length and other run length properties. Implementation procedures and some illustrations of these charts are also provided. Numerical results based on Monte Carlo analysis show that the new charts are superior to the Wilcoxon rank-sum chart for a class of non-normal distributions in detecting location shift. New charts also provide better control over false alarm when reference sample size is small.

One-Sided Shewhart-type Charts for Monitoring the Coefficient of Variation in Short Production Runs

Monitoring the coefficient of variation is an effective approach to Statistical Process Control when the process mean and standard deviation are not constant but their ratio is constant. Until now, research has not investigated the monitoring of the coefficient of variation for short production runs. Viewed under this perspective, this paper proposes a new method to monitor the coefficient of variation for a finite horizon production by means of one-sided Shewhart-type charts. Tables are provided for the statistical properties of the proposed charts when the shift size is deterministic. Two illustrative examples are given in order to illustrate the use of these charts on real data.

Control Charts for Monitoring Burr Type-X Percentiles

When the sampling distribution of a parameter estimator is unknown, using normality asymptotically, the Shewhart-type chart may provide improper control limits. To monitor Burr type-X percentiles, two parametric bootstrap charts (PBCs) are proposed and compared with the Shewhart-type chart via a Monte Carlo simulation. Simulation results exhibit that the proposed PBCs perform well with a short average run length to signal out-of-control when the process is out-of-control, and have more adequate control limits than the Shewhart-type chart in view of in-control false alarm rate. An example regarding single fiber strength is presented for illustrating the proposed PBCs.

Monitoring the covariance matrix with fewer observations than variables

Multivariate control charts are essential tools in multivariate statistical process control. In real applications, when a multivariate process shifts, it occurs in either location or scale. Several methods have been proposed recently to monitor the covariance matrix. Most of these methods deal with a full rank covariance matrix, i.e., in a situation where the number of rational subgroups is larger than the number of variables. When the number of features is nearly as large as, or larger than, the number of observations, existing Shewhart-type charts do not provide a satisfactory solution because the estimated covariance matrix is singular. A new Shewhart-type chart for monitoring changes in the covariance matrix of a multivariate process when the number of observations available is less than the number of variables is proposed. This chart can be used to monitor the covariance matrix with only one observation. The new control chart is based on using the graphical LASSO estimator of the covariance matrix instead of the traditional sample covariance matrix. The LASSO estimator is used here because of desirable properties such as being non-singular and positive definite even when the number of observations is less than the number of variables. The performance of this new chart is compared to that of several Shewhart control charts for monitoring the covariance matrix.

A Phase I nonparametric Shewhart-type control chart based on the median

A nonparametric Shewhart-type control chart is proposed for monitoring the location of a continuous variable in a Phase I process control setting. The chart is based on the pooled median of the available Phase I samples and the charting statistics are the counts (number of observations) in each sample that are less than the pooled median. An exact expression for the false alarm probability (FAP) is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits for a specified nominal FAP value (of 0.01, 0.05 and 0.10, respectively) and for some values of the sample size (n) and the number of Phase I samples (m). Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. A simulation study shows that the proposed chart performs as well as, and in some cases better than, an existing Shewhart-type chart based on the normal distribution. Numerical examples are given to demonstrate the implementation of the new chart.

A Distribution-Free Control Chart Based on Order Statistics

In this article, we introduce a new distribution-free Shewhart-type control chart that takes into account the location of a single order statistic of the test sample (such as the median) as well as the number of observations in that test sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution, and the average run length (ARL) are all derived. A key advantage of the chart is that, due to its nonparametric nature, the false alarm rate and in-control run length distribution are the same for all continuous process distributions, and so will be naturally robust. Tables are provided for the implementation of the chart for some typical ARL values and false alarm rates. The empirical study carried out reveals that the new chart is preferable from a robustness point of view in comparison to a classical Shewhart-type chart and also the nonparametric chart of Chakraborti et al. (2004).

Average Run Lengths in Shewhart Type Charts for 2-Order Autoregressive Process

In this paper the average run length is adopted as the tool to describe the performance of control charts. The respective methods for calculating the average run length of the modified Shewhart control chart and the Shewhart residual control chart for 2-order autoregressive process are derived and shown in detail. By the proposed approach some numerical results of average run lengths of both Shewhart type charts are formulated and discussed. We analyze and compare that the influence of the correlation coefficients of the 2-order autoregressive process on the performance of both charts based on the estimated data. Several clear and main points of the issue are summed up. Lastly, we give some recommendations for the choice of both Shewhart type control schemes.

A bootstrap control chart for inverse Gaussian percentiles

The conventional Shewhart-type control chart is developed essentially on the central limit theorem. Thus, the Shewhart-type control chart performs particularly well when the observed process data come from a near-normal distribution. On the other hand, when the underlying distribution is unknown or non-normal, the sampling distribution of a parameter estimator may not be available theoretically. In this case, the Shewhart-type charts are not available. Thus, in this paper, we propose a parametric bootstrap control chart for monitoring percentiles when process measurements have an inverse Gaussian distribution. Through extensive Monte Carlo simulations, we investigate the behaviour and performance of the proposed bootstrap percentile charts. The average run lengths of the proposed percentage charts are investigated.

A combined synthetic&X chart for monitoring the process mean

The Shewhart X chart (or X chart) is widely used to monitor the mean of a quality characteristic x. This chart decides the process status based on the magnitude of the sample mean x and is effective for detecting large mean shifts. The synthetic chart is also a Shewhart type chart for monitoring the process mean, but it utilises the information about the time interval between two nonconforming samples. Here a sample is nonconforming if its x value falls beyond the predetermined warning limits. Unlike the X chart, the synthetic chart is more powerful to detect small shifts. The applications of the X and synthetic charts cover a wide variety of manufacturing processes and production lines, e.g., the monitoring of the mean values of the inside diameter of a piston-ring, the viscosity of aircraft paint, the resistivity of silicon wafers. This article proposes a combined scheme, the Syn-X chart, that comprises a synthetic chart and an X chart. The results of the performance studies show that the Syn-X chart always outperforms the individual X chart and synthetic chart under different conditions. It is more effective than the X chart and synthetic chart by 47% and 20%, respectively, over the wide range of mean shift values in different experiment runs.

An economic design of double sampling [formula omitted] charts for correlated data using genetic algorithms

Double sampling X ¯ control chart (DS) which is a Shewhart-type chart can reduce sample size and detect small process shift fast. In process monitoring of real industry, observations may be interdependent and correlated, and an original DS design will occur high cost for the wrong determination of the process state. In this study, an economic design model of DS is developed based on Yang and Hancock’s assumption of correlation and Lorenzen and Vance’s cost model to determine sample size, sampling interval, and coefficients of control limits and warring limits. The genetic algorithms (GAs) are applied to solve this economic design model of DS for the determination of the optimal parameters. A real example of IC packaging process is given to illustrate the model application. Sensitivity analysis shows the influence of different model parameters on the DS chart designs.

Multivariate Process Variability Monitoring

Understanding multivariate variability is a difficult task because there is no single measure that can be properly used. This article presents a new measure that features good properties. If this measure is simultaneously used with generalized variance, it will give a better understanding of multivariate variability. It can also efficiently be used for large data sets with high dimensions. Furthermore, when it is used for constructing a Shewhart-type chart to monitor multivariate variability, the resulting chart has a much better out-of-control ARL than the generalized variance chart. An example illustrates its advantage.

EWMA Charts for Monitoring the Mean of Censored Weibull Lifetimes

A lower-sided and an upper-sided exponentially weighted moving average (EWMA) chart for detecting mean changes (decreases and increases) in processes characterized by Weibull distributions are developed in this paper when censoring occurs at a fixed level. We show that the lower-sided EWMA chart performs better than its counterpart, the Shewhart-type; that a Shewhart-type chart is not appropriate to detect mean increases when the censoring rate is high; and that the upper-sided EWMA chart works well in detecting mean increases.

Control charts based on linear combinations of order statistics

The last 20 years have seen an increasing emphasis on statistical process control as a practical approach to reducing variability in industrial applications. Control charts are used to detect problems such as outliers or excess variability in subgroup means that may have a special cause. We describe an approach to the computation of control limits for exponentially weighted moving average control charts where the usual statistics in classical charts are replaced by linear combinations of order statistics; in particular, the trimmed mean and Gini's mean difference instead of the mean and range, respectively. Control limits are derived, and simulated average run length experiments show the trimmed control charts to be less influenced by extreme observations than their classical counterparts, and lead to tighter control limits. An example is given that illustrates the benefits of the proposed charts. parameters; see, for example, Hunter (1986) and Montgomery (1996). On the other hand, EWMA charts have been shown to be more efficient than Shewharttype charts in detecting small shifts in the process mean; see, for example, Ng & Case (1989), Crowder (1989), Lucas & Saccucci (1990), Amin & Searcy (1991) and Wetherill & Brown (1991). In fact, the EWMA control chart has become popular for monitoring a process mean; see Hunter (1986) for a good discussion. More recently, EWMA charts have been developed for monitoring process variability;

Planning time-between-events Shewhart control charts

Shewhart charts for counts are widely used in statistical process control. Usually, they are designed with, as observation to be monitored, the number or the proportion of counts occurring in sampling intervals. However, in many production and administrative processes, as well as in health management and other applied areas, it may be more convenient to monitor the arrival time between successive counts. A unified methodology for planning one-sided and two-sided time-betweenevents Shewhart-type charts is presented. It can be readily applied to any underlying distribution of the counts, and shows how to select the control limits and evaluate the sensitivity of the chart. The procedure is illustrated in a situation of practical relevance in which the time-between-events is exponentially distributed. A numerical example is also given.

An economic study of the [formula omitted] chart design

Since the 1950s, various models have been proposed for a number of Shewhart-type charts including X and R charts employed jointly. Apart from such works, however, the design of a procedure to monitor and control process variability appears to have commanded very little attention. In 1989, Collani and Sheil first discussed the economic design of the s chart. However, they did not give a procedure to determine the optimal parameters of the s chart. This paper presents a procedure to economically determine the optimal parameters of the s chart. The results and the execution times of all numerical examples show that this procedure is quite accurate and efficient.