A control chart for monitoring process variation using multiple dependent state sampling

A control chart for monitoring process variation by using multiple dependent state (MDS) sampling is constructed in the present article. The operational formulas for in-control and out-of-control average run lengths (ARLs) are derived. Control constants are established by considering the target in-control ARL at a normal process. The extensive ARL tables are reported for various parameters and shifted values of process parameters. The performance of the proposed control chart has been evaluated with several existing charts in regard of ARLs, which empowered the presented chart and proved far better for timely detection of assignable causes. The application of the proposed concept is illustrated with a real-life industrial example and a simulation-based study to elaborate strength of the proposed chart over the existing concepts.     

An Evaluation of the Double Exponentially Weighted Moving Average Control Chart

In this article we perform a careful investigation of the double exponentially weighted moving average (DEWMA) chart performance for monitoring the process mean. We compare the performance of this chart to the usual EWMA control chart based on zero-state and worst-case average run length (ARL) measures. We also evaluate the signal resistance measure of the DEWMA chart and compare its maximum value to that of the EWMA chart. We show that the superiority of the DEWMA chart over the simpler standard EWMA chart based on zero-state ARL performance disappears when the smoothing constant of the EWMA chart is chosen to give weights to past observations closer to those given by the DEWMA chart. Moreover, our results show that the standard EWMA chart has much better performance than the DEWMA chart in terms of worst-case ARL values, especially when small smoothing constants are used. We also demonstrate using an illustrative example that the DEWMA chart can build up an exceedingly large amount of inertia when used to monitor the process mean.     

On the Construction of Families of Absolutely Continuous Copulas with Given Restrictions

ABSTRACT

We prove that the standard EWMA mean chart with asymptotic control limits and the EWMA mean chart with time-varying control limits for monitoring mean changes in a normal process with known mean and known variance are ARL-unbiased. Using the results derived we discuss the effects of estimation of the process mean on ARL.     

ARL Performance of the Tukey's Control Chart

A Tukey's control chart was designed to monitor single observation data. Its easy control-limits setting and simple statistical concept are the most important advantages. In this study, we setup the Tukey's control chart based on known probability distribution and construct its average run length calculator. ARL performance of the Tukey's control chart is evaluated under a normal distribution and non normal distributions, respectively. The comparison of ARL performance reveals that Tukey's control chart is not sensitive to shift detection when the process heavily violates the normal assumption. The number of observations needed for Tukey's control chart setup is less than Shewhart's control chart. Tukey's control chart is a better choice for the process mean monitoring.     

Shewhart x-charts with estimated process variance

Properties of the Shewhart X-chart for controlling the mean of a process with a normal distribution are investigated for the situation where the process variance Ó²must be estimated from initial sample data. The control limits of the X-chart depend on the estimate of Ó²and thus, unlike the case when Ó²is known, the X-chart is not equivalent to a sequence of independent tests. When Ó²is estimated the distribution of the run length is not geometric and cannot be characterized simply in terms of the probability of a signal at a given point. The average run length (ARL) for the X-chart is expressed in terms of an integral involving the normal cdf, and it is shown that the chart signals with

probability one, but the ARL may not be finite if the size of the 2 sample used to estimate Ó²is sufficiently small. In addition, certain bounds for the ARL are also derived. Numerical integration is use to show that the effect of using small sample sizes in estimating Ó²is to increase the ARL and the variance of the run length distribution     

Moving range EWMA control charts for monitoring the Weibull shape parameter

In this article, we propose an exponentially weighted moving average (EWMA) control chart for the shape parameter β of Weibull processes. The chart is based on a moving range when a single measurement is taken per sampling period. We consider both one-sided (lower-sided and upper-sided) and two-sided control charts. We perform simulations to estimate control limits that achieve a specified average run length (ARL) when the process is in control. The control limits we derive are ARL unbiased in that they result in ARL that is shorter than the stable-process ARL when β has shifted. We also perform simulations to determine Phase I sample size requirements if control limits are based on an estimate of β. We compare the ARL performance of the proposed chart to that of the moving range chart proposed in the literature.     

A new multivariate CUSUM chart using principal components with a revision of Crosier's chart

In this article, we introduce a new multivariate cumulative sum chart, where the target mean shift is assumed to be a weighted sum of principal directions of the population covariance matrix. This chart provides an attractive performance in terms of average run length (ARL) for large-dimensional data and it also compares favorably to existing multivariate charts including Crosier's benchmark chart with updated values of the upper control limit and the associated ARL function. In addition, Monte Carlo simulations are conducted to assess the accuracy of the well-known Siegmund's approximation of the average ARL function when observations are normal distributed. As a byproduct of the article, we provide updated values of upper control limits and the associated ARL function for Crosier's multivariate CUSUM chart.     

A control chart using an auxiliary variable and repetitive sampling for monitoring process mean

In this paper, a new control chart is proposed by using an auxiliary variable and repetitive sampling in order to enhance the performance of detecting a shift in process mean. The product-difference type estimator of the mean is plotted on the proposed control chart, which utilizes the information of an auxiliary variable correlated with the main quality variable. The proposed control chart is based on the outer and inner control limits so that repetitive sampling is allowed when the plotted statistic falls between the two limits. The average run length (ARL) of the proposed control chart is evaluated using the Monte Carlo simulation. The proposed control chart is compared with the Riaz M control chart and the results show the outperformance of the proposed control chart in terms of the ARL.     

A Simple Method for Computing the Arl of [ILM0001]-Charts with Zone Tests

In practice the [ILM0002]-chart has been augmented with one or more zone tests to improve the sensitivity of detecting small shifts in the process mean. The average run length (ARL) is one of the indices used to evaluate the performance of control chart procedures. Two unified patterns of transition probability matrices and four closed form expressions of the ARL are found based on the Markov Chain approach. These closed form expressions can be used for computing the ARL for both one-sided and two-sided [ILM0003]-charts with zone tests.     

Normalization of the origin-shifted exponential distribution for control chart construction

This study demonstrates that a location parameter of an exponential distribution significantly influences normalization of the exponential. The Kullback–Leibler information number is shown to be an appropriate index for measuring data normality using a location parameter. Control charts based on probability limits and transformation are compared for known and estimated location parameters. The probabilities of type II error (β-risks) and average run length (ARL) without a location parameter indicate an ability to detect an out-of-control signal of an individual chart using a power transformation similar to using probability limits. The β-risks and ARL of control charts with an estimated location parameter deviate significantly from their theoretical values when a small sample size of n≤50 is used. Therefore, without taking into account of the existence of a location parameter, the control charts result in inaccurate detection of an out-of-control signal regardless of whether a power or natural logarithmic transformation is used. The effects of a location parameter should be eliminated before transformation. Two examples are presented to illustrate these findings.     

Statistical quality control based on ranked set sampling

Different quality control charts for the sample mean are developed using ranked set sampling (RSS), and two of its modifications, namely median ranked set sampling (MRSS) and extreme ranked set sampling (ERSS). These new charts are compared to the usual control charts based on simple random sampling (SRS) data. The charts based on RSS or one of its modifications are shown to have smaller average run length (ARL) than the classical chart when there is a sustained shift in the process mean. The MRSS and ERSS methods are compared with RSS and SRS data, it turns out that MRSS dominates all other methods in terms of the out-of-control ARL performance. Real data are collected using the RSS, MRSS, and ERSS in cases of perfect and imperfect ranking. These data sets are used to construct the corresponding control charts. These charts are compared to usual SRS chart. Throughout this study we are assuming that the underlying distribution is normal. A check of the normality for our example data set indicated that the normality assumption is reasonable.     

Run length,average run length and false alarm rate of shewhart x-bar chart: exact derivations by conditioning

The effects of estimation of the control limits on the performance of the popular Shewhart X-bar chart are examined via the average run length and the probability of a false alarm, when one or both of the process mean and variance are unknown. Exact expressions for the run length, the average run length (ARL) and the false alarm rate are obtained, in each case, using expectation by conditioning. Applying Jensen's inequality, together with expectation by conditioning, a simple lower bound to the ARL is obtained. This could be useful in designing the charts. The expressions for the exact ARL and the exact probabilities of false alarm are evaluated, using simulations, for various numbers of subgroups and shift sizes. The calculations throw new light on the performance of the Shewhart X-bar chart. Some recommendations are given.     

Accurate ARL computation for EWMA-S<Superscript>2</Superscript> control charts

Originally, the exponentially weighted moving average (EWMA) control chart was developed for detecting changes in the process mean. The average run length (ARL) became the most popular performance measure for schemes with this objective. When monitoring the mean of independent and normally distributed observations the ARL can be determined with high precision. Nowadays, EWMA control charts are also used for monitoring the variance. Charts based on the sample variance S² are an appropriate choice. The usage of ARL evaluation techniques known from mean monitoring charts, however, is difficult. The most accurate method—solving a Fredholm integral equation with the Nyström method—fails due to an improper kernel in the case of chi-squared distributions. Here, we exploit the collocation method and the product Nyström method. These methods are compared to Markov chain based approaches. We see that collocation leads to higher accuracy than currently established methods.     

A Double Multivariate Exponentially Weighted Moving Average (dMEWMA) Control Chart for a Process Location Monitoring

A new control scheme, dMEWMA, for detecting shifts in the mean vector of multivariately normally distributed quality characteristics is presented. It is shown that the ARL performance of dMEWMA depends on the mean and variance-covariance matricies only through the non-centrality parameter value. Through Monte Carlo simulations, the performance of dMEWMA for detecting various shifts is compared to the competing control schemes, MEWMA and Hotelling's χ². It is concluded that dMEWMA outperforms MEWMA and Hotelling's χ² control schemes for small and larger shifts. In comparison to MEWMA control schemes, dMEWMA schemes are optimal for larger values of the smoothing parameter λ and perform much better for very small shifts in the process mean. Finally, an example to illustrate the construction of the dMEWMA control scheme is introduced.     

Control charts for attributes with maxima nominated samples

We develop quality control charts for attributes using the maxima nomination sampling (MNS) method and compare them with the usual control charts based on simple random sampling (SRS) method, using average run length (ARL) performance, the required sample size in detecting quality improvement, and non-existence region for control limits. We study the effect of the sample size, the set size, and nonconformity proportion on the performance of MNS control charts using ARL curve. We show that MNS control chart can be used as a better benchmark for indicating quality improvement or quality deterioration relative to its SRS counterpart. We consider MNS charts from a cost perspective. We also develop MNS attribute control charts using randomized tests. A computer program is designed to determine the optimal control limits for an MNS p-chart such that, assuming known parameter values, the absolute deviation between the ARL and a specific nominal value is minimized. We provide good approximations for the optimal MNS control limits using regression analysis. Theoretical results are augmented with numerical evaluations. These show that MNS based control charts can yield substantial improvement over the usual control charts based on SRS.     

A New Procedure to Monitor the Mean of a Quality Characteristic

The Shewhart, Bonferroni-adjustment, and analysis of means (ANOM) control charts are typically applied to monitor the mean of a quality characteristic. The Shewhart and Bonferroni procedure are utilized to recognize special causes in production process, where the control limits are constructed by assuming normal distribution for known parameters (mean and standard deviation), and approximately normal distribution regarding to unknown parameters. The ANOM method is an alternative to the analysis of variance method. It can be used to establish the mean control charts by applying equicorrelated multivariate non central t distribution. In this article, we establish new control charts, in phases I and II monitoring, based on normal and t distributions having as a cause a known (or unknown) parameter (standard deviation). Our proposed methods are at least as effective as the classical Shewhart methods and have some advantages.     

Nonparametric control charts based on order statistics: Some advances

ABSTRACT

In this article, we introduce new nonparametric Shewhart-type control charts that take into account the location of two order statistics of the test sample as well as the number of observations in that 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 new charts is that, due to its nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the proposed charts for some typical FAR and ARL values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations, while they seem preferable from a robustness point of view in comparison with the distribution-free control chart of Balakrishnan et al. (2009).     

Some Comments Regarding the Synthetic Chart

The steady-state average run length (ARL) is a function of the in-control probabilities of being in each nonabsorbing state. Davis and Woodall (2002) tabulated values that are significantly smaller than the steady-state ARLs, because they used the out-of-control probabilities. The synthetic chart signals when a second sample point falls beyond the control limits, no matter whether one of them falls above the centerline and the other falls below it. The side-sensitive version of the synthetic chart does not signal when the points beyond the control limits are on opposite sides. With this rule, the chart detects mean changes more quickly.