A EWMA control chart based on an auxiliary variable and repetitive sampling for monitoring process location

ABSTRACT

This article develops an exponentially weighted moving average (EWMA) control chart using an auxiliary variable and repetitive sampling for efficient detection of small to moderate shifts in location. A EWMA statistic of a product estimator of the average (which utilities the information of auxiliary variables as well as repetitive sampling) is plotted on the proposed chart. The control chart coefficients of the proposed EWMA chart are determined for two strategic limits known as outer and inner control limits for the target in-control average run length. The performance of the proposed EWMA chart is studied using average run length when a shift occurs in the process average. The efficiency of the developed chart is compared with the competitive existing control charts. The results of the study revealed that proposed EWMA chart is more efficient than others to detect small changes in process mean.     

An Examination of the Robustness to Non Normality of the EWMA Control Charts for the Dispersion

ABSTRACT

The EWMA control chart is used to detect small shifts in a process. It has been shown that, for certain values of the smoothing parameter, the EWMA chart for the mean is robust to non normality. In this article, we examine the case of non normality in the EWMA charts for the dispersion. It is shown that we can have an EWMA chart for dispersion robust to non normality when non normality is not extreme.     

The effect of autocorrelation on the ewma maxmin tolerance limits

Amin et al. (1999) developed an exponentially weighted moving average (EWMA) control chart, based on the smallest and largest observations in each sample. The resulting plot of the extremes suggests that the MaxMin EWMA may also be viewed as smoothed tolerance limits. Tolerance limits are limits that include a specific proportion of the population at a given confidence level. In the context of process control, they are used to make sure that production will not be outside specifications. Amin and Li (2000) provided the coverages of the MaxMin EWMA tolerance limits for independent data. In this article, it is shown how autocorrelation affects the confidence level of MaxMin tolerance limits, for a specified level of coverage of the population, and modified smoothed tolerance limits are suggested for autocorrelated processes.     

Design of a sign chart using a new EWMA statistic

Abstract

In this article, a new non parametric control chart based on the modified or controlled exponentially weighted moving average (EWMA) statistic is developed to monitor the process deviation from the target value. The proposed control chart is evaluated for different values of design parameters using the average run length as a performance criterion under various sample sizes. The proposed chart is compared with the existing non parametric EWMA sign control chart. It is observed that the proposed chart is better than the existing EWMA sign control chart in terms of run length characteristics. An empirical example is provided for the practical implementation of the proposed chart.     

Conditional design of the EWMA median chart with estimated parameters

The exponentially weighted moving average (EWMA) chart is often designed assuming the process parameters are known. In practice, the parameters are rarely known and need to be estimated from Phase I samples. Different Phase I samples are used when practitioners construct their own control chart's limits, which leads to the “Phase I between-practitioners” variability in the in-control average run length (ARL) of control charts. The standard deviation of the ARL (SDARL) is a good alternative to quantify this variability in control charts. Based on the SDARL metric, the performance of the EWMA median chart with estimated parameters is investigated in this paper. Some recommendations are given based on the SDARL metric. The results show that the EWMA median chart requires a much larger amount of Phase I data in order to reduce the variation in the in-control ARL up to a reasonable level. Due to the limitation of the amount of the Phase I data, the suggested EWMA median chart is designed with the bootstrap method which provides a good balance between the in-control and out-of-control ARL values.     

Robustness of the EWMA and the combined X¯–EWMA control charts with variable sampling intervals to non-normality

The exponentially weighted moving average (EWMA) control charts with variable sampling intervals (VSIs) have been shown to be substantially quicker than the fixed sampling intervals (FSI) EWMA control charts in detecting process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be true for some processes. In the present paper, the performances of the EWMA and combined X¯–EWMA control charts with VSIs are evaluated under non-normality. It is shown that adding the VSI feature to the EWMA control charts results in very substantial decreases in the expected time to detect shifts in process mean under both normality and non-normality. However, the combined X¯–EWMA chart has its false alarm rate and its detection ability is affected if the process data are not normally distributed.     

Ewma control chart under linear drift

An accurate numerical procedure is presented for computing the average run length (ARL) of an exponentially weighted moving average (EWMA) chart under a linear drift in the process mean. The performance of an EWMA chart is then evaluated under a linear drift in the mean. In processes where gradual linear drifts rather than abrupt changes in the mean model the shifts in the mean more accurately, an evaluation of the performance of an EWMA chart under a linear drift is more appropriate. Tables of optimal smoothing parameters and control chart limits are given which make the design of EWMA charts easy.     

Generally weighted moving average control charts using repetitive sampling

Abstract

Generally weighted moving average (GWMA) control charts have been validated for effective detection of small process shifts, and perform better than exponentially weighted moving average (EWMA) control charts. These charts are available based on single sampling; however, repetitive sampling charts have received less attention. Here, a GWMA control chart based on repetitive sampling (namely an RS-GWMA chart) is proposed for enhancing detectability of small process shifts. Simulations show that the proposed RS-GWMA chart with large design and small adjustment parameters outperforms existing hybrid EWMA (HEWMA) control charts based on repetitive sampling. An in-silico example is considered for demonstrating the applicability of the proposed RS-GWMA chart.     

A new approach for monitoring process variance

ABSTRACT

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having non-normal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used in such circumstances. In this paper, we propose a new variance chart based on a simple statistic to monitor process variance shifts. We explore the sampling properties of the new monitoring statistic and calculate the average run lengths (ARLs) of the proposed variance chart. Furthermore, an arcsine transformed exponentially weighted moving average (EWMA) chart is proposed because the ARLs of this modified chart are more intuitive and reasonable than those of the variance chart. We compare the out-of-control variance detection performance of the proposed variance chart with that of the non-parametric Mood variance (NP-M) chart with runs rules, developed by Zombade and Ghute [Nonparametric control chart for variability using runs rules. Experiment. 2014;24(4):1683–1691], and the nonparametric likelihood ratio-based distribution-free exponential weighted moving average (NLE) chart and the combination of traditional exponential weighted moving average (EWMA) mean and EWMA variance (CEW) control chart proposed by Zou and Tsung [Likelihood ratio-based distribution-free EWMA control charts. J Qual Technol. 2010;42(2):174–196] by considering cases in which the critical quality characteristic has a normal, a double exponential or a uniform distribution. Comparison results showed that the proposed chart performs better than the NP-M with runs rules, and the NLE and CEW control charts. A numerical example of service times with a right-skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the application of the proposed variance chart and of the arcsine transformed EWMA chart and to compare them with three existing variance (or standard deviation) charts. The proposed charts show better detection performance than those three existing variance charts in monitoring and detecting shifts in the process variance.     

Bayesian X̄ control limits for exponentially distributed measurements

This article deals with the construction of an X? control chart using the Bayesian perspective. We obtain new control limits for the X? chart for exponentially distributed data-generating processes through the sequential use of Bayes’ theorem and credible intervals. Construction of the control chart is illustrated using a simulated data example. The performance of the proposed, standard, tolerance interval, exponential cumulative sum (CUSUM) and exponential exponentially weighted moving average (EWMA) control limits are examined and compared via a Monte Carlo simulation study. The proposed Bayesian control limits are found to perform better than standard, tolerance interval, exponential EWMA and exponential CUSUM control limits for exponentially distributed processes.     

Statistical control charts for monitoring the mean of a stationary process

Recently statistical process control (SPC) methodologies have been developed to accommodate autocorrelated data. A primary method to deal with autocorrelated data is the use of residual charts. Although this methodology has the advantage that it can be applied to any autocorrelated data it needs time series modeling efforts. In addition for a X residual chart the detection capability is sometimes small compared to the X chart and EWMA chart. Zhang (1998) proposed the EWMAST chart which is constructed by charting the EWMA statistic for stationary processes to monitor the process mean. The performance of the EWMAST chart the X chart the X residual chart and other charts were compared in Zhang (1998). In this paper comparisons are made among the EWMAST chart the CUSUM residual chart and EWMA residual chart as well as the X residual chart and X chart via the average run length.     

Robustness to non-normality and autocorrelation of individuals control charts

This paper studies the effects of non-normality and autocorrelation on the performances of various individuals control charts for monitoring the process mean and/or variance. The traditional Shewhart X chart and moving range (MR) chart are investigated as well as several types of exponentially weighted moving average (EWMA) charts and combinations of control charts involving these EWMA charts. It is shown that the combination of the X and MR charts will not detect small and moderate parameter shifts as fast as combinations involving the EWMA charts, and that the performana of the X and MR charts is very sensitive to the normality assumption. It is also shown that certain combinations of EWMA charts can be designed to be robust to non-normality and very effective at detecting small and moderate shifts in the process mean and/or variance. Although autocorrelation can have a significant effect on the in-control performances of these combinations of EWMA charts, their relative out-of-control performances under independence are generally maintained for low to moderate levels of autocorrelation.     

The Robustness of the MaxEWMA Chart to Non-normality

Abstract

The MaxEWMA chart has recently been introduced as an improvement over the standard EWMA chart for detecting changes in the mean and/or standard deviation of a normally distributed process. Although this chart was originally developed for normally distributed process data, its robustness to violations of the normality assumption is the central theme of this study. For data distributions with heavy tails or displaying strong skewness, the in-control average run lengths (ARLs) for the MaxEWMA chart are shown to be significantly shorter than expected. On the other hand, out-of-control ARLs are comparable to normal theory values for a variety of symmetric non-normal distributions. The MaxEWMA chart is not robust to skewness.     

Exponentially weighted moving average control charts for three-level products

In this paper, exponentially weighted moving average (EWMA) control charts for multinomial data are developed with a three-level classification scheme. The lower and upper control limits of the proposed EWMA control chart are evaluated using Markov chain approximation. Compared with the three-level Shewhart control chart, numerical results indicate that the proposed EWMA control chart is relatively sensitive to small shifts in a three-level multinomial process. A figure and a table are provided for practitioners to select the value of chart limit coefficient that gives the desired in-control average run length.     

Control Chart for Monitoring Autocorrelated Process with Multiple Exogenous Inputs

We propose to use AR-Sieve Bootstrap in the construction of a control chart of an autocorrelated process influenced by multiple exogenous inputs. The control charts are compared with Exponentially Weighted Moving Average (EWMA) control chart through a simulation study. AR-Sieve bootstrap control limits are narrower than EWMA control limits. While the proposed method yields a higher rate of false alarms, it is quick in detecting even minimal structural changes.     

A multivariate exponentially weighted moving average control chart for monitoring process variability

This paper introduces a new multivariate exponentially weighted moving average (EWMA) control chart. The proposed control chart, called an EWMA V-chart, is designed to detect small changes in the variability of correlated multivariate quality characteristics. Through examples and simulations, it is demonstrated that the EWMA V-chart is superior to the |S|-chart in detecting small changes in process variability. Furthermore, a counterpart of the EWMA V-chart for monitoring process mean, called the EWMA M-chart is proposed. In detecting small changes in process variability, the combination of EWMA M-chart and EWMA V-chart is a better alternative to the combination of MEWMA control chart (Lowry et al. , 1992) and |S|-chart. Furthermore, the EWMA M- chart and V-chart can be plotted in one single figure. As for monitoring both process mean and process variability, the combined MEWMA and EWMA V-charts provide the best control procedure.     

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;     

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;     

Monitoring simple linear profiles using variable sample size schemes

ABSTRACT

Profile monitoring is one of the new research areas in statistical process control. Most of the control charts in this area are designed with fixed sampling rate which makes the control chart slow in detecting small to moderate shifts. In order to improve the performance of the conventional fixed control charts, adaptive features are proposed in which, one or more design parameters vary during the process. In this paper the variable sample size feature of EWMA₃ and MEWMA schemes are proposed for monitoring simple linear profiles. The EWMA₃ method is based on the combination of three exponentially weighted moving average (EWMA) charts for monitoring three parameters of a simple linear profile separately and the Multivariate EWMA (MEWMA) chart is based on the using a single chart to monitor the coefficients and variance of a general linear profile. Also a two-sided control chart is proposed for monitoring the standard deviation in the EWMA₃ method. The performance of the proposed charts is compared in terms of the average time to signal. Numerical examples show that using adaptive features increase the power of control charts in detecting the parameter shifts. Finally, the performance of the proposed variable sample size schemes is illustrated through a real case in the leather industry.     

A new adaptive EWMA control chart using auxiliary information for monitoring the process mean

The adaptive memory-type control charts, including the adaptive exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts, have gained considerable attention because of their excellent speed in providing overall good detection over a range of mean shift sizes. In this paper, we propose a new adaptive EWMA (AEWMA) chart using the auxiliary information for efficiently monitoring the infrequent changes in the process mean. The idea is to first estimate the unknown process mean shift using an auxiliary information based mean estimator, and then adaptively update the smoothing constant of the EWMA chart. Using extensive Monte Carlo simulations, the run length profiles of the AEWMA chart are computed and explored. The AEWMA chart is compared with the existing control charts, including the classical EWMA, CUSUM, synthetic EWMA and synthetic CUSUM charts, in terms of the run length characteristics. It turns out that the AEWMA chart performs uniformly better than these control charts when detecting a range of mean shift sizes. An illustrative example is also presented to demonstrate the working and implementation of the proposed and existing control charts.