An adaptive exponentially weighted moving average control chart for monitoring process variances

The exponentially weighted moving average (EWMA) control chart is efficient in detecting small changes in process parameters but less efficient when the changes are relatively large, due to what is known as the inertia problem. To diminish the inertia, an adaptive EWMA (AEWMA) chart has been proposed for monitoring process locations to improve over the traditional EWMA charts. The basic idea of the AEWMA scheme is to dynamically weight the past observations according to a suitable function of the current prediction error. This article extends the idea of the AEWMA chart for monitoring process locations to the case of monitoring process dispersion. A Markov chain model is established to analyze and design the suggested chart. It is shown that the AEWMA dispersion chart performs better than the EWMA and other dispersion charts in terms of its ability to perform relatively well at both small and large changes in process dispersion.     

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.     

Optimal design of the adaptive exponentially weighted moving average control chart over a range of mean shifts

The adaptive exponentially weighted moving average (AEWMA) control chart is a smooth combination of the Shewhart and exponentially weighted moving average (EWMA) control charts. This chart was proposed by Cappizzi and Masarotto (2003) to achieve a reasonable performance for both small and large shifts. Cappizzi and Masarotto (2003) used a pair of shifts in designing their control chart. In this study, however, the process mean shift is considered as a random variable with a certain probability distribution and the AEWMA control chart is optimized for a wide range of mean shifts according to that probability distribution and not just for a pair of shifts. Using the Markov chain technique, the results show that the new optimization design can improve the performance of the AEWMA control chart from an overall point of view relative to the various designs presented by Cappizzi and Masarotto (2003). Optimal design parameters that achieve the desired in-control average run length (ARL) are computed in several cases and formulas used to find approximately their values are given. Using these formulas, the practitioner can compute the optimal design parameters corresponding to any desired in-control ARL without the need to apply the optimization procedure. The results obtained by these formulas are very promising and would particularly facilitate the design of the AEWMA control chart for any in-control ARL value.     

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.     

Non parametric double generally weighted moving average sign charts based on process proportion

Non parametric control charts have received increasing attention in the field of statistical process control. This paper presents a non parametric double generally weighted moving average (DGWMA) sign chart for monitoring small deviations when the quality characteristics of a process are unknown. The statistical performance of the non parametric DGWMA sign chart is evaluated and compared with those of other charts, including the exponentially weighted moving average (EWMA), generally weighted moving average (GWMA), and double EWMA (DEWMA) sign charts. Simulation studies indicate that the non parametric DGWMA sign chart with a large design and median adjustment parameters is always more sensitive than other charts in detecting small changes.     

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.     

The Generally Weighted Moving Average Variance Chart

This article extends the generally weighted moving average (GWMA) technique for detecting changes in process variance. The proposed chart is called the generally weighted moving average variance (GWMAV) chart. Simulation is employed to evaluate the average run length (ARL) characteristics of the GWMAV and EWMA control charts. An extensive comparison of these control charts reveals that the GWMAV chart is more sensitive than the EWMA control charts for detecting small shifts in the variance of a process when the shifts are below 1.35 standard deviations. Additionally, the GWMAV control chart performs little better when the variance shifts are between 1.35 and 1.5 standard deviation, and the 2 charts performs similar when the variance shifts are above 1.5 standard deviation. The design of the GWMAV chart is also discussed.     

A mixed GWMA–CUSUM control chart for monitoring the process mean

The memory-type control charts are widely used in the process and service industries for monitoring the production processes. The reason is their sensitivity to quickly react against the small process disturbances. Recently, a new cumulative sum (CUSUM) chart has been proposed that uses the exponentially weighted moving average (EWMA) statistic, called the EWMA–CUSUM chart. Similarly, in order to further enhance the sensitivity of the EWMA–CUSUM chart, we propose a new CUSUM chart using the generally weighted moving average (GWMA) statistic, called the GWMA–CUSUM chart, for efficiently monitoring the process mean. The GWMA–CUSUM chart encompasses the existing CUSUM and EWMA–CUSUM charts. Extensive Monte Carlo simulations are used to explore the run length profiles of the GWMA–CUSUM chart. Based on comprehensive run length comparisons, it turns out that the GWMA–CUSUM chart performs substantially better than the CUSUM, EWMA, GWMA, and EWMA–CUSUM charts when detecting small shifts in the process mean. An illustrative example is also presented to explain the implementation and working of the EWMA–CUSUM and GWMA–CUSUM charts.     

Monitoring Process Mean and Variability with One Non-central Chi-square Chart

Traditionally, an X-chart is used to control the process mean and an R-chart to control the process variance. However, these charts are not sensitive to small changes in process parameters. A good alternative to these charts is the exponentially weighted moving average (EWMA) control chart for controlling the process mean and variability, which is very effective in detecting small process disturbances. In this paper, we propose a single chart that is based on the non-central chi-square statistic, which is more effective than the joint X and R charts in detecting assignable cause(s) that change the process mean and/or increase variability. It is also shown that the EWMA control chart based on a non-central chi-square statistic is more effective in detecting both increases and decreases in mean and/or variability.     

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.     

An EWMA chart for monitoring process mean

In the statistical process control literature, there exists several improved quality control charts based on cost-effective sampling schemes, including the ranked set sampling (RSS) and median RSS (MRSS). A generalized cost-effective RSS scheme has been recently introduced for efficiently estimating the population mean, namely varied L RSS (VLRSS). In this article, we propose a new exponentially weighted moving average (EWMA) control chart for monitoring the process mean using VLRSS, named the EWMA-VLRSS chart, under both perfect and imperfect rankings. The EWMA-VLRSS chart encompasses the existing EWMA charts based on RSS and MRSS (named the EWMA-RSS and EWMA-MRSS charts). We use extensive Monte Carlo simulations to compute the run length characteristics of the EWMA-VLRSS chart. The proposed chart is then compared with the existing EWMA charts. It is found that, with either perfect or imperfect rankings, the EWMA-VLRSS chart is more sensitive than the EWMA-RSS and EWMA-MRSS charts in detecting small to large shifts in the process mean. A real dataset is also used to explain the working of the EWMA-VLRSS chart.     

An Empirical-Likelihood-Based Multivariate EWMA Control Scheme

Nonparametric control charts are useful in statistical process control (SPC) when there is a lack of or limited knowledge about the underlying process distribution, especially when the process measurement is multivariate. This article develops a new multivariate SPC methodology for monitoring location parameter based on adapting a well-known nonparametric method, empirical likelihood (EL), to on-line sequential monitoring. The weighted version of EL ratio test is used to formulate the charting statistic by incorporating the exponentially weighted moving average control (EWMA) scheme, which results in a nonparametric counterpart of the classical multivariate EWMA (MEWMA). Some theoretical and numerical studies show that benefiting from using EL, the proposed chart possesses some favorable features. First, it is a data-driven scheme and thus is more robust to various multivariate non-normal data than the MEWMA chart under the in-control (IC) situation. Second, it is transformation-invariant and avoids the estimation of covariance matrix from the historical data by studentizing internally, and hence its IC performance is less deteriorated when the number of reference sample is small. Third, in comparison with the existing approaches, it is more efficient in detecting small and moderate shifts for multivariate non-normal process.     

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 New Adaptive Variable Sample Size Approach in EWMA Control Chart

Adaptive control charts have been developed for improving the capability of control charts in detecting small shifts. In this article, we propose a new exponential weighted moving average control chart with variable sample size, in which the sample size is determined as an integer linear function by EWMA statistic value. The performance of the proposed VSS EWMA control chart is compared with FSS EWMA as well as traditional VSS EWMA control charts. The results show the better performance of the proposed VSS strategy respect to the traditional one and fixed sample size.     

A new nonparametric synthetic EWMA control chart for monitoring process mean

The exponentially weighted moving average (EWMA) control charts are widely used in chemical and process industries because of their excellent speed in catching small to moderate shifts in the process target. In usual practice, many data come from a process where the monitoring statistic is non-normally distributed or it follows an unknown probability distribution. This necessitates the use of distribution-free/nonparametric control charts for monitoring the deviations from the process target. In this paper, we integrate the existing EWMA sign chart with the conforming run length chart to propose a new synthetic EWMA (SynEWMA) sign chart for monitoring the process mean. The SynEWMA sign chart encompasses the synthetic sign and EWMA sign charts. Monte Carlo simulations are used to compute the run length profiles of the SynEWMA sign chart. Based on a comprehensive comparison, it turns out that the SynEWMA sign chart is able to perform substantially better than the existing EWMA sign chart. Both real and simulated data sets are used to explain the working and implementation of existing and proposed control charts.     

Fast Initial Response Features for Poisson GWMA Control Charts

The Poisson GWMA (PGWMA) control chart is an extension model of Poisson EWMA chart. It is substantially sensitive to small process shifts for monitoring Poisson observations. Recently, some approaches have been proposed to modify EWMA charts with fast initial response (FIR) features. In this article, we employ these approaches in PGWMA charts and introduce a novel chart called Poisson double GWMA (PDGWMA) chart for comparison. Using simulation, various control schemes are designed and their average run lengths (ARLs) are computer and compared. It is shown that the PDGWMA chart is the first choice in detecting small shifts especially when the shifts are downward, and the PGWMA chart with adjusted time-varying control limits performs excellently in detecting great process shifts during the initial stage.     

Generally weighted moving average control charts with fast initial response features

The generally weighted moving average (GWMA) control chart is an extension model of exponentially weighted moving average (EWMA) control chart. Recently, some approaches have been proposed to modify EWMA charts with fast initial response (FIR) features. We introduce these approaches in GWMA-type charts. Via simulation, various control schemes are designed and then their average run lengths are computed and compared. Based on the overall performance, it is showed that the DGWMA chart is the best choice especially when the shift is moderate, and the GWMA charts provided with additional FIR feature have a good performance only in detecting large shifts during the initial stage.     

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.     

Monitoring the mean of autocorrelated observations with one generally weighted moving average control chart

Traditionally, using a control chart to monitor a process assumes that process observations are normally and independently distributed. In fact, for many processes, products are either connected or autocorrelated and, consequently, obtained observations are autocorrelative rather than independent. In this scenario, applying an independence assumption instead of autocorrelation for process monitoring is unsuitable. This study examines a generally weighted moving average (GWMA) with a time-varying control chart for monitoring the mean of a process based on autocorrelated observations from a first-order autoregressive process (AR(1)) with random error. Simulation is utilized to evaluate the average run length (ARL) of exponentially weighted moving average (EWMA) and GWMA control charts. Numerous comparisons of ARLs indicate that the GWMA control chart requires less time to detect various shifts at low levels of autocorrelation than those at high levels of autocorrelation. The GWMA control chart is more sensitive than the EWMA control chart for detecting small shifts in a process mean.     

On increasing the sensitivity of mixed EWMA–CUSUM control charts for location parameter

Control chart is an important statistical technique that is used to monitor the quality of a process. Shewhart control charts are used to detect larger disturbances in the process parameters, whereas cumulative sum (CUSUM) and exponential weighted moving average (EWMA) are meant for smaller and moderate changes. In this study, we enhanced mixed EWMA–CUSUM control charts with varying fast initial response (FIR) features and also with a runs rule of two out of three successive points that fall above the upper control limit. We investigate their run-length properties. The proposed control charting schemes are compared with the existing counterparts including classical CUSUM, classical EWMA, FIR CUSUM, FIR EWMA, mixed EWMA–CUSUM, 2/3 modified EWMA, and 2/3 CUSUM control charting schemes. A case study is presented for practical considerations using a real data set.