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.     

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.     

A Parameter-Tuned Genetic Algorithm for Economic-Statistical Design of Variable Sampling Interval X-Bar Control Charts for Non-Normal Correlated Samples

Among innovations and improvements that occurred in the past two decades on the techniques and tools used for statistical process control (SPC), adaptive control charts have shown to substantially improve the statistical and/or economical performances. Variable sampling intervals (VSI) control charts are one of the most applied types of the adaptive control charts and have shown to be faster than traditional Shewhart control charts in identifying small changes of concerned quality characteristics. While in the designing procedure of the VSI control charts the data or measurements are assumed independent normal observations, in real situations the validity of these assumptions is under question in many processes. This article develops an economic-statistical design of a VSI X-bar control chart under non-normality and correlation. Since the proposed design consists of a complex nonlinear cost model that cannot be solved using a classical optimization method, a genetic algorithm (GA) is employed to solve it. Moreover, to improve the performances, response surface methodology (RSM) is employed to calibrate GA parameters. The solution procedure, efficiency, and sensitivity analysis of the proposed design are demonstrated through a numerical illustration at the end.     

On the α-risks for shewhart control charts

The performance of the usual Shewhart control charts for monitoring process means and variation can be greatly affected by nonnormal data or subgroups that are correlated. Define the α_k-risk for a Shewhart chart to be the probability that at least one “out-of-control” subgroup occurs in k subgroups when the control limits are calculated from the k subgroups. Simulation results show that the α_k-risks can be quite large even for a process with normally distributed, independent subgroups. When the data are nonnormal, it is shown that the α_k-risk increases dramatically. A method is also developed for simulating an “in-control” process with correlated subgroups from an autoregressive model. Simulations with this model indicate marked changes in the α_k-risks for the Shewhart charts utilizing this type of correlated process data. Therefore, in practice a process should be investigated thoroughly regarding whether or not it is generating normal, independent data before out-of-control points on the control charts are interpreted to be due to some real assignable cause.     

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.     

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.     

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.     

The effect of Phase I sample size on the run length performance of control charts for autocorrelated data

Traditional control charts assume independence of observations obtained from the monitored process. However, if the observations are autocorrelated, these charts often do not perform as intended by the design requirements. Recently, several control charts have been proposed to deal with autocorrelated observations. The residual chart, modified Shewhart chart, EWMAST chart, and ARMA chart are such charts widely used for monitoring the occurrence of assignable causes in a process when the process exhibits inherent autocorrelation. Besides autocorrelation, one other issue is the unknown values of true process parameters to be used in the control chart design, which are often estimated from a reference sample of in-control observations. Performances of the above-mentioned control charts for autocorrelated processes are significantly affected by the sample size used in a Phase I study to estimate the control chart parameters. In this study, we investigate the effect of Phase I sample size on the run length performance of these four charts for monitoring the changes in the mean of an autocorrelated process, namely an AR(1) process. A discussion of the practical implications of the results and suggestions on the sample size requirements for effective process monitoring are provided.     

On designing new optimal synthetic Tukey’s control charts

Tukey’s control chart is generally used for monitoring the processes where the measurement process physically damages the product. It is based on single observation and robust to outliers. In this paper, two optimal synthetic Tukey’s control charts are proposed by integrating the conforming run length chart with the Tukey’s control chart and its modification. The performance comparison of the proposed charts with the existing Tukey’s control charts is made by using out-of-control average run length and extra quadratic loss as performance metrics. The proposed charts offer better protection against the process shifts as compare to the existing Tukey’s control charts when the underlying process distribution is symmetric or asymmetric. Simulation studies also establish the supremacy of the proposed control charts over the existing Tukey’s control charts. In the end, an illustrative example based on a real data set of the combined cycle power plant is provided for practical implementation.     

A mean deviation-based approach to monitor process variability

The study proposes a Shewhart-type control chart, namely an MD chart, based on average absolute deviations taken from the median, for monitoring changes (especially moderate and large changes – a major concern of Shewhart control charts) in process dispersion assuming normality of the quality characteristic to be monitored. The design structure of the proposed MD chart is developed and its comparison is made with those of two well-known dispersion control charts, namely the R and S charts. Using power curves as a performance measure, it has been observed that the design structure of the proposed MD chart is more powerful than that of the R chart and is very close competitor to that of the S chart, in terms of discriminatory power for detecting shifts in the process dispersion. The non-normality effect is also examined on design structures of the three charts, and it has been observed that the design structure of the proposed MD chart is least affected by departure from normality.     

Multivariate Control Chart Based on Multivariate Smirnov Test

Robust control charts are useful in statistical process control (SPC) when there is limited knowledge about the underlying process distribution, especially for multivariate observations. This article develops a new robust and self-starting multivariate procedure based on multivariate Smirnov test (MST), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on multivariate empirical distribution function (MEDF) and the change-point model. As expected, simulation results show that our proposed control chart is robust to nonnormally distributed data, and moreover, it is efficient in detecting process shifts, especially large shifts, which is one of the main drawbacks of most robust control charts in the literature. As it avoids the need for a lengthy data-gathering step, the proposed chart is particularly useful in start-up or short-run situations. Comparison results and a real data example show that our proposed chart has great potential for application.     

A robust R control chart based on a two-step estimator of the process dispersion

ABSTRACT

Control charts are the frequently used tools for monitoring and controlling the processes. Classical control charts are sensitive to existing contaminated data which may be presented in the data collected from the processes. Thus, these charts are not able to control the processes precisely when the data are contaminated. Robust control charts are those which are less sensitive to contamination. Some robust control charts for monitoring the process variability were proposed in the past which are robust to some sorts of contamination. In this paper a new robust R control chart is proposed which is less sensitive to wide range of contaminations, i.e. general and local contaminations. Simulation studies are performed to compare the performance of the proposed control chart with some classical and robust control charts, using ARL and MSD as criteria for comparisons purposes. The simulation results show a very good performance of the proposed chart when both types of contaminations exist.     

Study of a Markov model for a high-quality dependent process

For high-quality processes, non-conforming items are seldom observed and the traditional p (or np) charts are not suitable for monitoring the state of the process. A type of chart based on the count of cumulative conforming items has recently been introduced and it is especially useful for automatically collected one-at-a-time data. However, in such a case, it is common that the process characteristics become dependent as items produced one after another are inspected. In this paper, we study the problem of process monitoring when the process is of high quality and measurement values possess a certain serial dependence. The problem of assuming independence is examined and a Markov model for this type of process is studied, upon which suitable control procedures can be developed.     

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.     

A single-featured EWMA-X control chart for detecting shifts in process mean and standard deviation

The combined EWMA-X chart is a commonly used tool for monitoring both large and small process shifts. However, this chart requires calculating and monitoring two statistics along with two sets of control limits. Thus, this study develops a single-featured EWMA-X (called SFEWMA-X) control chart which has the ability to simultaneously monitor both large and small process shifts using only one set of statistic and control limits. The proposed SFEWMA-X chart is further extended to monitoring the shifts in process standard deviation. A set of simulated data are used to demonstrate the proposed chart's superior performance in terms of average run length compared with that of the traditional charts. The experimental examples also show that the SFEWMA-X chart is neater and easier to visually interpret than the original EWMA-X chart.     

One-sided synthetic control charts for monitoring the multivariate coefficient of variation

Shewhart's type control charts for monitoring the Multivariate Coefficient of Variation (MCV) have recently been proposed in order to monitor the relative variability compared with the mean. These approaches are known to be rather slow in the detection of small or moderate process shifts. In this paper, in order to improve the detection efficiency, two one-sided Synthetic charts for the MCV are proposed. A Markov chain method is used to evaluate the statistical performance of the proposed charts. Furthermore, computational experiments reveal that the proposed control charts outperform the Shewhart MCV control chart in terms of the average run length to detect an out-of-control state. Finally, the implementation of the proposed chart is illustrated with an example using steel sleeves data.     

Data driven choice of control charts

Standard control charts are often seriously in error when the distributional form of the observations differs from normality. Recently, control charts have been developed for larger parametric families. A third possibility is to apply a suitable (modified version of a) nonparametric control chart. This paper deals with the question when to switch from the control chart based on normality to a parametric control chart, or even to a nonparametric one. This model selection problem is solved by using the estimated model error as yardstick. It is shown that the new combined control chart asymptotically behaves as each of the specific control charts in their own domain. Simulations exhibit that the combined control chart performs very well under a great variety of distributions and hence it is recommended as an omnibus control chart, nicely adapted to the distribution at hand. The combined control chart is illustrated by an application on real data. The new modified nonparametric control chart is an attractive alternative and can be recommended as well.     

Average run lengths for cusum control charts applied to residuals

A common approach to building control charts for autocorrelated data is to apply classical SPC to the residuals from a time series model of the process. However, Shewhart charts and even CUSUM charts are less sensitive to small shifts in the process mean when applied to residuals than when applied to independent data. Using an approximate analytical model, we show that the average run length of a CUSUM chart for residuals can be reduced substantially by modifying traditional chart design guidelines to account for the degree of autocorrelation in the data.     

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.     

Poisson GWMA Control Chart

A generally weighted moving average (GWMA) control chart for monitoring Poisson observations is introduced. Using simulation, its average run lengths and standard deviations of run lengths are compared with those of other control charts for Poisson data. It is shown that the Poisson GWMA chart outperforms other control charts, especially when the process shift is small.