Economic and economic–statistical designs of VSI Bayesian control chart using Monte Carlo method and ABC algorithm

The usual practice in using a Bayesian control chart to monitor a process is done by taking samples from the process with fixed sampling intervals. Recent studies on traditional control charts have shown that variable sampling interval (VSI) scheme compared to classical scheme (fixed ratio sampling, FRS) helps practitioners to detect process shifts more quickly. In this paper, the effectiveness of VSI scheme on performance of Bayesian control chart has been studied, based on economic (ED) and economic–statistical designs (ESD). Monte Carlo method and artificial bee colony algorithm have been utilized to obtain optimal design parameters of Bayesian control chart (sample size, sampling intervals, warning limit and control limit) since the statistic of this approach does not have any specified distribution. Finally, VSI Bayesian control chart has been compared to FRS Bayesian and VSI X-bar approaches based on ED and ESD, separately. According to the results, it has been found that the performance of VSI Bayesian scheme is better than FRS Bayesian and VSI X-bar approaches.     

A Process Monitoring Scheme Controlling False Discovery Rate

A new process monitoring scheme is proposed by using the Storey procedure for controlling the positive false discovery rate in multiple testing. For the 2-span control scheme, it is shown numerically that the proposed method performs better than X-bar chart in terms of the average run length. Some simulations are accomplished to evaluate the performance of the proposed scheme in terms of the average run length and the conditional expected delay. The results are compared with those of the existing monitoring schemes including the X-bar chart. The false discovery rate is also estimated and compared with the target control level.     

A double exponentially weigiited moving average control procedure with variable sampling intervals

Shewhart, cumulative sum (CUSUM), and exponentially weighted moving average (EWMA) control procedures with variable sampling intervals (VSI) have been investigated in recent years for detecting shifts in the process mean. Such procedures have been shown to be more efficient when compared with the corresponding fixed sampling interval (FSI) charts with respect to the average time to signal (ATS) when the average run length (ARL) values of both types of procedures are held equal. Frequent switching between the different sampling intervals can be a complicating factor in the application of control charts with variable sampling intervals. In this article, we propose using a double exponentially weighted moving average control procedure with variable sampling intervals (VSI-DEWMA) for detecting shifts in the process mean. It is shown that the proposed VSI-DEWMA control procedure is more efficient when compared with the corresponding fixed sampling interval FSI-DEWMA chart with respect to the average time to signal (ATS) when the average run length (ARL) values of both types of procedures are held equal. It is also shown that the VSI-DEWMA procedure reduces the average number of switches between the sampling intervals and has similar ATS properties as compared to the VSI-EMTMA control procedure     

Nonparametric quality control charts based on the sign statistic

Nonparametric control chart are presented for the problem of detecting changes in the process median (or mean), or changes in the process variability when samples are taken at regular time intervals. The proposed procedures are based on sign-test statistics computed for each sample, and are used in Shewhart and cumulative sum control charts. When the process is in control the run length distributions for the proposed nonparametric control charts do not depend on the distribution of the observations. An additional advantage of the non-parametric control charts is that the variance of the process does not need to be established in order to set up a control chart for the mean. Comparisons with the corresponding parametric control charts are presented. It is also shown that curtailed sampling plans can considerably reduce the expected number of observations used in the Shewhart control schemes based on the sign statistic.     

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.     

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.     

Multivariate EWMA Control Chart with Adaptive Sample Sizes

Standard multivariate control charts usually employ fixed sample sizes at equal sampling intervals to monitor a process. In this study, a multivariate exponential weighted moving average (MEWMA) chart with adaptive sample sizes is investigated. Performance measure of the adaptive-sample-size MEWMA chart is obtained through a Markov chain approach. The performance of the adaptive-sample-size MEWMA chart is compared with the fixed-sample-size control chart in terms of steady-state average run length for different magnitude of shifts in the process mean. It is shown that the adaptive-sample-size chart is more efficient than the fixed-sample-size MEWMA control chart in detecting shifts in the process mean.     

A New Control Scheme Always Better Than X-Bar Chart

A new control scheme is proposed by borrowing the idea of the Benjamini–Hochberg procedure for controlling the false discovery rate in multiple testing. It is shown theoretically that the proposed 2-span control scheme outperforms the Shewhart X-bar chart in terms of the average run length under any size of mean shifts. Some simulations are carried out to demonstrate that the proposed scheme having various span sizes always outperforms the X-bar chart in terms of the average run lengths.     

An Improved Distribution-free EWMA Mean Chart

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 nonnormal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used here. In this article, we propose an improved asymmetric EWMA mean chart based on a simple statistic to monitor process mean shift. We explored the sampling properties of the new monitoring statistic and calculated the average run lengths of the proposed asymmetric EWMA mean chart. We recommend the proposed improved asymmetric EWMA mean chart because the average run lengths of the modified charts are more accurate and reasonable than those of the five existed mean charts. A numerical example of service times with a right skewed distribution from a service system of a bank branch is used to illustrate the application of the improved asymmetric EWMA mean chart and to compare it with the five existing mean charts. The proposed chart showed better detection performance than those of the five existing mean charts in monitoring and detecting shifts in the process mean.     

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.     

A median loss control chart for monitoring quality loss under skewed distributions

The quality and loss of products are crucial factors separating competitive companies in global market. Firms widely employ a loss function to measure the loss caused by a deviation of the quality variable from the target value. Monitoring this deviation from the process target value is important from the view of Taguchi’s philosophy. In reality, there are many situations where the distribution of the quality variable may not be normal but skewed. This paper aims at developing a median loss (ML) control chart for monitoring quality loss under skewed distributions. Both the cases with fixed and variable sampling intervals are considered. Numerical results show that the ML chart with (optimal) variable sampling intervals performs better than the ML chart in detecting small to moderate shifts in the process loss centre or in the difference of mean and target and/or variance of a process variable. The ML chart and the ML chart with variable sampling intervals also illustrate the best performance in detection out-of-control process for a process quality variable with a left-skewed distribution. A numerical example illustrates the application of the proposed control chart.     

Optimal Variable Sample Size and Sampling Interval MSE Chart Based on Improved Square Root Transformation

In the field of statistical process control (SPC), control charts for attributes are widely used to detect the out-of-control condition by checking the number of nondefective units or nondefective in a sample. In this article, we use the average time to signal (ATS) and the average number of observations to signal (ANOS) to evaluate the performance of the optimal variable sample size and sampling interval (VSSI) improved square root transformation (ISRT) mean square error (MSE) (VSSI_ ISRT_ MSE) control chart for attribute data. In addition, this control chart will be used to monitor: (1) the difference between the process mean and the target value, and (2) the process variance shifts. We found that the optimal VSSI_ ISRT_ MSE chart performs better than the specific VSSI, the optimal variable sampling interval (VSI), and the fixed parameters (FP) ISRT_MSE charts. An example is given to illustrate this new proposed approach.     

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.     

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.     

A new double sampling control chart for monitoring process mean using auxiliary information

Statistical quality control charts have been widely accepted as a potentially powerful process monitoring tool because of their excellent speed in tracking shifts in the underlying process parameter(s). In recent studies, auxiliary-information-based (AIB) control charts have shown superior run length performances than those constructed without using it. In this paper, a new double sampling (DS) control chart is constructed whose plotting-statistics requires information on the study variable and on any correlated auxiliary variable for efficiently monitoring the process mean, namely AIB DS chart. The AIB DS chart also encompasses the classical DS chart. We discuss in detail the construction, optimal design, run length profiles, and the performance evaluations of the proposed chart. It turns out that the AIB DS chart performs uniformly better than the DS chart when detecting different kinds of shifts in the process mean. It is also more sensitive than the classical synthetic and AIB synthetic charts when detecting a particular shift in the process mean. Moreover, with some realistic beliefs, the proposed chart outperforms the exponentially weighted moving average chart. An illustrative example is also presented to explain the working and implementation of the proposed chart.     

Maximum Chi-Square Generally Weighted Moving Average Control Chart for Monitoring Process Mean and Variability

In this article, we propose a new control chart called the maximum chi-square generally weighted moving average (MCSGWMA) control chart. This control chart can effectively combine two generally weighted moving average (GWMA) control charts into a single one and can detect both increases as well as decreases in the process mean and/or variability simultaneously. The average run length (ARL) characteristics of the MCSGWMA and maximum exponentially weighted moving average (MaxEWMA) charts are evaluated by performing computer simulations. The comparison of the ARLs shows that the MCSGWMA control chart performs better than the MaxEWMA control chart.     

Weighted adaptive multivariate CUSUM charts with variable sampling intervals

The adaptive multivariate CUSUM (AMCUSUM) chart has received considerable attention because of its superior sensitivity against a range of mean shift sizes than that of the conventional non-adaptive multivariate CUSUM (MCUSUM) chart. Recently, weighted AMCUSUM (WAMCUSUM) charts with a fixed sampling interval (FSI) have been proposed, called the WAMCUSUM-FSI charts, which provide more sensitivity than the AMCUSUM-FSI charts. In this paper, we extend this work and propose WAMCUSUM charts with variable sampling interval (VSI), named the WAMCUSUM-VSI charts, for efficiently monitoring the mean of a multivariate normally distributed process. The Monte Carlo simulation method is used to compute the average time to signal (ATS) and the adjusted ATS (AATS) profiles of the existing and proposed charts. It is found that the WAMCUSUM-VSI charts perform substantially and nearly uniformly better than the WAMCUSUM-FSI charts in terms of the ATS and AATS performance criterion. An example is given to explain the implementation of the WAMCUSUM charts with fixed and VSIs.     

Design of adaptive s control charts

The Shewhart s chart has been widely used to monitor the standard deviation of a process. However, the main disadvantage of an s chart is its slowness to signal small increases in the variability. In this paper, ideas of adaptive control charts are extended to the Shewhart s chart for improving the efficiency in signalling increases in the standard deviation. A Markov chain model is applied to evaluate its performances and compares its performances with combined double sampling and variable sampling intervals s chart, variable parameters (VP) R chart, exponentially weighted moving average and Cusum charts. The statistical performances show that the VP s chart is more sensitive to increases in standard deviation.     

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.     

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.