Monitoring the Weibull Shape Parameter with Type II Censored Data

In this article, we introduce a method for monitoring the Weibull shape parameter β with type II (failure) censored data. The control limits depend on the sample size, the number of censored observations, the target average run length, and the stable value of β. The method assumes that the scale parameter α is constant during each sampling period, which is true under rational subgrouping. The proposed method utilizes the relationship between Weibull and smallest extreme value distribution. We propose an unbiased estimator of σ = 1/β as the monitoring statistic. We derive the control limits for one‐sided and two‐sided charts for several stable process average run lengths. We discuss two schemes, namely, the control‐limits‐only scheme and the control‐limits‐with‐warning‐lines scheme. The stable process average run length performance of the proposed charts is studied and compared with those of other charts for monitoring β under similar assumptions. Copyright © 2014 John Wiley & Sons, Ltd.     

Effect of estimation under nonnormality on the phase II performance of linear profile monitoring approaches

The number of studies about control charts proposed to monitor profiles, where the quality of a process/product is expressed as function of response and explanatory variable(s), has been increasing in recent years. However, most authors assume that the in‐control parameter values are known in phase II analysis and the error terms are normally distributed. These assumptions are rarely satisfied in practice. In this study, the performance of EWMA‐R, EWMA‐3, and EWMA‐3(d²) methods for monitoring simple linear profiles is examined via simulation where the in‐control parameters are estimated and innovations have a Student's t distribution or gamma distribution. Instead of the average run length (ARL) and the standard deviation of run length, we used average and standard deviation of the ARL as performance measures in order to capture the sampling variation among different practitioners. It is seen that the estimation effect becomes more severe when the number of phase I profiles used in estimation decreases, as expected, and as the distribution deviates from normality to a greater extent. Besides, although the average ARL values get closer to the desired values as the amount of phase I data increases, their standard deviations remain far away from the acceptable level indicating a high practitioner‐to‐practitioner variability.     

Bivariate Dispersion Control Charts for Monitoring Non‐Normal Processes

Multivariate control charts are well known to be more sensitive to the occurrence of variation in processes with two or more correlated quality variables than univariate charts. The use of separate univariate control charts to monitor multivariate process can be misleading as it ignores the correlation between the quality characteristics. The application of multivariate control charts allows for the simultaneous monitoring of the quality characteristics by forming a single chart. The charts operate on the assumption that process observations are normally distributed, but in practice this is not always the case. In this study, we examine and present multivariate dispersion control charts for detecting shifts in the covariance matrix of normal and non‐normal bivariate processes. These control charts, referred to as SMAX, QMAX, MDMAX and MADMAX, rely on dispersion estimates, such as the sample standard deviation (S), interquartile range (Q), average absolute deviation from median (MD) and median absolute deviation (MAD), respectively. We compare the performances of these charts to the existing multivariate generalized variance |S| and RMAX charts for bivariate processes using normal and non‐normal parent distributions. The average run length (ARL) measure is used for the evaluation and comparison of the charts. A real life and simulated datasets are used to demonstrate the application of the charts. Copyright © 2016 John Wiley & Sons, Ltd.     

Efficient bivariate EWMA charts for monitoring process dispersion

To ensure high quality standards of a process, the application of control charts to monitor process performance has become a regular routine. Multivariate charts are a preferred choice in the presence of more than one process variable. In this article, we proposed a set of bivariate exponentially weighted moving average (EWMA) charts for monitoring the process dispersion. These charts are formulated based on a variety of dispersion statistics considering normal and non-normal bivariate parent distributions. The performance of the different bivariate EWMA dispersion charts is evaluated and compared using the average run length and extra quadratic loss criteria. For the bivariate normal process, the comparisons revealed that the EWMA chart based on the maximum standard deviation (SMAX_E) was the most efficient chart when the shift occurred in one quality variable. It also performed well when the sample size is small and the shift occurred in both quality variables. The EWMA chart based on the maximum average absolute deviation from median (MDMAX_E) performed better than the other charts in most situations when the shift occurred in the covariance matrix for the bivariate non-normal processes. An illustrative example is also presented to show the working of the charts.     

A New Synthetic Exponentially Weighted Moving Average Control Chart for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely recognized as a potentially powerful process monitoring tool of the statistical process control because of their excellent speed in detecting small to moderate shifts in the process parameters. Recently, new EWMA and synthetic control charts have been proposed based on the best linear unbiased estimator of the scale parameter using ordered ranked set sampling (ORSS) scheme, named EWMA‐ORSS and synthetic‐ORSS charts, respectively. In this paper, we extend the work and propose a new synthetic EWMA (SynEWMA) control chart for monitoring the process dispersion using ORSS, named SynEWMA‐ORSS chart. The SynEWMA‐ORSS chart is an integration of the EWMA‐ORSS chart and the conforming run length chart. Extensive Monte Carlo simulations are used to estimate the run length performances of the proposed control chart. A comprehensive comparison of the run length performances of the proposed and the existing powerful control charts reveals that the SynEWMA‐ORSS chart outperforms the synthetic‐R, synthetic‐S, synthetic‐D, synthetic‐ORSS, CUSUM‐R, CUSUM‐S, CUSUM‐ln S², EWMA‐ln S² and EWMA‐ORSS charts when detecting small shifts in the process dispersion. A similar trend is observed when the proposed control chart is constructed under imperfect rankings. An application to a real data is also provided to demonstrate the implementation and application of the proposed control chart. Copyright © 2014 John Wiley & Sons, Ltd.     

Monitoring the Weibull Shape Parameter by Control Charts for the Sample Range of Type II Censored Data

In this paper, we propose control charts to monitor the Weibull shape parameter β under type II (failure) censoring. This chart scheme is based on the sample ranges of smallest extreme value distributions derived from Weibull processes. We suggest one‐sided (high‐side or low‐side) and two‐sided charts, which are unbiased with respect to the average run length (ARL). The control limits for all types of charts depend on the sample size, the number of failures c under type II censoring, the desired stable‐process ARL, and the stable‐process value of β. This article also considers sample size requirements for phase I in retrospective charts. We investigate the effect of c on the out‐of‐control ARL. We discuss a simple approach to choosing c by cost minimization. The proposed schemes are then applied to data on the breaking strengths of carbon fibers. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

The average run length (ARL) is usually used as a sole measure of performance of a multivariate control chart. The Hotelling's T², multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts are commonly optimally designed based on the ARL. Similar to the case of univariate quality control, in multivariate quality control, the shape of the run length distribution changes in accordance to the magnitude of the shift in the mean vector, from highly skewed when the process is in‐control to nearly symmetric for large shifts. Because the shape of the run length distribution changes with the magnitude of the shift in the mean vector, the median run length (MRL) provides additional and more meaningful information about the in‐control and out‐of‐control performances of multivariate charts, not given by the ARL. This paper provides a procedure for optimal designs of the multivariate synthetic T² chart for the process mean, based on MRL, for both the zero and steady‐state modes. Two Mathematica programs, each for the zero state and steady‐state modes are given for a quick computation of the optimal parameters of the synthetic T² chart, designed based on MRL. These optimal parameters are provided in the paper, for the bivariate case with sample sizes, nin{4, 7, 10}. The MRL performances of the synthetic T², MEWMA and Hotelling's T² charts are also compared. Copyright © 2011 John Wiley & Sons, Ltd.     

Design and implementation of qth quantile‐unbiased tr‐chart for monitoring times between events

The times between events control charts have been proposed in literature for statistical monitoring of high‐yield processes by observing the waiting times up to r ^th (r ≥ 1  ) non‐conforming items or defects. The average run length (ARL) is the most widely used performance measure to evaluate the chart's performance, but in recent years, it has been subjected to criticisms. Because the run length distribution is highly skewed and hence, the ARL is not necessarily a typical value of the run length. Thus, evaluation of the control chart based on ARL alone could be misleading. In this paper, the quantiles of run length distribution are considered, instead of ARL, to design the t_r ‐chart. Further, we eliminate the bias in q ^th quantile function of the t_r ‐chart for both the known and unknown parameter case. In particular, the MRL‐unbiased t_r ‐chart is discussed in detail and compared with the ARL‐unbiased t_r ‐chart. It is found that the MRL‐unbiased t_r ‐chart outperforms than the corresponding ARL‐unbiased chart in unknown parameter case. It is also found that the proposed chart requires less phase I observations than that of the earlier studies has been suggested.     

A Synthetic Control Chart for Monitoring Process Variability

The control chart based on Downton's estimator (D chart) has recently been introduced in the literature for monitoring the process variability. The D chart is found to be equally efficient to the S chart in terms of detecting shifts in process variability. In this paper, salient features of D chart and the conforming run length chart are combined to produce synthetic D chart. The average run length performance of the synthetic D chart is investigated using simulation study and is compared with the originally proposed D chart and some other procedures proposed in the literature. It is found that it has an improved performance in comparison with the traditional control charts for process variability. Copyright © 2013 John Wiley & Sons, Ltd.     

A moving average control chart using a robust scale estimator for process dispersion

Control charts are one of the most powerful tools used to detect and control industrial process deviations in statistical process control. In this paper, a moving average control chart based on a robust scale estimator of standard deviation, namely, the sample median absolute deviation (MAD) statistic, for monitoring process dispersion, is proposed. A simulation study is conducted to evaluate the performance of the proposed moving average median absolute deviation (MA‐MAD) chart, in terms of average run length for various distributions. The results show that the moving average MAD chart performs well in detecting small and moderate shifts in process dispersion, especially when the normality assumption is violated. In addition, this chart is very efficient, especially when the quality characteristic follows a skewed distribution. Numerical and simulated examples are given at the end of the paper.     

DETECTING EFFECTIVENESS MEASURED BY AURL

This communication addresses the problem of comparing the effectiveness of different control-charting schemes. The measure of average unit run length (AURL) is used to compare the effectiveness of the x-bar charts and the R charts with different sampling frequencies and different sample sizes. Since the trade-off between the frequency of false alarm and the detecting effectiveness is the most critical issue in the design of the control charts, we adjust the control limits of the charts in order to conduct the fair comparisons based on equal frequency of false alarm. © 1997 by John Wiley & Sons, Ltd.     

An enhanced CUSUM‐t chart for process mean

In this paper, we propose an auxiliary‐information–based (AIB) Crosier cumulative sum (CCUSUM) t chart for monitoring the process mean, namely, the AIB‐CCUSUM‐t chart. The run length characteristics of the proposed chart are computed using Monte Carlo simulation. The optimal parameters for the AIB‐CCUSUM‐t chart to detect specific mean shifts are computed. The fast initial response (FIR) feature is also attached with the proposed chart. It is found that the AIB‐CCUSUM‐t and FIR‐AIB‐CCUSUM‐t charts perform uniformly and substantially better than the CCUSUM‐t and FIR‐CCUSUM‐t charts, respectively. An example is presented to support the theory.     

Nonparametric Progressive Mean Control Chart for Monitoring Process Target

Nonparametric control charts are widely used when the parametric distribution of the quality characteristic of interest is questionable. In this study, we proposed a nonparametric progressive mean control chart, namely the nonparametric progressive mean chart, for efficient detection of disturbances in process location or target. The proposed chart is compared with the recently proposed nonparametric exponentially weighted moving average and nonparametric cumulative sum charts using different run length characteristics such as the average run length, standard deviation of the run length, and the percentile points of the run length distribution. The comparisons revealed that the proposed chart outperformed recent nonparametric exponentially weighted moving average and nonparametric cumulative sum charts, in terms of detecting the shifts in process target. A real life example concerning the fill heights of soft drink beverage bottles is also provided to illustrate the application of the proposed nonparametric control chart. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal design of one-sided exponential cumulative sum charts with known and estimated parameters based on the median run length

As a useful tool in statistical process control (SPC), the exponential control chart is more and more popular for monitoring high-quality processes. Considering both known and estimated parameter cases, the one-sided exponential cumulative sum (CUSUM) charts are studied in this paper through a Markov chain approach. Because the shape of the run length (RL ) distribution of the one-sided exponential CUSUM charts is skewed and it also changes with the mean shift size and the number of Phase I samples used to estimate the process parameter, the median run length (MRL ) is employed as a good alternative performance measure for the charts. The optimal design procedures based on MRL of the one-sided exponential CUSUM charts with known and estimated parameters are discussed. By comparing the MRL performance of the chart with known parameters with the one of the chart with estimated parameters, we investigate the effect of estimated process parameters on the properties of the chart. Finally, an application is illustrated to show the implementation of the chart.     

A side-sensitive synthetic coefficient of variation chart

Control charts for monitoring the coefficient of variation (γ) are useful for processes with an inconsistent mean (μ) and a standard deviation (σ) which changes with μ, by monitoring the consistency in the ratio σ over μ. The synthetic-γ chart is one of the charts proposed to monitor γ, and its attractiveness lie in waiting until a second point to fall outside the control limits before a decision is made. However, existing synthetic-γ charts do not differentiate between the points falling outside the upper control limit (UCL) and lower control limit (LCL). Hence, this paper proposes a side-sensitive synthetic-γ chart, where successive nonconforming samples must either fall above the UCL or below the LCL. Formulae to compute the average run length (ARL), the standard deviation of the run length (SDRL) and expected average run length (EARL) are derived using the Markov chain approach, and the algorithms to obtain the optimal charting parameters are proposed. Subsequently, the optimal charting parameters, ARL, SDRL and EARL values for various numerical examples are shown. Comparisons show that the side-sensitive synthetic-γ chart consistently outperforms the existing synthetic-γ chart, especially for small shifts. The proposed chart also consistently outperforms the Shewhart-γ chart, while showing comparable or better performance than the Exponentially Weighted Moving Average (EWMA) chart for most shift sizes, except for very small shifts. Finally, this paper shows the implementation of the proposed chart on an industrial example.     

Efficient homogeneously weighted dispersion control charts with an application to distillation process

Monitoring disturbances in process dispersion using control chart is mostly based on the assumption that the quality characteristic follows normal distribution, which is not the case in many real-life situations. This paper proposes a set of new dispersion charts based on the homogeneously weighted moving average (HWMA) scheme, for efficient detection of shifts in process standard deviation (σ). These charts are based on a variety of σ estimators and are investigated for normal as well as heavy tailed symmetric and skewed distributions. The shift detection ability of the charts is evaluated using different run length characteristics, such as average run length (ARL), extra quadratic loss (EQL), and relative ARL measures. The performance of the proposed HWMA control charts is also compared with the existing EWMA dispersion charts, using different design parameters. Furthermore, an illustrative example is presented to monitor the vapor pressure in a distillation process.     

Practitioner advice: ERLDAT–A web-based tool to design and analyze EWMA control charts

The performance of a control chart is completely characterized by its run length distribution. Quality practitioners usually do not have access to the run length distribution but rely on the average run length (ARL) to design and evaluate the performance of an exponentially weighted moving average (EWMA) control chart. This article presents a web-based tool that provides users easy access to the Phase 2 (online or monitoring phase) run length distribution for a two-sided EWMA control chart with known parameters. The web-based tool calculates the run length distribution, percentiles of the run length distribution, as well as the mean (ARL) and variance (VRL) of the run length distribution. Additional functionality of the web-based tool includes plotting the run length distribution functions, building tables of the quantiles of the run length distribution, finding the smoothing parameter (λ) for an EWMA control chart for fixed control limit that satisfies ARL, VRL or percentile performance, and finding the control chart limit (k) for an EWMA control chart that satisfies ARL, VRL, or percentile performance. This tool and these techniques enable quality practitioners to better design and evaluate EWMA control charts.     

Computing the Percentage Points of the Run-Length Distributions of Multivariate CUSUM Control Charts

Multivariate CUSUM control charts are often used instead of the standard Hotelling's control charts in many practical problems when detection of small shifts in the process mean is important. However, design of multivariate CUSUM control charts are usually based on the average run length (ARL). In this work, we will compute the percentage points of the run-length distributions of two multivariate CUSUM control charts. It will be shown that interpretations based on ARL can be misleading since the in-control run-length distribution of a multivariate CUSUM is highly skewed. On the other hand, the percentage points of the run-length distribution provide additional information such as the median run length, early false out-of-control signals, and the skewness of the run-length distribution for a particular scheme. These extra information might provide quality control engineers further knowledge of a particular multivariate CUSUM control chart scheme.     

Moving range charts for monitoring the Weibull shape parameter with single observation samples

This paper proposes an approach to monitor shifts in the Weibull shape parameter bfβ via control charts based on the moving range of single‐point samples from a smallest extreme value distribution. The average run length (ARL) of the proposed charts are computed using Fredholm integral equations of the second kind. The derived control limits for one‐sided and two‐sided control charts are unbiased in the sense that the ARL when β has shifted is shorter than the desired stable‐process ARL. These control limits depend only on the desired stable‐process ARL and the stable value of β. The paper also discusses the sample size requirements for Phase I so that the run length distributions are similar under standards‐given scenario (β is given) and retrospective scenario (β is estimated from past data). The proposed methods are then applied to data on the breaking strengths of carbon fibers. The results suggest that one‐sided control charts can detect small shifts in β sooner than two‐sided charts. Copyright © 2011 John Wiley & Sons, Ltd.