Monitoring the Coefficient of Variation Using a Variable Sampling Interval Control Chart

The coefficient of variation (CV) is a quality characteristic that has several applications in applied statistics and is receiving increasing attention in quality control. Few papers have proposed control charts that monitor this normalized measure of dispersion. In this paper, an adaptive Shewhart control chart implementing a variable sampling interval (VSI) strategy is proposed to monitor the CV. Tables are provided for the statistical properties of the VSI CV chart, and a comparison is performed with a Fixed Sampling Rate Shewhart chart for the CV. An example illustrates the use of these charts on real data gathered from a casting process. Copyright © 2012 John Wiley & Sons, Ltd.     

Monitoring the Coefficient of Variation Using the Side Sensitive Group Runs Chart

Many quality characteristics have means and standard deviations that are not independent. Instead, the standard deviations of these quality characteristics are proportional to their corresponding means. Thus, monitoring the coefficient of variation (CV), for these quality characteristics, using a control chart has gained remarkable attention in recent years. This paper presents a side sensitive group runs chart for the CV (called the SSGR CV chart). The implementation and optimization procedures of the proposed chart are presented. Two optimization procedures are developed, i.e. (i) by minimizing the average run length (ARL) when the shift size is deterministic and (ii) by minimizing the expected average run length (EARL) when the shift size is unknown. An application of the SSGR CV chart using a real dataset is also demonstrated. Additionally, the SSGR CV chart is compared with the Shewhart CV, runs rules CV, synthetic CV and exponentially weighted moving average CV charts by means of ARLs and standard deviation of the run lengths. The performance comparison is also conducted using EARLs when the shift size is unknown. In general, the SSGR CV chart surpasses the other charts under comparison, for most upward and downward CV shifts. Copyright © 2015 John Wiley & Sons, Ltd.     

Side-sensitive modified group runs charts with and without measurement errors for monitoring the coefficient of variation

The coefficient of variation (CV) is used in process monitoring when the process mean and standard deviation are proportional to each other. In this work, a side-sensitive modified group runs CV (SSMGR CV) chart is proposed for monitoring the process CV. The run length performance of the SSMGR CV chart is compared to those of the existing CV charts in terms of the average and standard deviation of the run length criteria. The SSMGR CV chart is found to outperform the existing CV charts. In addition, the run length performance of the SSMGR CV chart is also evaluated in the presence of measurement errors, as these errors are not only unavoidable in practice but they also affect the sensitivity of a control chart in detecting an out-of-control situation. The results obtained show that the accuracy and precision errors affect the performance of the SSMGR CV chart in detecting an out-of-control situation.     

The economic performance of the Shewhart t chart

In the production of small batches of customized parts, high flexibility and frequent switching of production from one product variant to another could not allow for the implementation of a control chart to monitor the process. In fact, when a short‐run production should be started, the distribution parameters of the quality characteristics to be monitored are frequentlytextitunknown and the production run is too short to get sufficient Phase I samples. To overcome this problem, the statistical properties of Shewhart t charts monitoring a short production run have been recently discussed in literature. In this paper, we investigate their economic performance: the SPC inspection cost optimization is constrained by the manufacturing and the inspection activities configuration. The decision variables of the problem include the chart design parameters and the size of batches of parts to be worked and released to the local inspection area. A numerical analysis aimed at evaluating the economic performance of the Shewhart t chart vs the Shewhart chart with known parameters has been performed. The expected economic loss associated with the implementation of the Shewhart t chart is acceptable with respect to the ‘ideal’ condition of the control chart with known parameters when the cost optimization is achieved without a statistical constraint limiting the number of expected false alarms. Finally, the effect of an erroneous initial set‐up on the correctness of the inspection cost estimation has been investigated. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of X̄ control chart with modified limits in process control

The conventional Shewhart X̄ chart is developed based on the assumption that the within‐sample variation is due to the inherent process variation, and any significant variation between samples is attributed to the existence of assignable causes. In the manufacturing industry there are processes where there is variation between samples due to the inherent process variation. A straightforward application of the conventional Shewhart X̄ chart would thus result in more frequent false alarms. The problems associated with various Shewhart X̄ charts in monitoring such a process are discussed using a real data set from an integrated circuit (IC) assembly process. A Shewhart X̄ chart with modified limits is adapted for such a process. In addition to the usual ability to signal for assignable causes, the X̄ chart with modified limits is also developed as a tool to signal the need for adjustment of controllable process variables for improving the process capability. Practical application of this chart in monitoring an IC assembly process is discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

On the performance of coefficient of variation charts in the presence of measurement errors

In the literature, coefficient of variation control charts have been introduced under the assumption of no measurement errors. However, measurement errors always exist in practice, and they do affect the performance of control charts in the detection of an out‐of‐control situation. In this paper, we therefore study the performance of a coefficient of variation Shewhart‐type control chart (Shewhart‐CV chart) and also one‐sided coefficient of variation exponentially weighted moving average–type control charts (EWMA‐γ² charts) using a model with linear covariates. Moreover, we propose and study the performance of a two‐sided EWMA‐γ² chart using a model with linear covariates. Several figures and tables are provided and analyzed to evaluate the statistical performance of these control charts for different sources of measurement errors. The obtained results show that the precision and accuracy errors significantly affect the performance of both the Shewhart‐CV and EWMA‐γ² control charts. An example illustrating the use of this study is finally presented.     

Control chart for monitoring the coefficient of variation with an exponentially weighted moving average procedure

The coefficient of variation (CV) of a population is defined as the ratio of the population standard deviation to the population mean, which can be regarded as a measure of stability or uncertainty and can also indicate the relative dispersion of data to the population mean. This paper proposes a new exponentially weighted moving average chart for monitoring CV, which is constructed by truncating those negative normalized observations to 0 in the traditional exponentially weighted moving average CV statistics. The implementation and optimization procedures of the proposed chart are presented. The new chart is compared with some existing CV charts by means of average run length, and the comparison results show that the new chart outperforms other charts in most cases. Two examples illustrate the use of this chart on real data gathered from a metal sintering process and from a die casting hot chamber process.     

A Control Chart for the Multivariate Coefficient of Variation

Existing charts in the literature usually monitor either the mean or the variance of the process. However, in certain scenarios, the practitioner is not interested in the changes in the mean or the variance but is instead interested in monitoring the relative variability compared with the mean. This relative variability is called the coefficient of variation (CV). In the existing literature, none of the control charts that monitor the CV are applied for multivariate data. To fill this gap in research, this paper proposes a CV chart that monitors the CV for multivariate data. To the best of the authors' knowledge, this proposed chart is the first control chart for this purpose. The distributional properties of the sample CV for multivariate data and the procedures to implement the chart are presented in this paper. Formulae to compute the control limits, the average run length, the standard deviation of the run length, and the expected average run length for the case of unknown shift size are derived. From the numerical examples provided, the effects of the number of variables, the sample size, the shift size and the in‐control value of the CV are studied. Finally, we demonstrate the usefulness and applicability of the proposed chart on real data. Copyright © 2015 John Wiley & Sons, Ltd.     

Progressive Mean Control Chart for Monitoring Process Location Parameter

Control charts are widely used for process monitoring. They show whether the variation is due to common causes or whether some of the variation is due to special causes. To detect large shifts in the process, Shewhart‐type control charts are preferred. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are generally used to detect small and moderate shifts. Shewhart‐type control charts (without additional tests) use only current information to detect special causes, whereas CUSUM and EWMA control charts also use past information. In this article, we proposed a control chart called progressive mean (PM) control chart, in which a PM is used as a plotting statistic. The proposed chart is designed such that it uses not only the current information but also the past information. Therefore, the proposed chart is a natural competitor for the classical CUSUM, the classical EWMA and some recent modifications of these two charts. The conclusion of this article is that the performance of the proposed PM chart is superior to the compared ones for small and moderate shifts, and its performance for large shifts is better (in terms of the average run length). Copyright © 2012 John Wiley & Sons, Ltd.     

Combined Application of Shewhart and Cumulative Sum R Chart for Monitoring Process Dispersion

This study analyzes the performance of combined applications of the Shewhart and cumulative sum (CUSUM) range R chart and proposes modifications based on well‐structured sampling techniques, the extreme variations of ranked set sampling, for efficient monitoring of changes in the process dispersion. In this combined scheme, the Shewhart feature enables quick detection of large shifts from the target standard deviation while the CUSUM feature takes care of small to moderate shifts from the target value. We evaluate the numerical performance of the proposed scheme in terms of the average run length, standard deviation of run length, the average ratio average run length, and average extra quadratic loss. The results show that the combined scheme can detect changes in the process that were small or large enough to escape detection by the lone Shewhart R chart or CUSUM R chart, respectively. We present a comparison of the proposed schemes with several dispersion charts for monitoring changes in process variability. The practical application of the proposed scheme is demonstrated using real industrial data. Copyright © 2014 John Wiley & Sons, Ltd.     

A Distribution-Free Shewhart Quality Control Chart Based on Signed-Ranks

Since their inception by Walter Shewhart in the late 1920s, most control chart developments have been distribution-based procedures in the sense that the process output is assumed to follow a specified probability distribution (normal for continuous measurements and binomial or Poisson for attribute data). Due to Deming's influence and their widespread adoption as one of the seven basic tools of total quality management (TQM), control charts have been applied to processes where data may be markedly nonnormal. In this article, we propose a distribution-free (or nonparametric) statistical quality control chart for monitoring a process center. The proposed chart is of the Shewhart type and is based on the signed-ranks of grouped observations. The exact false alarm rate and the in-control average run length of the proposed chart are computed by using the null distribution of the well-known Wilcoxon signed-rank statistic. The out-of-control run lengths are computed exactly for normal underlying distributions and by simulation for uniform, double exponential, and Cauchy shift alternatives. Efficiency studies show that the proposed chart is more efficient than the traditional Shewhart X-bar chart under heavy-tailed distributions (the double exponential and the Cauchy) but is less efficient under light-tailed distributions (the uniform and the normal).     

Improving the Performance of Combined Shewhart–Cumulative Sum Control Charts

For an improved monitoring of process parameters, it is generally desirable to have efficient designs of control charting structures. The addition of Shewhart control limits to the cumulative sum (CUSUM) control chart is a simple monitoring scheme sensitive to wide range of mean shifts. To improve the detection ability of the combined Shewhart–CUSUM control chart to off‐target processes, we developed the scheme using ranked set sampling instead of the traditional simple random sampling. We investigated the run length properties of the Shewhart–CUSUM with ranked set samples and compared their performance with certain established control charts. It is revealed that the proposed schemes offer better protection against different types of mean shifts than the existing counterparts including classical Shewhart, classical CUSUM, classical combined Shewhart–CUSUM, adaptive CUSUM, double CUSUM, three simultaneous CUSUM, combined Shewhart‐weighted CUSUM, runs rules‐based CUSUM and the mixed exponentially weighted moving average‐CUSUM. Applications on real data sets are also given to demonstrate the implementation simplicity of the proposed schemes Copyright © 2012 John Wiley & Sons, Ltd.     

A New Chart for Monitoring Service Process Mean

Control charts are demonstrated effective in monitoring not only manufacturing processes but also service processes. In service processes, many data came from a process with nonnormal distribution or unknown distribution. Hence, the commonly used Shewhart variable control charts are not suitable because they could not be properly constructed. In this article, we proposed a new mean chart on the basis of a simple statistic to monitor the shifts of the process mean. We explored the sampling properties of the new monitoring statistic and calculated the average run lengths of the proposed chart. Furthermore, an arcsine transformed exponentially weighted moving average chart was proposed because the average run lengths of this modified chart are more intuitive and reasonable than those of the mean chart. We would recommend the arcsine transformed exponentially weighted moving average chart if we were concerned with the proper values of the average run length. A numerical example of service times with skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the proposed charts. Copyright © 2011 John Wiley & Sons, Ltd.     

Enhanced EWMA charts for monitoring the process coefficient of variation

The coefficient of variation (CV) is an important quality characteristic when the process variance is a function of the process mean for a production process. In this paper, we develop an auxiliary information–based (AIB) estimator for estimating the squared CV, along with its approximated mean and variance. This estimator is then used to devise new one-sided EWMA charts for monitoring the increases or decreases in the squared CV of a normal process, named the AIB-EWMA CV charts. In addition, the sensitivities of these control charts are also enhanced with the fast initial response feature. The Monte Carlo simulation method is used to compute the run length characteristics of the proposed CV charts. Based on detailed run length comparisons, it is found that the proposed AIB-EWMA CV charts are uniformly and substantially better than the existing EWMA CV charts when detecting different kinds of upward/downward shifts in the squared CV. The proposed charts are also applied to a real dataset to support the proposed theory.     

An adaptive EWMA control chart for monitoring the process mean in Bayesian theory under different loss functions

The Shewhart control chart is used for detecting the large shift and an exponentially weighted moving average (EWMA) control chart is used for detecting the small/moderate shift in the process mean. A scheme that combines both the Shewhart control chart and the EWMA control chart in a smooth way is called the adaptive EWMA (AEWMA) control chart. In this paper, we proposed a new AEWMA control chart for monitoring the process mean in Bayesian theory under different loss functions (LFs). We used informative (conjugate prior) under two different LFs: (1) squared error loss function and (2) linex loss function for posterior and posterior predictive distributions. We used the average run length and standard deviation of run length to measure the performance of the AEWMA control chart in the Bayesian theory. A comparative study is conducted for comparing the proposed AEWMA control chart in Bayesian theory with the existing Bayesian EWMA control chart. We conducted a Monte Carlo simulation study to evaluate the proposed AEWMA control chart. For the implementation purposes, we presented a real-data example.     

A New Distribution‐free Control Chart for Joint Monitoring of Unknown Location and Scale Parameters of Continuous Distributions

While the assumption of normality is required for the validity of most of the available control charts for joint monitoring of unknown location and scale parameters, we propose and study a distribution‐free Shewhart‐type chart based on the Cucconi 1 statistic, called the Shewhart‐Cucconi (SC) chart. We also propose a follow‐up diagnostic procedure useful to determine the type of shift the process may have undergone when the chart signals an out‐of‐control process. Control limits for the SC chart are tabulated for some typical nominal in‐control (IC) average run length (ARL) values; a large sample approximation to the control limit is provided which can be useful in practice. Performance of the SC chart is examined in a simulation study on the basis of the ARL, the standard deviation, the median and some percentiles of the run length distribution. Detailed comparisons with a competing distribution‐free chart, known as the Shewhart‐Lepage chart (see Mukherjee and Chakraborti 2 ) show that the SC chart performs just as well or better. The effect of estimation of parameters on the IC performance of the SC chart is studied by examining the influence of the size of the reference (Phase‐I) sample. A numerical example is given for illustration. Summary and conclusions are offered. Copyright © 2013 John Wiley & Sons, Ltd.     

Enhancing the Performance of Combined Shewhart‐EWMA Charts

The combination of Shewhart control charts and an exponentially weighted moving average (EWMA) control charts to simultaneously monitor shifts in the mean output of a production process has proven very effective in handling both small and large shifts. To improve the sensitivity of the control chart to detect off‐target processes, we propose a combined Shewhart‐EWMA (CSEWMA) control chart for monitoring mean output using a more structured sampling technique, i.e. ranked set sampling (RSS) instead of the traditional simple random sampling. We evaluated the performance of the proposed charts in terms of different run length (RL) properties including average RL, standard deviation of the RL, and percentile of the RL. Comparisons of these charts with some existing control charts designed for monitoring small, large, or both shifts revealed that the RSS‐based CSEWMA charts are more sensitive and offer better protection against all types of shifts than other schemes considered in this study. Copyright © 2012 John Wiley & Sons, Ltd.     

Shewhart and EWMA t control charts for short production runs

Short‐run productions are common in manufacturing environments like job shops, which are characterized by a high degree of flexibility and production variety. Owing to the limited number of possible inspections during a short run, often the Phase I control chart cannot be performed and correct estimates for the population mean and standard deviation are not available. Thus, the hypothesis of known in‐control population parameters cannot be assumed and the usual control chart statistics to monitor the sample mean are not applicable. t‐charts have been recently proposed in the literature to protect against errors in population standard deviation estimation due to the limitation of available sampling measures. In this paper the t‐charts are tested for implementation in short production runs to monitor the process mean and their statistical properties are evaluated. Statistical performance measures properly designed to test the chart sensitivity during short runs have been considered to compare the performance of Shewhart and EWMA t‐charts. Two initial setup conditions for the short run fixing the population mean exactly equal to the process target or, alternatively, introducing an initial setup error influencing the statistic distribution have been modelled. The numerical study considers several out‐of‐control process operating conditions including one‐step shifts for the population mean and/or standard deviation. The obtained results show that the t‐charts can be successfully implemented to monitor a short run. Finally, an illustrative example is presented to show the use of the investigated t charts. Copyright © 2010 John Wiley & Sons, Ltd.     

Poisson progressive mean control chart

In real life applications, many process‐monitoring problems in statistical process control are based on attribute data resulting from quality characteristics that cannot be measured on numerical or quantitative scales. For the monitoring of such data, a new attribute control chart has been proposed in this study, namely, the Poisson progressive Mean (PPM) control chart. The performance of the PPM chart is compared with the existing charts used for the monitoring of Poisson processes such as the Shewhart c‐chart, Poisson Exponentially Weighted Moving Average chart, Poisson double Exponentially Weighted Moving Average chart and the Poisson Cumulative Sum charts. The average run length comparison indicated the superior performance of the PPM chart in terms of shift detection ability. This study will help quality practitioners to choose an efficient attribute control chart.     

Using FIR to Improve CUSUM Charts for Monitoring Process Dispersion

Statistical process control deals with monitoring process to detect disturbances in the process. These disturbances may be from the process mean or variance. In this study, we propose some charts that are efficient for detecting early shifts in dispersion parameter, by applying the Fast Initial Response feature. Performance measures such as average run length, standard deviation of the run length, extra quadratic loss, relative average run length, and performance comparison index are used to compare the proposed charts with their existing counterparts, including the Shewhart R chart and the Shewhart S chart with and without warning lines. Others include the CUSUM R chart, the CUSUM S chart, the EWMA of ln S², the CUSUM of ln S², the P_σ CUSUM, the χ CUSUM, and the Change Point (CP) CUSUM charts. The proposed charts do not only detect early shifts in the process dispersion faster, but also have better overall performance than their existing counterparts. Copyright © 2016 John Wiley & Sons, Ltd.