EWMA Control Chart Performance with Estimated Parameters under Non‐normality

Exponentially weighted moving average (EWMA) control charts can be designed to detect shifts in the underlying process parameters quickly while enjoying robustness to non‐normality. Past studies have shown that performance of various EWMA control charts can be adversely affected when parameters are estimated or observations do not follow a normal distribution. To the best of our knowledge, simultaneous effect of parameter estimation and non‐normality has not been studied so far. In this paper, a Markov chain approach is used to model and evaluate performance of EWMA control charts when parameter estimation is subject to non‐normality using skewed and heavy‐tailed symmetric distributions. Using standard deviation of the run length (SDRL), average run length (ARL), and percentiles of run lengths for various phase I sample sizes, we show that larger phase I sample sizes do not necessarily lead to a better performance for non‐normal observations. Copyright © 2015 John Wiley & Sons, Ltd.     

A nonparametric CUSUM chart for process dispersion

In the present article, we propose a nonparametric cumulative sum control chart for process dispersion based on the sign statistic using in‐control deciles. The chart can be viewed as modified control chart due to Amin et al, 6 which is based on in‐control quartiles. An average run length performance of the proposed chart is studied using Markov chain approach. An effect of non‐normality on cumulative sum S² chart is studied. The study reveals that the proposed cumulative sum control chart is a better alternative to parametric cumulative sum S² chart, when the process distribution is non‐normal. We provide an illustration of the proposed cumulative sum control chart.     

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.     

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.     

Linear profiles monitoring in the presence of nonnormal random errors

Profile monitoring is a technique to test the stability of the relationship between a response variable and explanatory variables over time. The most relevant linear profile monitoring methods have been constructed using the normality assumption. However, the normality assumption could be violated in many quality control applications. In this study, we consider a situation in which the random errors in a linear profile model follow a skew‐normal distribution. The skew‐normal distribution is a generalized version of the normal distribution. A new Shewhart‐type chart and exponentially weighted moving average (EWMA) chart, named the Shewhart^R and EWMA^R charts, respectively, are constructed based on residuals to monitor the parameters of linear profile model. The simulation results show that the multivariate EWMA chart is sensitive to the normality assumption and that the proposed Shewhart^R and EWMA^R charts have good ability to detect big and small‐to‐moderate process shifts, respectively. An example using photo mask techniques in semiconductor manufacturing is provided to illustrate the applications of the Shewhart^R and EWMA^R charts.     

The Design of the S2 Control Charts Based on Conditional Performance via Exact Methods

In this paper, we consider the conditional performance of the equal‐tailed and average run lengths (ARL)‐unbiased two‐sided S² charts when the in‐control variance of a normal process is estimated. We derive the exact probability distributions of the conditional ARL for the two S² charts. Then we evaluate the performance of each S² chart in terms of the percentiles, mean and standard deviation of the conditional in‐control ARL distribution. Because the parameter estimation seriously affects the conditional performance of these S² charts, we propose an exact method to design the equal‐tailed and ARL‐unbiased S² charts with desired conditional in‐control performance. The results indicate that the new ARL‐unbiased S² chart has far smaller standard deviation ARL values and the unconditional ARL values are more close to the desired value than the corresponding new equal‐tailed S² chart. Copyright © 2017 John Wiley & Sons, Ltd.     

Monitoring Process Variance Using an ARL‐unbiased EWMA‐p Control Chart

Control charts are effective tools for signal detection in manufacturing processes. As much of the data in industries come from processes having non‐normal or unknown distributions, the commonly used Shewhart variable control charts cannot be appropriately used, because they depend heavily on the normality assumption. The average run length (ARL) is generally used to measure the detection performance of a process when using a control chart, but it is biased for the monitoring statistic with an asymmetric distribution. That is, the ARL‐biased control chart leads to take longer to detect the shifts in parameter than to trigger a false alarm. To overcome this problem, we herein propose an ARL‐unbiased exponentially weighted moving average proportion (EWMA‐p) chart to monitor the process variance for process data with non‐normal or unknown distributions. We further explore the procedure to determine the control limits and to investigate the out‐of‐control variance detection performance of the ARL‐unbiased EWMA‐p chart. With a numerical example involving non‐normal service times from a bank branch in Taiwan, we illustrate the applications of the proposed ARL‐unbiased EWMA‐p chart and also compare the out‐of‐control detection performance of the ARL‐unbiased EWMA‐p chart, the arcsin transformed symmetric EWMA variance, and other existing variance charts. The proposed ARL‐unbiased EWMA‐p chart shows superior detection performance. Thus, we recommend the ARL‐unbiased EWMA‐p chart for process data with non‐normal or unknown distributions. Copyright © 2015 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.     

Phase I control chart based on a kernel estimator of the quantile function

To measure the statistical performance of a control chart in Phase I applications, the in‐control average run length (ARL) is the most frequently used parameter. In typical start up situations, control limits must be computed without knowledge of the underlying distribution of the quality characteristic. Assumptions of an underlying normal distribution can increase the probability of false alarms when the underlying distribution is non‐normal, which can lead to unnecessary process adjustments. In this paper, a control chart based on a kernel estimator of the quantile function is proposed. Monte Carlo simulation was used to evaluate the in‐control ARL performance of this chart relative to that of the Shewhart individuals control chart. The results indicate that the proposed chart is more robust to deviations in the assumed underlying distribution (with respect to the in‐control ARL) and results in an alternative method of designing control charts for individual units. Copyright © 2011 John Wiley & Sons, Ltd.     

Statistical Performance of a Control Chart for Individual Observations Monitoring the Ratio of Two Normal Variables

Statistical Process Control monitoring of the ratio Z of two normal variables X and Y has received too little attention in quality control literature. Several applications dealing with monitoring the ratio Z can be found in the industrial sector, when quality control of products consisting of several raw materials calls for monitoring their proportions (ratios) within a product. Tables about the statistical performance of these charts are still not available. This paper investigates the statistical performance of a Phase II Shewhart control chart monitoring the ratio of two normal variables in the case of individual observations. The obtained results show that the performance of the proposed chart is a function of the distribution parameters of the two normal variables. In particular, the Shewhart chart monitoring the ratio Z outperforms the (p = 2) multivariate T² control chart when a process shift affects the in‐control mean of X or, alternatively, of Y and the correlation among X and Y is high and when the in‐control means of X and Y shift contemporarily to opposite directions. The sensitivity of the proposed chart to a shift of the in‐control dispersion has been investigated, too. We also show that the standardization of the two variables before computing their ratio is not a good practice due to a significant loss in the chart's statistical performance. An illustrative example from the food industry details the implementation of the ratio control chart. Copyright © 2013 John Wiley & Sons, Ltd.     

Robustness properties of multivariate EWMA control charts

In this paper, the robustness of the multivariate exponentially weighted moving average (MEWMA) control chart to non‐normal data is examined. Two non‐normal distributions of interest are the multivariate distribution and the multivariate gamma distribution. Recommendations for constructing MEWMA control charts when the normality assumption may be violated are provided. Copyright © 2002 John Wiley & Sons, Ltd.     

Design of a Phase II Control Chart for Monitoring the Ratio of two Normal Variables

In many industrial scenarios, on‐line monitoring of quality characteristics computed as the ratio of two normal random variables can be required. Potential industrial applications can include monitoring of processes where the correct proportion of a property between two ingredients or elements within a product should be maintained under statistical control; implementation of quality control procedures where the performance of a product is measured as a ratio before and after some specific operation, for example a chemical reaction following the introduction of an additive in a product and monitoring of a chemical or physical property of a product, which is itself defined and computed as a ratio. This paper considers Phase II Shewhart control charts with each subgroup consisting of n > 1 sample units. From one subgroup to another, the size of each sample unit, upon which a single measurement is made, can be changed. An approximation based on the normal distribution is used to efficiently handle the ratio distribution. Several tables are generated and commented to show the statistical performance of the investigated chart for known and random shift sizes affecting the in‐control ratio. An illustrative example from the food industry is given for illustration. Copyright © 2014 John Wiley & Sons, Ltd.     

A Comparison of Normal Approximation Rules for Attribute Control Charts

Control charts, known for more than 80 years, have been important tools for business and industrial manufactures. Among many different types of control charts, the attribute control chart (np‐chart or p‐chart) is one of the most popular methods to monitor the number of observed defects in products, such as semiconductor chips, automobile engines, and loan applications. The attribute control chart requires that the sample size n is sufficiently large and the defect rate p is not too small so that the normal approximation to the binomial works well. Some rules for the required values for n and p are available in the textbooks of quality control and mathematical statistics. However, these rules are considerably different, and hence, it is less clear which rule is most appropriate in practical applications. In this paper, we perform a comparison of five frequently used rules for n and p required for the normal approximation to the binomial. With this result, we also refine the existing rules to develop a new rule that has a reliable performance. Datasets are analyzed for illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

CUSUM charts for monitoring a zero‐inflated poisson process

A zero‐inflated Poisson (ZIP) process is different from a standard Poisson process in that it results in a greater number of zeros. It can be used to model defect counts in manufacturing processes with occasional occurrences of non‐conforming products. ZIP models have been developed assuming that random shocks occur independently with probability p, and the number of non‐conformities in a product subject to a random shock follows a Poisson distribution with parameter λ. In our paper, a control charting procedure using a combination of two cumulative sum (CUSUM) charts is proposed for monitoring increases in the two parameters of the ZIP process. Furthermore, we consider a single CUSUM chart for detecting simultaneous increases in the two parameters. Simulation results show that a ZIP‐Shewhart chart is insensitive to shifts in p and smaller shifts in λ in terms of the average number of observations to signal. Comparisons between the combined CUSUM method and the single CUSUM chart show that the latter's performance is worse when there are only increases in p, but better when there are only increases in λ or when both parameters increase. The combined CUSUM method, however, is much better than the single CUSUM chart when one parameter increases while the other decreases. Finally, we present a case study from the light‐emitting diode packaging industry. Copyright © 2011 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.     

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.     

Computing process capability indices for non‐normal data: a review and comparative study

When the distribution of a process characteristic is non‐normal, C_p and C_pk calculated using conventional methods often lead to erroneous interpretation of the process's capability. Though various methods have been proposed for computing surrogate process capability indices (PCIs) under non‐normality, there is a lack of literature that covers a comprehensive evaluation and comparison of these methods. In particular, under mild and severe departures from normality, do these surrogate PCIs adequately capture process capability, and which is the best method(s) in reflecting the true capability under each of these circumstances? In this paper we review seven methods that are chosen for performance comparison in their ability to handle non‐normality in PCIs. For illustration purposes the comparison is done through simulating Weibull and lognormal data, and the results are presented using box plots. Simulation results show that the performance of a method is dependent on its capability to capture the tail behaviour of the underlying distributions. Finally we give a practitioner's guide that suggests applicable methods for each defined range of skewness and kurtosis under mild and severe departures from normality. Copyright © 1999 John Wiley & Sons, Ltd.     

A double sampling scheme for process variability monitoring

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much of the data in service industries come from processes exhibiting nonnormal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, are not appropriately used here. This paper thus proposes a standardized asymmetric exponentially weighted moving average (EWMA) variance chart with a double sampling scheme (SDS EWMA‐AV chart) for monitoring process variability. We further explore the sampling properties of the new monitoring statistics and calculate the average run lengths when using the proposed SDS EWMA‐AV chart. The performance of the SDS EWMA‐AV chart and that of the single sampling EWMA variance (SS EWMA‐V) chart are then compared, with the former showing superior out‐of‐control detection performance versus the latter. We also compare the out‐of‐control variance detection performance of the proposed chart with those of nonparametric variance charts, the nonparametric Mood variance chart (NP‐M chart) with runs rules, and the nonparametric likelihood ratio‐based distribution‐free EWMA (NLE) chart and the combination of traditional EWMA (CEW) and the SS EWMA‐V control charts by considering cases in which the critical quality characteristic presents normal, double exponential, uniform, chi‐square, and exponential distributions. Comparison results show that the proposed chart always outperforms the NP‐M with runs rules, the NLE, CEW, and the SS EWMA‐V control charts. We hence recommend employing the SDS EWMA‐AV chart. Finally, a numerical example of a service system for a bank branch in Taiwan is used to illustrate the application of the proposed variability control chart.     

Location charts based on ranked set sampling for normal and non‐normal processes

Control charts, based on ranked set sampling schemes, had been proposed recently for efficient monitoring of process location. All the proposals in the literature are based on the ideal assumption of normally distributed quality characteristics. No study as of yet investigated the performance of location charts based on ranked set sampling for non‐normal processes. In this study, we investigated the location chart based on simple random sampling (SRS) and three well‐known rank‐based schemes ie, ranked set sampling (RSS), median ranked set sampling (MRSS), and extreme ranked set sampling (ERSS), considering normal and a variety of non‐normal parent distributions. Both heavy‐tailed symmetric and skewed cases have been considered in this study. The performance of the charts is evaluated using average run length (ARL), extra quadratic loss (EQL), and relative ARL (RARL) measures. A real life example is also presented that details the monitoring of pH levels in water for an experiment conducted to study the reproduction of Mysids. The study will help quality practitioners to choose the chart based on an efficient sampling scheme for normal and non‐normal processes.     

A new non‐parametric CUSUM mean chart

Not all data in practice came from a process with normal distribution. When the process distribution is non‐normal or unknown, the commonly used Shewhart control charts are not suitable. In this paper, a new non‐parametric CUSUM Mean Chart is proposed to monitor the possible small mean shifts in the process. The sampling properties of the new monitoring statistics are examined and the average run lengths of the proposed chart are examined. Two numerical examples are used to illustrate the proposed chart and compare with the two existing charts, assuming normality and Beta distribution, respectively. The CUSUM Mean Chart showed better detection ability than those two charts in monitoring and detecting small process mean shifts. Copyright © 2010 John Wiley & Sons, Ltd.