The Phase I Dispersion Charts for Bivariate Process Monitoring

Multivariate control charts are usually implemented in statistical process control to monitor several correlated quality characteristics. Process dispersion charts are used to determine the stability of process variation (which is typically done before monitoring the process location/mean). A Phase‐I study is generally used when population parameters are unknown. This article develops Phase‐I |S| and |G| control charts, to monitor the dispersion of a bivariate normal process. The charting constants are determined to achieve the required nominal false alarm probability (FAP₀). The performance of the proposed charts is evaluated in terms of (i) the attained false rate and (ii) the probability of signaling for out‐of‐control situations. The analysis shows that the proposed Phase‐I bivariate charts correctly control the FAP (the false alarm probability) and detect a shift occurring in the bivariate dispersion matrix with adequate probability. An example is given to explain the practical implementation of these charts. Copyright © 2015 John Wiley & Sons, Ltd.     

On the Performance of Phase I Dispersion Control Charts for Process Monitoring

Control charts are usually implemented in two phases: the retrospective phase (phase I) and the monitoring phase (phase II). The performance of any phase II control chart structure depends on the preciseness of the control limits obtained from the phase I analysis. In statistical process control, the performance of phase I dispersion charts has mainly been investigated for normal or contaminated normal distributions of the quality characteristic of interest. Little work has been carried out to investigate the performance of a wide range of possible phase I dispersion charts for processes following non‐normal distributions. The current study deals with the proper choice of a control chart for the evaluation of process dispersion in phase I. We have analyzed the performance of a wide range of dispersion control charts, including two distribution‐free structures. The performance of the control charts is evaluated in terms of probability to signal, under normal and non‐normal process setups. These results will be useful for quality control practitioners in their selection of a phase I control chart. Copyright © 2014 John Wiley & Sons, Ltd.     

EWMA Dispersion Control Charts for Normal and Non‐normal Processes

Process monitoring through control charts is a quite popular practice in statistical process control. This study is planned for monitoring the process dispersion parameter using exponentially weighted moving average (EWMA) control chart scheme. Most of the EWMA dispersion charts that have been proposed are based on the assumption that the parent distribution of the quality characteristic is normal, which is not always the case. In this study, we develop new EWMA charts based on a wide range of dispersion estimates for processes following normal and non‐normal parent distributions. The performance of all the charts is evaluated and compared using run length characteristics (such as the average run length). Extra quadratic loss, relative average run length, and performance comparison index measures are also used to examine the overall effectiveness of the EWMA dispersion charts.Copyright © 2014 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.     

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.     

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.     

Distribution‐free mixed cumulative sum‐exponentially weighted moving average control charts for detecting mean shifts

In this paper, we propose distribution‐free mixed cumulative sum‐exponentially weighted moving average (CUSUM‐EWMA) and exponentially weighted moving average‐cumulative sum (EWMA‐CUSUM) control charts based on the Wilcoxon rank‐sum test for detecting process mean shifts without any distributional assumption of the underlying quality process. The performances of the proposed charts are measured through the average run‐length, relative mean index, average extra quadratic loss, and average ratio of the average run‐length and performance comparison index. It is found that the proposed charts perform better than its counterparts considered in this paper under non‐normal distributions and outperform the classical mixed CUSUM‐EWMA and EWMA‐CUSUM charts in many cases under the normal distribution. The effect of the phase I sample size is also investigated on the phase II performance of the proposed charts. A numerical illustration is given to demonstrate the implementation and simplicity of the proposed charts.     

On Proper Choice of Variability Control Chart for Normal and Non‐normal Processes

Control charts are an important statistical process control tool used to monitor changes in process location and variability. This study addresses issues regarding the proper choice of control chart for efficient monitoring of process variability. The choice of the best estimator to be used for variability charts has not been made clear in literature. We have analyzed the performance of eight control chart structures, based on different estimates of process standard deviation. The performance of control charts is investigated under the existence and violation of ideal assumptions of normality. Control chart constants and factors required for computing probability limits, considering normal and different non‐normal parent distributions, are provided for all variability charts. This study aims at providing guidance to quality practitioners in choosing the appropriate variability control chart for normal and non‐normal processes. Copyright © 2011 John Wiley & Sons, Ltd.     

Approximate design and performance of the robust control charts for monitoring dispersion in Phase I

This article designs and studies the approximate performance of robust dispersion charts, namely, MAD chart, S_n chart, and Q_n chart, in Phase I analysis (recently developed in the literature). The proposed limits are based on false alarm probability for monitoring the dispersion of a process in Phase I analysis. The charting constants are determined to achieve the required nominal FAP (FAP₀). The performance of these structures is evaluated in (i) the attained false alarm rate and (ii) the probability of signals for out‐of‐control situations. The analysis shows that the proposed design of Phase I robust dispersion charts correctly controls the FAP and shows a good performance in detecting the shifts in the process variation. An illustrative example is used to explain the practical implementation of these limits.     

Using independent component analysis to monitor 2‐D geometric specifications

Functional data and profiles are characterized by complex relationships between a response and several predictor variables. Fortunately, statistical process control methods provide a solid ground for monitoring the stability of these relationships over time. This study focuses on the monitoring of 2‐dimensional geometric specifications. Although the existing approaches deploy regression models with spatial autoregressive error terms combined with control charts to monitor the parameters, they are designed based on some idealistic assumptions that can be easily violated in practice. In this paper, the independent component analysis (ICA) is used in combination with a statistical process control method as an alternative scheme for phase II monitoring of geometric profiles when non‐normality of the error term is present. The performance of this method is evaluated and compared with a regression‐ and PCA‐based approach through simulation of the average run length criterion. The results reveal that the proposed ICA‐based approach is robust against non‐normality in the in‐control analysis, and its out‐of‐control performance is on par with that of the PCA‐based method in case of normal and near‐normal error terms.     

Memory‐Type Control Charts for Monitoring the Process Dispersion

Control charts have been broadly used for monitoring the process mean and dispersion. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are memory control charts as they utilize the past information in setting up the control structure. This makes CUSUM and EWMA‐type charts good at detecting small disturbances in the process. This article proposes two new memory control charts for monitoring process dispersion, named as floating T ? S² and floating U ? S² control charts, respectively. The average run length (ARL) performance of the proposed charts is evaluated through a simulation study and is also compared with the CUSUM and EWMA charts for process dispersion. It is found that the proposed charts are better in detecting both positive as well as negative shifts. An additional comparison shows that the floating U ? S² chart has slightly smaller ARLs for larger shifts, while for smaller shifts, the floating T ? S² chart has better performance. An example is also provided which shows the application of the proposed charts on simulated datasets. Copyright © 2013 John Wiley & Sons, Ltd.     

Effect of non‐normality on the monitoring of simple linear profiles

In some statistical process control (SPC) applications, it is assumed that a quality characteristic or a vector of quality characteristics of interest follows a univariate or multivariate normal distribution, respectively. However, in certain applications this assumption may fail to hold and could lead to misleading results. In this paper, we study the effect of non‐normality when the quality of a process or product is characterized by a linear profile. Skewed and heavy‐tailed symmetric non‐normal distributions are used to evaluate the non‐normality effect numerically. The results reveal that the method proposed by Kimtextitet al. (J. Qual. Technol. 2003; 35 :317–328) can be designed to be robust to non‐normality for both highly skewed and heavy‐tailed distributions. Copyright © 2010 John Wiley & Sons, Ltd.     

A t‐chart for Monitoring Multi‐variety and Small Batch Production Run

Statistical process control is an important tool to monitor and control a process. It is used to ensure that the manufacturing process operates in the in‐control state. Multi‐variety and small batch production runs are common in manufacturing environments like flexible manufacturing systems and Just‐in‐Time systems, which are characterized by a wide variety of mixed products with small volume for each kind of production. It is difficult to apply traditional control charts efficiently and effectively in such environments. The method that control charts are plotted for each individual part is not proper, since the successive state of the manufacturing process cannot be reflected. In this paper, a proper t‐chart is proposed for implementation in multi‐variety and small batch production runs to monitor the process mean, and its statistical properties are evaluated. The run length distribution of the proposed t‐chart has been obtained by modelling the multi‐variety process. The ARL performance for various shifts, number of product types, and subgroup sizes has also been obtained. The results show that the t‐chart can be successfully implemented to monitor a multi‐variety production run. Finally, illustrative examples show that the proposed t‐chart is effective in multi‐variety and small batch manufacturing environment. Copyright © 2013 John Wiley & Sons, Ltd.     

A heuristic method for obtaining quasi ARL‐unbiased p‐Charts

It is known that control charts based on equal tail probability limits are ARL biased when the distribution of the plotted statistic is skewed. This is the case for p‐Charts that serve to monitor processes on the basis of the binomial distribution. For the particular case of the standard three‐sigma Shewhart p‐Chart, which is built on the basis of the binomial to normal distribution approximation, this ARL‐biased condition is particularly severe, and it greatly affects its monitoring capability. Surprisingly, in spite of this, the standard p‐Chart is still widely used and taught. Through a literature search, it was identified that several, simple to use, improved alternative p‐Charts had been proposed over the years; however, at first instance, it was not possible to determine which of them was the best. In order to identify the alternative that excelled, an ARL performance comparison was carried out in terms of their ARL bias severity level (ARL_BSL) and their In‐Control ARL (ARL₀). The results showed that even the best performing alternative charts would often be ARL‐biased or have nonoptimal ARL₀. To improve on the existing alternatives, the “Kmod p‐Chart” was developed; it offers easiness of use, superior ARL performance, and a simple and effective method for verifying its ARL‐bias condition.     

On improved dispersion control charts under ranked set schemes for normal and non‐normal processes

In manufacturing industries, control charts are the promising statistical tools used for an efficient monitoring of processes. These charts enhance the product quality by timely signaling for special variations at any stage of the process. There are two common concerns in statistical process monitoring, location and variability of the quality characteristic of interest. Besides location parameter, the monitoring of process dispersion remained a matter of concern for researchers. The conventional simple random sampling (SRS) is a usual practice; however, ranked set sampling (RSS) schemes are very effective methods of choosing sample values. This study intends to design and investigate dispersion control charts under different RSS strategies for normal and non‐normal processes. We have considered RSS, median ranked set sampling (MRSS), and extreme ranked set sampling (ERSS) schemes to design dispersion control charts. The performance of the existing and the proposed control charts is evaluated in terms of relative efficiency and power for normal and a variety of non‐normal distributions. The comparative analysis revealed that the proposed structures outperform the existing charts. The application of the proposed procedures is also shown for a bottles filling process for an efficient and timely signaling of any special causes in the process.     

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.     

On Enhanced Interquartile Range Charting for Process Dispersion

A careful monitoring of process dispersion is necessary to get optimum output from any process. Control charts are very important process monitoring tools of Statistical Process Control toolkit. Interquartile range (IQR) is a famous dispersion measure that is used to monitor process dispersion in quality control literature. In this study, a set of auxiliary information‐based Shewhart‐type IQR charts are proposed for efficient monitoring of process dispersion under bivariate normal, t and gamma distributed processes. The control structures of the proposed charts are developed, and their performances are evaluated in terms of Average Time to Signal (ATS), Average Extra Quadratic Loss and Average Ratio to ATS. Comparisons are made among different charts to establish their superiorities for a quicker detection of process changes. An illustrative example is also provided to elaborate the procedural details of the proposed IQR charts using a real data set. Copyright © 2013 John Wiley & Sons, Ltd.     

On Efficient Skewness Correction Charts Under Contamination and Non‐normality

Shewhart control charts are very popular in a variety of disciplines such as industry, agriculture and medical science. The design structure of the usual Shewhart charts depends on normality and one point decision rule. This makes the scope of these charts quite limited and not very efficient for small shifts. This study comes up with an intermediate solution by implementing runs rules schemes and adjusting the limits' coefficients for non‐normality using the idea of skewness correction. We have covered some commonly used location and dispersion charts, namely , R and S charts. We have investigated the performance of the proposals in terms of false alarm rate and signalling probability. We have observed that the proposals serve the dual purpose in terms of robustness and efficiency. The study also provides an application example using numerical dataset. Copyright © 2015 John Wiley & Sons, Ltd.     

A weighted variance capability index for general non‐normal processes

Process capability indices are considered to be one of the important quality measurement tools for the continuous improvement of quality and productivity. The most commonly used indices assume that process data are normally distributed. However, many studies have pointed out that the normally‐based indices are very sensitive to non‐normal processes. Therefore we propose a new process capability index applying the weighted variance control charting method for non‐normal processes to improve the measurement of process performance when the process data are non‐normally distributed. The main idea of the weighted variance method is to divide a skewed or asymmetric distribution into two normal distributions from its mean to create two new distributions which have the same mean but different standard deviations. In this paper we provide an example, a distribution generated from the Johnson family of distributions, to demonstrate how the weighted variance‐based process capability indices perform in comparison with another two non‐normal methods, namely the Clements and Johnson–Kotz–Pearn methods. This example shows that the weighted variance‐based indices are more consistent than the other two methods in estimating process fallout for non‐normal processes. Copyright © 1999 John Wiley & Sons, Ltd.     

An improved control chart structure for process location parameter

The article proposes a Shewhart‐type control chart, namely M_t chart, for an improved monitoring of the mean level of a quality characteristic of interest, say Y, in a process. The proposed control chart uses the information on a single auxiliary characteristic, say X, of the process and is based on product–difference‐type estimator, say M_t (in fact the spirit is to pool different styles of using information on auxiliary variable(s)). Assuming bivariate normality of the characteristic of interest Y and the auxiliary characteristic X, design structure of the proposed M_t chart is developed and its comparisons are made with those of some existing control charts used for the same purposes. The comparisons reveal an improved performance of the proposed M_t chart relative to its existing counterparts due to suggested merger of different approaches. Copyright © 2011 John Wiley & Sons, Ltd.