A Self‐Starting CUSUM Chart Combined with a Maximum Likelihood Estimator for the Time of a Detected Shift in the Process Mean

Detection of a special cause of variation and the identification of the time it occurs are two important activities in any quality improvement strategy. Detection of changes in a process can be done using control charts. One of these charts, the self‐starting CUSUM chart, was created to detect small sustained changes and be implemented without a Phase I or a priori knowledge of the parameters of the process. To estimate the time of a detected change, a CUSUM‐based change‐point estimator can be used, but experiments show that the corresponding MLE has smaller bias and standard error. This paper proposes the sequential use of the self‐starting CUSUM chart and the MLE of a change point in series of independent normal observations. Performance is studied with Monte Carlo simulations showing that the use of the MLE reduces the bias of the change‐point estimation. It is also shown how extra observations after a change is detected can be used to improve estimation of the change‐point time. Copyright © 2013 John Wiley & Sons, Ltd.     

Phase II monitoring of covariance stationary autocorrelated processes

Statistical process control charts are intended to assist operators in detecting process changes. If a process change does occur, the control chart should detect the change quickly. Owing to the recent advancements in data retrieval and storage technologies, today's industrial processes are becoming increasingly autocorrelated. As a result, in this paper we investigate a process‐monitoring tool for autocorrelated processes that quickly responds to process mean shifts regardless of the magnitude of the change, while supplying useful diagnostic information upon signaling. A likelihood ratio approach was used to develop a phase II control chart for a permanent step change in the mean of an ARMA (p, q) (autoregressive‐moving average) process. Monte Carlo simulation was used to evaluate the average run length (ARL) performance of this chart relative to that of the more recently proposed ARMA chart. Results indicate that the proposed chart responds more quickly to process mean shifts, relative to the ARMA chart, while supplying useful diagnostic information, including the maximum likelihood estimates of the time and the magnitude of the process shift. These crucial change point diagnostics can greatly enhance the special cause investigation. Copyright © 2010 John Wiley & Sons, Ltd.     

A New Cumulative Sum Quality Control Scheme for Monitoring the Process Mean

A control chart is a powerful statistical process monitoring tool that is frequently used in many industrial and service organizations to monitor in‐control and out‐of‐control performances of the manufacturing processes. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have been recognized as potentially powerful tool in quality and management control. These control charts are sensitive to both small and moderate changes in the process. In this paper, we propose a new CUSUM (NCUSUM) quality control scheme for efficiently monitoring the process mean. It is shown that the classical CUSUM control chart is a special case of the proposed controlling scheme. The NCUSUM control chart is compared with some of the recently proposed control charts by using characteristics of the distribution of run length, i.e. average run length, median run length and standard deviation of run length. It is worth mentioning that the NCUSUM control chart detects the random shifts in the process mean substantially quicker than the classical CUSUM, fast initial response‐based CUSUM, adaptive CUSUM with EWMA‐based shift, adaptive EWMA and Shewhart–CUSUM control charts. An illustrative example is given to exemplify the implementation of the proposed quality control scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

An efficient adaptive EWMA control chart for monitoring the process mean

In recent years, the memory‐type control charts—exponentially weighted moving average (EWMA) and cumulative sum (CUSUM)—along with the adaptive and dual control‐charting structures have received considerable attention because of their excellent ability in providing an overall good detection over a range of mean‐shift sizes. These adaptive memory‐type control charts include the adaptive exponentially weighted moving average (AEWMA), dual CUSUM, and adaptive CUSUM charts. In this paper, we propose a new AEWMA chart for efficiently monitoring the process mean. The idea is to first design an unbiased estimator of the mean shift using the EWMA statistic and then adaptively update the smoothing constant of the EWMA chart. The run length profiles of the proposed AEWMA chart are computed using extensive Monte Carlo simulations. Based on a comprehensive comparative study, it turns out that the proposed AEWMA chart performs better than the existing AEWMA, adaptive CUSUM, dual CUSUM, and Shewhart‐CUSUM charts, in terms of offering more balanced protection against mean shifts of different sizes. An example is also used to explain the working of the existing and proposed control charts.     

A note on GLR charts for monitoring count processes

Various generalized likelihood ratio (GLR) charts have been proposed to monitor count processes such as binomial, Bernoulli, Poisson, and multinomial processes. The advantages of GLR charts are that designing the chart is relatively easy, estimates of the process change‐point and shift size are available for post‐signal diagnosis, and they are effective in detecting a wide range of shifts in the process parameter. However, for some special cases of the observations, such as observing all defective items or all non‐defective items, the GLR chart statistic for monitoring a count process has been said to be undefined. We show that the GLR chart statistic is always well defined.     

A Generalized Likelihood Ratio Control Chart for Monitoring the Process Mean Subject to Linear Drifts

This article considers the problem of monitoring a normally distributed process variable when a special cause may produce a time‐varying linear drift in the mean. The design and application of a generalized likelihood ratio (GLR) control chart for drift detection are evaluated. The GLR drift chart does not require specification of any tuning parameters by the practitioner and has the advantage that, at the time of the signal, estimates of both the change point and the drift size are immediately available. An equation to accurately approximate the control limit is provided. The performance of the GLR drift chart is compared with that of other control charts such as a standard cumulative sum chart and a cumulative score chart designed for drift detection. We also compare the GLR chart designed for drift detection with the GLR chart designed for sustained shift detection because both of them require only a control limit to be specified. In terms of the expected time for detection and in terms of the bias and mean squared error of the change‐point estimators, the GLR drift chart has better performance for a wide range of drift rates relative to the GLR shift chart when the out‐of‐control process is truly a linear drift. Copyright © 2012 John Wiley & Sons, Ltd.     

Change Point Estimation Methods for Control Chart Postsignal Diagnostics: A Literature Review

Control charts are the most popular monitoring tools used to distinguish between special (assignable) and common causes of variation and to detect any changes in processes. The time that a control chart gives an out‐of‐control signal is not the real time of change. The actual time of the change is called the change point. Knowing the real time of the change will help and simplify finding the assignable causes of the signal, which may be the result of a shift in the process mean or change in process variability. This article gives an overview of change point estimation in control charts, provides a classification scheme, and describes the research that has previously appeared in the literature. In addition, a gap analysis in this area provides direction for future research. Copyright © 2011 John Wiley & Sons, Ltd.     

New GWMA‐CUSUM control chart for monitoring the process dispersion

The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have been widely accepted because of their fantastic speed in identifying small‐to‐moderate unusual variations in the process parameter(s). Recently, a new CUSUM chart has been proposed that uses the EWMA statistic, called the CS‐EWMA chart, for monitoring the process variability. On similar lines, in order to further improve the detection ability of the CS‐EWMA chart, we propose a CUSUM chart using the generally weighted moving average (GWMA) statistic, named the GWMA‐CUSUM chart, for monitoring the process dispersion. Monte Carlo simulations are used to compute the run length profiles of the GWMA‐CUSUM chart. On the basis of the run length comparisons, it turns out that the GWMA‐CUSUM chart outperforms the CUSUM and CS‐EWMA charts when identifying small variations in the process variability. A simulated dataset is also used to explain the working and implementation of the CS‐EWMA and GWMA‐CUSUM charts.     

Estimating the Change Point of a Poisson Rate Parameter with a Linear Trend Disturbance

Knowing when a process changed would simplify the search and identification of the special cause. In this paper, we compare the maximum likelihood estimator (MLE) of the process change point designed for linear trends to the MLE of the process change point designed for step changes when a linear trend disturbance is present. We conclude that the MLE of the process change point designed for linear trends outperforms the MLE designed for step changes when a linear trend disturbance is present. We also present an approach based on the likelihood function for estimating a confidence set for the process change point. We study the performance of this estimator when it is used with a cumulative sum (CUSUM) control chart and make direct performance comparisons with the estimated confidence sets obtained from the MLE for step changes. The results show that better confidence can be obtained using the MLE for linear trends when a linear trend disturbance is present. Copyright © 2005 John Wiley & Sons, Ltd.     

Estimating the time of step change with Poisson CUSUM and EWMA control charts

Knowing when a process has changed would simplify the search for and identification of the special cause. Consequently, having an estimate of the process change point following a control chart signal would be useful to process engineers. Much of the literature on change point models and techniques for statistical process control applications consider processes well modelled by the normal distribution. However, the Poisson distribution is commonly used in industrial quality control applications for modelling attribute-based process quality characteristics (e.g., counts of non-conformities). Some commonly used control charts for monitoring Poisson distributed data are the Poisson cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts. In this paper, we study the effect of changes in the design of the control chart on the performances of the change point estimators offered by these procedures. In particular, we compare root mean square error performances of the change point estimators offered by the Poisson CUSUM and EWMA control charts relative to that achieved by a maximum likelihood estimator for the process change point. Results indicate that the relative performance achieved by each change point estimator is a function of the corresponding control chart design. Relative mean index plots are provided to enable users of these control charts to choose a control chart design and change point estimator combination that will yield robust change point estimation performance across a range of potential change magnitudes.     

A clustering approach to identify the time of a step change in Shewhart control charts

Control charts are the most popular statistical process control tools used to monitor process changes. When a control chart indicates an out‐of‐control signal it means that the process has changed. However, control chart signals do not indicate the real time of process changes, which is essential for identifying and removing assignable causes and ultimately improving the process. Identifying the real time of the change is known as the change‐point estimation problem. Most of the traditional methods of estimating the process change point are developed based on the assumption that the process follows a normal distribution with known parameters, which is seldom true. In this paper, we propose clustering techniques to estimate Shewhart control chart change points. The proposed approach does not depend on the true values of the parameters and even the distribution of the process variables. Accordingly, it is applicable to both phase‐I and phase‐II of normal and non‐normal processes. At the end, we discuss the performance of the proposed method in comparison with the traditional procedures through extensive simulation studies. Copyright © 2008 John Wiley & Sons, Ltd.     

A new dual CUSUM mean chart

The conventional cumulative sum (CUSUM) chart is usually designed based on a known shift size. In usual practice, shift size is often unknown and can be assumed to vary within an interval. With such a range of shift size, the dual CUSUM (DCUSUM) chart provides more sensitivity than the CUSUM chart. In this paper, we propose dual Crosier CUSUM (DCCUSUM) charts with and without fast initial response features to efficiently monitor the infrequent changes in the mean of a normally distributed process. Monte Carlo simulations are used to compute the run length characteristics of one‐sided and two‐sided DCCUSUM charts. These run length characteristics are compared with those of the CUSUM, Crosier CUSUM, Shewhart‐CUSUM, and DCUSUM charts in terms of the integral relative average run length. It turns out that the proposed chart shows better performance when detecting a range of mean shift sizes. A real dataset is considered to illustrate the implementation of existing and proposed charts.     

Increasing the Sensitivity of Cumulative Sum Charts for Location

The cumulative sum (CUSUM) chart is a very effective control charting procedure used for the quick detection of small‐sized and moderate‐sized changes. It can detect small process shifts missed by the Shewhart‐type control chart, which is sensitive mainly to large shifts. To further enhance the sensitivity of the CUSUM control chart at detecting very small process disturbances, this article presents CUSUM control charts based on well‐structured sampling procedures, double ranked set sampling, median‐double ranked set sampling, and double‐median ranked set sampling. These sampling techniques significantly improve the overall performance of the CUSUM chart over the entire process mean shift range, without increasing the false alarm rate. The newly developed control schemes do not only dominate most of the existing charts but are also easy to design and implement as illustrated through an application example of real datasets. The control schemes used for comparison in this study include the conventional CUSUM chart, a fast initial response CUSUM chart, a 2‐CUSUM chart, a 3‐CUSUM chart, a runs rules‐based CUSUM chart, the enhanced adaptive CUSUM chart, the CUSUM chart based on ranked set sampling (RSS), and the single CUSUM and combined Shewhart–CUSUM charts based on median RSS. Copyright © 2014 John Wiley & Sons, Ltd.     

Mixed Exponentially Weighted Moving Average–Cumulative Sum Charts for Process Monitoring

The control chart is a very popular tool of statistical process control. It is used to determine the existence of special cause variation to remove it so that the process may be brought in statistical control. Shewhart‐type control charts are sensitive for large disturbances in the process, whereas cumulative sum (CUSUM)–type and exponentially weighted moving average (EWMA)–type control charts are intended to spot small and moderate disturbances. In this article, we proposed a mixed EWMA–CUSUM control chart for detecting a shift in the process mean and evaluated its average run lengths. Comparisons of the proposed control chart were made with some representative control charts including the classical CUSUM, classical EWMA, fast initial response CUSUM, fast initial response EWMA, adaptive CUSUM with EWMA‐based shift estimator, weighted CUSUM and runs rules–based CUSUM and EWMA. The comparisons revealed that mixing the two charts makes the proposed scheme even more sensitive to the small shifts in the process mean than the other schemes designed for detecting small shifts. Copyright © 2012 John Wiley & Sons, Ltd.     

Weighted adaptive multivariate CUSUM control charts

An adaptive multivariate cumulative sum (AMCUSUM) control chart has received considerable attention because of its ability to dynamically adjust the reference parameter whereby achieving a better performance over a range of mean shifts than the conventional multivariate cumulative sum (CUSUM) charts. In this paper, we introduce a progressive mean–based estimator of the process mean shift and then use it to devise new weighted AMCUSUM control charts for efficiently monitoring the process mean. These control charts are easy to design and implement in a computerized environment compared with their existing counterparts. Monte Carlo simulations are used to estimate the run‐length characteristics of the proposed control charts. The run‐length comparison results show that the weighted AMCUSUM charts perform substantially and uniformly better than the classical multivariate CUSUM and AMCUSUM charts in detecting a range of mean shifts. An example is used to illustrate the working of existing and proposed multivariate CUSUM control charts.     

Estimating the Change Point of the Process Fraction Non‐conforming with a Monotonic Change Disturbance in SPC

Knowing when a process has changed would simplify the search for and identification of the special cause. In this paper, we propose a maximum‐likelihood estimator for the change point of the process fraction non‐conforming without requiring knowledge of the exact change type a priori. Instead, we assume the type of change present belongs to a family of monotonic changes. We compare the proposed change‐point estimator to the maximum‐likelihood estimator for the process change point derived under a simple step change assumption. We do this for a number of monotonic change types and following a signal from a binomial cumulative sum (CUSUM) control chart. We conclude that it is better to use the proposed change point estimator when the type of change present is only known to be monotonic. The results show that the proposed estimator provides process engineers with an accurate and useful estimate of the time of the process change regardless of the type of monotonic change that may be present. Copyright © 2006 John Wiley & Sons, Ltd.     

New Synthetic EWMA and Synthetic CUSUM Control Charts for Monitoring the Process Mean

Exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts are potentially powerful statistical process monitoring tools because of their excellent speed in detecting small to moderate persistent process shifts. Recently, synthetic EWMA (SynEWMA) and synthetic CUSUM (SynCUSUM) control charts have been proposed based on simple random sampling (SRS) by integrating the EWMA and CUSUM control charts with the conforming run length control chart, respectively. These synthetic control charts provide overall superior detection over a range of mean shift sizes. In this article, we propose new SynEWMA and SynCUSUM control charts based on ranked set sampling (RSS) and median RSS (MRSS) schemes, named SynEWMA‐RSS and SynEWMA‐MRSS charts, respectively, for monitoring the process mean. Extensive Monte Carlo simulations are used to estimate the run length characteristics of the proposed control charts. The run length performances of these control charts are compared with their existing powerful counterparts based on SRS, RSS and MRSS schemes. It turns out that the proposed charts perform uniformly better than the Shewhart, optimal synthetic, optimal EWMA, optimal CUSUM, near‐optimal SynEWMA, near‐optimal SynCUSUM control charts based on SRS, and combined Shewhart‐EWMA control charts based on RSS and MRSS schemes. A similar trend is observed when constructing the proposed control charts based on imperfect RSS schemes. An application to a real data is also provided to demonstrate the implementations of the proposed SynEWMA and SynCUSUM control charts. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

New Synthetic Control Charts for Monitoring Process Mean and Process Dispersion

A statistical quality control chart is widely recognized as a potentially powerful tool that is frequently used in many manufacturing and service industries to monitor the quality of the product or manufacturing processes. In this paper, we propose new synthetic control charts for monitoring the process mean and the process dispersion. The proposed synthetic charts are based on ranked set sampling (RSS), median RSS (MRSS), and ordered RSS (ORSS) schemes, named synthetic‐RSS, synthetic‐MRSS, and synthetic‐ORSS charts, respectively. Average run lengths are used to evaluate the performances of the control charts. It is found that the synthetic‐RSS and synthetic‐MRSS mean charts perform uniformly better than the Shewhart mean chart based on simple random sampling (Shewhart‐SRS), synthetic‐SRS, double sampling‐SRS, Shewhart‐RSS, and Shewhart‐MRSS mean charts. The proposed synthetic charts generally outperform the exponentially weighted moving average (EWMA) chart based on SRS in the detection of large mean shifts. We also compare the performance of the synthetic‐ORSS dispersion chart with the existing powerful dispersion charts. It turns out that the synthetic‐ORSS chart also performs uniformly better than the Shewhart‐R, Shewhart‐S, synthetic‐R, synthetic‐S, synthetic‐D, cumulative sum (CUSUM) ln S², CUSUM‐R, CUSUM‐S, EWMA‐ln S², and change point CUSUM charts for detecting increases in the process dispersion. A similar trend is observed when the proposed synthetic charts are constructed under imperfect RSS schemes. Illustrative examples are used to demonstrate the implementation of the proposed synthetic charts. Copyright © 2014 John Wiley & Sons, Ltd.     

A GLR control chart for monitoring a multinomial process

The problem of detecting changes in the parameter p in a Bernoulli process with two possible categories for each observation has been extensively investigated in the SPC literature, but there is much less work on detecting changes in the vector parameter p in a multinomial process where there are more than two categories. A few papers have considered the case in which the direction of the change in p is known, but there is almost no work for the important case in which the direction of the change is unknown and individual observations are obtained. This paper proposes a multinomial generalized likelihood ratio (MGLR) control chart based on a likelihood ratio statistic for monitoring p when individual observations are obtained and the direction and size of the change in p are unknown. A set of 2‐sided Bernoulli cumulative sum (CUSUM) charts is proposed as a reasonable competitor of the MGLR chart. It is shown that the MGLR chart has better overall performance than the set of 2‐sided Bernoulli CUSUM charts over a wide range of unknown shifts. Equations are presented for obtaining the control limit of the MGLR chart when there are three or four components in p .