Time-between-events monitoring using nonhomogeneous Poisson process with power law intensity

The traditional process monitoring techniques used to study high-quality processes have several demerits, that is, high-false alarm rate and poor detection, etc. A recent and promising idea to monitor such processes is the use of time-between-events (TBE) control charts. However, the available TBE control charts have been developed in a nonadaptive fashion assuming the Poisson process. There are many situations where we need adaptive monitoring, for example, health, flood, food, system, or terrorist surveillance. Therefore, the existing control charts are not useful, especially in sequential monitoring. This article introduces new adaptive TBE control charts for high-quality processes based on the nonhomogeneous Poisson process by assuming the power law intensity. In particular, probability control limits are used to develop control charts. The proposed methodology allows us to get control limits that are dynamic and suitable for online process monitoring with an additional advantage to monitor a process where we believe the underlying failure rate may be changing over time. The average run length and coefficient of variation of the run length distribution are used to assess the performance of the proposed control charts. Besides simulation studies, we also discuss three examples to highlight the application of the proposed charts.     

Time between events control charts for gamma distribution

Modern and emerging techniques of technology have brought a revolution in quality inspection of products. When events in highly efficient production processes occur rarely, it requires to inspect and monitor the time between occurrence of these events (TBE). The exponential and gamma distributions are commonly used models for time between events (TBE) data. In this article, a new monitoring scheme has been established for TBE data based on exponential and gamma distributions. In a previous research, transformation-based control charts have been developed for TBE. The proposed study is aimed to use the exact probability distribution of charting statistic rather than applying transformations to data and this has remained still unaddressed. Average run length (ARL) and percentage decrease in ARL (ΔARL) have been calculated using Monte Carlo simulations and the proposed monitoring method has been compared with existing techniques applied to transformed data. The proposed scheme provides a simpler design structure and better performance on different sample sizes in identifying annoying process variations. Further, the technique has been applied to simulated and real-life data sets of time between manufacturing plant accidents to highlight the worth and particle applicability of the proposed work.     

Monitoring of time between events with a double generally weighted moving average control chart

Monitoring of time between events (TBE) instead of the number of events is used in high‐quality processes where the events occur rarely. This article presents a double generally weighted moving average control chart with a lower time‐varying control limit to monitor the TBE (regarded as DGWMA‐TBE chart). The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the DGWMA‐TBE chart is more effective than the DEWMA and GWMA charts in detecting moderate to large shifts. Furthermore, the DGWMA‐TBE chart is very robust for the same range of shifts when the TBE observations follow a Weibull or a lognormal distribution. Finally, examples are also presented to enhance the performance of the proposed chart.     

Probabilities on Algebraic Structures

Time-between-events (TBE) charts or T charts have attracted increasing research interest in statistical process control (SPC). These charts monitor TBE or the time interval T between the events. Currently, almost all studies on T charts are focused on applications under 100% inspection. However, due to limitations in resources and working conditions, sampling inspection has to be adopted for many SPC applications, especially when testing is destructive and/or expensive. The operational characteristics of T charts under sampling inspection could be quite different from those under 100% inspection. Specifically, some highly efficient techniques or methods, such as sequential analysis, may be adopted for sampling inspection. This article studies four T charts for sampling inspection: (1) a Shewhart T chart; (2) a CUSUM T chart and its variable sample size version; (3) a SA T chart (the T chart using sequential analysis); and (4) a curtailed SA T chart. It is the first time that sequential analysis and curtailment technique are adopted for TBE control charts. It is found that these SA-type charts, especially the curtailed chart, are significantly more effective than the Shewhart T chart, CUSUM T chart, and any other charts in current literature. This article has supplementary material online.     

Monitoring of Time‐Between‐Events with a Generalized Group Runs Control Chart

Control charting methods for time between events (TBE) is important in both manufacturing and nonmanufacturing fields. With the aim to enhance the speed for detecting shifts in the mean TBE, this paper proposes a generalized group runs TBE chart to monitor the mean TBE of a homogenous Poisson failure process. The proposed chart combines a TBE subchart and a generalized group conforming run length subchart. The zero‐state and steady‐state performances of the proposed chart were evaluated by applying a Markov chain method. Overall, it is found that the proposed chart outperforms the existing TBE charts, such as the T, T_r, EWMA‐T, Synth‐T_r, and GR‐T_r charts. Copyright © 2015 John Wiley & Sons, Ltd.     

A triple exponentially weighted moving average control chart for monitoring time between events

Time between events (TBE) charts are used in high-yield processes where the rate of occurrences is very low. In the current article, we propose a triple exponentially weighted moving average control chart to monitor TBE (regarded as triple exponentially weighted moving average TEWMA-TBE chart) modeled by a gamma distribution. One- and two-sided schemes of the proposed chart are designed and compared with the double EWMA DEWMA-TBE and EWMA-TBE charts. It is shown that the lower- and two-sided TEWMA-TBE charts outperform its competitors, especially for small to moderate downward shifts, while the upper-sided TEWMA-TBE chart has very good detection ability for small shifts. We also study the robustness of the proposed chart when the true distribution is a Weibull or a lognormal and it is found that the TEWMA-TBE chart has better robustness properties than its competitors, especially for small shifts. Two illustrative examples from airplane accidents and earthquakes are also provided to display the application of the proposed chart.     

Performance measures of discrete and continuous time-between-events control charts

An increasing number of applications in the chemical industry involve measuring nonconforming items, particularly in high-purity processes or high-yield processes. Dedicated monitoring tools such as the time-between-events (TBE) control charts have been developed for both discrete time (CCC-charts) and continuous time (t_r-charts) for detecting any shifts in the process defect rate. However, most common performance metrics used in literature are not always appropriate and may not suffice to describe the efficiency of a monitoring system. The definition of conditional performance measures described in the literature is extended to TBE charts. These metrics are computed for Shewhart-type TBE charts with run rules and used to compare these charts at different aggregation levels.     

Design of exponential control charts based on average time to signal using a sequential sampling scheme

Exponential charts based on time-between-events (TBE) data are widely investigated and applied in various fields. The average time to signal (ATS) is used instead of the average run length to evaluate the performance of TBE charts, since the ATS involves both the number and the time of samples inspected until a signal occurs. An ATS-unbiased exponential control chart is proposed when the in-control parameter is known. Considering the need in practice to start monitoring a production process as soon as possible, a sequential sampling scheme is adopted and the in-control parameter is estimated by an unbiased and consistent estimator. Some specific guidelines to stop updating control limits are obtained from the relationship between the phase I sample size and the actual false alarm rate. Finally, two real examples are given to illustrate the implementation and efficiency of the proposed method.     

Boxplot‐based phase I control charts for time between events

To monitor the quality/reliability of a (production) process, it is sometimes advisable to monitor the time between certain events (say occurrence of defects) instead of the number of events, particularly when the events occur rarely. In this case it is common to assume that the times between the events follow an exponential distribution. In this paper, we propose a one‐ and a two‐sided control chart for phase I data from an exponential distribution. The control charts are derived from a modified boxplot procedure. The charting constants are obtained by controlling the overall Type I error rate and are tabulated for some configurations. A numerical example is provided for illustration. The in‐control robustness and the out‐of‐control performance of the proposed charts are examined and compared with those of some existing charts in a simulation study. It is seen that the proposed charts are considerably more in‐control robust and have out‐control properties comparable to the competing charts. Copyright © 2011 John Wiley & Sons, Ltd.     

Properties of the exponential EWMA chart with parameter estimation

Count rates may reach very low levels in production processes with low defect levels. In such settings, conventional control charts for counts may become ineffective since the occurrence of many samples with zero defects would cause control statistic to be consistently zero. Consequently, the exponentially weighted moving average (EWMA) control chart to monitor the time between successive events (TBE) or counts has been introduced as an effective approach for monitoring processes with low defect levels. When the counts occur according to a Poisson distribution, the TBE observations are distributed as exponential. Although the assumption of exponential distribution is a reasonable choice as a model of TBE observations, its parameter, i.e. the mean (also the standard deviation), is rarely known in practice and its estimate is used in place of the unknown parameter when constructing the exponential EWMA chart. In this article, we investigate the effects of parameter estimation on the performance measures (average run length, standard deviation, and percentiles of the run length distribution) of the exponential EWMA control chart. A comprehensive analysis of the conditional performance measures of the chart shows that the effect of estimation can be serious, especially if small samples are used. An investigation of the marginal performance measures, which are calculated by averaging the conditional performance measures over the distribution of the parameter estimator, allows us to provide explicit sample size recommendations in constructing these charts to reach a satisfactory performance in both the in‐control and the out‐of‐control situation. Copyright © 2009 John Wiley & Sons, Ltd.     

Phase II Shewhart‐type Control Charts for Monitoring Times Between Events and Effects of Parameter Estimation

Monitoring times between events (TBE) is an important aspect of process monitoring in many areas of applications. This is especially true in the context of high‐quality processes, where the defect rate is very low, and in this context, control charts to monitor the TBE have been recommended in the literature other than the attribute charts that monitor the proportion of defective items produced. The Shewhart‐type t‐chart assuming an exponential distribution is one chart available for monitoring the TBE. The t‐chart was then generalized to the t_r‐chart to improve its performance, which is based on the times between the occurrences of r (≥1) events. In these charts, the in‐control (IC) parameter of the distribution is assumed known. This is often not the case in practice, and the parameter has to be estimated before process monitoring and control can begin. We propose estimating the parameter from a phase I (reference) sample and study the effects of estimation on the design and performance of the charts. To this end, we focus on the conditional run length distribution so as to incorporate the ‘practitioner‐to‐practitioner’ variability (inherent in the estimates), which arises from different reference samples, that leads to different control limits (and hence to different IC average run length [ARL] values) and false alarm rates, which are seen to be far different from their nominal values. It is shown that the required phase I sample size needs to be considerably larger than what has been typically recommended in the literature to expect known parameter performance in phase II. We also find the minimum number of phase I observations that guarantee, with a specified high probability, that the conditional IC ARL will be at least equal to a given small percentage of a nominal IC ARL. Along the same line, a lower prediction bound on the conditional IC ARL is also obtained to ensure that for a given phase I sample, the smallest IC ARL can be attained with a certain (high) probability. Summary and recommendations are given. Copyright © 2014 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.     

Attribute control charts using generalized zero‐inflated Poisson distribution

This paper presents a control charting technique to monitor attribute data based on a generalized zero‐inflated Poisson (GZIP) distribution, which is an extension of ZIP distribution. GZIP distribution is very flexible in modeling complicated behaviors of the data. Both the technique of fitting the GZIP model and the technique of designing control charts to monitor the attribute data based on the estimated GZIP model are developed. Simulation studies and real industrial applications illustrate that the proposed GZIP control chart is very flexible and advantageous over many existing attribute control charts. Copyright © 2008 John Wiley & Sons, Ltd.     

A double homogeneously weighted moving average control chart for monitoring of the process mean

In the service and manufacturing industry, memory-type control charts are extensively applied for monitoring the production process. These types of charts have the ability to efficiently detect disturbances, especially of smaller amount, in the process mean and/or dispersion. Recently, a new homogeneously weighted moving average (HWMA) chart has been proposed for efficient monitoring of smaller shifts. In this study, we have proposed a new double HWMA (DHWMA) chart to monitor the changes in the process mean. The run length profile of the proposed DHWMA chart is evaluated and compared with some existing control charts. The outcomes reveal that the DHWMA chart shows better performance over its competitor charts. The effect of non-normality (in terms of robustness) and the estimation of the unknown parameters on the performance of the DHWMA chart are also investigated as a part of this study. Finally, a real-life industrial application is offered to demonstrate the proposal for practical considerations.     

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.     

EWMA and DEWMA Control Charts for Poisson‐Exponential Distribution: Conditional Median Approach for Censored Data

Exponentially weighted moving average (EWMA) control charts are consistently used for the detection of small shifts contrary to Shewhart charts, which are commonly used for the detection of large shifts in the process. There are many interesting features of EWMA charts that have been studied for complete data in the literature. The aim of present study is to introduce and compare the double exponentially weighted moving average (DEWMA) and EWMA control charts under type‐I censoring for Poisson‐exponential distribution. The monitoring of mean level shifts using censored data is of a great interest in many applied problems. Moreover, a new idea of conditional median is introduced and further compared with the existing conditional expected values approach for monitoring the small mean level shifts. The performance of the DEWMA and EWMA charts is evaluated using the average run length, expected quadratic loss, and performance comparison index measures. The optimum sample size comparisons for the specified and unspecified parameters are also part of this study. Two applications for practical considerations are also discussed. It is observed that different censoring rates and the size of shifts significantly affect the performance of the EWMA and DEWMA charts. Copyright © 2016 John Wiley & Sons, Ltd.     

Exponential cumulative sums chart for detecting shifts in time-between-events

Time-between-events (TBE) charts use the time interval T between events to monitor process shifts (or failure rates λ). This paper presents a two-sided TBE cumulative sums (CUSUM) chart called a weighted CUSUM(WCUSUM)chart for detecting either a deterioration (decrease in T) or an improvement (increase in T) in the condition of a process. A new kind of WCUSUM chart that has an additional charting power parameter w is proposed here. A WCUSUM chart’s efficiency can be improved by using the parameter w, based on an estimated value of the mean shift. In addition, a methodology and optimal design are presented for minimising the average loss. Construction of the WCUSUM chart is illustrated by considering a random shift δ in λ (including both increasing and decreasing shifts) in the design.     

A predictive Bayesian approach to sequential time-between-events monitoring

A fundamental problem with all process monitoring techniques is the requirement of a large Phase-I data set to establish control limits and overcome estimation error. This assumption of having a large Phase-I data set is very restrictive and often problematic, especially when the sampling is expensive or not available, eg, time-between-events (TBE) settings. Moreover, with the advancement in technology, quality practitioners are now more interested in online process monitoring. Therefore, the Bayesian methodology not only provides a natural solution for sequential and adaptive learning but also addresses the problem of a large Phase-I data set for setting up a monitoring structure. In this study, we propose Bayesian control charts for TBE assuming homogenous Poisson process. In particular, a predictive approach is adopted to introduce predictive limit control charts. Beside the Bayesian predictive Shewhart charts with dynamic control limits, a comparison of the frequentist sequential charts, designed by using unbiased and biased estimator of the process parameter, is also a part of the present study. To assess the predictive TBE chart performance in the presence of practitioner-to-practitioner variability, we use the average of the average run length (AARL) and the standard deviation of the in-control run length (SDARL).     

Exponentially weighted control charts to monitor multivariate process variability for high dimensions

Multivariate monitoring of industrial or clinical procedures often involves more than three correlated quality characteristics and the status of the process is judged using a sample of size one. Majority of existing control charts for monitoring process variability for individual observations are capable of monitoring up to three characteristics. One of the hurdles in designing optimal control charts for large dimension data is the enormous computing resources and time that is required by simulation algorithm to estimate the charts parameters. This paper proposes a novel algorithm based on Parallelised Monte Carlo simulation to improve the ability of the Multivariate Exponentially Weighted Mean Squared Deviation and Multivariate Exponentially Weighted Moving Variance charts to monitor process variability for high dimensions in a computationally efficient way. Different techniques have been deployed to reduce computing space and execution time. The optimal control limits (L) to detect small, medium and large shifts in the covariance matrix of up to 15 characteristics are provided. Furthermore, utilising the large number of optimal L values generated by the algorithm enabled authors to develop exponential decay functions to predict L values. This eliminates the need for further execution of the parallelised Monte Carlo simulation.     

New memory‐type dispersion control charts

The exponentially weighted moving average (EWMA) control chart is a memory‐type process monitoring tool that is frequently used to monitor small and moderate disturbances in the process mean and/or process dispersion. In this study, we propose 2 new memory‐type control charts for monitoring changes in the process dispersion, namely, the generally weighted moving average and the hybrid EWMA charts. We use Monte Carlo simulations to compute the run length profiles of the proposed control charts. The run length comparisons of the proposed and existing charts reveal that the generally weighted moving average and hybrid EWMA charts provide better protection than the existing EWMA chart when detecting small to moderate shifts in the process dispersion. An illustrative dataset is also used to show the superiority of the proposed charts over the existing chart.