The computation of average run length and average time to signal: an overview

Control charts are widely used in industries to monitor a process for quality improvement. Evaluation of the average run length (ARL) or average time to signal (ATS) plays an important role in the design of control charts and performance comparison. In this paper, we review several basic and popular procedures, including the Markov chain and integral equation methods for computing ARL, ATS and associated run length distributions for cumulative sum charts, exponentially weighted moving average charts and combined control charts, respectively. Some important references and key formulations are provided for practitioners.     

EWNA charts for monitoring the mean and the autocovariances of stationary processes

In this paper various types of EWMA control charts are introduced for the simultaneous monitoring of the mean and the autocovariances. The target process is assumed to be a stationary process up to fourth-order or an ARMA process with heavy tailed innovations. The case of a Gaussian process is included in our results as well. The charts are compared within a simulation study. As a measure of the performance the average run length is taken. The target process is an ARMA (1,1) process with Student-t distributed innovations. The behavior of the charts is analyzed with respect to several out-of-control models. The best design parameters are determined for each chart. Our comparisons show that the multivariate EWMA chart applied to the residuals has the best overall performance.     

ARL-unbiased control charts for the monitoring of exponentially distributed characteristics based on type-II censored samples

In this paper, the problem of monitoring process data that can be modelled by exponential distribution is considered when observations are from type-II censoring. Such data are common in many practical inspection environment. An average run length unbiased (ARL-unbiased) control scheme is developed when the in-control scale parameter is known. The performance of the proposed control charts are investigated in terms of the ARL and standard deviation of the run length. The effects of parameter estimation on the proposed control charts are also evaluated. Then, we consider the design of the ARL-unbiased control charts when the in-control scale parameter is estimated. Finally, an example is used to illustrate the implementation of the proposed control charts.     

On designing new optimal synthetic Tukey’s control charts

Tukey’s control chart is generally used for monitoring the processes where the measurement process physically damages the product. It is based on single observation and robust to outliers. In this paper, two optimal synthetic Tukey’s control charts are proposed by integrating the conforming run length chart with the Tukey’s control chart and its modification. The performance comparison of the proposed charts with the existing Tukey’s control charts is made by using out-of-control average run length and extra quadratic loss as performance metrics. The proposed charts offer better protection against the process shifts as compare to the existing Tukey’s control charts when the underlying process distribution is symmetric or asymmetric. Simulation studies also establish the supremacy of the proposed control charts over the existing Tukey’s control charts. In the end, an illustrative example based on a real data set of the combined cycle power plant is provided for practical implementation.     

Zone control charts with estimated parameters for detecting prespecified changes in linear profiles

ABSTRACT

In profile monitoring, control charts are proposed to detect unanticipated changes, and it is usually assumed that the in-control parameters are known. However, due to the characteristics of a system or process, the prespecified changes would appear in the process. Moreover, in most applications, the in-control parameters are usually unknown. To overcome these issues, we develop the zone control charts with estimated parameters to detect small shifts of these prespecified changes. The effects of estimation error have been investigated on the performance of the proposed charts. To account for the practitioner-to-practitioner variability, the expected average run length (ARL) and the standard deviation of the average run length (SDARL) is used as the performance metrics. Our results show that the estimation error results in the significant variation in the ARL distribution. Furthermore, in order to adequately reduce the variability, more phase I samples are required in terms of the SDARL metric than that in terms of the expected ARL metric. In addition, more observations on each sampled profile are suggested to improve the charts' performance, especially for small phase I sample sizes. Finally, an illustrative example is given to show the performance of the proposed zone control charts.     

Improving the performance of the zone control chart

A general model for the zone control chart is presented. Using this model, it is shown that there are score vectors for zone control charts which result in superior average run length performance in comparison to Shewhart charts with common runs rules.

A fast initial response (FIR) feature for the zone control chart is also proposed. Average run lengths of the zone control chart with this feature are calculated. It is shown that the FIR feature improves zone control chart performance by providing significantly earlier signals when the process is out of control.     

A Class of Markov Chain Models for Average Run Length Computations for Autocorrelated Processes

The average run length (ARL) of conventional control charts is typically computed assuming temporal independence. However, this assumption is frequently violated in practical applications. Alternative ARL computations have often been conducted via time consuming and yet not necessarily very accurate simulations. In this article, we develop a class of Markov chain models for evaluating the run length performance of traditional control charts for autocorrelated processes. We show extensions from the univariate AR(1) model to the general multivariate VARMA(p, q) time series. The results of the proposed method are highly comparable to those of simulations and with significantly less computational overhead.     

Exceedance probabilities for parametric control charts

Common control charts assume normality and known parameters. Quite often, these assumptions are not valid and large relative errors result in the usual performance characteristics such as the false alarm rate or the average run length. A fully nonparametric approach can form an attractive alternative but requires more Phase I observations than usually available. Sufficiently general parametric families then provide realistic intermediate models. In this article, the performance of charts based on such families is considered. Exceedance probabilities of the resulting stochastic performance characteristics during in-control are studied. Corrections are derived to ensure that such probabilities stay within prescribed bounds. Attention is also devoted to the impact of the corrections for an out-of-control process. Simulations are presented both to illustrate and to demonstrate that the approximations obtained are sufficiently accurate for practical usage.     

Applying fast initial response features on GWMA control charts for monitoring autocorrelation data

ABSTRACT

A generally weighted moving average (GWMA) control chart with fast initial response (FIR) features is addressed to monitor an autoregressive process mean shift. Numerical simulations based on average run length (ARL) show that the GWMA control chart with additional FIR feature requires less time to detect small or moderate shifts than GWMA control chart at low level of autocorrelation; whereas these two control charts perform similarly at high level of autocorrelation. Regardless of any level of autocorrelation, GWMA control charts provided with additional FIR feature have a good performance in detecting large shifts during the initial stage.     

A class of distribution-free control charts

Summary.  A class of Shewhart-type distribution-free control charts is considered. A key advantage of these charts is that the in-control run length distribution is the same for all continuous process distributions. Exact expressions for the run length distribution and the average run length (ARL) are derived and properties of the charts are studied via evaluations of the run length distribution probabilities and the ARL. Tables are provided for implementation for some typical ARL values and false alarm rates. The charts proposed are preferable from a robustness point of view, have attractive ARL properties and would be particularly useful in situations where one uses a classical Shewhart   X -chart. A numerical illustration is given.     

A new nonparametric synthetic EWMA control chart for monitoring process mean

The exponentially weighted moving average (EWMA) control charts are widely used in chemical and process industries because of their excellent speed in catching small to moderate shifts in the process target. In usual practice, many data come from a process where the monitoring statistic is non-normally distributed or it follows an unknown probability distribution. This necessitates the use of distribution-free/nonparametric control charts for monitoring the deviations from the process target. In this paper, we integrate the existing EWMA sign chart with the conforming run length chart to propose a new synthetic EWMA (SynEWMA) sign chart for monitoring the process mean. The SynEWMA sign chart encompasses the synthetic sign and EWMA sign charts. Monte Carlo simulations are used to compute the run length profiles of the SynEWMA sign chart. Based on a comprehensive comparison, it turns out that the SynEWMA sign chart is able to perform substantially better than the existing EWMA sign chart. Both real and simulated data sets are used to explain the working and implementation of existing and proposed control charts.     

Effect of measurement error on exponentially weighted moving average control charts under ranked set sampling schemes

Control charts are a powerful statistical process monitoring tool often used to monitor the stability of manufacturing processes. In quality control applications, measurement errors adversely affect the performance of control charts. In this paper, we study the effect of measurement error on the detection abilities of the exponentially weighted moving average (EWMA) control charts for monitoring process mean based on ranked set sampling (RSS), median RSS (MRSS), imperfect RSS (IRSS) and imperfect MRSS (IMRSS) schemes. We also study the effect of multiple measurements and non-constant error variance on the performances of the EWMA control charts. The EWMA control chart based on simple random sampling is compared with the EWMA control charts based on RSS, MRSS, IRSS and IMRSS schemes. The performances of the EWMA control charts are evaluated in terms of out-of-control average run length and standard deviation of run lengths. It turns out that the EWMA control charts based on MRSS and IMRSS schemes are better than their counterparts for all measurement error cases considered here.     

Enhanced adaptive CUSUM charts for process mean

A statistical quality control chart is an important tool of the statistical process control, which is widely used to control and monitor a production process. The CUSUM chart is designed to detect a specific shift, provided that the shift size is known in advance. In practice, however, shift sizes are rarely known. It is then customary to use an adaptive CUSUM chart, which can effectively detect a range of shift sizes. In this paper, we enhance the sensitivities of the improved adaptive CUSUM mean charts using an auxiliary-information-based (AIB) mean estimator. The run length performances of the proposed charts are compared with those of the AIB adaptive and non-adaptive CUSUM charts in terms of the average run length (ARL), extra quadratic loss, and integral relative ARL. These run length comparisons reveal that the proposed charts are more sensitive than the existing charts when detecting different kinds of shift in the process mean. An example is given to demonstrate the implementation of existing and proposed charts.     

A new adaptive EWMA control chart using auxiliary information for monitoring the process mean

The adaptive memory-type control charts, including the adaptive exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts, have gained considerable attention because of their excellent speed in providing overall good detection over a range of mean shift sizes. In this paper, we propose a new adaptive EWMA (AEWMA) chart using the auxiliary information for efficiently monitoring the infrequent changes in the process mean. The idea is to first estimate the unknown process mean shift using an auxiliary information based mean estimator, and then adaptively update the smoothing constant of the EWMA chart. Using extensive Monte Carlo simulations, the run length profiles of the AEWMA chart are computed and explored. The AEWMA chart is compared with the existing control charts, including the classical EWMA, CUSUM, synthetic EWMA and synthetic CUSUM charts, in terms of the run length characteristics. It turns out that the AEWMA chart performs uniformly better than these control charts when detecting a range of mean shift sizes. An illustrative example is also presented to demonstrate the working and implementation of the proposed and existing control charts.     

Optimal CUSUM and adaptive CUSUM charts with auxiliary information for process mean

The CUSUM chart is good enough to detect small-to-moderate shifts in the process parameter(s) as it can be optimally designed to detect a particular shift size. The adaptive CUSUM (ACUSUM) chart provides good detection over a range of shift sizes because of its ability to update the reference parameter using the estimated process shift. In this paper, we propose auxiliary-information-based (AIB) optimal CUSUM (OCUSUM) and ACUSUM charts, named AIB-OCUSUM and AIB-ACUSUM charts, using a difference estimator of the process mean. The performance comparisons between existing and proposed charts are made in terms of the average run length (ARL), extra quadratic loss and integral relative ARL measures. It is found that the AIB-OCUSUM and AIB-ACUSUM charts are more sensitive than the AIB-CUSUM and ACUSUM charts, respectively. Moreover, the AIB-ACUSUM chart surpasses the AIB-OCUSUM chart when detecting a range of mean shift sizes. Illustrative examples are given to support the theory.     

Multivariate control charts based on a projection approach

Summary: In this paper the projection approach of Runger (1996) is applied to construct control charts for a multivariate process. It is assumed that a shift in the mean might only occur in a known subspace of the parameter space. The projection method permits a reduction of the dimensionality of the control problem.Several control schemes based on projections are introduced. We consider CUSUM type charts as well as EWMA schemes. The underlying variables are assumed to be independent and normally distributed. Using the average run length all control charts are compared with each other. Moreover, it is analyzed how sensitive the charts react on a false choice of the subspace.     

A new distribution-free control scheme based on order statistics

We establish a class of nonparametric Shewhart-type control charts based on a reference sample drawn from the process. The proposed nonparametric control chart takes advantage of the location of two different order statistics of the reference and test sample respectively. The decision rule of the new monitoring scheme is filled out by the number of test observations that are located between the control limits. The general setup of the new class of control charts is presented in detail, while the operating characteristic function is studied for both in- and out-of-control processes. Closed formulae for the evaluation of the alarm rate and the average run length are concluded for plausible shift in the underlying distribution to Lehmann alternatives. Several numerical results, displayed for the new family of nonparametric control charts, depict that the proposed control scheme attains competitive performance.     

A new synthetic control chart for monitoring process mean using auxiliary information

ABSTRACT

Quality control charts have been widely recognized as a potentially powerful statistical process monitoring tool in statistical process control because of their superior ability in detecting shifts in the process parameters. Recently, auxiliary-information-based control charts have been proposed and shown to have excellent speed in detecting process shifts than those based without it. In this paper, we design a new synthetic control chart that is based on a statistic that utilizes information from both the study and auxiliary variables. The proposed synthetic chart encompasses the classical synthetic chart. The construction, optimal design, run length profiles, and the performance evaluation of the new chart are discussed in detail. It turns out that the proposed synthetic chart performs uniformly better than the classical synthetic chart when detecting different kinds of shifts in the process mean under both zero-state and steady-state run length performances. Moreover, with reasonable assumptions, the proposed chart also surpasses the exponentially weighted moving average control chart. An application with a simulated data set is also presented to explain the implementation of the proposed control chart.     

Improved CUSUM charts for monitoring process mean

In this paper, we propose new cumulative sum (CUSUM) and Shewhart-CUSUM (SCUSUM) control charts for monitoring the process mean using ranked-set sampling (RSS) and ordered RSS (ORSS) schemes. The proposed CUSUM charts include the Crosier's CUSUM (CCUSUM) and Shewhart-CCUSUM (SCCUSUM) charts using RSS, and the CUSUM, CCUSUM, SCUSUM and SCCUSUM charts using ORSS. Moreover, fast initial response features are also attached with these CUSUM charts to improve their sensitivities for an initial out-of-control situation. Monte Carlo simulations are used to compute the run length characteristics of the proposed CUSUM charts. Upon comparing the run length performances of the CUSUM charts, it turns out that the proposed CUSUM charts are more sensitive than their existing counterparts. A real dataset is used to explain the implementation of the proposed CUSUM charts.     

An Adaptive Shiryaev–Roberts Procedure for Signalling Varying Location Shifts

This paper proposes a new adaptive chart based on the Shiryaev–Roberts procedure, by updating the reference value in an adaptive way to achieve the aim of overall good performance over a range of future expected but unknown mean shifts. A two-dimensional Markov chain model is developed to analyze the run length performance. The design guidelines are given. The comparisons of run length performance of the proposed scheme and other charts show that the proposed chart provides quite effective detecting ability over a range of mean shift sizes. The implementation of the new chart is illustrated by a real data example.