Globally applicable control chart for online monitoring of stability of process mean

The shape features of run chart patterns of the most recent m observations arising from stable and unstable processes are different. Using this fact, a new monitoring statistic is defined whose value for given m depends on the pattern parameters but not on the process parameters. A control chart for this statistic for given m, therefore, will be globally applicable to normal processes. The simulation study reveals that the proposed statistic approximately follows normal distribution. The performances of the globally applicable control chart in terms of average run lengths (ARLs) are evaluated and compared with the X chart. Both in-control ARL and out-of-control ARLs with respect to different abnormal process conditions are found to be larger than the X chart. However, the proposed concept is promising because it can eliminate the burden of designing separate control charts for different quality characteristics or processes in a manufacturing set-up.     

Accounting for phase I sampling variability in the performance of the MEWMA control chart with estimated parameters

In this article, we assess the performance of the multivariate exponentially weighted moving average (MEWMA) control chart with estimated parameters while considering the practitioner-to-practitioner variability. We evaluate the chart performance in terms of the in-control average run length (ARL) distributional properties; mainly the average (AARL), the standard deviation (SDARL), and some percentiles. We show through simulations that using estimates in place of the in-control parameters may result in an in-control ARL distribution that almost completely lies below the desired value. We also show that even with the use of larger amounts of historical data, there is still a problem with the excessive false alarm rates. We recommend the use of a recently proposed bootstrap-based design technique for adjusting the control limits. The technique is quite effective in controlling the percentage of short in-control ARLs resulting from the estimation error.     

CUSUM chart for detecting range shifts when monotonicity of likelihood ratio is invalid

It is often encountered in the literature that the log-likelihood ratios (LLR) of some distributions (e.g. the student t distribution) are not monotonic. Existing charts for monitoring such processes may suffer from the fact that the average run length (ARL) curve is a discontinuous function of control limit. It implies that some pre-specified in-control (IC) ARLs of these charts may not be reached. To guarantee the false alarm rate of a control chart lower than the nominal level, a larger IC ARL is usually suggested in the literature. However, the large IC ARL may weaken the performance of a control chart when the process is out-of-control (OC), compared with a just right IC ARL. To overcome it, we adjust the LLR to be a monotonic one in this paper. Based on it, a multiple CUSUM chart is developed to detect range shifts in IC distribution. Theoretical result in this paper ensures the continuity of its ARL curve. Numerical results show our proposed chart performs well under the range shifts, especially under the large shifts. In the end, a real data example is utilized to illustrate our proposed chart.     

A new S2 control chart using repetitive sampling

A new S² control chart is presented for monitoring the process variance by utilizing a repetitive sampling scheme. The double control limits called inner and outer control limits are proposed, whose coefficients are determined by considering the average run length (ARL) and the average sample number when the process is in control. The proposed control chart is compared with the existing Shewhart S² control chart in terms of the ARLs. The result shows that the proposed control chart is more efficient than the existing control chart in detecting the process shift.     

Optimal design of the adaptive exponentially weighted moving average control chart over a range of mean shifts

The adaptive exponentially weighted moving average (AEWMA) control chart is a smooth combination of the Shewhart and exponentially weighted moving average (EWMA) control charts. This chart was proposed by Cappizzi and Masarotto (2003) to achieve a reasonable performance for both small and large shifts. Cappizzi and Masarotto (2003) used a pair of shifts in designing their control chart. In this study, however, the process mean shift is considered as a random variable with a certain probability distribution and the AEWMA control chart is optimized for a wide range of mean shifts according to that probability distribution and not just for a pair of shifts. Using the Markov chain technique, the results show that the new optimization design can improve the performance of the AEWMA control chart from an overall point of view relative to the various designs presented by Cappizzi and Masarotto (2003). Optimal design parameters that achieve the desired in-control average run length (ARL) are computed in several cases and formulas used to find approximately their values are given. Using these formulas, the practitioner can compute the optimal design parameters corresponding to any desired in-control ARL without the need to apply the optimization procedure. The results obtained by these formulas are very promising and would particularly facilitate the design of the AEWMA control chart for any in-control ARL value.     

Determining Optimal Number of Samples for Constructing Multivariate Control Charts

Normally, an average run length (ARL) is used as a measure for evaluating the detecting performance of a multivariate control chart. This has a direct impact on the false alarm cost in Phase II. In this article, we first conduct a simulation study to calculate both in-control and out-of-control ARLs under various combinations of process shifts and number of samples. Then, a trade-off analysis between sampling inspection and false alarm costs is performed. Both the simulation results and trade-off analysis suggest that the optimal number of samples for constructing a multivariate control chart in Phase I can be determined.     

The effect of serial correlation on the in-control average run length of cumulative score charts

The cumulative sum (CUSUM) technique is well-established in theory and practice of process control. For a variant of the CUSUM technique, the cumulative score chart, we investigate the effect of serial correlation on the in-control average run length (ARL). The Shewhart chart is a special case of the cumulative score chart. Using the fact that the cumulative score statistic is a correlated random walk with a reflecting and an absorbing barrier, we derive an approximate but closed-form expression for the ARL of a control variable that follows a first-order autoregressive process with normally distributed disturbances. We also give an expression for the asymptotic (large in-control ARL) case. Our method of approximation gives ARL values that are in good agreement with Monte Carlo estimates of the true values. For positive serial correlation the ARL decreases with increasing value of the correlation coefficient. For increasing negative serial correlation, the ARL may decrease or increase depending on the choice of the parameters of the chart; parameterizations can be found which are rather insensitive for negative serial correlation. We use our results to give recommendations on how to modify the control chart procedure in the presence of serial correlation.     

Some Comments Regarding the Synthetic Chart

The steady-state average run length (ARL) is a function of the in-control probabilities of being in each nonabsorbing state. Davis and Woodall (2002) tabulated values that are significantly smaller than the steady-state ARLs, because they used the out-of-control probabilities. The synthetic chart signals when a second sample point falls beyond the control limits, no matter whether one of them falls above the centerline and the other falls below it. The side-sensitive version of the synthetic chart does not signal when the points beyond the control limits are on opposite sides. With this rule, the chart detects mean changes more quickly.     

A Robust Control Chart for Monitoring the Mean of an Autocorrelated Process

Residual control charts are frequently used for monitoring autocorrelated processes. In the design of a residual control chart, values of the true process parameters are often estimated from a reference sample of in-control observations by using least squares (LS) estimators. We propose a robust control chart for autocorrelated data by using Modified Maximum Likelihood (MML) estimators in constructing a residual control chart. Average run length (ARL) is simulated for the proposed chart when the underlying process is AR(1). The results show the superiority of the new chart under several situations. Moreover, the chart is robust to plausible deviations from assumed distribution of errors.     

CUSUM Control Charts for Multivariate Poisson Distribution

A cumulative sum control chart for multivariate Poisson distribution (MP-CUSUM) is proposed. The MP-CUSUM chart is constructed based on log-likelihood ratios with in-control parameters, Θ₀, and shifts to be detected quickly, Θ₁. The average run length (ARL) values are obtained using a Markov Chain-based method. Numerical experiments show that the MP-CUSUM chart is effective in detecting parameter shifts in terms of ARL. The MP-CUSUM chart with smaller Θ₁ is more sensitive than that with greater Θ₁ to smaller shifts, but more insensitive to greater shifts. A comparison shows that the proposed MP-CUSUM chart outperforms an existing MP chart.     

An improved mean deviation exponentially weighted moving average control chart to monitor process dispersion under ranked set sampling

Quality-control charts are widely used to monitor and detect shifts in the process mean and dispersion. Abbasi and Miller [MDEWMA chart: an efficient and robust alternative to monitor process dispersion, J Stat Comput Simul 2013;83:247–268] suggested a robust mean deviation exponentially weighted moving average (MDEWMA) control chart for monitoring process dispersion under simple random sampling. In this study, an improved MDEWMA (IMDEWMA) control chart is proposed under ranked set sampling to monitor process dispersion. Detailed Monte Carlo simulations are performed from symmetric and asymmetric populations to investigate the performances of the proposed and existing control charts in terms of average run length (ARL), median run length and standard deviation of run length. An application to real-life data is also presented to illustrate the use of the IMDEWMA control chart. It is observed that the IMDEWMA control chart indicates a shift in process dispersion substantially quicker than the MDEWMA control chart, while maintaining comparable ARLs when the process is in control.     

An overview of risk-adjusted charts

Summary.  The paper provides an overview of risk-adjusted charts, with examples based on two data sets: the first consisting of outcomes following cardiac surgery and patient factors contributing to the Parsonnet score; the second being age–sex-adjusted death-rates per year under a single general practitioner. Charts presented include the cumulative sum (CUSUM), resetting sequential probability ratio test, the sets method and Shewhart chart. Comparisons between the charts are made. Estimation of the process parameter and two-sided charts are also discussed. The CUSUM is found to be the least efficient, under the average run length (ARL) criterion, of the resetting sequential probability ratio test class of charts, but the ARL criterion is thought not to be sensible for comparisons within that class. An empirical comparison of the sets method and CUSUM, for binary data, shows that the sets method is more efficient when the in-control ARL is small and more efficient for a slightly larger range of in-control ARLs when the change in parameter being tested for is larger. The Shewart p -chart is found to be less efficient than the CUSUM even when the change in parameter being tested for is large.     

Synthetic exponential control charts with unknown parameter

The existing synthetic exponential control charts are based on the assumption of known in-control parameter. However, the in-control parameter has to be estimated from a Phase I dataset. In this article, we use the exact probability distribution, especially the percentiles, mean, and standard deviation of the conditional average run length (ARL) to evaluate the effect of parameter estimation on the performance of the Phase II synthetic exponential charts. This approach accounts for the variability in the conditional ARL values of the synthetic chart obtained by different practitioners. Since parameter estimation results in more false alarms than expected, we develop an exact method to design the adjusted synthetic charts with desired conditional in-control performance. Results of known and unknown in-control parameter cases show that the control limit of the conforming run length sub-chart of the synthetic chart should be as small as possible.     

A weighted likelihood ratio test-based chart for monitoring process mean and variability

In this paper, a new single exponentially weighted moving average (EWMA) control chart based on the weighted likelihood ratio test, referred to as the WLRT chart, is proposed for the problem of monitoring the mean and variance of a normally distributed process variable. It is easy to design, fast to compute, and quite effective for diverse cases including the detection of the decrease in variability and individual observation case. The optimal parameters that can be used as a design aid in selecting specific parameter values based on the average run length (ARL) and the sample size are provided. The in-control (IC) and out-of-control (OC) performance properties of the new chart are compared with some other existing EWMA-type charts. Our simulation results show that the IC run length distribution of the proposed chart is similar to that of a geometric distribution, and it provides quite a robust and satisfactory overall performance for detecting a wide range of shifts in the process mean and/or variability.     

Improved R and s control charts for monitoring the process variance

The Shewhart R control chart and s control chart are widely used to monitor shifts in the process spread. One fact is that the distributions of the range and sample standard deviation are highly skewed. Therefore, the R chart and s chart neither provide an in-control average run length (ARL) of approximately 370 nor guarantee the desired type I error of 0.0027. Another disadvantage of these two charts is their failure in detecting an improvement in the process variability. In order to overcome these shortcomings, we propose the improved R chart (IRC) and s chart (ISC) with accurate approximation of the control limits by using cumulative distribution functions of the sample range and standard deviation. Simulation studies show that the IRC and ISC perform very well. We also compare the type II error risks and ARLs of the IRC and ISC and found that the s chart is generally more efficient than the R chart. Examples are given to illustrate the use of the developed charts.     

The Robustness of the Three-Way Chart to Non Normality

In batch processing, the Three-Way control chart has been offered for controlling the mean of a process when the batch-to-batch variation is much greater than the within-batch variation. These two sources of variation are typically monitored along with usual batch sample means. Although the Three-Way chart was originally developed for normally distributed process data, its robustness to violations of the normality assumption is the central theme of this study. For data streams with heavy tails or displaying skewness, the in-control average run lengths (ARLs) for the Three-Way chart are seen to be significantly shorter than expected. On the other hand, out-of-control ARLs are much longer than the normal theory benchmarks for symmetric non-normal distributions. The Three-Way chart is not robust to moderate or strong skewness.     

On the run length of a Shewhart chart for correlated data

We consider an extension of the classical Shewhart control chart to correlated data which was introduced by Vasilopoulos/Stamboulis (1978). Inequalities for the moments of the run length are given under weak conditions. It is proved analytically that the average run length (ARL) in the in-control state of the correlated process is larger than that in the case of independent variables. The exact ARL is calculated for exchangeable normal variables and autoregressive processes (AR). Moreover, we compare this chart with residual charts. Especially, in the case of an AR(1)—process with positive coefficient, it turns out that the out-of-control ARL of the modified Shewhart chart is smaller than that of the Shewhart chart for the residuals.     

The Robustness of the MaxEWMA Chart to Non-normality

Abstract

The MaxEWMA chart has recently been introduced as an improvement over the standard EWMA chart for detecting changes in the mean and/or standard deviation of a normally distributed process. Although this chart was originally developed for normally distributed process data, its robustness to violations of the normality assumption is the central theme of this study. For data distributions with heavy tails or displaying strong skewness, the in-control average run lengths (ARLs) for the MaxEWMA chart are shown to be significantly shorter than expected. On the other hand, out-of-control ARLs are comparable to normal theory values for a variety of symmetric non-normal distributions. The MaxEWMA chart is not robust to skewness.     

An ARL-unbiased design of time-between-events control charts with runs rules

In this paper, we consider incorporating the runs rules into the cumulative quantity control (CQC) chart for monitoring time-between-events data. We propose a simple and effective procedure to design a CQC chart coupled with runs rules that can yield average run length (ARL)-unbiased performance and meet the required in-control ARL. The proposed design involves determining a relation between the upper side and lower side false alarm probabilities. A Markov chain approach is used to evaluate the ARL performance of various control schemes studied in this paper. An extensive numerical comparison shows that the proposed design approach can result in a significant reduction in ARL for detecting increases in the occurrence rate of the event in comparison with the basic CQC charts.     

Detecting mean increases in Poisson INAR(1) processes with EWMA control charts

Processes of serially dependent Poisson counts are commonly observed in real-world applications and can often be modeled by the first-order integer-valued autoregressive (INAR) model. For detecting positive shifts in the mean of a Poisson INAR(1) process, we propose the one-sided s exponentially weighted moving average (EWMA) control chart, which is based on a new type of rounding operation. The s-EWMA chart allows computing average run length (ARLs) exactly and efficiently with a Markov chain approach. Using an implementation of this procedure for ARL computation, the s-EWMA chart is easily designed, which is demonstrated with a real-data example. Based on an extensive study of ARLs, the out-of-control performance of the chart is analyzed and compared with that of a c chart and a one-sided cumulative sum (CUSUM) chart. We also investigate the robustness of the chart against departures from the assumed Poisson marginal distribution.