Monitoring the Weibull shape parameter by control charts for the sample range

In this paper, we propose control charts for monitoring changes in the Weibull shape parameter β. These charts are based on the range of a random sample from the smallest extreme value distribution. The control chart limits depend only on the sample size, the desired stable average run length (ARL), and the stable value of β. We derive control limits for both one‐ and two‐sided control charts. They are unbiased with respect to the ARL. We discuss sample size requirements if the stable value of βis estimated from past data. The proposed method is applied to data on the breaking strengths of carbon fibers. We recommend one‐sided charts for detecting specific changes in βbecause they are expected to signal out‐of‐control sooner than the two‐sided charts. Copyright © 2010 John Wiley & Sons, Ltd.     

Monitoring the Weibull Shape Parameter by Control Charts for the Sample Range of Type II Censored Data

In this paper, we propose control charts to monitor the Weibull shape parameter β under type II (failure) censoring. This chart scheme is based on the sample ranges of smallest extreme value distributions derived from Weibull processes. We suggest one‐sided (high‐side or low‐side) and two‐sided charts, which are unbiased with respect to the average run length (ARL). The control limits for all types of charts depend on the sample size, the number of failures c under type II censoring, the desired stable‐process ARL, and the stable‐process value of β. This article also considers sample size requirements for phase I in retrospective charts. We investigate the effect of c on the out‐of‐control ARL. We discuss a simple approach to choosing c by cost minimization. The proposed schemes are then applied to data on the breaking strengths of carbon fibers. Copyright © 2011 John Wiley & Sons, Ltd.     

Monitoring the Weibull Shape Parameter with Type II Censored Data

In this article, we introduce a method for monitoring the Weibull shape parameter β with type II (failure) censored data. The control limits depend on the sample size, the number of censored observations, the target average run length, and the stable value of β. The method assumes that the scale parameter α is constant during each sampling period, which is true under rational subgrouping. The proposed method utilizes the relationship between Weibull and smallest extreme value distribution. We propose an unbiased estimator of σ = 1/β as the monitoring statistic. We derive the control limits for one‐sided and two‐sided charts for several stable process average run lengths. We discuss two schemes, namely, the control‐limits‐only scheme and the control‐limits‐with‐warning‐lines scheme. The stable process average run length performance of the proposed charts is studied and compared with those of other charts for monitoring β under similar assumptions. Copyright © 2014 John Wiley & Sons, Ltd.     

Conditional Control Charts for Weibull Quantiles Under Type II Censoring

In this article, we propose control charts for the quantiles of the Weibull distribution, for type II censored data, based on the distribution of a pivotal quantity conditioned on ancillary statistics. These control charts must be considered as alternatives to bootstrap type control charts. We derive an analytical form of the conditional distribution function of the monitored statistic and we use this function to propose ARL‐unbiased control limits. We further demonstrate that the proposed conditional chart have a general analytical form for the ARL that can be evaluated numerically without use of simulations and we also show that these charts perform at least as well as the bootstrap type ones. We finally apply the conditional charts to a dataset on the strength of carbon fibers to detect shifts in a specified Weibull quantile. Copyright © 2014 John Wiley & Sons, Ltd.     

Phase II control charts for monitoring dispersion when parameters are estimated

Shewhart control charts are among the most popular control charts used to monitor process dispersion. To base these control charts on the assumption of known in-control process parameters is often unrealistic. In practice, estimates are used to construct the control charts and this has substantial consequences for the in-control and out-of-control chart performance. The effects are especially severe when the number of Phase I subgroups used to estimate the unknown process dispersion is small. Typically, recommendations are to use around 30 subgroups of size 5 each.

?We derive and tabulate new corrected charting constants that should be used to construct the estimated probability limits of the Phase II Shewhart dispersion (e.g., range and standard deviation) control charts for a given number of Phase I subgroups, subgroup size and nominal in-control average run-length (ICARL). These control limits account for the effects of parameter estimation. Two approaches are used to find the new charting constants, a numerical and an analytic approach, which give similar results. It is seen that the corrected probability limits based charts achieve the desired nominal ICARL performance, but the out-of-control average run-length performance deteriorate when both the size of the shift and the number of Phase I subgroups are small. This is the price one must pay while accounting for the effects of parameter estimation so that the in-control performance is as advertised. An illustration using real-life data is provided along with a summary and recommendations.     

Optimal control limits for CCC charts in the presence of inspection errors

The control chart based on cumulative count of conforming (CCC) items between the occurrence of two non‐conforming ones, or the CCC chart, has been shown to be very useful for monitoring high‐quality processes. However, as in the implementation of other Shewhart‐type control charts, it is usually assumed that the inspection is free of error. This assumption may not be valid and this may have a significant impact on the interpretation of the control chart and the setting of control limits. This paper first investigates the effect of inspection errors and discusses the setting of control limits in such cases. Even if inspection errors are considered, the average time to alarm increases in the beginning when the process deteriorates. Since this is undesirable, the control limits in the presence of inspection errors should be set so as to maximize the average run length when the process is at the normal level. A procedure is presented for solving this problem. Copyright © 2003 John Wiley & Sons, Ltd.     

Control charts for traffic intensity monitoring of Markovian multiserver queues

A number of recent research studies have applied queueing theory as an approximate modeling tool to mathematically describe industrial systems, which include manufacturing, distribution, and service, for instance. Among the main observable characteristics in queues, the number of users in the system can be controlled to keep waiting times as minimal as possible. The design of efficient control charts is an attempt to monitor and control such systems. Control charts are proposed to monitor infinite queues with Markovian arrivals, exponential service times, and s identical parallel servers. The proposed charts monitor traffic intensities, which are the ratio between the arrival rate and the service rate, estimated through the number of users in the queueing system at random epochs. The effectiveness and efficiency of the proposed approaches in terms of the average run lengths are established by a comprehensive set of Monte Carlo simulations.     

New EWMA control charts for monitoring process dispersion using auxiliary information

The exponentially weighted moving average (EWMA) control chart is a well‐known statistical process monitoring tool because of its exceptional pace in catching infrequent variations in the process parameter(s). In this paper, we propose new EWMA charts using the auxiliary information for efficiently monitoring the process dispersion, named the auxiliary‐information–based (AIB) EWMA (AIB‐EWMA) charts. These AIB‐EWMA charts are based on the regression estimators that require information on the quality characteristic under study as well as on any related auxiliary characteristic. Extensive Monte Carlo simulation are used to compute and study the run length profiles of the AIB‐EWMA charts. The proposed charts are comprehensively compared with a recent powerful EWMA chart—which has been shown to be better than the existing EWMA charts—and an existing AIB‐Shewhart chart. It turns out that the proposed charts perform uniformly better than the existing charts. An illustrative example is also given to explain the implementation and working of the AIB‐EWMA charts.     

Self-information-based weighted CUSUM charts for monitoring Poisson count data with varying sample sizes

In many applications, the Poisson count data with varying sample sizes are monitored using statistical process control charts. Among these applications, the weighted CUSUM charts are developed to deal with the effect of the varying sample sizes. However, some of them use limited information of the sample size or the count data while assigning the weights. To gain more information of the process, the self-information weight functions are developed based on both the sample size and the observed count data. Then, the weighted CUSUM charts are proposed with the self-information-based weight. Simulation studies show the self-information-based weighted CUSUM charts perform better than the benchmark methods in detecting small shifts. Moreover, the performance of proposed method with estimated parameters is investigated via simulation. Finally, an example is given to illustrate the application of the proposed weighted CUSUM charts.     

Control charts for monitoring a Poisson hidden Markov process

Monitoring stochastic processes with control charts is the main field of application in statistical process control. For a Poisson hidden Markov model (HMM) as the underlying process, we investigate a Shewhart individuals chart, an ordinary Cumulative Sum (CUSUM) chart, and two different types of log-likelihood ratio (log-LR) CUSUM charts. We evaluate and compare the charts' performance by their average run length, computed either by utilizing the Markov chain approach or by simulations. Our performance evaluation includes various out-of-control scenarios as well as different levels of dependence within the HMM. It turns out that the ordinary CUSUM chart shows the best overall performance, whereas the other charts' performance strongly depend on the particular out-of-control scenario and autocorrelation level, respectively. For illustration, we apply the HMM and the considered charts to a data set about weekly sales counts.     

Control charts for monitoring the mean and percentiles of Weibull processes with variance components

In this article, we study Shewhart and exponentially weighted moving average control charts for monitoring the mean or, equivalently, the percentiles of a Weibull process when additional sources of variation, also known as variance components, are present. We adopt a frailty model to describe the monitored process. We derive analytical properties for this model and use them to develop control charts. We consider charts for the sample mean and exponentially weighted moving averages. We compare their average run length performances to their traditional counterparts when they do not account for variance components.     

Percentile‐based control chart design with an application to Shewhart X̅ and S2 control charts

We present a method to design control charts such that in‐control and out‐of‐control run lengths are guaranteed with prespecified probabilities. We call this method the percentile‐based approach to control chart design. This method is an improvement over the classical and popular statistical design approach employing constraints on in‐control and out‐of‐control average run lengths since we can ensure with prespecified probability that the actual in‐control run length exceeds a desired magnitude. Similarly, we can ensure that the out‐of‐control run length is less than a desired magnitude with prespecified probability. Some numerical examples illustrate the efficacy of this design method.     

On a class of mixed EWMA-CUSUM median control charts for process monitoring

Monitoring of any manufacturing, production, or industrial process can be controlled and improved by removing these special cause of variations using control charts. Shewhart-type control charts are effective to control a large amount of special variations, whereas, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts detect small and moderate variations efficiently in the process parameters. Monitoring of location parameter can be done with mean control charts under the assumption that the parameters are known or correctly estimated from in-control samples and data are free from outliers (but in practice data occasionally have outliers). In this study, we have proposed generalized mixed EWMA-CUSUM median control charts structures for known and unknown parameters based on auxiliary variables for detecting shifts in process location parameter. The proposed charts are compared with the corresponding charts for the mean, based on contaminated and uncontaminated data. Different performance measures are used to evaluate the performance of proposed control charts and revealed through results that the median-based charts are more sensitive to detect a shift in process location parameter in the presence of outliers. An illustrative example using real data is also shown for practical consideration.     

Progressive Mean as a Special Case of Exponentially Weighted Moving Average

In the category of memory‐type control charts, progressive mean control chart was proposed recently, for monitoring the process location. Here we show, through the derivation, that the plotting statistic for the progressive mean control chart becomes a special case of exponentially weighted moving average when the sensitivity parameter becomes reciprocal of the sample number. 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.     

Memory‐type multivariate control charts with auxiliary information for process mean

In statistical process control, it is a common practice to increase the sensitivity of a control chart with the help of an efficient estimator of the underlying process parameter. In this paper, we consider an efficient estimator that requires information on several study variables along with one or more auxiliary variables when estimating the mean of a multivariate normally distributed process. Using this auxiliary‐information‐based (AIB) process mean estimator, we propose new multivariate EWMA (MEWMA), double MEWMA (DMEWMA), and multivariate CUSUM (MCUSUM) charts for monitoring the process mean, denoted by the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts, respectively. The run length characteristics of the proposed multivariate charts are computed using Monte Carlo simulations. The proposed charts are compared with their existing counterparts in terms of the run length characteristics. It turns out that the AIB‐MEWMA, AIB‐DMEWMA, and AIB‐MCUSUM charts are uniformly and substantially better than the MEWMA, DMEWMA, and MCUSUM charts, respectively, when detecting different shifts in the process mean. A real dataset is considered to explain the implementation of the proposed and existing multivariate control charts.     

Enhanced EWMA charts for monitoring the process coefficient of variation

The coefficient of variation (CV) is an important quality characteristic when the process variance is a function of the process mean for a production process. In this paper, we develop an auxiliary information–based (AIB) estimator for estimating the squared CV, along with its approximated mean and variance. This estimator is then used to devise new one-sided EWMA charts for monitoring the increases or decreases in the squared CV of a normal process, named the AIB-EWMA CV charts. In addition, the sensitivities of these control charts are also enhanced with the fast initial response feature. The Monte Carlo simulation method is used to compute the run length characteristics of the proposed CV charts. Based on detailed run length comparisons, it is found that the proposed AIB-EWMA CV charts are uniformly and substantially better than the existing EWMA CV charts when detecting different kinds of upward/downward shifts in the squared CV. The proposed charts are also applied to a real dataset to support the proposed theory.     

New CUSUM and Shewhart-CUSUM charts for monitoring the process mean

The CUmulative SUM (CUSUM) charts have sensitive nature against small and moderate shifts that occur in the process parameter(s). In this article, we propose the CUSUM and combined Shewhart-CUSUM charts for monitoring the process mean using the best linear unbiased estimator of the location parameter based on ordered double-ranked set sampling (RSS) scheme, where the CUSUM chart refers to the Crosier's CUSUM chart. The run-length characteristics of the proposed CUSUM charts are computed with the Monte Carlo simulations. The run-length profiles of the proposed CUSUM charts are compared with those of the CUSUM charts based on simple random sampling, RSS, and ordered RSS schemes. It is found that the proposed CUSUM charts uniformly outperform their existing counterparts when detecting all different kinds of shifts in the process mean. A real data set is also considered to explain the implementation of the proposed CUSUM charts.