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 Weibull Control Charts Using Multiple Linear Regression

Because in Weibull analysis, the key variable to be monitored is the lower reliability index (R(t)), and because the R(t) index is completely determined by both the lower scale parameter (η) and the lower shape parameter (β), then based on the direct relationships between η and β with the log‐mean (μ_x) and the log‐standard deviation (σ_x) of the analyzed lifetime data, a pair of control charts to monitor a Weibull process is proposed. Moreover, because the fact that in Weibull analysis, right censored data is common, and because it gives uncertainty to the estimated Weibull parameters, then in the proposed charts, μ_x and σ_x are estimated of the conditional expected times of the related Weibull family. After that both, μ_x and σ_x are used to monitor the Weibull process. In particular, μ_x was set as the lower control limit to monitor η, and σ_x was set as the upper control limit to monitor β. Numerical applications are used to show how the charts work. Copyright © 2016 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.     

Conditional Control Charts for Monitoring the Weibull Shape Parameter under Progressively Type II Censored Data

In this paper, (i) we propose new conditional Shewhart‐type control charts for monitoring the shape parameter of the Weibull distribution under a progressively type II censoring strategy, and (ii) we generalize the control charts proposed by Guo and Wang¹ for the progressively type II censoring case. We provide a comparison between these control charts in terms of the out‐of‐control average run length obtained by simulation for both the known and unknown parameter cases. A real example consisting of data from breaking stress of carbon fibers is also presented for illustration and comparison of the proposed control charts. Copyright © 2014 John Wiley & Sons, Ltd.     

Control Charts For Monitoring The Weibull Shape Parameter Based On Type‐II Censored Sample

In this paper, a new statistic is proposed to monitor the Weibull shape parameter when the sample is type II censored. The one‐sided and two‐sided average run length‐unbiased control charts are derived based on the new monitoring statistic. The control limits of the proposed control charts depend on the sample size, the failure number and the false alarm rate. Using Monte Carlo simulation, the performance of the proposed control charts is studied and compared with the range‐based charts proposed by Pascual and Li (2012), which is equivalent to the proposed control charts when r = 2. The simulation results show that the proposed control charts perform better than the ones of Pascual and Li (2012). This paper also evaluates the effects of parameter estimation on the proposed control charts. Finally, an example is used to illustrate the proposed control charts. Copyright © 2012 John Wiley & Sons, Ltd.     

Shewhart Control Charts for Monitoring Reliability with Weibull Lifetimes

In this paper, we present Shewhart‐type and S² control charts for monitoring individual or joint shifts in the scale and shape parameters of a Weibull distributed process. The advantage of this method is its ease of use and flexibility for the case where the process distribution is Weibull, although the method can be applied to any distribution. We illustrate the performance of our method through simulation and the application through the use of an actual data set. Our results indicate that and S² control charts perform well in detecting shifts in the scale and shape parameters. We also provide a guide that would enable a user to interpret out‐of‐control signals. Copyright © 2014 John Wiley & Sons, Ltd.     

Methods for estimating Weibull parameters for brittle materials

A Monte Carlo simulation is used to obtain the statistical properties of the Weibull parameters estimated by the linear regression, weighted linear regression, maximum likelihood and moments methods, respectively. Results reveal that the estimated Weibull modulus is always biased, which has a much lower accuracy than the scale parameter. The mean square error is adopted as a criterion for the comparison of the estimation methods. It is shown that both the probability estimators and the weight factors have great effects on the estimation precision of the Weibull modulus. The weighted linear regression with a weight factor of W _i =3.3P _i −27.5[1−(1−P _i )^0.025] and a probability estimator of P _i =(i−0.3)/(n+0.4) gives the most accurate estimation for sample sizes of 9–52. The maximum likelihood method recommended for any sample size by previous authors, comes first only for sample sizes larger than or equal to 53; furthermore, it is less conservative than the regression methods, and hence results in a lower safety in reliability predictions.     

Moving range charts for monitoring the Weibull shape parameter with single observation samples

This paper proposes an approach to monitor shifts in the Weibull shape parameter bfβ via control charts based on the moving range of single‐point samples from a smallest extreme value distribution. The average run length (ARL) of the proposed charts are computed using Fredholm integral equations of the second kind. The derived control limits for one‐sided and two‐sided control charts are unbiased in the sense that the ARL when β has shifted is shorter than the desired stable‐process ARL. These control limits depend only on the desired stable‐process ARL and the stable value of β. The paper also discusses the sample size requirements for Phase I so that the run length distributions are similar under standards‐given scenario (β is given) and retrospective scenario (β is estimated from past data). The proposed methods are then applied to data on the breaking strengths of carbon fibers. The results suggest that one‐sided control charts can detect small shifts in β sooner than two‐sided charts. Copyright © 2011 John Wiley & Sons, Ltd.     

Statistical monitoring of nonlinear product and process quality profiles

In many quality control applications, use of a single (or several distinct) quality characteristic(s) is insufficient to characterize the quality of a produced item. In an increasing number of cases, a response curve (profile) is required. Such profiles can frequently be modeled using linear or nonlinear regression models. In recent research others have developed multivariate T² control charts and other methods for monitoring the coefficients in a simple linear regression model of a profile. However, little work has been done to address the monitoring of profiles that can be represented by a parametric nonlinear regression model. Here we extend the use of the T² control chart to monitor the coefficients resulting from a parametric nonlinear regression model fit to profile data. We give three general approaches to the formulation of the T² statistics and determination of the associated upper control limits for Phase I applications. We also consider the use of non‐parametric regression methods and the use of metrics to measure deviations from a baseline profile. These approaches are illustrated using the vertical board density profile data presented in Walker and Wright (Comparing curves using additive models. Journal of Quality Technology 2002; 34:118–129). Copyright © 2007 John Wiley & Sons, Ltd.     

The joint economic‐statistical design of X and R charts for nonnormal data

We consider the joint economic‐statistical design of X and R control charts under the assumption that the quality measurement and the in‐control time have Johnson and Weibull distributions. The Johnson distribution is general in that it can be made to fit all possible values of skewness and kurtosis. The four parameters—the sample size n, time h between successive samples, and the control factors k₁ and k₂ for the X and R charts—are determined so that the mean hourly loss‐cost is minimized under constraints on the Type I and II error probabilities. We have generalized the Costa model to accommodate the Johnson and Weibull distributions. Sensitivity to nonnormality, shift, and Weibull scale parameter is considered in our analysis. Our sensitivity analysis shows that the optimal design parameters are sensitive to nonnormality. Comparisons of the fully economic and economic‐statistical designs are given. Copyright © 2010 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.     

An Individuals Generalized Likelihood Ratio Control Chart for Monitoring Linear Profiles

This paper investigates a generalized likelihood ratio (GLR) control chart for detecting sustained changes in the parameters of linear profiles when individual observations are sampled. The control charts usually used for monitoring linear profiles are based on taking a sample of n observations at each sampling time point, where n is large enough that a regression model can be fitted at each sampling point using these n observations. For this sampling scenario, it has been shown that a GLR control chart has many advantages over other control chart schemes in terms of convenience of design, fast detection of process changes, and useful diagnostic aids. However, in many applications, it may not be convenient or possible to take a sample larger than n = 1. Therefore, it is desirable to develop some control chart to monitor profile data with individual observations (n = 1) at each sampling point. In this paper, we consider a GLR control chart based on individual observations and show that it has certain advantages compared with the GLR chart based on groups of observations. An important advantage of GLR control charts is that the only design parameter that needs to be specified in order to use a GLR chart is the control limit, and here, control limits for linear profiles up to eight regression coefficients are provided for convenient use by practitioners. Copyright © 2013 John Wiley & Sons, Ltd.     

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 type‐2 censored reliability data in multistage processes

In this paper, we propose control charts to monitor the Weibull scale parameter of type‐2 censored reliability data in multistage processes. A cumulative sum control chart and 2 exponentially weighted moving average control charts based on conditional expected values are devised to detect decreases in the mean level of reliability‐related quality characteristic. The proposed control schemes are based on standard smallest extreme value distributions derived from Weibull processes to effectively account for the cascade property, which is the main characteristic of multistage processes. Subsequently, simulation study is conducted to evaluate the performance of the control charts using average run length criterion. Extra quadratic loss, performance comparison index, and relative average run length are also used to compare the detect ability of our proposed monitoring procedures. Moreover, sensitivity analysis is done to study the impact of failure number in the sample size and to investigate the robustness of the proposed monitoring procedures against the shift in the previous stage. Finally, a real case study in a glass bottle–making company is investigated to illustrate the performance of the competing control charts. The results reveal the superiority of the cumulative sum control chart.     

Lower Percentile Estimation of Accelerated Life Tests with Nonconstant Scale Parameter

Lower percentiles of product lifetime are useful for engineers to understand product failure, and avoiding costly product failure claims. This paper proposes a percentile re‐parameterization model to help reliability engineers obtain a better lower percentile estimation of accelerated life tests under Weibull distribution. A log transformation is made with the Weibull distribution to a smallest extreme value distribution. The location parameter of the smallest extreme value distribution is re‐parameterized by a particular 100pth percentile, and the scale parameter is assumed to be nonconstant. Maximum likelihood estimates of the model parameters are derived. The confidence intervals of the percentiles are constructed based on the parametric and nonparametric bootstrap method. An illustrative example and a simulation study are presented to show the appropriateness of the method. The simulation results show that the re‐parameterization model performs better compared with the traditional model in the estimation of lower percentiles, in terms of Relative Bias and Relative Root Mean Squared Error. Copyright © 2017 John Wiley & Sons, Ltd.     

An exponentially weighted moving average chart based on likelihood‐ratio test for monitoring Weibull mean and variance with subgroups

Monitoring changes in the Weibull mean and variance simultaneously is of interest in quality control. The mean and variance of a Weibull process are determined by its shape and scale parameters. Most studies are focused on monitoring the Weibull scale parameter with fixed shape parameter or the Weibull shape parameter with fixed scale parameter. In this paper, we propose an exponentially weighted moving average chart based on the likelihood‐ratio test and an inverse error function called ELR chart to monitor changes in the Weibull mean and variance simultaneously. The simulation approach is used to derive the average run length. We compare our proposed chart with other existing control charts for 3 cases, including scale parameter changes with fixed shape parameter, shape parameter changes with fixed scale parameter, and both parameters changes. The results show that the ELR chart outperforms the other control charts in terms of average run length in most cases. Two numerical examples are used to illustrate the applications of the proposed control chart.     

Effect of estimation under nonnormality on the phase II performance of linear profile monitoring approaches

The number of studies about control charts proposed to monitor profiles, where the quality of a process/product is expressed as function of response and explanatory variable(s), has been increasing in recent years. However, most authors assume that the in‐control parameter values are known in phase II analysis and the error terms are normally distributed. These assumptions are rarely satisfied in practice. In this study, the performance of EWMA‐R, EWMA‐3, and EWMA‐3(d²) methods for monitoring simple linear profiles is examined via simulation where the in‐control parameters are estimated and innovations have a Student's t distribution or gamma distribution. Instead of the average run length (ARL) and the standard deviation of run length, we used average and standard deviation of the ARL as performance measures in order to capture the sampling variation among different practitioners. It is seen that the estimation effect becomes more severe when the number of phase I profiles used in estimation decreases, as expected, and as the distribution deviates from normality to a greater extent. Besides, although the average ARL values get closer to the desired values as the amount of phase I data increases, their standard deviations remain far away from the acceptable level indicating a high practitioner‐to‐practitioner variability.     

A Shewhart‐type Control Scheme to Monitor Weibull Data without Subgrouping

This study proposes a Shewhart control scheme to simultaneously monitor the shape parameter and the scale parameter of Weibull data without subgrouping. The proposed control scheme comprises two charts: the X chart and the moving‐ratio (MRa) chart. The X chart plots individual observations to detect the shift of the scale parameter by assuming that the shape parameter is in‐control. In contrast, the MRa chart plots moving ratios, the minimum of two consecutive Weibull data divided by the maximum of them, to detect the shift of the shape parameter. This study models the transition process of the proposed control scheme as a Markov chain to calculate two performance measures: the average number of observations to signal and the average run length. Performance analysis shows that the proposed control scheme is effective in detecting the shift of parameters, especially for the downward shift of the shape parameter. Finally, the implementation of the proposed control scheme is illustrated in two skewed data sets. Copyright © 2013 John Wiley & Sons, Ltd.     

Control Charts for Monitoring Field Failure Data

One responsibility of the reliability engineer is to monitor failure trends for fielded units to confirm that pre‐production life testing results remain valid. This research suggests an approach that is computationally simple and can be used with a small number of failures per observation period. The approach is based on converting failure time data from fielded units to normal distribution data, using simple logarithmic or power transformations. Appropriate normalizing transformations for the classic life distributions (exponential, lognormal, and Weibull) are identified from the literature. Samples of size 500 field failure times are generated for seven different lifetime distributions (normal, lognormal, exponential, and four Weibulls of various shapes). Various control charts are then tested under three sampling schemes (individual, fixed, and random) and three system reliability degradations (large step, small step, and linear decrease in mean time between failures (MTBF)). The results of these tests are converted to performance measures of time to first out‐of‐control signal and persistence of signal after out‐of‐control status begins. Three of the well‐known Western Electric sensitizing rules are used to recognize the assignable cause signals. Based on this testing, the ―X‐chart with fixed sample size is the best overall for field failure monitoring, although the individual chart was better for the transformed exponential and another highly‐skewed Weibull. As expected, the linear decrease in MTBF is the most difficult change for any of the charts to detect. Copyright © 2005 John Wiley & Sons, Ltd.     

Monitoring simple linear profiles in the presence of generalized autoregressive conditional heteroscedasticity

In this paper, monitoring of simple linear profiles is investigated in the presence of nonequality of variances or heteroscedasticity, ie, generalized autoregressive conditional heteroscedasticity. In this condition, using of the common methods regardless of the heteroscedasticity leads to the fault interpretations. We consider a simple linear profile and assume that there is a generalized autoregressive conditional heteroscedasticity (GARCH) (1,1) model within the profiles. Here, we particularly focus on Phase II monitoring of simple linear regression. We studied the generalized autoregressive conditional heteroscedasticity effect, briefly GARCH effect, on the average run length criterion. As the remedial measures, the weighted least squares method to estimate the regression parameters and the heteroscedasticity‐consistent approaches to estimate the covariance matrix of regression parameters, are used to extract the GARCH effect. Two control chart methods namely T² and exponentially weighted moving average 3 are discussed to monitor the simple linear profiles. Their performances are evaluated by using the average run length criterion. Finally, a real case from an industry field is studied.