Monitoring reliability for a three-parameter Weibull distribution

Control charts are widely used to monitor production processes in the manufacturing industry and are also useful for monitoring reliability. A method to monitor reliability has recently been proposed when the distributions of inter-failure times are exponential and Weibull with known parameters. This method has also been extended to monitor the cumulative time elapsed between a fixed number of failures for the exponential distribution. In this paper, we consider a three-parameter Weibull distribution to model inter-failure times, use a robust estimation technique to estimate the unknown parameters, and extend the proposed method to monitor the cumulative time elapsed between r failures using the three-parameter Weibull distribution. Since the distribution of the sum of independent Weibull random variates is not known (except in specific cases with known parameters), we give two useful moment approximations to be able to apply their scheme. We show how effective the approximations are and the usefulness of the method in detecting a possible instability during production.     

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.     

An adaptive dimension reduction scheme for monitoring feedback‐controlled processes

Detecting dynamic mean shifts is particularly important in monitoring feedback‐controlled processes in which time‐varying shifts are usually observed. When multivariate control charts are being utilized, one way to improve performance is to reduce dimensions. However, it is difficult to identify and remove non‐informative variables statically in a process with dynamic shifts, as the contribution of each variable changes continuously over time. In this paper, we propose an adaptive dimension reduction scheme that aims to reduce dimensions of multivariate control charts through online variable evaluation and selection. The resulting chart is expected to keep only informative variables and hence maximize the sensitivity of control charts. Specifically, two sets of projection matrices are presented and dimension reduction is achieved via projecting process vectors into a low‐dimensional space. Although developed based on feedback‐controlled processes, the proposed scheme can be easily extended to monitor general multivariate applications. Copyright © 2008 John Wiley & Sons, Ltd.     

EFSIM: An integrated circuit early failure simulator

Early failures are the dominant concern as integrated circuit technology matures into consistently producing systems of high reliability. These failures are attributed to the presence of randomly occurring defects in elementary objects (contacts, vias, metal runs, gate oxides, bonds etc.) that result in extrinsic rather than intrinsic (wearout-related) mortality. A model relating system failure to failure at the elementary object level has been developed. Reliability is modelled as a function of circuit architecture, mask layout, material properties, life-test data, worst-case use-conditions and the processing environment. The effects of competing failure mechanisms, and the presence of redundant sub-systems are accounted for. Hierarchy is exploited in the analysis, allowing large scale designs to be simulated. Experimental validation of the modelling of oxide leakage related failure, based on a correlation between actual failures reported for a production integrated circuit and Monte Carlo simulations that incorporate wafer-level test results and process defect monitor data, is presented. The state of the art in IC reliability simulation is advanced in that a methodology that provides the capability to design-in reliability while accounting for early failures has been developed; applications include process qualification, design assessment and fabrication monitoring.     

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.     

A study of time‐between‐events control chart for the monitoring of regularly maintained systems

Owing to usage, environment and aging, the condition of a system deteriorates over time. Regular maintenance is often conducted to restore its condition and to prevent failures from occurring. In this kind of a situation, the process is considered to be stable, thus statistical process control charts can be used to monitor the process. The monitoring can help in making a decision on whether further maintenance is worthwhile or whether the system has deteriorated to a state where regular maintenance is no longer effective. When modeling a deteriorating system, lifetime distributions with increasing failure rate are more appropriate. However, for a regularly maintained system, the failure time distribution can be approximated by the exponential distribution with an average failure rate that depends on the maintenance interval. In this paper, we adopt a modification for a time‐between‐events control chart, i.e. the exponential chart for monitoring the failure process of a maintained Weibull distributed system. We study the effect of changes on the scale parameter of the Weibull distribution while the shape parameter remains at the same level on the sensitivity of the exponential chart. This paper illustrates an approach of integrating maintenance decision with statistical process monitoring methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Generalized likelihood ratio based risk-adjusted control chart for zero-inflated Poisson process

Nowadays, statistical process control has been widely used to monitor processes in various fields. To monitor processes with a large number of zero observations by control charts, the zero-inflated Poisson (ZIP) model has been adopted. Due to the heterogeneity of each sample in the process, several factors have been taken into account to predict values of two parameters in the ZIP model by risk adjustment. Instead of considering two parameters to be constant directly, risk-adjusted ZIP control charts can provide more reasonable monitoring results than traditional ones. However, existing methods ignored the interaction between parameters in the ZIP model, which leads to some risk-adjusted control charts unable to accurately estimate parameters to provide effective monitoring results. To address this problem, this paper presents a generalize likelihood ratio (GLR) based control chart to better monitor the risk-adjusted ZIP process with EWMA scheme, which can detect the random shift in both parameters efficiently. In the simulation study, the proposed control chart is compared with another two existing control charts and shows superior performance on detecting various types of shifts in parameters. Finally, the proposed control chart is applied to the Hong Kong influenza datasets and the flight delay datasets to illustrate its effectiveness and utility.     

Monitoring Two Dependent Process Steps Using Special Variable Sample Sizes and Sampling Intervals Cause‐Selecting Control Charts

Cause‐selecting control charts are effective statistical process control tools for monitoring multistage processes. In this article, an adaptive statistical process control scheme to monitor a process with two dependent steps is proposed. Two different policies based on a combination of two different sample sizes and sampling intervals are utilized. Adjusted average time to signal measure, calculated through Markov chain approach, is applied to evaluate performance of the proposed control scheme. Numerical results indicate that the proposed scheme has improved performance over the fixed sample sizes at fixed sampling intervals scheme. Finally, the optimal parameters of the proposed scheme with two different policies are recommended, and comparisons between the minimum adjusted average time to signal of the proposed charts and variable sample sizes and sampling intervals cause‐selecting control charts with three different sample sizes and sampling intervals are performed. It is shown that performance of the proposed scheme with four variable parameters is similar and even somewhat better than that of the scheme with six variable parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

Improving the Product Reliability in Multistage Manufacturing and Service Operations

Monitoring and improving the product reliability is of main concern in a large number of multistage manufacturing processes. The process output is commonly inspected under limited load conditions, and the tensile strength of reliability‐related quality characteristic is measured. This brings about censored observations that make the direct application of traditional control charts futile. The monitoring procedure becomes aggravated when the influence of variable competing risk is pronounced during the conducted test. To deal with this critical issue, we propose a regression‐adjusted cumulative sum (CUSUM) chart to effectively monitor a quality characteristic that may be right censored because of both fixed and variable competing risks. Moreover, two exponentially weighted moving average (EWMA) control charts on the basis of conditional expected values are devised to detect decreases in the tensile mean. The comparison of the three competing monitoring schemes confirms the superiority of the regression‐adjusted CUSUM procedure. Not only is the proposed control chart applicable to manufacturing processes with the aim of monitoring reliability‐related quality variables, it is also appropriate for monitoring similar quality measurements in service operations such as survivability measures in healthcare services. Copyright © 2011 John Wiley & Sons, Ltd.     

Phase I control charts for times between events

A count of the number of defects is often used to monitor the quality of a production process. When defects rarely occur in a process, it is often desirable to monitor the time between the occurrence of each defect rather than a count of the number of defects. An exponential distribution often provides a useful model of the time between defects. Phase I control charts for exponentially distributed processes are discussed. Methods for computing the control limits are given and the overall Type I error rates of these charts are evaluated. Copyright © 2002 John Wiley & Sons, Ltd.     

An economically designed, integrated quality and maintenance model using an adaptive Shewhart chart

This paper proposes a model for the economic design of a variable-parameter (Vp) Shewhart control chart used to monitor the mean in a process, where, apart from quality shifts, failures may also occur. Quality shifts result in poorer quality outcome, higher operational cost and higher failure rate. Thus, removal of such quality shifts, besides improving the quality of the outcome and reducing the quality cost, is also a preventive maintenance (PM) action since it reduces the probability of a failure and improves the equipment reliability. The proposed model allows the determination of the scheme parameters that minimize the total expected quality and maintenance cost of the procedure. The monitoring mechanism of the process employs an adaptive Vp-Shewhart control chart. To evaluate the effectiveness of the proposed model, its optimal expected cost is compared against the optimum cost of a fixed-parameter (Fp) chart.     

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.     

A two-stage neural network approach for process variance change detection and classification

Statistical process control charts (SPCC) have become one of the most commonly used tools for monitoring process variability in today's manufacturing environment. Meanwhile, neural networks have been gradually recommended as alternatives to SPCC due to their superior performances, especially in the case of monitoring process mean and unnatural patterns. Little attention has been given to the use of neural networks for monitoring the process variance. This paper describes a neural network approach to monitor process variance changes and to predict change-magnitudes. The performances of the proposed neural network monitoring scheme are compared to those of SPCC for a sample size of five and for individual observations. Simulation results show that the performance of the proposed method is comparable to that of SPCC in terms of average run lengths. In addition, the proposed neural network scheme has the capability to estimate the magnitude of the variance change by combining with a bootstrap resampling scheme. A robustness test is also applied to examine the performance of the proposed scheme for observations from a non-normal distribution.     

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.     

Using independent component analysis to monitor 2‐D geometric specifications

Functional data and profiles are characterized by complex relationships between a response and several predictor variables. Fortunately, statistical process control methods provide a solid ground for monitoring the stability of these relationships over time. This study focuses on the monitoring of 2‐dimensional geometric specifications. Although the existing approaches deploy regression models with spatial autoregressive error terms combined with control charts to monitor the parameters, they are designed based on some idealistic assumptions that can be easily violated in practice. In this paper, the independent component analysis (ICA) is used in combination with a statistical process control method as an alternative scheme for phase II monitoring of geometric profiles when non‐normality of the error term is present. The performance of this method is evaluated and compared with a regression‐ and PCA‐based approach through simulation of the average run length criterion. The results reveal that the proposed ICA‐based approach is robust against non‐normality in the in‐control analysis, and its out‐of‐control performance is on par with that of the PCA‐based method in case of normal and near‐normal error terms.     

Error Detection and Pattern Prediction Through Phase II Process Monitoring

The continuous monitoring of the machine is beneficial in improving its process reliability through reflected power function distribution. It is substantial for identifying and removing errors at the early stages of production that ultimately benefit the firms in cost-saving and quality improvement. The current study introduces control charts that help the manufacturing concerns to keep the production process in control. It presents an exponentially weighted moving average and extended exponentially weighted moving average and then compared their performance. The percentiles estimator and the modified maximum likelihood estimator are used to constructing the control charts. The findings suggest that an extended exponentially weighted moving average control chart based on the percentiles estimator performs better than exponentially weighted moving average control charts based on the percentiles estimator and modified maximum likelihood estimator. Further, these results will help the firms in the early detection of errors that enhance the process reliability of the telecommunications and financing industry.     

Quantitative effects of electrical and vibrational stresses on reliability

This article presents some details on how electrical and vibrational stresses exacerbate failures. This is one of a series of articles on similar subjects by this author. One electrical defect type is pin-holes in oxide layers in ICs. Electrical leakage current through a low insulation pin-hole could cause temperature rise and ultimately, when electrons of sufficient energy causing an avalanche, a thermal runaway condition could develop and maintain an electrical short. On the other hand electromigration could cause conductor opens. Aircraft data indicated that vibration related failures constituted more than 14 per cent of the total number of field failures. Fatigue failures can be directly related to S/N curves of stress to number of cycles to failure. Some measurements indicated that for a particular piece of equipment tested the time to failure varied inversely as the fourth power of vibrational acceleration; and failures of specific groups of component part types were sensitive to particular vibrational acceleration levels. Much information exists that gives quantitative measures on how stresses exacerbate failures. However, there is still a big gap in the relationship between the engineering fundamentals and the failures experienced. The author urges the readers to join force to develop a new reliability engineering foundation based on relationships of defects, failure mechanisms and stresses from which future reliability predictions and reliability analyses can be conducted.     

An Empirical Indicator of Product Appearance for Process Control

An innovative approach based on multiattribute utility theory and group processes was used to directly monitor and control cosmetic characteristics affecting product appearance to improve the quality of a product. The main advantage of this approach is its ability to provide a direct and reliable measure of visual appearance for products for which this is the critical quality characteristic. The paper is divided into two main parts. The first part briefly presents a systematic approach to develop an empirical indicator of product appearance, called the index of visual condition (IOVC). The second part presents a study of the applicability of the IOVC during process control to directly monitor and improve product appearance. This study was performed during industrial production of a ceramic product for a period of seven months. Several Shewhart charts were developed, including the first exploratory X-bar and R charts for product appearance (IOVC charts) and c-charts for number of defects. The performance of these charts was evaluated by their ability to detect special causes of variation and improve the product. The case study indicates that these two approaches complement each other. While the c-chart allows one to monitor the number of defects that impair product functionality, the IOVC chart brings a new level of capability, providing the ability to directly monitor product appearance considering defects that do not affect functionality. Both approaches were useful for process control and improvement.     

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.     

Enhancing the performance of the EWMA control chart for monitoring the process mean using auxiliary information

When using control charts to monitor manufacturing processes, the exponentially weighted moving average (EWMA) control chart is useful for detecting persistent shifts in the process parameter. This paper proposes enhancements to the applications of the EWMA control chart for those scenarios where the exact measurement of process units is difficult and expensive, but the visual ordering of the units can be done easily. The proposed charts use an auxiliary variable that is correlated with the process variable to provide efficient monitoring of shifts in the process mean and are formulated based on ranked set sampling (RSS) and median RSS schemes (MRSS). Simulation results showed that the proposed charting schemes are more efficient in detecting a shift in the process mean than the classical EWMA control chart and its modification. An example is provided to show the application of the proposed charts using a simulated benchmark process: the continuous stirred tank reactor (CSTR).