Ping Jiang

Reliability estimation in a Weibull lifetime distribution with zero-failure field data

The estimation of product reliability has attracted worldwide attention during the past several decades. The estimation procedure usually begins with parameter estimation based on test data. When there is no failure occurring in tests, traditional approaches like Maximum Likelihood Estimation (MLE) cannot be applied to estimate parameters. When product lifetime follows a Weibull distribution, to cope with this problem, we propose the modified MLE (MMLE) for estimating the parameters, based on the zero-failure data. In this paper, we also consider the prior reliability estimate from a similar product and make use of it by incorporating it with the MMLE to construct the shrinkage preliminary test estimator (SPTE). We present the calculation method of the shrinkage factor in the SPTE, by referring to the comparison of critical quality characteristics related to product reliability, between the current batch of products and the similar (or earlier version) batch of products. Restrictions for the shrinkage factor to ensure the performance of SPTE are also discussed. The example demonstrates that the proposed SPTE of the product reliability is an effective methodology to estimate the product reliability and improve the estimation performance of the MMLE, when only zero-failure data are available. Copyright

Real-time Reliability Evaluation for an Individual Product Based on Change-point Gamma and Wiener Process

Reliability evaluation based on degradation data has received significant attentions in recent years. However, existing works often assume that the degradation evolution over time is governed by a single stochastic process, which may not be realistic if change points exist. Here, for cases of degradation with change points, this paper attempts to capture the degradation process with a multi-phase degradation model and find the method to evaluate the real-time reliability of the product being monitored. Once new degradation information becomes available, the evaluation results are adaptively updated through the Bayesian method. In particular, for a two-stage degradation process of liquid coupling devices (LCDs), a model named as change-point gamma and Wiener process is developed, after which issues of real-time reliability evaluation and parameters’ estimation are addressed in detail. Finally, the proposed method is illustrated by a case study of LCDs, and the corresponding results indicate that trustful evaluation results depend on the fitting accuracy in cases of multi-phase degradation process. Copyright

Inference on the reliability of Weibull distribution with multiply Type-I censored data

In this paper, we focus on the reliability of Weibull distribution under multiply Type-I censoring, which is a general form of Type-I censoring. In multiply Type-I censoring in this study, all units in the life testing experiment are terminated at different times. Reliability estimation with the maximum likelihood estimate of Weibull parameters is conducted. With the delta method and Fisher information, we propose a confidence interval for reliability and compare it with the bias-corrected and accelerated bootstrap confidence interval. Furthermore, a scenario involving a few expert judgments of reliability is considered. A method is developed to generate extended estimations of reliability according to the original judgments and transform them to estimations of Weibull parameters. With Bayes theory and the Monte Carlo Markov Chain method, a posterior sample is obtained to compute the Bayes estimate and credible interval for reliability. Monte Carlo simulation demonstrates that the proposed confidence interval outperforms the bootstrap one. The Bayes estimate and credible interval for reliability are both satisfactory. Finally, a real example is analyzed to illustrate the application of the proposed methods.

Residual life estimation based on bivariate non-stationary gamma degradation process

Due to the growing importance in maintenance scheduling, the issue of residual life (RL) estimation for some high reliable products based on degradation data has been studied quite extensively. However, most of the existing work only deals with one-dimensional degradation data, which may not be realistic in some cases. Here, an adaptive method of RL estimation is developed based on two-dimensional degradation data. It is assumed that a product has two performance characteristics (PCs) and that the degradation of each PC over time is governed by a non-stationary gamma degradation process. From a practical consideration, it is further assumed that these two PCs are dependent and that their dependency can be characterized by a copula function. As the likelihood function in such a situation is complicated and computationally quite intensive, a two-stage method is used to estimate the unknown parameters of the model. Once new degradation information of the product being monitored becomes available, random effects are first updated by using the Bayesian method. Following that, the RL at current time is estimated accordingly. As the degradation data information accumulates, the RL can be re-estimated in an adaptive manner. Finally, a numerical example about fatigue cracks is presented in order to illustrate the proposed model and the developed inferential method.

Handling Uncertainties in Fault Tree Analysis by a Hybrid Probabilistic–Possibilistic Framework

Fault tree analysis is a method largely used in probabilistic risk assessment. Uncertainties should be properly handled in fault tree analyses to support a robust decision making. While many sources of uncertainties are considered, dependence uncertainties are not much explored. Such uncertainties can be labeled as ‘epistemic’ because of the way dependence is modeled. In practice, despite probability theory, alternative mathematical structures, including possibility theory and fuzzy set theory, for the representation of epistemic uncertainty can be used. In this article, a fuzzy β factor is considered to represent the failure dependence uncertainties among basic events. The relationship between β factor and system failure probability is analyzed to support the use of a hybrid probabilistic–possibilistic approach. As a result, a complete hybrid probabilistic–possibilistic framework is constructed. A case study of a high integrity pressure protection system is discussed. The results show that the proposed method provides decision makers a more accurate understanding of the system under analysis when failure dependencies are involved. Copyright

Schedule generation scheme for solving multi-mode resource availability cost problem by modified particle swarm optimization

The resource availability cost problem (RACP) (Möhring, Operations Research, 32:89–120, 1984) is commonly encountered in project scheduling. RACP aims to minimize the resource availability cost of a project by a given project deadline. In this study, RACP is extended from a single mode to a multi-mode called multi-mode RACP (MMRACP), which is more complicated than RACP but more convenient in practice. To solve MMRACP efficiently, forward activity list (FAL), a schedule generation scheme, is proposed. Heuristic algorithms are designed according to the characteristics of FAL to repair infeasible solutions and to improve the fitness of the solution. Modified particle swarm optimization (MPSO), which combines the advantages of particle swarm optimization and scatter search, is proposed to make the search for the best solution efficient. Computational experiments involving 180 instances are performed to validate the performance of the proposed algorithm. The results reveal that MPSO using FAL is a very effective method to solve MMRACP.

Weibull Failure Probability Estimation Based on Zero-Failure Data

Reliability testing is often carried out with small sample sizes and short duration because of increasing costs and the restriction of development time. Therefore, for highly reliable products, zero-failure data are often collected in such tests, which could not be used to evaluate reliability by traditional methods. To cope with this problem, the match distribution curve method was proposed by some researchers. The key step needed to exercise this method is to estimate the failure probability, which has yet to be solved in the case of the Weibull distribution. This paper presents a method to estimate the intervals of failure probability for the Weibull distribution by using the concavity or convexity and property of the distribution function. Furthermore, to use the method in practice, this paper proposes using the approximate value of the shape parameter determined by either engineering experience or by hypothesis testing through a p value. The estimation of the failure probability is thus calculated using a Bayesian approach. A numerical example is presented to validate the effectiveness and robustness of the method.

The determination method on products sample size under the condition of Bayesian sequential testing

In reliability engineering, especially in the context of reliability test design, it is important to determine how many product samples should be put into field test to evaluate product reliability level or verify whether the product reliability satisfy the predefined requirement in the development contract. In this paper, the determination method on sample size under the condition of Bayesian sequential testing is proposed. According to the sequential posterior odd testing (SPOT) method, the calculation on overall probability of decisions is firstly given out. Then the determination program of sample size used to test is presented by controlling probability level of correct decision-making. Finally, we make the determination of sample size on the mean of normal distribution under the condition of Bayesian sequential testing as an example to demonstrate the proposed determination process of sample size. The comparisons with classical methods are made to prove the effectiveness of the proposed method.

Residual life estimation fusing life data and expert information

A method fusing life data and expert information was proposed to estimate residual life of some kinds of single machine on satellite platform, which solved the problem that performance degradation information of these products could not be detected real-time and the life data collected were almost no-failure data. Firstly, expert information about reliability was transformed into prior distributions of parameters. Then life data were fused with expert information to determine posterior distribution of parameters. When ranges of parameters were determined, the residual life probability density distribution could be obtained. Besides, this paper took GPS receivers on the satellite platform as example, residual life estimation was given based on the method proposed in this paper. Lastly, a simulation method and a comparison with other methods were provided to prove the presented method effectiveness.

A robust optimization approach to postdisaster relief logistics planning under uncertainties

A stochastic model with bi-objective for post disaster relief logistics is proposed to decide the strategic planning on the mobilization levels of relief supplies, the initial deployment of vehicles and the transportation plans within the disaster region in a uncertain disaster environment. The robust optimization approach is introduced to cope with uncertainties and the robust counterpart of the proposed stochastic model is deduced. A lexicographic approach is utilized to convert the bi-objective robust model into two sequential single objective robust models. Numerical experiments show that the model can help post-disaster managers to determine the initial deployment of emergency resources, and the numerical results can express the tradeoff between optimization and robustness.