Reliability Analysis of Flexible Manufacturing Cells Based on Triangular Fuzzy Number

In most of flexible production schemes, the flexible manufacturing cells (FMCs) are more economical and feasible. So, an important task is to establish the correct reliability analysis model for the FMCs. However, with the increasing of system complexity, some reliability analysis modes can hardly describe the actual situation. Besides, due to the lack of test-data and field-data during the design stage of FMC' system, the reliability modeling will be more complicated. In order to deal with the deficient data and the uncertainty occurred from analysis and judgment, this article analyzes the reliability of FMCs system through the method of fuzzy fault tree, which is based on triangular fuzzy membership. At last, a practical example is illustrated. The reliability analysis model indicates that it can offer a diagnostic tool for FMCs system and improve the efficiency of operation and production in FMCs system.     

A Note on the Prior Distributions of Weibull Parameters for the Reliability Function

In Bayesian Inference it is often desirable to have a posterior density reflecting mainly the information from sample data. To achieve this purpose it is important to employ prior densities which add little information to the sample. We have in the literature many such prior densities, for example, Jeffreys (1967 Jeffreys , H. ( 1967 ). Theory of Probability , 3rd rev. ed. . London : Oxford University Press . [Google Scholar]), Lindley (1956 Lindley , D. V. ( 1956 ). On a measure of the information provided by an experiment . Ann. Mathemat. Statist. 27 : 986 – 1005 .[Crossref] , [Google Scholar]); (1961 Lindley , D. V. ( 1961 ). The use of prior probability distributions in statistical inference and decisions . In: Neyman , J. , ed. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability . Vol. 1. Berkeley : University of California Press , pp. 453 – 468 . [Google Scholar]), Hartigan (1964 Hartigan , J. ( 1964 ). Invariant priors distributions . Ann. Mathemat. Statist. 35 : 836 – 845 .[Crossref] , [Google Scholar]), Bernardo (1979 Bernardo , J. M. ( 1979 ). Reference posterior distributions for Bayesian inference . J. Roy. Statist. Soc. 41 ( 2 ): 113 – 147 . [Google Scholar]), Zellner (1984 Zellner , A. ( 1984 ). Maximal Data Information Prior Distributions, Basic Issues in Econometrics . Chicago : University of Chicago Press . [Google Scholar]), Tibshirani (1989 Tibshirani , R. ( 1989 ). Noninformative priors for one parameters of many . Biometrika 76 : 604 – 608 .[Crossref], [Web of Science ®] , [Google Scholar]), etc. In the present article, we compare the posterior densities of the reliability function by using Jeffreys, the maximal data information (Zellner, 1984 Zellner , A. ( 1984 ). Maximal Data Information Prior Distributions, Basic Issues in Econometrics . Chicago : University of Chicago Press . [Google Scholar]), Tibshirani's, and reference priors for the reliability function R(t) in a Weibull distribution.     

Some representation theorems in terms of mean residual lifetime and mean inactivity time of coherent systems

Abstract

In this paper, we consider a k-out-of-n system consisting of n identical components with independent lifetimes. We show that when the underlying distribution function F(t) is absolutely continuous, then it can be univocally determined by some particular mean residual lives or mean inactivity times of the system. It is then shown that these results may be extended to coherent (or mixed) systems.     

Numerical solution to stochastic partial differential equation with variable repair rate

ABSTRACT

In this paper, a numerical solution technique to stochastic partial differential equations in reliability engineering is presented. The method is based upon finite difference discretization of the governing equations for the Markovian reliability model. In realistic situations, the repair rates and failure rates of engineering system are variable. Such variable repair and failure rates are difficult to account in reliability modeling. The novelty in this work is to present a numerical method to easily take into consideration such variables and give an accurate prediction of reliability measures of engineering systems.     

Optimal preventive policies for imperfect maintenance models with replacement first and last

ABSTRACT

This paper proposes preventive replacement policies for an operating system which may continuously works for N jobs with random working times and is imperfectly maintained upon failure. As a failure occurs, the system suffers one of the two types of failures based on some random mechanism: type-I (repairable or minor) failure is rectified by a minimal repair, or type-II (non repairable or catastrophic) failure is removed by a corrective replacement. A notation of preventive replacement last model is considered in which the system is replaced before any type-II failure at an operating time T or at number N of working times, whichever occurs last. Comparisons between such a preventive replacement last and the conventional replacement first are discussed in detail. For each model, the optimal schedule of preventive replacement that minimizes the mean cost rate is presented theoretically and determined numerically. Because the framework and analysis are general, the proposed models extend several existing results.     

Monotone Log-Odds Rate Distributions in Reliability Analysis

Abstract

Monotone failure rate models [Barlow Richard, E., Marshall, A. W., Proschan, Frank. (1963 Barlow Richard, E., Marshall, A. W. and Proschan, Frank. 1963. Properties of probability distributions with monotone failure rate. Annals of Mathematical Statistics, 34: 375–389.  [Google Scholar]). Properties of probability distributions with monotone failure rate. Annals of Mathematical Statistics 34:375–389, and Barlow Richard, E., Proschan, Frank. (1965 Barlow Richard, E. and Proschan, Frank. 1965. Mathematical Theory of Reliability New York: John Wiley.  [Google Scholar]). Mathematical Theory of Reliability. New York: John Wiley & Sons, Barlow Richard, E., Proschan, Frank. (1966a Barlow Richard, E. and Proschan, Frank. 1966a. Tolerance and confidence limits for classes of distributions based on failure rate. Annals of Mathematical Statistics, 37(6): 1593–1601.  [Google Scholar]). Tolerance and confidence limits for classes of distributions based on failure rate. Annals of Mathematical Statistics 37(6):1593–1601, Barlow Richard, E., Proschan, Frank. (1966b Barlow Richard, E. and Proschan, Frank. 1966b. Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics, 37(6): 1574–1592.  [Google Scholar]). Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics 37(6):1574–1592, Barlow Richard, E., Proschan, Frank. (1975 Barlow Richard, E. and Proschan, Frank. 1975. Statistical Theory of Reliability and Life Testing New York: Holt, Rinehart and Winston, Inc..  [Google Scholar]). Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston, Inc.] have become one of the most important models of failure time for reliability practitioners to consider and use. The above authors also developed models and bounds for monotone increasing failure rates (IFR) and for monotone decreasing failure rates (DFR). The IFR models and bounds appear to be especially useful for describing and bounding the hazard of aging. This article extends a new model for time to failure based onthe log odds rate [Zimmer William, J., Wang Yao, Pathak, P. K. (1998 Zimmer William, J., Wang, F. K. and Keats, J. Bert. 1998. The Burr XII distribution in reliability analysis. Journal of Quality Technology, 30(4): 386–394.  [Google Scholar]). Log-odds rate and monotone log-odds rate distributions. Journal of Quality Technology 30(4):376–385.] which is comparable to the model based on the failure rate. It is shown that in the case of increasing log odds rate (ILOR) in terms of log time (ln t), the model is less stringent than the IFR model for aging. The characterization of distributions of failure time by log odds rate is also derived. It is shown that the logistic distribution has the property of constant log odds rate over time and that the log logistic distribution has the property of constant log odds rate with respect to ln t. Some other properties of ILOR distributions are presented and bounds based on the relationship to the log logistic distribution are provided for distributions which are ILOR with respect to ln t. Motivational examples are provided. The ILOR bounds are compared to the more stringent bounds based on the IFR model. Bounds on system reliability are also provided for certain systems.     

Statistical Analysis of General Degradation Path Model and Failure Time Data with Multiple Failure Modes

ABSTRACT

This article considers degradation and failure time models with multiple failure modes which used to study the problem of longevity and aging in survival analysis and reliability. Degradation process is modeled using general nonparametric, nonlinear path models. Semi-parametric models for the intensities of the traumatic failures are used supposing that these intensities depend on degradation level. Semi-parametric estimators of various reliability characteristics are proposed and asymptotic properties of the estimators are obtained. The theoretical results are illustrated using simulated data.     

The M-Bayesian Credible Limits of the Reliability Derived from Binomial Distribution

This article introduces a new method, M-Bayesian credible limit method, to estimate reliability derived from binomial distribution, in the case of zero-failure data. Firstly, the definition of one-sided and two-sided M-Bayesian credible limits are provided, and moreover, formulas of one-sided and two-sided M-Bayesian credible limits are also provided. secondly, properties of one-sided and two-sided M-Bayesian credible limits are discussed, and we will see that the M-Bayesian credible limit method is superior to the corresponding classical confidence limit method. Finally, the new estimation method is applied to a numerical example. Through the example the efficiency and easiness of operation of this method are commended.     

A new class of generalized logistic distribution

The logistic distribution and the S-shaped pattern of its cumulative distribution and quantile functions have been extensively used in many different spheres affecting human life. By far, the most well-known application of logistic distribution is in the logistic regression that is used for modeling categorical response variables. The exponentiated-exponential logistic distribution, a generalization of the logistic distribution, is obtained using the technique proposed by Alzaatreh et al. (2013 Alzaatreh, A., C. Lee, and F. Famoye. 2013. A new method for generating families of continuous distribution. Metron. 71:63–79.[Crossref] , [Google Scholar]) of mixing two distributions, hereafter called the EEL distribution. This distribution subsumes various types of logistic distribution. The structural analysis of the distribution in this paper includes limiting behavior, quantiles, moments, mode, skewness, kurtosis, order statistics, the large sample distributions of the sample maximum and the sample minimum, and the distribution of the sample median. For illustrative purposes, a real-life data set is considered as an application of the EEL distribution.     

Proportional hazards model with quantile functions

Proportional Hazards Model (PHM) introduced by Cox (1972) is extensively studied in literature. In this paper, we study reliability properties of the PHM using quantile functions. Some special properties of the quantile function, which are not shared by distribution function are explored to study various properties of the PHM. We discuss ageing properties and stochastic orders for the PHM. The quantile-based dynamic cumulative Kullback-Leibler divergence of PHM is studied. Characterizations of some important quantile densities using PHM are also proved.