FAILURE DISTRIBUTION AT COMPONENT LEVEL

Optimal maintenance policies for systems subject to a Markovian operating environment

A new three-parameter generalized distribution, namely, half-logistic generalized Weibull (HLGW) distribution, is proposed. The proposed distribution exhibits increasing, decreasing, bathtub-shaped, unimodal, and decreasing-increasing-decreasing hazard rates. The distribution is a compound distribution of type I half-logistic-G and Dimitrakopoulou distribution. The new model includes half-logistic Weibull distribution, half-logistic exponential distribution, and half-logistic Nadarajah-Haghighi distribution as submodels. Some distributional properties of the new model are investigated which include the density function shapes and the failure rate function, raw moments, moment generating function, order statistics, L-moments, and quantile function. The parameters involved in the model are estimated using the method of maximum likelihood estimation. The asymptotic distribution of the estimators is also investigated via Fisher’s information matrix. The likelihood ratio (LR) test is used to compare the HLGW distribution with its submodels. Some applications of the proposed distribution using real data sets are included to examine the usefulness of the distribution.

<div class="htmlview paragraph">The failure time of automobile and its components is typical random variable. In studying reliability automobile and its components some simple theoretical statistical models, such as Normal distribution, Lognormal distribution and Weibull's distribution, index distribution etc., are used to describe automobile reliability. These distributions can not often represent actual datum well, then more complex theoretical statistical model is needed. If using complex statistical theoretical model the more parameters must be determined. Methods of moment, Maximum likelihood method can not estimate the parameters of complex statistical models yet. With graph method though the type of random variable distribution can be recognized, the precision of estimated parameters is not high. Meanwhile there is not an objectively criterion to judge statistical model reasonability.</div> <div class="htmlview paragraph">Based on the adaptability of Weibull's distribution, multiple-mixed Weibull's distribution suits to describe the complex statistical distribution of random variable. To estimate the parameters of multiple Weibull's distribution correctly, optimization model for estimating parameters of multiple-mixed Weibull's distribution is presented, according to the principle of least-square.</div> <div class="htmlview paragraph">In order to obtain the optimal estimation, the Levebberg-Marquardt method with line-search and Gauss-Newton method are alternatively used. A safeguarded mixed quadratic and cubic polynomial interpolation and extrapolation are chosen as algorithm of line-search. The example shows that multiple Weibull's distribution has quite good adaptability in describing the complex statistical distribution of random variable. The model and algorithm of optimization for estimating multiple Weibull's distribution parameters have steady convergence.</div>

Purpose . This article describes the issues of planning testing scope for high-reliable objects. The development and manufacture of new samples of equipment is accompanied by a task to define their reliability characteristics. It is based on the fact that there are requirements related to the necessity to specify the above mentioned characteristics in certificates and technical descriptions of the products supplied to the market. The most objective way to define reliability characteristics of the products is a field test. But under the manufacture of complex expensive objects there is no opportunity to introduce a batch with lots of finished products for testing. Thus there is a task to define the duration of field testing and scope of products to be tested, provided there are requirements for the accuracy of estimations related to the objects’ reliability characteristics obtained as the result of testing. Planning of the scope is based on the requirements of a manufacturer related to a necessity to confirm the value of lower bound of reliable operation probability with a predefined confidence level. Two tasks are solved in this work. The first task is to define the scope of testing of a batch with finished products N 0 for a time moment t 0 , for which a customer’s requirement would be fulfilled related to the achievement of the lower bound of probability of reliable operation, specified with a confidence probability 1 - α . This task is solved using a поп-parametric approach. The second task is to define a required scope of test Nt1 of the equipment of this type for the time moment different from the moment of first studies t 1 ≠ t 0 . Here one solves the question: how are N t0 and N t1 correlated? The scope of tests N t1 is defined based on the determination of confidence levels providing with the same accuracy of indices as in point t 0 . This task is solved with a semiparametric approach. When solving the second task, the parameterization of mean time to failure distribution is used. Three types of distribution are studied: exponential law, Weibull distribution and distribution with linear function of a failure rate. The considered types of distribution laws help to study the behavior of the objects with a decreasing, constant and increasing function of failure rate. Methods. The formulas for calculation of test scope for different durations of a test-run are derived. Dependence of scope on the duration of a test-run and on a real level of probability of reliable operation is studied as well. Scope planning and respective studies are carried out for different behavior models of a failure rate of the product. Conclusion . Obtained results give the basis for a well-reasoned approach to the planning of scope of tests of high-reliable objects. The study results showed that the longer a test-run is, the fewer objects are required to be introduced for a test. Dependence is non-linear; it is specified by the parameterization of the failure rate function. Analogous dependence was also obtained for the probability of reliable operation: the higher the PRO is, the fewer objects are required to be tested.

<ns3:p>This paper introduces a novel non-classical probability distribution, termed the Logistic Map distribution, which is constructed by transforming a polynomial function derived from the second iteration of the logistic map. The logistic map a well-known discrete-time dynamical system has been extensively employed in diverse scientific domains, including population dynamics (to model bounded growth under environmental constraints), physics (to study nonlinear dynamics and deterministic chaos), and economics (to represent complex, nonlinear patterns in financial and economic time series). The proposed distribution is fully characterized by two parameters: a scale parameter and a shape parameter, with the constraint ensuring the non-negativity and integrability of the density. Within this valid parameter space, we rigorously derive and establish a comprehensive suite of statistical properties. These include the probability density function, cumulative distribution function, reliability (survival) function, and hazard (failure rate) function. Furthermore, we obtain analytical expressions for key descriptive measures such as the mode and median, as well as for higher-order characteristics including the moment generating function, factorial moment generating function, and characteristic function. The proposed distribution most closely application field in materials science specifically, the statistical modeling of particle or grain size distributions in industrial powder processing, metallurgy, and pharmaceutical manufacturing. The primary objective of this study is to formalize a new family of probability distributions grounded in the mathematical framework of dynamical systems, specifically leveraging the logistic function commonly encountered in differential and difference equations. By doing so, we bridge concepts from nonlinear dynamics and classical statistical theory. The secondary aim is to conduct a thorough investigation of the distribution’s mathematical structure and statistical behavior, thereby establishing its potential utility for modeling bounded, non-negative random phenomena in applied fields such as reliability engineering, survival analysis, and environmental statistics.</ns3:p>

This paper presents a probabilistic-approach-based reliability prediction model of semiconductor light emitting devices. Using this model with given initial light-emitting performance and degradation behavior otherwise determined by experiment, the reliability function of the devices is obtained, and the results correlate well with experimental results. (Modeling the initial light-emitting performance and the degradation behavior is still an on-going effort and is not included in this paper. Eventually, this model will include both parts of the modeling to provide complete analytical results of reliability prediction.) This study is a step to develop a complete physics-of-failure-based reliability prediction methodology for semiconductor light-emitting devices. It provides an approach and proves the feasibility of determining a reliability function based on fundamental parameters of device performance.

Optimal maintenance policy considering maintenance errors for systems operating under performance-based contracts

The stochastic version of Pontryagin's maximum principle is applied to determine an optimal maintenance policy of equipment subject to random deterioration. The deterioration of the equipment with age is modelled as a random process. Next the model is generalized to include random catastrophic failure of the equipment. The optimal maintenance policy is derived for two special probability distributions of time to failure of the equipment, namely, exponential and Weibull distributions Both the salvage value and deterioration rate of the equipment are treated as state variables and the maintenance as a control variable. The result is illustrated by an example.

Reliability growth assessment is an important part of the reliability growth management. According to the characteristic that the lifetime of electromechanical product obeys the Weibull distribution, the thesis provides a monotonous constraint model of reliability growth of Weibull distribution, including complete Samples and censored samples. The method proposed in this paper has been verified by actual data, and has good adaptability. The comprehensive evaluation results of MTBF are higher than those got through only utilizing the test data of the last stage.Background Information

In this article, we deal with optimal preventive maintenance policies based on online condition monitoring. The failure rate function is important for maintenance decisions. Two concepts of failure are proposed and the computing method based on condition monitoring data is given. Both un-repairable and repairable equipment are taken into consideration. For repairable equipment, the degree of degradation and failure rate will decrease after maintenance. The result of the simulation shows that taking the two types of failure rate functions into account will make the expected cost rate less than the classical method. So we draw a conclusion that the two types of failure rate functions are advantageous in maintenance decisions.

Structural engineering makes a giant stride forward with this symposium. This book represents an international sym­ posium held in Stockholm honoring Prof. W. Weibull. It drew participants from 21 countries with a total of 55 invited papers. There were 7 scientific sessions. Most of our studies in school revolved about deterministic happenings. Random loading becomes more and more important when we equate events in our world with physical and mathematical reasoning. Probabilistic failure model and stochastic fatigue crack growth play important roles in the safety of nuclear structures, random loading due to wind turbulence, earthquake spectra, offshore structural design, and aircraft design. The initial paper summarizes Prof. Weibull's life and ac­ complishments including a mention of Weibull distribution used in fatigue studies. The initial session deals with extreme value theory (EVT). The first paper reviewed applications of extreme value theory to material behavior and indicates the importance of the accompanying statistical theory. The next paper generalizes Weibull distribution in matrix form. The following papers proclaim (a) extreme value theory as com­ patible to random field theory, (b) incomplete samples can be transformed to a mixed-exponential survival for a more com­ prehensive Weibull distribution, (c) EVT in endurance testing of ball and roller bearings, (d) use of the Slepian model identi­ fying the behavior of Gaussian noise near or between its zero crossing, (e) plastic movement of SDOF elastic plastic oscillator subjected to stationary Gaussian process excitation, and (/) generalized Hermitian polynomial used in the pertur­ bation method for nonlinear random vibrations. The final paper reports on Weibull distribution in large earthquake modelling. Session 2 reports on fatigue crack growth. Beginning with an analysis of stochastic equation models of crack growth, it continues with stochastic models of fatigue crack growth and propagation, prediction of crack growth under spectrum loading employing a cycle-by-cycle technique and fatigue life distribution in a random loading sense with interacting failures. The final paper considers a new approach to fatigue crack propagation under random loading and the use of Monte Carlo simulation in correlating micro fracture process and fatigue crack propagation. This session indicates that more experimental work is required in crack propagation to reach the same level as crack initiation. Session 3 comprises a total of 13 papers and dwells upon probabilistic failure models. The initial paper proposes a generalized probabilistic model for the fatigue life and reliability prediction of a structure containing noninteracting cracks and employs Poisson random sets. The next group of papers focuses upon (a) structural reliability treated as a single stochastic model, (Jb) fatigue life and reliability estimates of mineral pipe lines using probability theory, (c) Weibull-based equation employed in predicting the failure of a stressed brittle material, (d) a preliminary suggestion that specimens failed in proof tests may be used to estimate the parameters of the reliability function for the survivors, and (e) brittle material design using three parametric Weibull distributions. The next set of papers considers ring-in-ring tests on strength of cladded glass using the three-parameter Weibull distribution, prob­ abilistic analysis of plastic plates utilizing yield-line theory plus lower bound reliability analysis of plastic plates evaluated by inclusion-exc lusion theory. The final set of papers ques­ tions the possibility of flaws in the present probabilistic models for unidirectional fibrous composites, strength rela­ tions for cracks in unidirectiona l long fiber composites employing Weibull distribution and a stochastic approach in studying the fracture and fatigue of concrete. Session 4 delves into probabilistic fracture mechanics. The initial paper estimates the probable failure of PWR pressure vessels employing Weibull distribution and updates the wellknown Marshall A(a) and B(a) functions. The final set of papers covers probabilistic assessment of structures with weld defects, probability of fracture in main coolant pipes of PWR, and statistical modelling of shaft predictions (as measured by Charpy impact curve) in the reference temperature of pressure vessel welds. The authors suggested that U.S. Nuclear Regulatory Guide 1.99 be updated to remove excessive conservatism.

A dynamic reliability prediction technique that has been applied to the DPS 7000, a mainframe computer system, is described. To construct a product reliability prediction that is applicable over the life cycle of the product, both the intrinsic technological characteristics and their evolutionary trends are considered. Improvements in such factors as organization, manufacturing, maintenance support, and customer familiarity can have a substantial influence on product reliability growth. Using observed measure of field performance, trends in reliability growth that are attributable to evolving product maturity can be identified and used to support dynamic adjustments to component reliability and reliability growth models, and thereby create dynamic reliability predictions. >

In machine learning, the reliability estimates for individual predictions provide more information about individual prediction error than the average accuracy of predictive model (e.g. relative mean squared error). Such reliability estimates may represent decisive information in the risk-sensitive applications of machine learning (e.g. medicine, engineering, and business), where they enable the users to distinguish between more and less reliable predictions. In the authors’ previous work they proposed eight reliability estimates for individual examples in regression and evaluated their performance. The results showed that the performance of each estimate strongly varies depending on the domain and regression model properties. In this paper they empirically analyze the dependence of reliability estimates’ performance on the data set and model properties. They present the results which show that the reliability estimates perform better when used with more accurate regression models, in domains with greater number of examples and in domains with less noisy data.

The model with failure rate function of bathtub-shaped is helpful in reliability analysis of any system and particularly in reliability associated privative maintenance. The usual Weibull distribution is, however, not capable to model the complete lifecycle of the any with a bathtub-shaped failure rate function. In this paper, failure rate and reliability analysis of the KOMATSU hydraulic excavator/shovel in surface mine is presented and also to improve the reliability and decrease the failure rate of each subsystem of the shovel based on the preventive maintenance. The model of the bathtub-shaped for shovel can also be seen as a simplification of the Weibull distribution.

The generalized Bilal (GB) distribution can be defined as the distribution of the median of three independent random variables drawn from the Weibull distribution. Its failure rate function can be monotonic (decreasing or increasing) or upside-down bathtub-shaped. In this study, we aim to reveal some important properties of the GB distribution that have not been considered before. The findings are both theoretical and practical. From the theoretical viewpoint, we present explicit expressions for both single and product moments of order statistics from the GB distribution. The L-moments are derived as well. From the practical viewpoint, the parameter estimations are accomplished using the maximum likelihood (ML) method, which is based on two different sampling schemes: simple random sampling (SRS) and ranked set sampling (RSS) schemes. Furthermore, the asymptotic confidence intervals for the SRS and RSS estimators are discussed. For the sake of comparison and illustration, a simulation study and a real data example are presented. Concluding remarks are given at the end.