A new efficient algorithm for computing the imprecise reliability of monotone systems

Reliability analysis of complex systems by partial information about reliability of components and by different conditions of independence of components may be carried out by means of the imprecise probability theory which provides a unified framework (natural extension, lower and upper previsions) for computing the system reliability. However, the application of imprecise probabilities to reliability analysis meets with a complexity of optimization problems which have to be solved for obtaining the system reliability measures. Therefore, an efficient simplified algorithm to solve and decompose the optimization problems is proposed in the paper. This algorithm allows us to practically implement reliability analysis of monotone systems under partial and heterogeneous information about reliability of components and under conditions of the component independence or the lack of information about independence. A numerical example illustrates the algorithm.     

Using random set theory to propagate epistemic uncertainty through a mechanical system

The Epistemic Uncertainty Project of Sandia National Laboratories (NM, USA) proposed two challenge problems intended to assess the applicability and the relevant merits of modern mathematical theories of uncertainty in reliability engineering and risk analysis. This paper proposes a solution to Problem B: the response of a mechanical system with uncertain parameters. Random Set Theory is used to cope with both imprecision and dissonance affecting the available information. Imprecision results in an envelope of CDFs of the system response bounded by an upper CDF and a lower CDF. Different types of parameter discretizations are introduced. It is shown that: (i) when the system response presents extrema in the range of parameters considered, it is better to increase the fineness of the discretization than to invoke a global optimization tool; (ii) the response expectation differed by less than 0.5% when the number of function calls was increased 15.7 times; (iii) larger differences (4–5%) were obtained for the lower tails of the CDFs of the response. Further research is necessary to investigate (i) parameter discretizations aimed at increasing the accuracy of the CDFs (lower) tails; (ii) the role of correlation in combining information.     

An approach to combining unreliable pieces of evidence and their propagation in a system response analysis

The paper describes an approach to representing, aggregating and propagating aleatory and epistemic uncertainty through computational models. The framework for the approach employs the theory of imprecise coherent probabilities. The approach is exemplified by a simple algebraic system, the inputs of which are uncertain. Six different uncertainty situations are considered, including mixtures of epistemic and aleatory uncertainty.     

A case study for quantifying system reliability and uncertainty

The ability to estimate system reliability with an appropriate measure of associated uncertainty is important for understanding its expected performance over time. Frequently, obtaining full-system data is prohibitively expensive, impractical, or not permissible. Hence, methodology which allows for the combination of different types of data at the component or subsystem levels can allow for improved estimation at the system level. We apply methodologies for aggregating uncertainty from component-level data to estimate system reliability and quantify its overall uncertainty. This paper provides a proof-of-concept that uncertainty quantification methods using Bayesian methodology can be constructed and applied to system reliability problems for a system with both series and parallel structures.     

Fault-tolerant embedded system design and optimization considering reliability estimation uncertainty

In this paper, we model embedded system design and optimization, considering component redundancy and uncertainty in the component reliability estimates. The systems being studied consist of software embedded in associated hardware components. Very often, component reliability values are not known exactly. Therefore, for reliability analysis studies and system optimization, it is meaningful to consider component reliability estimates as random variables with associated estimation uncertainty. In this new research, the system design process is formulated as a multiple-objective optimization problem to maximize an estimate of system reliability, and also, to minimize the variance of the reliability estimate. The two objectives are combined by penalizing the variance for prospective solutions. The two most common fault-tolerant embedded system architectures, N-Version Programming and Recovery Block, are considered as strategies to improve system reliability by providing system redundancy. Four distinct models are presented to demonstrate the proposed optimization techniques with or without redundancy. For many design problems, multiple functionally equivalent software versions have failure correlation even if they have been independently developed. The failure correlation may result from faults in the software specification, faults from a voting algorithm, and/or related faults from any two software versions. Our approach considers this correlation in formulating practical optimization models. Genetic algorithms with a dynamic penalty function are applied in solving this optimization problem, and reasonable and interesting results are obtained and discussed.     

A probabilistic approach for representation of interval uncertainty

In this paper, we propose a probabilistic approach to represent interval data for input variables in reliability and uncertainty analysis problems, using flexible families of continuous Johnson distributions. Such a probabilistic representation of interval data facilitates a unified framework for handling aleatory and epistemic uncertainty. For fitting probability distributions, methods such as moment matching are commonly used in the literature. However, unlike point data where single estimates for the moments of data can be calculated, moments of interval data can only be computed in terms of upper and lower bounds. Finding bounds on the moments of interval data has been generally considered an NP-hard problem because it includes a search among the combinations of multiple values of the variables, including interval endpoints. In this paper, we present efficient algorithms based on continuous optimization to find the bounds on second and higher moments of interval data. With numerical examples, we show that the proposed bounding algorithms are scalable in polynomial time with respect to increasing number of intervals. Using the bounds on moments computed using the proposed approach, we fit a family of Johnson distributions to interval data. Furthermore, using an optimization approach based on percentiles, we find the bounding envelopes of the family of distributions, termed as a Johnson p-box. The idea of bounding envelopes for the family of Johnson distributions is analogous to the notion of empirical p-box in the literature. Several sets of interval data with different numbers of intervals and type of overlap are presented to demonstrate the proposed methods. As against the computationally expensive nested analysis that is typically required in the presence of interval variables, the proposed probabilistic representation enables inexpensive optimization-based strategies to estimate bounds on an output quantity of interest.     

A generalized interval fuzzy mixed integer programming model for a multimodal transportation problem under uncertainty

A generalized interval fuzzy mixed integer programming model is proposed for the multimodal freight transportation problem under uncertainty, in which the optimal mode of transport and the optimal amount of each type of freight transported through each path need to be decided. For practical purposes, three mathematical methods, i.e. the interval ranking method, fuzzy linear programming method and linear weighted summation method, are applied to obtain equivalents of constraints and parameters, and then a fuzzy expected value model is presented. A heuristic algorithm based on a greedy criterion and the linear relaxation algorithm are designed to solve the model.     

A novel Bayesian approach to reliability modeling: The benefits of uncertainty evaluation in the model selection procedure

This paper proposes a different likelihood formulation within the Bayesian paradigm for parameter estimation of reliability models. Moreover, the assessment of the uncertainties associated with parameters, the goodness of fit, and the model prediction of reliability are included in a systematic framework for better aiding the model selection procedure. Two case studies are appraised to highlight the contributions of the proposed method and demonstrate the differences between the proposed Bayesian formulation and an existing Bayesian formulation.     

基于区间分析的不确定性结构动力学模型修正方法

提出了一种基于区间分析的不确定性有限元模型修正方法。在区间参数结构特征值分析理论和确定性有限元模型修正方法基础上,假设不确定性与初始有限元模型误差均较小,采用灵敏度方法推导了待修正参数区间中点值和不确定区间的迭代格式。以三自由度弹簧-质量系统和复合材料板为例,采用拉丁超立方抽样构造仿真试验模态参数样本,开展仿真研究。结果表明,当仿真试验样本能准确反映结构模态参数的区间特性时,方法的收敛精度和效率均较高;修正后计算模态参数能准确反映试验数据的区间特性。所提出方法适用于解决试验样本较少,仅能得到试验模态参数区间的有限元模型修正问题。     

Robust interval linear programming for environmental decision making under uncertainty

In this study, a robust interval linear programming (RILP) method is developed for dealing with uncertainties expressed as intervals with deterministic boundaries. An enhanced two-step method (ETSM) is also advanced to solve the RILP model. The developed RILP improves upon the conventional interval linear programming (ILP) method since it can generate solution intervals within a larger feasible zone. The decision space based on ETSM contains all feasible solutions, such that no useful information is neglected. Moreover, the RILP can guarantee the stability of the optimization model due to no violation for the best-case constraints. The results also suggest that the RILP is effective for practical environmental and engineering problems that involve uncertainties.     

Interpretations of alternative uncertainty representations in a reliability and risk analysis context

Probability is the predominant tool used to measure uncertainties in reliability and risk analyses. However, other representations also exist, including imprecise (interval) probability, fuzzy probability and representations based on the theories of evidence (belief functions) and possibility. Many researchers in the field are strong proponents of these alternative methods, but some are also sceptical. In this paper, we address one basic requirement set for quantitative measures of uncertainty: the interpretation needed to explain what an uncertainty number expresses. We question to what extent the various measures meet this requirement. Comparisons are made with probabilistic analysis, where uncertainty is represented by subjective probabilities, using either a betting interpretation or a reference to an uncertainty standard interpretation. By distinguishing between chances (expressing variation) and subjective probabilities, new insights are gained into the link between the alternative uncertainty representations and probability.     

A generic method for estimating system reliability using Bayesian networks

This study presents a holistic method for constructing a Bayesian network (BN) model for estimating system reliability. BN is a probabilistic approach that is used to model and predict the behavior of a system based on observed stochastic events. The BN model is a directed acyclic graph (DAG) where the nodes represent system components and arcs represent relationships among them. Although recent studies on using BN for estimating system reliability have been proposed, they are based on the assumption that a pre-built BN has been designed to represent the system. In these studies, the task of building the BN is typically left to a group of specialists who are BN and domain experts. The BN experts should learn about the domain before building the BN, which is generally very time consuming and may lead to incorrect deductions. As there are no existing studies to eliminate the need for a human expert in the process of system reliability estimation, this paper introduces a method that uses historical data about the system to be modeled as a BN and provides efficient techniques for automated construction of the BN model, and hence estimation of the system reliability. In this respect K2, a data mining algorithm, is used for finding associations between system components, and thus building the BN model. This algorithm uses a heuristic to provide efficient and accurate results while searching for associations. Moreover, no human intervention is necessary during the process of BN construction and reliability estimation. The paper provides a step-by-step illustration of the method and evaluation of the approach with literature case examples.     

含区间不确定性参数的机翼气动弹性优化

提出了一种具有区间不确定性的机翼颤振优化方法.采用拉丁超立方方法建立仿真试验表,基于MSC.Nastran平台进行颤振仿真分析.获得仿真数据之后,应用Kriging方法构造了包含区间不确定性参数的机翼颤振分析代理模型,并进行有效性检验.基于建立的代理模型并按照区间序数关系,将不确定性优化目标和约束条件转化为确定性表达形式,从面形成区间不确定性的结构优化设计方法.该方法将区间法优化和代理模型相结合,同时综合有限元仿真和遗传算法的优点,计算效率较高且应用范围较广.以某复杂机翼结构为例进行了含区间不确定性的颤振优化计算.分析结果表明了所提方法的正确性和可行性.     

A mathematical model for vehicle routing problem under endogenous uncertainty

In this study, a multistage stochastic programming (SP) model is presented for a variant of single-vehicle routing problem with stochastic demands from a dynamic viewpoint. It is assumed that the actual demand of a customer becomes known only when the customer is visited. This problem falls into the category of SP with endogenous uncertainty and hence, the scenario tree is decision-dependent. Therefore, nonanticipativity of decisions is ensured by conditional constraints making up a large portion of total constraints. Thus, a novel approach is proposed that considerably reduces the problem size without any effect on the solution space. Computational results on some test problems are reported.     

多源不确定性条件下气动弹性系统颤振可靠性分析方法

传统的气动弹性系统颤振分析模型大多是在确定性参数条件下建立的,当系统中存在不确定因素时,按确定性方法设计的气动弹性系统存在颤振失效风险.以概率和非概率区间模型为基础,建立了单源不确定性条件下颤振可靠性分析模型;在此基础上,针对含随机和区间多源不确定参数的气动弹性系统颤振可靠性分析问题,提出一种基于分步求解策略的新型混合...     

A structural reliability model for composite materials

This work is a part of a project aimed at developing a model and procedure to evaluate the structural reliability of a laminate whose laminae properties are statistically known. The inputs consist of the laminae's elastic properties (deterministic in that they are not as closely related to manufacturing technology) and the derived allowable strengths (aleatoric variables), as well as the sequence (number, orientation, position) of the laminae in the laminate. The output, the laminate's structural reliability, allows determination of the laminate's collapse process, with the most probable lamina failure sequence.     

Project management for uncertainty with multiple objectives optimisation of time,cost and reliability

This research adopts an approach that uses computer simulation and statistical analysis of uncertain activity time, activity cost, due date and project budget to address quality and the learning process with regard to project scheduling. Since the learning process affects the scheduling problem, a Cobb–Douglas multiplicative power model is used to represent the relationship between the dependent variable, which is the standard deviation of activity time, and the independent variables, which are the cumulative trials and the mean of activity time. The mean value and standard deviation are used to randomly generate activity times for project scheduling analysis. Response surface methodology (RSM) is used in order to develop a rationale of the time-cost trade-off problem. The solutions found with RSM are optimised only for a single objective, such as project completion time, total project cost, completion time probability and total cost probability. Thus, multiple objectives for further optimisation become necessary and a limited project budget, restricted completion time, allowable total cost probability and acceptable completion time probability have to be considered at the same time as the learning effect. With response functions from RSM, compromise programming is adopted in order to formulate the proposed project scheduling problem for multi-objective optimisation.     

Accounting for maintenance in a computer memory reliability prediction model

This paper reviews the benefit of periodic maintenance for improving the reliability of computer memory protected by error detection and correction (EDAC), and presents a new memory reliability model that accounts for such maintenance. Mean time to failure (MTTF) has been the traditional figure of merit for assessing the benefit of memory maintenance. This paper proposes failure probability calculated at maintenance intervals as a more meaningful figure of merit for periodically maintained systems and presents examples using the new model.     

A single-loop optimization method for reliability analysis with second order uncertainty

Reliability analysis may involve random variables and interval variables. In addition, some of the random variables may have interval distribution parameters owing to limited information. This kind of uncertainty is called second order uncertainty. This article develops an efficient reliability method for problems involving the three aforementioned types of uncertain input variables. The analysis produces the maximum and minimum reliability and is computationally demanding because two loops are needed: a reliability analysis loop with respect to random variables and an interval analysis loop for extreme responses with respect to interval variables. The first order reliability method and nonlinear optimization are used for the two loops, respectively. For computational efficiency, the two loops are combined into a single loop by treating the Karush–Kuhn–Tucker (KKT) optimal conditions of the interval analysis as constraints. Three examples are presented to demonstrate the proposed method.