Fault and Event Tree Analyses for Process Systems Risk Analysis: Uncertainty Handling Formulations

Quantitative risk analysis (QRA) is a systematic approach for evaluating likelihood, consequences, and risk of adverse events. QRA based on event (ETA) and fault tree analyses (FTA) employs two basic assumptions. The first assumption is related to likelihood values of input events, and the second assumption is regarding interdependence among the events (for ETA) or basic events (for FTA). Traditionally, FTA and ETA both use crisp probabilities; however, to deal with uncertainties, the probability distributions of input event likelihoods are assumed. These probability distributions are often hard to come by and even if available, they are subject to incompleteness (partial ignorance) and imprecision. Furthermore, both FTA and ETA assume that events (or basic events) are independent. In practice, these two assumptions are often unrealistic. This article focuses on handling uncertainty in a QRA framework of a process system. Fuzzy set theory and evidence theory are used to describe the uncertainties in the input event likelihoods. A method based on a dependency coefficient is used to express interdependencies of events (or basic events) in ETA and FTA. To demonstrate the approach, two case studies are discussed.     

A Fuzzy Decision Tree for Fault Classification

In plant accident management, the control room operators are required to identify the causes of the accident, based on the different patterns of evolution of the monitored process variables thereby developing. This task is often quite challenging, given the large number of process parameters monitored and the intense emotional states under which it is performed. To aid the operators, various techniques of fault classification have been engineered. An important requirement for their practical application is the physical interpretability of the relationships among the process variables underpinning the fault classification. In this view, the present work propounds a fuzzy approach to fault classification, which relies on fuzzy if-then rules inferred from the clustering of available preclassified signal data, which are then organized in a logical and transparent decision tree structure. The advantages offered by the proposed approach are precisely that a transparent fault classification model is mined out of the signal data and that the underlying physical relationships among the process variables are easily interpretable as linguistic if-then rules that can be explicitly visualized in the decision tree structure. The approach is applied to a case study regarding the classification of simulated faults in the feedwater system of a boiling water reactor.     

Vertices Contained in all or in no Minimum Paired-Dominating Set of a Tree

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. We characterize the set of vertices of a tree that are contained in all, or in no, minimum paired-dominating sets of the tree. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Finding a Length-Constrained Maximum-Density Path in a Tree

Let T = (V,E,w) be an undirected and weighted tree with node set V and edge set E, where w(e) is an edge weight function for e E. The density of a path, say e₁, e₂,..., e_k, is defined as ^k_{i = 1} w(e_i)/k. The length of a path is the number of its edges. Given a tree with n edges and a lower bound L where 1 L n, this paper presents two efficient algorithms for finding a maximum-density path of length at least L in O(nL) time. One of them is further modified to solve some special cases such as full m-ary trees in O(n) time.     

Vertex and Tree Arboricities of Graphs

This paper studies the following variations of arboricity of graphs. The vertex (respectively, tree) arboricity of a graph G is the minimum number va(G) (respectively, ta(G)) of subsets into which the vertices of G can be partitioned so that each subset induces a forest (respectively, tree). This paper studies the vertex and the tree arboricities on various classes of graphs for exact values, algorithms, bounds, hamiltonicity and NP-completeness. The graphs investigated in this paper include block-cactus graphs, series-parallel graphs, cographs and planar graphs.     

Combining Linear and Non-Linear Objectives in Spanning Tree Problems

A classical approach to multicriteria problems asks for the optimization of a suitable linear combination of the objectives. In this work we address such problems when one of the objectives is the linear function, the other is a non-linear one and we seek for a spanning tree of a given graph which optimizes the combination of the two functions. We consider both maximization and minimization problems and present the complexity status of 56 such problems, giving, whenever possible, polynomial solution algorithms.     

Constructing the Maximum Consensus Tree from Rooted Triples

We investigated the problem of constructing the maximum consensus tree from rooted triples. We showed the NP-hardness of the problem and developed exact and heuristic algorithms. The exact algorithm is based on the dynamic programming strategy and runs in O((m + n ²)3 ⁿ ) time and O(2 ⁿ ) space. The heuristic algorithms run in polynomial time and their performances are tested and shown by comparing with the optimal solutions. In the tests, the worst and average relative error ratios are 1.200 and 1.072 respectively. We also implemented the two heuristic algorithms proposed by Gasieniec et al. The experimental result shows that our heuristic algorithm is better than theirs in most of the tests.     

Evaluating Potential Distribution of High‐Risk Aquatic Invasive Species in the Water Garden and Aquarium Trade at a Global Scale Based on Current Established Populations

Aquatic non‐native invasive species are commonly traded in the worldwide water garden and aquarium markets, and some of these species pose major threats to the economy, the environment, and human health. Understanding the potential suitable habitat for these species at a global scale and at regional scales can inform risk assessments and predict future potential establishment. Typically, global habitat suitability models are fit for freshwater species with only climate variables, which provides little information about suitable terrestrial conditions for aquatic species. Remotely sensed data including topography and land cover data have the potential to improve our understanding of suitable habitat for aquatic species. In this study, we fit species distribution models using five different model algorithms for three non‐native aquatic invasive species with bioclimatic, topographic, and remotely sensed covariates to evaluate potential suitable habitat beyond simple climate matches. The species examined included a frog (Xenopus laevis), toad (Bombina orientalis), and snail (Pomacea spp.). Using a unique modeling approach for each species including background point selection based on known established populations resulted in robust ensemble habitat suitability models. All models for all species had test area under the receiver operating characteristic curve values greater than 0.70 and percent correctly classified values greater than 0.65. Importantly, we employed multivariate environmental similarity surface maps to evaluate potential extrapolation beyond observed conditions when applying models globally. These global models provide necessary forecasts of where these aquatic invasive species have the potential for establishment outside their native range, a key component in risk analyses.     

A Diagnostic Tree for Improving Production Line Performance

Improving performance of production systems is a critical but often unstructured activity. To help managers convert ad hoc or trial & error improvement efforts into efficient and systematic reviews, we develop a diagnostic tree which decomposes a performance improvement objective into successively more concrete sub‐objectives and finally into potential improvement strategies. Based on principles from the Operations Management literature, this tree is structured to enable a non‐specialist to better understand the links between corrective actions and performance. It also provides an important foundation for a principles‐based knowledge management system that couples the decision tree with a search engine for locating relevant documents within an intranet.     

Lognormal Approximations of Fault Tree Uncertainty Distributions

Fault trees are used in reliability modeling to create logical models of fault combinations that can lead to undesirable events. The output of a fault tree analysis (the top event probability) is expressed in terms of the failure probabilities of basic events that are input to the model. Typically, the basic event probabilities are not known exactly, but are modeled as probability distributions: therefore, the top event probability is also represented as an uncertainty distribution. Monte Carlo methods are generally used for evaluating the uncertainty distribution, but such calculations are computationally intensive and do not readily reveal the dominant contributors to the uncertainty. In this article, a closed‐form approximation for the fault tree top event uncertainty distribution is developed, which is applicable when the uncertainties in the basic events of the model are lognormally distributed. The results of the approximate method are compared with results from two sampling‐based methods: namely, the Monte Carlo method and the Wilks method based on order statistics. It is shown that the closed‐form expression can provide a reasonable approximation to results obtained by Monte Carlo sampling, without incurring the computational expense. The Wilks method is found to be a useful means of providing an upper bound for the percentiles of the uncertainty distribution while being computationally inexpensive compared with full Monte Carlo sampling. The lognormal approximation method and Wilks’s method appear attractive, practical alternatives for the evaluation of uncertainty in the output of fault trees and similar multilinear models.     

Simplified Containment Event Tree Methodology Applied to Peach Bottom SBO-ST Sequences1

The methodology presented here identifies an approach to accurately and economically analyze the effects on risk of various containment performance issues. Although this method facilitates the evaluation of potential containment improvements, it does so while utilizing the significant amount of information accumulated by the U.S. NRC Reactor Risk Reference Program. The use of hindsight and the acceptance of point estimate quantifications of risks allows the proposed methodology to be scrutable and understandable to the community as well as relatively simple and inexpensive to apply. A study of containment venting strategies was used to demonstrate the capabilities of the simplified containment event tree methodology. However, the methodology is flexible enough for a wide range of risk evaluations.     

Efficient Heuristics for Orientation Metric and Euclidean Steiner Tree Problems

We consider Steiner minimum trees (SMT) in the plane, where only orientations with angle , 0 i – 1 and an integer, are allowed. The orientations define a metric, called the orientation metric, , in a natural way. In particular, ₂ metric is the rectilinear metric and the Euclidean metric can beregarded as metric. In this paper, we provide a method to find an optimal SMT for 3 or 4 points by analyzing the topology of SMT's in great details. Utilizing these results and based on the idea of loop detection first proposed in Chao and Hsu, IEEE Trans. CAD, vol. 13, no. 3, pp. 303–309, 1994, we further develop an O(n²) time heuristic for the general SMT problem, including the Euclidean metric. Experiments performed on publicly available benchmark data for 12 different metrics, plus the Euclidean metric, demonstrate the efficiency of our algorithms and the quality of our results.     

Uncertainty Analysis in Fault Tree Models with Dependent Basic Events

In general, two types of dependence need to be considered when estimating the probability of the top event (TE) of a fault tree (FT): “objective” dependence between the (random) occurrences of different basic events (BEs) in the FT and “state‐of‐knowledge” (epistemic) dependence between estimates of the epistemically uncertain probabilities of some BEs of the FT model. In this article, we study the effects on the TE probability of objective and epistemic dependences. The well‐known Frèchet bounds and the distribution envelope determination (DEnv) method are used to model all kinds of (possibly unknown) objective and epistemic dependences, respectively. For exemplification, the analyses are carried out on a FT with six BEs. Results show that both types of dependence significantly affect the TE probability; however, the effects of epistemic dependence are likely to be overwhelmed by those of objective dependence (if present).     

Algorithm for the Cost Edge-Coloring of Trees

Let C be a set of colors, and let be a cost function which assigns a real number (c) to each color C in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors. In this paper we give an efficient algorithm to find an optimal edge-coloring of a given tree T, that is, an edge-coloring f of T such that the sum of costs (f(e)) of colors f(e) assigned to all edges e is minimum among all edge-colorings of T. The algorithm takes time O(n²) if n is the number of vertices and is the maximum degree of T.     

Allometric Scaling: Analysis of LD50 Data

The need to identify toxicologically equivalent doses across different species is a major issue in toxicology and risk assessment. In this article, we investigate interspecies scaling based on the allometric equation applied to the single, oral LD ₅₀ data previously analyzed by Rhomberg and Wolff.⁽ 1 ⁾ We focus on the statistical approach, namely, regression analysis of the mentioned data. In contrast to Rhomberg and Wolff's analysis of species pairs, we perform an overall analysis based on the whole data set. From our study it follows that if one assumes one single scaling rule for all species and substances in the data set, then β= 1 is the most natural choice among a set of candidates known in the literature. In fact, we obtain quite narrow confidence intervals for this parameter. However, the estimate of the variance in the model is relatively high, resulting in rather wide prediction intervals.     

Tree edge decomposition with an application to minimum ultrametric tree approximation

A k-decomposition of a tree is a process in which the tree is recursively partitioned into k edge-disjoint subtrees until each subtree contains only one edge. We investigated the problem how many levels it is sufficient to decompose the edges of a tree. In this paper, we show that any n-edge tree can be 2-decomposed (and 3-decomposed) within at most ⌈1.44 log n⌉ (and ⌈log n⌉ respectively) levels. Extreme trees are given to show that the bounds are asymptotically tight. Based on the result, we designed an improved approximation algorithm for the minimum ultrametric tree.     

Weighted Inverse Minimum Spanning Tree Problems Under Hamming Distance

In this paper, we consider weighted inverse minimum spanning tree problems under Hamming distance. Three models are considered: unbounded case, unbounded case with forbidden edges, and general bounded case. In terms of the method for minimum-weight node cover problem on bipartite graph, we present their respective strongly polynomial algorithms.This research is supported by TRAPOYT, National Natural Science Foundation of China (10271110, 60021201)     

Closed-Form Deterministic End-to-End Performance Bounds for the Generalized Processor Sharing Scheduling Discipline

The Generalized Processor Sharing (GPS) schedulingdiscipline is an important scheduling mechanism that can support both class isolation and bandwidth sharing among different service classes, thus making itan appealing choice for networks providing multiple services with Quality-of-Service guarantees. In this paper, we study a broad classof GPS networks known as Consistent Relative Session Treatment}(CRST) GPS networks and establish closed-form end-to-end performance boundsfor CRST GPS networks. This result generalizes the results of Parekhand Gallager (1994) where simple, closed-form end-to-end performancebounds are derived for a special sub-class of CRST GPS networks, theso-called Rate Proportional Processor Sharing (RPPS) GPS networks, but performance bounds for the general CRST GPS networks do not haveclosed-form. Our result is obtained through the notion of CRSTpartition, which in fact yields a broader class of CRST GPS networksthan the one originally defined in (Parekh and Gallager, 1993). Moreover,our approach is quite general. It not only applies to the deterministicanalysis of GPS networks, but can also be employed in the study of GPSnetworks in a stochastic setting.     

Morality De-Kanted or the Biological Roots of Moral Behavior

The ethical and moral behavior of Homo sapiens is no longer the exclusive domain of religion and philosophy because we recognize that such behavior affects the reproductive success of individuals within the species. We are a social species and therefore our survival is influenced by our capacity for cooperation and our willingness to take risks for kin. Emotions, some of which are found in other species, help to mediate our altruistic behavior. The reproductive benefits of helping kin, especially offspring, are readily seen. Helping non-kin can be beneficial if individuals can differentiate between ‘reciprocators’ and ‘non-reciprocators’ and direct altruistic behavior toward reciprocators. Also, if third parties are favorably impressed by observing altruistic behavior, the rewards need not come from the recipient of the altruistic behavior. This revised version was published online in July 2006 with corrections to the Cover Date.     

Relevance of Risk Predictions Derived from a Chronic Species Sensitivity Distribution with Cadmium to Aquatic Populations and Ecosystems

Criteria to protect aquatic life are intended to protect diverse ecosystems, but in practice are usually developed from compilations of single‐species toxicity tests using standard test organisms that were tested in laboratory environments. Species sensitivity distributions (SSDs) developed from these compilations are extrapolated to set aquatic ecosystem criteria. The protectiveness of the approach was critically reviewed with a chronic SSD for cadmium comprising 27 species within 21 genera. Within the data set, one genus had lower cadmium effects concentrations than the SSD fifth percentile‐based criterion, so in theory this genus, the amphipod Hyalella, could be lost or at least allowed some level of harm by this criteria approach. However, population matrix modeling projected only slightly increased extinction risks for a temperate Hyalella population under scenarios similar to the SSD fifth percentile criterion. The criterion value was further compared to cadmium effects concentrations in ecosystem experiments and field studies. Generally, few adverse effects were inferred from ecosystem experiments at concentrations less than the SSD fifth percentile criterion. Exceptions were behavioral impairments in simplified food web studies. No adverse effects were apparent in field studies under conditions that seldom exceeded the criterion. At concentrations greater than the SSD fifth percentile, the magnitudes of adverse effects in the field studies were roughly proportional to the laboratory‐based fraction of species with adverse effects in the SSD. Overall, the modeling and field validation comparisons of the chronic criterion values generally supported the relevance and protectiveness of the SSD fifth percentile approach with cadmium.