Weimin Dong

Fuzzy weighted averages and implementation of the extension principle

Abstract This paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and, in particular, to weighted average operations in risk and decision analysis. A computational algorithm based on the α-cut representation of fuzzy sets and interval analysis is described. The method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner. Examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non-linear programming. The algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers.

Vertex method for computing functions of fuzzy variables

Abstract This paper is to introduce a new approach — the vertex method — for computing functions of fuzzy variables. The method is based on the α-cut concept and interval analysis. The vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi-occurrence of variables in the functional expression by conventional interval analysis methods. The algorithm is very easy to implement and can be applied to many practical problems. Some examples are given to illustrate the applications.

Fuzzy computations in risk and decision analysis

Abstract This paper describes an algorithm for performing extended algebraic operations such as those encountered in risk and decision analysis under fuzzy conditions. The method makes use of the lambda-cut representations of fuzzy sets and interval analysis. It is an approximate computational technique but is highly efficient compared with the exact method of nonlinear programming, with an accuracy which is much better than the conventional discretization approach. The effectiveness and utility of the procedure are illustrated with examples of fuzzy risk and decision analysis taken from available literature.

Fuzzy information processing in seismic hazard analysis and decision making

Abstract This paper describes a methodology to incorporate vague information, based upon heuristic knowledge and expertise, into the conventional probabilistic approach for the seismic hazard analysis. The interval analysis method is introduced to process interval information with interpretation from Dempster and Shafers evidence theory. The Vertex Method is discussed to handle fuzzy information which is a generalization of interval information. These methods, along with the current approach of seismic hazard analysis, are used to assess the seismic hazard for the San Francisco Bay Area in California and to provide information for deciding strengthening policy of existing buildings.

Propagation of uncertainties in deterministic systems

Abstract This paper addresses the propagation of uncertainties in deterministic systems, i.e. the system definition is known but the system parameters and input to the system contain uncertain information. The effect of the uncertain information on the system response is to be assessed. Three models of uncertainties corresponding to differing degrees of knowledge about the uncertainty are considered: interval, fuzzy and random. A method to propagate uncertainties expressed as intervals is described; the method, called the Vertex method, is based on a generalization of combinatorial interval analysis techniques. It is shown how the Vertex method can be extended naturally to treat the propagation of uncertainties modeled as fuzzy sets. Finally, propagation of random uncertainties is described using the classical probabilistic technique of derived distribution functions. The computational implications of the three models of uncertainties and the corresponding methods of propagation are contrasted. It is suggested that when the available information is too crude to support a random definition, the interval or fuzzy model should be used to take advantage of the expediency with which interval and fuzzy uncertainties can be propagated and processed.

Dynamic response of structures with uncertain parameters: A comparative study of probabilistic and fuzzy sets models

Abstract This paper presents the results of a study on the response of structures with uncertain properties such as mass, stiffness and damping. The effect of the uncertain parameters on the response and the effect of the modelling of the uncertainties on the response are investigated. In particular, two types of uncertainties are distinguished: random and fuzzy uncertainties. Two kinds of models are studied: probabilistic and fuzzy set models. The two approaches to uncertainty modelling are compared with regard to their impacts on the analysis and on the uncertain structural response obtained. The study considers free vibration, forced vibration with deterministic excitation, and forced vibration with Gaussian white noise excitation. It is concluded that, in general, fuzzy models are much easier to implement and the associated analysis easier to perform than their probabilistic counterparts. When the available data on the structural parameters are crude and do not support a rigorous probabilistic model, the fuzzy set approach should be considered in view of its simplicity.

From uncertainty to approximate reasoning: part 1: conceptual models and engineering interpretations

Abstract Different models of uncertainties from the simplest interval representation to fuzzy sets and random numbers are reviewed. Mathematical theories spawned by these conceptual models are described, including interval analysis, possibility and fuzzy set theories, probability theory and the theory of evidence. The relationship among these theories is delineated from the perspective of understanding human reasoning. The discussion emphasizes conceptual understanding and physical intuition rather than mathematical theorems and axioms. It is intended to remove some of the mysteries surrounding existing viewpoints on uncertainties, which may have hindered wider understanding and acceptance of these viewpoints by practicing civil engineers. Subsequent parts of this paper will discuss the process of uncertainty propagation in a rules framework with reference to the various kinds of uncertainty representations.

From uncertainty to approximate reasoning: part 3: reasoning with conditional rules

Abstract This is the third part of a three-part paper. In part 1 (C/V Eng Syst. 1986, 3(3), 143-1 54), different models of uncertainties such as intervals, fuzzy sets, Dempster-Shafer evidence, and random numbers were compared. Then, in part 2 (Civ Eng Syst. 1986, 3(4), 192-202) reasoning based on an inference network of algorithmic rules was described. Algorithmic rules are the simplest inference rules in reasoning, but constitute only one extreme of the rule spectrum. Conditional rules are more general and correspond to less well-defined inference relations. The propagation and combination of uncertainties in a network of conditional rules are described in this part of the paper. Treatment of probabilistic uncertainties is described in detail, and the associated difficulties delineated. Finally, uncertain inference with other forms of uncertainties is presented to provide contrast, and to complete the discussion on the impact of uncertainty models on approximate reasoning.

Utilization of geophysical information in Bayesian seismic hazard model

Abstract This paper first describes the inferential structure of the Bayesian model and uses it to show why the empirical method using only short historical earthquake data cannot obtain a reliable hazard estimation. For improving the hazard prediction, newly developed information from the geophysical and geological studies should be incorporated with the historical data within the Bayesian framework. The paper, then, concentrates on the methodologies of how to use the energy flux concept, seismic moment and geological observation in the seismic hazard analysis. A refined Bayesian model, where some of the geophysical and geological input can be used flexibly and consistently, is suggested. Finally, numerical examples are presented for illustrating the application of the improved method.

From uncertainty to approximate reasoning: part 2: reasoning with algorithmic rules

Abstract This is the second part of a three-part paper. In the first part. Civ. Engng Syst. 1 986, 3(3), 143-1 54 different models of uncertainties from the simplest interval representation, to fuzzy sets and random numbers are described. This paper discusses uncertain inference or reasoning based on evidence-hypothesis rules. The rules are expressed in algorithmic, functional form. The process of uncertainty propagation, i.e. from uncertainties in the evidence to the uncertainty in the hypothesis, is discussed with reference to several kinds of uncertainty representations described in part 1. When a hypothesis is supported by several rules with differing uncertainties, ways to aggregate these uncertainties are described. As in part 1, the discussion emphasizes the commonality and differences among the various uncertainty representations and. in particular, theircomputational ramifications on inference within the narrow context of the rule-framework considered.