Alberto Bernardini

Determination of parameters range in rock engineering by means of Random Set Theory

Abstract Uncertainty arising from our relationship with the real world can often lead to imprecise probabilities, whose bounds are captured by Random Set Theory. Reasons leading to this kind of uncertainty in rock engineering are investigated, as well as the limitations of the probabilistic approach. Procedures are given to handle the usual information gained either in the field or from experts opinions, taking into account not only dissonance, but also non-specificity. Examples of applications are presented with special regard to event tree analysis, rock mass classifications, and reliability-based design of tunnels by means of the empirical recommendations of rock mass classifications.

Reliability analysis of rock mass response by means of Random Set Theory

Abstract When the parameters required to model a rock mass are known, the successive step is the calculation of the rock mass response based on these values of the parameters. If the latter are not deterministic, their uncertainty must be extended to the predicted behavior of the rock mass. In this paper, Random Set Theory is used to address two basic questions: (a) is it possible to conduct a reliable reliability analysis of a complex system such as a rock mass when a complex numerical model must be used? (b) is it possible to conduct a reliable reliability analysis that takes into account the whole amount of uncertainty experienced in data collection (i.e. both randomness and imprecision)? It is shown that, if data are only affected by randomness, the proposed procedures allow the results of a Monte Carlo simulation to be efficiently bracketed, drastically reducing the number of calculations required. This allows reliability analyses to be performed even when complex, non-linear numerical methods are adopted. If not only randomness but also imprecision affects input data, upper and lower bounds on the probability of predicted rock mass response are calculated with ease. The importance of imprecision (usually disregarded) turns out to be decisive in the prediction of the behavior of the rock mass. Applications are presented with reference to slope stability, the convergence-confinement method and the Distinct Element Method.

A RANDOM SET APPROACH TO THE OPTIMIZATION OF UNCERTAIN STRUCTURES

Abstract The single-objective optimization of structures, whose parameters are assigned as fuzzy numbers or fuzzy relations, is presented in this paper as a particular case of the random set theory and evidence theory approach to uncertainty. Some basic concepts concerning these theories are reviewed and the relationships among interval analysis, convex modeling, possibility theory and probability theory are pointed out. In this context a frequentistic view of fuzzy sets makes sense and it is possible to calculate bounds on the probability that the solution satisfies the constraints. Some special but useful cases illustrate in detail the meaning of the approach proposed and its links with a recent formulation conceived within the context of convex modeling. Some theorems allow a very efficient computational procedure to be set up in many real design situations. Two numerical examples illustrate the model presented.

Multiobjective optimization of uncertain structures through fuzzy set and random set theory

An interactive procedure to solve multicriteria optimization problems is proposed and discussed. A fuzzy set is used to model the engineers judgment on each objective function. The properties of the compromise solution obtained are investigated along with the links between the present method and those based on fuzzy logic. Uncertainty affecting the parameters is modeled by means of fuzzy relations or fuzzy numbers, whose probabilistic meaning is clarified by random set and possibility theory. Bounds to the probability that a solution satisfies a constraint can be calculated and procedures that consider the lower bound as a constraint or as an objective to be maximized are presented. Some theorems make the computational effort particularly limited in a vast class of practical problems. The relations with a recent formulation in the context of convex modeling are also stressed. Two examples show the effectiveness of the proposed approach.

Concept of Random Sets as Applied to the Design of Structures and Analysis of Expert Opinions for Aircraft Crash

Abstract The concept of random sets is exemplified for various applications pertinent to the education of students on the topic of uncertainty. In particular, several simple topics are discussed: reliability of a bar subjected to uncertain loads and the expert opinions on the failure of an aircraft. The examples elucidate the basic properties of the random sets.

Multiobjective optimization under uncertainty in tunneling: application to the design of tunnel support/reinforcement with case histories

In this paper, two questions common in tunnel design are addressed: (i) how to choose an optimum solution when more than one conflicting objective must be achieved; (ii) how to deal with data affected both by imprecision and randomness. Fuzzy Set Theory and Random Set Theory are used to develop a general interactive multiobjective procedure, which is then applied to the design of tunnel support/reinforcement. A case history illustrates how the procedure was successfully used in the preliminary design of a total of 40 km of tunnels in Central Italy. It is shown that the procedure allows the designer to become a knowledgeable decision maker because his interaction is required at the key points of the process, and because the trade-offs among the objective functions can be easily assessed. The designers personal input is valued and clearly defined in its impact on the solution. The case history demonstrates that, without an optimization procedure, it is extremely likely (probability of 99%) that a solution is chosen, which either increases the costs without increasing safety, or decreases the safety without decreasing the costs. Finally, it is shown that both imprecision and randomness can be easily taken into account in tunnel design.

Hybrid analysis of uncertainty: probability, fuzziness and anti-optimization

Abstract This paper elucidates that for rigorous analysis of many engineering structures, three seemingly unrelated concepts, namely probability, fuzziness and anti-optimization, ought to be utilized. This is in order to take into account the uncertainty in the actual values of the basic variables or on the models employed to evaluate the structure-expected behaviour. In this case, it is possible and reasonable to use the three paradigms in a combined manner. Some examples are given to illustrate these ideas.

Extreme probability distributions of random sets, fuzzy sets and p-boxes

The uncertain information given by a random set on a finite space of singletons determines a set of probability distributions defined by the convex hull of a finite set of extreme distributions. After placing random sets in the context of the theory of imprecise probabilities, algorithms are given to calculate these extreme distributions, and hence exact upper/lower bounds on the expectation of functions of the uncertain variable. Detailed applications are given to consonant random sets (or their equivalent fuzzy sets) and to p-boxes (non-consonant random sets). A procedure is presented to calculate the random set equivalent to a p-box and hence to derive extreme distributions from a p-box. A hierarchy of non-consonant (and eventually consonant) random sets ordered by the inclusion of the corresponding sets of probability distributions can yield the same upper and lower cumulative distribution functions of the p-box. Simple numerical examples illustrate the presented concepts and algorithms.

Bounding Uncertainty in Civil Engineering: Theoretical Background and Applications

The design of civil engineering constructions frequently involves a great uncertainty about loading conditions, material properties and their degradation in time, human errors in modeling, construction and successive management. These uncertainties seldom can be described by mapping probabilistic input variables through a deterministic model, to obtain the precise expectation of output parameters. In the last years researchers referred to the more general idea of imprecise probabilities to derive lower/upper bounds of output expectations. In this field, the theory of random sets appears as the most appropriate and relatively simple approach for many typical engineering problems, containing probabilistic methods, interval analyses and fuzzy sets as particular cases. The theoretical background and its connection to the more general theory of imprecise probabilities are briefly summarized. Finally the results of real world field applications of the theory to large scale urban building constructions in order to evaluate their seismic vulnerability are presented1.

Bounds on Previsions and Conditional Probabilities on Joint Finite Spaces Under the Assumption of Independence in Imprecise Probability

The primary difference between precise and imprecise probability theories lies in the allowance for imprecision, or a gap between upper and lower expectations (also called previsions) of bounded real functions. This gap generates a set of probability distributions or measures. As a result, in imprecise probabilities, the notion of independence on joint spaces is not unique; for example, notions of unknown interaction, epistemic irrelevance/independence and strong independence have been proposed in the literature. After introducing the three concepts of independence, various algorithms are proposed to calculate, through the different definitions of independence, both prevision and conditional probability bounds generated by marginal distributions over finite joint spaces. All algorithms are designed to accommodate two different types of constraints that define the sets of marginal distributions: previsions bounds or extreme distributions. Algorithms are applied to simple examples that show the role of the different quantities introduced and the equivalence of the two types of constraints. It is shown that, in epistemic irrelevance/independence, re-writing algorithms in terms of joint distributions turn quadratic optimization problems into linear ones.Copyright