Keywords: uncertainty modeling, fuzzy probabilities, fuzzy randomness, imprecise probabilities, interval probabilities, evidence theory.

The prediction of the behavior and reliability of engineering systems has frequently interfered with problems in the specification of the numerical model due to the characteristics of the available information. The available information frequently appears as imprecise, diffuse, fluctuating, incomplete, fragmentary, vague, ambiguous, dubious, or linguistic and may possess a data-based, expert-specified, objective, or subjective background. Information from maps, plans, measurements, observations, experience, expert knowledge, and from codes and standards has to be quantified and processed simultaneously. Changes of boundary and environmental conditions have to be taken into consideration. These characteristics of the available information impede the specification of certain numerical models and precise parameter values without an artificial introduction of unwarranted information. An appropriate mathematical modeling is required in accordance with the underlying real-world information. Shortcomings, in this regard, may lead to biased computational results with an unrealistic accuracy and, therefore, lead to wrong decisions with the potential for associated serious consequences.

Whilst traditional probabilistic models including Bayesian approaches are already well-established and recognized largely as applicable to real-world problems, non-traditional models are still in a virgin state. Controversial discussions and skepticism are frequently caused by the misinterpretation that the new models are seen as fighting against traditional probabilistics. These non-traditional models are, however, not developed as replacements or competitors, with respect to the established probabilistic models, but as supplementary elements which can be very useful in various cases. The non-traditional models can be combined with probabilistics in various manners to produce fruitful hybrid models of improved flexibility and adaptability to particular problems. Recent developments indicate that they emerge as a sound basis for decision-making.

Models for the simultaneous consideration of uncertainty and imprecision have attracted increasing attention over the past decade. The various developments can be summarized under the term imprecise probabilities. This framework includes models such as evidence theory, possibility theory, interval probabilities, random sets, sets of probability measures, coherent upper and lower bounds on probability, p-box approach, and fuzzy probability theory. Their common idea is to consider simultaneously an entire range, or set, of plausible traditional models. Applications have been reported for the solution of various problems in diverse engineering disciplines. With the new modeling, extended insights are obtained in sensitivities, and a worst-case analysis in terms of probability is performed intrinsically. The quality of the results is improved, and wrong decisions can be prevented.

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