Non Deterministic Approaches

The theory of fuzzy logic was introduced by Zadeh [21] in 1965, and has gained an increasing popularity during the last two decades. Its most important property is that it is capable of describing linguistic and, therefore, incomplete information in a non-probabilistic manner.

A fuzzy set can be interpreted as an extension of a classical set. Where a classical set clearly distinguishes between members and non-members of the set, the fuzzy set introduces a degree of membership, represented by the membership function. This membership function describes the grade of membership to the fuzzy set for each element in the domain. Figure 3 shows the membership functions of some typical normal fuzzy numbers.

Figure 3: Some typical membership functions that describe linguistic variables

While the concept of fuzzy logic was invented in 1967, it resulted mainly in practical applications during the last two decades. The works of Dubois and Prade [22, 23] contributed to a large extent to this evolution. The concept has been most successful in the application to controller design, known as fuzzy control [24].

Zadeh [25] extended the theory of fuzzy sets to a basis for reasoning with possibility. In this interpretation, the membership function is considered as a possibility distribution function, providing information on the values that the described quantity can adopt. More generally, the possibility is defined as a subjective measure that expresses the degree to which the analyst considers that an event can occur. It provides in a system of defining intermediate possibilities between strictly impossible and strictly possible events. Through this interpretation, the fuzzy concept has become a tool to model subjective knowledge numerically in a non-probabilistic concept. This has drawn the attention of the numerical community, since knowledge of uncertainties in a numerical model is commonly based on expert opinion. This has lead to the first attempts to use the fuzzy concept in a non-deterministic framework, resulting in some applications in structural optimisation under uncertainty [26,27]. Also, this has initiated the development of the Fuzzy Finite Element Method (FFEM) for numerical analysis of non-deterministic models [28].

However, the application of the fuzzy concept for non-deterministic numerical modelling is not straightforward. The main problem of the representation of a model property through a fuzzy set, is that the membership function does not relate to an objective measurable quantity. The level of membership that is assigned to different members of a fuzzy set is completely based on the subjective beliefs of the analyst. Therefore, also the fuzzy results obtained from the analysis will be biased with the subjective input. Hence, these results may only be interpreted in reference to the assumed fuzzy input. This poses an important restriction on the use of the fuzzy approach for numerical design validation purposes.

For a fuzzy representation of certain variabilities, the known PDF* has to be converted to a compatible membership function. A number of methods have been developed for this purpose [23,29]. The basic law for the conversion follows from the consistency principle, which states that the degree of possibility of an event is greater than or equal to its degree of probability. These conversion techniques always rely on some sort of subjective judgement. It is the authors' opinion that forcing the application of fuzzy sets into the domain of certain variabilities through a conversion of PDFs as described above is rather irrational. Available objective probabilistic data is replaced by a subjective description, resulting in the loss of very valuable information. This loss is generally unjustifiable. Therefore, the conversion of a PDF* to a membership function should not be done.

For uncertain variabilities, the fuzzy concept can be used for a hybrid uncertainty model. It stems from an alternative interpretation of a possibility distribution introduced by Dubois and Prade [30] based on the Evidence Theory. In this approach, a fuzzy number is used to represent a class of probability random quantities that have a cumulative distribution function (CDF*) in between boundaries derived directly from the possibility distribution. The left boundary on the compatible CDFs coincides with the increasing branch of the fuzzy number. The right boundary coincides with the complement of the decreasing branch of the fuzzy number. Figure 4 clarifies this approach. In this concept, the possibilistic approach becomes a tool to simultaneously examine the effect of a set of different PDFs in a single analysis. While the ability of this concept to model classes of probabilistic data seems extremely powerful, it has only been applied very rarely in uncertainty analysis.

Figure 4: Possibility distribution of a fuzzy number and corresponding lower and upper boundaries for CDF* compatible with the fuzzy number

Finally, an invariable uncertainty requires a fuzzy set that represents the subjective expectation of the analyst. When the invariable uncertainty represents an open design decision subject to optimisation, the analyst can express his preference of the quantity through the possibility distribution. Still, when interpreting the results, reference to the chosen input membership functions is imperative.

Considering the explicit subjective nature of a fuzzy set, it is concluded that it is most useful to describe uncertainties. The more objective information becomes available on a non-deterministic model property, the less the fuzzy concept is appropriate to describe it.