Fuzziness
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With the uncertainty model fuzziness subjective, incomplete and imprecise data can be described.
Fuzzy sets
In classical set theory the membership of elements in relation to a set is assessed in binary terms according to a crisp condition. An element either belongs or does not belong to the set, the boundary of the set is crisp. As a further development of classical set theory, fuzzy set theory permits the gradual assessment of the membership of elements in relation to a set; this is described with the aid of a membership function. A fuzzy set is defined as follows (see also Fig. 1):
If X represents a fundamental set and x are the elements of this fundamental set, to be assessed according to an (lexical or informal) uncertain proposition and assigned to a subset A of X, the set
is referred to as the uncertain set or fuzzy set on X.
µA(x) is the membership function of the fuzzy set and may be continuous, or the set contains only discrete elements assessed by membership values.
Fuzzy functions
A fuzzy function may be explained by extending the definition of a classical function. A classical function is a single-valued mapping of the elements t from the fundamental set T
where t 
T represents the arguments of the function and x X indicates the functional values or results. The set T is referred to as argument domain and X denotes the range of values of x(t).
The fuzzy function
(
) may thus also be interpreted as being a set of fuzzy results or fuzzy functional values 
F(X) belonging to specified 
F(T)
The fuzzy function
() may thus also be interpreted as being a set of fuzzy results or fuzzy functional values In structural analysis the fundamental set T may contain crisp parameters, e.g.
Figure 2: Fuzzy process
A multi-dimensional fuzzy function(t) may be formulated depending on fuzzy bunch parameters 
and crisp arguments t
Figure 3: Fuzzy field with bunch parameter representation
- the time coordinate τ
- the spartial coordinate θ
- and additional parameters φ
Figure 2: Fuzzy process
Bunch parameter representation
A multi-dimensional fuzzy function
(t) may be formulated depending on fuzzy bunch parameters For each crisp bunch parameter vector s with the assigned membership value µ(s) a crisp function
x(t) = x(s, t) with µ(x(t)) = µ(s) is obtained.
x(t) = x(s, t) with µ(x(t)) = µ(s) is obtained.
Figure 3: Fuzzy field with bunch parameter representation
References
- Beer, M (2004) Uncertain structural design based on nonlinear fuzzy analysis, Special Issue of ZAMM 84(10–11):740–753.
- Möller, B, and Beer, M (2004) Fuzzy Randomness - Uncertainty in Civil Engineering and Computational Mechanics, Springer, Berlin Heidelberg New York.
- Möller, B, and Beer, M (1998) Safety Assessment using Fuzzy Theory, In: Proceedings of the 1998 International Computing Congress in Civil Engineering. ASCE, Boston, pages 756–759.
- Möller, B, and Beer, M (1997) Uncertainty Analysis in Civil Engineering Using Fuzzy Modeling, In: 7-th International Conference on Computing in Civil and Building Engineering. , Seoul, pages 1425–1430.
- Zimmermann, H- (1992) Fuzzy set theory and its applications, Kluwer Academic Publishers, Boston London.
- Bandemer, H, and Gottwald, S (1989) Einführung in FUZZY-Methoden, Akademie-Verlag, Berlin.
- Rommelfanger, H (1988) Fuzzy Decision Support-Systeme, Springer, Berlin Heidelberg.
- Zadeh, LA (1965) Fuzzy sets, Information and Control 8:338–353.
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