α-level optimization supports the solution of the mapping of a vector of fuzzy input variables onto a vector of fuzzy result variables
The dependencies between crisp vectors x and z from the fundamental sets X and Z,
on which and are defined, are described by the
deterministic mapping model M. This mapping model can represent any
arbitrary numerical algorithm, e.g., for a structural analysis, for the
solution of an eigenvalue problem, or for describing general chemical
or physical phenomena. Special properties of the mapping model are not
As a basis for α-level optimization the concept of
α-discretization is applied to numerically represent fuzzy sets with
the aid of their α-level sets. The mapping of onto is then
formulated as an optimization problem and solved with a modified
evolution strategy. A post-computation is performed to improve the
performance of the procedure. This combination represents a
efficient tool for fuzzy analysis. The numerical cost approximately
linearly increases with the number of fuzzy input variables.
An α-level set of the fuzzy set is defined as a crisp set
associated with a selected real number αk ∈ (0, 1]. All α-level sets are crisp subsets of the support S(). For several α-level sets of the same fuzzy set the following inclusion property holds
If – in the one-dimensional case – the fuzzy set is convex, each α-level set is a connected interval with
The concept of α-discretization provides a numerically efficient representation of fuzzy sets. In contrast to the extension principle, the elements of a fuzzy set are not considered separately one after the other, which means a discretization of the support, but the membership scale is now discretized, and the associated α-level sets are considered. For a sufficiently high number of α-levels a fuzzy set can be completely represented as a set of its α-level sets, that is, by its α-discretization
For each α-level, the associated α-level sets of the fuzzy input variables constitute an n-dimensional crisp subspace of the x-space, see Fig. 2.
All fuzzy input variables are discretized using the same number of α-levels αk, k = 1, ..., r to form the associated crisp subspaces . Each point in the subspace represents a crisp input vector. With the aid of the mapping model
crisp input vectors x ∈ are transformed into the result space. The elements zj of the associated crisp result vectors z = (z1, ..., zm) = f(x) represent elements of the α-level sets of the fuzzy result variables on the α-level αk. The mapping of all elements of yields the crisp subspace of the z-space. Once the largest element and the smallest element of the α-level set have been found, two points of the membership function are known, see Fig. 3. These points are sufficient to completely describe convex fuzzy result variables.
The search for the smallest and largest result elements on each α-level represents an optimization problem and is thus referred to as α-level optimization. The objective functions
must be satisfied by the optimum points . As no requirements are formulated for the mapping model, the optimization problem has no special but very general properties. That is, the optimum points may be located arbitrarily in the input subspaces . To solve this general optimization problem a modified evolution strategy has been developed. If every crisp subspace is a connected set, and if the mapping model is continuous and single-valued, α-level optimization yields exact results as the fuzzy result variables are then convex fuzzy sets. If the mapping model is not continuous or single-valued, α-level optimization yields exact envelope curves of the actual membership functions of the fuzzy result variables, which is commonly sufficient in engineering.
The application of a modified evolution strategy to solving α-level optimization does not guarantee the detection of the global optima in any case. They are only found with a certain probability depending on the problem.
A post-computation is thus carried out in order to raise the success probability of finding the global optima on each α-level. After the completion of all optimizations for the selected α-levels, the computed optimum points xopt are rechecked for optimality. For this purpose, all of the computed crisp results zj at those points x, which comply with the respective constraint (x1, ..., xn) ∈ for the current α-level αk, are compared with the preliminary optima and . As the inclusion property of α-level sets also applies to input subspaces
this recheck includes not only the points x evaluated on the α-level αk under consideration but also all points from α-levels αi > αk and, additionally, several points from the α-levels αi < αk. If an improvement of a result or is observed, the optimization algorithm is restarted at the best point found for this particular optimum search.
At the end of these additional computations the post-computation is repeated. If no (significant) improvement is obtained, the α-level optimization is considered to be completed. Beyond increasing the success probability of finding global optima this post-computation also ensures that the fuzzy results are always obtained as convex fuzzy sets.
- Möller, B, and Beer, M (2004) Fuzzy Randomness - Uncertainty in Civil Engineering and Computational Mechanics, Springer, Berlin Heidelberg New York.
- Möller, B, Graf, W, and Beer, M (2000) Fuzzy structural analysis using alpha-level optimization, Computational Mechanics 26:547–565.
- Möller, B, and Beer, M (1997) Application of Fuzzy Modeling in Civil Engineering, In: Proceedings of the Second International ICSC Symposium on Fuzzy Logic and Applications ISFL 97, edited by N. Steele. ETH Zürich, pages 345–351.