α-level optimization
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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
required.
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
numerically
efficient tool for fuzzy analysis. The numerical cost approximately
linearly increases with the number of fuzzy input variables.
α-level sets
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
with
α-discretization
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
constitute an
n-dimensional crisp subspace 
of the x-space, see Fig. 2.
optimization problem
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 
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 
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.post-computation
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 subspacesthis 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.References
- 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.
