# Uncertainty in Engineering

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TU Dresden
Fakultät Bauingenieurwesen
Institut für Statik und Dynamik der Tragwerke
Prof. Dr.-Ing. habil. B. Möller
01062 Dresden
Germany

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Collaborative Research Center - SFB 528 granted by DFG
DFG Research Unit FOR500
Flood Risk Management Research Consortium

# α-level optimization

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α-level optimization is a general method for a numerical processing of fuzziness as a basis for fuzzy analysis.

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

Figure 1. α-level sets

If – in the one-dimensional case – the fuzzy set is convex, each α-level set is a connected  interval 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 of the fuzzy input variables constitute an n-dimensional crisp subspace of the x-space, see Fig. 2.

Figure 2. Constitution of crisp subspaces

### 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 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.

Figure 3. α-level optimization

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 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.