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Research in Engineering Design

Erschienen in:

01.03.2007 | Original Paper

Feature-based classifiers for design optimization

verfasst von: Haoyang Liu, T. Igusa

Erschienen in: Research in Engineering Design | Ausgabe 4/2007

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Abstract

We present a design optimization method for systems with high-dimensional parameter spaces using inductive decision trees. The essential idea is to map designs into a relatively low-dimensional feature space, and to derive a classifier to search for high-performing design alternatives within this space. Unlike learning classifier systems that were pioneered by Holland and Goldberg, classifiers defined by inductive decision trees were not originally developed for design optimization. In this paper, we explore modifications to such classifiers to make them more effective in the optimization problem. We expand the notions of feature space, generalize the tree construction heuristic beyond the original information-theoretic definitions, increase the reliance on domain expertise, and facilitate the transfer of design knowledge between related systems. There is a relatively small but rapidly growing body of work in the use of inductive trees for engineering design; the method presented herein is complementary to this research effort.

Vorheriger Artikel Quantifying uncertainty in multicriteria concept selection methods

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All columns have the same length and cross-section properties; the beams are 50% longer than the columns with a second moment of cross-section area that is twice that of the columns; the same elastic material is used for all beams and columns.

C _n ^j is the binomial coefficient of n and j and the reduction by the factor of 2 is due to symmetry.

Standard finite-element analysis is used to compute g.

We can set γ₀ = ∞ and γ_M = − ∞ so that classes C ₁ and C _M would correspond to the highest and lowest performing designs.

Here \({\overline g }\) and s _g are the sample average and standard deviation of the performance values g(x _j) of the training data set, and the multiplier of 0.8 was chosen so that the three classes had approximately the same number of supervised data.

This is also known as the naive Bayes classifier and is based on the maximum likelihood principle.

For relatively small m, the knowledge abstraction level and the associated evaluation effort of the feature functions are usually high; the corresponding features are termed high-level features (as opposed to low-level features). It is preferable, however, to separate any quantities that require substantial evaluation effort from the feature coordinates and to treat them separately in the knowledge modeling process. For instance, low-fidelity approximate models g _approx(x) which are simpler than the original performance function g(x) may be too complex to satisfy the simplicity attribute for the feature vector. It is shown in Sect. 4.2 how g _approx(x) can be used in the design problem.

This is simply the ratio of the column height over width. Other measures of slenderness will depend on the type of column being examined. For thin-walled steel members, such measures would be in terms of the first and second moment of cross-section area and other aggregated dimensional quantities (Liu et al. 2004).

The domain expert that we use is a novice structural engineer with knowledge of frame behavior gained primarily from a graduate-level course in structural mechanics. This level of expertise is sufficient for meaningful interaction with the design process as illustrated in Fig. 1.

For instance, reduced-error pruning repeatedly analyzes the classifier resolution, quantified by an average information entropy, at each non-leaf node. If the resolution of the subtree with the non-leaf node as the root is not much higher than that of the trivial subtree where the same node is replaced by a leaf, then the tree is pruned at this node.

It may be necessary to adjust the performance thresholds used in (2) so that the expanded training data would be more evenly distributed among the classes C _j.

Although the average information entropy in (10) could be viewed as a special case of the expected utility in (12) by setting u _ij = −log₂ P _ij, the utilities are not usually defined in terms of probabilities.

Statistically derived models (Rudnyi 1996; Buzas 1997; Haq and Kibria 1997; De La Cruz-Mesia and Marshall 2003) are not of interest herein because they are typically in terms of basis function expansions with non-informative coefficients.

In this method, it is assumed that the inflection points are at the midpoint of every beam and column, except for the lower columns.

The thresholds that define the classes C′_j used in the second system would have to be adjusted so that class 1 would still represent high-performing designs.

Our classifier approach has also been successfully applied to a much less intuitive design problem involving thin-walled steel columns (Liu et al. 2004). It is noted that the emphasis of that paper was on an exploration of a new nonlinear model for cold-formed steel columns; only a brief outline of a simpler form of the classifier approach was given.

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Titel: Feature-based classifiers for design optimization
verfasst von: Haoyang Liu
T. Igusa
Publikationsdatum: 01.03.2007
Verlag: Springer-Verlag
Erschienen in: Research in Engineering Design / Ausgabe 4/2007
Print ISSN: 0934-9839
Elektronische ISSN: 1435-6066
DOI: https://doi.org/10.1007/s00163-006-0024-4

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