Experimental comparison of probabilistic methods and fuzzy sets for designing under uncertainty
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Recently, probabilistic methods have been used extensively to model uncertainty in many design optimization problems. An alternative approach for modeling uncertainties is fuzzy sets. Fuzzy sets usually require much less information than probabilistic rnethods and they rely on expert opinion. In principle, probability theory should work better in problems involving only random uncertainties, if sufficient information is available to model these uncertainties accurately. However, because such information is rarely available, probabilistic models rely on a number of assumptions regarding the magnitude of the uncertainties and their distributions and correlations. Moreover, modeling errors can introduce uncertainty in the predicted reliability of the system. Because of these assumptions and inaccuracies it is not clear if a design obtained from probabilistic optimization will actually be more reliable than a design obtained using fuzzy set optimization. Therefore, it is important to compare probabilistic methods and fuzzy sets and determine the conditions under which each method provides more reliable designs. This research work aims to be a first step in that direction. The first objective is to understand how each approach maximizes reliability. The second objective is to experimentally compare designs obtained using each method.
A cantilevered truss structure is used as a test case. The truss is equipped with passive viscoelastic tuned dampers for vibration control. The structure is optimized by selecting locations for tuning masses added to the truss. The design requirement is that the acceleration at given points on the truss for a specified excitation be less than some upper limit. The properties of the dampers are the primary sources of uncertainty. They are described by their probability density functions in the probabilistic analysis. In the fuzzy set analysis, they are represented as fuzzy numbers.
Two pairs of alternate optimal designs are obtained from the probabilistic and the fuzzy set optimizations, respectively. The optimizations are performed using genetic algorithms. The probabilistic optimization minimizes the system probability of failure. Fuzzy set optimization minimizes the system possibility of failure. Problem parameters (e.g., upper limits on the acceleration) are selected in a way that the probabilities of failure of the alternate designs differ significantly, so that the difference can be measured with a relatively small number of experiments in the lab.
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