The first objective, constrained weight, is directly related to the minimization of the structure's raw material costs. In order to guarantee the structural mission compliance, different constraints are taken into account, including maximum allowable stresses, displacements of certain nodes and also the buckling effect.

The second objective, number of different cross-section types, is related to the construction costs and recently also with the life cycle cost. Its relevance grows with the structural size.

Two structural frame test cases of different size are compared. Real discrete standard cross section types are the variables which are used to encode the candidate chromosomes. So, the optimum solution set found by the multiobjective evolutionary algorithm represents practical structures which solve a real engineering design problem: each point is the structural design of minimum weight corresponding to a number of different cross-section types. The chromosome codification is done using the standard binary reflected gray code.

Among the evolutionary multiobjective algorithms, those belonging to the so called second generation are the most efficient. They are mainly characterized by the use of elitism and the growth of parameter independence compared with the first generation algorithms. Here, the DENSEA algorithm [1] is compared with a well-known second generation algorithm in the state of the art, NSGA2. Fifty executions of each case were run and a statistical analysis considering the hypervolume (S-metric [2]) is required to compare their relative performance. The DENSEA algorithm is designed specially to take profit of the reduced discrete functional search space in one of the objective functions, because the non-dominated front set is composed of isolated points and also the number of designs belonging to this set is low compared with the standard population size used in evolutionary algorithms. So, in this discrete search space problem with a low number of optimum solutions a proper treatment of population diversity is a key to success.

The results show a better quality of the final fronts achieved by DENSEA in terms of average and typical deviation of the achieved hypervolume compared with the NSGA2 in the structural test cases considered.

1

D. Greiner, J.M. Emperador, G. Winter, "Enhancing the multiobjective optimum design of structural trusses with evolutionary algorithms using DENSEA", 44th AIAA (American Institute of Aeronautics and Astronautics) Aerospace Sciences Meeting and Exhibit, paper AIAA-2006-1474, 2006.

2

E. Zitzler, L. Thiele, "Multiobjective Optimization using Evolutionary Algorithms - A comparative case study", In A.E. Eiben et al., (Editors), "Parallel Problem Solving from Nature", Lecture Notes in Computer Science, Vol. 1498, 292-301, 1998. doi:10.1007/BFb0056872

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