By Claes Sundin & Christer Bergström
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Example text
Figure 15 shows the results obtained in the design space together with the overlapping contour lines of the two objective functions. Figure 16 shows the solution obtained in the criterion space. As can be seen from the figures, the global Pareto front is composed of four distinct regions, one of which corresponds to the single point for which x ¼ (0, 0), the minimum of objective function F1 . The figures show that the NCþ method is capable of obtaining the complete global Pareto front, regardless of the symmetry problem.
Both main and total sensitivity indices are calculated. The discrepancy between these two indices represents the contribution to the output variance as a result of interaction effects among input variables. The main and total effects represent the minimum and maximum expected reduction of the output variance 54 M. D. GUENOV ET AL. if the input is fixed to a specific value. Therefore, freezing an input variable with a negligible total sensitivity index will not affect the output variance. It is proposed to remove from the original optimization formulation a variable with a negligible effect on all objectives and constraints [44].
Two main methods, Fourier amplitude sensitivity test (FAST) [40, 41] and Sobol [42, 43], have been developed to compute the different terms of the variance decomposition and are described in the two next sections. Based on two different approaches, they enable the computation of both main and total effect indices and lead to the same predictions. D. Reducing the Dimensionality of the Optimization Problem In the context of MDO, we assume that the model, which evaluates the objectives and constraints of the optimization problem, can be represented as a black box.