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There, y denotes an individual in objective space, and i and j have the same mean as y. Different function forms can be employed as an influence function, such as parabolic, square wave or Gaussian. A Gaussian influence function has the general form of: A Multi-Objective Genetic Algorithm Based on Density ψ (r ) = (1 (σ 2π )) e − r 2 17 2σ 2 (7) There σ is the standard deviation of the distribution. A very small σ can cause the influence function to decay very rapidly. In this case, the density function is insensitive to the distribution of the solution points.

0. f1 a1 i y l(i, y) A a2 l(j, y) j B f2 Fig. 1. Objective space with grids Fig. 2. Compensation to boundary effect Now let’s consider how to calculate the distance between two individuals. As a two-dimensional objective space shown in Fig. , the important thing is how to determine the values of a1 and a2. The grid size of each box becomes an indifferent region in which any solution points are considered to be the same after the quantities a1 and a2 are decided. In general, the appropriate grid size is problem-dependent and differs from one situation to another.

The reason for a Direct 70 break in the continuity in the Pareto-optimal 4 6 8 10 12 14 16 18 Launch Velocity (km/s) front is due to the availability of two different opportunity windows for an optimal mission. Fig. 5. Comparison of three transfer cases Next, we apply GOSpel on the same departure date window and obtain solutions with forced swing-bys. Figure 5 also shows the Pareto-optimal solutions for this case. Due to the swing-by option, the required launch velocity is much smaller than that obtained with the direct transfer.

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