design space pruning heuristics and global optimization method for ...

these two problems, the best known solutions were all located within the top 2% of the
ranked sequences in the branch-and-bound algorithm. Since the 7 th and 12 th best known
solutions (listed in Table 26) fall outside this range, the number of low-thrust
optimizations required to locate these two solutions is unknown.
Figure 57 plots the number of asteroid sequences that require low-thrust
optimization as a function of the number of asteroid sequences evaluated in the branchand-bound,
for the first 78,000 ranked sequences (plotted in red are the data points from
Table 26). In other words, Figure 57 illustrates the efficiency of the branch-and-bound
algorithm at pruning out branches of the tree using the relaxed two-impulse solutions. In
the range considered, approximately seven percent of the sequences in the branch-andbound
tree require low-thrust optimization – the rest are pruned out due to their optimal
two-impulse solution being less than the lower bound of 100 kg/yr. As the number of
sequences evaluated in the branch-and-bound tree increases, the percent that require lowthrust
optimization decreases, due to the fact that the optimal two-impulse solutions tend
to decrease as a function of their ranking in the branch-and-bound algorithm. This trend
was observed previously in Figure 53, which plots the evolution of the branch-and-bound
algorithm for the modified GTOC2 problem. The number of low-thrust optimizations
required as a function of the number of branch-and-bound sequences evaluated can also
be plotted for the modified GTOC2 problem, in order to illustrate this trend for a problem
where the entire branch-and-bound tree was evaluated. As expected, Figure 58 illustrates
how the percent of sequences that require low-thrust optimization decreases towards the
end of the branch-and-bound algorithm.
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