Progressively Interactive Evolutionary Multi-Objective Optimization ...

cating the robustness of our procedure.
6.4 ProblemDS1
Thisproblemhas10upperand10lowerlevelvariables. Thus,anoverallpopulationof
size Nu = 400 is used here. For this problem, we first consider τ = 1. Figure 19 shows
the obtained archive solutions for a typical run. It is worth mentioning here that this
problemwaspossibletobesolveduptoonlysixvariables(intotal)byourearlierfixed
BLEMOapproach(Deb and Sinha, 2009a). But here with our hybrid and self-adaptive
approach, we are able to solve 20-variable version of the problem. Later, we present
results with40variablesas well for this problem.
F2
1.2
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6
F1
0.8 1 1.2
Figure 19: Final archive solutions for
problemDS1.
F2
1.2
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6
F1
0.8 1 1.2
Figure 20: Attainment surfaces(0%, 50%
and100%)forproblemDS1from21runs.
The attainment surface plots in Figure 20 further show the robustness of the proposed
algorithm. The hypervolumes for the obtained attainment surfaces are 0.7812,
0.7984and0.7992,withamaximumdifferenceofabout2%. Theparametricstudywith
Nu in Figure 21 shows that Nu = 400 is the best choice in terms of achieving a similar
performance on the hypervolume measure using the smallest number of function
evaluations,thereby supporting our setting Nu = 20n.
Sincethis problemismore difficultcomparedtothepreviousproblems (TP1,TP2,
andTP4),we investigatethe effectofself-adaptivechanges inlower levelNSGA-IIparameters
(Nl and tl) with the generation counter. In the left side plot of Figure 22, we
show the average value of these two parameters for every upper level generation (T)
from a typical simulation run. It is interesting to note that, starting with an average
of 20 members in each lower level subpopulation, the number reduces with generation
counter, meaning that smaller population sizes are needed for later lower level
simulations. The average subpopulation size reduces to its lower permissible value
of four in 32 generations. Occasional increase in average Nl indicates that an upper
level variable vector may have been found near a previously undiscovered region of
the Pareto-optimal front. The hybrid algorithm increases its lower level population to
explorethe regionbetterinsuchoccasions.
Thevariationofnumberoflowerlevelgenerations(tl)beforeterminationalsofollows
a similar trend, except that at the end only 3-9 generations are required to fulfill
101