4 ARAVIND SESHADRI
Population Generations Po ol Size Tour Size η
c
η
m
200 500 50 2 20 20
Table 1. MOP1- Parameters for NSGA-II
3.5.2. Polynomial Mutation.
c
k
= p
k
+ (p
u
k
− p
l
k
)δ
k
where c
k
is the child and p
k
is the parent with p
u
k
being the upper bound
4
on
the parent compo nent, p
l
k
is the lower bound and δ
k
is small variation which is
calculated from a polynomial distribution by using
δ
k
=(2r
k
)
1
η
m
+ 1
− 1, if r
k
< 0.5
δ
k
=1 − [2(1 − r
k
)]
1
η
m
+ 1
if r
k
≥ 0.5
r
k
is an uniformly sampled random number between (0, 1) a nd η
m
is mutation
distribution index.
3.6. Recombination and Selection. The offspring population is combined with
the current generatio n population and selection is performed to set the individuals
of the next generation. Since all the previous and current best individuals are
added in the population, elitism is ensured. Population is now sor ted based on
non-domination. The new generation is filled by each front subsequently until the
population size exceeds the current population size. If by adding all the individuals
in front F
j
the population exceeds N then individuals in front F
j
are selected based
on their crowding distance in the descending order until the population size is N.
And hence the process repeats to generate the subsequent generations.
4. MOP1 (Example Problem - 1)
This problem is to find the global pareto front for a discontinuous function given
by objective function as in [3]
f
1
(x) =1 − e
−4x
1
sin
6
(6πx
1
)
f
2
(x) =g(x)(1 − (
f
1
(x)
g(x)
)
2
)
where,
g(x) =1 + 9(
6
X
i=1
x
i
4
)
0.25
subject to 0 ≤ x
i
≤ 1, i = 1, . . . , 6. This set of objective function resulted in the
following solution as shown in Figure 1.
4
The decision space upper bound and lower bound for that particular component.
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