Overview of Multiobjective Optimization - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 17

p ∈ P ,.where. P .is.the.population.with.size.| P |,.the.number.of.solutions.that.
dominate p .(called.the.domination.count. n p ).and.a.set.of.solutions.that. p domi-
nates.(denoted. S p ).are.computed..This.requires. O MN

2

.comparisons..Then,.

( )

solutions.with.their. n p 0.are.the.members.of.the.first.nondominated.front.

For. every. solution p . lying. on. the. first. front,. its. values. of. n q . and. S p . are.
decreased.by.one..If.there.exists.any.solution. q with.

=

n q 0,.it.is.stored.in.a.
separate. list. Q ,. which. contributes. to. the. second. nondominated. front.. This.
procedure.is.repeated.until.all.nondominated.fronts.are.acquired.

=

2.2.3.2 Crowded-Comparison Approach
Recognizing. the. difficulties. of. specifying. the. sharing. parameter. for. the.
preservation. of. population. diversity,. a. crowded-comparison. approach.
is. proposed. in. NSGA2.. It. involves. two. processes:. the. crowding-distance.
assignment.and.the.crowded-comparison.operation.

2.2.3.2.1 Crowding-DistanceAssignment
To.obtain.diversified.nondominated.solutions,.the.crowding-distance.assign-
ment.is.devised..Its.purpose.is.to.estimate.the.density.of.solutions.surround-
ing.a.particular.solution.within.the.population..Solutions.in.the.less-crowded.
area.will.be.chosen.on.the.basis.of.their.assigned.crowding.distances.
The.crowding.distance.measures.the.perimeter.of.the.cuboid.formed.using.
the.nearest.neighbors.as.the.vertices..An.example.is.depicted.in . Figure 2.6. .
The. crowding. distance. of. the. i th. nondominated. solution. marked. with. the.
solid.circle.is.the.average.side.length.of.the.cuboid.represented.by.a.dashed.

f 2

0

i - 1

( f 1 ' , f 2 ' )

i

( f 1 , f 2 )

i + 1

( f 1 " , f 2 " )

Cuboid

f 1

Figure 2.6
Crowding-distance.computation.