Overview of Multiobjective Optimization - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 27

where. N .is.the.total.number.of.ants,.and.∆τ i k .is.the.amount.of.pheromone.left.
on.the.path.from. i .to. j ,.relecting.the.quality.of.the.solution.of.ant. k ..It.can.be.
defined.as

Obj
Q

k

edge ij ( ) ∈ tour

k

k

∆τ =

. (2.15)

ij

0 otherwise

.

with.a.constant Q .

2.2.8 Particle Swarm Optimization

Particle.swarm.optimization.is.a.swarm.intelligence.algorithm.other.than.
ACO.. It. was. proposed. by. Ederhart. and. Kennedy. in. 1995. [46],. inspired. by.
the.social.action.of.searching.for.food.by.a.flock.of.birds..The.PSO.is.based.
on.a.simple.but.effective.mechanism.in.which.each.bird,.called.a.particle,.
adjusts.its.search.direction.according.to.three.factors:.its.own.velocity. v i ,.its.
own.best.previous.experience. pBest i

) ),.and.the.best.experience.in.the.flock.

(

( gBest ).[46].
The. PSO. has. been. successful. in. a. wide. variety. of. optimization. tasks..
However,.it.was.once.considered.unsuitable.to.deal.with.multiobjective.
optimizations.until.the.extension.made.by.Coello.Coello.[15]..By.apply-
ing.the.Pareto.ranking.and.keeping.the.historical.records.of.the.best.solu-
tions.found.by.the.particles.( pBest .and. gBest ).as.nondominated.solutions.
generated. in. the. past,. PSO. is. a. possible. solution. for. MOPs. . Figure 2.13 .
depicts.the.flowchart.of.a.multiobjective.PSO;.further.details.are.briefly.
described.next.
The.PSO.for.multiobjective.optimization.starts.with.a.population.of.ran-
dom. solutions. whose. itnesses. are. evaluated. by. the. objective. functions. of.
the.problem..Each.particle.flies.through.the.problem.space.with.a.velocity,.
which. is. constantly. updated. by. the. particle's. own. experience. and. the. best.
experience. of. its. neighbors.. In. each. iteration,. the. velocity. and. position. of.
each.particle.are.updated.by.the.following.equations:

( ) ( )

k

v g ( + 1) = ω v g ( ) + c r pBest − pos + c r gBest − pos g ( ) . (2.16)

i i 1 1 i i 2 2 i

.

pos g ( + 1) = pos g ( ) + v g ( + 1) . (2.17)

.

i i i

where. i = 1,2,…, N ,.ω.is.the.inertia.weight;. c 1 .and. c 2 .are.the.cognition.factor.
and. the. social-learning. factor,. respectively;. r 1 . and. r 2 . are. random. numbers.
between.0.and.1;. N .is.the.population.size;.and. v i ( g ).and. pos g

i ( ) .are.the.veloc-

ity.and.position.vectors.at.generation. g ,.respectively.