Performance Measures on Jumping Gene - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 90

. where. M .is.the.total.number.of.objective.functions,.and. f ( ) .and.

f m ( ) . are.the.values.of.the. m th.objective.function.of.the.nondomi-
nated.solution. I i .and.a.true.Pareto-optimal.solution P q ,.respectively.

m i

. 3.. Set. d = min( d ) .

i iq

. 4.. Compute.the.metric.α.by . Equation.(5.2) .

For.the.optimal.case,.the.metric.value.is.zero.when.all.the.nondominated.
solutions. are. exactly. equal. to. the. selected. true. Pareto-optimal. solutions..
Also,.a.smaller.metric.value.indicates.that.the.nondominated.front.is.closer.
to.the.true.Pareto-optimal.front,.that.is,.there.is.better.convergence.

5.2 ConvergenceMetric:DebandJainConvergenceMetric
For. most. real-world. applications,. the. huge. search. space. results. in. imprac-
ticality. for. inding. true. Pareto-optimal. solutions.. Therefore,. the. use. of. the.
generational. distance. convergence. metric. described. in. Section. 5.1. may. not.
be.practicable..Alternatively,.a.set.of.reference.solutions.is.used.to.form.the.
pseudo-Pareto-optimal. front.. This. has. been. employed. in. another. metric.
called.the.Deb.and.Jain.convergence.metric.[8].so.that.the.convergence.met-
ric.can.be.computed.
Assuming. that. a. group. of. algorithms. is. considered,. the. procedures. for.
producing.the.reference.solutions.are.as.follows:.(1).generate,.say,.50.sets.of.
nondominated.solutions.from.each.algorithm.and.(2).rank.all.nondominated.
solutions.obtained.by.these.algorithms..The.final.single.set.of.nondominated.
solutions.is.then.regarded.as.the.reference.solution.set.
The.mathematical.representation.of.this.metric.(denoted.β).is.the.same.as.
that.of.the.generational.distance..For. p .=.1,.one.has

N
∑

1

d i

β =

. (5.4)

N

.

i = 1

where. d i .is.the.normalized.distance.between.the. i th.nondominated.solution.
and.its.nearest.reference.solution,.and. N .is.the.total.number.of.solutions.in.
the.nondominated.set.
The.steps.to.calculate.β.are.as.follows:

. 1.. For.each.nondominated.solution. I i ,.calculate.the.normalized.Euclidean.

distances. d ir .between.the.solution. I i .and.each.reference.solution. R r .by

M
∑

2

f ( I ) −

−

f R ( )

m i m r

d

= . (5.5) ir

f R ( ) f R ( )

m max m min

.

m = 1