Performance Measures on Jumping Gene - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 91

. where. M .is.the.total.number.of.objective.functions;. f

m ( ) .
return.the. m th.objective.values.of.the.nondominated.solution. I i .and.
a.reference.solution. R r ,.respectively;. and. f R

) .and. f R m i (

min .are.
the. maximum. and. minimum. values. of. the. m th. objective. function.
within.all.solutions.in.the.reference.set,.respectively.

m ( max .and. f R

)

m (

)

. 2.. Set. d = min( d ) .

i ir

. 3.. Compute.the.metric.β.by.(5.4).

Similar.to.the.generational.distance,.a.smaller.value.of.β.reveals.that.the.non-
dominated.front.is.closer.to.the.reference.front,.that.is,.there.is.better.convergence.

5.3 DiversityMetric:Spread
The. spread. diversity. metric. evaluates. the. distribution. of. a. nondominated.
solution. set. spreading. along. the. true. Pareto-optimal. or. the. reference. front.
[7],.which.is.calculated.as

M N − 1

∑ ∑

e

d + d − d

m i

γ =

m = 1 i = 1

. (5.6)

M

∑

e

d + ( N − 1) ⋅ d

m

.

m = 1

where. M is.the.total.number.of.objective.functions;. d e .is.the.distance.between.
an.extreme.solution.in.the.true.Pareto-optimal.set.and.that.in.the.nondomi-
nated.set.for.the. m th.objective.function;. d i .is.the.distance.between.the. i th. an d.
( i .+.1)th.nondominated.solutions.(i.e.,.neighbor.nondominated.solutions);. d .is.
the.average.of.all. d i ;.and. N .is.the.total.number.of.solutions.in.the.nondomi-
nated.set.
The.procedures.for.calculating.γ.are.as.follows:

. 1.. Calculate.all.Euclidean.distances d i .for. i = 1,.2,.⋯,.( N −1):

M
∑ (

)

2

d = f ( I ) − f ( I ) . (5.7)

i m i m i + 1

.

m = 1

. where. M is.the.total.number.of.objective.functions,.and. f m i

) .and. (

1 + . return. the. m th objective. values. of. the. nondominated. solu-
tion. I i .and.the.nondominated.solution. I i +.1 ,.respectively.

f m i ( )