Performance Measures on Jumping Gene - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 92

. 2.. Calculate.the.average.value.of. d i :

N
∑

− 1

1

. (5.8)

d = d i

( N − 1)

.

i = 1

. 3.. Obtain.the.value.of. d e .for. m = 1,.2,.⋯,. M :

e

. (5.9)

d = f I − f P

( ) ( ) m ext m ext

.

. where. f

( ) . and. f m ex ( ) . are. the. values. of. the. m th objective. val-
ues. of. extreme. solutions. in. the. nondominated. set. and. in. the. true.
Pareto-optimal.set,.respectively.

m ext

. 4.. Compute.the.metric.γ.by . Equation.(5.6) .

If.the.nondominated.solutions.found.are.uniformly.distributed.[i.e.,. d

i = .
for. i = 1,. 2,.⋯,. ( N −1)]. and. the. nondominated. extreme. solutions. are. equal. to
the.true.Pareto-optimal.extreme.solutions.(i.e.,.

d

M

e

),.the.value.of.γ.will
be.zero..This.is.the.most.ideal.case.for.the.distribution.of.the.nondominated.
solutions.
Regarding.the.distribution.in.which. d

Σ d =

0 m

m =

1

= d ∀ , . Equation.(5.6) .b ecomes i

i

M

∑

e

d

m

γ =

m = 1

. (5.10)

M

∑

e

d + ( N − 1) ⋅ d

m

.

m = 1

M

e

If. the. nondominated. solutions. are. crowded. in. a. small. region,. Σ d ≠ ,.

0 m

m =

1

one.has.0.< γ < 1.
In.addition,.γ ≥ 1.represents.an.even-worse.distribution..A.smaller.value.
of. γ. means. that. the. nondominated. front. is. more. widely. spread. and. more.
uniformly.distributed.along.the.true.Pareto-optimal.or.reference.front.(i.e.,.
there.is.better.diversity).

5.4 DiversityMetric:ExtremeNondominated
SolutionGeneration
The.“knee”.of.the.nondominated.front.and.its.extremes.should.be.equally.
important..However,.the.extreme.solutions.are.preferable.in.some.appli-
cations,.and.their.existence.is.critical.for.multiobjective.optimization..For.
example,. suppose. that. there. are. two. contradicting. objectives,. cost. and.