Performance Measures on Jumping Gene - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 89

5

Performanc e Measures on Jumping Gene

5.1 ConvergenceMetric:GenerationalDistance
The. generational. distance. convergence. metric. α measures. the. degree. of.
closeness. between. the. sets. of. the. nondominated. solutions. and. the. true.
Pareto-optimal.solutions..Its.mathematical.representation.[18].is.given.by

1

N

p

1

∑

d i p

. (5.1)

α =

N

.

i = 1

where. d i .is.the.distance.between.the. i th.nondominated.solution.and.its.near-
est.true.Pareto-optimal.solution,.and. N .is.the.total.number.of.solutions.in.the.
nondominated.set.
It.was.recommended.in.Zitzler.[20].that.a.value.of. p = 1.is.more.appropriate.
due.to.its.simplicity.and.averaging.characteristics.[7]..Then,. Equation (5.1) .
becomes

N
∑

1

α = d i . (5.2)

N

.

i = 1

The.steps.for.calculating.α.are.as.follows:

. 1.. Find. a. set. of,. say,. 500. uniformly. spaced. nondominated. solutions.

lying.on.the.true.Pareto-optimal.front.

. 2.. For.each.nondominated.solution. I i ,.ind.the.Euclidean.distances. d iq .

between. the. solution. i . and. each. true. Pareto-optimal. solution. P q . in.
the.space.of.the.objective.values.using.the.following.formula:

M
∑ (

2

)

. (5.3)

d = f ( I ) − f P ( )

iq m i m q

.

m = 1

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