Overview of Multiobjective Optimization - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 18

box.. Assuming. that. the. nondominated. solutions i ,. ( i −. 1),. and. ( i +. 1). in.
Figure 2.6 . possess.the.objective.values.of. f

2 ,.and. (

)

1 2 ,.respec-
tively,.the.crowding.distance. I [ ] distance .of.the.solution. i .in.the.nondominated.
set. I .is.given.by

( , f ), . ( f ′ , f ′ ) f ′′ , f ′′

1 2 1

I i [ ] distance = f ′ − f + f ′′− f + f ′′− f

2 .

′ (2.7)

1 1 1 1 2

.

The.lowchart.of.the.crowding-distance.assignment.is.shown.in . Figure 2.7 ..
Its.procedures.are.as.follows:

. 1.. Initialize.the.crowding.distance. I [ ] distance .of.each.nondominated.solu-

tion.to.zero.

. 2.. Set.the.counter.of.objective.function.to.one,. m = 1.

. 3.. Sort. the. population. according. to. each. objective. value. of. the. m th.

objective.function.in.ascending.order.

. 4.. For. the. m th. objective. function,.assign. a. nominated. relatively. large.

value.to. I [ ] distance .of.the.extreme.solutions.of.the.nondominated.front,.
that.is,.the.smallest.and.largest.function.values..This.value.must.be.
greater.than.the.crowding.distance.values.of.other.solutions.within.
the.same.nondominated.front.

START I = sort( I,m ) m = m + 1

No

I [1] distance = ∞

l = |I |

m = M ?

Yes

I [ l ] distance = ∞

Set the solution
counter i = 1

END

i = 2

I [ i ] distance = 0

I [ i ] distance = I [ i ] distance + I [ i + 1] m - I [ i - 1] m

i = i + 1

f m max - f m min

No

i = l ?

i = i + 1

Yes

No Yes

Set the solution
counter m = 1

i = l - 1?

Figure 2.7
Flowchart.of.the.crowding-distance.assignment.