Comparative Study on Multi-Objective Genetic Algorithms for Seismic Response Controls of Structures - Design Optimization of Active and Passive Structural Control Systems - page 342

= + ∑

f 1 s (8)

• Stores the non-dominated solutions found

so far in an external population.

i i

i i ,  j

• Uses the concept of the Pareto dominance

in order to assign scalar fitness values to
individuals.

where f

j ∈ [ , 1 . When the size of the archive
set is more than the defined size, the clustering
analysis is carried out and then prunes the infe-
rior individuals. The average linkage method
(Morse 1980) is suggested as the clustering
analysis. The first step is initialing cluster set C ,
and the next step is calculating the distance d of
two cluster c 1 and c 2 . The equation of d is ex-
pressed (Zitzler and Thiele 1999) as

N

• Performs clustering to reduce the number

of non-dominated solutions stored without
destroying the characteristics of the trad-
eoff front.

The originalities of the SPEA method are as
follows:

• It combines the above three techniques as a

single algorithm.

1

∑

d i i

= ⋅ − (9)

1 2

c ⋅ c

• Irrespective of the dominancy of members

of the population, the individual fitness is
calculated only from the solutions of ex-
ternal sets.

i ∈ c i , ∈ c

1 2 1 1 2 2

where the metric ⋅ is the distance between two
individuals i 1 and i 2 . Two clusters c 1 and c 2 are
determined with a minimal distance d ; and then,
the chosen clusters are added to the larger cluster;
finally, the reduced non-dominated set is com-
puted by selecting a representative individual per
each cluster (Zitzler and Thiele 1999).

• The individuals of the external set will par-

ticipate in the selection process.

• The Pareto-based new niching method

which does not rely on any sharing or niche
radius is developed to sustain diversity in
the population (Zitzler and Thiele 1999).

The non-dominated sorting and assigning fit-
ness are a little bit different with the previously
reviewed GAs. The individuals in the archive
set are ranked and then the individuals in current
population are evaluated. The fitness of the archive
set is defined as (Zitzler and Thiele 1999)

Strength Pareto Evolutionary
Algorithm 2 (SPEA2)

A modified version of the SPEA (i.e., SPEA2)
is suggested by Zitzler et al. (2001, 2002). The
main improvements are (Zitzler et al. 2001, 2002):

n

s

i =

(7)

• A fine-grained fitness assignment strategy

N +1

• Density estimation technique

• Enhanced archive truncation method

where n is the number of individuals in P that are
covered by i , and N is the size of P . The fitness of
the current population is calculated by summing
the strengths of all the external non-dominated
solutions i that cover j . The equation is expressed
as (Zitzler and Thiele 1999)

In the fitness assignment of this algorithm,
both non-dominated individuals and dominated
individuals are considered simultaneously to avoid
the situation that individuals are dominated by the
same archived members as shown in Figure 5. As