Geology Reference
In-Depth Information
O
d
m
X
= −
(
1
β
)
X
+
β
X
(9)
optimization some of researchers (Canyurt &
Hajela, 2005; Rajasekaran, 2001; Gholizadeh &
Salajegheh, 2010b) have combined the concepts
of CA and GA to create cellular genetic algorithms
(CGA). The CGA presented in this chapter is a
modified version of the CGA proposed by Gholiza-
deh and Salajegheh (2010b) and is denoted as
modified cellular genetic algorithm (MCGA). In
the MCGA, as well as the CGA, individuals of a
selected population are set on discrete locations
of a 2D grid. The state variables of each site are
the design variables. In the MCGA, the evolution
process is accomplished locally, with probabilistic
interaction rules applied synchronously to each
central site, and using information only from
members of its Moore neighborhood. When the
population has been updated, the evolutionary
rules of the automaton are repeated until one of
the stopping criteria is met. In both the CGA and
MCGA, the objective function of the optimization
problem is employed to define the fitness of each
design vector. In the MCGA the evolutionary rule
of the automaton includes the cellular crossover
operation (CCO) and the mutation operation (MO)
applied to the sites.
In the GA type evolutionary algorithms cross-
over operation creates one or more offspring from
the selected parents. Many different methods have
been proposed for crossing over in the GA such as
point crossover method. In this simple method one
or more points in the chromosome are randomly
selected as the crossover points. Then the vari-
ables between these points are merely swapped
between two parents. The problem with these point
crossover methods is that no new information is
introduced. But the blending methods mitigate this
difficulty by finding ways to combine variables
values from the two parents into new variable
values in the offspring. For instance in (Haupt &
Haupt, 2004) such method has been proposed so
that a single offspring variable value
X
O
is pro-
duced from combination of the two corresponding
parents' variable values as follows:
where
β
is a random number on interval [0, 1],
X
d
and
X
m
are variables in the father and mother
chromosomes, respectively.
These types of crossovers can not provide suf-
ficient information for a comprehensive search of
the design space in the complex and large-scaled
problems. Therefore the found solutions are
usually local optima. In the MCGA a powerful
crossover operation is employed that provides
sufficient information for comprehensive and fast
exploration of the design space.
In the MCGA model, a population of potential
designs is structured in a 2D grid. In this case, each
site contains a real-valued string describing of a
design and therefore the state of the cellular au-
tomaton in each site is a vector of design variables.
T
s
→ =
X
{ ,
x x
, ...,
x
} ,
i
=
1 2
,
, ...,
n
i
i
1
2
n
c
(10)
The CCO acts on the design variables and
combines the information available at the central
sites and their immediate neighbors. In this case, a
virtual individual is produced by using the fitness
indices of the individual in the immediate neigh-
bors of each central site as follows (Gholizadeh
& Salajegheh, 2010b):
n
n
c
c
∑
∑
X
X
best
(
(
X
best
Xi f
) ) /
f
=
+
−
i
i
i j
,
j
j
j
=
1
j
=
1
(11)
where
X
best
is the best solution found up to current
iteration.
X
i
best
is the best individual in immediate
neighbors of
i
ith central cell.
X
i
,
j
is the
j
th indi-
vidual in immediate neighbors of
i
ith central cell
and
f
j
is its fitness
v
alue.
In creation of
X
i
the effect of individuals hav-
ing better fitness values is higher and vice versa.
In each discrete time step, the CCO combines the
central cell with the virtual individual and pro-
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