Hybrid Approach for Optimal Nesting
Using a Genetic Algorithmand
a Local Minimization Algorithm
Kikuo Fujita,Shinsuke Akagi and Noriyasu Hirokawa
Department of Mechanical Engineering
for Industrial Machinery and Systems,
Osaka University
Suita,Osaka,JAPAN
Abstract
In layout design problems including blank nesting,the positions
and directions of layout elements must be determined so as to
minimize the total space.It is difcult and computationally
timeconsuming to nd the optimal solution for such layout
problems,because they include a lot of underlying combinational
conditions.In this paper,we present an approach for
optimal nesting by combining a genetic algorithm and a local
minimization algorithm.In the approach,the genetic algorithm
is used for handling the combinations which are represented
in the string,and the local minimization algorithm is used for
determining the embodiment layout under the xed combinations
so as to minimize the scrap volume which is corresponding to
the tness value in the genetic algorithm.And we present an
example for showing the effective nesting result produced by this
approach.
1 Introduction
Layout problems are found in various kinds of design problems
such as blank nesting problems and component placement
problems in plant design and VLSI design.Those layout
problems,in general,include many combinational conditions and
it is difcult to nd an optimal solution by means of traditional
hillclimbing methods of mathematical programming techniques.
Therefore,heuristic search procedures are applied for many
cases,and recently some probabilistic approaches such as
simulated annealing methods (SA,[van Laarhoven,1987]) and
genetic algorithms (GA,[Goldberg,1989][Michalewicz,1992])
are used.As for the nesting problems,Cheok et al.introduced
some heuristics [Cheok,1991] and Jain et al.applied simulated
annealing [Jain,1990].Kr¨oger et al.used a genetic algorithm
for the case that the pieces are rectangular [Kr¨oger,1991].
Among these probabilistic approaches,the interests on genetic
algorithms are going to be grown as a novel and effective
approach in the elds of combinational optimization and global
optimization problems.
The origin of genetic algorithms was found in the studies for
simulating the mechanism of the natural evolution and selection
by John Holland [Goldberg,1989].In genetic algorithms,
the design variables to be optimized are coded as a string
corresponding to a gene,and it is optimized through the
repeating manipulations with the genetic operators for a set of
strings.Its characteristics are summarized as follows:(1) It
is a probabilistic approach using a set of tentative solutions,
which is called a population of individuals in the terminology
of genetic algorithms.(2) Binary or character strings are used
for representing a solution.And,(3) how subparts of strings,that
are corresponding to building blocks,effect on the optimality is
important for better search performance.
As aforementioned,Kr¨oger et al.used a genetic
algorithm for the rectangular nesting problems [Kr¨oger,1991].
And,Shahookar et al.applied another genetic algorithm
to the standard cell placement problem in VLSI design
[Shahookar,1990].However,they dealt with simple problems
that layouts could be represented only with discrete or
combinational variables without continuous variables.Their
approaches can not be immediately applied to the cases that the
pieces with free shapes should be arranged,in which the layout
of a piece must be represented directly with its position and
direction in a coordinate system.
In this paper,we propose a hybrid approach combining a
1
h
B
Plate
X
Y
i
( )
i
x
i
, y
O
Figure 1:Nesting problem
genetic algorithm and a local minimization algorithm for such
hard nesting problems.In the approach,the nesting problem
is considered as a global optimization problem which includes
a number of multiplepeaks,and the description of a layout is
hierarchically separated into the two parts;the combinational
part and the continuous part.Accordingly,a genetic algorithm
for combinational optimization is used for the former and a local
minimization algorithm in a continuous subspace is used for the
latter,respectively.
2 Formulation of Nesting Problem
Firstly,the nesting problem and its formulation as an
optimization problem are described.
In this paper,we consider the nesting problem that the plural
pieces,Piece
i
,(i = 1;:::;n),are arranged into a plate with a
certain breadth,B,so as to minimize the scrap volume (Fig.1).
The formulation of this kind of nesting problem is shown in the
following.
Besides,the shapes of pieces are assumed to be convex
polygons in order to reduce the computation time in the
computational example which will be shown in the later section.
2.1 Design Variables
The position of the key point in a coordinate system;(x
i
;y
i
),and
the direction of the key axis;
i
,for the respective piece,Piece
i
,
are taken as the design variables.The position of the plate is
determined so as to t its leftside edge to the piece which is
located in the most left position and to t its lowerside edge to
the piece which is located in the lowest position under the xed
positions and directions of the pieces.After that,the necessary
height of the plate,h,and the overhang value,,can be calculated
by referring the determined position of the plate.
2.2 Constraints
The two following constraints must be considered.
The constraint in order to avoid any overlaps among the
pieces:The overlap between Piece
i
and Piece
j
is denoted
with overlap
i;j
.Then,the total of the overlaps are
represented with the following formula:
Overlap =
n−1
i=1
n
j=i+1
overlap
i;j
(1)
Then,the constraint is formulated as follows.
Overlap =0 (2)
Theconstraint inorder that all ofpiecesarelocatedwithin
the breadth,B,of the plate:This constraint is dened as
follows by using the overhang value,.
=0 (3)
2.3 Objective Function
The objective is to minimize the scrap volume,Scrap,as
mentioned above.This can be dened as the following equation
by using the variables shown in Fig.1.
min:Scrap =(B+ )h−
n
i=1
a
i
(4)
where,a
i
is the area of Piece
i
.
Besides,the reason why the part,(B+ ) in Equation (4) is not
represented simply as B is that the overhang constraint dened
with Equation (3) may be violated in the process of the local
minimization,which will mentioned in the later section,though
it should be satised at the end of the process.
3 Hybrid Approach for Nesting Problem
3.1 Building Block Hypothesis in Genetic Algo
rithms
Genetic algorithms are the search algorithm using population
of strings,and how subparts of strings effect on the optimality
and how they are preserved and produced through the genetic
operators are important for better search performance,as
mentioned in Introduction.The efciency and advantage of
genetic algorithms is theoretically explained with building block
hypothesis (e.g.,[Goldberg,1989]).
The concept called schemata is usually introduced for
discussing superiority or inferiority of the subparts of strings.In
genetic algorithms,the respective string itself is not meaningful,
2
h
e
c
g
j
a
b
i
f
l
d
k
h
e
c
g
j
a
b
d
f
k
i
l
Parent 1:
Parent 2:
Child:
Scrap
= 13.30
Scrap
= 19.43
Scrap = 9.99
Building
Block
Building Block
`****jabdf***'
`he*********'
j
a
b
d
f
h
e
h
e
c
g
j
a
b
d
f
k
i
l
Figure 2:Examples of building blocks in a nesting problem
but the set of strings which share the substructure each other is
important.The substructure sharing over the subset of strings is
represented with a schema.For convenience,a search problem
is assumed to be coded with character strings like`hecgabfd',
and this type of coding is used for traveling salesman problem
etc.In order to represent sharing substructures,the don't care
symbol,`*',which is matched with any characters is introduced.
For example,a schema,`**cga***'is a representation of a
subset of similar strings including`hecga
bfd',`ehcga
fbd',
`decga
bfh'etc.,and its dening length is 3 corresponding
to the length of its subpart;`cga'.How the schemata are
preserved and produced from generation to generation depends
on the number of similar strings,the tness values of the similar
strings,the dening length of the schema,the coding method and
the genetic operators.Among these factors,the latter two ones
are related to the actual implementation of genetic algorithms
for the respective problem,and it is important to select suitable
coding method and genetic operators for better performance.As
for the coding method,the symbols in the close positions of a
string should share some meanings for better tness value each
other.As for the genetic operators,their operation must preserve
the effective schemata and have the ability of enlarging them.
In these contexts,the better substructures included in the string
are called`building blocks'.
3.2 Concept of Hybrid Approach
When the building block hypothesis is contrasted with the nesting
problem,the following context emerges:In the nesting problem,
there are a variety of substructures of layouts which are effective
and meaningful for reducing the scrap volume.If some of
them could be well merged each other into the whole layout,
better layouts would be produced.Figure 2 gives an example
of building blocks in the nesting problem mentioned in the
later section,where a new layout called`Child'is produced by
merging two layouts called`Parent 1'and`Parent 2'through
a crossover operator.Child inherits some substructures;for
example, h and e from Parent 1,and j,a,b,d and f from
Parent 2.These substructures are corresponding to building
blocks.Moreover,the respective substructure of plural pieces can
be recognized to be composed of a set of pairs of pieces,which
we call meaningful neighboring relationships.For example,
the ordered substructure, j,a,b,d and f,is broken into the
pairs, j and a, a and b, b and d,and d and f.In this
concept,the pair not only means a list of two pieces but also
includes the dimensional relations;in which directions they are
touching each other.This point is different from the traditional
coding method used for solving the traveling salesman problem
with genetic algorithms.
Accordingly,we dene the string for nesting problems based
on the meaningful neighboring relationships.That is,the chained
list of the pairs forms an ordered list of pieces corresponding
to the string used in genetic algorithms.However,traditional
genetic algorithms manipulate only the combinations within the
ordered list.And it can not directly manipulate the embodiment
positions and directions of the pieces,which are represented with
a large number of continuous variables.Therefore,we combine
a local minimization algorithm with a genetic algorithm in order
to handle them.
This hybrid concept summarizes as the following approach for
the nesting problem:
Genetic algorithm The neighboring structures of the
pieces are manipulated with the genetic operators.
Local minimization algorithm The embodiment posi
tions and directions of the pieces are determined by numer
ical search algorithm under the neighboring relationships
xed with the genetic algorithm.
3.3 Formulation for Hybrid Approach
The formulation of the nesting problem shown in the previous
section must be arranged for the above hybrid approach.The
arrangement is shown in the following.
The neighboring relationships,i.e.,the ordered list of the
pieces,Piece
i
(i = 1;:::;n),are restricted in the order,Piece
s
1
,
Piece
s
2
,Piece
s
3
,:::,Piece
s
n
,which is used as a string in the
genetic algorithm.In addition to the order,the string includes the
relatively positional relationships between the two pieces which
are in the neighbor.
Corresponding to the string representation,in order to
maintain the neighboring relationships in the process of
local minimization,we introduce the following supplementary
objective function in addition to the constraints and objective
function which were mentioned in Section 2.
min:Dist2 =
n−1
i=1
distance
s
i
;s
i+1
2
(5)
where,distance
i;j
is the distance between Piece
i
and
Piece
j
.
3
X
Y
String:
Layout:
. . . . .
. . . . .
( 0, 0 )
.
.
.
.
.
.
.
.
O
2
S
X
1
S
1
S
O
1
S
Piece
1
S
l
2
S
Y
2
S
Y
1
S
X
2
S
l
i
S
X
i
S
O
i
S
S
Piece
i
X
i  1
S
Y
i
S
Y
i  1
S
S
Piece
i
S
Piece
i  1
Piece
2
S
Piece
1
S
Piece
2
S
r
i  1
S
+
2
S
S
Piece
i  1
O
i  1
S
i
S
r
i
S

i  1
S
1
S
Figure 3:Representation of a layout
In the equation,the square on the distance is introduced in order
to t its dimension with the one of Equation (4).
On the other hand,the relatively positional relationships
between pieces corresponding to their embodiment positions and
directions are represented in the following way under the order
of the pieces dened in the string.
For the respective piece Piece
s
i
,the key point and key direction
are dened,and the local coordinate system,O
s
i
−X
s
i
Y
s
i
,that the
key point is its origin and that the key direction is its x axis is
introduced as shown in Fig.3.Based on the local coordinate
systems,the position of the key point and the angle of the key
direction of Piece
s
i
are dened on the local coordinate system
of Piece
s
i−1
with a polar coordinate expression as followings:
First,a line segment between the origins,O
s
i−1
and O
s
i
,of
the two local coordinate systems is introduced.And then,the
differential angle between the line segment and both of the x
axes of the local coordinate systems are dened as
s
i−1
and
s
i
,respectively,as shown in the gure.The length of the line
segment is also represented with three variables,r
+
s
i−1
,l
s
i
and
r
−
s
i
,which are respectively the part on Piece
s
i−1
,the part in the
outside of both pieces,and the part on Piece
s
i
.Among these three
length variables,r
+
s
i−1
and r
−
s
i
are respectively depended on the
angles,
s
i−1
and
s
i
,and the shapes of the pieces.Consequently,
the relatively positional relationship between two pieces can be
represented with the three independent variables,
s
i−1
,l
s
i
and
s
i
.In addition,as for the rst piece Piece
s
1
in the string,
the position of its key point,O
s
1
,is unied with the origin of
the global coordinate system,O−XY,and its direction,
s
1
is
dened as the gradient from the x axis of the global coordinate
system.This denotation method is effective for preserving how
each piece is neighboring with other piece on applying genetic
operators,which will be mentioned in the next section.
With the above denotation,the design variable vector u,which
is manipulated with the local minimization algorithm,is dened
as follows:
u =(
s
1
;
s
1
;l
s
2
;
s
2
;
s
2
;:::;l
s
n
;
s
n
) (6)
Besides,the key point of the respective piece is set to the
center point of the circumscribed circle which has the minimum
radius of its various circumscribed circles for the convenience of
computation.
4 HybridAlgorithmfor Nesting Problem
The hybrid algorithm for the nesting problems based on the
concept mentioned in the previous section is shown in this
section.
4.1 Outline of Hybrid Algorithm
Figure 4 shows the outline of the algorithm,in which the
local minimization process is included in the evaluation step
of individuals in the genetic algorithm.The genetic algorithm
is composed of evaluation,scaling,selection,crossover and
mutation steps as similar with traditional genetic algorithms.
The local optimization algorithm is applied for determining the
embodiment layout based on the string and the initial values of
the design variables u which are determined through the above
genetic operation steps.
The respective steps of the approach are explained in the
following subsections.Besides,as for the genetic operations,
there are many versions and it is important to select effective
ones from various ones,as mentioned in the previous section.
The following genetic operators used in our algorithm are
circumspectly selected for the nesting problem.
4.2 Population Exchange
In the genetic algorithm,as shown in Fig.4,rst of all,the
initial population which consists of N individuals is randomly
generated.After that,the next population is produced from
the current population through the various genetic operations.
And these population exchanges are repeated until the repetition
number reaches sufcient iterations.
4.3 Evaluation
The tness value f
i
of an individual,which is corresponding to
the objective function to be maximized in traditional optimization
4
Local
Minimization
Algorithm
Genetic
Algorithm
Quasi
Newton
method
gen = 0
start
Generating Initial Population
Evaluation
Generating New Population
Scaling
Selection
Crossover
Mutation
Local Minimization
False
gen = gen + 1
True
gen > GEN
end
String &
Initial u
Optimal
u
Figure 4:Outline of hybrid algorithm
techniques,is calculated from the scrap volume Scrap
i
of the
layout with the following equation (e.g.,[Rao,1991]).
f
i
=
Scrap
max
−Scrap
i
Scrap
max
−Scrap
min
2
(i =1;:::;N) (7)
where,Scrap
max
and Scrap
min
are the maximum and minimum
value of the scrap volumes of the individuals in a population,
respectively.Through the above equation,the tness value for a
respective individual can be translated into the relative one.
4.4 Scaling
Next,the following operations are applied for the tness values
determined by Equation (7) in order to effectively execute the
selection which is mentioned in the next subsection.
truncation (e.g.,[Goldberg,1989]) Based on the
average f
avg
and the standard deviation of the tness
values,the individuals which satisfy the equation;f
i
<
f
avg
−c ,are eliminated from the population pool for
the selection which is mentioned in the next subsection.
Besides,a c = 2:0 is used in the computational example
mentioned in the next section.
Linear Scaling (e.g.,[Goldberg,1989]) Moreover,the
tness value of the rest of individuals are rearranged with
the following equation.
f
0
i
= a f
i
+ b (8)
where,the coefcients a and b are determined so as to
satisfy the equations;f
0
max
= C
mult
f
0
avg
,f
0
avg
= f
avg
.
Besides,a C
mult
=1:6 is used in the computational example
mentioned in the next section.
These rearrangements are expected to effect on maintaining the
variety of the individuals in a respective population and nding
globally optimal layouts.
4.5 Selection
After rearranging the tness values,the pairs of individuals,
which are the parents for mating,are selected fromthe population
pool based on the tness values,f
0
i
,until the number of pairs
reaches the necessary number.For this selection step,we use
the two strategies; expected value plan and elitist plan (e.g.,
[Goldberg,1989]).With the expected value plan,the number
of the copies of a respective individual selected is certainly
determined according to the expected number calculated fromthe
tness value,and this plan is effective for reducing the stochastic
errors of selection.With the elitist plan,the individual with the
highest tness value,i.e,the best layout in the parent population,
is directly copied into the child population without applying the
crossover and mutation operators.
4.6 Crossover
Two child individuals are generated by mating a selected pair
of parent individuals with the crossover operator which is called
order crossover (e.g.,[Shahookar,1990]) under the crossover
probability P
c
.
Figure 5 illustrates the order crossover.It operates as follows:
(i).The individuals of the selected pair is named`Parent 1'and
`Parent 2',respectively.
(ii).The crossover point is selected randomly.
(iii).As for the left part of the crossover point,the substring is
simply inherited from Parent 1 to Child 1.And as for the
right part,the genes,i.e.,pieces,which are not included in
the left part of Parent 1 are arranged into Child 1 in the
sequence that they are included in Parent 2.With these
procedures,a new full string,Child 1,is produced.
(iv).The operation (iii) is repeated by exchanging the role of the
Parent 1 and Parent 2 for producing Child 2.
5
Parent 2:
Parent 1:
cj fda i b gh e
+a +b +e +f +h+c +d +i +j
Crossover point
X
Child 1 :+a +b +c +d +e +f h=j i g
+g
Figure 5:Order crossover
On the other hand,the relatively positional relationship with the
previous and behind pieces,i.e.,initial values of l
s
i
,
s
i
,and
s
i
,
is inherited fromthe parents in the following way:
Fundamentally,each piece simply inherits the relatively
positional relationship from its parent.That is,each
piece Piece
i
is reconnected with other pieces,taking the
variables,l
i
,
i
and
i
,with it.
In the case that the piece,Piece
k
,was the rst one in the
parent string and that is going to be moved to the other
position in the child string,the initial values of l
k
and
k
are set up randomly.
In Fig.5,the symbols;`+',`'and`=',represent from which
parent the relationship is inherited.the symbol`+'represents
that it is inherited from Parent 1,the symbol`'represents that
it is inherited from Parent 2,and the symbol`='represents that
it is set up randomly.
The advantage of this crossover operator is that the
substructures,i.e.,the chains of meaningful neighboring
relationships,included in the left parts of the strings are likely
to be preserved and the effective and suitable substrings,i.e.,
building blocks,are likely to be gradually grown to the longer
substrings from left to right.
An example of the order crossover in which it acts effectively
for reducing scrap volume will be shown in Fig.9 in the
computational example.
4.7 Mutation
Furthermore,the mutation operator named remove and reinsert
(e.g.,[Manderick,1992]) is applied to a respective string under
the mutation probability P
m
.
Figure 6 is an example of the mutation.First a piece is
randomly selected from the string and it is removed from the
string.Second the removed piece is inserted into the other
position which is also randomly selected,involving the relatively
positional relationship in the similar way of the crossover
operator.The purpose of introducing this kind of operator is to
Child :+a +b +c +e +f =j i g
+d
Child' :
+a +b +c i g+e +f =j +dh
h
removed piece
reinsert point
Figure 6:Mutation remove and reinsert
avoid rapidly losing the variety of the schemata and to nd the
globally optimal solutions.
4.8 Realizing the Layout Applying Local
Minimization
Finally,the respective individual which is determined as a
string through the above genetic operations is embodied into
a actual layout.That is,the layout is determined with a
local minimization procedure under the layout structure dened
in the string,which are constrained with the neighboring
relationships dened with Equation (5) and the initial values
of the design variables u inherited from the strings of its
parents.In this paper,we use QuasiNewton method as a local
minimization algorithm.In order to formulate the problem as
an unconstrained minimization problem,the following objective
function is dened by including the supplementary objective
function;Equation (5),and the constraints;Equations (2) and
(3),into the primary objective function;Equation (4),as penalty
terms.
min:Obj = Scrap +k
1
Dist2
+k
2
2
+k
3
Overlap
2
(9)
where,k
1
,k
2
and k
3
are weighting factors.
Besides,k
1
= 250 and k
2
= k
3
= 500 are used in the
computational example mentioned in the next section.
These are the operations used in the hybrid algorithm.The
population exchanges involving these operations are iterated in
several tens times and then desirable layout will be obtained.
5 A Computational Example
In this section,a computational example where 12 pieces are
arranged into a plate is demonstrated.In this example,the
parameters on the genetic algorithm is set as follows;the
crossover probability;P
c
= 0:6,the mutation probability;P
m
=
0:03,the population size;N =31.Incidentally,these values and
other setting mentioned in the previous section are roughly set up
based on references and they are not sufciently adjusted.
6
g
a
f
i
j
b
l
k
c
h
d
e
Generation 13
Scrap = 10.29
g
a
f
i
j
b
l
k
c
h
e
d
Generation 25
Scrap = 9.38
g
a
f
i
j
b
l
k
c
h
d
e
Generation 28
Scrap = 9.12
g
a
f
i
j
b
l
k
c
h
e
d
Generation 32
Scrap = 9.08
g
a
f
i
j
b
l
k
c
h
e
d
Generation 44
Scrap = 9.05
Generation 0
Scrap = 17.47
l
g
c
d
e
i
j
b
h
f
a
k
Generation 3
Scrap = 16.63
g
a
f
i
j
b
l
k
c
d
h
e
Generation 4
Scrap = 14.81
g
a
f
i
j
b
l
k
c
d
h
e
Generation 5
Scrap = 14.37
g
a
f
i
j
b
l
k
c
h
e
d
Generation 6
Scrap = 14.07
g
a
f
i
j
b
l
k
c
d
h
e
Generation 10
Scrap = 13.80
g
a
f
i
j
b
l
k
c
h
d
e
Generation 11
Scrap = 13.75
g
a
f
i
j
b
l
k
c
h
e
d
Generation 12
Scrap = 12.29
j
c
g
b
d
k
a
l
f
e
i
h
Figure 8:History of best layouts
Maximum
Average
Minimum
0 10 20 30 40 50
Generation
20.0
40.0
60.0
80.0
100.0
Scrap
Figure 7:Convergence of GA
Figure 7 shows the converging history of the algorithm.The
vertical axis of the graph is scrap volume,the horizontal axis is
generations.In the gure,the history of minimum,average and
maximumvalues of scrap volumes in the population of respective
generations are shown.It is recognized that the set of solutions
were gradually converging,and that after the 32nd generation the
variety of the strings has been almost lost,and the solutions have
been converged.
Figure 8 shows the history of the best layouts in some
generations corresponding to Fig.7.In the respective layout,the
alphabetical symbols are the indexes for the respective pieces.
Because the elitist plan is used in the selection step of the genetic
algorithm as mentioned in the previous section,the best layouts
in the generations are not always exchanged.From the history
of the best layouts,it is ascertained that the substructures of the
layouts are gradually grown into the better layouts according to
the aim of using the order crossover.
Figure 9 shows an example of the crossover operations
between the 24th and 25th generations,which produced the
best layout in the population of the latter generation.In the
example,the parents resemble each other in their strings,but
they are slightly different in their embodiment layouts.The child
is produced by mating them at the crossover point.Though
the initial layout of the child has two overlap areas and a little
overhang value,it is arranged into a better layout than both of the
parents by the local minimization procedure.
The above calculation takes about 12.5 hours with a Sun
SPARC station 10/Model 41.However,the computation time
for calculating the overlaps in Equation (1) and the distances
in Equation (5) is very large,and if it could be reduced,the
calculation time would be much improved.
7
Local
Minimization
Child:
Scrap = 9.38
Overhang:
Crossover point
+g+a+f+i+j+b
Parent 1:
Scrap
= 13.90
+g+a+f+i+j+b
Crossover point
+l+k+c+h+d+e
g
a
f
ij
b
l
k
c
h
d
e
Parent 2:
Scrap
= 14.48
gafijb
Crossover point
lkched
g
a
f
i
j
b
l
k
c
h
e
d
g
a
f
i
j
b
l
k
c
h
e
d
g
a
f
i
j
b
l
k
c
h
e
d
lkched
Overlap
Figure 9:An example of crossover operations
6 Summary
This paper presents a hybrid approach for the nesting problems
by combining the genetic algorithm and the local minimization
algorithm.The characteristic of the approach is that the
combinational conditions of layouts are represented with a form
of ordered lists,which representation enables the hybridization
of the two algorithms.The computational example shows the
effectiveness and potential of the approach.However,since
the property of solutions and convergence are dependent on
the kinds of genetic operators such as crossover and mutation,
their various parameters,the population size,etc.,which
is generally found in the applications of genetic algorithms
(e.g.,[Goldberg,1989][Starkweather,1992]),it is necessary to
improve the performance of the hybrid algorithm by comparing
the kinds of genetic operators and the values of parameters.
Acknowledgments
We would like to acknowledge the support provided by The
Ministry of Education,Science and Culture of Japan through
GrantinAid for General Scientic Research 044521319.
References
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8
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