# Genetic Algorithm, Theory

Τεχνίτη Νοημοσύνη και Ρομποτική

23 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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Genetic Algorithm, Theory

There are so many books and so many resources on the Web about Genetic
Algorithms. The best that I can do is quote some nice descriptions from my
preferred sites.

"Genetic algorithms are a part of evolutionary computing, which is
a rapidly growing
area of artificial intelligence. As you can guess, genetic algorithms are inspired by
Darwin's theory about evolution. Simply said, solution to a problem solved by
genetic algorithms is evolved."

The GA begins, like any other optimization

algorithm, by defining the optimization
variables. It ends like other optimization algorithms too, by testing for convergence.
A path through the components of the GA is shown as a flowchart in Figure 1.

Fig. 1

-

A path through the components of the GA

Define cost function and cost

For each problem there is a cost function. For example, maximum of a 3D surface
with peaks and valleys when displayed in variable spa
ce. Cost, a value for fitness, is
assigned to each solution.

Chromosomes and genes

A gene is a number between 0 to n
-
1. A chromosome is an array of these genes. It
could be an answer. Population in each generation has determined the number of
chromosomes.

Create a random initial population

An initial population is created from a random selection of chromosomes. The
number of generations needed for convergence depends on the random initial
population.

Decode the chromosome and find the cost

To find the assig
ned cost for each chromosome a cost function is defined. The result
of the cost function called is called cost value. Finally, the average of cost values of
each generation converges to the desired answer.

Mating and next generation

Those chromosomes with
a higher fitness (lesser cost) value are used to produce
the next generation. The offsprings are a product of the father and the mother,
whose composition consists of a combination of genes from them (this process is
known as "crossing over"). If the new g
eneration contains a chromosome that
produces an output that is close enough or equal to the desired answer then the
problem has been solved. If this is not the case, then the new generation will go
through the same process as their parents did. This will
continue until a solution is
reached.

In chess, a queen can move as far as she pleases, horizontally, vertically, or
diagonally. A chess board has 8 rows and 8 columns. The standard 8 by 8 queen's
problem asks how to place 8 quee
ns on an ordinary chess board so that none of
them can hit any other in one move. Here we solve this problem with a genetic
algorithm for a n (n is between 8 and 30) queen problem.

Using the code

We first describe the variables and the functions:

Variables
:

int

ChromosomeMatrix[
30
][
1000
]
: This variable is a matrix of
numbers between 0 to
ChessBoardLenght
-
1. For example, if
ChessBoardLength
=8,
ChromosomeMatrix

could be
{4,6,0,2,7,5,3,1} so that the first number defines the place of the queen in
the first row
, the second number defines the place of the queen in the second
row and so on.

int

CostMatrix[
1000
]
: For each chromosome matrix, a cost value is
defined. The best value is 0 and when no queen can take the other one.

int

CrossOverMatrix[
30
][
1000
]
: For ge
nerating children from
parents, a crossover matrix is needed that decides which gene must be
selected from the two parents.

int

Population
,
int

Iteration
,
float

Mutationrate
: These
variables are genetic algorithm parameters.

int

ChessBoardLength
: This de
termines how many rows and columns
a chessboard has.

int

area[
30
][
30
]
: This variable is a chess board matrix.

Functions:

void

CGAQueen::Clear()

{

// to Reset All cells

for

(
int

i=
0

for

(
int

j=
0

area[i][j]=
0
;

}

This function is used to clear the chess board (
area[][]
) with 0s.

void

CGAQueen::InitialPopulation()

{

int

random;

bool

bCheck;

for

(
int

index=
0
;index<=Population
-
1
;index++)

for

(
int

a=
0

{

random=rand();

bCheck=
1
;

for

(
int

b=
0
;b<a;b++)

if

bCheck=
0
;

if

(bCheck)

ChromosomeMatrix[a][index]=random%ChessBor

else

a
--
;

}

}

This function is used to generate the initial population. This generation is a purely
random generation. With this function,
ChromosomeMatrix[][]

is valued with
random numbers between 0 and
ChessBoardLe
ngth
-
1. The number of
ChromosomeMatrix

is equal to
Population
.

void

CGAQueen::FillArea(
int

index)

{

Clear();

for

(
int

i=
0

area[i][ChromosomeMatrix[i][index]]=
1
;

}

This function is used to fill the chess board with a
chromosome. For example, if
ChromosomeMatrix

= {3,6,8,5,1,4,0,7,9,2}, the first number defines the
column of the queen in the first row, the second number defines the column of the
queen in the second row and so on. The area is shown in fig. 2

here
Chess
BoardLength

=10.

Fig. 2

-

The chess board with
ChromosomeMatrix
={ 3,6,8,5,1,4,0,7,9,2}

Collapse

int

CGAQueen::CostFunc(
int

index)

{

int

costValue=
0
;

int

m,n;

int

i,j;

for
(i=
0

{

j=ChromosomeMatrix[i][index];

m=i+
1
;

n=j
-
1
;

while
n>=
0
)

{

if
(area[m][n]==
1
)

costValue++;

m++;

n
--
;

}

m=i+
1
;

n=j+
1
;

while

{

if
(area[m][n]==
1
)

costValue++;

m++;

n++;

}

m=i
-
1
;

n=j
-
1
;

while
(m>=
0

&& n>=
0
)

{

if
(area[m][n]==
1
)

costValue++;

m
--
;

n
--
;

}

m=i
-
1
;

n=j+
1
;

while
(m>=
0

{

if
(area[m][n]==
1
)

costValue++;

m
--
;

n++;

}

}

return

costValue;

}

This functio
n is used to determines the cost value for each
ChromosomeMatrix[][]

and keeps it in
CostMatrix[]
. For example, for
chromosome = { 2,6,9,3,5,0,4,1,7,8 }, the cost value will be 2. (See fig. 3.)

Fig. 3

-

Because of these two queens that hit each other, the cost value is 2.

void

CGAQueen::PopulationSort()

{

bool

k=
1
;

int

Temp;

while

(k)

{

k=
0
;

for

(
int

i=
0
;i<=Population
-
2
;i++)

{

if

(CostMatrix[i]>CostMatrix[i+
1
])

{

Temp=CostMatrix[i];

CostMatrix[i]=CostMatrix[i+
1
];

CostMatrix[i+
1
]=Temp;

for

(
int

j=
0

{

Temp=ChromosomeMatrix[j][i];

ChromosomeMatrix[j][i]=ChromosomeMatrix[j][i+
1
];

ChromosomeMatrix[j][i+
1
]=Temp;

}

k=
1
;

}

}

}

}

This functi
on is used to sort the new generation according to their cost value. The
best (minimum) is placed on top and the worst (maximum) is placed in the bottom.

void

CGAQueen::GenerateCrossOverMatrix()

{

int

randomCrossOver;

for

(
int

index=
0
;index<=Popula
tion
-
1
;index++)

for

(
int

a=
0

{

randomCrossOver=rand();

CrossOverMatrix[a][index]=randomCrossOver%
2
;

}

}

This function is used to generate the cross over matrix. This matrix contains 0s

and
1s.

void

CGAQueen::Mating()

{

int

TempMatrix[
30
][
2
];

int

TempMatrix0[
30
],TempMatrix1[
30
];

int

Temp,j,k;

}

This function is used to generate children from parents using
CrossOverMatrix
.
It is a way to do this job. Genes are drawn fro
m
p
0

and
p
1
. A gene is drawn from one
parent and it is appended to the offspring (child) chromosome. The corresponding
gene is deleted in the other parent (see figure 4). This step is repeated until both
parent chromosomes are empty and the offspring conta
ins all genes involved.

Fig. 4

The Cross Over method

void

CGAQueen::ApplyMutation()

{

int

randomChromosome;

int

randomGen0,randomGen1;

int

Temp;

int

NumberOfMutation=
int
(MutationRate*(Population
-
1

for
(
int

k=
0
;k<=NumberOfMutation;k++)

{

randomChromosome=
0
;

while
((randomChromosome=rand()%Population)==
00
;

w
hile

Temp=ChromosomeMatrix[randomGen0][randomChromosome];

ChromosomeMatrix[randomGen0][randomChromosome]=

ChromosomeMatrix[randomGen1][randomChromosome];

ChromosomeMatrix[
randomGen0][randomChromosome]=Temp;

}

}

This function is used to apply mutation to the current generation. First of all, a
random chromosome is selected but the first (best) one in the list. Then, two
random genes of this chromosome are selected and re
placed with each other.
Increasing the number of mutations increases the algorithm’s freedom to search
outside the current region of chromosome space.

There is a demo for you to enjoy the genetic algorithm.

Have fun.