# 7. Genetic Algorithms - Iran University of Science and ... Algorithms... · PDF fileGenetic Algorithms: Part 3 Representation of Individuals Depends on the problem The most used encodings:

• View
232

1

Embed Size (px)

### Text of 7. Genetic Algorithms - Iran University of Science and ... Algorithms... · PDF...

• 7. Genetic Algorithms

7.3 The Component of Genetic Algorithms

Fall 2010

Instructor: Dr. Masoud Yaghini

• Genetic Algorithms: Part 3

Outline

Representation of Individuals

Mutation

Recombination

Population Models

Parent Selection

Survivor Selection

Glossary

References

• Representation of Individuals

• Genetic Algorithms: Part 3

Representation of Individuals

Depends on the problem

The most used encodings:

Binary representation

Integer representation

Real-valued representation Real-valued representation

Permutations representation

• Genetic Algorithms: Part 3

Binary Representations

Simplest and most common

Chromosome: string of bits

genes: 0 / 1

Example: binary representation of an integer

3: 00011

15: 01111

16: 10000

• Genetic Algorithms: Part 3

Binary Representation

For those problem concerning Booleandecision variables, the genotype-

phenotype mapping is natural

Example: knapsack problem

Many optimization problems involve

integer or real numbers and bit string can

be used to encode these numbers

• Genetic Algorithms: Part 3

Mapping Integer Values on Bit Strings

Integer values can be also binary coded: x = 5 00101

• Genetic Algorithms: Part 3

Mapping real values on bit strings

Real values can be also binary coded

z [x, y] represented by {a1,,aL} {0,1}L

[x, y] {0,1}L must be invertible (one phenotype per genotype)

: {0,1}L [x, y] defines the representation

Only 2L values out of infinite are represented

L determines possible maximum precision of solution

High precision long chromosomes (slow evolution)

],[)2(12

),...,(1

0

1 yxaxy

xaaj

L

j

jLLL

+=

=

• Genetic Algorithms: Part 3

Mapping real values on bit strings

Example: z [1, 10]

We use L = 8 bits to represent real values in the domain [1, 10] , i.e., we use 256 numbers

1 00000000 (0)

10 11111111 (255)10 11111111 (255)

Convert 00111100, to real value:1 + ((10-1) / (256-1)) * [(0 * 20) + (0 * 21) + (1 * 22) + (1 * 23) + (1 *

24) + (1 * 25) + (0 * 26) + (0 * 27)] = 3.12

Convert n = 3.14 to bit string:n = 3.14 (3.14-1)*255/(10-1) = 60 00111100

0011 1100 3.12 (better precision requires more bits)

• Genetic Algorithms: Part 3

Binary Representations

Hamming distance:

Number of bits that have to be changed to map one

string into another one

E.g. 000 and 001 distance = 1

Problem: Hamming distance between Problem: Hamming distance between consecutive integers may be > 1

example: 5 bit binary representation

14: 01110 15: 01111 16: 10000

Probability of changing 15 into 16 by independent bit

flips (mutation) is not same as changing it into 14!

• Genetic Algorithms: Part 3

Binary encoding problem

Remember: small changes in genotype should cause small changes in phenotype

Gray coding solves problem.

• Genetic Algorithms: Part 3

Binary Representation

Binary coding of 0-7 :

0102

0011

0000 Hamming distance, e.g.:

000 (0) and 001 (1)

1106

1117

1015

1004

0113

0102 000 (0) and 001 (1)

Distance = 1 (optimal)

011 (3) and 100 (4)

Distance = 3 (max possible)

• Genetic Algorithms: Part 3

Binary Representation

Gray coding of 0-7:

Gray coding is a representation

that ensures that consecutive

integers always have Hamming

distance one.0103

0112

0011

0000

Gray coding is a mapping that

means that small changes in the

genotype cause small changes in

the phenotype (unlike binary

coding).

1016

1007

1115

1104

0103

• Genetic Algorithms: Part 3

Binary Representation

Nowadays it is generally accepted that it is

better to encode numerical variables directly

as:

Integers

Floating point variables

• Genetic Algorithms: Part 3

Integer representations

Some problems naturally have integer variables

e.g. for optimization of a function with integer

variables

Values may be Values may be

unrestricted (all integers)

restricted to a finite set

e.g. {0,1,2,3}

e.g. {North,East,South,West}

• Genetic Algorithms: Part 3

Integer representations

Any natural relations between possible values?

obvious for ordinal attributes 2 is more like 3 than it is 389

small < medium < large small < medium < large

hot > mild > cool

maybe no natural ordering for cardinal attributes

e.g. set of compass points

e.g. employee ID

• Genetic Algorithms: Part 3

Real-valued Representation

Many problems occur as real valued problems,

e.g. continuous parameter optimization

Example: a chromosome can be a pair

(x, y)

Vector of real values Vector of real values

floating point numbers

Genotype for solution becomes the vector with xi

• Genetic Algorithms: Part 3

Permutation Representations

Deciding on sequence of events

most natural representation is permutation of a set of integers

In ordinary GA numbers may occur more than

once on chromosomeonce on chromosome

invalid permutations!

New variation operators needed

• Genetic Algorithms: Part 3

Permutation Representations

Two classes of problems

based on order of events

e.g. scheduling of jobs

Job-Shop Scheduling Problem

e.g. Travelling Salesperson Problem (TSP)

finding a complete tour of minimal length between n cities, visiting each city only once

• Genetic Algorithms: Part 3

Permutation Representations

Two ways to encode a permutation

i th element represents event that happens in that

location in a sequence

value of i th element denotes position in sequence

in which i th event occursin which i th event occurs

Example (TSP): 4 cities A,B,C,D and permutation [3,1, 2, 4] denotes the tours:

first encoding type: [CABD]

• Mutation

• Genetic Algorithms: Part 3

Mutation

Mutation is a variation operator

Create one offspring from one parent

Occurs at a mutation rate: pm behaviour of a GA depends on pm

• Genetic Algorithms: Part 3

Bitwise Mutation

flips bits

01 and 10

setting of pm depends on nature of problem,

typically between:

1 / pop_size1 / pop_size

1 / chromosome_length

• Genetic Algorithms: Part 3

Mutation for Integer Representation

For integers there are two principal forms of

mutation:

Random resetting

Creep Mutation

Both of them, mutate each gene independently

with user-defined probability pm

• Genetic Algorithms: Part 3

Random Resetting

A new value is chosen with from the set of permissible integer values

Most suitable for cardinal attributes

• Genetic Algorithms: Part 3

Creep Mutation

Add small (positive / negative) integer to gene

value

random value

sampled from a distribution

symmetric around 0 symmetric around 0

with higher probability of small changes

Designed for ordinal attributes

Step size is important

controlled by parameters

setting of parameters important

• Genetic Algorithms: Part 3

Mutation for Floating-Point Representation

Allele values come from a continuous distribution

Change allele values randomly within its

domain

upper and lower boundaries, U and L respectively upper and lower boundaries, Ui and Li respectively

[ ]iiiinn

ULxxwhere

xxxxxx

,,

,...,,,...,, 2121

>

• Genetic Algorithms: Part 3

Mutation for Floating-Point Representation

According to the probability distribution from

which the new gene values are drawn, there

are two types of floating-point mutation:

Uniform Mutation

Non-Uniform Mutation with a Fixed Distribution Non-Uniform Mutation with a Fixed Distribution

• Genetic Algorithms: Part 3

Uniform Mutation

Values of xi drawn uniformly randomly from the

[Li,Ui], Similar to

bit flipping for binary representations

random resetting for integer representations

usually used with positionwise mutation usually used with positionwise mutation

probability

• Genetic Algorithms: Part 3

Non-Uniform Mutation with a Fixed Distribution

Most common form

Similar to creep mutation for integer

representations

Add an amount to gene value

Amount randomly drawn from a distribution Amount randomly drawn from a distribution

• Genetic Algorithms: Part 3

Non-Uniform Mutation with a Fixed Distribution

Gaussian distribution (normal distribution)

with mean 0

user-specified standard deviation

may have to adjust to interval [Li,Ui]

2/3 of samples lie within one standard deviation of 2/3 of samples lie within one standard deviation of

mean (- to + )

Documents
Documents
Education
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Education
Education
Documents
Documents
Documents
Documents
Documents
Technology
Documents
Documents
Documents
Documents
Engineering
Technology
Documents
Documents