introduction to ga

8/19/2019 Introduction to GA

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I. Introduction

First Words

Genetic algorithms (GA) 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 of

evolution. Simply said, problems are solved by an evolutionary process

resulting in a best (fittest) solution (survivor) - in other words, the solution is

evolved.

History

Evolutionary computing was introduced in the 1960s by I. Rechenberg in his

work

Evolution strategies

(

Evolutionsstrategie

in original). His idea was

then developed by other researchers. Genetic Algorithms (GAs) were

invented by John Holland and developed by him and his students and

colleagues. This lead to Holland's book Adaption in Natural and Artificial

Systems published in 1975.

In 1992 John Koza has used genetic algorithm to evolve programs to perform certain

tasks. He called his method genetic programming (GP). LISP programs were used,

because programs in this language can expressed in the form of a parse tree , which is

the object the GA works on.


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2. Genetic Algorithm

Basic Description

Genetic algorithms are inspired by Darwin's theory of evolution. Solution to a

problem solved by genetic algorithms uses an evolutionary process (it is

evolved).

Algorithm begins with a set of solutions (represented by chromosomes)

called population. Solutions from one population are taken and used to form

a new population. This is motivated by a hope, that the new population will

be better than the old one. Solutions which are then selected to form new

solutions (offspring) are selected according to their fitness - the more

suitable they are the more chances they have to reproduce.

This is repeated until some condition (for example number of populations or

improvement of the best solution) is satisfied.

Example

As you already know from the chapter about search space, problem solving

can be often expressed as looking for the extreme of a function. We solve

exactly this problem here - a function is given and GA tries to find the

minimum of the function.

Try to run genetic algorithm in the following applet by pressing the Start

button. The graph represents a search space and vertical lines represent

solutions points in search space). The red line is the best solution, green

lines are the other ones.

The Start button starts the algorithm, Step button performs one step i.e.

forming one new generation), Stop button stops the algorithm and Reset

button resets the population.

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Outline of the Basic Genetic Algorithm

1. [Start] Generate random population of n chromosomes (suitable

solutions for the problem)

2. [Fitness] Evaluate the fitness f x) of each chromosome x in the

population

3. [New population] Create a new population by repeating following

steps until the new population is complete

1. [Selection] Select two parent chromosomes from a population

according to their fitness (the better fitness, the bigger chance

to be selected)

2. [Crossover] With a crossover probability cross over the parents

to form new offspring (children). If no crossover was performed,

offspring is the exact copy of parents.

3. [Mutation] With a mutation probability mutate new offspring at

each locus (position in chromosome).

4. [Accepting] Place new offspring in the new population

4. [Replace] Use new generated population for a further run of the

algorithm

5. [Test] If the end condition is satisfied, stop, and return the best

solution in population

6. [Loop] Go to step 2

Some Comments

As you can see, the outline of the Basic GA is very general. There are many

parameters and settings that can be implemented differently in various

problems.

The first question to ask is how to create chromosomes and what type of

encoding to choose. We then address Crossover and Mutation, the two

basic operators of GA. Encoding, crossover and mutation are introduced in

next chapter.


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The next question is how to select parents for crossover. This can be done

in many ways, but the main idea is to select the better parents (best

survivors) in the hope that the better parents will produce better offspring.

You may think that generating populations from only from two parents may

cause you to loose the best chromosome from the last population. This is

true, and so elitism is often used. This means, that at least one of a

generation's best solution is copied without changes to a new population,

so the best solution can survive to the succeeding generation.

You might be wandering, why genetic algorithms work. It can be partially

explained by the Schema Theorem (Holland), however, this theorem has

been criticized in recent times

3. Operators of GA

Overview

As you can see from the genetic algorithm outline, the crossover and

mutation are the most important parts of the genetic algorithm. The

performance is influenced mainly by these two operators. Before we can

explain more about crossover and mutation, some information about

chromosomes will be given.

Encoding of a Chromosome

A chromosome should in some way contain information about solution that

it represents. The most used way of encoding is a binary string. A

chromosome then could look like this:

Chromosome 1 1101100100110110

Chromosome 2 1101111000011110

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Each chromosome is represented by a binary string. Each bit in the string

can represent some characteristics of the solution. Another possibility is that

the whole string can represent a number - this has been used in the basic

GA applet.

Of course, there are many other ways of encoding. The encoding depends

mainly on the solved problem. For example, one can encode directly integer

or real numbers, sometimes it is useful to encode some permutations and

so on.

Crossover

After we have decided what encoding we will use, we can proceed to

crossover operation. Crossover operates on selected genes from parent

chromosomes and creates new offspring. The simplest way how to do that

is to choose randomly some crossover point and copy everything before this

point from the first parent and then copy everything after the crossover point

from the other parent.

Crossover can be illustrated as follows: ( | is the crossover point):

Chromosome 1

1101100100110110

(D936h)

Chromosome 2

1101111000011110

(DE1Eh)

Chromosome 1 11011 | 00100110110

Chromosome 2 11011 | 11000011110

Offspring 1 11011 | 11000011110

Offspring 2 11011 | 00100110110

There are other ways how to make crossover, for example we can choose

more crossover points. Crossover can be quite complicated and depends

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mainly on the encoding of chromosomes. Specific crossover made for a

specific problem can improve performance of the genetic algorithm.

Mutation

After a crossover is performed, mutation takes place. Mutation is intended to

prevent falling of all solutions in the population into a local optimum of the

solved problem. Mutation operation randomly changes the offspring resulted

from crossover. In case of binary encoding we can switch a few randomly

chosen bits from 1 to 0 or from 0 to 1. Mutation can be then illustrated as

follows:

Original offspring 1 1101111000011110

Original offspring 2 1101100100110110

Mutated offspring 1 1100111000011110

Mutated offspring 2 1101101100110100

The technique of mutation (as well as crossover) depends mainly on the

encoding of chromosomes. For example when we are encoding

permutations, mutation could be performed as an exchange of two genes.

3-1. Selection

Introduction

As you already know from the GA outline, chromosomes are selected from

the population to be parents for crossover. The problem is how to select

these chromosomes. According to Darwin's theory of evolution the best

ones survive to create new offspring. There are many methods in selecting

the best chromosomes. Examples are roulette wheel selection, Boltzman

selection, tournament selection, rank selection, steady state selection and

some others.

Roulette wheel selection will be described in this chapter.

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Roulette Wheel Selection

Parents are selected according to their fitness. The better the chromosomes

are, the more chances to be selected they have. Imagine a roulette wheel

where all the chromosomes in the population are placed. The size of the

section in the roulette wheel is proportional to the value of the fitness

function of every chromosome - the bigger the value is, the larger the

section is. See the following picture for an example.

A marble is thrown in the roulette wheel and the chromosome where it stops

is selected. Clearly, the chromosomes with bigger fitness value will be

selected more times.

This process can be described by the following algorithm.

1. [Sum] Calculate the sum of all chromosome fitnesses in population -

sum S.

2. [Select] Generate random number from the interval 0,S) - r.

3. [Loop] Go through the population and sum the fitnesses from 0 -

sum s. When the sum s is greater then r, stop and return the

chromosome where you are.

Of course, the step 1 is performed only once for each population.

3-2. Crossover and Mutation

Introduction


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Crossover and mutation are two basic operators of GA. Performance of GA

depends on them very much. The type and implementation of operators

depends on the encoding and also on the problem.

There are many ways how to perform crossover and mutation. In this

chapter we briefly describe some examples and suggestions how to perform

them several encoding.

Binary Encoding

Crossover

Single point crossover - one crossover point is selected, binary string from

the beginning of the chromosome to the crossover point is copied from the

first parent, the rest is copied from the other parent

11001011+11011111 = 11001111

Two point crossover - two crossover points are selected, binary string from

the beginning of the chromosome to the first crossover point is copied from

the first parent, the part from the first to the second crossover point is

copied from the other parent and the rest is copied from the first parent

again

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11001011 + 11011111 = 11011111

Uniform crossover - bits are randomly copied from the first or from the

second parent

11001011 + 11011101 = 11011111

Arithmetic crossover - some arithmetic operation is performed to make a

new offspring

11001011 + 11011111 = 11001001 (AND)

Mutation

Bit inversion - selected bits are inverted

11001001 => 10001001


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A Genetic Algorithm Example

As a simple example, imagine a population of four strings, each with five

bits. Also imagine an objective function f(x)=2x. The goal is to optimize (in

this case maximize) the objective function over the domain . Now

imagine a population of the four strings in Table 1, generated at random

before GA execution. The corresponding fitness values and percentages

come from the objective function f(x).

Table 1: Four strings and their fitness values.

The values in the `` '' column provide the probability of each string's

selection. So initially 11000 has a 38.1% chance of selection, 00101 has an

7.9% chance, and so on. The results of the selections are given in the

``Actual Count'' column of Table 1. As expected, these values are similar to

those in the ``Expected Count'' column.

After selecting the strings, the GA randomly pairs the newly selected

members and looks at each pair individually. For each pair (e.g.

A

= 11000

and B = 10110), the GA decides whether or not to perform crossover. If it

does not, then both strings in the pair are placed into the population with

possible mutations (described below). If it does, then a random crossover

point is selected and crossover proceeds as follows:

A = 1 1 | 0 0 0 B = 1 0 | 1 1 0

are crossed and become

A

' = 1 1 1 1 0

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B' = 1 0 0 0 0.

Then the children A' and B' are placed in the population with possible

mutations. The GA invokes the mutation operator on the new bit strings very

rarely (usually on the order of probability), generating a random

number for each bit and flipping that particular bit only if the random number

is less than or equal to the mutation probability.

After the current generation's selections, crossovers, and mutations are

complete, the new strings are placed in a new population representing the

next generation, as shown in Table 2. In this example generation, average

fitness increased by approximately 30% and maximum fitness increased by

25%. This simple process would continue for several generations until a

stopping criterion is met.

Table 2: The population after selection and crossover.
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