GEAIO_lab3

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Slide 1

Gesti Aeroporturia i del Espai Aeri

iInvestigaci Operativa

Genetic Algorithm Genetic Algorithm ToolBox

University of Sheffield

05/11/2012


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Slide 2

Genetic Algorithm

Introduction

Basic algorithm

Population representation and initialization Objective and fitness functions

Selection

Crossover

Mutation

Reinsertion

Termination of the GA

Data structures

Multiple population

Simple GA

Multi-population GA

Problem

[Chipperfield, 1994]


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Introduction

[Weise, 2009]


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Chromosomes

Genotypes chromosome values

Phenotype meaning of the genotype

Different codification possibilities:

Real values

Binary representation

Introduction

0 0 1 1 0 1 0 0 1 0 0

x1 x2

Different length might imply

different precision

Once decoded it is possible to assess performance fitness (Objective function)


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Reproduction phase

Probability of being chosen in function fitness

Reproduction uses genes

Simplest combination: One point crossover. Done with a probability (Px)

Introduction

X1a X2a X3a X4a X5a X1n X2n X3n X4n X5nSa Sn

One point crossover

X1n X2n X3n X4a X5a


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Reproduction phase

Probability of being chosen in function fitness

Reproduction uses genes

Simplest combination: One point cross-over. Done with a probability (Px)

Mutation applied with a probability to ensure search in all the search space.

Introduction

Mutation

X1b X2b X3b X4b X5bSb

X1b X2b X3b X4b X5b


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Some differences with other optimization algorithms

Search with a population not with a single point

Dont require derivate information. Only objective function and fitness levels

Probabilistic behavior, not deterministic

Works with encoded parameters not with the parameters themselves

Ends with a set of solutions. If Pareto might find all of them

Introduction


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Basic Algorithm

Begin

t=0;

initialize P(t);

evaluate P(t);

while not finished do

begin

t=t+1;

select P(t) from P(t-1);

reproduce pairs in P(t);

evaluate P(t);

end

end


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Population representation and init ialization

Population representation


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Usually population between 30 and 100 individuals

Most common representation binary string

Other representations:

Gray coding instead of binary coding

Logarithmic coding

Real-value coding

Population representation

Binary, Integer and floating point chromosome representationGA

toolbox

Dec Gray Binary0 000 000

1 001 001

2 011 010

3 010 011

4 110 100

5 111 101

6 101 1107 100 111


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Initialization


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Random

Using problem information to maximize the search space used and avoid local minima

Initialization

crtbpcreate binary population

ctrbasebuilds a vector describing integer representation used

crtrpinitialize real-value population

bs2rvconversion between binary string and real values. Support Gray and logarithmic

scale codification.

GA

toolbox


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Objective and fitness functions

Objective function


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Used to provide a measure of how individuals have performed in the problem domain.

In a minimization problem the most fit the individual the lowest value of the objective

function

Used as an intermediate stage in determining the relative performance of individuals in

GA

Objective function

Must be created by the user. Example functions starts with objGA

toolbox


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Fitness function


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Used to transform the objective function value into a measure of relative fitness

Fitness function

f: objective function

g: transform the objective function in non-negative value

F: the result of the fitness

This mapping is always needed when minimizing as minimal objective value implies

maximum fitness value

The probability of having offspring is generally proportional to the fitness value.

Therefore F(x) can be computed relative to the whole population:

Nind: the population size

xi: the phenotypic value of individual i

Ensure all probability reproducing but problem with negative values scaling

O f f


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Linear transformation prior to fitting to ensure non-negative value and

minimization/maximization

Fitness function

a: positive scaling factor if maximizing, negative value if

minimizing

b: ensure non-negative value

Problem with rapid convergence. Best individuals too good

Obj ti d fit f ti


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Fitness function

Option to avoid rapid convergence: Limit the reproduce range of individuals. Fitting

according to rank in population instead of just performance. MAX used to determine

selective pressure

MIN = 2.0 MAX

INC = 2.0 x (MAX-1.0)/Nind

LOW = INC/2.0

MIN is the lower bound, INC is the difference between fitness of adjacent individuals,

LOW is expected number of times selected. MAX between [1.1, 2.0].

Obj ti d fit f ti


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Fitness function

ranking supports linear and non-linear ranking methods

scaling a simple linear scaling function. Linear scaling is not suitable for use with objective

functions that return negative values.

GAtoolbox

S l ti


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Selection

Selection

S l ti


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Selection

Process to determine the number of times, trials, a particular individual is chosen for

reproduction.

It is done in two different processes:

1. Determination of the number of trials an individual can expect to receive

2. Conversion of the expected number of trials into a discrete number of offspring

The first part consist on the transformation of the fitness into a probability to reproduce.

The second part is the probabilistic selection of individuals based on the fitness of

individuals relative to one another: sampling

Selection

S l ti


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Selection

Create an interval between [0,Sum] where Sum is determined by the fitness of the

individuals.

The size of each individual interval correspond to the fitness value associated to each

individual.

To select an individual, a random number is generated in the interval [0,Sum]

Roulette wheel selection methods

S l ti


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Selection

Stochastic Sampling with Replacement (SSR): Segment size and selection probability

remains the same during the selection. Any individual with segment size > 0 could fill

the next population.

Stochastic Sampling with Partial Replacement (SSPR): resize individuals segment

when selected by reducing it by 1.0.

Remainder Stochastic Sampling with Replacement (RSSR): Select deterministically by

their expected trials and then the remaining use a SSR.

Roulette wheel selection methods

Selection


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Selection

Stochastic Universal Sampling (SUS): single-phase sampling algorithm. Generate N

pointers on the population by dividing Sum/N then shuffle the individuals and randomly

select a points. Select the individuals pointed by the pointers spaced by 1.

Stochastic Universal Sampling

4 2 3 7 1 5 8 6

Selection


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Selection

rws Stochastic Sampling with Replacement algorithm

sus Stochastic Universal Sampling function

GA

toolbox

Crossover


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Crossover

Crossover


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Crossover

Single-point Crossover

X1a X2a X3a X4a X5a X1n X2n X3n X4n X5nSa Sn

One point crossover

X1n X2n X3n X4a X5a

Crossover


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Crossover

Parts of the chromosome representation that contribute to the most to the performance

of an individual might not be adjacent substrings

Multi-point Crossover

Crossover


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Crossover

Randomly create a mask to do the cross over

Allow the crossing of any loci

Uniform Crossover

Crossover


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Crossover

A single cross-point is selected, but before exchanging the bits they are randomly

shuffled in both parents. The bits are unshuffled after the recombination.

Shuffle

Ensure always produce new individuals.

Reduced surrogate

Crossover


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Crossover

With a real-valued encoding of the chromosome generate offspring with the rule:

Intermediate recombination

Each variable in the offspring is the results of combining the variables in the parents

with a new

for each pair of genes.

: scaling factor chosen randomly at an interval i.e. [-0.25,

1.25]

P1, P2: parent chromosomes

Crossover


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Crossover

Intermediate recombination

10 8 7 10 4 6

4 3 9 11 3 8

P1

P2

0.01 0.2 1.1 -0.1 0.8 0.75 randomly from

[-0.25,1.25]

-6 -5 2 1 -1 2P2-P1

-0,06 -1 2,2 -0,1 -0,8 1,5(P2-P1)

-0,6 -8 15,4 -1 -3,2 9O=P1 x (P2-P1)

Crossover


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Crossover

Similar to Intermediate Recombination but just an used in the recombination.

Line Recombination

Crossover


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Crossover

xovsp single-point crossover. Can operate on any chromosome representation.

xovdp double-point crossover. Can operate on any chromosome representation. xovsh shuffle crossover. Can operate on any chromosome representation.

xovsprs reduced surrogate crossover with single-point crossover.

xovdprs reduced surrogate crossover with double-point crossover.

xovshrs reduced surrogate crossover with shuffle crossover.

xovmp general multi-point crossover routine.

recdis discrete recombination, crossover on real-valued individuals similar to crossover

operators.

reclin line recombination for real-valued individuals.

recint intermediate recombination for real-valued individuals.

recombinhigh-level entry function to all of the crossover operators.

GA

toolbox

Mutation


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Mutation

Mutation


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Mutation

Usually applied with a low probability (between 0.001 and 0.01)

Guarantee that all the search space is potentially searched

In non-binary representations, mutation can be achieved by perturbing the gene values

Options to mutate more low fitted individuals or mutate with higher probability weak

genes in an individual

mut binary mutation.

mutbga real-valued mutation.

mutatehigh-level entry function mutation operators.

GA

toolbox

Reinsertion


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Reinsertion

Reinsertion


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Reinsertion

The population should be kept with the same number after each iteration

Generation gap = How extra individuals are created in each iteration

New individuals have to be reinserted in old population

Randomly

Always pass the best fitted (elitist strategy)

Remove the oldest

reinsdo the reinsertion into population. Parameters allow to use uniform random or fitness-based reinsertion. Can be used to reinsert fewer offspring than generated after

recombination.

GA

toolbox



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As GA is a stochastic search method, it is difficult to specify convergence criteria.

Terminate the GA after a number of generations

If best individual not very good reinitialize the algorithm and restart.

When genes are similar then stop

Data structures


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Data structures

Data to be represented:

Chromosomes

Phenotypes

Objective function values

Fitness values

Data structures


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Data structures

GA

toolbox

Chromosomes data structure stores a whole population in a matrix Nind x Lind

Nind number of individuals

Lind length of the genotypic representation of the individuals

Each row corresponds to an individuals genotype, of base-n, typically binary

Chromosomes

Only restriction: all chromosomes are of the same length.

Need a suitable decoding function to transform chromosomes to phenotypes.

Data structures


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GA

toolbox

The decision variables, phenotype, is obtained by applying a mapping between a

chromosome into the variable space.

Each chromosome decodes into a row vector of order Nvar

The decision variables are stored in a numerical matrix of size Nind x Nvar

Phenotypes

The mapping between chromosome and phenotype depends on the DECODE function

implemented.

Data structures


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GA

toolbox

The objective function is used to evaluate the phenotypes in the problem domain.

Objective functions can be scalar or vectorial

Objective values are not necessarily the same as the fitness values

Objectives values are stored in a numerical matrix of size Nind x Nobj, where Nobj is

the number of objectives.

Objective funct ion values

OBJFUN is an arbitrary objective function.

Data structures


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GA

toolbox

Fitness values are derived from objective function values through a scaling or ranking

function.

Fitness are non-negative scalars and are stored in column vectors of Nind.

Fitness values

FITNESS is an arbitrary fitness function.

For multi-objective functions, the fitness of an individual is a function of a vector of

objective function values.

Multiple population


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p p p

Possibility to have more than one population in parallel and migrate individuals between

populations

Multiple population


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p p p

GA toolbox supports multi-population with the functions:

mutate mutation operators.

recombin crossover and recombination operators.

resins uniform random and fitness-based reinsertion.

select independent subpopulation selection.

migrate

to migrate individuals between populations

GA

toolbox

Simple GA


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p

Optimize the De Jongs function:

With n dimensions.

In this problem n=20

Minimum at xi=0

Simple GA


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p

Optimize the De Jongs function:

N individuals = 40

Max Generations = 300

Number of variables = 20

Precision = 20 bits per variable

Generation gap = 0.9

Gray codification

Arithmetic

Maximum value all variables 512

Minimum value all variables -512

Use stochastic universal sampling

Single-point crossover

Fitness-based reinsertion

To create the FieldD use matlab function rep

The project


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X1

Y1

X2

Y2

X3

Y3

X4

Y4

X5

Y5

Sectorisation defined by coordinates of Voronoi points


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References

[Weise, 2009] T. Weise, Global Optimization Algorithms Theory and

Applications, 2ndEdition, electronic book available at: www.it-weise.de

[Chipperfield, 1994] A. Chipperfield et al, Genetic Algorithm ToolBox Users

Guide, version 1.2
http://www.it-weise.de/http://www.it-weise.de/

Documents

GEAIO_lab3