Download Genetic Algorithms: A Tutorial

Genetic Algorithm
“Genetic Algorithms are
good at taking large,
potentially huge search
spaces and navigating
them, looking for optimal
combinations of things,
solutions you might not
otherwise find in a
lifetime.”
Salvatore Mangano
Computer Design, May 1995
Structure of Biological Gen
Page 1
Genetic Algorithm
Biological Background (cell)
• Every animal cell is a complex structure where many small
“factories” are working together
• The center of this all is the cell nucleus
• The nucleus contains the genetic information
Page 2
Genetic Algorithm
Biological Background (chromosomes)
• The genetic information is stored in the chromosomes
• Each chromosome is build of DNA (deoxyribonucleic acid).
• Chromosomes in humans form are pairs.
• There are 23 pairs (in the human cell).
• The chromosome is divided
into parts: genes.
• Genes code for properties.
• The posibilities of the genes for
one property is called: allele.
• Every gene has an unique position
on the chromosome: locus.
Page 3
Genetic Algorithm
Biological Background (chromosomes)
On the picture we can see a human cell with all metaphase chromosome ordered
by size and an other attribute
metaphase: the state of gen dividing when the chromosomes are situated in the plane of the equator of
cell.
Page 4
Genetic Algorithm
Biological Background (reproduction)
• Reproduction of genetical information
• Mitosis
• Meiosis
• Mitosis is copying the same
genetic information to new
offspring: there is no
exchange of information
• Mitosis is the normal way of
growing of multicell structures,
like organs.
Page 5
Genetic Algorithm
Biological Background (reproduction)
• Meiosis is the basis of sexual
reproduction
• After meiotic division 2 gametes
appear in the process
• In reproduction two gametes
conjugate to a zygote wich
will become the new individual
• Hence genetic information is
shared between the parents in
order to create new offspring
Page 6
Genetic Algorithm
Biological Background (reproduction)
• During reproduction “errors” occur
• Due to these “errors” genetic variation exists
• Most important “errors” are:
• Recombination (cross-over)
• Mutation
Page 7
Genetic Algorithm
Biological Background
(natural secection)
• The origin of species: “Preservation of favourable
variations and rejection of unfavourable variations.”
• There are more individuals born than can survive, so there
is a continuous struggle for life.
• Individuals with an advantage have a greater chance for
survive: survival of the fittest.
Page 8
Genetic Algorithm
Genetic Algoritm
(in Technical Tasks)
●
●
Directed search algorithms based on the
mechanics of biological evolution.
Developed by John Holland, University of
Michigan (1970’s)
♦
♦
Page 9
To understand the adaptive processes of natural
systems
To design artificial systems software that retains
the robustness of natural systems
Genetic Algorithm
Genetic Algoritm
(in Technical Tasks)
●
●
Provide efficient, effective techniques for
optimization and machine learning applications
Widely-used today in business, scientific and
engineering circles
Page 10
Genetic Algorithm
Classes of Search Techniques
Search techniques
Calculus-based techniques
Direct methods
Finonacci
Indirect methods
Newton
Evolutionary algorithms
Evolutionary strategies
Centralized
Simulated annealing
Dynamic programming
Genetic algorithms
Parallel
Page 11
Enumerative techniques
Guided random search techniques
Distributed
Sequential
Steady-state
Generational
Genetic Algorithm
Genetic Algoritm (Components)
A problem to solve, and ...
● Encoding technique
(gene, chromosome)
● Initialization procedure
(creation)
● Evaluation function
(environment)
● Selection of parents
(reproduction)
● Genetic operators
(mutation, recombination)
● Parameter settings
(practice and art)
Page 12
Genetic Algorithm
Simple Genetic Algorithm
{
_initialize population;
_evaluate population;
while Termination_Criteria_Not_Satisfied
{
_select parents for reproduction;
_perform recombination and mutation;
_evaluate population;
}
}
Page 13
Genetic Algorithm
Genetic Algoritm; Cycle of Reproduction
reproduction
selected
parents
modified
children
parents
initialization
population
deleted
members
modification
evaluated
children
evaluation
discard
Page 14
Genetic Algorithm
Cycle of Reproduction; Population
initialization
population
Elements of population (chromosomes) could be:
♦
♦
♦
♦
♦
♦
Bit strings
Real numbers
Permutations of element
Lists of rules
Program elements
... any data structure ...
Page 15
(0110001011010011100)
(43.2, -33.1, ... 0.0, 89.2)
(E11, E3, E7, ... E1, E15)
(R1, R2, R3, ... R22, R23)
(genetic, programming)
Genetic Algorithm
Bit string chromosome
parameters
chromosome (bit string)
a gen = 1 bit
range of integer number
linear decoding
parameter range
Page 16
x=26+13*(4/31) = 27.68
Genetic Algorithm
Bit string cromosome
Page 17
Genetic Algorithm
Real numbers chromosome
x0
x1
a gen = one real number
Real number chromosome = vector of real numbers
Page 18
Genetic Algorithm
Permutation of elements cromosome
chromosome (permutation of elements)
a gen = an order number at a given position
representation by graph
Page 19
Genetic Algorithm
Cycle of Reproduction; Reproduction
reproduction
selected
parents
parents
population
Parents are selected at random with selection
chances biased in relation to chromosome
evaluations (fitness function).
Page 20
Genetic Algorithm
Reproduction; Fitness Function
Bit string chromosome
Real numbers chromosome
Page 21
Fitness(x,y) is any calculated
value
Genetic Algorithm
Reproduction; Fitness Function
chromosome (permutation of elements)
Fitness ( cromosome ) for
example is the distance
between the points
represented positions
Page 22
Genetic Algorithm
Reproduction; Rulett Whell Selection
”Cake” of fitness functions
in a population
F[pv4]
F[pv5]
F[pv6]
F[pv3]
F[pv2
]
F[pv1]
F[pv7]
F[pv8]
area = fitness(x7,y7)
Page 23
Genetic Algorithm
Reproduction; Rank Selection
●
●
●
●
●
●
Ranking is a parent selection method based on
the rank of chromosomes.
Each chromosome is ranked by its fitness value.
r1 is assigned to the worst; r2 to second worst;
and so on.
Higher fitness value has the higher ranking, which
means it will be chosen with higher probability.
Calculate the sum of ranks: result is Rsum.
Parent selection: Random number generating
between 0..Rsum
Page 24
Genetic Algorithm
Reproduction; Tournament Selection
●
●
●
Tournament selection is one of many methods of
selection in genetic algorithms which runs a
"tournament" among a few individuals chosen at
random from the population and selects the
winner (the one with the best fitness) for
crossover.
If the tournament size is higher, weak individuals
have a smaller chance to be selected.
A 1-way tournament selection is equivalent to
random selection.
Page 25
Genetic Algorithm
Cycle of Reproduction; Modification
selected parents
modification
modified children
Modifications are stochastically triggered
● Operator types are:
♦ Crossover (recombination) (probability of crossover)
♦ Mutation (probability of mutation)
Page 26
Genetic Algorithm
Modification; Crossover
cut
P1
P2
(0 1 1 0 1 0 0 0)
(1 1 0 1 1 0 1 0)
(1 1 0 0 1 0 0 0)
(0 1 1 1 1 0 1 0)
Ch1
Ch2
Crossover is a critical feature of genetic algorithms:
• It greatly accelerates search early in evolution of
a population
• It leads to effective combination of schemata
(sub solutions on different chromosomes)
Page 27
Genetic Algorithm
Modification; Mutation
random selected positions
Bit string chromosome
Before:
(1 0 1 1 0 1 1 0)
After:
(0 1 1 0 0 1 1 0)
Real numbers chromosome
●
●
Before:
(1.38 -69.4 326.44 0.1)
After:
(1.38 -67.5 326.44 0.1)
Causes movement in the search space
(local or global)
Restores lost information to the population
Page 28
Genetic Algorithm
Cycle of Reproduction; Evaluation
evaluated
children
●
●
modified
children
evaluation
The evaluator decodes a chromosome and
assigns it a fitness measure
The evaluator is the only link between a classical
GA and the problem it is solving
Page 29
Genetic Algorithm
Cycle of Reproduction; Deletion
population
discarded
members
discard
●
●
Generational GA:
entire populations (fully) replaced with each iteration
Steady-state GA:
a few members replaced each generation
Page 30
Genetic Algorithm
Cycle of Reproduction
Page 31
Genetic Algorithm
“The Gene is by far the most sophisticated program around.”
- Bill Gates, Business Week, June 27, 1994
Page 32
Genetic Algorithm
A Simple Example
The Travelling Salesman Problem (TSP):
We search the summa of distance between the
towns with the next conditions:
♦ we can visit every town only ones
♦ we have to minimize the full distance
(summa distance between the towns in a
given sequence)
Page 33
Genetic Algorithm
Representation of TSP
Representation is an ordered list of city
numbers known as an order-based GA.
1) Oulu
3) Velence
2) Budapest 4) Singapur
CityList1
CityList2
Page 34
5) Peking
6) London
7) Tokio
8) New York
(3 5 7 2 1 6 4 8)
(2 5 7 6 8 1 3 4)
Genetic Algorithm
Crossover of TSP
Crossover combines inversion and recombination:
random selected positions
Parent2
*
*
(3 5 7 2 1 6 4 8)
(2 5 7 6 8 1 3 4)
Child
(5 8 7 2 1 6 3 4)
Parent1
Page 35
Genetic Algorithm
Mutation of TSP
Mutation involves reordering of the list:
random selected positions
Before:
*
*
(5 8 7 2 1 6 3 4)
After:
(5 8 6 2 1 7 3 4)
Page 36
Genetic Algorithm
Traveling Salesman Problem
Page 37
Genetic Algorithm
Rucksack (Backpack) Packing
The problem:
We have N pieces of objects. Every object has a mass and a face value.
We have a rucksack (backpack) which has X kg weight-bearing capacity.
The task is selecting a suitable subset of the objects, where the face
value is maximal and the sum mass of objects are limited to X kg.
We solve the problem applying the genetic algoritm with bitstring
chromosome. The coding in the chromosome gives whether the object is
selected or not.
The upper matrix containing the parent chromosomes the lower the
crossovered and mutated chromosomes. The numbers beside the
matrixes are the sum of face values of selected objects with other words
the fitness function.
Page 38
Genetic Algorithm
Rucksack (Backpack) Packing
Page 39
Genetic Algorithm
Maximum Value of Function
Finding the maximum value of a function
(MVFN):
We finding the maximal value of an explicit function,
(the global maximum):
♦
the input parameter ranges are given
for the search
Page 40
Genetic Algorithm
Representation of MVFN
The chromosome is coded with a vector of real
numbers.
x = { -20.0, +12.0 }
y = { -30.0, +100.0 }
range of search x
range of search y
c(i) = [ x(i), y(i) ]
the form of chromosome
c(1) = [ 3.234, 5.111 ]
c(2) = [ 10.12, -3.9234 ]
chromosome(1)
chromosome(2)
Page 41
Genetic Algorithm
Crossover of MVFN
Parent(1)= [x(1), y(1)];
Parent(2) = [x(2), y(2)];
a = random number between {0..1}
Child(1) = [ a*x(1) +
a*y(1) +
Child(2) = [ {1-a}*x(1)
{1-a}*y(1)
Page 42
{1-a}*x(2),
{1-a}*y(2) ]
+ a*x(2),
+ a*y(2) ]
/ type!!! /
/ ranges!!! /
/ ranges!!! /
Genetic Algorithm
Mutation of MVFN
The mutation a randomly selected element value
changing randomly:
a,b = random numbers with range {0..1} / type!!! /
c = constant { for example = 0.25 }
Before: [ x1, y1 ]
After: [ x1 + (x1max - x1min)*c*a,
y1 + (y1max - y1min)*c*a ]
Page 43
/ ranges!!! /
/ ranges!!! /
Genetic Algorithm
Type of Random Numbers:
Uniform Distribution
 1

f ( x ) =  (b − a )
0

Page 44
if a ≤ x ≤ b
otherwise
Genetic Algorithm
Type of Random Numbers:
Exponential Distribution
λ ⋅ e − λ ⋅ x
f (x) = 
0
Page 45
for 0 ≤ x
otherwise
Genetic Algorithm
Type of Random Numbers:
Normal Distribution
1

 −y
f ( x ) = 
 ⋅ e
 σ⋅ 2⋅π 
( x − µ) 2
y=
2 ⋅ σ2
Page 46
Genetic Algorithm
Type of Random Numbers:
Triangular Distribution
checkpoint = ( usual -minimum ) /
( maximum - minimum );
triangledistribution = ( random <=
checkpoint )?
sqrt ( random * ( maximum - minimum
) * (usual - minimum) )+minimum :
maximum - sqrt ((1random)*(minimummaximum)*(usual-maximum)) ;
Page 47
Genetic Algorithm
Maximum Value of Function
The function:
f(x, y) = 20 - (x1 - x0)^2 + (y1 - y0)^2
where x0 and y0 are constants
We can find also the minimum
value, in this case we
maximazing the - f(x,y) function.
Page 48
Genetic Algorithm
Maximum Value of Function
2
(
f ( x ) = 20 − ∑ x[ i ] − 5.0
i =1
Page 49
)
2
Genetic Algorithm
Maximum Value of Function
f ( x ) = 20 − ∑ ( x[ i ] − x[ i0 ] )
N
2
i =1
Page 50
Genetic Algorithm
Search the Structure of a Function
We have big number of measuring data where the
input parameter(s) is(are) known to the measured
data.
For example the measuring data is the area of a circle
and the input parameter the radius of the circle.
We want to find a suitable structure of function
is giving the connection between the input parameter
and measured data.
Page 51
Genetic Algorithm
Search the Structure of a Function
Page 52
Genetic Algorithm
Issues for GA Practitioners
●
Choosing basic implementation issues:
♦
♦
♦
♦
●
●
●
representation of information
population size, mutation rate, ...
selection, deletion policies
crossover, mutation operators
Termination Criteria
Performance, scalability
Solution is only as good as the evaluation
function (often hardest part)
Page 53
Genetic Algorithm
Benefits of Genetic Algorithms
●
●
●
●
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Concept is easy to understand
Modular, separate from application
Supports multi-objective optimization
Good for “noisy” environments
Always have an answer; answer became
better and better with time
Inherently parallel; easily distributed
Page 54
Genetic Algorithm
Benefits of Genetic Algorithms
●
●
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Many ways to speed up and improve a GAbased application as knowledge about problem
domain is gained.
Easy to exploit previous or alternate solutions.
Flexible building blocks for hybrid applications.
Substantial history and range of use.
Page 55
Genetic Algorithm
When to Use a GA
●
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Alternate solutions are too slow or overly
complicated
Need an exploratory tool to examine new
approaches
Problem is similar to one that has already been
successfully solved by using a GA
Want to hybridize with an existing solution
Benefits of the GA technology meet key problem
requirements
Page 56
Genetic Algorithm
Some GA Application Types
Domain
Application Type
Control
gas, pipeline, pole balancing, missile evasion, pursuit
Design
Scheduling
semiconductor layout, aircraft design, keyboard
configuration, communication networks
manufacturing, facility scheduling, resource allocation
Robotics
trajectory planning
Machine Learning
Signal Processing
designing neural networks, improving classification
algorithms, classifier systems
filter design
Game Playing
poker, checkers, prisoner’s dilemma
Combinatorical
Optimization
set covering, traveling salesman, routing, bin packing,
graph coloring and partitioning
Page 57
Genetic Algorithm
Thank you for your attention
Questions?
Page 58
Genetic Algorithm