Soft computing06

Soft Computing
Lecture 06: Introduction “Genetic Algorithms are good at
to Genetic Algorithms 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

GENETIC ALGORITHM

A biologically inspired model of intelligence and the
principles of biological evolution are applied to find solutions
to difficult problems

The problems are not solved by reasoning logically about
them; rather populations of competing candidate solutions
are spawned and then evolved to become better solutions
through a process patterned after biological evolution

Less worthy candidate solutions tend to die out, while those
that show promise of solving a problem survive and
reproduce by constructing new solutions out of their
components

GENETIC ALGORITHM

GA begin with a population of candidate problem solutions

Candidate solutions are evaluated according to their ability
to solve problem instances: only the fittest survive and
combine with each other to produce the next generation of
possible solutions

Thus increasingly powerful solutions emerge in a Darwinian
universe

Learning is viewed as a competition among a population of
evolving candidate problem solutions

This method is heuristic in nature and it was introduced by
John Holland in 1975

GENETIC ALGORITHM

Basic Algorithm

begin
set time t = 0;
initialise population P(t) = {x1t, x2t, …, xnt} of solutions;
while the termination condition is not met do
begin
evaluate fitness of each member of P(t);
select some members of P(t) for creating offspring;
produce offspring by genetic operators;
replace some members with the new offspring;
set time t = t + 1;
end
end

GENETIC ALGORITHM

The Evolutionary Cycle

parents
selection modification
modified
offspring
initiate &
population evaluation
evaluate evaluated offspring
deleted
members

discard

GENETIC ALGORITHM

Representation of Solutions: The Chromosome

Gene: A basic unit, which represents one characteristic of the
individual. The value of each gene is called an allele

Chromosome: A string of genes; it represents an individual
i.e. a possible solution of a problem. Each chromosome
represents a point in the search space

Population: A collection of chromosomes

An appropriate chromosome representation is important for
the efficiency and complexity of the GA

GENETIC ALGORITHM


The classical representation scheme for chromosomes is
binary vectors of fixed length

In the case of an I-dimensional search space, each
chromosome consists of I variables with each variable
encoded as a bit string

GENETIC ALGORITHM

Example: Cookies Problem

Two parameters sugar and flour (in kgs). The range for both
is 0 to 9 kgs. Therefore a chromosome will comprise of two
genes called sugar and flour

5 1 Chromosome # 01

2 4 Chromosome # 02

GENETIC ALGORITHM

Example: Expression satisfaction Problem

F = (¬a ∨ c) ∧ (¬a ∨ c ∨ ¬e)
∧ (¬b ∨ c ∨ d ∨ ¬e) ∧ (a ∨ ¬b ∨ c)
∧ (¬e ∨ f)

Chromosome: Six binary genes abcdef e.g. 100111

GENETIC ALGORITHM

Chromosomes have either binary or real valued genes

In binary coded chromosomes, every gene has two alleles

In real coded chromosomes, a gene can be assigned any value
from a domain of values

GENETIC ALGORITHM

Model Learning

Use GA to learn the concept Yes Reaction from the Food
Allergy problem’s data

GENETIC ALGORITHM

Chromosomes Encoding

A potential model of the data can be represented as a
chromosome with the genetic representation:

Gene # 1 Gene # 2 Gene # 3 Gene # 4
Restaurant Meal Day Cost

The alleles of genes are:

Restaurant gene: Sam, Lobdell, Sarah, X
Meal gene: breakfast, lunch, X
Day gene: Friday, Saturday, Sunday, X
Cost gene: cheap, expensive, X

GENETIC ALGORITHM

Chromosomes Encoding (Hypotheses Representation)

Hypotheses are often represented by bit strings (because they
can be easily manipulated by genetic operators), but other
numerical and symbolic representations are also possible

Set of if-then rules:
Specific sub-strings are allocated for encoding each
rule pre-condition and post-condition

Example: Suppose we have an attribute “Outlook”
which can take on values: Sunny, Overcast or Rain

GENETIC ALGORITHM


We can represent it with 3 bits:
100 would mean the value Sunny,
010 would mean Overcast &
001 would mean Rain

110 would mean Sunny or Overcast
111 would mean that we don’t care about its value

The pre-conditions and post-conditions of a rule are
encoding by concatenating the individual representation of
attributes

GENETIC ALGORITHM


Example:

If (Outlook = Overcast or Rain) and Wind = strong
then PlayTennis = No

can be encoded as 0111001

Another rule
If Wind = Strong
then PlayTennis = Yes

can be encoded as 1111010

GENETIC ALGORITHM


An hypothesis comprising of both of these rules can be
encoded as a chromosome

01110011111010

Note that even if an attribute does not appear in a rule, we
reserve its place in the chromosome, so that we can have
fixed length chromosomes

GENETIC ALGORITHM

Variable size chromosomes

Sometimes we need a variable size chromosome; e.g. to
represent a set of rules

Example:
Suppose we are representing a set of rules by a chromosome

If a1 = T and a2 = F then c = T
If a2 = T then c = F

The chromosome would be 10 01 1 11 10 0
where a1 = T is represented by 10,
a2 = F by 01,
and so on

GENETIC ALGORITHM
Evaluation/Fitness Function

It is used to determine the fitness of a chromosome

Creating a good fitness function is one of the challenging
tasks of using GA

GENETIC ALGORITHM

Example: Cookies Problem

Two parameters sugar and flour (in kgs). The range for both
is 0 to 9 kgs. Therefore a chromosome will comprise of two
genes called sugar and flour

5 1

2 4

The fitness function for a chromosome is the taste of the
resulting cookies; range of 1 to 9

GENETIC ALGORITHM

Example: Expression satisfaction Problem

F = (¬a ∨ c) ∧ (¬a ∨ c ∨ ¬e)
∧ (¬b ∨ c ∨ d ∨ ¬e) ∧ (a ∨ ¬b ∨ c)
∧ (¬e ∨ f)

Chromosome: Six binary genes abcdef e.g. 100111

Fitness function: No of clauses having truth value of 1
e.g. 010010 has fitness 2

GENETIC ALGORITHM

Model Learning

Use GA to learn the concept Yes Reaction from the Food
Allergy problem’s data

The fitness function can be the number of training samples
correctly classified by a chromosome (model)

GENETIC ALGORITHM
Population Size

Number of individuals present and competing in an iteration
(generation)

If the population size is too large, the processing time is high
and the GA tends to take longer to converge upon a
solution (because less fit members have to be selected to
make up the required population)

If the population size is too small, the GA is in danger of
premature convergence upon a sub-optimal solution (all
chromosomes will soon have identical traits). This is
primarily because there may not be enough diversity in
the population to allow the GA to escape local optima

GENETIC ALGORITHM

Selection Operators (Algorithms)

They are used to select parents from the current population

The selection is primarily based on the fitness. The better the
fitness of a chromosome, the greater its chance of being
selected to be a parent

GENETIC ALGORITHM

Selection Operators: Random Selection

Individuals are selected randomly with no reference to fitness
at all

All the individuals, good or bad, have an equal chance of
being selected

GENETIC ALGORITHM

Selection Operators: Proportional Selection

Chromosomes are selected based on their fitness relative to
the fitness of all other chromosomes

For this all the fitness are added to form a sum S and each
chromosome is assigned a relative fitness (which is its fitness
divided by the total fitness S)

A process similar to spinning a roulette wheel is adopted to
choose a parent; the better a chromosome’s relative fitness,
the higher its chances of selection

GENETIC ALGORITHM


The selection of only the most fittest chromosomes may result
in the loss of a correct gene value which may be present in a
less fit member (and then the only chance of getting it back is
by mutation)

One way to overcome this risk is to assign probability of
selection to each chromosome based on its fitness

In this way even the less fit members have some chance of
surviving into the next generation
Chromosomes are selected based on their fitness relative to
the fitness of all other chromosomes

GENETIC ALGORITHM


For this all the fitness are added to form a sum S and each
chromosome is assigned a relative fitness (which is its fitness
divided by the total fitness S)

A process similar to spinning a roulette wheel is adopted to
choose a parent; the better a chromosome’s relative fitness,
the higher its chances of selection

GENETIC ALGORITHM

The probability of selection of a chromosome “i” may be
calculated as

pi = fitnessi / ∑ j fitnessj

Example

Chromosome Fitness Selection Probability
1 7 7/14
2 4 4/14
3 2 2/14
4 1 1/14

GENETIC ALGORITHM

GENETIC ALGORITHM
Selection Operators: Proportional Selection Algorithm

1. [Sum] Calculate sum of all chromosome fitnesses in population - sum S.
2. [Assign] Assign a range to each chromosome over a line ranging from 0-S.
3. [Select] Generate random number from interval (0,S) - r.
4. [Select] Select the chromosome belongs to r
Of course, step 1 is performed only once for each population.

GENETIC ALGORITHM

Selection Operators: Rank based selection

Rank based selection uses the rank ordering of the fitness
values to determine the probability of selection and not the
fitness values themselves

This means that the selection probability is independent of
the actual fitness value

Ranking therefore has the advantage that a highly fit
individual will not dominate in the selection process as a
function of the magnitude of its fitness

GENETIC ALGORITHM
Proportional Selection have problems when the fitnesses differs very much. For
example, if the best chromosome fitness is 90% of all the roulette wheel then
the other chromosomes will have very few chances to be selected.
Rank selection first ranks the population and then every chromosome receives
fitness from this ranking. The worst will have fitness 1, second worst 2 etc. and
the best will have fitness N (number of chromosomes in population)
You can see in following picture, how the situation changes after changing
fitness to order number.

Before ranking After ranking

GENETIC ALGORITHM


The population is sorted from best to worst according to the
fitness

Each chromosome is then assigned a new
fitness based on a linear ranking function

New Fitness = (P – r) + 1

where P = population size, r = fitness rank of the chromosome
If P = 11, then a chromosome of rank 1 will have a New
Fitness of 10 + 1 = 11 & a chromosome of rank 6 will have 6

GENETIC ALGORITHM


A user adjusted slope can also be incorporated

New Fitness = {(P – r) (max - min)/(P – 1)} + min

where max and min are set by the user to determine the slope
(max - min)/(P – 1) of the function

Let P = 11, max = 8, min = 3,
then a chromosome of rank 1 will have a New fitness of
10*5/10 + 3 = 8
& a chromosome of rank 6 will have 5*5/10 + 3 = 5.5

GENETIC ALGORITHM


Once the new fitness is assigned, parents are selected by the
same roulette wheel procedure used in proportionate
selection

GENETIC ALGORITHM

Selection Operators: Tournament Selection
Extracts k individuals from the population with uniform
probability (without re-insertion) and makes them play a
“tournament”, where the probability for an individual to
win is generally proportional to its fitness

GENETIC ALGORITHM

Reproduction Operators

Genetic operators are applied to chromosomes that are
selected to be parents, to create offspring

Basically of two types: Crossover and Mutation

Crossover operators create offspring by recombining the
chromosomes of selected parents

Mutation is used to make small random changes to a
chromosome in an effort to add diversity to the population

GENETIC ALGORITHM

Reproduction Operators: Crossover

Crossover operation takes two candidate solutions and
divides them, swapping components to produce two new
candidates

GENETIC ALGORITHM


Figure illustrates crossover on bit string patterns of length 8

The operator splits them and forms two children whose initial
segment comes from one parent and whose tail comes from
the other

Input Bit Strings
11#0101# #110#0#1

Resulting Strings
11#0#0#1 #110101#

GENETIC ALGORITHM


Two genes sugar and flour (in kgs)
Crossover operation on chromosomes

5 1 5 4

2 4 2 1

GENETIC ALGORITHM


The place of split in the candidate solution is an arbitrary
choice. This split may be at any point in the solution

This splitting point may be randomly chosen or changed
systematically during the solution process

Crossover can unite an individual that is doing well in one
dimension with another individual that is doing well in the
other dimension

GENETIC ALGORITHM

Two types: Single point crossover & Uniform crossover

Single type crossover
This operator takes two parents and randomly selects a
single point between two genes to cut both chromosomes
into two parts (this point is called cut point)
The first part of the first parent is combined with the
second part of the second parent to create the first child
The first part of the second parent is combined with the
second part of first parent to create the second child

1000010 1000001
1110001 1110010

GENETIC ALGORITHM

Uniform crossover
The value of each gene of an offspring’s chromosome is
randomly taken from either parent
This is equivalent to multiple point crossover

1000010
1110001 1010010

GENETIC ALGORITHM

Reproduction Operators: Crossover (Variable size chromosomes)

Sometimes we need a variable size chromosome; e.g. to
represent a set of rules

Example:
Suppose we are representing a set of rules by a chromosome

If a1 = T and a2 = F then c = T
If a2 = T then c = F

The chromosome would be 10 01 1 11 10 0
where a1 = T is represented by 10,
a2 = F by 01,
and so on

GENETIC ALGORITHM


The sub-strings can be considered as a single entity during
cross-over (i.e. crossover point is not allowed in the middle of
the sub-string)

Another way can be to allow all possible crossovers, but assign
a very low fitness to resulting chromosomes which have
undesirable sub-string meaning(s)

e.g. 01110011111011
would mean, we do not care whether we play tennis or
not

GENETIC ALGORITHM


For such chromosomes we use a modified cross-over operator

To perform a crossover operation on two parents, two
crossover points are first chosen at random in one of the
parents

Example: Let the two parents be
10 01 1 11 10 0
and 01 11 0 10 01 0

GENETIC ALGORITHM


Suppose the crossover points chosen randomly for the first
parent are after bit position 1 and 8

1st parent 10 01 1 11 10 0
2nd parent 01 11 0 10 01 0

Let d1 denote the distance from the leftmost of these crossover
points to the rule boundary immediately to the left
d1 = 1
Let d2 denote the distance from the rightmost of these
crossover points to the rule boundary immediately to the left
d2 = 3

GENETIC ALGORITHM


The crossover points in the second parent are now randomly
chosen, subject to the constraint that they must have the same
d1 and d2 values

1st parent 10 01 1 11 10 0 d1 = 1, d2 = 3
2nd parent 01 11 0 10 01 0

The possible crossover points for the 2nd parent are at bit
positions (1, 3), (1, 8), and (6, 8)

2nd parent 01 11 0 10 01 0 d1 = 1, d2 = 3
01 11 0 10 01 0
01 11 0 10 01 0

GENETIC ALGORITHM


Suppose crossover points (1, 3) happen to be chosen for the 2nd
parent

1st parent 10 01 1 11 10 0
2nd parent 01 11 0 10 01 0

The resulting two offspring would be
11 10 0
and
00 01 1 11 11 0 10 01 0

GENETIC ALGORITHM

Reproduction Operators: Mutation

Mutation is another important genetic operator

Mutation takes a single candidate and randomly changes
some aspect (gene) of it

For example, mutation may randomly select a bit in the
pattern and change it, switching a 1 to a 0 or to # (don’t care)

GENETIC ALGORITHM


Mutation is important in that the initial population may
exclude an essential component of a solution

For example, if no member of the initial population has a 1 in
the first position, then crossover in the middle, cannot
produce a child that could become a solution

GENETIC ALGORITHM

Each gene of each offspring is mutated with a given
mutation rate pµ (say 0.01)

It is hence possible that no gene may be mutated for many
generations. On the other hand more than one gene may
be mutated in the same generation (or even in the same
chromosome)

For real valued genes, the value is selected randomly
from the alleles

If the rate is too low, new traits will appear too slowly in
the population. If the rate is too high, each generation will
be unrelated to the previous generation

GENETIC ALGORITHM

Q: Is it some kind of learning technique like neural networks ?

A: No

Q: Then, what is it ?

GENETIC ALGORITHM
Search Techniqes

Calculus Base Enumerative
Guided random search Techniques
Techniques techniques

Fibonacci Sort DFS Dynamic BFS
Programming

Tabu Search Hill Simulated Evolutionary
Climbing Annealing Algorithms

Genetic Genetic
Programming Algorithms

Figure: Taxonomy of searching techniques

GENETIC ALGORITHM

GA Quick Overview

• Developed: USA in the 1970’s
• Early names: J. Holland, K. DeJong, D. Goldberg
• Typically applied to:
– discrete optimization
• Attributed features:
– not too fast
– good heuristic for combinatorial problems
• Special Features:
– Traditionally emphasizes combining information from good
parents (crossover)
– many variants, e.g., reproduction models, operators

GENETIC ALGORITHM

The MAXONE problem

Suppose we want to maximize the number
of ones in a string of l binary digits
Is it a trivial problem?

It may seem so because we know the answer
in advance
However, we can think of it as maximizing
the number of correct answers, each
encoded by 1, to l yes/no difficult questions`

GENETIC ALGORITHM

The MAXONE problem

• An individual is encoded (naturally) as a
string of l binary digits
• The fitness f of a candidate solution to the
MAXONE problem is the number of ones in
its genetic code
• We start with a population of n random
strings. Suppose that l = 10 and n = 6

GENETIC ALGORITHM

The MAXONE problem (initialization step)

We toss a fair coin 60 times and get the
following initial population:
s1 = 1111010101 f (s1) = 7
s2 = 0111000101 f (s2) = 5
s3 = 1110110101 f (s3) = 7
s4 = 0100010011 f (s4) = 4
s5 = 1110111101 f (s5) = 8
s6 = 0100110000 f (s6) = 3

GENETIC ALGORITHM

The MAXONE problem (selection step)
Next we apply fitness proportionate selection with
the roulette wheel method; the individual i have
the probability to chose: f (i )

∑ f (i )
i
We repeat the extraction Area is
1 2
as many times as the n
Proportional
to fitness
number of individuals we value

need to have the same 3
parent population size 4
(6 in our case)

GENETIC ALGORITHM

The MAXONE problem (selection step)

Suppose that, after performing selection, we
get the following population:
s1` = 1111010101 (s1)
s2` = 1110110101 (s3)
s3` = 1110111101 (s5)
s4` = 0111000101 (s2)
s5` = 0100010011 (s4)
s6` = 1110111101 (s5)

GENETIC ALGORITHM

The MAXONE problem (crossover step)
Next we mate strings for crossover. For each
couple we decide according to crossover
probability (for instance 0.6) whether to
actually perform crossover or not
Suppose that we decide to actually perform
crossover only for couples (s1`, s2`) and (s5`,
s6`). For each couple, we randomly extract a
crossover point, for instance 2 for the first
and 5 for the second

GENETIC ALGORITHM

The MAXONE problem (crossover step)

Before crossover:
s1` = 1111010101 s5` = 0100010011
s2` = 1110110101 s6` = 1110111101

After crossover:

s1`` = 1110110101 s5`` = 0100011101
s2`` = 1111010101 s6`` = 1110110011

GENETIC ALGORITHM

The MAXONE problem (mutation step)
The final step is to apply random mutation: for each bit
that we are to copy to the new population we allow a
small probability of error (for instance 0.1)
Before applying mutation:
s1`` = 1110110101
s2`` = 1111010101
s3`` = 1110111101
s4`` = 0111000101
s5`` = 0100011101
s `` = 1110110011

GENETIC ALGORITHM

The MAXONE problem (mutation step)

After applying mutation:
s1``` = 1110100101 f (s1``` ) = 6
s2``` = 1111110100 f (s2``` ) = 7
s3``` = 1110101111 f (s3``` ) = 8
s4``` = 0111000101 f (s4``` ) = 5
s5``` = 0100011101 f (s5``` ) = 5
s6``` = 1110110001 f (s6``` ) = 6

GENETIC ALGORITHM

The MAXONE problem (example end)

In one generation, the total population
fitness changed from 34 to 37, thus
improved by ~9%

At this point, we go through the same
process all over again, until a stopping
criterion is met

GENETIC ALGORITHM

Short Assignment:
Design a genetic algorithm to learn conjunctive
classification rules for the Play-Tennis problem.
Describe precisely the bit-string encoding of hypotheses
and a set of crossover operators.
Due Date:
24-05-2012

Hint: see Chapter 09, Machine Learning book by Tom. Mitchell

GENETIC ALGORITHM

Short Assignment Data:

GENETIC ALGORITHM

Major Assignment (10 marks) :
Task: you have to analyse some research paper and give
your critical analysis in the form of a report. The report
will be critically reviewed and accordingly marked.
Deliverables: A report + Presentation
Relevant Information: The assignment will be prepared
within groups. However, the marks will be assigned based
on individual performance.
Research Topics: the topics will be uploaded on the group

GENETIC ALGORITHM

The New Generation

The new offspring can replace the old population without any
fitness comparison

or

only the better ones from the new and old make it to the new
generation (more processing needed)

GENETIC ALGORITHM

New Generation: Elitism

Elitism is a value between 0 and 1, which represents the
fraction of the individuals of a population that will be
duplicated to the next generation

Example P = 20 and elitism = 0.1, then 2 individuals of
current population do not get replaced

The elite chromosome selection may be based on highest
fitness values

If highest fitness valued chromosomes are carried over to the
next generation, we ensure that the maximum fitness does not
decrease from one generation to next

GENETIC ALGORITHM

New Generation: Generation Gap

The number of individuals replaced in the next generation is
called generation gap

A generation gap of 100% will mean that whole of the
population comprises of new chromosomes

We may have a generation gap which is not fixed. In this
method, the fittest P chromosomes will be selected from the
set of current population plus new children, and will form the
new generation

GENETIC ALGORITHM
New Generation: Number of Duplicates Allowed

Duplicates are Chromosomes that are same

If they are allowed then that chromosome has higher
probability of producing an offspring, and may probably
create many offspring

Eliminating duplicates increases the efficiency of the genetic
search and reduces the danger of premature convergence

Eliminating duplicates means that if an offspring is created
which is a duplicate of a chromosome of the current
population, we terminate it immediately and create a new
one. It increases the processing time in large populations

GENETIC ALGORITHM

Termination Requirement

The GA continues until some termination requirement is met,
such as

- having a solution whose fitness exceeds some threshold
- the fitness of solutions becomes stable & stops improving

GENETIC ALGORITHM

References

Engelbrecht Chapter 8 & 9

Soft computing06

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