11
Sep

Lecture 32: Genetic Algorithms


Ah. So, in our course; Selected Topics in
Decision Modeling, now we are in our thirty second lecture that is on genetic algorithms,
right. Now ah in our previous lecture, we have seen the Metaheuristics and one of the
very important Metaheuristics are the evolutionary computing methods.
And they are all essentially coming from the genetic algorithms, right. So, the; as the
name suggests, the genetic algorithm really coming from the essential idea of genetics.
So, they are all nature inspired algorithms. So, here is a comparison, you can see that
environment in nature in genetic algorithm is the optimization problem.
The feasible solutions are the individuals, the solution quality and individuals adaptation
ah to the environment are comparable set of feasible solutions and the population species
and the various operators the selection recombination and reproduction.
So, ah what really happening is that the genetic algorithms are intelligent search technique
maintaining a population of candidate solutions for a given problem and search the solution
space by applying variation operators. See ah what really happens? The biological
evolution process is one of survival of the fittest. So, what exactly it means? It means
that assume that in a particular place ah you know the there are a large number of trees.
So, assume that place is very near to a hilly region. So, what happens in a hilly region
the the sun’s rays are coming, but sun’s rays are not coming uniformly to all the trees,
it is the tall trees which gets the sun’s rays, but this smaller trees are not getting
the sun’s rays. So, what will happen out there? You see a over generations, you will find
the that particular place in a hilly region, you only find tall trees and you will not
find small trees; what is the reason for it? The reason is that it is not that there were
not all kinds of trees we had all kinds of trees, but it is the tall trees which survives
over generations and the small trees ah they die out because nature selection does not
favor them. The same thing about you know the animals
you find in the hilly regions; ah you know ah who really are well equipped to face the
winter ah you know the kind of animals, you find in the plane region ah if they really
go without far ah to the hilly regions, they will not survive.
So, this is the process of natural selection, the tall trees grow near the mountains the
animals with far; ah they are in the wintry regions the summer regions, we do not find
animals with lot of far. So, these are the things ah that we actually observed in the
natural situations in nature the question is that how nature does this selection; what
is the essential process of nature the essential process of nature is that you know that ah
the parents who was selected in a natural process are by a process called survival of
the fittest the fittest. The fittest species, you know the those ah
specific what you call parents are selected for reproduction, who are ah more fit, then
the others alright and then the chromosomes, they join each other by processes like cross
over and mutation to really produce offspring, is it alright? So, that is the essential biological
process the genetic algorithms mimics it right. So, mimics it, but since it is a it is an
algorithm; obviously, we can apply these algorithm in the best possible manner. So, that ah we
have the best possible solutions. So, this is the essential idea of the genetic algorithm.
So, ah as you can see, inspired by natural evolution, gas involved direct manipulation
of the coding achieved by cross over and mutation it begins the search from many points not
from a single point. So, therefore, it contains population of feasible solutions to the problem,
is it alright? So, supposing I have a solutions space suppose this is our solution space,
right and in this solution space, there are many points. Now question is out of all those
different points that are possible solutions, the gas randomly starts with points which
are really covering you know the entire search space entire search space.
So, you see; we we define a population; what is a population supposing my population size
is 8, then 1, 2, 3, 4, 5, 6, 7, 8; these are the 8 ah particles or may be points which
we have selected ah to define as our first generation. Now what we do; ah you know, I
I ah have out of these 8, I select may be a few may be say, 4, I select based on which
which of these 8 has the highest fitness what is the fitness; fitness is the objective function
in that case the function x fx. So, supposing there is a function fx which we are maximizing.
So, these are like x each individual point is an x. So, if I know x, I know fx, right.
So, that is our what is known as the fitness function.
So, based on this fitness function value, you know I choose out of these 8 may be 4
of them, then by the process of cross over and mutation which I ah I will discuss ah
you know from the 4 parents, we obtain the 8 offspring, right, sometimes we do not really
have 8 offspring, we may have let us say 6 offspring and ah the best two parents we also
add may be. So, you see we can ah have something like this either 8 offspring or 6 offspring
plus 2 parents who are the best in in terms of fitness value that becomes our second generation.
So, you see from my first generation of 8 points to second generation of another 8 points;
what is the fitness of these 8 and what is the fitness of these 8? If I compare, I will
find the second generation ah you know the ah points, we will have a higher fitness value,
why it happened because we have chosen out of the 8, 4 for reproduction which are of
higher fitness value and the off springs fitness are very near to the parents not exactly same
because they are not same, but then they are very near to them because it is.
So, you know I have reason to believe that the average fitness has really gone up is
it alright. So, that is the essential idea about genetic algorithm as I move from population
to population may be generation to generation. So, from second generation, when I go to third
generation to fourth generation to fifth generation, you know, I have a set of ah points with higher
fitness, right. Again ah looking at it, while we define the
population, we should see that the population diversity; that means, representing all types
of points and selection pressure; that means, the points are having higher fitness value
ah they are maintained, is it alright. So, that is the essential idea of genetic algorithm.
The advantage it is an answer based method, they do not need auxiliary information like
gradients search via sampling. So, no advance calculations and therefore, calculation is
easy the iterative process is also simple, is it alright, ah the search by stochastic
operations not by deterministic rules. So, it is by a random choice ah to guide highly
exploitative search, it is alright that is the essential idea of ah genetic algorithm.
Let us give another example, ah supposing you know there is a large city and you really
want to find ah persons of a certain type, is it alright; persons of a certain type ah
see you know there are some people who are highly knowledgeable on a given area, but
they are very rare you know you really do not know how to get there is it alright and
do you know in a city there are. So, many of them may be in a city of ah several crores
people ah you have may be 20 of them, you need the best one out of that who has knowledge
on the particular field. But then these 20 people; they are not together
this twenty people are dispersed over the city in a random manner and how do you search
them supposing you go to the city and you knock at every door that do you have this
knowledge ah it is a it is a foolish method, you just cannot really get that particular
person ah by this method, is it all right? But assume you hold a conference in that conference
you randomly invite say 50 people, is it alright and these 50 people ah you tell that you go
to several parts of the city and hold a another conference of fifty people, is it alright?
So, you see from a 50 people a the next you know you are having another ah set of 50.
So, 50 into 50; 2500 people are explored ah by second generation itself. So, that is the
you know ah advantage of genetic algorithms, the search space is actually exploited in
an exponential manner, is it alright, ah how it is done that that initial population of
50 that you choose randomly that is like holding the conference for the first generation.
But then out of these 50, you create through cross over and mutation if you know the next
generation and the process is inherently such that it actually explores a much higher solutions
space, is it alright. So, when you go from several ah generations you have actually exploited
almost the entire search space. So, chances are if you are continuously evaluating the
fitness function value ah you are going to reach at the optimum or at least near optimum
the essential idea. So, inspired by natural evolution, ah you
know the gas are direction manipulation. So, already we have discussed this.
So, let us see our next slide that the idea of GA the process is such that we do an encoding,
then select on the basis of fitness, then employ recombination operators that is selection
cross over and mutation repeatedly to preserve the good portion of the string the string
is the chromosome here the good portion of the strings or chromosomes usually lead to
an optimal or near optimal solution the method is applied over a desired number of generations
if well designed the population will converge faster.
So, let us look at the ah GA evolutionary process. So, you see, what really happen.
We we we begin here that is we initiate and evaluate the population. So, initially we
initiate the population, supposing you know, we we are asked that find the number between
1 to 100, just take a very simple problem find the number between 1 to 2 you know which
fits a functional value fx, right. That means it returns a maximum value of this
function fx. So, we randomly generate through random numbers may be some arbitrary number
like 29, 49, 87, 92, arbitrarily, suppose we choose these 4 numbers and for each, I
find out the functional value what are they; obviously, they will be f of 29, f of 49 and
f of 92, supposing these are my functional values arbitrarily; I am writing the functional
ah value returns these kind of numbers right suppose these are my functional values.
Now, look here this f 49 and f 87 are best possible parents. So, based on these fitness
function, we select, right, we select out of these generation of 4, you know I select
two of them which one. So, I select basically 49 and I select 87. So, this is where the
selection is done they are my parents who are the parents the two which are selected,
is it alright, then the process of cross over is really done. So, I will explain what is
the process of cross over. So, through the process of cross over the 49 and 87 will become
you know from these two, I can get 4 different numbers right suppose in those 4 different
numbers are 41; say 42, 55, 78 and say 89. So, how these number are obtained you know,
it will depend on how these are ah you know these chromosome structures are written.
So, these 4 numbers could be generated by the process of cross over and mutation of
these two numbers. So, this is how the modification is done, then this is to be evaluated again
that evaluated means again for each of them, I have to obtain the f 42, f 55, f 74 and
f 89, right. So, after this evaluation, we might delete some members or I generate the
next population. So, from this first population, I get the second population and then iterate
till my fitness value is improved, right. So, that is the essential process of the genetic
algorithm right So, we will discuss in more detail in our
next class, but here I am just giving a overview. So, what is a chromosome? Now look here, a
chromosome can be represented in many different ways one of the ah way is by binary encoding.
So, what is binary encoding binary encoding is by writing the number by zeros and ones.
Now it is not always that ah you know binary encoding simply means ah a number it could
be something else also supposing ah in this example can be seen that supposing there are
seven switches right the each switch could be 0 or 1.
Or assume ah you know in a in a knapsack problem where I have seven things, it is a 0-1 knapsack
the knapsack basically is that there are seven things and a particular thing could be either
there or not there each knapsack has a value right how to fill up the knapsack in the certain
manner. So, that the value could be maximized and there would be some weight restriction
also. So, that is how it is. So, now, here you see these are the seven things a particular
chromosome could be 1 0 0 1 1 1 0; that means, first thing is there second and third not
there fourth fifth and sixth is there and seventh is not there that is what the chromosome
means in the context of that problem, I hope it is clear that a 0-1 knapsack, there are
seven things a chromosome here means which ones are there which ones are not there
If it is a number, it can also represent a number, right. So, each bit corresponds to
a gene and the value for a given genes are called alleles. So, what is an allele? A allele
is the number that is the value, is it alright. So, that is the value is an allele and the
chromosome is essentially ah you know the in in case of binary encoding; that is the
entire thing and population is a set of chromosomes right a set of chromosomes can be called a
population. So, that is the essential idea of how these things are represented, right.
Now, having known this, now ah this is how the GA ah over the generations happens already
we discussed that initialize selection cross over mutation insertion into the population
and stop otherwise finish. So, that is how the Goldberg ah you know who has written a
very important book on genetic algorithms ah gives the GA over generations, right.
Now, encoding as I was saying, there are different kinds of encoding. What are they? It could
be binary numbers, it could be real numbers, it could be a permutation encoding, see a
example could be a travelling salesman problem supposing there are 5 cities. So, ah you have
to travel to each city exactly once. So, here the encoding could be say 2, 3, 4, 5, 1; you
see from 2 to 3 to 4 to 5 to 1; right that could be your encoding.
Now, you cannot really have 2 3 3 5 1, this is a mistake, why, it is a mistake because
in case of a travelling salesman problem, you have to visit every city exactly once
here where is city 4, right. So, this is not a good encoding whereas, this is a good permutation
encoding right. So, in a permutation encoding every number should be exactly once which
represents the order in which in a ts problem cities are visited then we could have value
encoding where actual values could be used as genes in a chromosomes.
There are tree encoding also which we shall discuss ah later where ah the chromosome is
a tree structure, they have got very unique uses alright the tree encoding ah. So, discussion
ah I will have discussions later, right. So, that is how the different ah kind of encoding
processes that are available ah for chromosome .
Now, question is; how do you select? The selection is also a very interesting process it could
be roulette wheel selection with or without scaling then a stochastic tournament selection
with a tournament size of two a remainder stochastic sampling with replacement remainder
stochastic sampling with and without replacement and elitism is, alright.
So, you see what is really done ah the as I said that the initial population has got
ah each member has got a fitness value a fitness function value. So, based on supposing I I
make a roulette wheel, ah supposing, I make a roulette wheel in this manner, I make a
wheel and supposing there are 8 members in a population. So, I divide this population
into 8 regions. So, in this case; 1, 2, 3, 4, 5, 6. So, there could be seven and 8 also.
So, you see these are my 8 regions, as you can see those 8 regions are not equal the
regions are ah suppose region 1, 2, 3, 4, 5, 6, 7, 8, each region is proportional to
fitness value alright, it is it is it is actually proportional to the fitness value. So, chromosome
one has got higher fitness. So, it is given higher region three or seven has less fitness.
So, they have been given less amount of region right. So, that is the roulette wheel.
Now, you rotate the roulette wheel and you know it just rotates and when it stops may
be there is a pointer here that is a pointer. So, pointer will select ah one of them. So,
suppose the pointer selects 5. So, 5 is chosen that is 50 is selected, right. So, so that
is that is the essential idea of roulette wheel ah selection method. So, suppose you
turn every time you turn may be you get 4, next time you get 1, next time and you get
8 next time. So, you see the chances of getting selected ah directly depends on the fitness.
So, if you have higher fitness ah there is more chance of getting selected right.
Scaling is something where you actually modify the fitness values ah based on certain criteria
is, alright, sometimes, we we do select two them and then ah make a tournament the better
1, we select right remainder stochastic is basically ah divide the fitness value by average
fitness and any value where the integer portion of numbers automatically are selected the
roulette wheel is constituted based on the remainders, is it alright? An elitism is a
very specific ones where the parent directly goes to the next generation not all parents
some of the parents are directly selected in the next generation.
Now, every selection scheme has got pros and cons see, specifically, it is known that for
genetic algorithm, it is important that I have a balance between population diversity
with; that means, that whether I represent points from the entire region and selection
pressure; that means higher fitness value. So, balance has to be obtained between the
population diversity and selection pressure that is very important for the genetic algorithms,
right. So, having said that let us move over and
look at the cross over and mutation processes. See cross over essentially look at, suppose,
I have two parents 101 110, 001 111. So, there is a cross over point. So, these line is basically
is a cross over point. So, these line is like a cross over point. So, what happens the at
the cross over points these hundred one is here and these 110 ah sorry these 111 is here
alright. So, these portion and these portion together they become a child and these portion
and these portion because these should be first that is the first three and these are
the last 3. So, these becomes another child right. So, that is how ah the cross over ah
can be done these are called single point cross over, right.
And there is a two point cross over you see this is one gene ah sorry chromosome this
is another chromosome. So, there are two cross over points. So, one is 10, then these 01
and then these 01 and then this 00, this is 1 and 11 these eleven these 11 and these eleven.
So, these becomes another, right. So, this is how the cross over is made and ah from
two parents we get two childs, is it alright. So, that is how we we we we actually obtain
ah the future generation from a set of parents. Sometimes a mutation is also done may be a
bit is inverted a zero is converted to one ah as of to make mutation, is it alright.
So, these are cross over and mutation examples. So, there are different parameters which we
will also discuss in due time there is a size of the population how big should be taken
20, 30 or 50,000, its problem specific the cross over probability whether to do cross
over or not a usually a high, probability is taken the low probability is taken for
mutation rate and cross over and mutation type, there are large number of cross over
the mutation types which are again problem dependent, is it alright?
So, they are all used ah here are some benefits the benefits, you know, they are modular in
structure good for noisy environment solutions are obtained all the time distributed parallel.
So, parallel computing can be made use of easy to exploit flexible building blocks can
be used in multi objective optimizations. So, those are the benefits, right.
And when to use a genetic algorithm when alternate solutions are slow need and exploratory tool
problem is similar to one that has already been solved when we can hybridize with an
existing salutations and where near optimal solution adequate computational power is available
and problem converges that is when the benefits are maximum.
Finally here are some GA applications, right, they could be used in control design scheduling
robotics machine learning signal processing game playing combinatorial optimization and
many many other ah situations, see as I said the optimization problems are nowadays more
and more dependent on nature inspired algorithms where GA is probably the pioneering one and
many others like pso sco and many other algorithms they are essentially derived from the basic
genetic algorithm process. So, if you understand the basic genetic algorithm
process, ah you can understand other variations also. Now, a since we are increasingly going
to have large optimization problems, it is essential that we solve them and quickly solve
them. So, that we get a solution if not the optimal at least the near optimal. So, it
is always better than not solving at all, is it alright?
So, we shall more details about the GA process in our next lecture, right so.
Thanks for patiently hearing .

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