Genitic Algorithm:

Aim

• Write a code in MATLAB to optimise the stalagmite function and find the global maxima of the function.
• Clearly expalin the concept of genetic algorithm in your own words and also explain the syntax for ga in MATLAB in your report.
• Make sure that your code gives the same output, even if it is made to run several times.
• Plot graphs for all 3 studies and for F maximum vs no. of iterations. Title and axes labels are a must, legends could be shown if necessary.

Theory

A genetic algorithm is a search heuristic that is inspired by Charles Darwin’s theory of natural evolution. This algorithm reflects the process of natural selection where the fittest individuals are selected for reproduction in order to produce offspring of the next generation.

Notion of Natural Selection

The process of natural selection starts with the selection of fittest individuals from a population. They produce offspring which inherit the characteristics of the parents and will be added to the next generation. If parents have better fitness, their offspring will be better than parents and have a better chance at surviving. This process keeps on iterating and at the end, a generation with the fittest individuals will be found.

This notion can be applied for a search problem. We consider a set of solutions for a problem and select the set of best ones out of them.

Five phases are considered in a genetic algorithm.

1. Initial population
2. Fitness function
3. Selection
4. Crossover
5. Mutation

Initial Population

The process begins with a set of individuals which is called a Population. Each individual is a solution to the problem you want to solve.

An individual is characterized by a set of parameters (variables) known as Genes. Genes are joined into a string to form a Chromosome (solution).

In a genetic algorithm, the set of genes of an individual is represented using a string, in terms of an alphabet. Usually, binary values are used (string of 1s and 0s). We say that we encode the genes in a chromosome.

Fitness Function

The fitness function determines how fit an individual is (the ability of an individual to compete with other individuals). It gives a fitness score to each individual. The probability that an individual will be selected for reproduction is based on its fitness score.

Selection

The idea of selection phase is to select the fittest individuals and let them pass their genes to the next generation.

Two pairs of individuals (parents) are selected based on their fitness scores. Individuals with high fitness have more chance to be selected for reproduction.

Crossover

Crossover is the most significant phase in a genetic algorithm. For each pair of parents to be mated, a crossover point is chosen at random from within the genes.

For example, consider the crossover point to be 3 as shown below.

Crossover point

Offspring are created by exchanging the genes of parents among themselves until the crossover point is reached.

Exchanging genes among parents

The new offspring are added to the population.

New offspring

Mutation

In certain new offspring formed, some of their genes can be subjected to a mutation with a low random probability. This implies that some of the bits in the bit string can be flipped.

Mutation: Before and After

Mutation occurs to maintain diversity within the population and prevent premature convergence.

Termination

The algorithm terminates if the population has converged (does not produce offspring which are significantly different from the previous generation). Then it is said that the genetic algorithm has provided a set of solutions to our problem.

Comments

The population has a fixed size. As new generations are formed, individuals with least fitness die, providing space for new offspring.

The sequence of phases is repeated to produce individuals in each new generation which are better than the previous generation.

Program

clear all

close all

clc






x=linspace(0,1,100);

y=linspace(0,1,100);




[xx yy]=meshgrid(x,y);




for i=1:length(xx)

    for j= 1:length(yy)

        input(1)=xx(i,j);

        input(2)=yy(i,j);

        f(i,j)=stalagmite(input);

        [Input,fval(i)]=ga(@stalagmite,2);

        x1(i)=Input(1);

      y1(i)= Input(2);

        

    end

end

figure(1)

surfc(xx,yy,f)

hold on

plot3(x1,y1,fval)

hold off

figure(2)

plot(fval)

shading interp

[Input,fval]=ga(@stalagmite,2);




for i=1:length(xx)

   for j= 1:length(yy)

        input(1)=xx(i,j);

       input(2)=yy(i,j);

        f(i,j)=stalagmite(input);

       [Input,fval(i)]=ga(@stalagmite,2,[],[],[],[],[0:0],[0.9:0.9]);    %%

       x2(i)=Input(1);

      y2(i)= Input(2);

        

   end

end

figure(3)

surfc(xx,yy,f)

hold on

plot3(x1,y1,fval)

hold off

figure(4)

plot(fval)

shading interp




options=optimoptions(\'ga\')

options=optimoptions(options,\'PopulationSize\',400);

tic;




for i=1:length(xx)

    for j= 1:length(yy)

        input(1)=xx(i,j);

        input(2)=yy(i,j);

        f(i,j)=stalagmite(input);

        [Input,fval(i)]=ga(@stalagmite,2,[],[],[],[],[0:0],[0.2:0.2],[],[],options);

        x3(i)=Input(1);

      y3(i)= Input(2);

        

    end

end

toc;

v=toc;

figure(5)

surfc(xx,yy,f)

hold on

plot3(x3,y3,fval)

shading interp

hold off

figure(6)

plot(fval)

Stalagmite function





function [F]= stalagmite(Input)

x=Input(1);

y=Input(2);

f1=(sin((5.1*pi*x)+0.5))^6;

f2=((sin(5.1*pi*y)+0.5))^6;

f3= exp((-4*log(2)*((x-0.0667)^2)/0.64));

f4=exp((-4*log(2)*((y-0.0667)^2)/0.64));

F=(f1*f2*f3*f4*-1);

end

Result:

In this we have done three type of analysis

1.  Ga general which will give generic results. Here for this we can see that the have many maxima and minmas with this we cannot find which is maxima and which is minima

1. The below graph show the reduction in lines by giving the upper and lower bound.  So we are selecting a range of values with in which the GA is occurring and finding the result.

1. The third study includes the definition of population size along with the bound

Here we can see that the GA value converges to the point nothing but our global maxima.