Week 4.1 - Genetic Algorithm

Genetic Algorithm

• This algorithm reflects the process of natural selection, where the fittest individuals are selected for reproduction in order to produce offspring’s of the next generation.
• This can be applied to any engineering fields by doing many number if iterations to the final best solution.

Main steps of genetic algorithm:

1. Initialization or initial population
2. Fitness function
3. Selection
4. Cross over (Re combination)
6. Terminating or stopping criterian

Initialization or initial population:

• Initial population consists of all the probable solutions of the given problem. Here an individual is characterized by a set of parameters known as genes.
• Genes are joined in a string to form a chromosome; chromosomes are basically strings of DNA.

Fitness function:

• After the initialization of population we determine the ability of the individual to compete with other individuals by using the fitness function.
• F(x)=-1/4x² +2x+5 where x is the fitness function.

Selection:

• The selected individuals are then arranged in pairs of two to enhance reproduction, these individuals pass on their next genes to the next generation.
• The objective of this phase is to establish the region with high chances of generating the best solution to the problem.

Cross over:

• It is the most significant phase of the genetic algorithm, for each of parents to be mated; a cross over point is chosen at random form with in the genes.

Mutation:

• During their growth, there is some change in the gene of the children which makes them different from their parents.

Termination:

• If there is no improvement in the population for over X iterations.
• If we already have predefined an absolute number of generations to our algorithm.
• If our fitness function has reached a predefined value.

Syntax for Genetic algorithm (ga):

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options)

where,

fun- the objective function.

nvar- number of variables.

A,b – the linear equalities.

Aeq,beq – if the nonlinear in equalities exist.

Ub,Lb – the upper and lower bounds.

nonlcon-minimization of constrains defined using nonlcon.

Options- sets the population size using optimoptions.

clear all %clears all previous values in workspace.
close all  % closes all the previous results.
clc        % clears the command window.
%define the search space for the function
num_cases=50;  %numer if cases is defined to be 50.
x=linspace(0,0.6,150);  %x value is assigned to be 150 values from 0 to 0.6
y=linspace(0,0.6,150);  %y value is assigned to be 150 values from 0 to 0.6
%creating 2D mesh
[xx,yy]=meshgrid(x,y); % meshgeid command is used to multiply vectors
%evaluating the stalagmite function.
for i=1:length(xx) % for loop i from 1 to length of xx
    for j=1:length(yy) % for loop j from 1 to length of yy
        input_vector(1)=xx(i,j); % input vector1 is being assigned as xx
        input_vector(2)=yy(i,j); % input vector2 is being assigned as yy
        %call the function
        f(i,j)=stalagmite(input_vector); %call stalagmite function
    end
end    

%% study-1 satistical behaviour with unbounded inputs
tic   % tic-toc command to find time taken.
num_cases=50;
for i=1:num_cases % for loop i from 1 to 50
    [inputs1,fopt1(i)]=ga(@stalagmite,2); %staligmate is called 
    xopt1(i)=inputs1(1);  % xopt1 is assigned to input 1(1)
    yopt1(i)=inputs1(2); % yopt1 is assigned to input 1(2)
end
time_taken1=toc
%ploptting
figure (1);
subplot(2,1,1); % plots figure in top of same page
hold on     % waits for next graph to save on same page
surfc(xx,yy,f); % surface of xx,yy, anf function is made
shading interp % to avoid the shaded lines in the plot
plot3(xopt1,yopt1,fopt1,'marker','+','markersize',5) % 3d plot is drawn
title('Behaviour with unbounder inputs'); % title is given to figure
subplot(2,1,2); % plots figure in bottom of same page
plot(fopt1); % function is being plotted
xlabel('Iterations');   %xlabel is being given
ylabel('Function minimum'); %ylabel is being given

%% study-2 Statistical behaviour with upper and lower bounds
tic 
for i=1:num_cases % for loop i from 1 to 50
    [inputs2,fopt2(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6]); %staligmate is called with upper and lower bounds 
    xopt2(i)=inputs2(1);  % xopt2 is assigned to input 2(1)
    yopt2(i)=inputs2(2); % yopt2 is assigned to input 2(2)
end
time_taken2=toc
%plotting 2
figure(2)
subplot(2,1,1) % plots figure in top of same page
hold on % waits for next graph to save on same page
surfc(xx,yy,f); % surface of xx,yy, anf function is made
shading interp % to avoid the shaded lines in the plot
plot3(xopt2,yopt2,fopt2,'marker','+','markersize',3);  % 3d plot is drawn
title('With bounded inputs')  % title is given to figure
subplot(2,1,2) % plots figure in bottom of same page
plot(fopt2) % function is being plotted
xlabel('Iterations'); %xlabel is being given
ylabel('Function minimum'); %ylabel is being given

%% study-3 Introducing Population size
%have to use optimoptions to set the population size and curent search
%space is to 200
options =optimoptions('ga');
options=optimoptions(options,'populationsize',200)
tic
for i=1:num_cases
    [inputs3,fopt3(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[0.6;0.6],[],[],options);  %staligmate is called with upper and lower bounds and populations
     xopt3(i)=inputs3(1); % xopt3 is assigned to input 3(1)
    yopt3(i)=inputs3(2); % yopt3 is assigned to input 3(2)
end
time_taken3=toc
figure(3)
subplot(2,1,1) % plots figure in top of same page
hold on    % waits for next graph to save on same page
surfc(xx,yy,f); % surface of xx,yy, anf function is made
shading interp % to avoid the shaded lines in the plot
plot3(xopt3,yopt3,fopt3,'marker','+','markersize',3);  % 3d plot is drawn
title('With bounded inputs and population size')  % title is given to figure
subplot(2,1,2) % plots figure in bottom of same page
plot(fopt3); % function is being plotted
xlabel('Iterations');  %xlabel is being given
ylabel('Function minimum'); %ylabel is being given