Week 4.1 - Genetic Algorithm

WORKING OF GENETIC ALGORITHM :

The working of genetic algorithm is as follows,

1. The algorithm begins by creating a random initial population.

2. 1. Scores each member of the current population by computing its fitness value. These values are called the raw fitness scores.

   2. Scales the raw fitness scores to convert them into a more usable range of values. These scaled values are called expectation values.

   3. Selects members, called parents, based on their expectation.

   4. Some of the individuals in the current population that have lower fitness are chosen as elite. These elite individuals are passed to the next population.

   5. Produces children from the parents. Children are produced either by making random changes to a single parent—mutation—or by combining the vector entries of a pair of parents—crossover.

   6. Replaces the current population with the children to form the next generation.

   Syntax for GA:

   x = ga(fun,nvars) finds a local unconstrained minimum, x, to the objective function, fun. nvars is the dimension (number of design variables) of fun.

   x = ga(fun,nvars,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that a solution is found in the range lb ≤ x ≤ ub. (Set Aeq=[] and beq=[] if no linear equalities exist.)

   x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the default optimization parameters replaced by values in options. (Set nonlcon=[] if no nonlinear constraints exist.) Create options using optimoptions

CODE EXPLANATION : Coding and minimizing the fitness function using genetic algorithm.

STEP 1 :FITNESS FUNCTION CODE

A fitness function must take one input x and input y where x and y is a row vector with as many elements as number of variables in the problem. The fitness function computes the value of the function and returns that scalar value in its one return argument f.

function[f]= stalagmite(input_vector)
% inputs
x=input_vector(1);
y=input_vector(2);

% defining the term
term1=(sin((5.1*pi*x)+0.5))^6 ;
term2=exp((-4)*log(2)*((x-0.0667)^2)/0.64);
term3=(sin((5.1*pi*y)+0.5))^6 ;
term4=exp((-4)*log(2)*((y-0.0667)^2)/0.64);

 f=-(term1*term2*term3*term4);
end

STEP 2: MINIMIZE USING GA:

STUDY 1 statistical behavior shows the minimization using the ga

To minimize the fitness function using ga pass a function handle to the fitness function as well as the number of variables in the problem .x = ga(fun,nvars) finds a local unconstrained minimum, x, to the objective function, fun. nvars is the dimension (number of design variables) of fun. here fun is defined as stalagmite and nvars as 2.

%study 1--statistical behavior
tic
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study_time1=toc
%plotting
figure(1)
subplot(2,1,1) % used to plot multiple plots on same figure
hold on
surfc(x,y,f)    %  creates a three-dimensional surface plot with a contour plot underneath
shading interp    % varies the color in each line segment and face by interpolating the colormap index or true color value across the line or face
xlabel('x values')
ylabel('y values')
title('statistical behavior with unbound inputs')
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'MarkerFaceColor','r') %plots coordinates in 3-D space
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('Fuction maximum')

STEP 3:MINIMIZING USING THE ADDITIONAL PARAMETER:

STUDY 2 statistical behavior with upper and lower bounds represents minimizing using the additional parameters.To minimize the fitness function using ga , pass a function handle to the fitness function as well as the number of variables in the problem. To have ga examine the relevant region, include bounds lower bound as [0 0] and the upper bounds[1 1] Pass the bounds as the fifth and sixth arguments after numberofvariable.

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables,x, so that a solution is found in the range lb ≤ x ≤ ub. (Set Aeq=[] and beq=[] if no linear equalities exist.)

% study 2-- statistical behavior with upper and lower bounds
tic
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study_time2=toc
figure(2)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
xlabel('x values')
ylabel('y values')
title('statistical behavior with upper and lower bounds')
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'MarkerFaceColor','r')
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('Fuction maximum')

STEP 3: SPECIFY POPULATION OPTION:

Population options can have a large effect on solver performance. The speed of each iteration depends on the population size: a larger population leads to slower iterations. Conversely, a larger population leads to ga exploring more thoroughly, so can lead to a better solution. Similarly, a wider initial range can lead to more thorough exploration, but can require a larger population to explore the wider range with a similar thoroughness.

Specify Population Size

ga creates a default initial population by using a uniform random number generator. The default population size used by ga is 50 when the number of decision variables is less than 5, and 200 otherwise. The default size might not work well for some problems; for example, a smaller population size can be sufficient for smaller problems. Since the current problem has only two variables, specify a population size of 220. Set the value of the option populationsize to 220 in the existing options,opts.

Modify the value of the option polulationsize in the existing options,opts.

opts.populationsize = [220];

Run the ga solver.

[x,Fval,exitFlag,Output] = ga(FitnessFunction,numberOfVariables,[],[],[], ...
    [],[],[],[],opts);

optimoptions syntax:

% study 3--statistical behavior with increasing the iteration of Ga
options=optimoptions('ga')
options=optimoptions(options,'PopulationSize',220);
tic
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study_time3=toc
figure(3)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
xlabel('x values')
ylabel('y values')
title('statistical behavior with increasing iteration GA')
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'MarkerFaceColor','r')
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('Fuction maximum')

STEP 4:

Reproduce Results

By default,ga starts with a random initial population created using MATLAB random number generators. The solver produces the next generation using ga operators that also use these same random number generators. Every time a random number is generated, the state of the random number generators changes. So, even if you do not change any options, you can get different results when you run the solver again.

Reset the stream and rerun ga . The results are identical to the previous run.as rng(thestate).

STEP 5:

CONCULSION :The best function value can improve or get worse based on the specified operators. Experimenting with different operators is often the best way to determine which set of operators works best for your problem.

MATLAB CODE :

MAIN CODE

clear
close all
clc
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);

% creating 2-D mesh
[xx,yy]=meshgrid(x,y);

num_cases=50;


for i=1:length(xx)
    for j=1:length(yy)
        input_vector(1)=xx(i,j);
        input_vector(2)=yy(i,j);
f(i,j)=stalagmite(input_vector);
    end
end

%study 1--statistical behavior
tic
for i=1:num_cases
 
    [inputs,fopt(i)]=ga(@stalagmite,2);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study_time1=toc
%plotting
figure(1)
subplot(2,1,1) % used to plot multiple plots on same figure
hold on
surfc(x,y,f)    %  creates a three-dimensional surface plot with a contour plot underneath
shading interp    % varies the color in each line segment and face by interpolating the colormap index or true color value across the line or face
xlabel('x values')
ylabel('y values')
title('statistical behavior with unbound inputs')
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'MarkerFaceColor','r') %plots coordinates in 3-D space
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('Fuction maximum')

% study 2-- statistical behavior with upper and lower bounds
tic
for i=1:num_cases
    [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study_time2=toc
figure(2)
subplot(2,1,1)
hold on
surfc(x,y,f)
shading interp
xlabel('x values')
ylabel('y values')
title('statistical behavior with upper and lower bounds')
plot3(xopt,yopt,fopt,'marker','o','markersize',5,'MarkerFaceColor','r')
subplot(2,1,2)
plot(fopt)
xlabel('Iterations')
ylabel('Fuction maximum')

% study 3--statistical behavior with increasing the iteration of Ga
options=optimoptions('ga')
options=optimoptions(options,'PopulationSize',220);
tic
for i=1:num_cases
    rng(thestate);
    [inputs,fopt(i)]=ga(@stalagmite,2,[],[],[],[],[0;0],[1;1],[],[],options);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end
study_time3=toc
figure(3)
subplot(2,1,1)
hold on

RESULT/OUTPUT :

1. plot for the study 1 statistical behavior with nobounds

2. Plot for the study 2 statistical behavior with upper and lower bounds.

3.Plot for study 3 statistical behavior increasing the population size to get the maximum point.

4.Result showing the time study.

REFERENCE :https://www.mathworks.com/help/matlab/