Week 4.1 - Genetic Algorithm

• The idea of the 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.

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

• The new offspring are added to the population.

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 occurs to maintain diversity within the population and maintain 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.

2.2.3 METHODOLOGY:

2.3 GENETIC ALGORITHM IN MATLAB:

2.3.1 SYNTAX:

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

• ga finds a local minimum x to fun, subject to the linear inequalities A*x ≤ b. ga evaluates the matrix product A*x as if x is transposed (A*x'), subject to the linear equalities Aeq*x = beq and A*x ≤ b. (Set A=[] and b=[] if no linear inequalities exist.) ga evaluates the matrix product Aeq*x as if x is transposed (Aeq*x'), 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.), subjects the minimization to the constraints defined in nonlcon. The function nonlcon accepts x and returns vectors C and Ceq, representing the nonlinear inequalities and equalities respectively. ga minimizes the fun such that C(x) ≤ 0 and Ceq(x) = 0. (Set lb=[] and ub=[] if no bounds exist.) and minimizes with the default optimization parameters replaced by values in options. (Set nonlcon=[] if no nonlinear constraints exist.) Create options using optimoptions.

2.4 STALAGMITE:

• A stalagmite is a type of rock formation that rises from the floor of a cave due to the accumulation of material deposited on the floor from ceiling drippings. Stalagmites are typically composed of calcium carbonate but may consist of lava, mud, peat, pitch, sand, sinter, and amber.

2.4.1 STALAGMITE MATHEMATICAL MODEL:

3. MATLAB CODE :

3.1. MAIN CODE:

clear all 
close all
clc
x=linspace(0,0.6,150);
y=linspace(0,0.6,150);
num_cases=50;

[f ]=get_stalagmite(x,y);

% study1 - statistical behaviour without bounds
for i=1:num_cases
    [inputs fopt(i)]= ga(@stalagmite,2);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end

figure(1)
subplot(2,1,1)
hold
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','markersize',5 ,'markerfacecolor','r')
title('unbounded inputs')


subplot(2,1,2)
plot(fopt)
xlabel('no_of_iterations')
ylabel('minimum_of_function') 


% study 2 - statistical behaviour with lower and upper bounds
for i=1:num_cases
    [inputs fopt(i)]= ga(@stalagmite,2,[],[],[],[],[0,0],[1,1]);
    xopt(i)=inputs(1);
    yopt(i)=inputs(2);
end

figure(2)
subplot(2,1,1)
hold
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','markersize',5 ,'markerfacecolor','r')
title('bounded inputs')


subplot(2,1,2)
plot(fopt)
xlabel('no_of_iterations')
ylabel('minimum_of_function') 


%study 3 - increasing ga iterations by increasing population size
options = optimoptions(@ga,'PopulationSize',300);

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

figure(3)
subplot(2,1,1)
hold
surfc(x,y,f)
shading interp
plot3(xopt,yopt,fopt,'marker','o','markersize',5 ,'markerfacecolor','r')
title('increased population size')


subplot(2,1,2)
plot(fopt)
xlabel('no_of_iterations')
ylabel('minimum_of_function') 

3.2 GET STALAGMITE FUNCTION CODE:

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

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

3.3 STALAGMITE FUNCTION CODE:

function [f]=stalagmite(a)
    x=a(1);
    y=a(2);
    f1x=(sin((5.1*pi.*x )+0.5)).^6;
    f1y=(sin((5.1*pi*y)+0.5)).^6;
    a1=-4*log(2).*((x-0.0667).^2 )/0.64;
    a2=-4*log(2).*((y-0.0667).^2) /0.64;
    f2x=exp(a1);
    f2y=exp(a2);
    f=-(f1x.*f1y.*f2x.*f2y);
end

4. OBSERVATIONS:

PLOT 1

PLOT 2

PLOT 3

5. CONCLUSION:

• CASE 1: From plot 1, it is observed that the output predicted by the genetic algorithm is not the same in every iteration. Because genetic algorithm deals with the random sampling of the population. The sample population would be different in each iteration. ga finds the best solution in the given sample of the population only. so that the different outputs are obtained.
• CASE 2: In the second plot, it is found that the efficiency of the ga has been increased with bounded inputs. So that accuracy is more than that of the case with unbounded inputs.
• CASE 3: In the third study, the accuracy of the genetic algorithm is improved as expected, since the population size is increased. Because the number of sample data points taken by the ga is gone up.

6. REFERENCES:

1. https://in.mathworks.com/help/gads/how-the-genetic-algorithm-works.html
2. https://skill-lync.com/knowledgebase/genetic-algorithm-explained
3. https://in.mathworks.com/help/gads/ga.html
4. https://in.mathworks.com/help/gads/options-in-genetic-algorithm.html
5. https://in.mathworks.com/help/gads/genetic-algorithm-options.html
6. https://www.iitg.ac.in/rkbc/CE602/CE602/Genetic%20Algorithms.pdf
7. https://www.linkedin.com/pulse/introduction-optimization-genetic-algorithm-ahmed-gad