x = optimum solution

f = ga(fun,nvars,A,b,[],[],lb,ub,nonlcon,Intcon,options)

fun = fitness function

nvars = number of variables

A,b = Linear inequality constraints

Aeq,beq = Linear equality constraints

lb = lower bound limit

ub = upper bound limit

nonlcon = Use of nonlinear constraint

Intcon = to get solution in integer

Options = To set population size minimize with non-default options.

STEPS INVOLVED IN GA STALAGMITE CODING :

• Stalagmite equation for genetic algorithm

• Another step is converting these equation in function in MATLAB for GA solving
• To find global maxima we have to invert the equations, so to do that just multiply with [-1]

f = -(f1*f2*f3*f4)

CODING FITNESS FUNCTION :

The main thing while writing the function code is that we have to save the function in the same name of the function, For here it is "stalagmite.m"

% Equation input
 
 function A = stalagmite(input_vector)
            x = input_vector(1);
            y = input_vector(2);
 
            f1 = (sin(5.1*pi*x+0.5)).^6;
            f2 = exp((-4*log(2)*((x-0.0667).^2))/0.64);
            f3 = (sin(5.1*pi*y+0.5)).^6;
            f4 = exp((-4*log(2)*((y-0.0667).^2))/0.64);
            
 A = -(f1*f2*f3*f4);
 
 end

• The above stalagmite equation for GA is coded in stalagmite.m having 2 variables x&y.
• These function will be called in the program.

GET_STALAGMITE CODE :

function[f] = get_stalagmite(x,y)

% Creating a 2Dimensional mesh 
[xx yy] = meshgrid(x,y);

% Evaluating the stalagmite function 
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

• For creating 2D mesh we will write a meshgrid between particular two coordinates & for code, it will be x,y
• Evaluating the stalagmite function using for loop and we called the main funtion which is stored in name of stalagmite.m

MAIN CODE :

clear all
close all
clc

% Inputs
x = linspace(0,0.6,150);
y = linspace(0,0.6,150);
num_cases = 50
f = get_stalagmite(x,y);

tic
% Study1 - Unbounded statistical behavior
for i = 1:num_cases
    [inputs,fopt(i)] = ga(@stalagmite,2);
    xopt(i) = inputs(1);
    yopt(i) = inputs(2);
end
study1_time = toc

% Plotting 
figure(1)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
xlabel('x value')
ylabel('y value')
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('Unbounded inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('Iterations')
ylabel('Function maximum')

tic
% Study2 - Statistical behavior with lower and upper bound
for i = 1:num_cases
    [inputs,fopt(i)] = ga(@stalagmite,2,[],[],[],[],[0;0],[1;1]);
    xopt(i) = inputs(1);
    yopt(i) = inputs(2);
end

study2_time = toc

figure(2)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
xlabel('x value')
ylabel('y value')
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('Bounded inputs')
subplot(2,1,2)
plot(-fopt)
xlabel('Iterations')
ylabel('Function maximum')

% Study3 - Increasing GA iterations
options = optimoptions('ga')
options = optimoptions(options,'populationsize',250);

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

study3_time = toc

figure(3)
subplot(2,1,1)
hold on
surfc(x,y,-f)
shading interp
xlabel('x value')
ylabel('y value')
plot3(xopt,yopt,-fopt,'marker','o','markersize',5,'markerfacecolor','r')
title('Bounded inputs with modified population size')
subplot(2,1,2)
plot(-fopt)
xlabel('Iterations')
ylabel('Function maximum')
 

1. Firstly, clear all, close all &clc command were used to clear workspace close all figures and erased all other.
2. The search space of ga is define using linspace command the variable lies between 0 to 0.6 with 150 values in between.
3. The number of cases means these were the iterations i.e. ga will run for 50 times.
4. The get_stalagmite.m function was called
5. For loop was used to give inputs to get_stalagmite function inputs which means while running 50 iterations the value changes.
6. We had executed 3 different studies ; 1. Unbounded statistical behavior; 2. Bounded statistical behavior; 3. Bounded with increase population size.
7. Study1 :- In this the optimum values for inputs taken for each iterations with the help of a for loop.
8. Study timers commands tic & toc are used to record the study time.
9. The 'ga' syntax used in first study (Unbounded statistical behavior) ; X = ga(fun,nvars)
10. Plotting commands : Figure command is used for opening a figure window and subplot command is used to get multiple plot in one figure window.
11. Now for creating a 2D surface for stalagmite, we have to write "surfc(x,y,f)"
12. Proper shading the surface we have to write a code for that and that will be "shading interp". If we will not provide shading then it will create mesh structure on stalagmite.
13. plot3 command was used for getting surface as 3D
14. Then the title command is used for getting title for plot and label the plot.
15. Study2 :- Sa,e parameters were used only difference is in using the syntax of 'ga', x = ga(fun,nvars,A,b,Aeq,beq,lb,ub) i.e. using lower and upper bound limit.
16. Study3 :- The plotting and other command were as same as above but syntax for ga changed by using optimoptions and increades iterations/populationsize is 250.
17. The 'ga' syntax for 3rd study is : x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,Intcon,options)

STUDY TIME :

• study1_time = 2.4117
• study2_time = 5.3328
• study3_time = 9.6320

OUTPUTS OBTAINED :

Output of Study1 :

Output of Study2:

Output of Study3 :

CONCLUSION :

1. There were different types of syntax used for diffrent equations.
2. The perfect global maxima can be obtained by increasing the population size.
3. The 'ga' is used to find minimum value for maximum value we need to inverse it that's why we multiply formula by [-1] in stalagmite.m
4. The study time increases with respect to population size.

REFERENCES :

https://en.wikipedia.org/wiki/Genetic_algorithm

https://in.mathworks.com/help/gads/what-is-the-genetic-algorithm.html

https://in.mathworks.com/help/gads/what-is-the-genetic-algorithm.html

https://towardsdatascience.com/introduction-to-genetic-algorithms-including-example-code-e396e98d8bf3