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<h1>Main Function</h1>
<introduction>
<p>Main program to run the NSGA-II MOEA. Read the corresponding documentation to learn more about multiobjective optimization
using evolutionary algorithms. initialize_variables has two arguments; First being the population size and the second the
problem number. '1' corresponds to MOP1 and '2' corresponds to MOP2.
</p>
</introduction>
<h2>Contents</h2>
<div>
<ul>
<li><a href="#1">Initialize the variables</a></li>
<li><a href="#2">Sort the initialized population</a></li>
<li><a href="#3">Start the evolution process</a></li>
<li><a href="#4">Result</a></li>
<li><a href="#5">Visualize</a></li>
</ul>
</div>
<h2>Initialize the variables<a name="1"></a></h2>
<p>Declare the variables and initialize their values pop - population gen - generations pro - problem number</p><pre class="codeinput">pop = 200;
gen = 1;
pro = 1;
<span class="keyword">switch</span> pro
<span class="keyword">case</span> 1
<span class="comment">% M is the number of objectives.</span>
M = 2;
<span class="comment">% V is the number of decision variables. In this case it is</span>
<span class="comment">% difficult to visualize the decision variables space while the</span>
<span class="comment">% objective space is just two dimensional.</span>
V = 6;
<span class="keyword">case</span> 2
M = 3;
V = 12;
<span class="keyword">end</span>
<span class="comment">% Initialize the population</span>
chromosome = initialize_variables(pop,pro);
</pre><h2>Sort the initialized population<a name="2"></a></h2>
<p>Sort the population using non-domination-sort. This returns two columns for each individual which are the rank and the crowding
distance corresponding to their position in the front they belong.
</p><pre class="codeinput">chromosome = non_domination_sort_mod(chromosome,pro);
</pre><h2>Start the evolution process<a name="3"></a></h2><pre class="codeinput"><span class="comment">% The following are performed in each generation</span>
<span class="comment">% Select the parents</span>
<span class="comment">% Perfrom crossover and Mutation operator</span>
<span class="comment">% Perform Selection</span>
<span class="keyword">for</span> i = 1 : gen
<span class="comment">% Select the parents</span>
<span class="comment">% Parents are selected for reproduction to generate offspring. The</span>
<span class="comment">% original NSGA-II uses a binary tournament selection based on the</span>
<span class="comment">% crowded-comparision operator. The arguments are</span>
<span class="comment">% pool - size of the mating pool. It is common to have this to be half the</span>
<span class="comment">% population size.</span>
<span class="comment">% tour - Tournament size. Original NSGA-II uses a binary tournament</span>
<span class="comment">% selection, but to see the effect of tournament size this is kept</span>
<span class="comment">% arbitary, to be choosen by the user.</span>
pool = round(pop/2);
tour = 2;
parent_chromosome = tournament_selection(chromosome,pool,tour);
<span class="comment">% Perfrom crossover and Mutation operator</span>
<span class="comment">% The original NSGA-II algorithm uses Simulated Binary Crossover (SBX) and</span>
<span class="comment">% Polynomial crossover. Crossover probability pc = 0.9 and mutation</span>
<span class="comment">% probability is pm = 1/n, where n is the number of decision variables.</span>
<span class="comment">% Both real-coded GA and binary-coded GA are implemented in the original</span>
<span class="comment">% algorithm, while in this program only the real-coded GA is considered.</span>
<span class="comment">% The distribution indeices for crossover and mutation operators as mu = 20</span>
<span class="comment">% and mum = 20 respectively.</span>
mu = 20;
mum = 20;
offspring_chromosome = genetic_operator(parent_chromosome,pro,mu,mum);
<span class="comment">% Intermediate population</span>
<span class="comment">% Intermediate population is the combined population of parents and</span>
<span class="comment">% offsprings of the current generation. The population size is almost 1 and</span>
<span class="comment">% half times the initial population.</span>
[main_pop,temp] = size(chromosome);
[offspring_pop,temp] = size(offspring_chromosome);
intermediate_chromosome(1:main_pop,:) = chromosome;
intermediate_chromosome(main_pop + 1 : main_pop + offspring_pop,1 : M+V) = <span class="keyword">...</span>
offspring_chromosome;
<span class="comment">% Non-domination-sort of intermediate population</span>
<span class="comment">% The intermediate population is sorted again based on non-domination sort</span>
<span class="comment">% before the replacement operator is performed on the intermediate</span>
<span class="comment">% population.</span>
intermediate_chromosome = <span class="keyword">...</span>
non_domination_sort_mod(intermediate_chromosome,pro);
<span class="comment">% Perform Selection</span>
<span class="comment">% Once the intermediate population is sorted only the best solution is</span>
<span class="comment">% selected based on it rank and crowding distance. Each front is filled in</span>
<span class="comment">% ascending order until the addition of population size is reached. The</span>
<span class="comment">% last front is included in the population based on the individuals with</span>
<span class="comment">% least crowding distance</span>
chromosome = replace_chromosome(intermediate_chromosome,pro,pop);
<span class="keyword">if</span> ~mod(i,10)
fprintf(<span class="string">'%d\n'</span>,i);
<span class="keyword">end</span>
<span class="keyword">end</span>
</pre><h2>Result<a name="4"></a></h2>
<p>Save the result in ASCII text format.</p><pre class="codeinput">save <span class="string">solution.txt</span> <span class="string">chromosome</span> <span class="string">-ASCII</span>
</pre><h2>Visualize<a name="5"></a></h2>
<p>The following is used to visualize the result for the given problem.</p><pre class="codeinput"><span class="keyword">switch</span> pro
<span class="keyword">case</span> 1
plot(chromosome(:,V + 1),chromosome(:,V + 2),<span class="string">'*'</span>);
title(<span class="string">'MOP1 using NSGA-II'</span>);
xlabel(<span class="string">'f(x_1)'</span>);
ylabel(<span class="string">'f(x_2)'</span>);
<span class="keyword">case</span> 2
plot3(chromosome(:,V + 1),chromosome(:,V + 2),chromosome(:,V + 3),<span class="string">'*'</span>);
title(<span class="string">'MOP2 using NSGA-II'</span>);
xlabel(<span class="string">'f(x_1)'</span>);
ylabel(<span class="string">'f(x_2)'</span>);
zlabel(<span class="string">'f(x_3)'</span>);
<span class="keyword">end</span>
</pre><p class="footer"><br>
Published with MATLAB® 7.0<br></p>
<!--
##### SOURCE BEGIN #####
%% Main Function
% Main program to run the NSGA-II MOEA.
% Read the corresponding documentation to learn more about multiobjective
% optimization using evolutionary algorithms.
% initialize_variables has two arguments; First being the population size
% and the second the problem number. '1' corresponds to MOP1 and '2'
% corresponds to MOP2.
%% Initialize the variables
% Declare the variables and initialize their values
% pop - population
% gen - generations
% pro - problem number
pop = 200;
gen = 1;
pro = 1;
switch pro
case 1
% M is the number of objectives.
M = 2;
% V is the number of decision variables. In this case it is
% difficult to visualize the decision variables space while the
% objective space is just two dimensional.
V = 6;
case 2
M = 3;
V = 12;
end
% Initialize the population
chromosome = initialize_variables(pop,pro);
%% Sort the initialized population
% Sort the population using non-domination-sort. This returns two columns
% for each individual which are the rank and the crowding distance
% corresponding to their position in the front they belong.
chromosome = non_domination_sort_mod(chromosome,pro);
%% Start the evolution process
% The following are performed in each generation
% Select the parents
% Perfrom crossover and Mutation operator
% Perform Selection
for i = 1 : gen
% Select the parents
% Parents are selected for reproduction to generate offspring. The
% original NSGA-II uses a binary tournament selection based on the
% crowded-comparision operator. The arguments are
% pool - size of the mating pool. It is common to have this to be half the
% population size.
% tour - Tournament size. Original NSGA-II uses a binary tournament
% selection, but to see the effect of tournament size this is kept
% arbitary, to be choosen by the user.
pool = round(pop/2);
tour = 2;
parent_chromosome = tournament_selection(chromosome,pool,tour);
% Perfrom crossover and Mutation operator
% The original NSGA-II algorithm uses Simulated Binary Crossover (SBX) and
% Polynomial crossover. Crossover probability pc = 0.9 and mutation
% probability is pm = 1/n, where n is the number of decision variables.
% Both real-coded GA and binary-coded GA are implemented in the original
% algorithm, while in this program only the real-coded GA is considered.
% The distribution indeices for crossover and mutation operators as mu = 20
% and mum = 20 respectively.
mu = 20;
mum = 20;
offspring_chromosome = genetic_operator(parent_chromosome,pro,mu,mum);
% Intermediate population
% Intermediate population is the combined population of parents and
% offsprings of the current generation. The population size is almost 1 and
% half times the initial population.
[main_pop,temp] = size(chromosome);
[offspring_pop,temp] = size(offspring_chromosome);
intermediate_chromosome(1:main_pop,:) = chromosome;
intermediate_chromosome(main_pop + 1 : main_pop + offspring_pop,1 : M+V) = ...
offspring_chromosome;
% Non-domination-sort of intermediate population
% The intermediate population is sorted again based on non-domination sort
% before the replacement operator is performed on the intermediate
% population.
intermediate_chromosome = ...
non_domination_sort_mod(intermediate_chromosome,pro);
% Perform Selection
% Once the intermediate population is sorted only the best solution is
% selected based on it rank and crowding distance. Each front is filled in
% ascending order until the addition of population size is reached. The
% last front is included in the population based on the individuals with
% least crowding distance
chromosome = replace_chromosome(intermediate_chromosome,pro,pop);
if ~mod(i,10)
fprintf('%d\n',i);
end
end
%% Result
% Save the result in ASCII text format.
save solution.txt chromosome -ASCII
%% Visualize
% The following is used to visualize the result for the given problem.
switch pro
case 1
plot(chromosome(:,V + 1),chromosome(:,V + 2),'*');
title('MOP1 using NSGA-II');
xlabel('f(x_1)');
ylabel('f(x_2)');
case 2
plot3(chromosome(:,V + 1),chromosome(:,V + 2),chromosome(:,V + 3),'*');
title('MOP2 using NSGA-II');
xlabel('f(x_1)');
ylabel('f(x_2)');
zlabel('f(x_3)');
end
##### SOURCE END #####
-->
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