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.

Contents

• Initialize the variables
• Sort the initialized population
• Start the evolution process
• Result
• Visualize

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

Published with MATLAB® 7.0