NSGA - II: A multi-objective optimization algorithm

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19 Mar 2006 (Updated )

A function for multi-objective optimization using evolutionary algorithms

function f = genetic_operator(parent_chromosome, M, V, mu, mum, l_limit, u_limit)

function f = genetic_operator(parent_chromosome, M, V, mu, mum, l_limit, u_limit)

This function is utilized to produce offsprings from parent chromosomes. The genetic operators corssover and mutation which are carried out with slight modifications from the original design. For more information read the document enclosed.

parent_chromosome - the set of selected chromosomes. M - number of objective functions V - number of decision varaiables mu - distribution index for crossover (read the enlcosed pdf file) mum - distribution index for mutation (read the enclosed pdf file) l_limit - a vector of lower limit for the corresponding decsion variables u_limit - a vector of upper limit for the corresponding decsion variables

The genetic operation is performed only on the decision variables, that is the first V elements in the chromosome vector.

[N,m] = size(parent_chromosome);

clear m
p = 1;
% Flags used to set if crossover and mutation were actually performed.
was_crossover = 0;
was_mutation = 0;


for i = 1 : N
    % With 90 % probability perform crossover
    if rand(1) < 0.9
        % Initialize the children to be null vector.
        child_1 = [];
        child_2 = [];
        % Select the first parent
        parent_1 = round(N*rand(1));
        if parent_1 < 1
            parent_1 = 1;
        end
        % Select the second parent
        parent_2 = round(N*rand(1));
        if parent_2 < 1
            parent_2 = 1;
        end
        % Make sure both the parents are not the same.
        while isequal(parent_chromosome(parent_1,:),parent_chromosome(parent_2,:))
            parent_2 = round(N*rand(1));
            if parent_2 < 1
                parent_2 = 1;
            end
        end
        % Get the chromosome information for each randomnly selected
        % parents
        parent_1 = parent_chromosome(parent_1,:);
        parent_2 = parent_chromosome(parent_2,:);
        % Perform corssover for each decision variable in the chromosome.
        for j = 1 : V
            % SBX (Simulated Binary Crossover).
            % For more information about SBX refer the enclosed pdf file.
            % Generate a random number
            u(j) = rand(1);
            if u(j) <= 0.5
                bq(j) = (2*u(j))^(1/(mu+1));
            else
                bq(j) = (1/(2*(1 - u(j))))^(1/(mu+1));
            end
            % Generate the jth element of first child
            child_1(j) = ...
                0.5*(((1 + bq(j))*parent_1(j)) + (1 - bq(j))*parent_2(j));
            % Generate the jth element of second child
            child_2(j) = ...
                0.5*(((1 - bq(j))*parent_1(j)) + (1 + bq(j))*parent_2(j));
            % Make sure that the generated element is within the specified
            % decision space else set it to the appropriate extrema.
            if child_1(j) > u_limit(j)
                child_1(j) = u_limit(j);
            elseif child_1(j) < l_limit(j)
                child_1(j) = l_limit(j);
            end
            if child_2(j) > u_limit(j)
                child_2(j) = u_limit(j);
            elseif child_2(j) < l_limit(j)
                child_2(j) = l_limit(j);
            end
        end
        % Evaluate the objective function for the offsprings and as before
        % concatenate the offspring chromosome with objective value.
        child_1(:,V + 1: M + V) = evaluate_objective(child_1, M, V);
        child_2(:,V + 1: M + V) = evaluate_objective(child_2, M, V);
        % Set the crossover flag. When crossover is performed two children
        % are generate, while when mutation is performed only only child is
        % generated.
        was_crossover = 1;
        was_mutation = 0;
    % With 10 % probability perform mutation. Mutation is based on
    % polynomial mutation.
    else
        % Select at random the parent.
        parent_3 = round(N*rand(1));
        if parent_3 < 1
            parent_3 = 1;
        end
        % Get the chromosome information for the randomnly selected parent.
        child_3 = parent_chromosome(parent_3,:);
        % Perform mutation on eact element of the selected parent.
        for j = 1 : V
           r(j) = rand(1);
           if r(j) < 0.5
               delta(j) = (2*r(j))^(1/(mum+1)) - 1;
           else
               delta(j) = 1 - (2*(1 - r(j)))^(1/(mum+1));
           end
           % Generate the corresponding child element.
           child_3(j) = child_3(j) + delta(j);
           % Make sure that the generated element is within the decision
           % space.
           if child_3(j) > u_limit(j)
               child_3(j) = u_limit(j);
           elseif child_3(j) < l_limit(j)
               child_3(j) = l_limit(j);
           end
        end
        % Evaluate the objective function for the offspring and as before
        % concatenate the offspring chromosome with objective value.
        child_3(:,V + 1: M + V) = evaluate_objective(child_3, M, V);
        % Set the mutation flag
        was_mutation = 1;
        was_crossover = 0;
    end
    % Keep proper count and appropriately fill the child variable with all
    % the generated children for the particular generation.
    if was_crossover
        child(p,:) = child_1;
        child(p+1,:) = child_2;
        was_cossover = 0;
        p = p + 2;
    elseif was_mutation
        child(p,:) = child_3(1,1 : M + V);
        was_mutation = 0;
        p = p + 1;
    end
end
f = child;

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