NSGA2

0. 主函数代码

function nsga_2_optimization
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%此处可以更改
%更多机器学习内容请访问omegaxyz.com
pop = 200; %种群数量
gen = 500; %迭代次数
M = 2; %目标函数数量
V = 30; %维度（决策变量的个数）
min_range = zeros(1, V); %下界 生成1*30的个体向量 全为0
max_range = ones(1,V); %上界 生成1*30的个体向量 全为1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
chromosome = initialize_variables(pop, M, V, min_range, max_range);%初始化种群
chromosome = non_domination_sort_mod(chromosome, M, V);%对初始化种群进行非支配快速排序和拥挤度计算


for i = 1 : gen
    pool = round(pop/2);%round() 四舍五入取整 交配池大小
    tour = 2;%竞标赛  参赛选手个数
    parent_chromosome = tournament_selection(chromosome, pool, tour);%竞标赛选择适合繁殖的父代
    mu = 20;%交叉和变异算法的分布指数
    mum = 20;
    offspring_chromosome = genetic_operator(parent_chromosome,M, V, mu, mum, min_range, max_range);%进行交叉变异产生子代 该代码中使用模拟二进制交叉和多项式变异 采用实数编码
    [main_pop,~] = size(chromosome);%父代种群的大小
    [offspring_pop,~] = size(offspring_chromosome);%子代种群的大小
    
    clear temp
    intermediate_chromosome(1:main_pop,:) = chromosome;
    intermediate_chromosome(main_pop + 1 : main_pop + offspring_pop,1 : M+V) = offspring_chromosome;%合并父代种群和子代种群
    intermediate_chromosome = non_domination_sort_mod(intermediate_chromosome, M, V);%对新的种群进行快速非支配排序
    chromosome = replace_chromosome(intermediate_chromosome, M, V, pop);%选择合并种群中前N个优先的个体组成新种群
    if ~mod(i,100)
        clc;
        fprintf('%d generations completed\n',i);
    end
end


if M == 2
    plot(chromosome(:,V + 1),chromosome(:,V + 2),'*');
    xlabel('f_1'); ylabel('f_2');
    title('Pareto Optimal Front');
elseif M == 3
    plot3(chromosome(:,V + 1),chromosome(:,V + 2),chromosome(:,V + 3),'*');
    xlabel('f_1'); ylabel('f_2'); zlabel('f_3');
    title('Pareto Optimal Surface');
end

1. 初始化代码

function f = initialize_variables(N, M, V, min_range, max_range)
%f是一个由种群个体组成的矩阵
    min = min_range;
    max = max_range;
    K = M + V;
    %%K是数组的总元素个数。为了便于计算，决策变量和目标函数串在一起形成一个数组。  
    %对于交叉和变异，利用目标变量对决策变量进行选择
    for i = 1 : N
        for j = 1 : V
            f(i,j) = min(j) + (max(j) - min(j))*rand(1);
            %f(i j)表示的是种群中第i个个体中的第j个决策变量，
            %这行代码为每个个体的所有决策变量在约束条件内随机取值
        end
        f(i,V + 1: K) = evaluate_objective(f(i,:), M, V); 
        % M是目标函数数量 V是决策变量个数
        %为了简化计算将对应的目标函数值储存在染色体的V + 1 到 K的位置。
        end

2. 快速非支配排序和拥挤度计算代码

    %% 对初始种群开始排序 快速非支配排序
    % 使用非支配排序对种群进行排序。该函数返回每个个体对应的排序值和拥挤距离，是一个两列的矩阵。  
    % 并将排序值和拥挤距离添加到染色体矩阵中 
    function f = non_domination_sort_mod(x, M, V)
    [N, ~] = size(x);% N为矩阵x的行数，也是种群的数量
    clear m
    front = 1;
    F(front).f = [];
    individual = [];
     
    for i = 1 : N
        individual(i).n = 0;%n是个体i被支配的个体数量
        individual(i).p = [];%p是被个体i支配的个体集合
        for j = 1 : N
            dom_less = 0;
            dom_equal = 0;
            dom_more = 0;
            for k = 1 : M        %判断个体i和个体j的支配关系
                if (x(i,V + k) < x(j,V + k))  
                    dom_less = dom_less + 1;
                elseif (x(i,V + k) == x(j,V + k))
                    dom_equal = dom_equal + 1;
                else
                    dom_more = dom_more + 1;
                end
            end
            if dom_less == 0 && dom_equal ~= M % 说明i受j支配，相应的n加1
                individual(i).n = individual(i).n + 1;
            elseif dom_more == 0 && dom_equal ~= M % 说明i支配j,把j加入i的支配合集中
                individual(i).p = [individual(i).p j];
            end
        end   
        if individual(i).n == 0 %个体i非支配等级排序最高，属于当前最优解集，相应的染色体中携带代表排序数的信息
            x(i,M + V + 1) = 1;
            F(front).f = [F(front).f i];%等级为1的非支配解集
        end
    end
    %上面的代码是为了找出等级最高的非支配解集
    %下面的代码是为了给其他个体进行分级
    while ~isempty(F(front).f)
       Q = []; %存放下一个front集合
       for i = 1 : length(F(front).f)%循环当前支配解集中的个体
           if ~isempty(individual(F(front).f(i)).p)%个体i有自己所支配的解集
            	for j = 1 : length(individual(F(front).f(i)).p)%循环个体i所支配解集中的个体
                	individual(individual(F(front).f(i)).p(j)).n = ...%...表示的是与下一行代码是相连的， 这里表示个体j的被支配个数减1
                    	individual(individual(F(front).f(i)).p(j)).n - 1;
            	   	 if individual(individual(F(front).f(i)).p(j)).n == 0% 如果q是非支配解集，则放入集合Q中
                   		x(individual(F(front).f(i)).p(j),M + V + 1) = ...%个体染色体中加入分级信息
                            front + 1;
                        Q = [Q individual(F(front).f(i)).p(j)];
                    end
                end
           end
       end
       front =  front + 1;
       F(front).f = Q;
    end
     
    [temp,index_of_fronts] = sort(x(:,M + V + 1));%对个体的代表排序等级的列向量进行升序排序 index_of_fronts表示排序后的值对应原来的索引
    for i = 1 : length(index_of_fronts)
        sorted_based_on_front(i,:) = x(index_of_fronts(i),:);%sorted_based_on_front中存放的是x矩阵按照排序等级升序排序后的矩阵
    end
    current_index = 0;
     
    %% Crowding distance 计算每个个体的拥挤度
     
    for front = 1 : (length(F) - 1)%这里减1是因为代码55行这里，F的最后一个元素为空，这样才能跳出循环。所以一共有length-1个排序等级
        distance = 0;
        y = [];
        previous_index = current_index + 1;
        for i = 1 : length(F(front).f)
            y(i,:) = sorted_based_on_front(current_index + i,:);%y中存放的是排序等级为front的集合矩阵
        end
        current_index = current_index + i;%current_index =i
        sorted_based_on_objective = [];%存放基于拥挤距离排序的矩阵
        for i = 1 : M
            [sorted_based_on_objective, index_of_objectives] = ...
                sort(y(:,V + i));%按照目标函数值排序
            sorted_based_on_objective = [];
            for j = 1 : length(index_of_objectives)
                sorted_based_on_objective(j,:) = y(index_of_objectives(j),:);% sorted_based_on_objective存放按照目标函数值排序后的x矩阵
            end
            f_max = ...
                sorted_based_on_objective(length(index_of_objectives), V + i);%fmax为目标函数最大值 fmin为目标函数最小值
            f_min = sorted_based_on_objective(1, V + i);
            y(index_of_objectives(length(index_of_objectives)),M + V + 1 + i)...%对排序后的第一个个体和最后一个个体的距离设为无穷大
                = Inf;
            y(index_of_objectives(1),M + V + 1 + i) = Inf;
             for j = 2 : length(index_of_objectives) - 1%循环集合中除了第一个和最后一个的个体
                next_obj  = sorted_based_on_objective(j + 1,V + i);
                previous_obj  = sorted_based_on_objective(j - 1,V + i);
                if (f_max - f_min == 0)
                    y(index_of_objectives(j),M + V + 1 + i) = Inf;
                else
                    y(index_of_objectives(j),M + V + 1 + i) = ...
                         (next_obj - previous_obj)/(f_max - f_min);
                end
             end
        end
        distance = [];
        distance(:,1) = zeros(length(F(front).f),1);
        for i = 1 : M
            distance(:,1) = distance(:,1) + y(:,M + V + 1 + i);
        end
        y(:,M + V + 2) = distance;
        y = y(:,1 : M + V + 2);
        z(previous_index:current_index,:) = y;
    end
    f = z();%得到的是已经包含等级和拥挤度的种群矩阵 并且已经按等级排序排序

3. 竞标赛选择代码

function f = tournament_selection(chromosome, pool_size, tour_size)
[pop, variables] = size(chromosome);%获得种群的个体数量和决策变量数量
rank = variables - 1;%个体向量中排序值所在位置
distance = variables;%个体向量中拥挤度所在位置
%竞标赛选择法，每次随机选择两个个体，优先选择排序等级高的个体，如果排序等级一样，优选选择拥挤度大的个体
for i = 1 : pool_size
    for j = 1 : tour_size
        candidate(j) = round(pop*rand(1));%随机选择参赛个体
        if candidate(j) == 0
            candidate(j) = 1;
        end
        if j > 1
            while ~isempty(find(candidate(1 : j - 1) == candidate(j)))%防止两个参赛个体是同一个
                candidate(j) = round(pop*rand(1));
                if candidate(j) == 0
                    candidate(j) = 1;
                end
            end
        end
    end
    for j = 1 : tour_size% 记录每个参赛者的排序等级 拥挤度
        c_obj_rank(j) = chromosome(candidate(j),rank);
        c_obj_distance(j) = chromosome(candidate(j),distance);
    end
    min_candidate = ...
        find(c_obj_rank == min(c_obj_rank));%选择排序等级较小的参赛者，find返回该参赛者的索引
    if length(min_candidate) ~= 1%如果两个参赛者的排序等级相等 则继续比较拥挤度 优先选择拥挤度大的个体
        max_candidate = ...
        find(c_obj_distance(min_candidate) == max(c_obj_distance(min_candidate)));
        if length(max_candidate) ~= 1
            max_candidate = max_candidate(1);
        end
        f(i,:) = chromosome(candidate(min_candidate(max_candidate)),:);
    else
        f(i,:) = chromosome(candidate(min_candidate(1)),:);
    end
end

4.交叉变异代码

    function f  = genetic_operator(parent_chromosome, M, V, mu, mum, l_limit, u_limit)
    [N,m] = size(parent_chromosome);%N是交配池中的个体数量
     
    clear m
    p = 1;
    was_crossover = 0;%是否交叉标志位
    was_mutation = 0;%是否变异标志位
     
    for i = 1 : N%这里虽然循环N次，但是每次循环都会有概率产生2个或者1个子代，所以最终产生的子代个体数量大约是2N个
        if rand(1) < 0.9%交叉概率0.9
            child_1 = [];
            child_2 = [];
            parent_1 = round(N*rand(1));
            if parent_1 < 1
                parent_1 = 1;
            end
            parent_2 = round(N*rand(1));
            if parent_2 < 1
                parent_2 = 1;
            end
            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
            parent_1 = parent_chromosome(parent_1,:);
            parent_2 = parent_chromosome(parent_2,:);
            for j = 1 : V
                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
                child_1(j) = ...
                    0.5*(((1 + bq(j))*parent_1(j)) + (1 - bq(j))*parent_2(j));
                child_2(j) = ...
                    0.5*(((1 - bq(j))*parent_1(j)) + (1 + bq(j))*parent_2(j));
                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
            child_1(:,V + 1: M + V) = evaluate_objective(child_1, M, V);
            child_2(:,V + 1: M + V) = evaluate_objective(child_2, M, V);
            was_crossover = 1;
            was_mutation = 0;
        else%if >0.9
            parent_3 = round(N*rand(1));
            if parent_3 < 1
                parent_3 = 1;
            end
            child_3 = parent_chromosome(parent_3,:);
            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
               child_3(j) = child_3(j) + delta(j);
               if child_3(j) > u_limit(j) % 条件约束
                   child_3(j) = u_limit(j);
               elseif child_3(j) < l_limit(j)
                   child_3(j) = l_limit(j);
               end
            end 
            child_3(:,V + 1: M + V) = evaluate_objective(child_3, M, V);
            was_mutation = 1;
            was_crossover = 0;
        end% if <0.9
        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;

5. 生成新的种群（精英策略）

    function f  = replace_chromosome(intermediate_chromosome, M, V,pop)%精英选择策略
     
    [N, m] = size(intermediate_chromosome);
    [temp,index] = sort(intermediate_chromosome(:,M + V + 1));
     
    clear temp m
    for i = 1 : N
        sorted_chromosome(i,:) = intermediate_chromosome(index(i),:);
    end
     
    max_rank = max(intermediate_chromosome(:,M + V + 1));
     
    previous_index = 0;
    for i = 1 : max_rank
        current_index = max(find(sorted_chromosome(:,M + V + 1) == i));
        if current_index > pop
            remaining = pop - previous_index;
            temp_pop = ...
                sorted_chromosome(previous_index + 1 : current_index, :);
            [temp_sort,temp_sort_index] = ...
                sort(temp_pop(:, M + V + 2),'descend');
            for j = 1 : remaining
                f(previous_index + j,:) = temp_pop(temp_sort_index(j),:);
            end
            return;
        elseif current_index < pop
            f(previous_index + 1 : current_index, :) = ...
                sorted_chromosome(previous_index + 1 : current_index, :);
        else
            f(previous_index + 1 : current_index, :) = ...
                sorted_chromosome(previous_index + 1 : current_index, :);
            return;
        end
        previous_index = current_index;
    end