hey i wanna ask about these evaluation metric, what is the best for gd,igd,Spacing, Spread  and hv? should be bigger number or lower:


clc;
clear;
close all;

%% Parameters
NPop = 300;              % Population size
T = 500;               % Number of iterations
dim = 30;              % Number of decision variables
lb = zeros(1, dim);    % Lower bound
ub = ones(1, dim);     % Upper bound
num_runs = 10;          % Number of runs for statistical evaluation

% Initialize metrics storage
GD_values = zeros(num_runs, 1);
IGD_values = zeros(num_runs, 1);
Spacing_values = zeros(num_runs, 1);
Spread_values = zeros(num_runs, 1);
Hypervolume_values = zeros(num_runs, 1);

% Reference point for hypervolume calculation
reference_point = [1.1, 1.1];  % Adjust based on your problem

%% Run MOEMA Multiple Times
for run = 1:num_runs
    fprintf('Run #%d/%d\n', run, num_runs);
    
    % Run MOEMA
    [paretoFront, paretoSet, Trajectories, fitness_history, position_history] = ...
        MOEMA(NPop, T, lb, ub, dim, @zdt2);
    
    % Get True Pareto Front
    true_pareto = get_true_pareto();
    
    % Calculate all metrics
    [GD_values(run), IGD_values(run), Spacing_values(run), Spread_values(run)] = ...
        calculate_metrics(paretoFront, true_pareto);
    
    % Calculate hypervolume
    Hypervolume_values(run) = calculate_hypervolume(paretoFront, reference_point);
end

%% Calculate Statistics
metrics = {GD_values, IGD_values, Spacing_values, Spread_values, Hypervolume_values};
metric_names = {'GD', 'IGD', 'Spacing', 'Spread', 'Hypervolume'};

% Print results
fprintf('\nPerformance Metrics over %d runs:\n', num_runs);
for i = 1:length(metrics)
    fprintf('\n%s Statistics:\n', metric_names{i});
    fprintf('  Best    = %.4f\n', min(metrics{i}));
    fprintf('  Average = %.4f\n', mean(metrics{i}));
    fprintf('  Median  = %.4f\n', median(metrics{i}));
    fprintf('  Worst   = %.4f\n', max(metrics{i}));
    fprintf('  Std Dev = %.4f\n', std(metrics{i}));
end

%% Plot Results
% Plot Pareto Front
figure;
plot(paretoFront(:,1), paretoFront(:,2), 'ro', 'MarkerSize', 6, 'MarkerFaceColor', 'r');
hold on;
plot(true_pareto(:,1), true_pareto(:,2), 'b--', 'LineWidth', 1.5);
xlabel('f1');
ylabel('f2');
title('Pareto Front for zdt2 using MOEMA');
legend('Obtained Pareto Front', 'True Pareto Front');
grid on;
hold off;

function [paretoFront, paretoSet, Trajectories, fitness_history, position_history] = MOEMA(NPop, T, lb, ub, dim, FITNESSFCN)
    % Preallocation
    fitness_history = zeros(NPop, T, 2);
    position_history = zeros(NPop, T, dim);
    Trajectories = zeros(dim, T);
    
    % Parameters
    archive_size_max = 100;
    p_mutation = 0.1;
    
    % Generation of initial population with better spread
    pop = initialization(NPop, dim, ub, lb);
    pop(:,1) = linspace(0, 1, NPop);  % Initialize x1 with uniform spread
    
    pop_dad = pop(1:NPop/2, :);
    pop_mum = pop(NPop/2+1:end, :);
    
    % Evaluation of objective function for initialization
    obj_dad = evaluate_population(pop_dad, FITNESSFCN);
    obj_mum = evaluate_population(pop_mum, FITNESSFCN);
    obj = [obj_dad; obj_mum];
    
    % Initialize archive
    archive = update_archive([], [], pop, obj);
    
    for gen = 1:T
        k1 = randperm(NPop/2);
        k2 = randperm(NPop/2);
        k1x = pop_dad(k1, :);
        k2x = pop_mum(k2, :);
        
        for k = 1:NPop/2
            % Mating process
            Imates = 1 + (var(k1x(k, :)) - var(k2x(k, :)));
            p = randn(1, dim);
            r = rand();
            
            if Imates >= 0
                temp1 = p .* k1x(k, :);
                temp2 = (1-p) .* k2x(k, :);
                pop_child = temp1 + temp2;
            else
                temp1 = p .* k2x(k, :);
                temp2 = (1-p) .* k1x(k, :);
                pop_child = temp1 + temp2;
            end
            
            % Impact of best environment
            if ~isempty(archive.pop)
                best_pop = archive.pop(randi(size(archive.pop, 1)), :);
                K = rand(1, dim) < 0.8;
                pop_child = best_pop .* (1-K) + pop_child .* K;
            end
            
            % Mutation operator
            if rand() < p_mutation
                % Special handling for x1
                if rand() < 0.5
                    pop_child(1) = pop_child(1) + randn() * 0.1;
                    pop_child(1) = max(0, min(1, pop_child(1)));
                end
                % Regular mutation for other variables
                mutation_idx = rand(1, dim-1) < 0.1;
                pop_child(2:end) = pop_child(2:end) + mutation_idx .* randn(1, dim-1) * 0.1;
            end
            
            % Boundary handling
            pop_child = max(min(ub, pop_child), lb);
            
            % Evaluate and update population
            obj_child = FITNESSFCN(pop_child);
            [pop_dad, obj_dad] = update_population(pop_dad, obj_dad, pop_child, obj_child, k1(k));
            [pop_mum, obj_mum] = update_population(pop_mum, obj_mum, pop_child, obj_child, k2(k));
            
            % Exploration process
            if r < 0.2 && ~isempty(archive.pop)
                temp1 = archive.pop(randi(size(archive.pop, 1)), :);
                pop_child = rand(1, dim) .* temp1;
                pop_child = max(min(ub, pop_child), lb);
                obj_child = FITNESSFCN(pop_child);
                
                [pop_dad, obj_dad] = update_population(pop_dad, obj_dad, pop_child, obj_child, randi(NPop/2));
                [pop_mum, obj_mum] = update_population(pop_mum, obj_mum, pop_child, obj_child, randi(NPop/2));
            end
        end
        
        % Combine updated populations
        pop = [pop_dad; pop_mum];
        obj = [obj_dad; obj_mum];
        
        % Update archive
        archive = update_archive(archive.pop, archive.obj, pop, obj);
        
        % Record history
        fitness_history(:, gen, :) = obj;
        position_history(:, gen, :) = pop;
        
        % Update Trajectories
        if ~isempty(archive.pop)
            Trajectories(:, gen) = archive.pop(randi(size(archive.pop, 1)), :)';
        end
        
        % Display progress
        if mod(gen, 10) == 0
            disp(['Iteration #', num2str(gen), ' completed. Archive size: ', num2str(size(archive.pop, 1))]);
        end
    end
    
    % Return final results
    paretoFront = archive.obj;
    paretoSet = archive.pop;
end

function pop = initialization(NPop, dim, ub, lb)
    pop = rand(NPop, dim) .* (ub - lb) + lb;
end

function obj = evaluate_population(pop, FITNESSFCN)
    obj = zeros(size(pop, 1), 2);
    for i = 1:size(pop, 1)
        obj(i, :) = FITNESSFCN(pop(i, :));
    end
end

function [pop, obj] = update_population(pop, obj, child, child_obj, index)
    if dominates(child_obj, obj(index, :))
        pop(index, :) = child;
        obj(index, :) = child_obj;
    elseif ~dominates(obj(index, :), child_obj)
        current_spread = std(pop(:,1));
        new_pop = pop;
        new_pop(index, :) = child;
        new_spread = std(new_pop(:,1));
        
        accept_prob = 0.5;
        if new_spread > current_spread
            accept_prob = 0.8;
        end
        
        if rand() < accept_prob
            pop(index, :) = child;
            obj(index, :) = child_obj;
        end
    end
end

function archive = update_archive(archive_pop, archive_obj, pop, obj)
    combined_pop = [archive_pop; pop];
    combined_obj = [archive_obj; obj];
    
    [is_dominated, ~] = pareto_dominance(combined_obj);
    non_dominated = ~is_dominated;
    
    archive_candidates_pop = combined_pop(non_dominated, :);
    archive_candidates_obj = combined_obj(non_dominated, :);
    
    [~, sort_idx] = sort(archive_candidates_pop(:,1));
    archive.pop = archive_candidates_pop(sort_idx, :);
    archive.obj = archive_candidates_obj(sort_idx, :);
    
    if size(archive.pop, 1) > 100
        keep_idx = round(linspace(1, size(archive.pop, 1), 100));
        archive.pop = archive.pop(keep_idx, :);
        archive.obj = archive.obj(keep_idx, :);
    end
end

function [is_dominated, domination_count] = pareto_dominance(objectives)
    [n_solutions, ~] = size(objectives);
    is_dominated = false(n_solutions, 1);
    domination_count = zeros(n_solutions, 1);
    
    for i = 1:n_solutions
        for j = i+1:n_solutions
            if dominates(objectives(i, :), objectives(j, :))
                is_dominated(j) = true;
                domination_count(i) = domination_count(i) + 1;
            elseif dominates(objectives(j, :), objectives(i, :))
                is_dominated(i) = true;
                domination_count(j) = domination_count(j) + 1;
            end
        end
    end
end

function result = dominates(obj1, obj2)
    result = all(obj1 <= obj2) && any(obj1 < obj2);
end

function f = zdt2(x)
    f1 = x(1);
    g = 1 + 9 * sum(x(2:end)) / (length(x) - 1);
    h = 1 - (f1 / g)^2;
    f2 = g * h;
    f = [f1, f2];
end

% ------------------- True Pareto Front Function -------------------
function true_pareto = get_true_pareto()
    % Generate the true Pareto front for ZDT2
    num_points = 1000;
    f1 = linspace(0, 1, num_points)';
    f2 = 1 - (f1).^2;
    true_pareto = [f1, f2];
end

%% Performance Metrics Functions
function [GD, IGD, Spacing, Spread] = calculate_metrics(obtained_pf, true_pf)
    GD = generational_distance(obtained_pf, true_pf);
    IGD = inverted_generational_distance(obtained_pf, true_pf);
    Spacing = spacing_metric(obtained_pf);
    Spread = spread_metric(obtained_pf, true_pf);
end

function gd = generational_distance(obtained_pf, true_pf)
    if isempty(obtained_pf)
        gd = Inf;
        return;
    end
    
    distances = zeros(size(obtained_pf, 1), 1);
    for i = 1:size(obtained_pf, 1)
        dist_to_true = sqrt(sum((true_pf - obtained_pf(i,:)).^2, 2));
        distances(i) = min(dist_to_true);
    end
    
    p = 2;
    gd = (sum(distances.^p)/size(obtained_pf, 1))^(1/p);
end

function igd = inverted_generational_distance(obtained_pf, true_pf)
    if isempty(obtained_pf)
        igd = Inf;
        return;
    end
    
    distances = zeros(size(true_pf, 1), 1);
    for i = 1:size(true_pf, 1)
        dist_to_obtained = sqrt(sum((obtained_pf - true_pf(i,:)).^2, 2));
        distances(i) = min(dist_to_obtained);
    end
    
    p = 1;
    igd = (sum(distances.^p)/size(true_pf, 1))^(1/p);
end

function spacing = spacing_metric(obtained_pf)
    if size(obtained_pf, 1) < 2
        spacing = Inf;
        return;
    end
    
    n = size(obtained_pf, 1);
    distances = zeros(n, 1);
    
    for i = 1:n
        min_dist = Inf;
        for j = 1:n
            if i ~= j
                dist = sqrt(sum((obtained_pf(i,:) - obtained_pf(j,:)).^2));
                min_dist = min(min_dist, dist);
            end
        end
        distances(i) = min_dist;
    end
    
    d_bar = mean(distances);
    spacing = sqrt(sum((distances - d_bar).^2)/(n-1));
end

function spread = spread_metric(obtained_pf, true_pf)
    if size(obtained_pf, 1) < 2
        spread = Inf;
        return;
    end
    
    [~, idx] = sort(obtained_pf(:,1));
    obtained_pf = obtained_pf(idx,:);
    
    df = zeros(1, 2);
    for i = 1:2
        obtained_extreme = obtained_pf(i,:);
        true_extreme = true_pf(i,:);
        df(i) = sqrt(sum((obtained_extreme - true_extreme).^2));
    end
    
    distances = sqrt(sum(diff(obtained_pf).^2, 2));
    d_bar = mean(distances);
    
    spread = (sum(df) + sum(abs(distances - d_bar))) / (sum(df) + size(obtained_pf,1) * d_bar);
end

function hypervolume = calculate_hypervolume(obtained_pf, reference_point)
    if isempty(obtained_pf)
        hypervolume = 0;
        return;
    end
    
    [obtained_pf, ~] = sortrows(obtained_pf);
    
    hypervolume = 0;
    for i = 1:size(obtained_pf, 1)
        if i == 1
            width = reference_point(1) - obtained_pf(i,1);
        else
            width = obtained_pf(i,1) - obtained_pf(i-1,1);
        end
        height = reference_point(2) - obtained_pf(i,2);
        hypervolume = hypervolume + width * height;
    end
end