how to plot the run 1 and run 2 pareto plot:
% MOEMA_PerformanceMetrics_Full_ZDT2.m
% Main script to run MOEMA optimization and evaluate performance metrics for ZDT2

clc;
clear;
close all;

%% Parameters
NPop = 50;            % Population size
T = 200;              % Number of iterations
dim = 30;             % Number of decision variables
lb = zeros(1, dim);   % Lower bound
ub = ones(1, dim);    % Upper bound

num_runs = 3;        % Number of runs for statistical evaluation

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

Convergence_values = zeros(num_runs,1);
Diversity_values = zeros(num_runs,1);

%% 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 Generational Distance (GD)
    GD = generational_distance(paretoFront, true_pareto);
    
    % Calculate Inverted Generational Distance (IGD)
    IGD = inverted_generational_distance(paretoFront, true_pareto);
    
    % Calculate Convergence Metric (using GD as an example)
    Convergence = GD; % Modify if a different convergence metric is desired
    
    % Calculate Diversity Metric
    Diversity = diversity_metric(paretoFront, true_pareto);
    
    % Store metrics
    GD_values(run) = GD;
    IGD_values(run) = IGD;
    Convergence_values(run) = Convergence;
    Diversity_values(run) = Diversity;
end
%% Compute Statistics for GD
best_GD_value = min(GD_values);
average_GD_value = mean(GD_values);
median_GD_value = median(GD_values);
worst_GD_value = max(GD_values);

%% Compute Statistics for IGD
best_IGD_value = min(IGD_values);
average_IGD_value = mean(IGD_values);
median_IGD_value = median(IGD_values);
worst_IGD_value = max(IGD_values);

%% Display Metrics
fprintf('\nPerformance Metrics over %d runs:\n', num_runs);

% Generational Distance (GD) Statistics
fprintf('Generational Distance (GD):\n');
fprintf('  Best    = %.4f\n', best_GD_value);
fprintf('  Average = %.4f\n', average_GD_value);
fprintf('  Median  = %.4f\n', median_GD_value);
fprintf('  Worst   = %.4f\n', worst_GD_value);

% Inverted Generational Distance (IGD) Statistics
fprintf('Inverted Generational Distance (IGD):\n');
fprintf('  Best    = %.4f\n', best_IGD_value);
fprintf('  Average = %.4f\n', average_IGD_value);
fprintf('  Median  = %.4f\n', median_IGD_value);
fprintf('  Worst   = %.4f\n', worst_IGD_value);

% Convergence Metric Statistics
fprintf('Convergence Metric (GD as example):\n');
fprintf('  Mean = %.4f, Std = %.4f\n', mean(Convergence_values), std(Convergence_values));

% Diversity Metric Statistics
fprintf('Diversity Metric:\n');
fprintf('  Mean = %.4f, Std = %.4f\n', mean(Diversity_values), std(Diversity_values));

%% Plot Pareto Front from the Last Run
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(sprintf('Pareto Front for ZDT1 using MOEMA (Run #%d)', num_runs));
legend('Obtained Pareto Front', 'True Pareto Front');
grid on;
hold off;

%% FUNCTIONS

% ------------------- Main MOEMA Function -------------------
function [paretoFront, paretoSet, Trajectories, fitness_history, position_history] = MOEMA(NPop, T, lb, ub, dim, FITNESSFCN)
    % Multi-Objective Evolutionary Mating Algorithm (MOEMA)
    % Optimize the multi-objective problem defined by FITNESSFCN
    %
    % Inputs:
    %   NPop      - Population size
    %   T         - Number of iterations
    %   lb        - Lower bounds (1 x dim)
    %   ub        - Upper bounds (1 x dim)
    %   dim       - Number of decision variables
    %   FITNESSFCN - Handle to the objective function
    %
    % Outputs:
    %   paretoFront    - Set of non-dominated solutions in objective space
    %   paretoSet      - Set of non-dominated solutions in decision space
    %   Trajectories   - Best solution trajectory over iterations
    %   fitness_history - History of fitness values
    %   position_history - History of population positions
    
    % Preallocation
    fitness_history = zeros(NPop, T, 2);  % Assuming 2 objectives
    position_history = zeros(NPop, T, dim);
    Trajectories = zeros(dim, T);
    
    % Generation of initial population
    pop = initialization(NPop, dim, ub, lb);
    pop_dad = pop(1:NPop/2, :);      % Parent group 1
    pop_mum = pop(NPop/2+1:end, :);  % Parent group 2
    
    % 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 with non-dominated solutions
    archive = update_archive([], [], pop, obj);
    
    for gen = 1:T  % Start iterations
        k1 = randperm(NPop/2);
        k2 = randperm(NPop/2);
        k1x = pop_dad(k1, :);
        k2x = pop_mum(k2, :);
        
        for k = 1:NPop/2
            % Mating process
            I_mates = 1 + (var(k1x(k, :)) - var(k2x(k, :)));
            p = randn(1, dim);
            r = rand();
            
            if I_mates >= 0  % Upper mating strategy
                temp1 = p .* k1x(k, :);
                temp2 = (1-p) .* k2x(k, :);
                pop_child = temp1 + temp2;
            else              % Lower mating strategy
                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
            
            % Bounding of the solution
            pop_child = max(min(ub, pop_child), lb);
            
            % Evaluate child
            obj_child = FITNESSFCN(pop_child);
            
            % Update population using non-dominated sorting
            [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
                if ~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);
                    
                    % Update population using non-dominated sorting
                    [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
        end
        
        % Combine updated dad and mum pools
        pop = [pop_dad; pop_mum];
        obj = [obj_dad; obj_mum];
        
        % Update archive
        archive = update_archive(archive.pop, archive.obj, pop, obj);
        
        % Record the history at each iteration
        fitness_history(:, gen, :) = obj;
        position_history(:, gen, :) = pop;
        
        % Update Trajectories (using a random non-dominated solution)
        if ~isempty(archive.pop)
            Trajectories(:, gen) = archive.pop(randi(size(archive.pop, 1)), :)';
        end
        
        fprintf('  Iteration #%d completed. Archive size: %d\n', gen, size(archive.pop, 1));
    end
    
    % Return Pareto front and Pareto set
    paretoFront = archive.obj;
    paretoSet = archive.pop;
end

% ------------------- Initialization Function -------------------
function pop = initialization(NPop, dim, ub, lb)
    % Initialize population within the given bounds
    pop = rand(NPop, dim) .* (ub - lb) + lb;
end

% ------------------- Evaluate Population Function -------------------
function obj = evaluate_population(pop, FITNESSFCN)
    % Evaluate objective functions for a population
    num_sol = size(pop, 1);
    obj = zeros(num_sol, 2);  % Assuming 2 objectives
    parfor i = 1:num_sol  % Parallel loop for efficiency
        obj(i, :) = FITNESSFCN(pop(i, :));
    end
end

% ------------------- Update Population Function -------------------
function [pop, obj] = update_population(pop, obj, child, child_obj, index)
    % Update the population with a child if it dominates or is non-dominated
    if dominates(child_obj, obj(index, :))
        pop(index, :) = child;
        obj(index, :) = child_obj;
    elseif ~dominates(obj(index, :), child_obj)
        if rand() < 0.5
            pop(index, :) = child;
            obj(index, :) = child_obj;
        end
    end
end

% ------------------- Update Archive Function -------------------
function archive = update_archive(archive_pop, archive_obj, pop, obj)
    % Update the archive with non-dominated solutions from the combined population
    combined_pop = [archive_pop; pop];
    combined_obj = [archive_obj; obj];
    
    [is_dominated, ~] = pareto_dominance(combined_obj);
    non_dominated = ~is_dominated;
    
    archive.pop = combined_pop(non_dominated, :);
    archive.obj = combined_obj(non_dominated, :);
end

% ------------------- Pareto Dominance Function -------------------
function [is_dominated, domination_count] = pareto_dominance(objectives)
    % Determine dominated solutions
    [n_solutions, ~] = size(objectives);
    is_dominated = false(n_solutions, 1);
    domination_count = zeros(n_solutions, 1); % Optional: count of dominations
    
    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

% ------------------- Dominance Check Function -------------------
function result = dominates(obj1, obj2)
    % Check if obj1 dominates obj2
    result = all(obj1 <= obj2) && any(obj1 < obj2);
end

% ------------------- True Pareto Front Function -------------------
% ------------------- True Pareto Front Function for ZDT2 -------------------
function true_pareto = get_true_pareto()
    % Generate the true Pareto front for ZDT2
    num_points = 100; % Number of points in the true Pareto front
    f1 = linspace(0, 1, num_points)';
    f2 = 1 - f1.^2;
    true_pareto = [f1, f2];
end
% ------------------- Generational Distance (GD) Function -------------------
function gd = generational_distance(obtained_pf, true_pf)
    % Calculate Generational Distance (GD)
    distances = sqrt(sum((obtained_pf - closest_point(obtained_pf, true_pf)).^2, 2));
    gd = mean(distances);
end

% ------------------- Inverted Generational Distance (IGD) Function -------------------
function igd = inverted_generational_distance(obtained_pf, true_pf)
    % Calculate Inverted Generational Distance (IGD)
    distances = sqrt(sum((true_pf - closest_point(true_pf, obtained_pf)).^2, 2));
    igd = mean(distances);
end

% ------------------- Diversity Metric Function -------------------
function spread = diversity_metric(obtained_pf, true_pf)
    % Calculate Diversity Metric (Spread)
    % Normalize the obtained Pareto front based on the true Pareto front
    
    sorted_pf = sortrows(obtained_pf, 1); % Sort based on f1
    num_obtained = size(sorted_pf, 1);
    
    % Handle edge case with insufficient points
    if num_obtained < 2
        spread = Inf;
        return;
    end
    
    % Calculate gaps between consecutive points
    gaps = sqrt(sum(diff(sorted_pf).^2, 2));
    spread = sum(gaps) / (sqrt(2) * (num_obtained - 1));
end

% ------------------- Closest Point Helper Function -------------------
function closest = closest_point(src, target)
    % Find the closest point in target for each point in src
    num_src = size(src, 1);
    num_target = size(target, 1);
    closest = zeros(size(src));
    
    for i = 1:num_src
        distances = sqrt(sum((target - src(i,:)).^2, 2));
        [~, idx] = min(distances);
        closest(i,:) = target(idx,:);
    end
end

% ------------------- ZDT2 Objective Function -------------------
function f = zdt2(x)
    % ZDT2 Test Problem
    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