here is my introduction, can you change my intro based on my code:

intro:
Multi-objective optimization presents a critical challenge across diverse scientific and engineering domains, particularly in scenarios involving conflicting objectives. This challenge is especially evident in smart building systems, where the simultaneous optimization of occupant comfort and energy consumption creates inherently competing goals. While numerous nature-inspired algorithms have emerged to tackle these challenges, significant limitations persist in handling truly multi-objective optimization scenarios, particularly in balancing the delicate trade-off between maintaining optimal comfort levels and minimizing energy usage.
The No-Free Lunch theorem for optimization fundamentally underscores the ongoing need for novel algorithmic approaches. This theorem logically demonstrates that no single optimization algorithm can universally solve all optimization problems, thereby creating continuous opportunities for innovative methodological developments. In the context of smart buildings, this becomes particularly relevant as traditional optimization methods often struggle to effectively balance the complex relationship between comfort indices (including temperature, humidity, and air quality) and energy consumption patterns.
Building upon this understanding, this research introduces a novel Multi-Objective Evolutionary Mating Algorithm (MOEMA) designed to address existing gaps in multi-objective optimization techniques, with specific application to smart building systems. The proposed MOEMA distinguishes itself through two key innovative strategies. First, it integrates an enhanced crowding distance function inspired by the Non-dominated Sorting Genetic Algorithm (NSGA), enabling more nuanced solution differentiation. Second, it incorporates a Levy flight approach to significantly augment exploration behavior, potentially discovering more diverse and optimal solution sets that better balance comfort parameters with energy efficiency goals.


code:

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