Solving the hub location problem

Problem description:

When solving the Hub Location Problem (HLP), a genetic algorithm was used. Below is the genetic algorithm code for the entire model, but when running the main program `data.m`, the following error message keeps appearing. Despite multiple modifications, the same error persists. Please carefully review all the code for solving the HLP and ensure that the code is corrected to produce the correct results.

Error:

Error using evaluateFitness (line 12)

Individual length does not match the expected total length. Expected: 186, Got: 100

Error in data (line 146)

[obj1, obj2] = evaluateFitness(individual, num_hubs, num_manufacturers, num_customers,

...

data.m (main program file)

beta = 0.7;
num_hubs = 6; 
num_manufacturers = 4; 
num_customers = 20; 

% 1. distance matrix
manufacturers = [113.5494,22.7076; 113.6798,22.6487; 
    113.3497,22.8446; 113.3735,22.9538]; % Manufacturers' coordinates
hubs = [113.2734,23.1352; 113.3235,23.0990; 113.3685,23.1304; 
    106.6296,26.6836; 113.2266,23.1875; 113.5936,22.8079]; % Hub centers' coordinates
customers = [113.2702,23.1671; 113.2910,23.2179; 113.2630,23.2053; 
    113.3101,23.2404; 113.2760,23.0623; 113.2226,23.2340;
    113.2947,23.2567; 113.3622,23.2983; 113.4110,23.3917;
    113.2777,23.1401; 113.3057,23.1285; 113.2864,23.1507; 
    113.3144,23.1049; 113.2791,23.0815; 113.2316,23.1262; 
    113.2508,23.0613; 113.3601,23.1581; 113.3235,23.1685;
    113.3934,23.1393; 113.4756,23.1078]; % Customers' coordinates

% Matrix of the distance between the manufacturer and the hub center d_ki
d_ki = [0.5089 0.4519 0.4599 7.9807 0.3643 0.1767; 
    0.5784 0.1096 0.6339 0.5742 0.2466 7.7066; 
    0.5735 8.1232 0.7041 0.1810 0.2072 0.2760;
    0.3004 0.2557 0.2864 7.7394 0.1536 0.2641];

% The distance matrix between hub centers d_ii
d_ii = [0 0.0618 0.0952 7.5320 0.0702 0.4579;
    0.0618 0 0.0549 7.5933 0.1312 0.3971;
    0.0952 0.0549 0 7.6183 0.1530 0.3933;
    7.5320 7.5933 7.6183 0 7.4661 7.9698;
    0.0702 0.1312 0.1530 7.4661 0 0.5280;
    0.4579 0.3971 0.3933 7.9698 0.5280 0];

% The distance matrix of customers to the hub center d_ij
d_ij = [0.0321 0.0846 0.0709 0.1114 0.0729 0.1111 0.1234 0.1857 0.2911 0.0065 0.0330 0.0202 0.0510 0.0540 0.0428 0.0773 0.0897 0.0602 0.1201 0.2040; 
    0.0865 0.1233 0.1223 0.1420 0.0600 0.1685 0.1603 0.2030 0.3055 0.0615 0.0345 0.0636 0.0108 0.0477 0.0958 0.0819 0.0695 0.0695 0.0807 0.1524; 
    0.1049 0.1169 0.1294 0.1245 0.1149 0.1789 0.1463 0.1680 0.2647 0.0913 0.0628 0.0846 0.0598 0.1019 0.1370 0.1365 0.0289 0.0590 0.0264 0.1095;
    7.5142 7.5090 7.4900 7.5156 7.5689 7.4409 7.4945 7.5358 7.5382 7.5335 7.5637 7.5362 7.5825 7.5625 7.4994 7.5473 7.5979 7.5607 7.6362 7.7236; 
    0.0481 0.0712 0.0405 0.0988 0.1346 0.0467 0.0971 0.1751 0.2751 0.0697 0.0987 0.0702 0.1205 0.1183 0.0615 0.1285 0.1367 0.0987 0.1736 0.2614; 
    0.4833 0.5096 0.5169 0.5171 0.4069 0.5650 0.5392 0.5423 0.6117 0.4584 0.4309 0.4603 0.4076 0.4169 0.4820 0.4263 0.4209 0.4505 0.3872 0.3223];

% 2. traffic matrix
h_ki = [30 35 60 90 120 100;
    25 36 58 70 150 85;
    15 20 38 50 95 55;
    90 115 160 260 320 190];
g_ii = [0 2 5 7 10 15;
    7 0 6 8 12 10;
    15 10 0 6 18 12;
    6 25 10 0 5 9;
    6 8 7 4 0 10;
    2 8 10 12 4 0];
s_ij = [3 4 5 8 10 9 2 5 4 6 25 15 2 6 4 5 6 8 4 6;
    8 7 12 14 10 12 14 25 18 15 6 7 5 1 5 7 8 9 6 7;
    5 6 4 7 6 8 10 14 3 6 2 15 17 15 1 2 5 8 25 10;
    3 6 7 9 13 11 1 2 5 7 10 14 5 4 2 6 8 10 3 2; 
    1 4 5 8 8 7 9 6 10 15 5 4 2 4 6 10 7 4 5 6; 
    14 12 5 2 3 4 10 9 3 4 3 2 10 15 9 2 3 8 20 12];
% 3. Transportation cost matrix
c_ki = [0.3448 0.3384 0.3393 1.1840 0.3527 0.3000;
    0.3589 0.3522 0.3521 1.2000 0.3668 0.3080;
    0.3214 0.3164 0.3199 1.1569 0.3286 0.3154;
    0.3110 0.3049 0.3075 1.1532 0.3187 0.3174];
c_ii = [0.0000 0.2054 0.2084 0.8615 0.2062 0.2402;
    0.2054 0.0000 0.2048 0.8669 0.2115 0.2349;
    0.2084 0.2048 0.0000 0.8691 0.2134 0.2345;
    0.8615 0.8669 0.8691 0.0000 0.8558 0.9000;
    0.2062 0.2115 0.2134 0.8558 0.0000 0.2464;
    0.2402 0.2349 0.2345 0.9000 0.2464 0.0000];
c_ij = [0.3030 0.3093 0.3115 1.1756 0.3049 0.3556 0.3091 0.3136 0.3129 1.1750 0.3075 0.3587 0.3075 0.3135 0.3143 1.1728 0.3040 0.3595 0.3122 0.3158; 
    0.3138 1.1757 0.3108 0.3595 0.3077 0.3062 0.3126 1.1820 0.3149 0.3467 0.3122 0.3189 0.3201 1.1670 0.3047 0.3651 0.3136 0.3179 0.3163 1.1733; 
    0.3106 0.3621 0.3209 0.3229 0.3188 1.1781 0.3197 0.3625 0.3332 0.3349 0.3301 1.1784 0.3313 0.3706 0.3000 0.3064 0.3099 1.1778 0.3074 0.3527;
    0.3031 0.3033 0.3066 1.1814 0.3108 0.3495 0.3016 0.3067 0.3091 1.1781 0.3074 0.3529 0.3052 0.3005 0.3062 1.1835 0.3133 0.3468 0.3055 0.3048; 
    0.3111 1.1812 0.3130 0.3479 0.3042 0.3104 0.3152 1.1739 0.3064 0.3555 0.3083 0.3088 0.3152 1.1794 0.3142 0.3490 0.3097 0.3073 0.3026 1.1853; 
    0.3152 0.3483 0.3063 0.3073 0.3061 1.1810 0.3108 0.3495 0.3132 0.3087 0.3023 1.1898 0.3195 0.3444 0.3230 0.3170 0.3120 1.2000 0.3297 0.3368];

% 4. Fixed costs of alternative hub constructionf_i
f_i = [5400 5500 6500 7200 8000 7500];
% 5. Matrix of time
t_ij = [3.2100 5.4650 2.6470 6.8142 4.8100 5.8330 5.4650 3.6330 2.8945 6.6090 4.7120 5.6960 3.8090 5.2230 2.4050 6.4900 4.4050 5.6690 5.8114 3.2020; 
    2.6450 6.8156 4.9880 5.6710 3.6290 5.4060 2.8149 6.6890 4.6746 5.4650 5.2340 3.8603 2.2463 6.4945 4.8710 5.8923 5.6510 3.4560 2.8445 6.6358; 
    4.7510 5.4230 3.2911 5.3055 2.6470 6.6382 4.2751 5.6117 3.0650 5.6150 2.9130 6.5335 4.6970 5.4584 3.3300 5.3450 2.6280 6.5637 4.9870 5.4309;
    3.2020 5.6360 2.8460 6.5362 4.7020 5.4603 3.5100 5.1080 2.5980 6.5825 4.2050 5.4076 3.5400 5.4770 2.1019 6.5625 4.1830 5.4169 3.4280 5.9580; 
    2.1370 6.4994 4.6150 5.4820 3.7730 5.8190 2.1365 6.5473 4.2850 5.4263 3.8970 5.6950 2.2890 6.5979 4.3670 5.4209 3.6020 5.6950 2.5900 6.5607; 
    4.9870 5.4505 3.2010 5.8070 2.2640 6.6362 4.7360 5.3872 3.2040 5.1524 2.1095 6.7236 4.2614 5.3223 3.3015 5.2678 4.3567 6.8574 4.2587 5.2698];

% The maximum tolerable time that the customer can accept T_j
T_j = [4.7965 4.8349 4.6668 4.8555 4.7798 4.7664 4.7936 4.6281 4.7137 4.7141 4.7304 4.5685 4.5469 4.6866 4.6645 4.7111 4.8142 4.6719 4.4622 4.8853];

% 6. Binary variables
% variable x_ki: whether each manufacturer k is connected to each hub i
x_ki = zeros(num_manufacturers, num_hubs); % The matrix initialized to zero

% Variable y i: whether each candidate hub center i is selected
y_i = zeros(num_hubs, 1); % The matrix initialized to zero

% Variable y ii: Whether there is a connection between each pair of hub centers i and i
y_ii = zeros(num_hubs, num_hubs); %The matrix initialized to zero

% Variable z ij: Whether each customer j is served by hub i
z_ij = zeros(num_hubs, num_customers); % The matrix initialized to zero

% parameter setting
popSize = 100; % population size
numGenerations = 150; % iterations
crossoverRate = 0.8; % crossover rate
mutationRate = 0.01; % mutation rate

% Calculate the length of each individual
individual_length = (num_manufacturers * num_hubs) + num_hubs + (num_hubs * num_hubs) + (num_hubs * num_customers);

% Initialize the population (each individual is a binary vector representing the hub center selection)
population = randi([0 1], popSize, individual_length); % Generate a 100x186 0/1 matrix

% The fitness of each generation was recorded
obj1_values = zeros(numGenerations, 1); % Save the value of the optimal obj1 for each generation
obj2_values = zeros(numGenerations, 1); % Save the value of the optimal obj2 for each generation

% 2D floor plan display and fitness change mapping
figure;
hold on;

% Plot the scatter of manufacturers and customers
scatter(manufacturers(:, 1), manufacturers(:, 2), 'b', 'filled'); % manufactures
scatter(customers(:, 1), customers(:, 2), 'g', 'filled'); % customers
xlabel('X axis');
ylabel('Y axis');
title('Distribution of manufacturers and customers');

% Initializes the graphics store
fitnessHistory_1 = zeros(numGenerations, 1); % Used to store the fitness of obj 1 for each generation
fitnessHistory_2 = zeros(numGenerations, 2); % Used to store the fitness of obj 2 for each generation

% Main loop of genetic algorithm
for gen = 1:numGenerations
    % For each individual in the population, two objective values are calculated
    fitness = zeros(popSize, 2); % Used to store two target values for each individual
    
    for i = 1:popSize
        individual = population(i, :); % The ith individual is selected from the population
        [obj1, obj2] = evaluateFitness(individual, num_hubs, num_manufacturers, num_customers, ...
            d_ki, d_ii, d_ij, h_ki, g_ii, s_ij, ...
            c_ki, c_ii, c_ij, f_i, t_ij, T_j);
        
        fitness(i, :) = [obj1, obj2]; % Storage fitness
    end
    
    % Save the best fitness value for this generation
    fitnessHistory(gen, :) = [min(fitness(:, 1)), max(fitness(:, 2))]; %The fitness values of each generation were recorded
    
    % Select operation (according to fitness)
    selected = rouletteWheelSelection(fitness, popSize);
    
    % Crossover operation
    offspring = crossover(selected, crossoverRate, num_hubs);
    
    %Mutation operation
    offspring = mutation(offspring, mutationRate);
    
    % Renewal of population
    population = offspring;
    
    % Draw fitness changes (every two generations to reduce the number of images)
    if mod(gen, 2) == 0
        subplot(2, 1, 1);
        plot(1:gen, fitnessHistory(1:gen, 1), 'b-o', 'DisplayName', 'total cost');
        xlabel('iterations');
        ylabel('total cost');
        title('Change in fitness: total cost');
        legend('Location', 'best');
        subplot(2, 1, 2);
        plot(1:gen, fitnessHistory(1:gen, 2), 'g-o', 'DisplayName', 'customer satisfaction');
        xlabel('iterations');
        ylabel('customer satisfaction');
        title('Change in fitness: customer satisfaction');
        legend('Location', 'best');
        drawnow; % Dynamic update image
    end
end

Error using solution>evaluateFitness (line 194)
Individual length does not match the expected total length. Expected: 186, Got: 100

% Output the selected hub center
selected_hubs = find(population(bestIndexObj1, 1:num_hubs) == 1); % Save the selected hub center number
fprintf('The selected hub center number is:\n');
disp(selected_hubs);

evaluateFitness.m

function [obj1, obj2] = evaluateFitness(individual, num_hubs, num_manufacturers, num_customers, ...
    d_ki, d_ii, d_ij, h_ki, g_ii, s_ij, ...
    c_ki, c_ii, c_ij, f_i, t_ij, T_j)
% Calculate the expected length in evaluateFitness.m
expected_length = (num_manufacturers * num_hubs) + num_hubs + (num_hubs * num_hubs) + (num_hubs * num_customers);

% Get the length of individual
individual_length = length(individual);

% Length check
if individual_length ~= expected_length
    error('Individual length does not match the expected total length. Expected: %d, Got: %d', expected_length, individual_length);
end

% Extract binary variables from individual
x_ki = reshape(individual(1:num_manufacturers * num_hubs), num_manufacturers, num_hubs); % Manufacturer to hub connections
y_i = individual(num_manufacturers * num_hubs + 1:num_manufacturers * num_hubs + num_hubs); % Hub center open status

% Calculate y_ii
start_idx_y_ii = num_manufacturers * num_hubs + num_hubs + 1;
y_ii = reshape(individual(start_idx_y_ii:start_idx_y_ii + (num_hubs * num_hubs) - 1), num_hubs, num_hubs); % Hub-to-hub connections

% Calculate z_ij
start_idx_z_ij = start_idx_y_ii + (num_hubs * num_hubs);
z_ij = reshape(individual(start_idx_z_ij:end), num_hubs, num_customers); % Customer service status

% Initialize objective functions
obj1 = 0;  % Objective function 1: Total cost of hubs
obj2 = 0;  % Objective function 2: Customer satisfaction

% Initialize penalty
penalty = 0;

% Objective function 1 - Total cost of hubs
for k = 1:num_manufacturers
    if sum(x_ki(k, :)) ~= 1  % Check constraint: each manufacturer is connected to exactly one hub
        penalty = penalty + 1000;  % Add penalty
    end
    for i = 1:num_hubs
        if y_i(i) == 1 && x_ki(k, i) == 1  % Only consider open hubs
            obj1 = obj1 + d_ki(k, i) * c_ki(k, i) * h_ki(k, i);  % Transportation cost
        end
    end
end

% Cost of building hub centers
for i = 1:num_hubs
    if y_i(i) == 1  % Only calculate fixed cost for open hub centers
        obj1 = obj1 + f_i(i);
    end
end

% Transportation cost between hub centers
for i = 1:num_hubs
    for i_prime = 1:num_hubs
        if i ~= i_prime && y_i(i) == 1 && y_ii(i, i_prime) == 1  % Only consider open hubs
            obj1 = obj1 + d_ii(i, i_prime) * c_ii(i, i_prime) * g_ii(i, i_prime);
        end
    end
end

% Transportation cost from hub centers to customers
for i = 1:num_hubs
    for j = 1:num_customers
        if y_i(i) == 1 && z_ij(i, j) == 1  % Only consider customers served by open hubs
            obj1 = obj1 + d_ij(i, j) * c_ij(i, j) * s_ij(i, j);
        end
    end
end

% Objective function 2 - Customer satisfaction
for i = 1:num_hubs
    for j = 1:num_customers
        if y_i(i) == 1 && z_ij(i, j) == 1  % Only consider open hubs and their served customers
            % Customer satisfaction function
            if t_ij(i, j) > 0 && t_ij(i, j) <= 1
                satisfaction = 1.00;
            elseif t_ij(i, j) > 1 && t_ij(i, j) <= 2
                satisfaction = 0.85;
            elseif t_ij(i, j) > 2 && t_ij(i, j) <= 3
                satisfaction = 0.75;
            elseif t_ij(i, j) > 3 && t_ij(i, j) <= 4
                satisfaction = 0.45;
            elseif t_ij(i, j) > 4 && t_ij(i, j) <= 5
                satisfaction = 0.20;
            else
                satisfaction = 0.00;  % Out-of-range time, satisfaction is 0
            end
            obj2 = obj2 + satisfaction;  % Accumulate customer satisfaction
        end
    end
end

% Check constraints and add penalties
for j = 1:num_customers
    if sum(z_ij(:, j)) ~= 1
        penalty = penalty + 1000;  % Add penalty, no customer should be served by multiple hubs
    end
end

for k = 1:num_manufacturers
    for i = 1:num_hubs
        if x_ki(k, i) > y_i(i)
            penalty = penalty + 1000;  % Add penalty, manufacturer connected to a non-open hub
        end
    end
end

for i = 1:num_hubs
    for j = 1:num_customers
        if z_ij(i, j) > y_i(i)
            penalty = penalty + 1000;  % Add penalty, customer should not be served by a non-open hub
        end
    end
end

% Flow balance constraint
for i = 1:num_hubs
    if sum(h_ki(:, i) .* x_ki(:, i) .* y_i(i)) ~= sum(s_ij(i, :) .* z_ij(i, :) .* y_i(i))
        penalty = penalty + 1000;  % Add penalty, flow balance mismatch
    end
end

% Maximum allowable time for customers
for i = 1:num_hubs
    for j = 1:num_customers
        if t_ij(i, j) * z_ij(i, j) * y_i(i) > T_j(j)
            penalty = penalty + 1000;  % Add penalty, customer service time exceeds maximum allowable time
        end
    end
end

% Symmetry constraint
for i = 1:num_hubs
    for i_prime = 1:num_hubs
        if y_ii(i, i_prime) ~= y_ii(i_prime, i)
            penalty = penalty + 1000;  % Add penalty, symmetry mismatch
        end
    end
end

% Constraint 7: y_ii <= min(y_i, y_i')
for i = 1:num_hubs
    for i_prime = 1:num_hubs
        if y_ii(i, i_prime) > min(y_i(i), y_i(i_prime))
            penalty = penalty + 1000;  % Add penalty, constraint mismatch
        end
    end
end

% Ensure all binary variables are 0 or 1
if any([x_ki(:); y_i(:); z_ij(:); y_ii(:)] ~= round([x_ki(:); y_i(:); z_ij(:); y_ii(:)]))
    penalty = penalty + 1000;  % Add penalty
end

% Add penalty to obj1 and obj2
obj1 = obj1 + penalty;
obj2 = obj2 - penalty;  % Adjust the impact of penalty on obj2 as needed
end

generateIndividual.m

function individual = generateIndividual(num_manufacturers, num_hubs, num_customers)
% Calculate the expected length of the individual
expected_length = (num_manufacturers * num_hubs) + num_hubs + (num_hubs * num_hubs) + (num_hubs * num_customers);

% Generate the individual
individual = zeros(expected_length, 1);

% Fill in each part
individual(1:num_manufacturers * num_hubs) = randi([0, 1], num_manufacturers * num_hubs, 1); % Manufacturer to hub connections
individual(num_manufacturers * num_hubs + 1:num_manufacturers * num_hubs + num_hubs) = randi([0, 1], num_hubs, 1); % Hub selection
individual(num_manufacturers * num_hubs + num_hubs + 1:num_manufacturers * num_hubs + num_hubs + num_hubs * num_hubs) = randi([0, 1], num_hubs * num_hubs, 1); % Connections between hubs
individual(num_manufacturers * num_hubs + num_hubs + num_hubs * num_hubs + 1:end) = randi([0, 1], num_hubs * num_customers, 1); % Connections from hubs to customers
end

selection.m

% Selection function
function selected = selection(pop, obj1, obj2, popSize, alpha)
% alpha is a parameter controlling the weight between objective function 1 and objective function 2, where 0 <= alpha <= 1
% Calculate the combined fitness of each individual
fitness = alpha * obj1 + (1 - alpha) * obj2;
% Roulette wheel selection
prob = exp(fitness - max(fitness)); % Prevent numerical overflow
prob = prob / sum(prob);
selected = pop(randsample(size(pop, 1), popSize, true, prob), :);
end

rouletteWheelSelection.m

function selected = rouletteWheelSelection(fitness, popSize)
% Calculate the total fitness of all individuals
totalFitness = sum(fitness);

% Calculate the cumulative probability distribution
cumProb = cumsum(fitness) / totalFitness;

% Initialize the selection result
selected = zeros(1, popSize);

for i = 1:popSize
    r = rand();  % Generate a random number in the interval [0, 1]
    index = find(cumProb >= r, 1);  % Find the first individual with cumulative probability >= r
    
    if ~isempty(index)
        selected(i) = index;
    else
        % Handle the case where no suitable individual is found
        selected(i) = length(cumProb);  % Choose the last individual as a default value
    end
end
end

crossover.m

function offspring = crossover(selected, crossoverRate, num_hubs)
numSelected = size(selected, 1);      % Get the number of selected individuals
numGenes = size(selected, 2);         % Get the number of genes
popSize = size(selected, 1);          % Get the population size
offspring = zeros(popSize, numGenes); % Initialize the offspring matrix with size popSize x numGenes

% Perform crossover based on the crossover rate
for i = 1:2:popSize
    if rand <= crossoverRate
        % Randomly select parents
        parent1 = selected(randi(numSelected), :);
        parent2 = selected(randi(numSelected), :);
        
        % Randomly choose a crossover point
        crossoverPoint = randi(numGenes - 1);
        
        % Generate offspring
        offspring(i, :) = [parent1(1:crossoverPoint), parent2(crossoverPoint+1:end)];
        if i < popSize
            offspring(i+1, :) = [parent2(1:crossoverPoint), parent1(crossoverPoint+1:end)];
        end
    else
        % If no crossover, copy directly
        offspring(i, :) = selected(i, :);
        if i < popSize
            offspring(i+1, :) = selected(i+1, :);
        end
    end
end
end

mutation.m

function mutated = mutation(offspring, mutationRate)
numOffspring = size(offspring, 1);
numGenes = size(offspring, 2);

mutated = offspring;
for i = 1:numOffspring
    for j = 1:numGenes
        if rand < mutationRate
            mutated(i, j) = 1 - mutated(i, j); % Flip the bit
        end
    end
end
end

Answer 1

[Torsten]

You set

    % Renewal of population
    population = offspring;    

and offspring has only size 1x100 instead of 1x186.

Thus in your loop

% Main loop of genetic algorithm
for gen = 1:numGenerations
    % For each individual in the population, two objective values are calculated
    fitness = zeros(popSize, 2); % Used to store two target values for each individual
    
    for i = 1:popSize
        individual = population(i, :); % The ith individual is selected from the population
        [obj1, obj2] = evaluateFitness(individual, num_hubs, num_manufacturers, num_customers, ...
            d_ki, d_ii, d_ij, h_ki, g_ii, s_ij, ...
            c_ki, c_ii, c_ij, f_i, t_ij, T_j);
        
        fitness(i, :) = [obj1, obj2]; % Storage fitness
    end
    

when gen = 2, population(i, :) has changed size, and evaluateFitness errors.

[马清]

First of all, thank you for your response. However, could you please give me a specific program file to solve the hub location problem or modify the program based on the above program so that these program can output a correct result? Additionally, do you consider the model I constructed to contain errors?

[Torsten]

I have no knowledge about the background of the hub location problem.

If you give a mathematical description of what you are trying to solve, maybe someone is able to help. Else I would contact the authors of the code and ask what went wrong.

[马清]

I'm so sorry for responding to your valuable answer about solving the HLP so late, as I have had to handle many issues regarding my company in recent days. Since the previous HLP file is not related to my research field, I need to modify the HLP file. In the new file, I have added specific details about the solved HLP (e.g., the background of the problem, some relevant assumptions) so that readers can better understand and solve the HLP. Additionally, I still hope to use the genetic algorithm (GA) to solve the model. Therefore, could you please provide me with detailed MATLAB programming for GA to handle the HLP model? I would be extremely grateful! I'm looking forward to your response.