Implementation of Optimization Software in MATLAB

optimizationWe selected MATLAB as our optimization software due to its robust capabilities for solving linear prog

% MATLAB script for urban transportation optimization

% Step 1: Define Input Data

% Cost coefficients

c_v = [10; 15; 20]; % Vehicle costs per hour

c_d = [5; 7; 6]; % Driver costs per hour

% Minimum required vehicles and drivers for each route

R = [2; 3; 1]; % Minimum vehicles for routes

D = [3; 2; 2]; % Minimum drivers for routes

% Maximum working hours per vehicle and driver

H = 8; % Maximum hours allowed

% Emission rates per vehicle for each route

e = [1; 2; 1]; % Emission per hour for vehicles

E = [10; 12; 10]; % Maximum emissions allowed for each route

% Step 2: Formulate the Linear Programming Problem

% Objective function coefficients

f = [c_v; c_d]; % Combine vehicle and driver costs

% Initialize constraints

A = []; % Coefficients for inequality constraints

b = []; % Right-hand side values for constraints

% Minimum route coverage constraints

for j = 1:length(R)

A = [A; zeros(1, length(c_v)), eye(1, length(c_d))]; % For drivers

b = [b; R(j)];

A = [A; eye(1, length(c_v)), zeros(1, length(c_d))]; % For vehicles

b = [b; D(j)];

end

% Maximum working hours constraints

for i = 1:length(c_v)

A = [A; eye(1, length(c_v)), zeros(1, length(c_d))]; % For vehicles

b = [b; H];

A = [A; zeros(1, length(c_v)), eye(1, length(c_d))]; % For drivers

b = [b; H];

end

% Environmental constraints

for j = 1:length(E)

A = [A; e(j) * eye(1, length(c_v)), zeros(1, length(c_d))];

b = [b; E(j)];

end

% Variable bounds

lb = zeros(length(c_v) + length(c_d), 1); % Lower bounds for decision variables

ub = []; % Upper bounds can be defined as needed

% Step 3: Solve the Linear Programming Problem

% Call linprog to solve the problem

options = optimoptions('linprog','Display','iter'); % Display output during optimization

[x, fval] = linprog(f, A, b, [], [], lb, ub, options);

% Step 4: Display Results

disp('Optimal Allocation of Vehicles and Drivers (hours):');

disp(x);

disp('Minimum Total Cost:');

disp(fval);