Travel salesman problem--ant colony optimization algorithm solution (matlab implementation)

Today, let's share how matlab achieves an ant colony optimization algorithm to solve the traveler problem.
Ant Colony Optimization (ACO) algorithm is described in detail in the last blog post. A little buddy who needs it can see it

1. Initialize city information
     1.1 Random city coordinates
     1.2 Find the distance between cities
2. Initialization parameters
3. Iterate to find the best path
     3.1 Randomly Generate Ant Start Cities
     3.2 Iterative Calculation of Selected City Probability
     3.3 roulette to pick the next city
     3.4 Update Shortest Path
     3.5 Update Pheromones
4. Result
5. All Codes

1. Initialize city information

The first step initializes information about the city, such as quantity, coordinates, distance, and so on.

1.1 Random city coordinates

First, the coordinates of n cities are randomly generated, then the horizontal and vertical coordinates of n cities are generated using randperm(100,n). randperm(100,n) indicates that n integers of 1-100 (including 1 and 100) are randomly generated, and plot functions are used to plot points, for example:

1.2 Find the distance between cities

Calculate the distance between cities, stored i n a matrix of n*n. Since the distance formula behind is placed on the denominator, the distance from city I to city I cannot be 0. This is set to eps, which is a very small number.

The diagonal line is the distance from city I to i, replaced by a small number.

2. Initialization parameters

Before iterating to find the best path, initialize some necessary parameters, such as pheromone importance factor, heuristic importance factor, pheromone volatile factor, heuristic function, and so on.

3. Iterate to find the best path

Once you are ready, the following is to iteratively find the best path to cycle m ants and n cities. The core is two formulas of ant colony optimization algorithm, one is to calculate the probability of selecting a city, the other is to update the pheromone.

3.1 Randomly Generate Ant Start Cities

The preferred option is to randomly place ants i n n cities. Of course, instead of actually taking ants here, it randomly generates a number of 1-n, which represents the initial city of the first ant.

3.2 Iterative Calculation of Selected City Probability

To calculate the probability that an ant will select a city that has not been visited at present, it is necessary to remove the visited city beforehand. The formula for calculating the probability of selecting a city is as follows:

3.3 roulette to pick the next city

After calculating the selection probability for ants to choose other cities that they have not visited, they can not simply choose the one with the highest probability. This makes it easy to fall into the local optimal solution. This way, they use the roulette method to select. Students who are not familiar with roulette can use this blog for reference.

3.4 Update Shortest Path

Because ultimately the shortest path is required, this step is to sum all the paths after choosing the city above to get the path and (plus the distance from the last node to the starting point) of each ant on this tour.
***It is important to note that the calculated shortest path for this time is not necessarily smaller than the last one, and the convergence trajectory of the resulting shortest path will fluctuate significantly, such as the following

So here we can improve it, the basic idea is: if the shortest path size in this tour is larger than the last one, then the shortest path size in this tour will be changed to the same as the last one, so that there will be no shortest distance in the later iteration than the previous one, so the treatment will be more in line with people's ideas, more reasonable, and modified as follows

3.5 Update Pheromones

All ants begin their next week after the next one. They update pheromones before the next one. Pheromones = last pheromone* (1-volatile factor) + last increase of pheromones. The more ants walk on the path, the more pheromones increase, the stronger than the pheromones.

4. Result

Write the code based on the above steps, taking 75 ants and 30 cities as examples, and get the following results, Shortest Path Roadmap

Below are the shortest and average distance convergence curves

Command Line Output Results

5. All Codes

The code draws on two videos, and you are interested and patient to see them. I think both of them are good. The second video has a postscript video that talks about tuning.

t0=clock;   %For timing
%% Initialize city information
n=30;                    %Define the number of cities
city=[(randperm(100,n))',(randperm(100,n))'];  %Getting city coordinates
plot(city(:,1),city(:,2),'o');  %Tracing
for i=1:n
    text(city(i,1)+0.5,city(i,2),num2str(i));       %Grade
%Set the range of the horizontal and vertical coordinates
grid on     %Gridlines
hold on

%% Find the distance between cities
D=zeros(n,n);      %Create a new one n*n Matrix storage distance of
for i=1:n
   for j=1:n
       if i~=j
           D(i,j)=eps;     %Set to a small number, but not 0, as used in the following denominator

%% Initialization parameters
m=75;                   %Ant number
alpha = 1;              %Pheromone Importance Factor
beta = 5;               %Heuristic Importance Factor
rho = 0.2;              %Pheromone Volatile Factor
Q=10;                   %constant coefficient
Eta = 1./D;             %Heuristics(inverse distance)
Tau = ones(n,n);        %Pheromone Matrix(Initially, each road's pheromone is set to 1)
Table = zeros(m,n);     %Path Record Table, Record m Path taken by ants
iter = 1;               %Initial value of iteration number
iter_max = 100;         %Maximum number of iterations
Route_best = zeros(iter_max,n);     %Best Routes for Generations
Length_best = zeros(iter_max,1);    %Length of the best path for each generation
Length_ave = zeros(iter_max,1);     %Average length of each generation path

%% Iterate to find the best path
for iter=1:iter_max
    %Starting City for Random Ants
    for i=1:m
    citys_index=1:n;        %Define all city indexes
    for i=1:m               %Traverse all ants
       for j=2:n            %Traverse all other cities
           tabu=Table(i,1:(j-1));      %Visited City(Prohibit access to tables)
           allow=citys_index(~ismember(citys_index,tabu)); %Remove Visited Cities(City to visit)
           P=allow;         %Store probability, defined as anything, as long as and allow Same length
           %Calculating the probability of selection between cities
           for k=1:length(allow)
               %tabu(end)Represents the city where the ant is now located. allow(k)The city where the ant is going
           P=P/sum(P);      %Find the set of probabilities for ultimately going to another city
           Pc=cumsum(P);    %Cumulative probability
           %Roulette Select Next City
           target_index=find(Pc>=rand);     %Find cases larger than random values from cumulative probability
           target=allow(target_index(1));   %Take out the first result of the roulette
           Table(i,j)=target;               %Determine the i The first day an ant goes j-1 City 
    %Calculate the length of each ant's path once each ant has traveled
    for i=1:m
        for j=1:(n-1)
        %Last plus the distance from the last point to the starting point
    %% Calculating Shortest Path Distance and Average Distance
    [min_length,min_index]=min(Length);     %Take out the shortest path and its subscript
    if iter==1
        Length_best(iter)=min_length;           %Save this shortest path distance
        Length_best(iter)=min(Length_best(iter-1),min_length);  %Select the shortest path that is smaller this time and last time
    Route_best(iter,:)=Table(min_index,:);  %Save this shortest route
    Length_ave(iter)=mean(Length);          %Save the average distance for this route
    %% Update pheromone
    Tau_Ant=Q./Length;          %Pheromone concentration left by each ant along the path
    for i=1:m
        for j=1:n-1
            %Update pheromone
        %Update last node to start pheromone
    Tau=(1-rho)*Tau+Tau_Temp;   %Adding new pheromones after evaporation of pheromones
    Table = zeros(m,n);     %Clear the route for the next round

%% Command Line Window Displays Results
[min_length,min_index]=min(Length_best);         %Take out the shortest path and its subscript
disp(['The shortest distance is:' num2str(min_length)]);
disp(['The shortest path is:' num2str(Finnly_Route)]);
disp(['Runtime:' num2str(last_time) 'second']);

%% Mapping
     [city(Finnly_Route,2);city(Finnly_Route,2)],'bo-');  %Tracing
text(city(Finnly_Route(1),1),city(Finnly_Route(1),2),'    Starting point');
text(city(Finnly_Route(end),1),city(Finnly_Route(end),2),'    End');

subplot(1,2,1);     %First column displayed in one row and two columns of images
xlabel('Number of iterations');
ylabel('Minimum distance');
title('Shortest distance convergence curve');

subplot(1,2,2);     %The second column displayed in one row and two columns of images
xlabel('Number of iterations');
ylabel('Average Distance');
title('Average Distance Convergence Curve');

Keywords: MATLAB Algorithm AI

Added by Vivid Lust on Thu, 03 Mar 2022 20:55:58 +0200