How to Solve Travelling Salesman Problem (TSP) using Optimization Solver in Matlab

clc
close all
clear all
%Original code source: https://www.mathworks.com/help/optim/ug/travelling-salesman-problem.html

figure;
load('usborder.mat','x','y','xx','yy');
rng(3,'twister') % makes a plot with stops in Maine & Florida, and is reproducible
nStops = 140; % you can use any number, but the problem size scales as N^2
fprintf('Number of cities in this instance: %d\n',nStops);
stopsLon = zeros(nStops,1); % allocate x-coordinates of nStops
stopsLat = stopsLon; % allocate y-coordinates
n = 1;
while (n <= nStops)
    xp = rand*1.5;
    yp = rand;
    if inpolygon(xp,yp,x,y) % test if inside the border
        stopsLon(n) = xp;
        stopsLat(n) = yp;
        n = n+1;
    end
end
plot(x,y,'Color','red'); % draw the outside border
hold on
% Add the stops to the map
plot(stopsLon,stopsLat,'*b')
idxs = nchoosek(1:nStops,2);
dist = hypot(stopsLat(idxs(:,1)) - stopsLat(idxs(:,2)), ...
             stopsLon(idxs(:,1)) - stopsLon(idxs(:,2)));
lendist = length(dist);
Aeq = spones(1:length(idxs)); % Adds up the number of trips
beq = nStops;
Aeq = [Aeq;spalloc(nStops,length(idxs),nStops*(nStops-1))]; % allocate a sparse matrix
for ii = 1:nStops
    whichIdxs = (idxs == ii); % find the trips that include stop ii
    whichIdxs = sparse(sum(whichIdxs,2)); % include trips where ii is at either end
    Aeq(ii+1,:) = whichIdxs'; % include in the constraint matrix
end
beq = [beq; 2*ones(nStops,1)];
intcon = 1:lendist;
lb = zeros(lendist,1);
ub = ones(lendist,1);
opts = optimoptions('intlinprog','Display','off');
[x_tsp,costopt,exitflag,output] = intlinprog(dist,intcon,[],[],Aeq,beq,lb,ub,opts);
segments = find(x_tsp); % Get indices of lines on optimal path
lh = zeros(nStops,1); % Use to store handles to lines on plot
lh = updateSalesmanPlot(lh,x_tsp,idxs,stopsLon,stopsLat);
tours = detectSubtours(x_tsp,idxs);
numtours = length(tours); % number of subtours
%fprintf('# of subtours: %d\n',numtours);
A = spalloc(0,lendist,0); % Allocate a sparse linear inequality constraint matrix
b = [];
while numtours > 1 % repeat until there is just one subtour
    % Add the subtour constraints
    b = [b;zeros(numtours,1)]; % allocate b
    A = [A;spalloc(numtours,lendist,nStops)]; % a guess at how many nonzeros to allocate
    for ii = 1:numtours
        rowIdx = size(A,1)+1; % Counter for indexing
        subTourIdx = tours{ii}; % Extract the current subtour
%         The next lines find all of the variables associated with the
%         particular subtour, then add an inequality constraint to prohibit
%         that subtour and all subtours that use those stops.
        variations = nchoosek(1:length(subTourIdx),2);
        for jj = 1:length(variations)
            whichVar = and((sum(idxs==subTourIdx(variations(jj,1)),2)),(sum(idxs==subTourIdx(variations(jj,2)),2)));
            A(rowIdx,whichVar) = 1;
        end
        b(rowIdx) = length(subTourIdx)-1; % One less trip than subtour stops
    end
    % Try to optimize again
    [x_tsp,costopt,exitflag,output] = intlinprog(dist,intcon,A,b,Aeq,beq,lb,ub,opts);
    % Visualize result
    lh = updateSalesmanPlot(lh,x_tsp,idxs,stopsLon,stopsLat);
    % How many subtours this time?
    tours = detectSubtours(x_tsp,idxs);
    numtours = length(tours); % number of subtours
end
title('Final solution');
hold off
fprintf('The solution gap = ')
disp(output.absolutegap)
fprintf('Note: If the solution gap = 0, the global optimal solution is found')