MTSP_GA Multiple Traveling Salesmen Problem (M-TSP) Genetic Algorithm (GA). Finds a (near) optimal solution to the M-TSP by setting up a GA to search for the shortest route (least distance needed for the salesmen to travel to each city exactly once and return to their starting locations)
% MTSP_GA Multiple Traveling Salesmen Problem (M-TSP) Genetic Algorithm (GA)
% Finds a (near) optimal solution to the M-TSP by setting up a GA to search
% for the shortest route (least distance needed for the salesmen to travel
% to each city exactly once and return to their starting locations)
%
% Summary:
% 1. Each salesman travels to a unique set of cities and completes the
% route by returning to the city he started from
% 2. Each city is visited by exactly one salesman
%
% Input:
% XY (float) is an Nx2 matrix of city locations, where N is the number of cities
% DMAT (float) is an NxN matrix of city-to-city distances or costs
% NSALESMEN (scalar integer) is the number of salesmen to visit the cities
% MINTOUR (scalar integer) is the minimum tour length for any of the salesmen
% POPSIZE (scalar integer) is the size of the population (should be divisible by 8)
% NUMITER (scalar integer) is the number of desired iterations for the algorithm to run
% SHOWPROG (scalar logical) shows the GA progress if true
% SHOWRESULT (scalar logical) shows the GA results if true
%
% Output:
% OPTROUTE (integer array) is the best route found by the algorithm
% OPTBREAK (integer array) is the list of route break points (these specify the indices
% into the route used to obtain the individual salesman routes)
% MINDIST (scalar float) is the total distance traveled by the salesmen
%
% Route/Breakpoint Details:
% If there are 10 cities and 3 salesmen, a possible route/break
% combination might be: rte = [5 6 9 1 4 2 8 10 3 7], brks = [3 7]
% Taken together, these represent the solution [5 6 9][1 4 2 8][10 3 7],
% which designates the routes for the 3 salesmen as follows:
% . Salesman 1 travels from city 5 to 6 to 9 and back to 5
% . Salesman 2 travels from city 1 to 4 to 2 to 8 and back to 1
% . Salesman 3 travels from city 10 to 3 to 7 and back to 10
%
% Example:
% n = 35;
% xy = 10*rand(n,2);
% nSalesmen = 5;
% minTour = 3;
% popSize = 80;
% numIter = 5e3;
% a = meshgrid(1:n);
% dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),n,n);
% [optRoute,optBreak,minDist] = mtsp_ga(xy,dmat,nSalesmen,minTour, ...
% popSize,numIter,1,1);
%
% Example:
% n = 50;
% phi = (sqrt(5)-1)/2;
% theta = 2*pi*phi*(0:n-1);
% rho = (1:n).^phi;
% [x,y] = pol2cart(theta(:),rho(:));
% xy = 10*([x y]-min([x;y]))/(max([x;y])-min([x;y]));
% nSalesmen = 5;
% minTour = 3;
% popSize = 80;
% numIter = 1e4;
% a = meshgrid(1:n);
% dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),n,n);
% [optRoute,optBreak,minDist] = mtsp_ga(xy,dmat,nSalesmen,minTour, ...
% popSize,numIter,1,1);
%
% Example:
% n = 35;
% xyz = 10*rand(n,3);
% nSalesmen = 5;
% minTour = 3;
% popSize = 80;
% numIter = 5e3;
% a = meshgrid(1:n);
% dmat = reshape(sqrt(sum((xyz(a,:)-xyz(a',:)).^2,2)),n,n);
% [optRoute,optBreak,minDist] = mtsp_ga(xyz,dmat,nSalesmen,minTour, ...
% popSize,numIter,1,1);
%
%
function varargout = mtsp_ga(xy,dmat,nSalesmen,minTour,popSize,numIter,showProg,showResult)
% Process Inputs and Initialize Defaults
nargs = 8;
for k = nargin:nargs-1
switch k
case 0
xy = 10*rand(40,2);
case 1
N = size(xy,1);
a = meshgrid(1:N);
dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),N,N);
case 2
nSalesmen = 5;
case 3
minTour = 3;
case 4
popSize = 80;
case 5
numIter = 5e3;
case 6
showProg = 1;
case 7
showResult = 1;
otherwise
end
end
% Verify Inputs
[N,dims] = size(xy);
[nr,nc] = size(dmat);
if N ~= nr || N ~= nc
error('Invalid XY or DMAT inputs!')
end
n = N;
% Sanity Checks
nSalesmen = max(1,min(n,round(real(nSalesmen(1)))));
minTour = max(1,min(floor(n/nSalesmen),round(real(minTour(1)))));
popSize = max(8,8*ceil(popSize(1)/8));
numIter = max(1,round(real(numIter(1))));
showProg = logical(showProg(1));
showResult = logical(showResult(1));
% Initializations for Route Break Point Selection
nBreaks = nSalesmen-1;
dof = n - minTour*nSalesmen; % degrees of freedom
addto = ones(1,dof+1);
for k = 2:nBreaks
addto = cumsum(addto);
end
cumProb = cumsum(addto)/sum(addto);
% Initialize the Populations
popRoute = zeros(popSize,n); % population of routes
popBreak = zeros(popSize,nBreaks); % population of breaks
popRoute(1,:) = (1:n);
popBreak(1,:) = rand_breaks();
for k = 2:popSize
popRoute(k,:) = randperm(n);
popBreak(k,:) = rand_breaks();
end
% Select the Colors for the Plotted Routes
pclr = ~get(0,'DefaultAxesColor');
clr = [1 0 0; 0 0 1; 0.67 0 1; 0 1 0; 1 0.5 0];
if nSalesmen > 5
clr = hsv(nSalesmen);
end
% Run the GA
globalMin = Inf;
totalDist = zeros(1,popSize);
distHistory = zeros(1,numIter);
tmpPopRoute = zeros(8,n);
tmpPopBreak = zeros(8,nBreaks);
newPopRoute = zeros(popSize,n);
newPopBreak = zeros(popSize,nBreaks);
if showProg
pfig = figure('Name','MTSP_GA | Current Best Solution','Numbertitle','off');
end
for iter = 1:numIter
% Evaluate Members of the Population
for p = 1:popSize
d = 0;
pRoute = popRoute(p,:);
pBreak = popBreak(p,:);
rng = [[1 pBreak+1];[pBreak n]]';
for s = 1:nSalesmen
d = d + dmat(pRoute(rng(s,2)),pRoute(rng(s,1)));
for k = rng(s,1):rng(s,2)-1
d = d + dmat(pRoute(k),pRoute(k+1));
end
end
totalDist(p) = d;
end
% Find the Best Route in the Population
[minDist,index] = min(totalDist);
distHistory(iter) = minDist;
if minDist < globalMin
globalMin = minDist;
optRoute = popRoute(index,:);
optBreak = popBreak(index,:);
rng = [[1 optBreak+1];[optBreak n]]';
if showProg
% Plot the Best Route
figure(pfig);
for s = 1:nSalesmen
rte = optRoute([rng(s,1):rng(s,2) rng(s,1)]);
if dims > 2, plot3(xy(rte,1),xy(rte,2),xy(rte,3),'.-','Color',clr(s,:));
else plot(xy(rte,1),xy(rte,2),'.-','Color',clr(s,:)); end
title(sprintf('Total Distance = %1.4f, Iteration = %d',minDist,iter));
hold on
end
hold off
end
end
% Genetic Algorithm Operators
randomOrder = randperm(popSize);
for p = 8:8:popSize
rtes = popRoute(randomOrder(p-7:p),:);
brks = popBreak(randomOrder(p-7:p),:);
dists = totalDist(randomOrder(p-7:p));
[ignore,idx] = min(dists); %#ok
bestOf8Route = rtes(idx,:);
bestOf8Break = brks(idx,:);
routeInsertionPoints = sort(ceil(n*rand(1,2)));
I = routeInsertionPoints(1);
J = routeInsertionPoints(2);
for k = 1:8 % Generate New Solutions
tmpPopRoute(k,:) = bestOf8Route;
tmpPopBreak(k,:) = bestOf8Break;
switch k
case 2 % Flip
tmpPopRoute(k,I:J) = tmpPopRoute(k,J:-1:I);
case 3 % Swap
tmpPopRoute(k,[I J]) = tmpPopRoute(k,[J I]);
case 4 % Slide
tmpPopRoute(k,I:J) = tmpPopRoute(k,[I+1:J I]);
case 5 % Modify Breaks
tmpPopBreak(k,:) = rand_breaks();
case 6 % Flip, Modify Breaks
tmpPopRoute(k,I:J) = tmpPopRoute(k,J:-1:I);
tmpPopBreak(k,:) = rand_breaks();
case 7 % Swap, Modify Breaks
tmpPopRoute(k,[I J]) = tmpPopRoute(k,[J I]);
tmpPopBreak(k,:) = rand_breaks();
case 8 % Slide, Modify Breaks
tmpPopRoute(k,I:J) = tmpPopRoute(k,[I+1:J I]);
tmpPopBreak(k,:) = rand_breaks();
otherwise % Do Nothing
end
end
newPopRoute(p-7:p,:) = tmpPopRoute;
newPopBreak(p-7:p,:) = tmpPopBreak;
end
popRoute = newPopRoute;
popBreak = newPopBreak;
end
if showResult
% Plots
figure('Name','MTSP_GA | Results','Numbertitle','off');
subplot(2,2,1);
if dims > 2, plot3(xy(:,1),xy(:,2),xy(:,3),'.','Color',pclr);
else plot(xy(:,1),xy(:,2),'.','Color',pclr); end
title('City Locations');
subplot(2,2,2);
imagesc(dmat(optRoute,optRoute));
title('Distance Matrix');
subplot(2,2,3);
rng = [[1 optBreak+1];[optBreak n]]';
for s = 1:nSalesmen
rte = optRoute([rng(s,1):rng(s,2) rng(s,1)]);
if dims > 2, plot3(xy(rte,1),xy(rte,2),xy(rte,3),'.-','Color',clr(s,:));
else plot(xy(rte,1),xy(rte,2),'.-','Color',clr(s,:)); end
title(sprintf('Total Distance = %1.4f',minDist));
hold on;
end
subplot(2,2,4);
plot(distHistory,'b','LineWidth',2);
title('Best Solution History');
set(gca,'XLim',[0 numIter+1],'YLim',[0 1.1*max([1 distHistory])]);
end
% Return Outputs
if nargout
varargout{1} = optRoute;
varargout{2} = optBreak;
varargout{3} = minDist;
end
% Generate Random Set of Break Points
function breaks = rand_breaks()
if minTour == 1 % No Constraints on Breaks
tmpBreaks = randperm(n-1);
breaks = sort(tmpBreaks(1:nBreaks));
else % Force Breaks to be at Least the Minimum Tour Length
nAdjust = find(rand < cumProb,1)-1;
spaces = ceil(nBreaks*rand(1,nAdjust));
adjust = zeros(1,nBreaks);
for kk = 1:nBreaks
adjust(kk) = sum(spaces == kk);
end
breaks = minTour*(1:nBreaks) + cumsum(adjust);
end
end
end