MAT之GA：遗传算法（GA）解决M-TSP多旅行商问题

导读
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)

输出结果

实现代码

输出结果

实现代码

% 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-1switch kcase 0xy = 10*rand(40,2);case 1N = size(xy,1);a = meshgrid(1:N);dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),N,N);case 2nSalesmen = 5;case 3minTour = 3;case 4popSize = 80;case 5numIter = 5e3;case 6showProg = 1;case 7showResult = 1;otherwiseend
end% Verify Inputs
[N,dims] = size(xy);
[nr,nc] = size(dmat);
if N ~= nr || N ~= ncerror('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:nBreaksaddto = 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:popSizepopRoute(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 > 5clr = 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 showProgpfig = figure('Name','MTSP_GA | Current Best Solution','Numbertitle','off');
end
for iter = 1:numIter% Evaluate Members of the Populationfor p = 1:popSized = 0;pRoute = popRoute(p,:);pBreak = popBreak(p,:);rng = [[1 pBreak+1];[pBreak n]]';for s = 1:nSalesmend = d + dmat(pRoute(rng(s,2)),pRoute(rng(s,1)));for k = rng(s,1):rng(s,2)-1d = d + dmat(pRoute(k),pRoute(k+1));endendtotalDist(p) = d;end% Find the Best Route in the Population[minDist,index] = min(totalDist);distHistory(iter) = minDist;if minDist < globalMinglobalMin = minDist;optRoute = popRoute(index,:);optBreak = popBreak(index,:);rng = [[1 optBreak+1];[optBreak n]]';if showProg% Plot the Best Routefigure(pfig);for s = 1:nSalesmenrte = 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,:)); endtitle(sprintf('Total Distance = %1.4f, Iteration = %d',minDist,iter));hold onendhold offendend% Genetic Algorithm OperatorsrandomOrder = randperm(popSize);for p = 8:8:popSizertes = popRoute(randomOrder(p-7:p),:);brks = popBreak(randomOrder(p-7:p),:);dists = totalDist(randomOrder(p-7:p));[ignore,idx] = min(dists); %#okbestOf8Route = rtes(idx,:);bestOf8Break = brks(idx,:);routeInsertionPoints = sort(ceil(n*rand(1,2)));I = routeInsertionPoints(1);J = routeInsertionPoints(2);for k = 1:8 % Generate New SolutionstmpPopRoute(k,:) = bestOf8Route;tmpPopBreak(k,:) = bestOf8Break;switch kcase 2 % FliptmpPopRoute(k,I:J) = tmpPopRoute(k,J:-1:I);case 3 % SwaptmpPopRoute(k,[I J]) = tmpPopRoute(k,[J I]);case 4 % SlidetmpPopRoute(k,I:J) = tmpPopRoute(k,[I+1:J I]);case 5 % Modify BreakstmpPopBreak(k,:) = rand_breaks();case 6 % Flip, Modify BreakstmpPopRoute(k,I:J) = tmpPopRoute(k,J:-1:I);tmpPopBreak(k,:) = rand_breaks();case 7 % Swap, Modify BreakstmpPopRoute(k,[I J]) = tmpPopRoute(k,[J I]);tmpPopBreak(k,:) = rand_breaks();case 8 % Slide, Modify BreakstmpPopRoute(k,I:J) = tmpPopRoute(k,[I+1:J I]);tmpPopBreak(k,:) = rand_breaks();otherwise % Do NothingendendnewPopRoute(p-7:p,:) = tmpPopRoute;newPopBreak(p-7:p,:) = tmpPopBreak;endpopRoute = newPopRoute;popBreak = newPopBreak;
endif showResult
% Plotsfigure('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); endtitle('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:nSalesmenrte = 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,:)); endtitle(sprintf('Total Distance = %1.4f',minDist));hold on;endsubplot(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 nargoutvarargout{1} = optRoute;varargout{2} = optBreak;varargout{3} = minDist;
end% Generate Random Set of Break Pointsfunction breaks = rand_breaks()if minTour == 1 % No Constraints on BreakstmpBreaks = randperm(n-1);breaks = sort(tmpBreaks(1:nBreaks));else % Force Breaks to be at Least the Minimum Tour LengthnAdjust = find(rand < cumProb,1)-1;spaces = ceil(nBreaks*rand(1,nAdjust));adjust = zeros(1,nBreaks);for kk = 1:nBreaksadjust(kk) = sum(spaces == kk);endbreaks = minTour*(1:nBreaks) + cumsum(adjust);endend
end

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MAT之GA：遗传算法（GA）解决M-TSP多旅行商问题

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