MAT之GA：遺傳算法（GA）解決M-TSP多旅行商問題

導讀
? ? ? 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)

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% 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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