【TSP】基于matlab遗传算法求解旅行商问题【含Matlab源码 1337期】

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一、TSP简介

旅行商问题,即TSP问题(Traveling Salesman Problem)又译为旅行推销员问题、货郎担问题,是数学领域中著名问题之一。假设有一个旅行商人要拜访n个城市,他必须选择所要走的路径,路径的限制是每个城市只能拜访一次,而且最后要回到原来出发的城市。路径的选择目标是要求得的路径路程为所有路径之中的最小值。 TSP的数学模型 在这里插入图片描述

二、遗传算法简介

1 引言 在这里插入图片描述 在这里插入图片描述 2 遗传算法理论 2.1 遗传算法的生物学基础 在这里插入图片描述 在这里插入图片描述 2.2 遗传算法的理论基础 在这里插入图片描述 在这里插入图片描述 在这里插入图片描述 在这里插入图片描述 2.3 遗传算法的基本概念 在这里插入图片描述 在这里插入图片描述 在这里插入图片描述 在这里插入图片描述 在这里插入图片描述 在这里插入图片描述 2.4 标准的遗传算法 在这里插入图片描述 在这里插入图片描述 2.5 遗传算法的特点 在这里插入图片描述 在这里插入图片描述 2.6 遗传算法的改进方向 在这里插入图片描述 3 遗传算法流程 在这里插入图片描述 在这里插入图片描述 在这里插入图片描述 4 关键参数说明 在这里插入图片描述

三、部分源代码

function [ globalMin, optRoute, optCentroid] = bdtsp_ga_basic(popSize, numIter, xy, alpha, range  )
% FUNCTION: mclust_ga_basic calculates the minimum of the sum of the max
% distances for several clusters.  In essence, it is a clustering tool
% that minimizes the max distances between each cluster and the farthest
% member of that cluster. Genetic algorithm minimizes the sum of the max 
% distance of each of the centroid (cluster centers) then calculates the
% TSP distance from each centroid to each centroid.
% Inputs: 
%        population size and iterations
%        xy coordinates
%        speed of drone as a factor of blimp speed
%        range of drone
% Outputs:
%        number of centroids
%        assignement of xy coordinates to centroids
%        (note that one or more coordinates will become centroids)
% GA uses uses a tournament approach mutation type genetic algorithm.  
%  Initialize:
%  (1) Calculate the distance from each xy coordinate to every other xy
%  coordinate as distance matrix dmat.
%  Body:
%  (1) Randomly generate populations 
%  (2) Find min-cost of all pop (trials); keep best pop member and plot.
%  (3) Shuffle (reshuffle) pop for a new tournament
%  (4) Sub-group pop into groups of 4.
%      Find the best of the 4; Overwrite worst of 4 from sub-group pop
%  (5) Mutate the best of 4 (winner) in each sub-group
%  (6) Insert best of 4 (winner) and all mutations back into population
%  (7) If iteration budget remains, go to step 4, else terminate.
%  Termination: based on iteration budget.
% 
%  Example of inputs:
%  nStops  =   30;   % Number of delivery stops for blimp-drone
%  popSize =  500;   % Size of the population of trials.
%  numIter = 2500;   % Number of iterations of GA; iteration budget.
%  alpha   =    2;   % Speed of drone as a factor of blimp
%  range   =   10;   % Range of drone (i.e. 10 km)
%  xy      =   50*rand([nStops,2]);
%  bdtsp_ga_basic(popSize, numIter, xy, alpha, range )

% init variables
  minCost=inf; iter=0; 
  costHistory=[];costIteration=[];
% If null arguments to function call 
  showprogress=true;
if nargin < 5  
  showprogress=true;
  nStops=40;  popSize=800; numIter=4000; alpha=2; range=10;
  xy=45*rand([nStops,2]);
end
  
  % initialize distance matrix
    [nPoints , ~]=size(xy);
    meshg = meshgrid(1:nPoints); % calc. distance
    dmat = reshape(sqrt(sum((xy(meshg,:)-xy(meshg',:)).^2,2)),nPoints,nPoints);
  % Initialize the Population
    [n, ~]=  size(xy);
    pop  = zeros(popSize,n);
    popc = zeros(popSize,n);
    pop(1,:) = (1:n);
    for k = 2:popSize
        % random trials of of blimp-drone routing
        pop(k,:) = randperm(n);
    end
        
 % Run the GA
    globalMin     = Inf;
    totalCost     = zeros(1,popSize);
    costHistory   = zeros(1,numIter);
    costIteration = zeros(1,numIter);
    tmpPop        = zeros(4,n);
    newPop        = zeros(popSize,n);
  
 for iter = 1:numIter
        % Evaluate Each Population Member
        for p = 1:popSize
           % first point is always a centroid
             c = pop(p,1); popc(p,:)=zeros([1 n]);
              popc(p,1)=c; % first centroid
              sumDmax = dmat(c,pop(p,2)); % get dmax 
              dMax=sumDmax;
              for k=2:n  % Get max distances
                     % what is the distance of this from centroid
                     d = dmat(c,pop(p,k));
                   if  d>dMax || d>range%.4
                        % assign as a new centroid 
                         if k<n 
                          c  = pop(p,k);
                          dMax = dmat(c,pop(p,k+1));
                          popc(p,k)=c; % current centroid
                         else
                           c = pop(p,k-1);
                           dMax = dmat(c,pop(p,k));  
                           popc(p,k-1)=c; % current centroid
                         end
                         sumDmax = sumDmax + dMax;  % get next dmax 
                   end 
              end
              poptsp = popc(p,:).*(popc(p,:)>0);
              poptsp = poptsp(poptsp>0);
              ln=length(poptsp);
              d2= dmat(poptsp(ln),poptsp(1));
              for k=1:ln-1
                  d2 = d2 + dmat(poptsp(k),poptsp(k+1));
              end
              totalCost(p) = (2*sumDmax/alpha) + d2 ; 
        end
        
        % Find and keep the best layout in the population
        [minCost,index] = min(totalCost);
        costHistory(iter) = minCost;
        costIteration(iter) = iter;
        if minCost < globalMin
            globalMin = minCost;
            optRoute  = pop(index,:);   
            optCentroid = popc(index,:);
            layout      = optRoute(1:n);  % best layout
            centroids   = optCentroid(1:n); 
            if showprogress==true
              call_plot(xy,layout, centroids);
            end
        end
        
        % Genetic Algorithm Operators: tournament mutations
        % randomly reshuffle population for tournament - play different
        % teams on each iteration
        randomOrder = randperm(popSize);

        for p = 4:4:popSize
            % random reshuffle population, group by 4     
            laytes = pop(randomOrder(p-3:p),:); 
            csts   = totalCost(randomOrder(p-3:p));
                % what is the min layout?
            [~,idx] = min(csts); 
                % what is the best layout of 4
            bestOf4Layout = laytes(idx,:);
                % randomly select two layout insertion points and sort
            routeInsertionPoints = sort(ceil(n*rand(1,2)));
                I = routeInsertionPoints(1);
                J = routeInsertionPoints(2);
            for k = 1:4 % Mutate the best layout to get three new layouts; keep orig.
                % a small matrix of 4 rows of best layout
                tmpPop(k,:) = bestOf4Layout;
                switch k
                       % flip segment between two of the departments
                    case 2 % Flip
                        tmpPop(k,I:J) = tmpPop(k,J:-1:I);
                    case 3 % Swap departments
                        tmpPop(k,[I J]) = tmpPop(k,[J I]);
                    case 4 % Slide departments down
                       tmpPop(k,I:J) = tmpPop(k,[I+1:J I]);
                    otherwise % Do Nothing
                end
            end
             % using the original population, create a new population
            newPop(p-3:p,:) = tmpPop;
        end
        pop = newPop;
 end
 
     function call_plot( xy, ~,~)
      subplot(1,2,1)
      plot(xy(:,1), xy(:,2),'k.');
      hold on;
      idx2=(centroids>0).*centroids;
      idx2=idx2(idx2>0);
      cz = centroids; cnt=0;
      for kk= 1:length(cz)
          c=centroids(kk);
          if c>0
              cc=c;
              cnt=cnt+1;
              %RC=[ rand rand rand];
              RC=[  0  0 0];
          else
          ly=layout(kk);
          plot(xy([cc ly],1), xy([cc ly],2),'k');
          plot(xy(ly,1),xy(ly,2),...
           'MarkerSize',15,...
           'MarkerEdgeColor','black',...
           'MarkerFaceColor',RC);
          plot( xy(cc,1), xy(cc,2),'ks');
          end
      end
      idx2=[idx2 idx2(1)];
      plot(xy(idx2,1),xy(idx2,2),'k--+');
      hold off;
      title('Blimp Route/Drone Spokes');
      xlabel('x-coordinate'); ylabel('y-coordinate'); 
      drawnow;
      if iter>0
      subplot(1,2,2)
      dH=costHistory(costHistory>0);
      dI=costIteration(costIteration>0);
      plot(dI, dH,'k-'); 
      title(sprintf('Min-time: %1.1f',minCost));
      xlabel('iter'); ylabel('Cost');
      end
     end
end
    


四、运行结果

在这里插入图片描述

五、matlab版本及参考文献

1 matlab版本 2014a

2 参考文献 [1] 包子阳,余继周,杨杉.智能优化算法及其MATLAB实例(第2版)[M].电子工业出版社,2016. [2]张岩,吴水根.MATLAB优化算法源代码[M].清华大学出版社,2017.

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