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
二、遗传算法简介
1 引言
2 遗传算法理论
2.1 遗传算法的生物学基础
2.2 遗传算法的理论基础
2.3 遗传算法的基本概念
2.4 标准的遗传算法
2.5 遗传算法的特点
2.6 遗传算法的改进方向
3 遗传算法流程
4 关键参数说明
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
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.
