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function [pTS,fmin]=grTravSale(C)% Function [pTS,fmin]=grTravSale(C) solve the nonsymmetrical% traveling salesman problem.% Input parameter:% C(n,n) - matrix of distances between cities,% maybe, nonsymmetrical;% n - number of cities.% Output parameters:% pTS(n) - the order of cities;% fmin - length of way.% Uses the reduction to integer LP-problem:% Look: Miller C.E., Tucker A. W., Zemlin R. A.% Integer Programming Formulation of Traveling Salesman Problems. % J.ACM, 1960, Vol.7, p. 326-329.% Needed other products: MIQP.M.% This software may be free downloaded from site:% http://control.ee.ethz.ch/~hybrid/miqp/% Author: Sergiy Iglin% e-mail: siglin@yandex.ru% personal page: http://iglin.exponenta.ru% ============= Input data validation ==================if nargin<1,error('There are no input data!')endif ~isnumeric(C),error('The array C must be numeric!')endif ~isreal(C),error('The array C must be real!')ends=size(C); % size of array Cif length(s)~=2,error('The array C must be 2D!')endif s(1)~=s(2),error('Matrix C must be square!')endif s(1)<3,error('Must be not less than 3 cities!')end% ============ Size of problem ====================n=s(1); % number of vertexesm=n*(n-1); % number of arrows% ============ Parameters of integer LP problem ========Aeq=[]; % for the matrix of the boundary equationsfor k1=1:n,z1=zeros(n);z1(k1,:)=1;z2=[z1;eye(n)];Aeq=[Aeq z2([1:2*n-1],setdiff([1:n],k1))];endAeq=[Aeq zeros(2*n-1,n-1)];A=[]; % for the matrix of the boundary inequationsfor k1=2:n,z1=[];for k2=1:n,z2=eye(n)*(n-1)*(k2==k1);z1=[z1 z2(setdiff([2:n],k1),setdiff([1:n],k2))];endz2=-eye(n);z2(:,k1)=z2(:,k1)+1;A=[A;[z1 z2(setdiff([2:n],k1),2:n)]];endbeq=ones(2*n-1,1); % the right parts of the boundary equations b=ones((n-1)*(n-2),1)*(n-2); % the right parts of the boundary inequationsC1=C'+diag(ones(1,n)*NaN);C2=C1(:);c=[C2(~isnan(C2));zeros(n-1,1)]; % the factors for objective functionvlb=[zeros(m,1);-inf*ones(n-1,1)]; % the lower boundsvub=[ones(m,1);inf*ones(n-1,1)]; % the upper boundsH=zeros(n^2-1); % Hessian% ============= We solve the MIQP problem ========== [xmin,fmin]=MIQP(H,c,A,b,Aeq,beq,[1:m],vlb,vub);% ============= We return the results ==============eik=round(xmin(1:m)); % the arrows of the waye1=[zeros(1,n-1);reshape(eik,n,n-1)];e2=[e1(:);0]; % we add zero to a diagonale3=(reshape(e2,n,n))'; % the matrix of the waypTS=[1 find(e3(1,:))]; % we build the waywhile pTS(end)>1, % we add the city to the waypTS=[pTS find(e3(pTS(end),:))];endreturn。
多旅行商问题的m a t l a b程序Document serial number【NL89WT-NY98YT-NC8CB-NNUUT-NUT108】%多旅行商问题的m a t l a b程序function varargout =mtspf_ga(xy,dmat,salesmen,min_tour,pop_size,num_iter,show_prog,show_r es)% MTSPF_GA Fixed Multiple Traveling Salesmen Problem (M-TSP) Genetic Algorithm (GA)% Finds a (near) optimal solution to a variation of the M-TSP by setting% up a GA to search for the shortest route (least distance needed for% each salesman to travel from the start location to individual cities% and back to the original starting place)%% Summary:% 1. Each salesman starts at the first point, and ends at thefirst% point, but travels to a unique set of cities in between% 2. Except for the first, each city is visited by exactly one salesman%% Note: The Fixed Start/End location is taken to be the first XY point%% 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% SALESMEN (scalar integer) is the number of salesmen to visit the cities% MIN_TOUR (scalar integer) is the minimum tour length for any of the% salesmen, NOT including the start/end point% POP_SIZE (scalar integer) is the size of the population (should be divisible by 8)% NUM_ITER (scalar integer) is the number of desired iterations for the algorithm to run% SHOW_PROG (scalar logical) shows the GA progress if true% SHOW_RES (scalar logical) shows the GA results if true%% Output:% OPT_RTE (integer array) is the best route found by the algorithm% OPT_BRK (integer array) is the list of route break points (these specify the indices% into the route used to obtain the individual salesman routes)% MIN_DIST (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 4 2 8 10 3 7], brks = [3 7] % Taken together, these represent the solution [1 5 6 9 1][1 4 2 8 1][1 10 3 7 1],% which designates the routes for the 3 salesmen as follows:% . Salesman 1 travels from city 1 to 5 to 6 to 9 and back to 1% . Salesman 2 travels from city 1 to 4 to 2 to 8 and back to 1% . Salesman 3 travels from city 1 to 10 to 3 to 7 and back to 1%% 2D Example:% n = 35;% xy = 10*rand(n,2);% salesmen = 5;% min_tour = 3;% pop_size = 80;% num_iter = 5e3;% a = meshgrid(1:n);% dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),n,n);% [opt_rte,opt_brk,min_dist] =mtspf_ga(xy,dmat,salesmen,min_tour, ...% pop_size,num_iter,1,1);%% 3D Example:% n = 35;% xyz = 10*rand(n,3);% salesmen = 5;% min_tour = 3;% pop_size = 80;% num_iter = 5e3;% a = meshgrid(1:n);% dmat = reshape(sqrt(sum((xyz(a,:)-xyz(a',:)).^2,2)),n,n);% [opt_rte,opt_brk,min_dist] =mtspf_ga(xyz,dmat,salesmen,min_tour, ...% pop_size,num_iter,1,1);%% See also: mtsp_ga, mtspo_ga, mtspof_ga, mtspofs_ga, mtspv_ga, distmat%% Author: Joseph Kirk% Email% Release:% Release Date: 6/2/09% Process Inputs and Initialize Defaultsnargs = 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 2salesmen = 5;case 3min_tour = 2;case 4pop_size = 80;case 5num_iter = 5e3;case 6show_prog = 1;case 7show_res = 1;otherwiseendend% Verify Inputs[N,dims] = size(xy);[nr,nc] = size(dmat);if N ~= nr || N ~= ncerror('Invalid XY or DMAT inputs!')endn = N - 1; % Separate Start/End City% Sanity Checkssalesmen = max(1,min(n,round(real(salesmen(1)))));min_tour = max(1,min(floor(n/salesmen),round(real(min_tour(1))))); pop_size = max(8,8*ceil(pop_size(1)/8));num_iter = max(1,round(real(num_iter(1))));show_prog = logical(show_prog(1));show_res = logical(show_res(1));% Initializations for Route Break Point Selectionnum_brks = salesmen-1;dof = n - min_tour*salesmen; % degrees of freedomaddto = ones(1,dof+1);for k = 2:num_brksaddto = cumsum(addto);endcum_prob = cumsum(addto)/sum(addto);% Initialize the Populationspop_rte = zeros(pop_size,n); % population of routespop_brk = zeros(pop_size,num_brks); % population of breaksfor k = 1:pop_sizepop_rte(k,:) = randperm(n)+1;pop_brk(k,:) = randbreaks();end% Select the Colors for the Plotted Routesclr = [1 0 0; 0 0 1; 0 1; 0 1 0; 1 0];if salesmen > 5clr = hsv(salesmen);end% Run the GAglobal_min = Inf;total_dist = zeros(1,pop_size);dist_history = zeros(1,num_iter);tmp_pop_rte = zeros(8,n);tmp_pop_brk = zeros(8,num_brks);new_pop_rte = zeros(pop_size,n);new_pop_brk = zeros(pop_size,num_brks);if show_progpfig = figure('Name','MTSPF_GA | Current BestSolution','Numbertitle','off');endfor iter = 1:num_iter% Evaluate Members of the Populationfor p = 1:pop_sized = 0;p_rte = pop_rte(p,:);p_brk = pop_brk(p,:);rng = [[1 p_brk+1];[p_brk n]]';for s = 1:salesmend = d + dmat(1,p_rte(rng(s,1))); % Add Start Distancefor k = rng(s,1):rng(s,2)-1d = d + dmat(p_rte(k),p_rte(k+1));endd = d + dmat(p_rte(rng(s,2)),1); % Add End Distanceendtotal_dist(p) = d;end% Find the Best Route in the Population[min_dist,index] = min(total_dist);dist_history(iter) = min_dist;if min_dist < global_minglobal_min = min_dist;opt_rte = pop_rte(index,:);opt_brk = pop_brk(index,:);rng = [[1 opt_brk+1];[opt_brk n]]';if show_prog% Plot the Best Routefigure(pfig);for s = 1:salesmenrte = [1 opt_rte(rng(s,1):rng(s,2)) 1];if dims == 3, 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 = %, Iteration= %d',min_dist,iter));hold onendif dims == 3, plot3(xy(1,1),xy(1,2),xy(1,3),'ko');else plot(xy(1,1),xy(1,2),'ko'); endhold offendend% Genetic Algorithm Operatorsrand_grouping = randperm(pop_size);for p = 8:8:pop_sizertes = pop_rte(rand_grouping(p-7:p),:);brks = pop_brk(rand_grouping(p-7:p),:);dists = total_dist(rand_grouping(p-7:p));[ignore,idx] = min(dists);best_of_8_rte = rtes(idx,:);best_of_8_brk = brks(idx,:);rte_ins_pts = sort(ceil(n*rand(1,2)));I = rte_ins_pts(1);J = rte_ins_pts(2);for k = 1:8 % Generate New Solutionstmp_pop_rte(k,:) = best_of_8_rte;tmp_pop_brk(k,:) = best_of_8_brk;switch kcase 2 % Fliptmp_pop_rte(k,I:J) = fliplr(tmp_pop_rte(k,I:J)); case 3 % Swaptmp_pop_rte(k,[I J]) = tmp_pop_rte(k,[J I]);case 4 % Slidetmp_pop_rte(k,I:J) = tmp_pop_rte(k,[I+1:J I]); case 5 % Modify Breakstmp_pop_brk(k,:) = randbreaks();case 6 % Flip, Modify Breakstmp_pop_rte(k,I:J) = fliplr(tmp_pop_rte(k,I:J)); tmp_pop_brk(k,:) = randbreaks();case 7 % Swap, Modify Breakstmp_pop_rte(k,[I J]) = tmp_pop_rte(k,[J I]);tmp_pop_brk(k,:) = randbreaks();case 8 % Slide, Modify Breakstmp_pop_rte(k,I:J) = tmp_pop_rte(k,[I+1:J I]); tmp_pop_brk(k,:) = randbreaks();otherwise % Do Nothingendendnew_pop_rte(p-7:p,:) = tmp_pop_rte;new_pop_brk(p-7:p,:) = tmp_pop_brk;endpop_rte = new_pop_rte;pop_brk = new_pop_brk;endif show_res% Plotsfigure('Name','MTSPF_GA | Results','Numbertitle','off');subplot(2,2,1);if dims == 3, plot3(xy(:,1),xy(:,2),xy(:,3),'k.');else plot(xy(:,1),xy(:,2),'k.'); endtitle('City Locations');subplot(2,2,2);imagesc(dmat([1 opt_rte],[1 opt_rte]));title('Distance Matrix');subplot(2,2,3);rng = [[1 opt_brk+1];[opt_brk n]]';for s = 1:salesmenrte = [1 opt_rte(rng(s,1):rng(s,2)) 1];if dims == 3, 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 = %',min_dist));hold on;endif dims == 3, plot3(xy(1,1),xy(1,2),xy(1,3),'ko');else plot(xy(1,1),xy(1,2),'ko'); endsubplot(2,2,4);plot(dist_history,'b','LineWidth',2);title('Best Solution History');set(gca,'XLim',[0 num_iter+1],'YLim',[0 *max([1 dist_history])]); end% Return Outputsif nargoutvarargout{1} = opt_rte;varargout{2} = opt_brk;varargout{3} = min_dist;end% Generate Random Set of Break Pointsfunction breaks = randbreaks()if min_tour == 1 % No Constraints on Breakstmp_brks = randperm(n-1);breaks = sort(tmp_brks(1:num_brks));else % Force Breaks to be at Least the Minimum Tour Lengthnum_adjust = find(rand < cum_prob,1)-1;spaces = ceil(num_brks*rand(1,num_adjust));adjust = zeros(1,num_brks);for kk = 1:num_brksadjust(kk) = sum(spaces == kk);endbreaks = min_tour*(1:num_brks) + cumsum(adjust); endendend。
问题:已知n个城市之间的相互距离，现有一个推销员必须遍访这n个城市，并且每个城市只能访问一次，最后又必须返回出发城市。
如何安排他对这些城市的访问次序，可使其旅行路线的总长度最短？分析:用离散数学或图论的术语来说，假设有一个图g=(v,e)，其中v是顶点集，e 是边集，设d=(dij)是由顶点i和顶点j之间的距离所组成的距离矩阵，旅行商问题就是求出一条通过所有顶点且每个顶点只通过一次的具有最短距离的回路。
这个问题可分为对称旅行商问题(dij=dji,,任意i,j=1,2,3，…,n)和非对称旅行商问题(dij≠dji,,任意i,j=1,2,3，…,n)。
若对于城市v={v1,v2,v3,…,vn}的一个访问顺序为t=(t1,t2,t3,…,ti,…,tn),其中ti∈v(i=1,2,3,…,n)，且记tn+1= t1，则旅行商问题的数学模型为：min l=σd(t(i),t(i+1)) （i=1,…,n）旅行商问题是一个典型的组合优化问题，并且是一个np难问题，其可能的路径数目与城市数目n是成指数型增长的，所以一般很难精确地求出其最优解，本文采用遗传算法求其近似解。
遗传算法：初始化过程：用v1,v2,v3,…,vn代表所选n个城市。
定义整数pop-size作为染色体的个数，并且随机产生pop-size个初始染色体，每个染色体为1到18的整数组成的随机序列。
适应度f的计算：对种群中的每个染色体vi，计算其适应度，f=σd(t(i),t(i+1)).评价函数eval(vi)：用来对种群中的每个染色体vi设定一个概率，以使该染色体被选中的可能性与其种群中其它染色体的适应性成比例，既通过轮盘赌，适应性强的染色体被选择产生后台的机会要大，设alpha∈(0,1)，本文定义基于序的评价函数为eval(vi)=alpha*(1-alpha).^(i-1) 。
[随机规划与模糊规划]选择过程：选择过程是以旋转赌轮pop-size次为基础，每次旋转都为新的种群选择一个染色体。
M A T L A B多旅行商问题源代码Company Document number：WTUT-WT88Y-W8BBGB-BWYTT-19998MATLAB多旅行商问题源代码function varargout =mtspf_ga(xy,dmat,salesmen,min_tour,pop_size,num_iter,show_prog,show_res)% MTSPF_GA Fixed Multiple Traveling Salesmen Problem (M-TSP) Genetic Algorithm (GA)% Finds a (near) optimal solution to a variation of the M-TSP by setting% up a GA to search for the shortest route (least distance needed for% each salesman to travel from the start location to individual cities% and back to the original starting place)%% Summary:% 1. Each salesman starts at the first point, and ends at the first% point, but travels to a unique set of cities in between% 2. Except for the first, each city is visited by exactly one salesman%% Note: The Fixed Start/End location is taken to be the first XY point%% 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% SALESMEN (scalar integer) is the number of salesmen to visit the cities% MIN_TOUR (scalar integer) is the minimum tour length for any of the% salesmen, NOT including the start/end point% POP_SIZE (scalar integer) is the size of the population (should be divisible by 8) % NUM_ITER (scalar integer) is the number of desired iterations for the algorithm to run% SHOW_PROG (scalar logical) shows the GA progress if true% SHOW_RES (scalar logical) shows the GA results if true%% Output:% OPT_RTE (integer array) is the best route found by the algorithm% OPT_BRK (integer array) is the list of route break points (these specify the indices% into the route used to obtain the individual salesman routes)% MIN_DIST (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 4 2 8 10 3 7], brks = [3 7]% Taken together, these represent the solution [1 5 6 9 1][1 4 2 8 1][1 10 3 7 1],% which designates the routes for the 3 salesmen as follows:% . Salesman 1 travels from city 1 to 5 to 6 to 9 and back to 1% . Salesman 2 travels from city 1 to 4 to 2 to 8 and back to 1% . Salesman 3 travels from city 1 to 10 to 3 to 7 and back to 1%% 2D Example:% n = 35;% xy = 10*rand(n,2);% salesmen = 5;% min_tour = 3;% pop_size = 80;% num_iter = 5e3;% a = meshgrid(1:n);% dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),n,n);% [opt_rte,opt_brk,min_dist] = mtspf_ga(xy,dmat,salesmen,min_tour, ... % pop_size,num_iter,1,1);%% 3D Example:% n = 35;% xyz = 10*rand(n,3);% salesmen = 5;% min_tour = 3;% pop_size = 80;% num_iter = 5e3;% a = meshgrid(1:n);% dmat = reshape(sqrt(sum((xyz(a,:)-xyz(a',:)).^2,2)),n,n);% [opt_rte,opt_brk,min_dist] = mtspf_ga(xyz,dmat,salesmen,min_tour, ... % pop_size,num_iter,1,1);%% See also: mtsp_ga, mtspo_ga, mtspof_ga, mtspofs_ga, mtspv_ga, distmat %% Author: Joseph Kirk% Release:% Release Date: 6/2/09% Process Inputs and Initialize Defaultsnargs = 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 2salesmen = 5;case 3min_tour = 2;case 4pop_size = 80;case 5num_iter = 5e3;case 6show_prog = 1;case 7show_res = 1;otherwiseendend% Verify Inputs[N,dims] = size(xy);[nr,nc] = size(dmat);if N ~= nr || N ~= ncerror('Invalid XY or DMAT inputs!')endn = N - 1; % Separate Start/End City% Sanity Checkssalesmen = max(1,min(n,round(real(salesmen(1)))));min_tour = max(1,min(floor(n/salesmen),round(real(min_tour(1))))); pop_size = max(8,8*ceil(pop_size(1)/8));num_iter = max(1,round(real(num_iter(1))));show_prog = logical(show_prog(1));show_res = logical(show_res(1));% Initializations for Route Break Point Selectionnum_brks = salesmen-1;dof = n - min_tour*salesmen; % degrees of freedomaddto = ones(1,dof+1);for k = 2:num_brksaddto = cumsum(addto);endcum_prob = cumsum(addto)/sum(addto);% Initialize the Populationspop_rte = zeros(pop_size,n); % population of routespop_brk = zeros(pop_size,num_brks); % population of breaksfor k = 1:pop_sizepop_rte(k,:) = randperm(n)+1;pop_brk(k,:) = randbreaks();end% Select the Colors for the Plotted Routesclr = [1 0 0; 0 0 1; 0 1; 0 1 0; 1 0];if salesmen > 5clr = hsv(salesmen);end% Run the GAglobal_min = Inf;total_dist = zeros(1,pop_size);dist_history = zeros(1,num_iter);tmp_pop_rte = zeros(8,n);tmp_pop_brk = zeros(8,num_brks);new_pop_rte = zeros(pop_size,n);new_pop_brk = zeros(pop_size,num_brks);if show_progpfig = figure('Name','MTSPF_GA | Current Best Solution','Numbertitle','off'); endfor iter = 1:num_iter% Evaluate Members of the Populationfor p = 1:pop_sized = 0;p_rte = pop_rte(p,:);p_brk = pop_brk(p,:);rng = [[1 p_brk+1];[p_brk n]]';for s = 1:salesmend = d + dmat(1,p_rte(rng(s,1))); % Add Start Distancefor k = rng(s,1):rng(s,2)-1d = d + dmat(p_rte(k),p_rte(k+1));endd = d + dmat(p_rte(rng(s,2)),1); % Add End Distanceendtotal_dist(p) = d;end% Find the Best Route in the Population[min_dist,index] = min(total_dist);dist_history(iter) = min_dist;if min_dist < global_minglobal_min = min_dist;opt_rte = pop_rte(index,:);opt_brk = pop_brk(index,:);rng = [[1 opt_brk+1];[opt_brk n]]';if show_prog% Plot the Best Routefigure(pfig);for s = 1:salesmenrte = [1 opt_rte(rng(s,1):rng(s,2)) 1];if dims == 3, 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 = %, Iteration = %d',min_dist,iter)); hold onendif dims == 3, plot3(xy(1,1),xy(1,2),xy(1,3),'ko');else plot(xy(1,1),xy(1,2),'ko'); endhold offendend% Genetic Algorithm Operatorsrand_grouping = randperm(pop_size);for p = 8:8:pop_sizertes = pop_rte(rand_grouping(p-7:p),:);brks = pop_brk(rand_grouping(p-7:p),:);dists = total_dist(rand_grouping(p-7:p));[ignore,idx] = min(dists);best_of_8_rte = rtes(idx,:);best_of_8_brk = brks(idx,:);rte_ins_pts = sort(ceil(n*rand(1,2)));I = rte_ins_pts(1);J = rte_ins_pts(2);for k = 1:8 % Generate New Solutionstmp_pop_rte(k,:) = best_of_8_rte;tmp_pop_brk(k,:) = best_of_8_brk;switch kcase 2 % Fliptmp_pop_rte(k,I:J) = fliplr(tmp_pop_rte(k,I:J));case 3 % Swaptmp_pop_rte(k,[I J]) = tmp_pop_rte(k,[J I]);case 4 % Slidetmp_pop_rte(k,I:J) = tmp_pop_rte(k,[I+1:J I]);case 5 % Modify Breakstmp_pop_brk(k,:) = randbreaks();case 6 % Flip, Modify Breakstmp_pop_rte(k,I:J) = fliplr(tmp_pop_rte(k,I:J));tmp_pop_brk(k,:) = randbreaks();case 7 % Swap, Modify Breakstmp_pop_rte(k,[I J]) = tmp_pop_rte(k,[J I]);tmp_pop_brk(k,:) = randbreaks();case 8 % Slide, Modify Breakstmp_pop_rte(k,I:J) = tmp_pop_rte(k,[I+1:J I]);tmp_pop_brk(k,:) = randbreaks();otherwise % Do Nothingendendnew_pop_rte(p-7:p,:) = tmp_pop_rte;new_pop_brk(p-7:p,:) = tmp_pop_brk;endpop_rte = new_pop_rte;pop_brk = new_pop_brk;endif show_res% Plotsfigure('Name','MTSPF_GA | Results','Numbertitle','off');subplot(2,2,1);if dims == 3, plot3(xy(:,1),xy(:,2),xy(:,3),'k.');else plot(xy(:,1),xy(:,2),'k.'); endtitle('City Locations');subplot(2,2,2);imagesc(dmat([1 opt_rte],[1 opt_rte]));title('Distance Matrix');subplot(2,2,3);rng = [[1 opt_brk+1];[opt_brk n]]';for s = 1:salesmenrte = [1 opt_rte(rng(s,1):rng(s,2)) 1];if dims == 3, 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 = %',min_dist));hold on;endif dims == 3, plot3(xy(1,1),xy(1,2),xy(1,3),'ko');else plot(xy(1,1),xy(1,2),'ko'); endsubplot(2,2,4);plot(dist_history,'b','LineWidth',2);title('Best Solution History');set(gca,'XLim',[0 num_iter+1],'YLim',[0 *max([1 dist_history])]); end% Return Outputsif nargoutvarargout{1} = opt_rte;varargout{2} = opt_brk;varargout{3} = min_dist;end% Generate Random Set of Break Pointsfunction breaks = randbreaks()if min_tour == 1 % No Constraints on Breakstmp_brks = randperm(n-1);breaks = sort(tmp_brks(1:num_brks));else % Force Breaks to be at Least the Minimum Tour Length num_adjust = find(rand < cum_prob,1)-1;spaces = ceil(num_brks*rand(1,num_adjust));adjust = zeros(1,num_brks);for kk = 1:num_brksadjust(kk) = sum(spaces == kk);endbreaks = min_tour*(1:num_brks) + cumsum(adjust);endendend。
基于MATLAB的蚁群算法解决旅行商问题(附带源程序、仿真) ..摘要：旅行商问题的传统求解方法是遗传算法，但此算法收敛速度慢，并不能获得问题的最优化解。
蚁群算法是受自然界中蚁群搜索食物行为启发而提出的一种智能优化算法，通过介绍蚁群觅食过程中基于信息素的最短路径的搜索策略，给出基于MATLAB的蚁群算法在旅行商问题中的应用，对问题求解进行局部优化。
经过计算机仿真结果表明，这种蚁群算法对求解旅行商问题有较好的改进效果。
关键词：蚁群算法；旅行商问题；MATLAB；优化一、意义和目标旅行商问题是物流领域中的典型问题，它的求解具有十分重要的理论和现实意义。
采用一定的物流配送方式，可以大大节省人力物力，完善整个物流系统。
已被广泛采用的遗传算法是旅行商问题的传统求解方法，但遗传算法收敛速度慢，具有一定的缺陷。
本文采用蚁群算法，充分利用蚁群算法的智能性，求解旅行商问题，并进行实例仿真。
进行仿真计算的目标是，该算法能够获得旅行商问题的优化结果，平均距离和最短距离。
二、国内外研究现状仿生学出现于XXXX年代中期，人们从生物进化机理中受到启发，提出了遗传算法、进化规划、进化策略等许多用以解决复杂优化问题的新方法。
这些以生物特性为基础的演化算法的发展及对生物群落行为的发现引导研究人员进一步开展了对生物社会性的研究，从而出现了基于群智能理论的蚁群算法，并掀起了一股研究的热潮。
XXXX年代意大利科学家M.Dorigo M最早提出了蚁群优化算法——蚂蚁系统（Ant system, AS），在求解二次分配、图着色问题、车辆调度、集成电路设计以及通信网络负载问题的处理中都取得了较好的结果。
旅行商问题（TSP, Traveling Salesman Problem）被认为是一个基本问题，是在1859年由威廉·汉密尔顿爵士首次提出的。
所谓TSP问题是指：有N个城市，要求旅行商到达每个城市各一次，且仅一次，并回到起点，且要求旅行路线最短。
基于MATLAB的蚁族算法求解旅行商问题作者：李艳平来源：《计算机光盘软件与应用》2013年第14期摘要：目前求解旅行商问题效果最好的混合算法是最大最小蚂蚁算法和局部搜索算法，本文对蚁群算法的仿真学原理进行概要介绍，蚁群算法是受自然界中蚁群搜索食物行为启发而提出的一种智能多目标优化算法，通过蚁群觅食过程中最短路径的搜索策略，给出基于MATLAB的蚁群算法在旅行商问题中的应用，并通过实例仿真结果表明，此算法有一定优越性。
关键词：蚁群算法；旅行商问题；仿真；多目标优化中图分类号：TP301.6旅行商问题（TSP）是一个经典的组合优化问题。
TSP可以描述为：一个商品推销员要去若干个城市推销商品，该推销员从一个城市出发，需要经过所有城市后，回到出发地。
应如何选择行进路线，以使总的行程最短。
从图论的角度来看，该问题实质是在一个带权完全无向图中，找一个权值最小的Hamilton回路。
由于该问题的可行解是所有顶点的全排列，随着顶点数的增加，会产生组合爆炸，它是一个N P完全问题。
随着问题规模的增大，人们对复杂事物和复杂系统建立数学模型并进行求解的能力是有限的，目标函数和约束条件往往不能以明确的函数关系表达，或因函数带有随机参、变量，导致基于数学模型的优化方法在应用于实际生产时，有其局限性甚至不适用。
基于仿真的优化（Simulation Based Optimization，SBO）方法正是在这样的背景下发展起来的。
近年来应用蚁群算法求解旅行商问题，由于其并行性与分布性，特别适用于大规模启发式搜索，实验结果表明这种研究方法是可行的。
1 蚁群算法的仿生学原理蚁群算法最早是由意大利学者M.Dorigo提出来的，它的灵感来源于蚂蚁在寻找食物过程中发现路径的行为，蚂蚁集体寻找路径时，利用称为“外激素”的生物信息激素选择后继行为的智能过程。
蚂蚁是一种群居昆虫，在觅食等活动中，彼此依赖、相互协作共同完成特定的任务。
蚁群的行为是整体协作，相互分工，以一个整体去解决一些对单个蚂蚁来说不可能完成的任务。
1.遗传算法解决TSP 问题（附matlab源程序）2.知n个城市之间的相互距离，现有一个推销员必须遍访这n个城市，并且每个城市3.只能访问一次，最后又必须返回出发城市。
如何安排他对这些城市的访问次序，可使其4.旅行路线的总长度最短？5.用图论的术语来说，假设有一个图g=(v,e)，其中v是顶点集，e是边集，设d=(dij)6.是由顶点i和顶点j之间的距离所组成的距离矩阵，旅行商问题就是求出一条通过所有顶7.点且每个顶点只通过一次的具有最短距离的回路。
8.这个问题可分为对称旅行商问题(dij=dji,,任意i,j=1,2,3，…,n)和非对称旅行商9.问题(dij≠dji,,任意i,j=1,2,3，…,n)。
10.若对于城市v={v1,v2,v3,…,vn}的一个访问顺序为t=(t1,t2,t3,…,ti,…,tn),其中11.ti∈v(i=1,2,3,…,n)，且记tn+1= t1，则旅行商问题的数学模型为：12.min l=σd(t(i),t(i+1)) （i=1,…,n）13.旅行商问题是一个典型的组合优化问题，并且是一个np难问题，其可能的路径数目14.与城市数目n是成指数型增长的，所以一般很难精确地求出其最优解，本文采用遗传算法15.求其近似解。
16.遗传算法：17.初始化过程：用v1,v2,v3,…,vn代表所选n个城市。
定义整数pop-size作为染色体的个数18.，并且随机产生pop-size个初始染色体，每个染色体为1到18的整数组成的随机序列。
19.适应度f的计算：对种群中的每个染色体vi，计算其适应度，f=σd(t(i),t(i+1)).20.评价函数eval(vi)：用来对种群中的每个染色体vi设定一个概率，以使该染色体被选中21.的可能性与其种群中其它染色体的适应性成比例，既通过轮盘赌，适应性强的染色体被22.选择产生后台的机会要大，设alpha∈(0,1)，本文定义基于序的评价函数为eval(vi)=al23.pha*(1-alpha).^(i-1) 。
clearclca 0.99温度衰减函数的参数t0 97初始温度tf 3终⽌温度t t0;Markov_length 10000Markov链长度load data.txt128 x(:);228 y(:);7040;x, y];data;coordinates [1565.0575.0225.0185.03345.0750.0;4945.0685.05845.0655.06880.0660.0;725.0230.08525.01000.09580.01175.0;10650.01130.0111605.0620.0121220.0580.0;131465.0200.0141530.0 5.015845.0680.0;16725.0370.017145.0665.018415.0635.0;19510.0875.020560.0365.021300.0465.0;22520.0585.023480.0415.024835.0625.0;25975.0580.0261215.0245.0271320.0315.0;281250.0400.029660.0180.030410.0250.0;31420.0555.032575.0665.0331150.01160.0;34700.0580.035685.0595.036685.0610.0;37770.0610.038795.0645.039720.0635.0;40760.0650.041475.0960.04295.0260.0;43875.0920.044700.0500.045555.0815.0;46830.0485.0471170.065.048830.0610.0;49605.0625.050595.0360.0511340.0725.0;521740.0245.0;];coordinates(:,1 [];amount 1城市的数⽬通过向量化的⽅法计算距离矩阵dist_matrix zeros(amount,amount);coor_x_tmp1 11,amount);coor_x_tmp2 ';21,amount);coor_y_tmp2 ';22); sol_new 1产⽣初始解，sol_new是每次产⽣的新解sol_current sol_current是当前解sol_best sol_best是冷却中的最好解E_current E_current是当前解对应的回路距离E_best E_best是最优解p 1;rand('state', sum(clock));for110000sol_current [randperm(amount)];E_current 0;for11)E_current 1));endif E_bestsol_best sol_current;E_best E_current;endendwhile tffor1Markov链长度产⽣随机扰动if0.5)两交换ind1 0;ind2 0;while ind2)ind1 amount);ind2 amount);endtmp1 sol_new(ind1);sol_new(ind1) sol_new(ind2);sol_new(ind2) tmp1;else三交换ind13));ind sort(ind);sol_new 11112131231:end]); end检查是否满⾜约束计算⽬标函数值（即内能）E_new 0;for11)E_new 1));end再算上从最后⼀个城市到第⼀个城市的距离E_new 1));if E_currentE_current E_new;sol_current sol_new;if E_bestE_best E_new;sol_best sol_new;endelse若新解的⽬标函数值⼤于当前解，则仅以⼀定概率接受新解if t)E_current E_new;sol_current sol_new;elsesol_new sol_current;endendendt 控制参数t（温度）减少为原来的a倍endE_best 1));disp('最优解为:');disp(sol_best);disp('最短距离:');disp(E_best);data1 2 );for1:length(sol_best)data1(i, :) 1,i), :);enddata1 11),:)];figureplot(coordinates(:,1', coordinates(:,2)''*k'1', data1(:, 2)''r');title( [ '近似最短路径如下，路程为'clc;clear;close all;coordinates [225.0185.03345.0750.0;4945.0685.05845.0655.06880.0660.0;725.0230.08525.01000.09580.01175.0;10650.01130.0111605.0620.0121220.0580.0;131465.0200.0141530.0 5.015845.0680.0;16725.0370.017145.0665.018415.0635.0;19510.0875.020560.0365.021300.0465.0;22520.0585.023480.0415.024835.0625.0;25975.0580.0261215.0245.0271320.0315.0;281250.0400.029660.0180.030410.0250.0;31420.0555.032575.0665.0331150.01160.0;34700.0580.035685.0595.036685.0610.0;37770.0610.038795.0645.039720.0635.0;40760.0650.041475.0960.04295.0260.0; 43875.0920.044700.0500.045555.0815.0; 46830.0485.0471170.065.048830.0610.0; 49605.0625.050595.0360.0511340.0725.0; 521740.0245.0;];coordinates(:,1 [];data coordinates;读取数据load data.txt;128 x(:);228 y(:);x 1);y 2);start 565.0575.0];data [start; data;start];[start; x, y;start];180;计算距离的邻接表count 1));d zeros(count);for11for1:count1122))...22));d(i,j)20.5 ;6370acos(temp);endendd '; % 对称 i到j==j到iS0存储初值Sum存储总距离rand('state', sum(clock));求⼀个较为优化的解，作为初值for110000S 112), count];temp 0;for11temp 1));endif SumS0 S;Sum temp;endende 0.140终⽌温度L 2000000最⼤迭代次数at 0.999999降温系数T 2初温退⽕过程for1:L产⽣新解c 1112));c sort(c);c1 12);if1c1 1;endif1c2 1;end计算代价函数值df 11...(d(S0(c111)));接受准则if0S0 1111:count)];Sum df;elseif exp(1)S0 1111:count)];Sum df;endT at;if ebreak;endenddata1 2, count);[start; x, y; start];for1:countdata1(:, i) 1';endfigureplot(x, y, 'o'12'r');title( [ '近似最短路径如下，路程为' , num2str( Sum ) ] ) ; disp(Sum);S0。
摘要：TSP 是一个典型的NPC 问题。
本文首先介绍旅行商问题和粒子群优化算法的基本概念。
然后构造一种基于交换子和交换序[1]概念的粒子群优化算法，通过控制学习因子1c 和2c 、最大速度max V ，尝试求解旅行商问题。
本文以中国31个省会城市为例，通过MATLAB 编程实施对旅行商问题的求解,得到了一定优化程度的路径，是粒子群优化算法在TSP 问题中运用的一次大胆尝试。
关键字：TSP 问题；粒子群优化算法；MATLAB ；中国31个城市TSP 。
粒子群优化算法求解旅行商问题1.引言1.旅行商问题的概述旅行商问题，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问题是一个组合优化问题，其描述非常简单，但最优化求解非常困难，若用穷举法搜索，对N个城市需要考虑N！种情况并两两对比，找出最优，其算法复杂性呈指数增长，即所谓“指数爆炸”。
所以，寻求和研究TSP问题的有效启发式算法，是问题的关键。
2.粒子群优化算法的概述粒子群优化算法(Particle Swarm optimization,PSO)又翻译为粒子群算法、微粒群算法、或微粒群优化算法。
是通过模拟鸟群觅食行为而发展起来的一种基于群体协作的随机搜索算法。
通常认为它是群集智能(Swarm intelligence, SI)的一种。
它可以被纳入多主体优化系统 (Multiagent Optimization System, MAOS). 粒子群优化算法是由Eberhart博士和Kennedy 博士发明。
PSO模拟鸟群的捕食行为。
一群鸟在随机搜索食物，在这个区域里只有一块食物。
所有的鸟都不知道食物在那里。
%多旅行商问题的matlab程序function varargout = mtspf_gaxy,dmat,salesmen,min_tour,pop_size,num_iter,show_prog,show_res % MTSPF_GA Fixed Multiple Traveling Salesmen Problem M-TSP Genetic Algorithm GA% Finds a near optimal solution to a variation of the M-TSP by setting% up a GA to search for the shortest route least distance needed for% each salesman to travel from the start location to individual cities% and back to the original starting place%% Summary:% 1. Each salesman starts at the first point, and ends at the first% point, but travels to a unique set of cities in between% 2. Except for the first, each city is visited by exactly one salesman%% Note: The Fixed Start/End location is taken to be the first XY point%% 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% SALESMEN scalar integer is the number of salesmen to visit the cities% MIN_TOUR scalar integer is the minimum tour length for any of the% salesmen, NOT including the start/end point% POP_SIZE scalar integer is the size of the population should be divisible by 8% NUM_ITER scalar integer is the number of desired iterations for the algorithm to run% SHOW_PROG scalar logical shows the GA progress if true% SHOW_RES scalar logical shows the GA results if true%% Output:% OPT_RTE integer array is the best route found by the algorithm% OPT_BRK integer array is the list of route break points these specify the indices% into the route used to obtain the individual salesman routes% MIN_DIST 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 4 2 8 10 3 7, brks = 3 7% Taken together, these represent the solution 1 5 6 9 11 4 2 8 11 10 3 7 1,% which designates the routes for the 3 salesmen as follows:% . Salesman 1 travels from city 1 to 5 to 6 to 9 and back to 1% . Salesman 2 travels from city 1 to 4 to 2 to 8 and back to 1% . Salesman 3 travels from city 1 to 10 to 3 to 7 and back to 1%% 2D Example:% n = 35;% xy = 10randn,2;% salesmen = 5;% min_tour = 3;% pop_size = 80;% num_iter = 5e3;% a = meshgrid1:n;% dmat = reshapesqrtsumxya,:-xya',:.^2,2,n,n;% opt_rte,opt_brk,min_dist = mtspf_gaxy,dmat,salesmen,min_tour, ... % pop_size,num_iter,1,1;%% 3D Example:% n = 35;% xyz = 10randn,3;% salesmen = 5;% min_tour = 3;% pop_size = 80;% num_iter = 5e3;% a = meshgrid1:n;% dmat = reshapesqrtsumxyza,:-xyza',:.^2,2,n,n;% opt_rte,opt_brk,min_dist = mtspf_gaxyz,dmat,salesmen,min_tour, ... % pop_size,num_iter,1,1;%% See also: mtsp_ga, mtspo_ga, mtspof_ga, mtspofs_ga, mtspv_ga, distmat %% Author: Joseph Kirk% Email: jdkirk630gmail% Release: 1.3% Release Date: 6/2/09% Process Inputs and Initialize Defaultsnargs = 8;for k = nargin:nargs-1switch kcase 0xy = 10rand40,2;case 1N = sizexy,1;a = meshgrid1:N;dmat = reshapesqrtsumxya,:-xya',:.^2,2,N,N;case 2salesmen = 5;case 3min_tour = 2;case 4pop_size = 80;case 5num_iter = 5e3;case 6show_prog = 1;case 7show_res = 1;otherwiseendend% Verify InputsN,dims = sizexy;nr,nc = sizedmat;if N ~= nr || N ~= ncerror'Invalid XY or DMAT inputs'endn = N - 1; % Separate Start/End City% Sanity Checkssalesmen = max1,minn,roundrealsalesmen1;min_tour = max1,minfloorn/salesmen,roundrealmin_tour1; pop_size = max8,8ceilpop_size1/8;num_iter = max1,roundrealnum_iter1;show_prog = logicalshow_prog1;show_res = logicalshow_res1;% Initializations for Route Break Point Selectionnum_brks = salesmen-1;dof = n - min_toursalesmen; % degrees of freedom addto = ones1,dof+1;for k = 2:num_brksaddto = cumsumaddto;endcum_prob = cumsumaddto/sumaddto;% Initialize the Populationspop_rte = zerospop_size,n; % population of routes pop_brk = zerospop_size,num_brks; % population of breaks for k = 1:pop_sizepop_rtek,: = randpermn+1;pop_brkk,: = randbreaks;end% Select the Colors for the Plotted Routesclr = 1 0 0; 0 0 1; 0.67 0 1; 0 1 0; 1 0.5 0;if salesmen > 5clr = hsvsalesmen;end% Run the GAglobal_min = Inf;total_dist = zeros1,pop_size;dist_history = zeros1,num_iter;tmp_pop_rte = zeros8,n;tmp_pop_brk = zeros8,num_brks;new_pop_rte = zerospop_size,n;new_pop_brk = zerospop_size,num_brks;if show_progpfig = figure'Name','MTSPF_GA | Current Best Solution','Numbertitle','off'; endfor iter = 1:num_iter% Evaluate Members of the Populationfor p = 1:pop_sized = 0;p_rte = pop_rtep,:;p_brk = pop_brkp,:;rng = 1 p_brk+1;p_brk n';for s = 1:salesmend = d + dmat1,p_rterngs,1; % Add Start Distancefor k = rngs,1:rngs,2-1d = d + dmatp_rtek,p_rtek+1;endd = d + dmatp_rterngs,2,1; % Add End Distanceendtotal_distp = d;end% Find the Best Route in the Populationmin_dist,index = mintotal_dist;dist_historyiter = min_dist;if min_dist < global_minglobal_min = min_dist;opt_rte = pop_rteindex,:;opt_brk = pop_brkindex,:;rng = 1 opt_brk+1;opt_brk n';if show_prog% Plot the Best Routefigurepfig;for s = 1:salesmenrte = 1 opt_rterngs,1:rngs,2 1;if dims == 3, plot3xyrte,1,xyrte,2,xyrte,3,'.-','Color',clrs,:;else plotxyrte,1,xyrte,2,'.-','Color',clrs,:; endtitlesprintf'Total Distance = %1.4f, Iteration = %d',min_dist,iter;hold onendif dims == 3, plot3xy1,1,xy1,2,xy1,3,'ko';else plotxy1,1,xy1,2,'ko'; endhold offendend% Genetic Algorithm Operatorsrand_grouping = randpermpop_size;for p = 8:8:pop_sizertes = pop_rterand_groupingp-7:p,:;brks = pop_brkrand_groupingp-7:p,:;dists = total_distrand_groupingp-7:p;ignore,idx = mindists;best_of_8_rte = rtesidx,:;best_of_8_brk = brksidx,:;rte_ins_pts = sortceilnrand1,2;I = rte_ins_pts1;J = rte_ins_pts2;for k = 1:8 % Generate New Solutionstmp_pop_rtek,: = best_of_8_rte;tmp_pop_brkk,: = best_of_8_brk;switch kcase 2 % Fliptmp_pop_rtek,I:J = fliplrtmp_pop_rtek,I:J;case 3 % Swaptmp_pop_rtek,I J = tmp_pop_rtek,J I;case 4 % Slidetmp_pop_rtek,I:J = tmp_pop_rtek,I+1:J I;case 5 % Modify Breakstmp_pop_brkk,: = randbreaks;case 6 % Flip, Modify Breakstmp_pop_rtek,I:J = fliplrtmp_pop_rtek,I:J;tmp_pop_brkk,: = randbreaks;case 7 % Swap, Modify Breakstmp_pop_rtek,I J = tmp_pop_rtek,J I;tmp_pop_brkk,: = randbreaks;case 8 % Slide, Modify Breakstmp_pop_rtek,I:J = tmp_pop_rtek,I+1:J I;tmp_pop_brkk,: = randbreaks;otherwise % Do Nothingendendnew_pop_rtep-7:p,: = tmp_pop_rte;new_pop_brkp-7:p,: = tmp_pop_brk;endpop_rte = new_pop_rte;pop_brk = new_pop_brk;endif show_res% Plotsfigure'Name','MTSPF_GA | Results','Numbertitle','off';subplot2,2,1;if dims == 3, plot3xy:,1,xy:,2,xy:,3,'k.';else plotxy:,1,xy:,2,'k.'; endtitle'City Locations';subplot2,2,2;imagescdmat1 opt_rte,1 opt_rte;title'Distance Matrix';subplot2,2,3;rng = 1 opt_brk+1;opt_brk n';for s = 1:salesmenrte = 1 opt_rterngs,1:rngs,2 1;if dims == 3, plot3xyrte,1,xyrte,2,xyrte,3,'.-','Color',clrs,:;else plotxyrte,1,xyrte,2,'.-','Color',clrs,:; endtitlesprintf'Total Distance = %1.4f',min_dist;hold on;endif dims == 3, plot3xy1,1,xy1,2,xy1,3,'ko';else plotxy1,1,xy1,2,'ko'; endsubplot2,2,4;plotdist_history,'b','LineWidth',2;title'Best Solution History';setgca,'XLim',0 num_iter+1,'YLim',0 1.1max1 dist_history; end% Return Outputsif nargoutvarargout{1} = opt_rte;varargout{2} = opt_brk;varargout{3} = min_dist;end% Generate Random Set of Break Pointsfunction breaks = randbreaksif min_tour == 1 % No Constraints on Breakstmp_brks = randpermn-1;breaks = sorttmp_brks1:num_brks;else % Force Breaks to be at Least the Minimum Tour Length num_adjust = findrand < cum_prob,1-1;spaces = ceilnum_brksrand1,num_adjust;adjust = zeros1,num_brks;for kk = 1:num_brksadjustkk = sumspaces == kk;endbreaks = min_tour1:num_brks + cumsumadjust;endendend。
MATLAB关于旅⾏商问题遗传算法的研究基于遗传算法对TSP问题的研究摘要：作为⼀种模拟⽣物⾃然遗传与进化过程的优化⽅法,遗传算法(GA)因其具有隐并⾏性、不需⽬标函数可微等特点,常被⽤于解决⼀些传统优化⽅法难以解决的问题。
旅⾏商问题(TSP)是典型的NP难题组合优化问题之⼀,且被⼴泛应⽤于许多领域,所以研究遗传算法求解TSP具有重要的理论意义和应⽤价值。
关键字：遗传算法旅⾏商问题Abstract：Genetic algorithm(GA) which has the characteristic of latent parallelism, non-differentiability of objective function and so on, as a optimization method of simulating the process of natural biotic inherit and evolution, is used to solve some problems which are difficult to solve by the traditional optimization method. Travel salesman problem(TSP) is a typical NF s combination and optimization problem, and is widely used in many fields. So the genetic algorithm to solve TSP has important theoretical significance and application value.Keywords：genetic algorithm TSP⼀、引⾔在过去,⼈们往往只能够处理⼀些简单的问题，对于⼤型复杂系统的优化和⾃适应仍然⽆能为⼒。
用MATLAB实现中国旅行商问题的求解
李明海;邢桂华
【期刊名称】《网络新媒体技术》
【年(卷),期】2004(025)002
【摘要】本文利用遗传算法的全局搜索能力进行组合优化问题求解,针对中国旅行商问题(CTSP),设计一种大比例的优秀个体保护的大变异遗传算法,并使用MATLAB 语言进行了实际的编程求解,编程中的各个模块分别实现了复制、交叉、变异等关键环节.用编制的程序快速求解出了满意的结果,用本文设计的大变异遗传算法的思路和编制程序是正确的,而且本文算法的求解速度是非常快的.
【总页数】5页(P218-222)
【作者】李明海;邢桂华
【作者单位】南京师范大学化学与环境科学学院,南京,210097;南京师范大学数学与计算机学院,南京,210097
【正文语种】中文
【中图分类】TP3
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