MTSP问题

问题描述：m个旅行商去旅游 n个城市，规定都必须从同一个出发点出发，而且返回原出发点，需要将所有的城市遍历完毕，每个城市只能游历一次，但是为了路径最短可以路过这个城市多次。这个就是多旅行商问题。是在TSP问题的基础上进行了扩展。

问题解决方案：

明确M-TSP与TSP的区别在哪里？

　　TSP指的是单个旅行商遍历一圈，将所有城市旅行一遍，

　　 MTSP指的是将城市群划分成M个组，每组采用TSP得到最短的旅行路线，所以问题的关键在于如何确定城市群的分组。

改良圈算法——可得到优化解但不是最优解

先得到一个哈密顿圈，然后修改权值路径，得到新的哈密顿圈，如果新的哈密顿圈路径权值小于初始的圈，就替换掉原来的圈，直到路径权值最小。

用于得到一个较好的初始化种群；然后用到遗传算法中

代码：

clear,clc
load sj.txt;
x=sj(:,1:2:8);x=x(:);%将25*4矩阵变为100*1矩阵
y=sj(:,2:2:8);y=y(:);
sj=[x,y];
d1=[70,40];%初始起飞基地
sj0=[d1;sj;d1];%102个基地
%计算距离矩阵d
sj=sj0*pi/180;
d=zeros(102);
for i=1:101
    for j=i+1:102
        temp=cos(sj(i,1)-sj(j,1))*cos(sj(i,2))*cos(sj(j,2))+sin(sj(i,2))*sin(sj(j,2));
        d(i,j)=6370*acos(temp);
    end
end
d=d+d';%对称矩阵
L=102;w=50;dai=100;
%通过改良圈算法选取优良父代A
for k=1:w
    c=randperm(100);%把1到100这些数随机打乱得到的一个数字序列
    c1=[1,c+1,102];%染色体
    flag=1;
    while flag>0
        flag=0;
        for m=1:L-3
            for n=m+2:L-1
                if(d(c1(m),c1(n))+d(c1(m+1),c1(n+1))<d(c1(m),c1(m+1))+d(c1(n),c1(n+1)))
                    flag=1;
                    c1(m+1:n)=c1(n:-1:m+1);
                end
            end
        end
     end
    J(k,c1)=1:102;
end
J=J/102;
J(:,1)=0;J(:,102)=1;
rand('state',sum(clock));
%遗传算法实现过程
A=J;
for k=1:dai  %产生 0～1 间随机数列进行编码
    %交配产生子代 B
    B=A;
    c=randperm(w);  %产生1~50随机数
    for i=1:2:w     %从1到50依次两两配对，即i与(i+1)配对
        F=2+floor(100*rand(1)); %随机产生交叉点
        temp=B(c(i),F:102);
        B(c(i),F:102)=B(c(i+1),F:102);
        B(c(i+1),F:102)=temp;     %交叉更换完毕
    end
    %变异产生子代 C
    by=find(rand(1,w)<0.1); %返回随机数<0.1的位置
    if length(by)==0     %如果上一步找不到，则随机产生一个变异点
        by=floor(w*rand(1))+1;
    end
    C=A(by,:);
    L3=length(by);
    for j=1:L3
        bw=2+floor(100*rand(1,3));    %随机选取三个整数
        bw=sort(bw);    %满足1<u<v<w<102
        C(j,:)=C(j,[1:bw(1)-1,bw(2)+1:bw(3),bw(1):bw(2),bw(3)+1:102]); %把u，v之间（包括u和v）的基因段插到w后面
    end
    G=[A;B;C];    %获得父代、交叉子代、变异子代合集G
    %在父代和子代中选择优良品种作为新的父代
    TL=size(G,1);
    [dd,IX]=sort(G,2);%dd为升序后的G，IX为索引
    temp(1:TL)=0;
    for j=1:TL
        for i=1:101
            temp(j)=temp(j)+d(IX(j,i),IX(j,i+1));   %按照新的序列重新获得距离矩阵
        end
    end
    [DZ,IZ]=sort(temp);
    A=G(IZ(1:w),:); %选择目标函数值最小的w个个体进化到下一代
end
path=IX(IZ(1),:) ;
long=DZ(1) ;
%toc
xx=sj0(path,1);
yy=sj0(path,2);
plot(xx,yy,'-o');
---------------------
作者：越溪
来源：CSDN
原文：https://blog.csdn.net/longxinghaofeng/article/details/77504212
版权声明：本文为博主原创文章，转载请附上博文链接！

　　经典的求解MTSP问题的（起始点为同一点）的matlab代码为：

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
% Email: jdkirk630@gmail.com
% Release: 1.3
% Release Date: 6/2/09

% Process Inputs and Initialize Defaults
nargs = 8;
for k = nargin:nargs-1
    switch k
        case 0
            xy = 10*rand(40,2);
        case 1
            N = size(xy,1);
            a = meshgrid(1:N);
            dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),N,N);
        case 2
            salesmen = 5;
        case 3
            min_tour = 5;
        case 4
            pop_size = 160;
        case 5
            num_iter = 5e3;
        case 6
            show_prog = 1;
        case 7
            show_res = 1;
        otherwise
    end
end

% Verify Inputs
[N,dims] = size(xy);
[nr,nc] = size(dmat);
if N ~= nr || N ~= nc
    error('Invalid XY or DMAT inputs!')
end
n = N - 1; % Separate Start/End City

% Sanity Checks
salesmen = 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 Selection
num_brks = salesmen-1;
dof = n - min_tour*salesmen;          % degrees of freedom
addto = ones(1,dof+1);
for k = 2:num_brks
    addto = cumsum(addto);
end
cum_prob = cumsum(addto)/sum(addto);

% Initialize the Populations
pop_rte = zeros(pop_size,n);          % population of routes
pop_brk = zeros(pop_size,num_brks);   % population of breaks
for k = 1:pop_size
    pop_rte(k,:) = randperm(n)+1;
    pop_brk(k,:) = randbreaks();
end

% Select the Colors for the Plotted Routes
clr = [1 0 0; 0 0 1; 0.67 0 1; 0 1 0; 1 0.5 0];
if salesmen > 5
    clr = hsv(salesmen);
end

% Run the GA
global_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_prog
    pfig = figure('Name','MTSPF_GA | Current Best Solution','Numbertitle','off');
end
for iter = 1:num_iter
    % Evaluate Members of the Population
    for p = 1:pop_size
        d = 0;
        p_rte = pop_rte(p,:);
        p_brk = pop_brk(p,:);
        rng = [[1 p_brk+1];[p_brk n]]';
        for s = 1:salesmen
            d = d + dmat(1,p_rte(rng(s,1))); % Add Start Distance
            for k = rng(s,1):rng(s,2)-1
                d = d + dmat(p_rte(k),p_rte(k+1));
            end
            d = d + dmat(p_rte(rng(s,2)),1); % Add End Distance
        end
        total_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_min
        global_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 Route
            figure(pfig);
            for s = 1:salesmen
                rte = [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,:)); end
                title(sprintf('Total Distance = %1.4f, Iteration = %d',min_dist,iter));
                hold on
            end
            if dims == 3, plot3(xy(1,1),xy(1,2),xy(1,3),'ko');
            else plot(xy(1,1),xy(1,2),'ko'); end
            hold off
        end
    end

    % Genetic Algorithm Operators
    rand_grouping = randperm(pop_size);
    for p = 8:8:pop_size
        rtes = 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 Solutions
            tmp_pop_rte(k,:) = best_of_8_rte;
            tmp_pop_brk(k,:) = best_of_8_brk;
            switch k
                case 2 % Flip
                    tmp_pop_rte(k,I:J) = fliplr(tmp_pop_rte(k,I:J));
                case 3 % Swap
                    tmp_pop_rte(k,[I J]) = tmp_pop_rte(k,[J I]);
                case 4 % Slide
                    tmp_pop_rte(k,I:J) = tmp_pop_rte(k,[I+1:J I]);
                case 5 % Modify Breaks
                    tmp_pop_brk(k,:) = randbreaks();
                case 6 % Flip, Modify Breaks
                    tmp_pop_rte(k,I:J) = fliplr(tmp_pop_rte(k,I:J));
                    tmp_pop_brk(k,:) = randbreaks();
                case 7 % Swap, Modify Breaks
                    tmp_pop_rte(k,[I J]) = tmp_pop_rte(k,[J I]);
                    tmp_pop_brk(k,:) = randbreaks();
                case 8 % Slide, Modify Breaks
                    tmp_pop_rte(k,I:J) = tmp_pop_rte(k,[I+1:J I]);
                    tmp_pop_brk(k,:) = randbreaks();
                otherwise % Do Nothing
            end
        end
        new_pop_rte(p-7:p,:) = tmp_pop_rte;
        new_pop_brk(p-7:p,:) = tmp_pop_brk;
    end
    pop_rte = new_pop_rte;
    pop_brk = new_pop_brk;
end

if show_res
    % Plots
    figure('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.'); end
    title(' 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:salesmen
        rte = [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,:)); end
        title(sprintf('Total time = %1.4f',min_dist));
        hold on;
    end
    if dims == 3, plot3(xy(1,1),xy(1,2),xy(1,3),'ko');
    else plot(xy(1,1),xy(1,2),'ko'); end
    subplot(2,2,4);
plot(dist_history,'b','LineWidth',2);
    title('Best Solution History');
    set(gca,'XLim',[0 num_iter+1],'YLim',[0 1.1*max([1 dist_history])]);
end

% Return Outputs
if nargout
    varargout{1} = opt_rte;
    varargout{2} = opt_brk;
    varargout{3} = min_dist;
end

    % Generate Random Set of Break Points
    function breaks = randbreaks()
        if min_tour == 1 % No Constraints on Breaks
            tmp_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_brks
                adjust(kk) = sum(spaces == kk);
            end
            breaks = min_tour*(1:num_brks) + cumsum(adjust);
        end
    end
end