蚁群算法模拟退火算法贪心算法遗传算法无线可充电传感器问题

%% 蚁群算法及Matlab实现——TSP问题
% 《MATLAB数学建模方法与实践》(《MATLAB在数学建模中的应用》升级版)，北航出版社，卓金武、王鸿钧编著. 
%% 数据准备
% 清空环境变量
clear all
clc

% 程序运行计时开始
t0 = clock;

%导入数据
citys=xlsread('C题附件1.xlsx', 'B2:C31');
x=xlsread('C题附件1.xlsx','B2:B31');
y=xlsread('C题附件1.xlsx','C2:C31');

%% 计算城市间相互距离
n = size(citys,1);
D = zeros(n,n);
r=6371;
for i = 1:n
    for j = 1:n
        if i ~= j
            D(i,j) =  r*acos(cos(y(i))*cos(y(j))*cos(x(i)-x(j))+sin(y(i))*sin(y(j)));
        else
            D(i,j) = 1e-4;      %设定的对角矩阵修正值
        end
    end    
end

%% 初始化参数
m = 75;                              % 蚂蚁数量
alpha = 1;                           % 信息素重要程度因子
beta = 5;                            % 启发函数重要程度因子
vol = 0.2;                           % 信息素挥发(volatilization)因子
Q = 10;                               % 常系数
Heu_F = 1./D;                        % 启发函数(heuristic function)
Tau = ones(n,n);                     % 信息素矩阵
Table = zeros(m,n);                  % 路径记录表
iter = 1;                            % 迭代次数初值
iter_max = 100;                      % 最大迭代次数 
Route_best = zeros(iter_max,n);      % 各代最佳路径       
Length_best = zeros(iter_max,1);     % 各代最佳路径的长度  
Length_ave = zeros(iter_max,1);      % 各代路径的平均长度  
Limit_iter = 0;                      % 程序收敛时迭代次数

%% 迭代寻找最佳路径
while iter <= iter_max
    % 随机产生各个蚂蚁的起点城市
      start = zeros(m,1);
      for i = 1:m
          temp = 1;
          start(i) = temp(1);
      end
      Table(:,1) = start; 
      % 构建解空间
      citys_index = 1:n;
      % 逐个蚂蚁路径选择
      for i = 1:m
          % 逐个城市路径选择
         for j = 2:n
             tabu = Table(i,1:(j - 1));           % 已访问的城市集合(禁忌表)
             allow_index = ~ismember(citys_index,tabu);    % 参加说明1（程序底部）
             allow = citys_index(allow_index);  % 待访问的城市集合
             P = allow;
             % 计算城市间转移概率
             for k = 1:length(allow)
                 P(k) = Tau(tabu(end),allow(k))^alpha * Heu_F(tabu(end),allow(k))^beta;
             end
             P = P/sum(P);
             % 轮盘赌法选择下一个访问城市
            Pc = cumsum(P);     %参加说明2(程序底部)
            target_index = find(Pc >= rand); 
            target = allow(target_index(1));
            Table(i,j) = target;
         end
      end
      % 计算各个蚂蚁的路径距离
      Length = zeros(m,1);
      for i = 1:m
          Route = Table(i,:);
          for j = 1:(n - 1)
              Length(i) = Length(i) + D(Route(j),Route(j + 1));
          end
          Length(i) = Length(i) + D(Route(n),Route(1));
      end
      % 计算最短路径距离及平均距离
      if iter == 1
          [min_Length,min_index] = min(Length);
          Length_best(iter) = min_Length;  
          Length_ave(iter) = mean(Length);
          Route_best(iter,:) = Table(min_index,:);
          Limit_iter = 1; 
          
      else
          [min_Length,min_index] = min(Length);
          Length_best(iter) = min(Length_best(iter - 1),min_Length);
          Length_ave(iter) = mean(Length);
          if Length_best(iter) == min_Length
              Route_best(iter,:) = Table(min_index,:);
              Limit_iter = iter; 
          else
              Route_best(iter,:) = Route_best((iter-1),:);
          end
      end
      % 更新信息素
      Delta_Tau = zeros(n,n);
      % 逐个蚂蚁计算
      for i = 1:m
          % 逐个城市计算
          for j = 1:(n - 1)
              Delta_Tau(Table(i,j),Table(i,j+1)) = Delta_Tau(Table(i,j),Table(i,j+1)) + Q/Length(i);
          end
          Delta_Tau(Table(i,n),Table(i,1)) = Delta_Tau(Table(i,n),Table(i,1)) + Q/Length(i);
      end
      Tau = (1-vol) * Tau + Delta_Tau;
    % 迭代次数加1，清空路径记录表
    iter = iter + 1;
    Table = zeros(m,n);
end

%% 结果显示
[Shortest_Length,index] = min(Length_best);
Shortest_Route = Route_best(index,:);
Time_Cost=etime(clock,t0);
disp(['最短距离:' num2str(Shortest_Length)]);
disp(['最短路径:' num2str([Shortest_Route Shortest_Route(1)])]);
disp(['收敛迭代次数:' num2str(Limit_iter)]);
disp(['程序执行时间:' num2str(Time_Cost) '秒']);

%% 绘图
figure(1)
plot([citys(Shortest_Route,1);citys(Shortest_Route(1),1)],...  %三点省略符为Matlab续行符
     [citys(Shortest_Route,2);citys(Shortest_Route(1),2)],'o-');
grid on
for i = 1:size(citys,1)
    text(citys(i,1),citys(i,2),['   ' num2str(i)]);
end
text(citys(Shortest_Route(1),1),citys(Shortest_Route(1),2),'       起点');
text(citys(Shortest_Route(end),1),citys(Shortest_Route(end),2),'       终点');
xlabel('城市位置横坐标')
ylabel('城市位置纵坐标')
title(['ACA最优化路径(最短距离:' num2str(Shortest_Length) ')'])
figure(2)
plot(1:iter_max,Length_best,'b')
legend('最短距离')
xlabel('迭代次数')
ylabel('距离')
title('算法收敛轨迹')
%--------------------------------------------------------------------------
%% 程序解释或说明
% 1. ismember函数判断一个变量中的元素是否在另一个变量中出现，返回0-1矩阵；
% 2. cumsum函数用于求变量中累加元素的和，如A=[1, 2, 3, 4, 5], 那么cumsum(A)=[1, 3, 6, 10, 15]。

运行结果

最短距离:433.5578
最短路径:1 2 8 3 10 7 9 11 12 13 14 15 16 17 28 6 4 5 29 23 22 25 24 30 27 26 20 19 18 21 1
收敛迭代次数:5
程序执行时间:9.502秒

　退火算法　

clear	
	clc
    num=xlsread('C题附件1.xlsx', 'B2:C31');
    x=xlsread('C题附件1.xlsx','B2:B31');
    y=xlsread('C题附件1.xlsx','C2:C31');
	a = 0.99;	% 温度衰减函数的参数
	t0 = 97; tf = 3; t = t0;
	Markov_length = 10000;	% Markov链长度
	coordinates = [num,x,y];
	coordinates(:,1) = [];
	amount = size(coordinates,1); 	% 城市的数目
	
%% 计算城市间相互距离
n = size(num,1);
D = zeros(n,n);
r=6371;
for i = 1:n
    for j = 1:n
        if i ~= j
            D(i,j) =  r*acos(cos(y(i))*cos(y(j))*cos(x(i)-x(j))+sin(y(i))*sin(y(j)));
        else
            D(i,j) = 1e-4;      %设定的对角矩阵修正值
        end
    end    
end

	sol_new = 1:amount;         % 产生初始解
% sol_new是每次产生的新解；sol_current是当前解；sol_best是冷却中的最好解；
	E_current = inf;E_best = inf; 		% E_current是当前解对应的回路距离；
% E_new是新解的回路距离；
% E_best是最优解的
	sol_current = sol_new; sol_best = sol_new;          
	p = 1;

	while t>=tf
		for r=1:Markov_length		% Markov链长度
			% 产生随机扰动
			if (rand < 0.5)	% 随机决定是进行两交换还是三交换
				% 两交换
				ind1 = 0; ind2 = 0;
				while (ind1 == ind2)
					ind1 = ceil(rand.*amount);
					ind2 = ceil(rand.*amount);
				end
				tmp1 = sol_new(ind1);
				sol_new(ind1) = sol_new(ind2);
				sol_new(ind2) = tmp1;
			else
				% 三交换
				ind1 = 0; ind2 = 0; ind3 = 0;
				while (ind1 == ind2) || (ind1 == ind3) ...
					|| (ind2 == ind3) || (abs(ind1-ind2) == 1)
					ind1 = ceil(rand.*amount);
					ind2 = ceil(rand.*amount);
					ind3 = ceil(rand.*amount);
				end
				tmp1 = ind1;tmp2 = ind2;tmp3 = ind3;
				% 确保ind1 < ind2 < ind3
				if (ind1 < ind2) && (ind2 < ind3)
					;
				elseif (ind1 < ind3) && (ind3 < ind2)
					ind2 = tmp3;ind3 = tmp2;
				elseif (ind2 < ind1) && (ind1 < ind3)
					ind1 = tmp2;ind2 = tmp1;
				elseif (ind2 < ind3) && (ind3 < ind1) 
					ind1 = tmp2;ind2 = tmp3; ind3 = tmp1;
				elseif (ind3 < ind1) && (ind1 < ind2)
					ind1 = tmp3;ind2 = tmp1; ind3 = tmp2;
				elseif (ind3 < ind2) && (ind2 < ind1)
					ind1 = tmp3;ind2 = tmp2; ind3 = tmp1;
				end
				
				tmplist1 = sol_new((ind1+1):(ind2-1));
				sol_new((ind1+1):(ind1+ind3-ind2+1)) = ...
					sol_new((ind2):(ind3));
				sol_new((ind1+ind3-ind2+2):ind3) = ...
					tmplist1;
			end

			%检查是否满足约束
			
			% 计算目标函数值（即内能）
			E_new = 0;
			for i = 1 : (amount-1)
				E_new = E_new + ...
					D(sol_new(i),sol_new(i+1));
			end
			% 再算上从最后一个城市到第一个城市的距离
			E_new = E_new + ...
				D(sol_new(amount),sol_new(1));
			
			if E_new < E_current
				E_current = E_new;
				sol_current = sol_new;
				if E_new < E_best
% 把冷却过程中最好的解保存下来
					E_best = E_new;
					sol_best = sol_new;
				end
			else
				% 若新解的目标函数值小于当前解的，
				% 则仅以一定概率接受新解
				if rand < exp(-(E_new-E_current)./t)
					E_current = E_new;
					sol_current = sol_new;
				else	
					sol_new = sol_current;
				end
			end
		end
		t=t.*a;		% 控制参数t（温度）减少为原来的a倍
	end

	disp('最优解为：')
	disp(sol_best)
	disp('最短距离：')
	disp(E_best)

　　运行结果：

最优解为：
1 至 22 列

1 5 18 21 19 20 26 27 30 24 25 22 23 29 4 6 11 13 14 17 28 16 15 12

23 至 30 列

9 7 10 3 8 2

最短距离：
405.2805

贪心算法

n = 30 ;                                   %用于记录点数
best = 1:1:n;                              %生成一个用来存储点顺序的矩阵
handle = 1:1:n;

x=xlsread('C题附件1.xlsx','B2:B31');
y=xlsread('C题附件1.xlsx','C2:C31');
r=6371;
D = zeros(n) ;
for i = 1 : n 
    for j = 1 : n
        D(i,j) =  r*acos(cos(y(i))*cos(y(j))*cos(x(i)-x(j))+sin(y(i))*sin(y(j)));        %距离矩阵
    end
end

best(1) = 1;                             %默认起点
num = 1; 
for a = 1:(n-2)                         %需要n-2次判断
    handle(:,1)=[];                     %上一次最优点的数据裁掉
    dis = zeros(1,(n-a));               %用来存剩下各个点的距离
    for b = 1:(n-a)                     %用来获取剩下各个点的距离
        dis(b) = D (num  , handle(b));      
    end
    num1 = find( dis == min(dis) );     %得到最优点所在检索
    
    t = handle(1);                      %将最优点与最前面的点位置进行交换  
    handle(1) = handle(num1);
    handle(num1) = t;
    
    num = handle(1);                    %获取下次进行操作的数
    best(a+1) = handle(1);              %将最优点存入best数组
end

best(n) = handle(num1);                 %补上最后一个点

plot(x(best),y(best),'-+') ;            %用'+'标出点并用实线连接得到最优路径
grid on
disp(best)

　　运行结果：

最短路径：

1 至 22 列

1 2 8 10 3 4 6 11 9 7 12 13 14 17 16 15 28 5 29 23 22 24

23 至 30 列

25 20 26 30 27 19 21 18

最短距离：484.4739