蚁群算法（Ant Colony Optimization, ACO）是一种启发式搜索算法，用于寻找优化路径问题的近似解，如旅行商问题（TSP）、调度问题等。这里我将给出一个简单的旅行商问题（TSP）的蚁群算法实现示例。
在TSP中，我们的目标是找到一条最短的路径，使得一个旅行商可以访问n个城市一次并返回起点，且总行程最短。
以下是一个简化的Matlab实现：
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| | function aco_tsp(numCities, numAnts, numIters, alpha, beta, rho, Q) |
| | % 初始化参数 |
| | numCities = numCities; % 城市数量 |
| | numAnts = numAnts; % 蚂蚁数量 |
| | numIters = numIters; % 迭代次数 |
| | alpha = alpha; % 信息素重要程度因子 |
| | beta = beta; % 启发式信息重要程度因子 |
| | rho = rho; % 信息素挥发系数 |
| | Q = Q; % 信息素强度 |
| | |
| | % 随机生成城市坐标 |
| | cityLocations = rand(numCities, 2) * 100; |
| | |
| | % 初始化距离矩阵 |
| | D = squareform(pdist(cityLocations)); |
| | |
| | % 初始化信息素矩阵 |
| | pheromone = ones(numCities, numCities); |
| | |
| | % 开始迭代 |
| | for iter = 1:numIters |
| | % 每只蚂蚁的路径 |
| | allPaths = cell(numAnts, 1); |
| | % 每只蚂蚁的总距离 |
| | allDistances = zeros(numAnts, 1); |
| | |
| | % 蚂蚁构建路径 |
| | for ant = 1:numAnts |
| | % 随机选择起始城市 |
| | startCity = randi(numCities); |
| | path = [startCity]; |
| | unvisited = setdiff(1:numCities, startCity); |
| | |
| | % 构建路径 |
| | while ~isempty(unvisited) |
| | probabilities = zeros(1, length(unvisited)); |
| | for j = unvisited |
| | % 计算启发式信息 |
| | heuristic = 1 / D(path(end), j); |
| | % 计算选择概率 |
| | probabilities(j) = (pheromone(path(end), j) .^ alpha) * (heuristic .^ beta); |
| | end |
| | probabilities = probabilities / sum(probabilities); |
| | % 轮盘赌选择下一个城市 |
| | nextCity = unvisited(randsrc(1, 1, [unvisited; probabilities(:)]')); |
| | path = [path, nextCity]; |
| | unvisited(unvisited == nextCity) = []; |
| | end |
| | |
| | % 计算路径总距离 |
| | allDistances(ant) = sum(D(sub2ind(size(D), path(1:end-1), path(2:end)))); |
| | allPaths{ant} = path; |
| | end |
| | |
| | % 更新信息素 |
| | deltaPheromone = zeros(size(pheromone)); |
| | for ant = 1:numAnts |
| | % 只在蚂蚁走过的路径上增加信息素 |
| | for i = 1:length(allPaths{ant})-1 |
| | start = allPaths{ant}(i); |
| | end = allPaths{ant}(i+1); |
| | deltaPheromone(start, end) = deltaPheromone(start, end) + Q / allDistances(ant); |
| | end |
| | end |
| | |
| | % 信息素挥发和更新 |
| | pheromone = (1 - rho) * pheromone + deltaPheromone; |
| | |
| | % 可选：打印迭代信息 |
| | fprintf('Iteration %d: Best Distance = %.2f\n', iter, min(allDistances)); |
| | end |
| | |
| | % 找到最佳路径 |
| | [minDist, bestIdx] = min(allDistances); |
| | bestPath = allPaths{bestIdx}; |
| | disp(['Best Path: ', num2str(bestPath)]); |
| | disp(['Best Distance: ', num2str(minDist)]); |
| | end |
使用方法
可以通过调用aco_tsp函数并传入适当的参数来运行蚁群算法。例如：
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| | aco_tsp(50, 20, 100, 1, 5, 0.5, 100); |
这里设置了50个城市，20只蚂蚁，迭代100次，信息素重要程度因子为1，启发式信息重要程度因子为5，信息素挥发系数为0.5，信息素强度为100。