运用Matlab免疫算法解决31城市旅行商问题的步骤及代码详解

一、简介

旅行商问题（TSP）是一个经典的优化问题，其目标是找到一条最短的路径，使路径经过给定的城市一次且仅一次。TSP在许多领域都有着广泛的应用，例如物流配送、车辆调度、电路板布线等。

免疫算法（AIA）是一种新型的智能算法，其灵感来源于生物免疫系统。AIA模拟了生物免疫系统中抗体和抗原之间的相互作用，从而来解决优化问题。AIA具有鲁棒性强、收敛速度快、易于并行化等优点，因此近年来在TSP领域得到了广泛的应用。

二、Matlab免疫算法求解TSP的步骤

1. 初始化种群

首先，我们需要初始化一个种群，即一组候选解。种群中的每个个体都表示一条可能的路径。我们可以使用随机生成或启发式方法来初始化种群。

2. 计算个体的适应度

接下来，我们需要计算每个个体的适应度。适应度函数衡量个体解决TSP问题的优劣程度。通常，适应度函数取路径长度的倒数。

3. 选择操作

选择操作根据个体的适应度来选择出优秀的个体，这些个体会进入下一代种群。我们可以使用各种选择算法，例如轮盘赌选择、锦标赛选择、随机选择等。

4. 交叉操作

交叉操作将两个父代个体结合起来，生成一个新的子代个体。我们可以使用各种交叉算法，例如单点交叉、双点交叉、均匀交叉等。

5. 变异操作

变异操作对子代个体进行随机扰动，以增加种群的多样性。我们可以使用各种变异算法，例如插入变异、交换变异、反转变异等。

6. 重复步骤2-5

重复步骤2-5，直到满足终止条件。终止条件可以是达到最大迭代次数、找到最优解或适应度达到某个阈值等。

7. 输出结果

最后，我们将输出最优解和最优解对应的路径。

三、Matlab免疫算法求解TSP的代码

% 初始化参数
numCities = 31;
maxIterations = 1000;
populationSize = 100;
crossoverProbability = 0.8;
mutationProbability = 0.1;

% 初始化种群
population = InitializePopulation(populationSize, numCities);

% 计算个体的适应度
fitness = CalculateFitness(population, numCities);

% 重复步骤2-5
for iteration = 1:maxIterations
    % 选择操作
    selectedParents = SelectParents(population, fitness);
    
    % 交叉操作
    newPopulation = Crossover(selectedParents, crossoverProbability);
    
    % 变异操作
    newPopulation = Mutate(newPopulation, mutationProbability);
    
    % 计算新种群的适应度
    fitness = CalculateFitness(newPopulation, numCities);
    
    % 更新种群
    population = newPopulation;
end

% 输出结果
bestIndividual = GetBestIndividual(population);
bestTour = GetTourFromIndividual(bestIndividual, numCities);
disp(['最优路径长度：', num2str(bestTourLength)]);
disp(['最优路径：', num2str(bestTour)]);

% 初始化种群
function population = InitializePopulation(populationSize, numCities)
    population = zeros(populationSize, numCities);
    for i = 1:populationSize
        population(i, :) = randperm(numCities);
    end
end

% 计算个体的适应度
function fitness = CalculateFitness(population, numCities)
    fitness = zeros(size(population, 1), 1);
    for i = 1:size(population, 1)
        tourLength = CalculateTourLength(population(i, :), numCities);
        fitness(i) = 1 / tourLength;
    end
end

% 选择操作
function selectedParents = SelectParents(population, fitness)
    selectedParents = zeros(size(population, 1), 2);
    for i = 1:size(population, 1)
        parent1 = TournamentSelect(population, fitness);
        parent2 = TournamentSelect(population, fitness);
        selectedParents(i, :) = [parent1, parent2];
    end
end

% 交叉操作
function newPopulation = Crossover(selectedParents, crossoverProbability)
    newPopulation = zeros(size(selectedParents, 1), size(selectedParents, 2));
    for i = 1:size(selectedParents, 1)
        parent1 = selectedParents(i, 1);
        parent2 = selectedParents(i, 2);
        if rand() < crossoverProbability
            newPopulation(i, :) = OrderedCrossover(parent1, parent2);
        else
            newPopulation(i, :) = parent1;
        end
    end
end

% 变异操作
function newPopulation = Mutate(population, mutationProbability)
    newPopulation = population;
    for i = 1:size(population, 1)
        if rand() < mutationProbability
            newPopulation(i, :) = SwapMutation(population(i, :));
        end
    end
end

% 获取最优个体
function bestIndividual = GetBestIndividual(population)
    [~, index] = max(population(:, end));
    bestIndividual = population(index, :);
end

% 获取路径
function tour = GetTourFromIndividual(individual, numCities)
    tour = zeros(1, numCities);
    for i = 1:numCities
        tour(i) = individual(i);
    end
end

% 计算路径长度
function tourLength = CalculateTourLength(tour, numCities)
    tourLength = 0;
    for i = 1:numCities-1
        tourLength = tourLength + Distance(tour(i), tour(i+1));
    end
    tourLength = tourLength + Distance(tour(numCities), tour(1));
end

% 计算两个城市之间的距离
function distance = Distance(city1, city2)
    distance = sqrt((city1(1) - city2(1))^2 + (city1(2) - city2(2))^2);
end

% 锦标赛选择
function selectedIndividual = TournamentSelect(population, fitness)
    selectedIndividuals = zeros(size(population, 1), 2);
    for i = 1:size(population, 1)
        index1 = randi(size(population, 1));
        index2 = randi(size(population, 1));
        if fitness(index1) > fitness(index2)
            selectedIndividuals(i, 1) = index1;
        else
            selectedIndividuals(i, 1) = index2;
        end
    end
    for i = 1:size(population, 1)
        index1 = randi(size(population, 1));
        index2 = randi(size(population, 1));
        if fitness(index1) > fitness(index2)
            selectedIndividuals(i, 2) = index1;
        else
            selectedIndividuals(i, 2) = index2;
        end
    end
    selectedIndividual = selectedIndividuals(randi(size(selectedIndividuals, 1)), :);
end

% 有序交叉
function offspring = OrderedCrossover(parent1, parent2)
    offspring = zeros(1, size(parent1, 2));
    crossoverPoints = sort(randperm(size(parent1, 2), 2));
    offspring(crossoverPoints(1):crossoverPoints(2)) = parent1(crossoverPoints(1):crossoverPoints(2));
    remainingGenes = setdiff(parent2, offspring);
    offspring([1:crossoverPoints(1)-1, crossoverPoints(2)+1:end]) = remainingGenes;
end

% 交换变异
function offspring = SwapMutation(parent)
    offspring = parent;
    mutationPoints = randi(size(parent, 2), 1, 2);
    temp = offspring(mutationPoints(1));
    offspring(mutationPoints(1)) = offspring(mutationPoints(2));