TSP：遗传算法解决旅行商问题，代码获取与详细步骤

2023-11-07 23:02:29

TSP简介

旅行商问题（Traveling Salesman Problem，TSP）是一个经典的组合优化问题。它涉及一个旅行商，他需要访问一组城市，并且希望以最短的总距离完成旅行。TSP是一个NP难问题，这意味着它没有多项式时间的解决方案。但是，可以使用启发式算法来找到TSP的近似解。

遗传算法的基本原理

遗传算法是一种进化计算方法，它模拟了自然选择和遗传的机制。遗传算法从一组随机生成的解开始。然后，这些解被评估，并根据它们的适应度进行选择。适应度较高的解更有可能被选择，并且它们更有可能产生后代。

遗传算法的每个迭代都包括以下步骤：

选择：选择一组解，这些解将用于产生后代。
交叉：将两个解组合在一起，形成一个新的解。
变异：对新的解进行随机修改。
评估：评估新的解，并根据它们的适应度进行选择。

遗传算法将重复这些步骤，直到找到一个令人满意的解。

使用matlab实现遗传算法

我们可以使用matlab来实现遗传算法。首先，我们需要创建一个表示旅行商问题的类。这个类将包含城市的位置、旅行商当前的位置以及旅行商访问过的城市。

classdef TSP
    properties
        cities;
        currentCity;
        visitedCities;
    end

    methods
        function obj = TSP(cities)
            obj.cities = cities;
            obj.currentCity = 1;
            obj.visitedCities = [];
        end

        function tourLength = tourLength(obj)
            tourLength = 0;
            for i = 1:length(obj.visitedCities) - 1
                tourLength = tourLength + norm(obj.cities(obj.visitedCities(i), :) - obj.cities(obj.visitedCities(i + 1), :));
            end
            tourLength = tourLength + norm(obj.cities(obj.visitedCities(end), :) - obj.cities(obj.visitedCities(1), :));
        end

        function nextCity = selectNextCity(obj)
            unvisitedCities = setdiff(1:length(obj.cities), obj.visitedCities);
            probabilities = 1 ./ pdist2(obj.cities(obj.currentCity, :), obj.cities(unvisitedCities, :));
            probabilities = probabilities / sum(probabilities);
            nextCity = randsample(unvisitedCities, 1, true, probabilities);
        end

        function visitCity(obj, city)
            obj.visitedCities = [obj.visitedCities, city];
            obj.currentCity = city;
        end
    end
end

接下来，我们需要创建一个表示遗传算法的类。这个类将包含种群大小、突变率以及遗传算法的最大迭代次数。

classdef GeneticAlgorithm
    properties
        populationSize;
        mutationRate;
        maxIterations;
    end

    methods
        function obj = GeneticAlgorithm(populationSize, mutationRate, maxIterations)
            obj.populationSize = populationSize;
            obj.mutationRate = mutationRate;
            obj.maxIterations = maxIterations;
        end

        function bestTour = solve(obj, cities)
            population = obj.initializePopulation(cities);
            for i = 1:obj.maxIterations
                population = obj.select(population);
                population = obj.crossover(population);
                population = obj.mutate(population);
                [bestTour, bestTourLength] = obj.getBestTour(population);
                disp(['Iteration ', num2str(i), ': Best tour length = ', num2str(bestTourLength)]);
            end
        end

        function population = initializePopulation(obj, cities)
            population = [];
            for i = 1:obj.populationSize
                population = [population, TSP(cities)];
            end
        end

        function population = select(obj, population)
            fitnessValues = zeros(1, length(population));
            for i = 1:length(population)
                fitnessValues(i) = 1 / population(i).tourLength();
            end
            fitnessValues = fitnessValues / sum(fitnessValues);
            newPopulation = [];
            for i = 1:obj.populationSize
                parent1 = randsample(population, 1, true, fitnessValues);
                parent2 = randsample(population, 1, true, fitnessValues);
                newPopulation = [newPopulation, obj.crossover(parent1, parent2)];
            end
            population = newPopulation;
        end

        function population = crossover(obj, parent1, parent2)
            crossoverPoint = randi([1, length(parent1.visitedCities) - 1]);
            child1 = TSP(parent1.cities);
            child2 = TSP(parent2.cities);
            child1.visitedCities = [parent1.visitedCities(1:crossoverPoint), parent2.visitedCities(crossoverPoint + 1:end)];
            child2.visitedCities = [parent2.visitedCities(1:crossoverPoint), parent1.visitedCities(crossoverPoint + 1:end)];
            population = [child1, child2];
        end

        function population = mutate(obj, population)
            for i = 1:length(population)
                if rand < obj.mutationRate
                    mutationPoint1 = randi([1, length(population(i).visitedCities)]);
                    mutationPoint2 = randi([1, length(population(i).visitedCities)]);
                    temp = population(i).visitedCities(mutationPoint1);
                    population(i).visitedCities(mutationPoint1) = population(i).visitedCities(mutationPoint2);
                    population(i).visitedCities(mutationPoint2) = temp;
                end
            end
        end

        function [bestTour, bestTourLength] = getBestTour(obj, population)
            bestTourLength = Inf;
            bestTour = [];
            for i = 1:length(population)
                tourLength = population(i).tourLength();
                if tourLength < bestTourLength
                    bestTourLength = tourLength;
                    bestTour = population(i);
                end
            end
        end
    end
end

最后，我们可以使用这些类来求解TSP。以下是一个示例：

% 城市的位置
cities = [
    1, 2;
    3, 4;
    5, 6;
    7, 8;
    9, 10
];

% 遗传算法的参数
populationSize = 100;
mutationRate = 0.1;
maxIterations = 100;

% 创建遗传算法对象
ga = GeneticAlgorithm(populationSize, mutationRate, maxIterations);

% 求解TSP
bestTour = ga.solve(cities);

% 打印最佳解
disp(['最佳路径：', num2str(bestTour.visitedCities)]);
disp(['最佳路径长度：', num2str(bestTour.tourLength())]);

% 绘制最佳路径
figure;
plot(cities(:, 1), cities(:, 2), 'bo');
hold on;
plot(cities(bestTour.visitedCities, 1), cities(bestTour.visitedCities, 2), 'r-');
hold off;
legend('城市', '最佳路径');