巡回置换算法例题详解

巡回置换问题是指在一个完全图中，每个顶点之间都有一条边，求一条经过每个顶点一次且仅一次的回路，使得回路的总长度最小。这个问题是一个NP难问题，因此通常使用启发式算法来解决。其中一种启发式算法是遗传算法。

遗传算法是一种模拟自然进化过程的优化算法。在巡回置换问题中，可以将每个可能的回路看作一个个体，通过交叉、变异等操作来产生新的个体，并通过适应度函数来评估每个个体的优劣程度。经过多代进化，最终得到一个较优的回路。

以下是巡回置换问题的遗传算法的Python实现：

import random
# 生成初始种群
def generate_population(city_num, pop_size):
    population = []
    for i in range(pop_size):
        chromosome = list(range(city_num))
        random.shuffle(chromosome)
        population.append(chromosome)
    return population

# 计算路径长度
def get_distance(city1, city2):
    return ((city1[0] - city2[0]) ** 2 + (city1[1] - city2[1]) ** 2) ** 0.5

def get_path_length(path, cities):
    length = 0
    for i in range(len(path) - 1):
        length += get_distance(cities[path[i]], cities[path[i+1]])
    length += get_distance(cities[path[-1]], cities[path[0]])
    return length

# 选择操作
def selection(population, cities):
    fitness_list = [1 / get_path_length(chromosome, cities) for chromosome in population]
    total_fitness = sum(fitness_list)
    probability_list = [fitness / total_fitness for fitness in fitness_list]
    selected_population = []
    for i in range(len(population)):
        selected_population.append(random.choices(population, probability_list)[0])
    return selected_population

# 交叉操作
def crossover(parent1, parent2):
    child = [-1] * len(parent1)
    start = random.randint(0, len(parent1) - 1)
    end = random.randint(0, len(parent1) - 1)
    if start > end:
        start, end = end, start
    for i in range(start, end+1):
        child[i] = parent1[i]
    j = 0
    for i in range(len(parent2)):
        if child[j] == -1:
            if parent2[i] not in child:
                child[j] = parent2[i]
                j += 1
        else:
            j += 1
    return child

# 变异操作
def mutation(chromosome):
    index1 = random.randint(0, len(chromosome) - 1)
    index2 = random.randint(0, len(chromosome) - 1)
    chromosome[index1], chromosome[index2] = chromosome[index2], chromosome[index1]
    return chromosome

# 遗传算法主函数
def genetic_algorithm(city_list, pop_size, generation_num, crossover_rate, mutation_rate):
    population = generate_population(len(city_list), pop_size)
    for i in range(generation_num):
        population = selection(population, city_list)
        new_population = []
        for j in range(pop_size):
            parent1 = random.choice(population)
            if random.random() < crossover_rate:
                parent2 = random.choice(population)
                child = crossover(parent1, parent2)
            else:
                child = parent1
            if random.random() < mutation_rate:
                child = mutation(child)
            new_population.append(child)
        population = new_population
    best_path = min(population, key=lambda x: get_path_length(x, city_list))
    best_length = get_path_length(best_path, city_list)
    return best_path, best_length

# 测试
if __name__ == '__main__':
    city_list = [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
    best_path, best_length = genetic_algorithm(city_list, 100, 1000, 0.8, 0.1)
    print('最优路径：', best_path)
    print('最短路径长度：', best_length)