模拟退火求解att48.tsp

TSP（Traveling Salesman Problem，旅行商问题）是一个经典的组合优化问题，目标是找到一条经过所有城市的最短路径。att48.tsp 是一个包含48个城市的 TSP 问题，我们可以使用模拟退火算法来求解该问题。

首先，我们需要读取 att48.tsp 文件并计算出任意两个城市之间的距离。可以使用以下代码实现：

import math

# 读取 TSP 文件并计算城市之间的距离
def read_tsp_file(filename):
    with open(filename, 'r') as f:
        lines = f.readlines()

        # 读取城市坐标
        cities = []
        for line in lines:
            if line[0].isdigit():
                city_info = line.split()
                city_id = int(city_info[0])
                x = float(city_info[1])
                y = float(city_info[2])
                cities.append((city_id, x, y))

        # 计算城市之间的距离
        distances = {}
        for i in range(len(cities)):
            for j in range(i+1, len(cities)):
                city1 = cities[i]
                city2 = cities[j]
                distance = math.sqrt((city1[1]-city2[1])**2 + (city1[2]-city2[2])**2)
                distances[(city1[0], city2[0])] = distance
                distances[(city2[0], city1[0])] = distance

        return cities, distances

# 测试
cities, distances = read_tsp_file('att48.tsp')
print("城市坐标：", cities)
print("城市之间的距离：", distances)

接下来，我们可以实现一个 TSP 的目标函数，即计算给定路径的总距离：

# 计算路径的总距离
def total_distance(path, distances):
    distance = 0
    for i in range(len(path)-1):
        distance += distances[(path[i], path[i+1])]
    distance += distances[(path[-1], path[0])]
    return distance

然后，我们可以使用模拟退火算法来求解 TSP 问题。以下是一个求解 att48.tsp 的模拟退火算法的代码实现：

import random
import math

# 模拟退火算法
def simulated_annealing_tsp(distances, n_iterations, initial_temp, temp_reduction):
    # 随机生成一个初始解
    current_path = list(distances.keys())
    random.shuffle(current_path)
    current_fitness = total_distance(current_path, distances)

    # 记录温度
    current_temp = initial_temp

    # 迭代优化
    for i in range(n_iterations):
        # 随机生成一个新解
        candidate_path = list(current_path)
        index1, index2 = sorted(random.sample(range(len(candidate_path)), 2))
        candidate_path[index1:index2+1] = reversed(candidate_path[index1:index2+1])

        # 计算新解的适应度
        candidate_fitness = total_distance(candidate_path, distances)

        # 计算适应度提高的程度
        delta_fitness = candidate_fitness - current_fitness

        # 判断是否接受新解
        if delta_fitness < 0 or math.exp(-delta_fitness / current_temp) > random.random():
            current_path = candidate_path
            current_fitness = candidate_fitness

        # 降低温度
        current_temp *= temp_reduction

    return current_path, current_fitness

# 测试
cities, distances = read_tsp_file('att48.tsp')
best_path, best_fitness = simulated_annealing_tsp(distances, 10000, 100, 0.95)
print("最优路径：", best_path)
print("最优路径的总距离：", best_fitness)

在该代码中，我们随机生成一个初始解，然后在每次迭代中随机生成一个新解，并根据一定的概率接受该解或者继续保留当前解。随着迭代次数的增加和温度的降低，算法最终收敛到一个局部最优解。我们最终得到的最优路径和总距离都可以通过调用 `simulated_annealing_tsp` 函数来获得。