举一个模拟退火算法的例子

以下是一个模拟退火算法的例子，用于解决旅行商问题（Traveling Salesman Problem，TSP）：

import random
import math

def calculate_distance(city1, city2):
    # 计算两个城市之间的距离
    x_diff = city1[0] - city2[0]
    y_diff = city1[1] - city2[1]
    distance = math.sqrt(x_diff**2 + y_diff**2)
    return distance

def calculate_total_distance(cities, order):
    # 计算给定顺序下的总路程
    total_distance = 0
    num_cities = len(order)
    for i in range(num_cities):
        city1 = cities[order[i]]
        city2 = cities[order[(i+1) % num_cities]]
        distance = calculate_distance(city1, city2)
        total_distance += distance
    return total_distance

def simulated_annealing(cities, initial_order, initial_temperature, cooling_rate):
    current_order = initial_order
    current_temperature = initial_temperature
    best_order = current_order.copy()
    best_distance = calculate_total_distance(cities, current_order)

    while current_temperature > 1:
        # 生成新解
        new_order = current_order.copy()
        index1 = random.randint(0, len(new_order)-1)
        index2 = random.randint(0, len(new_order)-1)
        new_order[index1], new_order[index2] = new_order[index2], new_order[index1]

        # 计算当前和新解的总路程
        current_distance = calculate_total_distance(cities, current_order)
        new_distance = calculate_total_distance(cities, new_order)

        # 判断是否接受新解
        if new_distance < current_distance or random.random() < math.exp((current_distance - new_distance) / current_temperature):
            current_order = new_order
            current_distance = new_distance

        # 更新最优解
        if current_distance < best_distance:
            best_order = current_order.copy()
            best_distance = current_distance

        # 降温
        current_temperature *= cooling_rate

    return best_order, best_distance

# 城市坐标
cities = [(0, 0), (1, 5), (3, 2), (4, 7), (6, 3)]

# 初始顺序
initial_order = [0, 1, 2, 3, 4]

# 初始温度和降温率
initial_temperature = 100
cooling_rate = 0.99

# 运行模拟退火算法
best_order, best_distance = simulated_annealing(cities, initial_order, initial_temperature, cooling_rate)

print("最优顺序:", best_order)
print("最短路程:", best_distance)

这个例子中，我们随机生成了5个城市的坐标，然后使用模拟退火算法来寻找这些城市之间的最短路径。算法通过不断交换城市的顺序，并根据一定的概率接受劣解，从而逐渐收敛到全局最优解。最后输出最优顺序和最短路程。请注意，这只是一个简单的示例，实际的旅行商问题可能涉及更多的城市和复杂的优化目标。