【建模算法】基于模拟退火算法求解TSP问题（Python实现）

【建模算法】基于模拟退火算法求解TSP问题（Python实现）

TSP (traveling salesman problem，旅行商问题)是典型的NP完全问题，即其最坏情况下的时间复杂度随着问题规模的增大按指数方式增长，到目前为止还未找到一个多项式时间的有效算法。本文探讨了基于模拟退火算法求解TSP问题的Python实现。

一、问题描述

​ 本案例以31个城市为例，假定31个城市的位置坐标如表1所列。寻找出一条最短的遍历31个城市的路径。

城市编号	X坐标	Y坐标	城市编号	X坐标	Y坐标
1	1.304	2.312	17	3.918	2.179
2	3.639	1.315	18	4.061	2.37
3	4.177	2.244	19	3.78	2.212
4	3.712	1.399	20	3.676	2.578
5	3.488	1.535	21	4.029	2.838
6	3.326	1.556	22	4.263	2.931
7	3.238	1.229	23	3.429	1.908
8	4.196	1.044	24	3.507	2.376
9	4.312	0.79	25	3.394	2.643
10	4.386	0.57	26	3.439	3.201
11	3.007	1.97	27	2.935	3.24
12	2.562	1.756	28	3.14	3.55
13	2.788	1.491	29	2.545	2.357
14	2.381	1.676	30	2.778	2.826
15	1.332	0.695	31	2.37	2.975
16	3.715	1.678

二、模拟退火算法思想

模拟退火算法的(Simulated Annealing，SA)是一种基于概率的全局寻优方法，已在理论上被证明以概率l 收敛于全局最优解。模拟退火算法模拟物理退火过程，从某一较高初温出发，随着温度的不断下降，以一定概率突跳在全局进行寻优，并最终趋于全局最优，搜索过程中趋于零概率的突跳特性可有效避免算法陷入局部最优。模拟退火算法依赖现有求解规则，是一种对已有规则进行改造的算法，它的解与初始值无关；其核心思想是以1概率接受较优解，以较小概率接受裂解（Metropolis准则）。

三、求解结果

初始路线与初始距离：

最优路线与最优值：

最优轨迹图：

四、实现代码

#模拟退火算法求解TSP问题完整代码：
#31座城市TSP问题
import math
import random
import numpy as np 
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import sys
from numpy.matlib import rand
from matplotlib.artist import getp
import copy

#构建初始参考距离矩阵
def getdistance():
    for i in range(n):
        for j in range(n):
            x = pow(city_x[i] - city_x[j], 2)
            y = pow(city_y[i] - city_y[j], 2)
            distance[i][j] = pow(x + y, 0.5)
    for i in range(n):
        for j in range(n):
            if distance[i][j] == 0:
                distance[i][j] = sys.maxsize

#计算总距离
def cacl_best(rou):
    sumdis = 0.0
    for i in range(n-1):
        sumdis += distance[rou[i]][rou[i+1]]
    sumdis += distance[rou[n-1]][rou[0]]     
    return sumdis

#得到新解
def getnewroute(route, time):
    #如果是偶数次，二变换法
    '''
    注意：数组直接复制是复制地址
    例如， current = route
    想要得到一个新的有同样内容的数组，应该用： current = copy.copy(route) 
    '''
    current = copy.copy(route)  
    
    if time % 2 == 0:
        u = random.randint(0, n-1)
        v = random.randint(0, n-1)
        temp = current[u]
        current[u] = current[v]
        current[v] = temp
    #如果是奇数次，三变换法 
    else:
        temp2 = random.sample(range(0, n), 3)
        temp2.sort()
        u = temp2[0]
        v = temp2[1]
        w = temp2[2]
        w1 = w + 1
        temp3 = [0 for col in range(v - u + 1)]
        j =0
        for i in range(u, v + 1):
            temp3[j] = current[i]
            j += 1
        
        for i2 in range(v + 1, w + 1):
            current[i2 - (v-u+1)] = current[i2]
        w = w - (v-u+1)
        j = 0
        for i3 in range(w+1, w1):
            current[i3] = temp3[j]
            j += 1
    
    return current
    
def draw(best):
    result_x = [0 for col in range(n+1)]
    result_y = [0 for col in range(n+1)]
    
    for i in range(n):
        result_x[i] = city_x[best[i]]
        result_y[i] = city_y[best[i]]
    result_x[n] = result_x[0]
    result_y[n] = result_y[0]
    plt.rcParams['font.sans-serif'] = 'SimHei'  # 设置中文显示
    plt.rcParams['axes.unicode_minus'] = False
    plt.xlim(0, 5)  # 限定横轴的范围
    plt.ylim(0, 4)  # 限定纵轴的范围
    plt.plot(result_x, result_y, marker='>', mec='r', mfc='w',label=u'路线')
    plt.legend()  # 让图例生效
    plt.margins(0)
    plt.subplots_adjust(bottom=0.15)
    for i in range(len(best)):
        plt.text(result_x[i] + 0.05, result_y[i] + 0.05, str(best[i]+1), color='red')
    plt.xlabel('横坐标')
    plt.ylabel('纵坐标')
    plt.title('轨迹图')
    plt.show()
     
def print_route(route):
    result_cur_best=[]
    for i in route:
        result_cur_best+=[i]
    for i in range(len(result_cur_best)):
        result_cur_best[i] += 1
    result_path = result_cur_best
    result_path.append(result_path[0])
    return result_path    
    
def solve():
    #得到距离矩阵
    getdistance()
    #得到初始解以及初始距离
    route = random.sample(range(0, n), n) 
    total_dis = cacl_best(route)
    print("初始路线：", print_route(route))
    print("初始距离：", total_dis)
    draw(route)
    #新解
    newroute = []
    new_total_dis = 0.0
    best = route
    best_total_dis = total_dis
    t = T0
    
    while True:
        if t <= Tend:
            break
        #令温度为初始温度
        for rt2 in range(L):
            newroute = getnewroute(route, rt2)
            new_total_dis = cacl_best(newroute)
            delt = new_total_dis - total_dis
            if delt <= 0:
                route = newroute
                total_dis = new_total_dis
                if best_total_dis > new_total_dis:
                    best = newroute
                    best_total_dis = new_total_dis
            elif delt > 0:
                p = math.exp(-delt / t)
                ranp = random.uniform(0, 1)
                if ranp < p:
                    route = newroute
                    total_dis = new_total_dis
        t = t * a
    print("现在温度为：", t)
    print("最优路线：", print_route(best))
    print("最优值：", best_total_dis)  
    draw(best)   
if __name__=="__main__":
    #读取31座城市坐标
    coord = []
    with open("data.txt", "r") as lines:
        lines = lines.readlines()
    for line in lines:
        xy = line.split()
        coord.append(xy)
    coord = np.array(coord)
    w, h = coord.shape
    coordinates = np.zeros((w, h), float)
    for i in range(w):
        for j in range(h):
            coordinates[i, j] = float(coord[i, j])
    city_x=coordinates[:,0]
    city_y=coordinates[:,1]
    #城市数量
    n = coordinates.shape[0]
    distance = [[0 for col in range(n)] for raw in range(n)]
    #初始温度 结束温度
    T0 = 31
    Tend = 1e-8
    #循环控制常数
    L = 10
    #温度衰减系数
    a = 0.98
    solve()