螞蟻幾乎沒有視力，但他們卻能夠在黑暗的世界中找到食物，而且能夠找到一條從洞穴到食物的最短路徑。它們是如何做到的呢？
簡介
由來
蟻群算法是一種用來尋找優化路徑的概率型算法。它由Marco Dorigo於1992年在他的博士論文中提出，其靈感來源於螞蟻在尋找食物過程中發現路徑的行為。
這種算法具有分布計算、信息正反饋和啟發式搜索的特征，本質上是進化算法中的一種啟發式全局優化算法。
思想
將蟻群算法應用於解決優化問題的基本思路為：用螞蟻的行走路徑表示待優化問題的可行解，整個螞蟻群體的所有路徑構成待優化問題的解空間。路徑較短的螞蟻釋放的信息素量較多，隨着時間的推進，較短的路徑上累積的信息素濃度逐漸增高，選擇該路徑的螞蟻個數也愈來愈多。最終，整個螞蟻會在正反饋的作用下集中到最佳的路徑上，此時對應的便是待優化問題的最優解。
應用
蟻群算法應用廣泛，如旅行商問題(traveling salesman problem,簡稱TSP)、指派問題、Job-shop調度問題、車輛路徑問題（vehicle routing problem）、圖着色問題(graph coloring problem)和網絡路由問題（network routing problem）等等，下面以TSP的求解為例。
scikit-opt庫實現
import numpy as np from scipy import spatial import pandas as pd import matplotlib.pyplot as plt num_points = 25 points_coordinate = np.random.rand(num_points, 2) # generate coordinate of points distance_matrix = spatial.distance.cdist(points_coordinate, points_coordinate, metric='euclidean') def cal_total_distance(routine): num_points, = routine.shape return sum([distance_matrix[routine[i % num_points], routine[(i + 1) % num_points]] for i in range(num_points)]) from sko.ACA import ACA_TSP aca = ACA_TSP(func=cal_total_distance, n_dim=num_points, size_pop=50, max_iter=200, distance_matrix=distance_matrix) best_x, best_y = aca.run() # %% Plot fig, ax = plt.subplots(1, 2) best_points_ = np.concatenate([best_x, [best_x[0]]]) best_points_coordinate = points_coordinate[best_points_, :] ax[0].plot(best_points_coordinate[:, 0], best_points_coordinate[:, 1], 'o-r') pd.DataFrame(aca.y_best_history).cummin().plot(ax=ax[1]) plt.show()
Python實現
# -*- coding: utf-8 -*- import random import copy import time import sys import math import tkinter #//GUI模塊 import threading from functools import reduce # 參數 ''' ALPHA:信息啟發因子，值越大，則螞蟻選擇之前走過的路徑可能性就越大 ，值越小，則蟻群搜索范圍就會減少，容易陷入局部最優 BETA:Beta值越大，蟻群越就容易選擇局部較短路徑，這時算法收斂速度會 加快，但是隨機性不高，容易得到局部的相對最優 ''' (ALPHA, BETA, RHO, Q) = (1.0,2.0,0.5,100.0) # 城市數，蟻群 (city_num, ant_num) = (50,50) distance_x = [ 178,272,176,171,650,499,267,703,408,437,491,74,532, 416,626,42,271,359,163,508,229,576,147,560,35,714, 757,517,64,314,675,690,391,628,87,240,705,699,258, 428,614,36,360,482,666,597,209,201,492,294] distance_y = [ 170,395,198,151,242,556,57,401,305,421,267,105,525, 381,244,330,395,169,141,380,153,442,528,329,232,48, 498,265,343,120,165,50,433,63,491,275,348,222,288, 490,213,524,244,114,104,552,70,425,227,331] #城市距離和信息素 distance_graph = [ [0.0 for col in range(city_num)] for raw in range(city_num)] pheromone_graph = [ [1.0 for col in range(city_num)] for raw in range(city_num)] #----------- 螞蟻 ----------- class Ant(object): # 初始化 def __init__(self,ID): self.ID = ID # ID self.__clean_data() # 隨機初始化出生點 # 初始數據 def __clean_data(self): self.path = [] # 當前螞蟻的路徑 self.total_distance = 0.0 # 當前路徑的總距離 self.move_count = 0 # 移動次數 self.current_city = -1 # 當前停留的城市 self.open_table_city = [True for i in range(city_num)] # 探索城市的狀態 city_index = random.randint(0,city_num-1) # 隨機初始出生點 self.current_city = city_index self.path.append(city_index) self.open_table_city[city_index] = False self.move_count = 1 # 選擇下一個城市 def __choice_next_city(self): next_city = -1 select_citys_prob = [0.0 for i in range(city_num)] #存儲去下個城市的概率 total_prob = 0.0 # 獲取去下一個城市的概率 for i in range(city_num): if self.open_table_city[i]: try : # 計算概率：與信息素濃度成正比，與距離成反比 select_citys_prob[i] = pow(pheromone_graph[self.current_city][i], ALPHA) * pow((1.0/distance_graph[self.current_city][i]), BETA) total_prob += select_citys_prob[i] except ZeroDivisionError as e: print ('Ant ID: {ID}, current city: {current}, target city: {target}'.format(ID = self.ID, current = self.current_city, target = i)) sys.exit(1) # 輪盤選擇城市 if total_prob > 0.0: # 產生一個隨機概率,0.0-total_prob temp_prob = random.uniform(0.0, total_prob) for i in range(city_num): if self.open_table_city[i]: # 輪次相減 temp_prob -= select_citys_prob[i] if temp_prob < 0.0: next_city = i break # 未從概率產生，順序選擇一個未訪問城市 # if next_city == -1: # for i in range(city_num): # if self.open_table_city[i]: # next_city = i # break if (next_city == -1): next_city = random.randint(0, city_num - 1) while ((self.open_table_city[next_city]) == False): # if==False,說明已經遍歷過了 next_city = random.randint(0, city_num - 1) # 返回下一個城市序號 return next_city # 計算路徑總距離 def __cal_total_distance(self): temp_distance = 0.0 for i in range(1, city_num): start, end = self.path[i], self.path[i-1] temp_distance += distance_graph[start][end] # 回路 end = self.path[0] temp_distance += distance_graph[start][end] self.total_distance = temp_distance # 移動操作 def __move(self, next_city): self.path.append(next_city) self.open_table_city[next_city] = False self.total_distance += distance_graph[self.current_city][next_city] self.current_city = next_city self.move_count += 1 # 搜索路徑 def search_path(self): # 初始化數據 self.__clean_data() # 搜素路徑，遍歷完所有城市為止 while self.move_count < city_num: # 移動到下一個城市 next_city = self.__choice_next_city() self.__move(next_city) # 計算路徑總長度 self.__cal_total_distance() #----------- TSP問題 ----------- class TSP(object): def __init__(self, root, width = 800, height = 600, n = city_num): # 創建畫布 self.root = root self.width = width self.height = height # 城市數目初始化為city_num self.n = n # tkinter.Canvas self.canvas = tkinter.Canvas( root, width = self.width, height = self.height, bg = "#EBEBEB", # 背景白色 xscrollincrement = 1, yscrollincrement = 1 ) self.canvas.pack(expand = tkinter.YES, fill = tkinter.BOTH) self.title("TSP蟻群算法(n:初始化 e:開始搜索 s:停止搜索 q:退出程序)") self.__r = 5 self.__lock = threading.RLock() # 線程鎖 self.__bindEvents() self.new() # 計算城市之間的距離 for i in range(city_num): for j in range(city_num): temp_distance = pow((distance_x[i] - distance_x[j]), 2) + pow((distance_y[i] - distance_y[j]), 2) temp_distance = pow(temp_distance, 0.5) distance_graph[i][j] =float(int(temp_distance + 0.5)) # 按鍵響應程序 def __bindEvents(self): self.root.bind("q", self.quite) # 退出程序 self.root.bind("n", self.new) # 初始化 self.root.bind("e", self.search_path) # 開始搜索 self.root.bind("s", self.stop) # 停止搜索 # 更改標題 def title(self, s): self.root.title(s) # 初始化 def new(self, evt = None): # 停止線程 self.__lock.acquire() self.__running = False self.__lock.release() self.clear() # 清除信息 self.nodes = [] # 節點坐標 self.nodes2 = [] # 節點對象 # 初始化城市節點 for i in range(len(distance_x)): # 在畫布上隨機初始坐標 x = distance_x[i] y = distance_y[i] self.nodes.append((x, y)) # 生成節點橢圓，半徑為self.__r node = self.canvas.create_oval(x - self.__r, y - self.__r, x + self.__r, y + self.__r, fill = "#ff0000", # 填充紅色 outline = "#000000", # 輪廓白色 tags = "node", ) self.nodes2.append(node) # 顯示坐標 self.canvas.create_text(x,y-10, # 使用create_text方法在坐標（302，77）處繪制文字 text = '('+str(x)+','+str(y)+')', # 所繪制文字的內容 fill = 'black' # 所繪制文字的顏色為灰色 ) # 順序連接城市 #self.line(range(city_num)) # 初始城市之間的距離和信息素 for i in range(city_num): for j in range(city_num): pheromone_graph[i][j] = 1.0 self.ants = [Ant(ID) for ID in range(ant_num)] # 初始蟻群 self.best_ant = Ant(-1) # 初始最優解 self.best_ant.total_distance = 1 << 31 # 初始最大距離 self.iter = 1 # 初始化迭代次數 # 將節點按order順序連線 def line(self, order): # 刪除原線 self.canvas.delete("line") def line2(i1, i2): p1, p2 = self.nodes[i1], self.nodes[i2] self.canvas.create_line(p1, p2, fill = "#000000", tags = "line") return i2 # order[-1]為初始值 reduce(line2, order, order[-1]) # 清除畫布 def clear(self): for item in self.canvas.find_all(): self.canvas.delete(item) # 退出程序 def quite(self, evt): self.__lock.acquire() self.__running = False self.__lock.release() self.root.destroy() print (u"\n程序已退出...") sys.exit() # 停止搜索 def stop(self, evt): self.__lock.acquire() self.__running = False self.__lock.release() # 開始搜索 def search_path(self, evt = None): # 開啟線程 self.__lock.acquire() self.__running = True self.__lock.release() while self.__running: # 遍歷每一只螞蟻 for ant in self.ants: # 搜索一條路徑 ant.search_path() # 與當前最優螞蟻比較 if ant.total_distance < self.best_ant.total_distance: # 更新最優解 self.best_ant = copy.deepcopy(ant) # 更新信息素 self.__update_pheromone_gragh() print (u"迭代次數：",self.iter,u"最佳路徑總距離：",int(self.best_ant.total_distance)) # 連線 self.line(self.best_ant.path) # 設置標題 self.title("TSP蟻群算法(n:隨機初始 e:開始搜索 s:停止搜索 q:退出程序) 迭代次數: %d" % self.iter) # 更新畫布 self.canvas.update() self.iter += 1 # 更新信息素 def __update_pheromone_gragh(self): # 獲取每只螞蟻在其路徑上留下的信息素 temp_pheromone = [[0.0 for col in range(city_num)] for raw in range(city_num)] for ant in self.ants: for i in range(1,city_num): start, end = ant.path[i-1], ant.path[i] # 在路徑上的每兩個相鄰城市間留下信息素，與路徑總距離反比 temp_pheromone[start][end] += Q / ant.total_distance temp_pheromone[end][start] = temp_pheromone[start][end] # 更新所有城市之間的信息素，舊信息素衰減加上新迭代信息素 for i in range(city_num): for j in range(city_num): pheromone_graph[i][j] = pheromone_graph[i][j] * RHO + temp_pheromone[i][j] # 主循環 def mainloop(self): self.root.mainloop() #----------- 程序的入口處 ----------- if __name__ == '__main__': print (u""" -------------------------------------------------------- 程序：蟻群算法解決TPS問題程序 作者：許彬 日期：2015-12-10 語言：Python 2.7 -------------------------------------------------------- """) TSP(tkinter.Tk()).mainloop()
C實現
#include<bits/stdc++.h> using namespace std; const int INF = 0x3f3f3f3f; #define sqr(x) ((x)*(x)) #define eps 1e-8 string file_name; int type;// type == 1 全矩陣, type == 2 二維歐拉距離 int N;//城市數量 double **dis;//城市間距離 double **pheromone;//信息素 double **herustic;//啟發式值 double **info;// info = pheromone ^ delta * herustic ^ beta double pheromone_0;//pheromone初始值，這里是1 / (avg * N)其中avg為圖網中所有邊邊權的平均數。 int m;//種群數量 int delta, beta;//參數 double alpha; int *r1, *s, *r;//agent k的出發城市，下一個點，當前點。 int MAX, iteration;//最大迭代次數，迭代計數變量 set<int> empty, *J; struct vertex{ double x, y;// 城市坐標 int id;// 城市編號 int input(FILE *fp){ return fscanf(fp, "%d %lf %lf", &id, &x, &y); } }*node; typedef pair<int, int> pair_int; struct Tour{//路徑 vector<pair_int> path;//path[i]，存儲一條邊(r,s) double L; void clean(){ L = INF; path.clear(); path.shrink_to_fit(); }//清空 void calc(){ L = 0; int sz = path.size(); for (int i = 0; i < sz; i ++){ L += dis[path[i].first][path[i].second]; } }//計算長度 void push_back(int x, int y){ path.push_back(make_pair(x, y)); } int size(){ return (int)path.size(); } int r(int i){ return path[i].first; } int s(int i){ return path[i].second; } void print(FILE *fp){ int sz = path.size(); for (int i = 0; i < sz; i ++){ fprintf(fp, "%d->", path[i].first + 1); } fprintf(fp, "%d\n", path[sz - 1].second + 1); } bool operator <(const Tour &a)const{ return L < a.L; }//重載 } *tour, best_so_far; double EUC_2D(const vertex &a, const vertex &b){ return sqrt(sqr(a.x - b.x) + sqr(a.y - b.y)); } void io(){//輸入 printf("input file_name and data type\n"); cin >> file_name >> type; FILE *fp = fopen(file_name.c_str(), "r"); fscanf(fp, "%d", &N); node = new vertex[N + 5]; dis = new double*[N + 5]; double tmp = 0; int cnt = 0; if (type == 1){ for (int i = 0; i < N; i ++){ dis[i] = new double[N]; for (int j = 0; j < N; j ++){ fscanf(fp, "%lf", &dis[i][j]); tmp += i != j ? dis[i][j] : 0;// i == j的時候dis不存在，所以不考慮。 cnt += i != j ? 1 : 0;// i == j的時候dis不存在，所以不考慮。 } } }else{ for (int i = 0; i < N; i ++) node[i].input(fp); for (int i = 0; i < N; i ++){ dis[i] = new double[N]; for (int j = 0; j < N; j ++){ dis[i][j] = EUC_2D(node[i], node[j]);// 計算距離 tmp += i != j ? dis[i][j] : 0;// i == j的時候 dis不存在，所以不考慮。 cnt += i != j ? 1 : 0;// i == j的時候dis不存在，所以不考慮。 } } } pheromone_0 = (double)cnt / (tmp * N);//pheromone初始值，這里是1 / (avg * N)其中avg為圖網中所有邊邊權的平均數。 fclose(fp); return; } void init(){//初始化 alpha = 0.1;//evaporation parameter，揮發參數，每次信息素要揮發的量 delta = 1; beta = 6;// delta 和 beta分別表示pheromones和herustic的比重 m = N; pheromone = new double*[N + 5]; herustic = new double*[N + 5]; info = new double*[N + 5]; r1 = new int[N + 5]; r = new int[N + 5]; s = new int[N + 5]; J = new set<int>[N + 5]; empty.clear(); for (int i = 0; i < N; i ++){ pheromone[i] = new double[N + 5]; herustic[i] = new double[N + 5]; info[i] = new double[N + 5]; empty.insert(i); for (int j = 0; j < N; j ++){ pheromone[i][j] = pheromone_0; herustic[i][j] = 1 / (dis[i][j] + eps);//加一個小數eps，防止被零除 } } best_so_far.clean(); iteration = 0; MAX = N * N; } double power(double x, int y){//快速冪，計算x ^ y，時間復雜度O(logn),感興趣可以百度 double ans = 1; while (y){ if (y & 1) ans *= x; x *= x; y >>= 1; } return ans; } void reset(){ tour = new Tour[m + 5]; for (int i = 0; i < N; i ++){ tour[i].clean(); r1[i] = i;//初始化出發城市， J[i] = empty; J[i].erase(r1[i]);//初始化agent i需要訪問的城 市 r[i] = r1[i];//當前在出發點 } for (int i = 0; i < N; i ++) for (int j = 0; j < N; j ++){ info[i][j] = power(pheromone[i][j], delta) * power(herustic[i][j], beta); }//選擇公式 } int select_next(int k){ if (J[k].empty()) return r1[k]; //如果J是空的，那么返回出發點城市 double rnd = (double)(rand()) / (double)RAND_MAX;//產生0..1的隨機數 set<int>::iterator it = J[k].begin(); double sum_prob = 0, sum = 0; for (; it != J[k].end(); it ++){ sum += info[r[k]][*it]; }//計算概率分布 rnd *= sum; it = J[k].begin(); for (; it != J[k].end(); it ++){ sum_prob += info[r[k]][*it]; if (sum_prob >= rnd){ return *it; } }//依照概率選取下一步城市 } void construct_solution(){ for (int i = 0; i < N; i ++){ for (int k = 0; k < m; k ++){ int next = select_next(k);//選擇下一步的最優決策 J[k].erase(next); s[k] = next; tour[k].push_back(r[k], s[k]); r[k] = s[k]; } } } void update_pheromone(){ Tour now_best; now_best.clean();//初始化 for (int i = 0; i < m; i ++){ tour[i].calc(); if (tour[i] < now_best) now_best = tour[i];//尋找當前迭代最優解 } if (now_best < best_so_far){ best_so_far = now_best;//更新最優解 } for (int i = 0; i < N; i ++) for (int j = 0; j < N; j ++) pheromone[i][j] *= (1 - alpha);//信息素揮發 int sz = now_best.size(); for (int i = 0; i < sz; i ++){ pheromone[now_best.r(i)][now_best.s(i)] += 1. / (double)now_best.L; pheromone[now_best.s(i)][now_best.r(i)] = pheromone[now_best.r(i)][now_best.s(i)];// 對稱 }//更新信息素含量 } int main(){ srand((unsigned) time(0));//初始化隨機種子 io(); init(); double last = INF; int bad_times = 0; for (; iteration < MAX; iteration ++){ if (bad_times > N) break;//進入局部最優 reset();//初始化agent信息 construct_solution();//對於所有的agent構造一個完整的tour update_pheromone();//更新信息素 printf("iteration %d:best_so_far = %.2f\n", iteration, best_so_far.L); if (last > best_so_far.L) last = best_so_far.L, bad_times = 0; else bad_times ++;//記錄當前未更新代數，若迭代多次未更新，認為進入局部最優 } printf("best_so_far = %.2f\n", best_so_far.L);// 輸出目標值 best_so_far.print(stdout);//輸出路徑 } 算例演示 例一 滿秩矩陣式（type = 1） 輸入文件格式為： File_name File_type salesman.in 1 5 0 1 2 2 3 2 0 3 4 2 3 2 0 4 1 3 4 5 0 5 2 4 1 4 0 輸出結果為： opt_solution: 11 例二 二維坐標式（type = 2） 輸入文件格式為： File_name File_type KroA100.tsp 2 100 1 1380 939 2 2848 96 3 3510 1671 4 457 334 5 3888 666 6 984 965 7 2721 1482 8 1286 525 9 2716 1432 10 738 1325 11 1251 1832 12 2728 1698 13 3815 169 14 3683 1533 15 1247 1945 16 123 862 17 1234 1946 18 252 1240 19 611 673 20 2576 1676 21 928 1700 22 53 857 23 1807 1711 24 274 1420 25 2574 946 26 178 24 27 2678 1825 28 1795 962 29 3384 1498 30 3520 1079 31 1256 61 32 1424 1728 33 3913 192 34 3085 1528 35 2573 1969 36 463 1670 37 3875 598 38 298 1513 39 3479 821 40 2542 236 41 3955 1743 42 1323 280 43 3447 1830 44 2936 337 45 1621 1830 46 3373 1646 47 1393 1368 48 3874 1318 49 938 955 50 3022 474 51 2482 1183 52 3854 923 53 376 825 54 2519 135 55 2945 1622 56 953 268 57 2628 1479 58 2097 981 59 890 1846 60 2139 1806 61 2421 1007 62 2290 1810 63 1115 1052 64 2588 302 65 327 265 66 241 341 67 1917 687 68 2991 792 69 2573 599 70 19 674 71 3911 1673 72 872 1559 73 2863 558 74 929 1766 75 839 620 76 3893 102 77 2178 1619 78 3822 899 79 378 1048 80 1178 100 81 2599 901 82 3416 143 83 2961 1605 84 611 1384 85 3113 885 86 2597 1830 87 2586 1286 88 161 906 89 1429 134 90 742 1025 91 1625 1651 92 1187 706 93 1787 1009 94 22 987 95 3640 43 96 3756 882 97 776 392 98 1724 1642 99 198 1810 100 3950 1558 輸出結果為： best_known_solution: 21282
參考鏈接：
1. 百度百科-蟻群算法
4. CSDN-fanxin_i-蟻群算法原理及實現（Python）
5. 知乎-數據魔法師-十分鍾快速get蟻群算法（附C代碼）