Python編程實現蟻群演算法詳解，python演算法詳解

Python編程實現蟻群演算法詳解，python演算法詳解

簡介

蟻群演算法(ant colony optimization, ACO)，又稱螞蟻演算法，是一種用來在圖中尋找最佳化路徑的機率型演算法。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。蟻群演算法是一種類比進化演算法，初步的研究表明該演算法具有許多優良的性質。針對PID控制器參數最佳化設計問題，將蟻群演算法設計的結果與遺傳演算法設計的結果進行了比較，數值模擬結果表明，蟻群演算法具有一種新的類比進化最佳化方法的有效性和應用價值。
定義

各個螞蟻在沒有事先告訴他們食物在什麼地方的前提下開始尋找食物。當一隻找到食物以後，它會向環境釋放一種揮發性分泌物pheromone (稱為資訊素,該物質隨著時間的推移會逐漸揮發消失，資訊素濃度的大小表徵路徑的遠近)來實現的，吸引其他的螞蟻過來，這樣越來越多的螞蟻會找到食物。有些螞蟻並沒有像其它螞蟻一樣總重複同樣的路，他們會另闢蹊徑，如果另開闢的道路比原來的其他道路更短，那麼，漸漸地，更多的螞蟻被吸引到這條較短的路上來。最後，經過一段時間運行，可能會出現一條最短的路徑被大多數螞蟻重複著。

解決的問題

三維地形中，給出起點和重點，找到其最優路徑。

作圖源碼：

from mpl_toolkits.mplot3d import proj3dfrom mpl_toolkits.mplot3d import Axes3Dimport numpy as npheight3d = np.array([[2000,1400,800,650,500,750,1000,950,900,800,700,900,1100,1050,1000,1150,1300,1250,1200,1350,1500],          [1100,900,700,625,550,825,1100,1150,1200,925,650,750,850,950,1050,1175,1300,1350,1400,1425,1450],          [200,400,600,600,600,900,1200,1350,1500,1050,600,600,600,850,1100,1200,1300,1450,1600,1500,1400],          [450,500,550,575,600,725,850,875,900,750,600,600,600,725,850,900,950,1150,1350,1400,1450],          [700,600,500,550,600,550,500,400,300,450,600,600,600,600,600,600,600,850,1100,1300,1500],          [500,525,550,575,600,575,550,450,350,475,600,650,700,650,600,600,600,725,850,1150,1450],          [300,450,600,600,600,600,600,500,400,500,600,700,800,700,600,600,600,600,600,1000,1400],          [550,525,500,550,600,875,1150,900,650,725,800,700,600,875,1150,1175,1200,975,750,875,1000],          [800,600,400,500,600,1150,1700,1300,900,950,1000,700,400,1050,1700,1750,1800,1350,900,750,600],          [650,600,550,625,700,1175,1650,1275,900,1100,1300,1275,1250,1475,1700,1525,1350,1200,1050,950,850],          [500,600,700,750,800,1200,1600,1250,900,1250,1600,1850,2100,1900,1700,1300,900,1050,1200,1150,1100],          [400,375,350,600,850,1200,1550,1250,950,1225,1500,1750,2000,1950,1900,1475,1050,975,900,1175,1450],          [300,150,0,450,900,1200,1500,1250,1000,1200,1400,1650,1900,2000,2100,1650,1200,900,600,1200,1800],          [600,575,550,750,950,1275,1600,1450,1300,1300,1300,1525,1750,1625,1500,1450,1400,1125,850,1200,1550],          [900,1000,1100,1050,1000,1350,1700,1650,1600,1400,1200,1400,1600,1250,900,1250,1600,1350,1100,1200,1300],          [750,850,950,900,850,1000,1150,1175,1200,1300,1400,1325,1250,1125,1000,1150,1300,1075,850,975,1100],          [600,700,800,750,700,650,600,700,800,1200,1600,1250,900,1000,1100,1050,1000,800,600,750,900],          [750,775,800,725,650,700,750,775,800,1000,1200,1025,850,975,1100,950,800,900,1000,1050,1100],          [900,850,800,700,600,750,900,850,800,800,800,800,800,950,1100,850,600,1000,1400,1350,1300],          [750,800,850,850,850,850,850,825,800,750,700,775,850,1000,1150,875,600,925,1250,1100,950],          [600,750,900,1000,1100,950,800,800,800,700,600,750,900,1050,1200,900,600,850,1100,850,600]])fig = figure()ax = Axes3D(fig)X = np.arange(21)Y = np.arange(21)X, Y = np.meshgrid(X, Y)Z = -20*np.exp(-0.2*np.sqrt(np.sqrt(((X-10)**2+(Y-10)**2)/2)))+20+np.e-np.exp((np.cos(2*np.pi*X)+np.sin(2*np.pi*Y))/2)ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap='cool')ax.set_xlabel('X axis')ax.set_ylabel('Y axis')ax.set_zlabel('Z')ax.set_title('3D map')point0 = [0,9,Z[0][9]] point1 = [20,7,Z[20][7]]ax.plot([point0[0]],[point0[1]],[point0[2]],'r',marker = u'o',markersize = 15)ax.plot([point1[0]],[point1[1]],[point1[2]],'r',marker = u'o',markersize = 15)x0,y0,_ = proj3d.proj_transform(point0[0],point0[1],point0[2], ax.get_proj())x1,y1,_ = proj3d.proj_transform(point1[0],point1[1],point1[2], ax.get_proj())label = pylab.annotate(  "start",   xy = (x0, y0), xytext = (-20, 20),  textcoords = 'offset points', ha = 'right', va = 'bottom',  bbox = dict(boxstyle = 'round,pad=0.5', fc = 'yellow', alpha = 1),  arrowprops = dict(arrowstyle = '->', connectionstyle = 'arc3,rad=0'),fontsize=15)label2 = pylab.annotate(  "end",   xy = (x1, y1), xytext = (-20, 20),  textcoords = 'offset points', ha = 'right', va = 'bottom',  bbox = dict(boxstyle = 'round,pad=0.5', fc = 'yellow', alpha = 1),  arrowprops = dict(arrowstyle = '->', connectionstyle = 'arc3,rad=0'),fontsize=15)def update_position(e):  x2, y2, _ = proj3d.proj_transform(point0[0],point0[1],point0[2],ax.get_proj())  label.xy = x2,y2  label.update_positions(fig.canvas.renderer)  x1,y1,_ = proj3d.proj_transform(point1[0],point1[1],point1[2],ax.get_proj())  label2.xy = x1,y1  label2.update_positions(fig.canvas.renderer)  fig.canvas.draw()fig.canvas.mpl_connect('button_release_event', update_position)

基本原理

螞蟻k根據各個城市間連結路徑上的資訊素濃度決定其下一個訪問城市，設Pkij(t)表示t時刻螞蟻k從城市i轉移到矩陣j的機率，其計算公式為

計算完城市間的轉移機率後，採用與遺傳演算法中一樣的輪盤賭方法選擇下一個待訪問的城市。

當所有的螞蟻完成一次迴圈後，各個城市間連結路徑上的資訊素濃度需進行更新，計算公式為

其中，Δτkij表示第k只螞蟻在城市i與城市j串連路徑上釋放的資訊素濃度；Δτij表示所有螞蟻在城市i與城市j串連路徑上釋放的資訊素濃度之和。

螞蟻釋放資訊素的模型

程式碼：

import numpy as npimport matplotlib.pyplot as plt%pylabcoordinates = np.array([[565.0,575.0],[25.0,185.0],[345.0,750.0],[945.0,685.0],[845.0,655.0],            [880.0,660.0],[25.0,230.0],[525.0,1000.0],[580.0,1175.0],[650.0,1130.0],            [1605.0,620.0],[1220.0,580.0],[1465.0,200.0],[1530.0, 5.0],[845.0,680.0],            [725.0,370.0],[145.0,665.0],[415.0,635.0],[510.0,875.0],[560.0,365.0],            [300.0,465.0],[520.0,585.0],[480.0,415.0],[835.0,625.0],[975.0,580.0],            [1215.0,245.0],[1320.0,315.0],[1250.0,400.0],[660.0,180.0],[410.0,250.0],            [420.0,555.0],[575.0,665.0],[1150.0,1160.0],[700.0,580.0],[685.0,595.0],            [685.0,610.0],[770.0,610.0],[795.0,645.0],[720.0,635.0],[760.0,650.0],            [475.0,960.0],[95.0,260.0],[875.0,920.0],[700.0,500.0],[555.0,815.0],            [830.0,485.0],[1170.0, 65.0],[830.0,610.0],[605.0,625.0],[595.0,360.0],            [1340.0,725.0],[1740.0,245.0]])def getdistmat(coordinates):  num = coordinates.shape[0]  distmat = np.zeros((52,52))  for i in range(num):    for j in range(i,num):      distmat[i][j] = distmat[j][i]=np.linalg.norm(coordinates[i]-coordinates[j])  return distmatdistmat = getdistmat(coordinates)numant = 40 #螞蟻個數numcity = coordinates.shape[0] #城市個數alpha = 1  #資訊素重要程度因子beta = 5  #啟發函數重要程度因子rho = 0.1  #資訊素的揮發速度Q = 1iter = 0itermax = 250etatable = 1.0/(distmat+np.diag([1e10]*numcity)) #啟發函數矩陣，表示螞蟻從城市i轉移到矩陣j的期望程度pheromonetable = np.ones((numcity,numcity)) # 資訊素矩陣pathtable = np.zeros((numant,numcity)).astype(int) #路徑記錄表distmat = getdistmat(coordinates) #城市的距離矩陣lengthaver = np.zeros(itermax) #各代路徑的平均長度lengthbest = np.zeros(itermax) #各代及其之前遇到的最佳路徑長度pathbest = np.zeros((itermax,numcity)) # 各代及其之前遇到的最佳路徑長度while iter < itermax:  # 隨機產生各個螞蟻的起點城市  if numant <= numcity:#城市數比螞蟻數多    pathtable[:,0] = np.random.permutation(range(0,numcity))[:numant]  else: #螞蟻數比城市數多，需要補足    pathtable[:numcity,0] = np.random.permutation(range(0,numcity))[:]    pathtable[numcity:,0] = np.random.permutation(range(0,numcity))[:numant-numcity]  length = np.zeros(numant) #計算各個螞蟻的路徑距離  for i in range(numant):    visiting = pathtable[i,0] # 當前所在的城市    #visited = set() #已訪問過的城市，防止重複    #visited.add(visiting) #增加元素    unvisited = set(range(numcity))#未訪問的城市    unvisited.remove(visiting) #刪除元素    for j in range(1,numcity):#迴圈numcity-1次，訪問剩餘的numcity-1個城市      #每次用輪盤法選擇下一個要訪問的城市      listunvisited = list(unvisited)      probtrans = np.zeros(len(listunvisited))      for k in range(len(listunvisited)):        probtrans[k] = np.power(pheromonetable[visiting][listunvisited[k]],alpha)\            *np.power(etatable[visiting][listunvisited[k]],alpha)      cumsumprobtrans = (probtrans/sum(probtrans)).cumsum()      cumsumprobtrans -= np.random.rand()      k = listunvisited[find(cumsumprobtrans>0)[0]] #下一個要訪問的城市      pathtable[i,j] = k      unvisited.remove(k)      #visited.add(k)      length[i] += distmat[visiting][k]      visiting = k    length[i] += distmat[visiting][pathtable[i,0]] #螞蟻的路徑距離包括最後一個城市和第一個城市的距離  #print length  # 包含所有螞蟻的一個迭代結束後，統計本次迭代的若干統計參數  lengthaver[iter] = length.mean()  if iter == 0:    lengthbest[iter] = length.min()    pathbest[iter] = pathtable[length.argmin()].copy()     else:    if length.min() > lengthbest[iter-1]:      lengthbest[iter] = lengthbest[iter-1]      pathbest[iter] = pathbest[iter-1].copy()    else:      lengthbest[iter] = length.min()      pathbest[iter] = pathtable[length.argmin()].copy()    # 更新資訊素  changepheromonetable = np.zeros((numcity,numcity))  for i in range(numant):    for j in range(numcity-1):      changepheromonetable[pathtable[i,j]][pathtable[i,j+1]] += Q/distmat[pathtable[i,j]][pathtable[i,j+1]]    changepheromonetable[pathtable[i,j+1]][pathtable[i,0]] += Q/distmat[pathtable[i,j+1]][pathtable[i,0]]  pheromonetable = (1-rho)*pheromonetable + changepheromonetable  iter += 1 #迭代次數指標+1  #觀察程式執行進度，該功能是非必須的  if (iter-1)%20==0:     print iter-1# 做出平均路徑長度和最優路徑長度    fig,axes = plt.subplots(nrows=2,ncols=1,figsize=(12,10))axes[0].plot(lengthaver,'k',marker = u'')axes[0].set_title('Average Length')axes[0].set_xlabel(u'iteration')axes[1].plot(lengthbest,'k',marker = u'')axes[1].set_title('Best Length')axes[1].set_xlabel(u'iteration')fig.savefig('Average_Best.png',dpi=500,bbox_inches='tight')plt.close()#作出找到的最優路徑圖bestpath = pathbest[-1]plt.plot(coordinates[:,0],coordinates[:,1],'r.',marker=u'$\cdot$')plt.xlim([-100,2000])plt.ylim([-100,1500])for i in range(numcity-1):#  m,n = bestpath[i],bestpath[i+1]  print m,n  plt.plot([coordinates[m][0],coordinates[n][0]],[coordinates[m][1],coordinates[n][1]],'k')plt.plot([coordinates[bestpath[0]][0],coordinates[n][0]],[coordinates[bestpath[0]][1],coordinates[n][1]],'b')ax=plt.gca()ax.set_title("Best Path")ax.set_xlabel('X axis')ax.set_ylabel('Y_axis')plt.savefig('Best Path.png',dpi=500,bbox_inches='tight')plt.close()

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