1、基本概念

遺傳演算法（GA）是最早由美國Holland教授提出的一種基於自然界的“適者生存，優勝劣汰”基本法則的智慧搜尋演算法。該法則很好地詮釋了生物進化的自然選擇過程。遺傳演算法也是借鑑該基本法則，通過基於種群的思想，將問題的解通過編碼的方式轉化為種群中的個體，並讓這些個體不斷地通過選擇、交叉和變異運算元模擬生物的進化過程，然後利用“優勝劣汰”法則選擇種群中適應性較強的個體構成子種群，然後讓子種群重複類似的進化過程，直到找到問題的最優解或者到達一定的進化（運算）時間。

基因：在GA演算法中，基因代表了具體問題解的一個決策變數，問題解和染色體中基因的對應關係如下所示：

種群：多個個體即組成一個種群。GA演算法中，一個問題的多組解即構成了問題的解的種群。

2、主要步驟

GA演算法的基本步驟如下：

Step 1. 種群初始化。選擇一種編碼方案然後在解空間內通過隨機生成的方式初始化一定數量的個體構成GA的種群。

Step 2. 評估種群。利用啟發式演算法對種群中的個體（矩形件的排入順序）生成排樣圖並依此計算個體的適應函式值（利用率），然後儲存當前種群中的最優個體作為搜尋到的最優解。

Step 3. 選擇操作。根據種群中個體的適應度的大小，通過輪盤賭或者期望值方法，將適應度高的個體從當前種群中選擇出來。

Step 4. 交叉操作。將上一步驟選擇的個體，用一定的概率閥值Pc控制是否利用單點交叉、多點交叉或者其他交叉方式生成新的交叉個體。

Step 5. 變異操作。用一定的概率閥值Pm控制是否對個體的部分基因執行單點變異或多點變異。

Step 6. 終止判斷。若滿足終止條件，則終止演算法，否則返回Step 2。

流程圖如下所示：

3、主要操作介紹

3.1 種群初始化

種群的初始化和具體的問題有關。比如一個問題有n個決策變數{x1,x2,…,xn}。每個決策變數有取值範圍:下界{L1,L2,Ln}和上界{U1,U2,Un}，則種群中個體的初始化即隨機地在決策變數的取值範圍內生成各個決策變數的值：Xj={x1,...,xn}，其中xi屬於範圍(Li，Ui)內。所有的個體即構成種群。當每個個體都初始化後，即種群完成初始化。

3.2 評價種群

種群的評價即計算種群中個體的適應度值。假設種群population有popsize個個體。依次計算每個個體的適應度值及評價種群。

3.3 選擇操作

GA演算法中常見的選擇操作有輪盤賭方式：種群中適應度值更優的個體被選擇的概率越大。假設popsize=4，按照如下表達式計算各個個體的被選擇概率的大小，然後用圓餅圖表示如下。

P(Xj) = fit(Xj)/(fit(X1)+fit(X2)+fit(X3)+fit(X4)),j=1,2,3,4

當依據輪盤賭方式進行選擇時，則概率越大的越容易被選擇到。

3.4 交叉操作

交叉操作也有許多種：單點交叉，兩點交叉等。此處僅講解一下兩點交叉。首先利用選擇操作從種群中選擇兩個父輩個體parent1和parent2,然後隨機產生兩個位置pos1和pos2，將這兩個位置中間的基因位資訊進行交換，便得到下圖所示的off1和off2兩個個體，但是這兩個個體中一般會存在基因位資訊衝突的現象（整數編碼時），此時需要對off1和off2個體進行調整：off1中的衝突基因根據parent1中的基因調整為parent2中的相同位置處的基因。如off1中的“1”出現了兩次，則第二處的“1”需要調整為parent1中“1”對應的parent2中的“4”，依次類推處理off1中的相沖突的基因。需要注意的是，調整off2，則需要參考parent2。

3.5 變異操作

變異操作的話，根據不同的編碼方式有不同的變異操作。

如果是浮點數編碼，則變異可以就染色體中間的某一個基因位的資訊進行變異（重新生成或者其他調整方案）。

如果是採用整數編碼方案，則一般有多種變異方法：位置變異和符號變異。

位置變異：

符號變異：

4、Python程式碼

#-*- coding:utf-8 -*-

import random
import math
from operator import itemgetter

class Gene:
  '''
  This is a class to represent individual(Gene) in GA algorithom
  each object of this class have two attribute: data,size
  '''
  def __init__(self,**data):
    self.__dict__.update(data)    
    self.size = len(data['data'])#length of gene
                
    
class GA:
  '''
  This is a class of GA algorithm. 
  '''
  def __init__(self,parameter):
    '''
    Initialize the pop of GA algorithom and evaluate the pop by computing its' fitness value .
    The data structure of pop is composed of several individuals which has the form like that:
    
    {'Gene':a object of class Gene,'fitness': 1.02(for example)}

    Representation of Gene is a list: [b s0 u0 sita0 s1 u1 sita1 s2 u2 sita2]
    
    '''
    #parameter = [CXPB,MUTPB,NGEN,popsize,low,up]
    self.parameter = parameter

    low = self.parameter[4]
    up = self.parameter[5]
    
    self.bound = []
    self.bound.append(low)
    self.bound.append(up)
    
    pop = []
    for i in range(self.parameter[3]):
      geneinfo = []
      for pos in range(len(low)):
        geneinfo.append(random.uniform(self.bound[0][pos],self.bound[1][pos]))#initialise popluation
        
      fitness = evaluate(geneinfo)#evaluate each chromosome
      pop.append({'Gene':Gene(data = geneinfo),'fitness':fitness})#store the chromosome and its fitness
      
    self.pop = pop
    self.bestindividual = self.selectBest(self.pop)#store the best chromosome in the population
    
  def selectBest(self,pop):
    '''
    select the best individual from pop
    '''
    s_inds = sorted(pop,key = itemgetter("fitness"),reverse = False)
    return s_inds[0]
    
  def selection(self,individuals,k):
    '''
    select two individuals from pop
    '''
    s_inds = sorted(individuals,reverse=True)#sort the pop by the reference of 1/fitness 
    sum_fits = sum(1/ind['fitness'] for ind in individuals) #sum up the 1/fitness of the whole pop
    
    chosen = []
    for i in xrange(k):
      u = random.random() * sum_fits#randomly produce a num in the range of [0,sum_fits]
      sum_ = 0
      for ind in s_inds:
        sum_ += 1/ind['fitness']#sum up the 1/fitness
        if sum_ > u:
          #when the sum of 1/fitness is bigger than u,choose the one,which means u is in the range of [sum(1,n-1),sum(1,n)] and is time to choose the one,namely n-th individual in the pop
          chosen.append(ind)
          break
    
    return chosen  


  def crossoperate(self,offspring):
    '''
    cross operation
    '''
    dim = len(offspring[0]['Gene'].data)

    geninfo1 = offspring[0]['Gene'].data#Gene's data of first offspring chosen from the selected pop
    geninfo2 = offspring[1]['Gene'].data#Gene's data of second offspring chosen from the selected pop
    
    pos1 = random.randrange(1,dim)#select a position in the range from 0 to dim-1,pos2 = random.randrange(1,dim)

    newoff = Gene(data = [])#offspring produced by cross operation
    temp = []
    for i in range(dim):
      if (i >= min(pos1,pos2) and i <= max(pos1,pos2)):
        temp.append(geninfo2[i])
        #the gene data of offspring produced by cross operation is from the second offspring in the range [min(pos1,pos2),max(pos1,pos2)]
      else:
        temp.append(geninfo1[i])
        #the gene data of offspring produced by cross operation is from the frist offspring in the range [min(pos1,pos2)]
    newoff.data = temp
    
    return newoff


  def mutation(self,crossoff,bound):
    '''
    mutation operation
    '''
    
    dim = len(crossoff.data)

    pos = random.randrange(1,dim)#chose a position in crossoff to perform mutation.

    crossoff.data[pos] = random.uniform(bound[0][pos],bound[1][pos])
    return crossoff
  
  def GA_main(self):
    '''
    main frame work of GA
    '''
    
    popsize = self.parameter[3]
    
    print("Start of evolution")
    
    # Begin the evolution
    for g in range(NGEN):
      
      print("-- Generation %i --" % g)   
           
      #Apply selection based on their converted fitness
      selectpop = self.selection(self.pop,popsize)  

      nextoff = []  
      while len(nextoff) != popsize:   
        # Apply crossover and mutation on the offspring      
                
        # Select two individuals
        offspring = [random.choice(selectpop) for i in xrange(2)]
        
        if random.random() < CXPB: # cross two individuals with probability CXPB
          crossoff = self.crossoperate(offspring)
          fit_crossoff = evaluate(self.xydata,crossoff.data)# Evaluate the individuals      
          
          if random.random() < MUTPB: # mutate an individual with probability MUTPB
            muteoff = self.mutation(crossoff,self.bound)
            fit_muteoff = evaluate(self.xydata,muteoff.data)# Evaluate the individuals
            nextoff.append({'Gene':muteoff,'fitness':fit_muteoff})
            
      # The population is entirely replaced by the offspring
      self.pop = nextoff
      
      # Gather all the fitnesses in one list and print the stats
      fits = [ind['fitness'] for ind in self.pop]
        
      length = len(self.pop)
      mean = sum(fits) / length
      sum2 = sum(x*x for x in fits)
      std = abs(sum2 / length - mean**2)**0.5
      best_ind = self.selectBest(self.pop)

      if best_ind['fitness'] < self.bestindividual['fitness']:
        self.bestindividual = best_ind

      print("Best individual found is %s,%s" % (self.bestindividual['Gene'].data,self.bestindividual['fitness']))
      print(" Min fitness of current pop: %s" % min(fits))
      print(" Max fitness of current pop: %s" % max(fits))
      print(" Avg fitness of current pop: %s" % mean)
      print(" Std of currrent pop: %s" % std)
    
    print("-- End of (successful) evolution --")  

if __name__ == "__main__":

  CXPB,popsize = 0.8,0.3,50,100#control parameters
  
  up = [64,64,64]#upper range for variables
  low = [-64,-64,-64]#lower range for variables
  parameter = [CXPB,up]
  
  run = GA(parameter)
  run.GA_main()

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