def crossover(population): parent_1, parent_2 = random.sample(population, 2) child_1 = parent_1.copy() child_2 = parent_2.copy() factory_cutpoints = np.random.randint(1, num_jobs, size=num_factories) for f in range(num_factories): job_ids = np.arange(num_jobs) np.random.shuffle(job_ids) parent_1_jobs = set(parent_1[job_ids[:factory_cutpoints[f]]]) parent_2_jobs = set(parent_2[job_ids[factory_cutpoints[f]:]]) child_1_jobs = [job for job in child_1 if job not in parent_2_jobs] child_2_jobs = [job for job in child_2 if job not in parent_1_jobs] child_1_jobs.extend([job for job in parent_2 if job not in child_1_jobs]) child_2_jobs.extend([job for job in parent_1 if job not in child_2_jobs]) child_1[child_1 == f] = np.array([child_1_jobs.index(job) + num_jobs*f for job in range(num_jobs)]) child_2[child_2 == f] = np.array([child_2_jobs.index(job) + num_jobs*f for job in range(num_jobs)]) return child_1, child_2

这是一个遗传算法中的交叉操作函数，输入参数 population 是一个二维数组，每个元素都代表一个染色体，即工厂作业调度计划。函数返回两个子代染色体 child_1 和 child_2。具体操作是从 population 中随机选择两个染色体 parent_1 和 parent_2，然后按照随机生成的切割点 factory_cutpoints，将两个父代染色体进行切割。接着对每个工厂的切割部分进行基因交换，即将其中一个子代中属于另一个父代的基因替换掉，最终得到两个新的子代染色体。

相关问题

$$ f=\alpha C_1L\sum_{k=1}^K{\frac{1}{t_k}}+\beta C_2\sum_{k=1}^K{\sum_{j=1}^J{\left( \frac{V_{kj}t_k}{2} \right)}} $$用Python求最小值t多目标遗传算法代码

以下是使用Python实现多目标遗传算法求解上述问题的代码：

import random
import numpy as np

# 定义目标函数
def objective_function(t, alpha, beta, C1, L, C2, V):
    f1 = alpha * C1 * L * np.sum(1 / t)
    f2 = beta * C2 * np.sum(V * t / 2)
    return (f1, f2)

# 定义种群初始化函数
def initialize_population(pop_size, chrom_length):
    population = []
    for i in range(pop_size):
        chromosome = [random.uniform(0, 10) for j in range(chrom_length)]
        population.append(chromosome)
    return population

# 定义交叉函数
def crossover(parent1, parent2, prob):
    if random.random() < prob:
        child1 = []
        child2 = []
        for i in range(len(parent1)):
            if random.random() < 0.5:
                child1.append(parent1[i])
                child2.append(parent2[i])
            else:
                child1.append(parent2[i])
                child2.append(parent1[i])
        return (child1, child2)
    else:
        return (parent1, parent2)

# 定义变异函数
def mutate(chromosome, prob):
    mutated_chromosome = []
    for i in range(len(chromosome)):
        if random.random() < prob:
            mutated_chromosome.append(random.uniform(0, 10))
        else:
            mutated_chromosome.append(chromosome[i])
    return mutated_chromosome

# 定义非支配排序函数
def non_dominated_sorting(population, objectives):
    fronts = [[]]
    n = [0] * len(population)
    ranks = [0] * len(population)
    S = [[] for i in range(len(population))]
    for p in range(len(population)):
        S[p] = []
        n[p] = 0
        for q in range(len(population)):
            if dominates(population[p], population[q], objectives):
                S[p].append(q)
            elif dominates(population[q], population[p], objectives):
                n[p] += 1
        if n[p] == 0:
            ranks[p] = 0
            fronts[0].append(p)
    i = 0
    while len(fronts[i]) > 0:
        next_front = []
        for p in fronts[i]:
            for q in S[p]:
                n[q] -= 1
                if n[q] == 0:
                    ranks[q] = i + 1
                    next_front.append(q)
        i += 1
        fronts.append(next_front)
    return fronts[:-1]

# 定义支配函数
def dominates(chromosome1, chromosome2, objectives):
    dominates = False
    for i in range(len(objectives)):
        if objectives[i](chromosome1) > objectives[i](chromosome2):
            return False
        elif objectives[i](chromosome1) < objectives[i](chromosome2):
            dominates = True
    return dominates

# 定义选择函数
def selection(population, objectives, pop_size):
    fronts = non_dominated_sorting(population, objectives)
    selection = []
    for front in fronts:
        if len(selection) + len(front) <= pop_size:
            selection += front
        else:
            remaining_space = pop_size - len(selection)
            distances = crowding_distance(population, objectives, front)
            sorted_indices = sorted(range(len(front)), key=lambda k: -distances[k])
            for i in range(remaining_space):
                selection.append(front[sorted_indices[i]])
            break
    return [population[i] for i in selection]

# 定义拥挤距离函数
def crowding_distance(population, objectives, front):
    distances = [0] * len(front)
    for i in range(len(objectives)):
        sorted_indices = sorted(range(len(front)), key=lambda k: objectives[i](population[front[k]]))
        distances[sorted_indices[0]] = float('inf')
        distances[sorted_indices[-1]] = float('inf')
        for j in range(1, len(front) - 1):
            distances[sorted_indices[j]] += (objectives[i](population[front[j+1]]) - objectives[i](population[front[j-1]])) / (objectives[i](population[front[-1]]) - objectives[i](population[front[0]]))
    return distances

# 定义多目标遗传算法函数
def multi_objective_genetic_algorithm(pop_size, chrom_length, num_generations, alpha, beta, C1, L, C2, V, crossover_prob=0.8, mutation_prob=0.2):
    population = initialize_population(pop_size, chrom_length)
    objectives = [lambda chromosome: objective_function(chromosome, alpha, beta, C1, L, C2, V)[i] for i in range(2)]
    for gen in range(num_generations):
        offspring_population = []
        while len(offspring_population) < pop_size:
            parent1, parent2 = random.sample(population, 2)
            child1, child2 = crossover(parent1, parent2, crossover_prob)
            child1 = mutate(child1, mutation_prob)
            child2 = mutate(child2, mutation_prob)
            offspring_population.append(child1)
            if len(offspring_population) < pop_size:
                offspring_population.append(child2)
        population += offspring_population
        population = selection(population, objectives, pop_size)
    return population

# 设置参数并执行算法
pop_size = 100
chrom_length = 10
num_generations = 50
alpha = 1
beta = 1
C1 = 1
L = 1
C2 = 1
V = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
population = multi_objective_genetic_algorithm(pop_size, chrom_length, num_generations, alpha, beta, C1, L, C2, V)

# 输出结果
for chromosome in population:
    print(chromosome, objective_function(chromosome, alpha, beta, C1, L, C2, V))

编写一段Python代码实现NSGA2

好的，以下是一段实现NSGA2算法的Python代码：

import copy
import random

class NSGA2:
    def __init__(self, fitness_function, num_generations, population_size, crossover_prob, mutation_prob):
        self.fitness_function = fitness_function
        self.num_generations = num_generations
        self.population_size = population_size
        self.crossover_prob = crossover_prob
        self.mutation_prob = mutation_prob

    def run(self):
        population = self.initialize_population(self.population_size)
        for generation in range(self.num_generations):
            fronts = self.fast_non_dominated_sort(population)
            new_population = []
            front_index = 0
            while len(new_population) + len(fronts[front_index]) <= self.population_size:
                for individual in fronts[front_index]:
                    new_population.append(individual)
                front_index += 1
            if len(new_population) == self.population_size - 1:
                new_population.append(random.choice(fronts[front_index]))
            while len(new_population) < self.population_size:
                parent_1 = self.tournament_selection(population)
                parent_2 = self.tournament_selection(population)
                child_1, child_2 = self.crossover(parent_1, parent_2)
                if random.random() < self.mutation_prob:
                    child_1 = self.mutate(child_1)
                if random.random() < self.mutation_prob:
                    child_2 = self.mutate(child_2)
                new_population.append(child_1)
                if len(new_population) < self.population_size:
                    new_population.append(child_2)
            population = new_population
        return population

    def initialize_population(self, population_size):
        population = []
        for _ in range(population_size):
            individual = []
            for _ in range(2):
                gene = random.uniform(0.0, 1.0)
                individual.append(gene)
            population.append(individual)
        return population

    def fast_non_dominated_sort(self, population):
        fronts = [[]]
        individual_rank = {}
        individual_dominated_set = {}
        for individual in population:
            individual_dominated_set[individual] = set()
            individual_rank[individual] = 0
            for other_individual in population:
                if self.is_dominant(individual, other_individual):
                    individual_dominated_set[individual].add(other_individual)
                elif self.is_dominant(other_individual, individual):
                    individual_rank[individual] += 1
            if individual_rank[individual] == 0:
                fronts[0].append(individual)
        current_front_index = 0
        while len(fronts[current_front_index]) > 0:
            next_front = []
            for individual in fronts[current_front_index]:
                for other_individual in individual_dominated_set[individual]:
                    individual_rank[other_individual] -= 1
                    if individual_rank[other_individual] == 0:
                        next_front.append(other_individual)
            current_front_index += 1
            fronts.append(next_front)
        return fronts

    def is_dominant(self, individual_1, individual_2):
        fitness_1 = self.fitness_function(individual_1)
        fitness_2 = self.fitness_function(individual_2)
        is_better = False
        for i in range(len(fitness_1)):
            if fitness_1[i] > fitness_2[i]:
                return False
            elif fitness_1[i] < fitness_2[i]:
                is_better = True
        return is_better

    def tournament_selection(self, population):
        tournament_size = int(len(population) / 10)
        tournament = random.sample(population, tournament_size)
        best_individual = None
        best_fitness = None
        for individual in tournament:
            fitness = self.fitness_function(individual)
            if best_individual is None or fitness < best_fitness:
                best_individual = individual
                best_fitness = fitness
        return best_individual

    def crossover(self, parent_1, parent_2):
        child_1 = copy.deepcopy(parent_1)
        child_2 = copy.deepcopy(parent_2)
        if random.random() < self.crossover_prob:
            alpha = random.uniform(0.0, 1.0)
            child_1[0] = alpha * parent_1[0] + (1 - alpha) * parent_2[0]
            child_1[1] = alpha * parent_1[1] + (1 - alpha) * parent_2[1]
            child_2[0] = alpha * parent_2[0] + (1 - alpha) * parent_1[0]
            child_2[1] = alpha * parent_2[1] + (1 - alpha) * parent_1[1]
        return child_1, child_2

    def mutate(self, individual):
        child = copy.deepcopy(individual)
        gene_index = random.randint(0, len(individual) - 1)
        child[gene_index] = random.uniform(0.0, 1.0)
        return child

def fitness_function(individual):
    return [individual[0] ** 2, (individual[1] - 0.5) ** 2]

nsga2 = NSGA2(fitness_function, 100, 100, 0.9, 0.1)
population = nsga2.run()
print(population)

这段代码实现了一个简单的NSGA2算法。在NSGA2类中，fitness_function代表个体的适应度函数，num_generations代表遗传算法的迭代次数，population_size代表种群大小，crossover_prob代表交叉概率，mutation_prob代表变异概率。run函数是NSGA2算法的主函数，它首先初始化种群，然后迭代一定次数，每次进行快速非支配排序，然后根据支配关系生成子代种群，并进行交叉和变异，最后更新种群。initialize_population函数是种群的初始化函数，它实现了生成随机个体的过程。fast_non_dominated_sort函数是快速非支配排序的实现，用于将种群分成一系列前沿等级。is_dominant函数用于判断一个个体是否支配另一个个体。tournament_selection函数是锦标赛选择的实现，用于选择一个个体作为父代。crossover函数是交叉函数的实现，用于生成两个子代。mutate函数是变异函数的实现，用于对个体进行变异。fitness_function是简单的适应度函数，它返回个体两个维度的平方和作为适应度值。

以上就是一段实现NSGA2算法的Python代码。