$$ f=\alpha C_1L\sum_{k=1}^K{\frac{1}{t_k}}+\beta C_2\sum_{k=1}^K{\sum_{j=1}^J{\left( \frac{V_{kj}t_k}{2} \right)}} $$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))