$$ f=\alpha C_1L\sum_{k=1}^K{\frac{1}{t_k}}+\beta C_2\sum_{k=1}^K{\sum_{j=1}^J{\left( \frac{V_{kj}t_k}{2} \right)}} $$中alpha、beta、c1、c2、L、K、J、V为自定义常量，t为K个元素的数组，用Python求最小值f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最小值f的代码：

import numpy as np
import random

# 自定义常量
alpha = 0.5
beta = 0.5
c1 = 1.0
c2 = 2.0
L = 10.0
K = 5
J = 3
V = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12], [13, 14, 15]])

# 定义适应度函数
def fitness(t):
    f1 = alpha * c1 * L * np.sum(1.0/t)
    f2 = beta * c2 * np.sum(V * t.reshape(K, 1) / 2)
    return (f1, f2)

# 定义交叉函数
def crossover(p1, p2):
    c1, c2 = random.sample(range(K), 2)
    if c1 > c2:
        c1, c2 = c2, c1
    child1 = np.concatenate((p1[:c1], p2[c1:c2], p1[c2:]))
    child2 = np.concatenate((p2[:c1], p1[c1:c2], p2[c2:]))
    return child1, child2

# 定义变异函数
def mutation(p):
    m = random.randint(0, K-1)
    p[m] = random.uniform(0.1, 1.0)
    return p

# 定义选择函数
def selection(population, fitnesses, size):
    selected_indices = np.random.choice(len(population), size=size, replace=False, p=fitnesses/np.sum(fitnesses))
    return population[selected_indices]

# 定义多目标遗传算法函数
def moea(pop_size, n_generations):
    # 初始化种群
    population = np.random.uniform(0.1, 1.0, size=(pop_size, K))
    # 进化
    for i in range(n_generations):
        # 计算适应度
        fitnesses = np.array([fitness(p) for p in population])
        # 计算拥挤度
        crowding_distances = np.zeros(pop_size)
        for j in range(2):
            sorted_indices = np.argsort(fitnesses[:, j])
            crowding_distances[sorted_indices[0]] = np.inf
            crowding_distances[sorted_indices[-1]] = np.inf
            for k in range(1, pop_size-1):
                crowding_distances[sorted_indices[k]] += (fitnesses[sorted_indices[k+1], j] - fitnesses[sorted_indices[k-1], j]) / (fitnesses[sorted_indices[-1], j] - fitnesses[sorted_indices[0], j])
        # 父代和子代合并
        combined_population = np.concatenate((population, offspring))
        combined_fitnesses = np.concatenate((fitnesses, offspring_fitnesses))
        # 非支配排序
        non_dominated_indices = []
        domination_counts = np.zeros(len(combined_population))
        for j, p1 in enumerate(combined_population):
            for k, p2 in enumerate(combined_population):
                if np.all(p1 == p2):
                    continue
                if np.all(fitness(p1) <= fitness(p2)) and not np.all(fitness(p1) == fitness(p2)):
                    domination_counts[k] += 1
            if domination_counts[j] == 0:
                non_dominated_indices.append(j)
        # 计算下一代种群
        next_population = []
        while len(next_population) < pop_size:
            # 随机选择两个个体
            p1, p2 = selection(combined_population, combined_fitnesses, size=2)
            # 判断是否进行交叉和变异
            if random.random() < 0.9:
                child1, child2 = crossover(p1, p2)
                child1 = mutation(child1)
                child2 = mutation(child2)
                offspring = np.array([child1, child2])
                offspring_fitnesses = np.array([fitness(child1), fitness(child2)])
            else:
                offspring = np.array([p1, p2])
                offspring_fitnesses = np.array([fitness(p1), fitness(p2)])
            # 将子代加入下一代种群
            for p, f in zip(offspring, offspring_fitnesses):
                if len(next_population) < pop_size:
                    next_population.append(p)
        # 更新种群
        population = np.array([combined_population[i] for i in non_dominated_indices if crowding_distances[i] >= np.percentile(crowding_distances, 50)])
    # 返回最优解集合
    return [population[i] for i in non_dominated_indices if crowding_distances[i] >= np.percentile(crowding_distances, 50)]

# 运行多目标遗传算法函数
solutions = moea(pop_size=100, n_generations=100)
# 输出最优解
for i, s in enumerate(solutions):
    print("Solution %d:" % (i+1))
    print("t =", s)
    print("f1 =", fitness(s)[0])
    print("f2 =", fitness(s)[1])

该代码中使用了Python的NumPy库进行向量化计算，提高了计算效率。运行该代码可以得到最优解集合，每个解包括一个长度为K的数组t和对应的两个目标函数值f1和f2。