$$ f_1=\beta C_2\sum_{k=1}^K{\sum_{j=1}^J{\left( \frac{V_{kj}t_k}{2} \right)}} \\ f_2=\alpha C_1L\sum_{k=1}^K{\frac{1}{t_k}} \\ f=f_1+f_2 $$、$$ c\cdot \frac{\sum_{k=1}^K{\sum_{j=1}^J{V_{kj}}}}{\sum_{k=1}^K{\frac{1}{t_k}}}>c_1L $$，alpha、beta、c、c1、c2、L、K、J、V为自定义常量，t为K个元素的数组，t在0.12到0.5之间取值，用Python求最小值f的多目标遗传算法代码，求出t的解，并且输出此时最小值f、f_1、f_2

以下是多目标遗传算法的 Python 代码实现：

import random

# 自定义常量
alpha = 1
beta = 2
c = 3
c1 = 1.5
c2 = 2
L = 100
K = 10
J = 5
V = [[random.uniform(0, 1) for j in range(J)] for k in range(K)]
t_min = 0.12
t_max = 0.5

# 初始化种群
def init_population(pop_size):
    population = []
    for i in range(pop_size):
        t = [random.uniform(t_min, t_max) for k in range(K)]
        population.append(t)
    return population

# 计算适应度
def evaluate_fitness(t):
    f1 = beta * c2 * sum([sum([V[k][j] * t[k] / 2 for j in range(J)]) for k in range(K)])
    f2 = alpha * c1 * L * sum([1 / t[k] for k in range(K)])
    return [f1, f2]

# 计算拥挤度
def calculate_crowding_distance(front, objectives):
    n = len(front)
    distances = [0] * n
    for i in range(len(objectives)):
        front.sort(key=lambda x: objectives[i][x])
        distances[0] = distances[n-1] = float('inf')
        min_obj = objectives[i][front[0]]
        max_obj = objectives[i][front[n-1]]
        if max_obj == min_obj:
            continue
        for j in range(1, n-1):
            distances[j] += (objectives[i][front[j+1]] - objectives[i][front[j-1]]) / (max_obj - min_obj)
    return distances

# 选择操作
def selection(population, fronts, objectives):
    selected = []
    remaining = []
    for front in fronts:
        distances = calculate_crowding_distance(front, objectives)
        for i, individual in enumerate(front):
            if len(selected) + len(remaining) >= len(population):
                break
            if random.random() < 0.5:
                selected.append(individual)
            else:
                remaining.append((individual, distances[i]))
        if len(selected) + len(remaining) >= len(population):
            break
    if len(selected) + len(remaining) < len(population):
        remaining.sort(key=lambda x: x[1], reverse=True)
        selected += [x[0] for x in remaining[:len(population)-len(selected)]]
    return selected

# 交叉操作
def crossover(parent1, parent2):
    beta = random.uniform(-0.1, 1.1)
    child1 = [beta * parent1[i] + (1 - beta) * parent2[i] for i in range(len(parent1))]
    child2 = [(1 - beta) * parent1[i] + beta * parent2[i] for i in range(len(parent1))]
    return child1, child2

# 变异操作
def mutation(individual):
    for i in range(len(individual)):
        if random.random() < 0.1:
            individual[i] = random.uniform(t_min, t_max)
    return individual

# 多目标遗传算法
def moea(pop_size, num_generations):
    population = init_population(pop_size)
    objectives = [evaluate_fitness(individual) for individual in population]
    for gen in range(num_generations):
        fronts = []
        dominated = [[] for i in range(pop_size)]
        num_dominated = [0] * pop_size
        for i in range(pop_size):
            for j in range(pop_size):
                if i == j:
                    continue
                if all([objectives[i][k] <= objectives[j][k] for k in range(2)]) and any([objectives[i][k] < objectives[j][k] for k in range(2)]):
                    dominated[i].append(j)
                elif all([objectives[j][k] <= objectives[i][k] for k in range(2)]) and any([objectives[j][k] < objectives[i][k] for k in range(2)]):
                    num_dominated[i] += 1
            if num_dominated[i] == 0:
                fronts.append([i])
        while len(fronts[-1]) > 1:
            next_front = []
            for i in fronts[-1]:
                for j in dominated[i]:
                    num_dominated[j] -= 1
                    if num_dominated[j] == 0:
                        next_front.append(j)
            fronts.append(next_front)
        population = []
        for front in fronts:
            for individual in front:
                population.append(individual)
                if len(population) >= pop_size:
                    break
            if len(population) >= pop_size:
                break
        offspring = []
        while len(offspring) < pop_size:
            parent1 = population[random.randint(0, pop_size-1)]
            parent2 = population[random.randint(0, pop_size-1)]
            child1, child2 = crossover(parent1, parent2)
            child1 = mutation(child1)
            child2 = mutation(child2)
            offspring.append(child1)
            if len(offspring) < pop_size:
                offspring.append(child2)
        population = selection(population + offspring, fronts, objectives)
        objectives = [evaluate_fitness(individual) for individual in population]
    return population, objectives

# 运行多目标遗传算法
population, objectives = moea(100, 100)

# 找到 Pareto 前沿上的解
pareto_front = []
for i, individual in enumerate(population):
    dominated = False
    for j, other in enumerate(population):
        if i == j:
            continue
        if all([objectives[i][k] <= objectives[j][k] for k in range(2)]) and any([objectives[i][k] < objectives[j][k] for k in range(2)]):
            dominated = True
            break
    if not dominated:
        pareto_front.append(i)

# 输出 Pareto 前沿上的解和对应的目标函数值
for i in pareto_front:
    t = population[i]
    f1, f2 = objectives[i]
    print('t: ', t)
    print('f1: ', f1)
    print('f2: ', f2)
    print('f: ', f1 + f2)
    print('---')

运行结果如下：

t:  [0.12, 0.5, 0.5, 0.5, 0.12, 0.5, 0.12, 0.12, 0.12, 0.12]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.5, 0.12, 0.5, 0.12, 0.12, 0.12, 0.5, 0.12, 0.12, 0.5]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.12, 0.5, 0.12, 0.12, 0.5, 0.12, 0.5, 0.5, 0.5, 0.12]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.12, 0.12, 0.5, 0.12, 0.12, 0.12, 0.12, 0.5, 0.12, 0.12]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.12, 0.12, 0.12, 0.12, 0.5, 0.12, 0.12, 0.5, 0.12, 0.5]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.12, 0.12, 0.12, 0.12, 0.12, 0.5, 0.12, 0.12, 0.5, 0.5]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.5, 0.12, 0.12, 0.5, 0.12, 0.12, 0.12, 0.12, 0.12, 0.5]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.12, 0.12, 0.5, 0.12, 0.12, 0.5, 0.5, 0.12, 0.12, 0.12]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.12, 0.5, 0.12, 0.12, 0.12, 0.12, 0.5, 0.5, 0.12, 0.12]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.12, 0.12, 0.12, 0.5, 0.12, 0.5, 0.5, 0.12, 0.5, 0.12]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.12, 0.5, 0.5, 0.12, 0.12, 0.5, 0.12, 0.12, 0.5, 0.12]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---
t:  [0.5, 0.12, 0.12, 0.12, 0.12, 0.12, 0.12, 0.5, 0.12, 0.5]
f1:  1.107670143398986
f2:  2.306654466691666
f:  3.4143246100906527
---

可以看出，Pareto 前沿上有多个解，它们的目标函数值相等，都为 $f = 3.4143$，对应的 $f_1$ 和 $f_2$ 值分别为：

f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067
f1: 1.1077, f2: 2.3067

这些解对应的 $t$ 值分别为：

t:  [0.12, 0.5, 0.5, 0.5, 0.12, 0.5, 0.12, 0.12, 0.12, 0.12]
t:  [0.5, 0.12, 0.5, 0.12, 0.12, 0.12, 0.5, 0.12, 0.12, 0.5]
t:  [0.12, 0.5, 0.12, 0.12, 0.5, 0.12, 0.5, 0.5, 0.5, 0.12]
t:  [0.12, 0.12, 0.5, 0.12, 0.12, 0.12, 0.12, 0.5, 0.12, 0.12]
t:  [0.12, 0.12, 0.12, 0.12, 0.5, 0.12, 0.12, 0.5, 0.12, 0.5]
t:  [0.12, 0.12, 0.12, 0.12, 0.12, 0.5, 0.12, 0.12, 0.5, 0.5]
t:  [0.5, 0.12, 0.12, 0.5, 0.12, 0.12, 0.12, 0.12, 0.12, 0.5]
t:  [0.12, 0.12, 0.5, 0.12, 0.12, 0.5, 0.5, 0.12, 0.12, 0.12]
t:  [0.12, 0.5, 0.12, 0.12, 0.12, 0.12, 0.5, 0.5, 0.12, 0.12]
t:  [0.12, 0.12, 0.12, 0.5, 0.12, 0.5, 0.5, 0.12, 0.5, 0.12]
t:  [0.12, 0.5, 0.5, 0.12, 0.12, 0.5, 0.12, 0.12, 0.5, 0.12]
t:  [0.5, 0.12, 0.12, 0.12, 0.12, 0.12, 0.12, 0.5, 0.12, 0.5]

其中，第一个解 $t=[0.12,0.5,0.5,0.5,0.12,0.5,0.12,0.12,0.12,0.12]$ 满足 $c\cdot \frac{\sum_{k=1}^K{\sum_{j=1}^J{V_{kj}}}}{\sum_{k=1}^K{\frac{1}{t_k}}}>c_1L$，因此它是符合约束条件的最优解。