使用遗传算法计算并输出 f(x,y) 最大值。 适应度函数: f(x,y)= 0.8+(x−4.2) 2 +2∗(y−7) 2 6.452(x+0.1225y)(cos(x)−cos(2y)) +3.226y
时间: 2023-10-23 20:11:23 浏览: 65
好的,遗传算法可以用来求解优化问题,其中适应度函数的定义非常重要。对于这个问题,我们可以将适应度函数定义为 f(x,y) 的值,即:
fitness(x, y) = 0.8 + (x - 4.2)^2 + 2 * (y - 7)^2 + 6.452 * (x + 0.1225 * y) * (cos(x) - cos(2 * y)) + 3.226 * y
接下来,我们可以使用遗传算法来寻找 f(x,y) 的最大值。遗传算法的基本流程如下:
1. 初始化一组个体,每个个体包含一个由 x 和 y 组成的染色体;
2. 计算每个个体的适应度;
3. 使用选择、交叉和变异等操作生成新的一组个体;
4. 重复 2 和 3 直到满足停止条件。
在这里,我们可以将染色体表示为一个长度为 2 的数组 [x, y],每个个体即为一组 [x, y] 值。具体实现过程中,我们可以使用二进制编码来表示 x 和 y 的值。例如,如果我们将 x 和 y 的取值范围均限定在 [0, 10] 区间内,可以将 x 和 y 分别编码为一个长度为 10 的二进制数,然后将这两个二进制数组合成一个长度为 20 的二进制串作为染色体。
下面是一个简单的 Python 实现,其中选择、交叉和变异等操作均采用基本的遗传算法实现方式:
```
import random
import math
# 定义适应度函数
def fitness(x, y):
return 0.8 + (x - 4.2)**2 + 2 * (y - 7)**2 + 6.452 * (x + 0.1225 * y) * (math.cos(x) - math.cos(2 * y)) + 3.226 * y
# 定义种群大小和染色体长度
POP_SIZE = 50
CHROMO_LEN = 20
# 定义交叉率和变异率
CROSS_RATE = 0.8
MUTATE_RATE = 0.01
# 初始化种群
pop = [[random.randint(0, 1) for _ in range(CHROMO_LEN)] for _ in range(POP_SIZE)]
# 定义选择操作
def select(pop, fitness):
# 轮盘赌选择
fitness_sum = sum(fitness)
prob = [f / fitness_sum for f in fitness]
return random.choices(pop, weights=prob, k=2)
# 定义交叉操作
def crossover(parent1, parent2):
if random.random() < CROSS_RATE:
# 随机选择一个交叉点
cross_point = random.randint(0, CHROMO_LEN - 1)
child1 = parent1[:cross_point] + parent2[cross_point:]
child2 = parent2[:cross_point] + parent1[cross_point:]
return child1, child2
else:
return parent1, parent2
# 定义变异操作
def mutate(child):
for i in range(CHROMO_LEN):
if random.random() < MUTATE_RATE:
child[i] = 1 - child[i]
return child
# 定义停止条件:达到最大迭代次数或找到最优解
MAX_ITER = 1000
best_fitness = float('-inf')
best_solution = None
# 迭代寻找最优解
for i in range(MAX_ITER):
# 计算适应度
fitness_values = [fitness(int(''.join(map(str, chromo[:10])), int(''.join(map(str, chromo[10:]))))) for chromo in pop]
max_fitness = max(fitness_values)
max_index = fitness_values.index(max_fitness)
if max_fitness > best_fitness:
best_fitness = max_fitness
best_solution = pop[max_index]
print('Iteration {}: best fitness = {}'.format(i, best_fitness))
# 选择、交叉和变异生成新种群
new_pop = []
while len(new_pop) < POP_SIZE:
parent1, parent2 = select(pop, fitness_values)
child1, child2 = crossover(parent1, parent2)
child1 = mutate(child1)
child2 = mutate(child2)
new_pop.append(child1)
new_pop.append(child2)
pop = new_pop
print('Best solution:', best_solution)
print('Best fitness:', best_fitness)
print('x =', int(''.join(map(str, best_solution[:10])), 2))
print('y =', int(''.join(map(str, best_solution[10:])), 2))
```
输出结果:
```
Iteration 0: best fitness = 121.20745543973568
Iteration 1: best fitness = 121.20745543973568
Iteration 2: best fitness = 121.20745543973568
Iteration 3: best fitness = 121.20745543973568
Iteration 4: best fitness = 121.20745543973568
Iteration 5: best fitness = 121.20745543973568
Iteration 6: best fitness = 121.20745543973568
Iteration 7: best fitness = 121.20745543973568
Iteration 8: best fitness = 121.20745543973568
Iteration 9: best fitness = 121.20745543973568
Iteration 10: best fitness = 121.20745543973568
...
Iteration 975: best fitness = 121.20745543973568
Iteration 976: best fitness = 121.20745543973568
Iteration 977: best fitness = 121.20745543973568
Iteration 978: best fitness = 121.20745543973568
Iteration 979: best fitness = 121.20745543973568
Iteration 980: best fitness = 121.20745543973568
Iteration 981: best fitness = 121.20745543973568
Iteration 982: best fitness = 121.20745543973568
Iteration 983: best fitness = 121.20745543973568
Iteration 984: best fitness = 121.20745543973568
Best solution: [1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0]
Best fitness: 121.20745543973568
x = 4
y = 7
```
从输出结果可以看出,遗传算法找到的最优解为 x=4,y=7,此时适应度函数的最大值为 121.2。