遗传算法（三）—— 旅行商问题TSP

遗传算法 (GA) 算法最主要的就是我们要想明白什么是他的 DNA 和怎么样对个体进行评估 (他们的 Fitness).

Fitness和DNA

这次的编码 DNA 方式又不一样, 我们可以尝试对每一个城市有一个 ID, 那经历的城市顺序就是按 ID 排序咯. 比如说商人要经过3个城市, 我们就有

• 0-1-2
• 0-2-1
• 1-0-2
• 1-2-0
• 2-0-1
• 2-1-0

这6种排列方式. 每一种排列方式我们就能把它当做一种 DNA 序列, 用 numpy 产生这种 DNA 序列的方式很简单.

>>> np.random.permutation(3)
# array([1, 2, 0])

计算 fitness 的时候, 我们只要将 DNA 中这几个城市连成线, 计算一下总路径的长度, 根据长度, 我们定下规则, 越短的总路径越好, 下面的 fitness0 就用来计算 fitness 啦. 因为越短的路径我们更要价大幅度选择, 所以这里我用到了 fitness1 这种方式.

fitness0 = 1/total_distance
fitness1 = np.exp(1/total_distance)

交叉和变异

我们要注意的是在 crossover 和 mutate 的时候有一点点不一样, 因为对于路径点, 我们不能随意变化. 比如 如果按平时的 crossover, 可能会是这样的结果:

p1=[0,1,2,3] (爸爸)

p2=[3,2,1,0] (妈妈)

cp=[m,b,m,b] (交叉点, m: 妈妈, b: 爸爸)

c1=[3,1,1,3] (孩子)

那么这样的 c1 要经过两次城市 3, 两次城市1, 而没有经过 2, 0. 显然不行. 所以我们 crossover 以及 mutation 都要换一种方式进行. 其中一种可行的方式是这样. 同样是上面的例子.

p1=[0,1,2,3] (爸爸)

cp=[_,b,_,b] (选好来自爸爸的点)

c1=[1,3,_,_] (先将爸爸的点填到孩子的前面)

此时除开来自爸爸的 1, 3. 还有0, 2 两个城市, 但是0,2 的顺序就按照妈妈 DNA 的先后顺序排列. 也就是 p2=[3,2,1,0] 的 0, 2 两城市在 p2 中是先有 2, 再有 0. 所以我们就按照这个顺序补去孩子的 DNA.

c1=[1,3,2,0]

按照这样的方式, 我们就能成功避免在 crossover 产生的问题: 访问多次通过城市的问题了. 用 Python 的写法很简单.

if np.random.rand() < self.cross_rate:
    i_ = np.random.randint(0, self.pop_size, size=1)                        # select another individual from pop
    cross_points = np.random.randint(0, 2, self.DNA_size).astype(np.bool)   # choose crossover points
    keep_city = parent[cross_points]                                       # find the city number
    swap_city = pop[i_, np.isin(pop[i_].ravel(), keep_city, invert=True)]   # 找到与爸爸不同的城市
    parent[:] = np.concatenate((keep_city, swap_city))

在 mutate 的时候, 也是找到两个不同的 DNA 点, 然后交换这两个点就好了.

for point in range(self.DNA_size):
    if np.random.rand() < self.mutate_rate:
        swap_point = np.random.randint(0, self.DNA_size)
        swapA, swapB = child[point], child[swap_point]
        child[point], child[swap_point] = swapB, swapA

完整代码：

"""
Visualize Genetic Algorithm to find the shortest path for travel sales problem.
Visit my tutorial website for more: https://morvanzhou.github.io/tutorials/
"""
import matplotlib.pyplot as plt
import numpy as np

N_CITIES = 20  # DNA size
CROSS_RATE = 0.1
MUTATE_RATE = 0.02
POP_SIZE = 500
N_GENERATIONS = 500


class GA(object):
    def __init__(self, DNA_size, cross_rate, mutation_rate, pop_size, ):
        self.DNA_size = DNA_size
        self.cross_rate = cross_rate
        self.mutate_rate = mutation_rate
        self.pop_size = pop_size

        self.pop = np.vstack([np.random.permutation(DNA_size) for _ in range(pop_size)])

    def translateDNA(self, DNA, city_position):     # get cities‘ coord in order
        line_x = np.empty_like(DNA, dtype=np.float64)
        line_y = np.empty_like(DNA, dtype=np.float64)
        for i, d in enumerate(DNA):
            city_coord = city_position[d]
            line_x[i, :] = city_coord[:, 0]
            line_y[i, :] = city_coord[:, 1]
        return line_x, line_y

    def get_fitness(self, line_x, line_y):
        total_distance = np.empty((line_x.shape[0],), dtype=np.float64)
        for i, (xs, ys) in enumerate(zip(line_x, line_y)):
            total_distance[i] = np.sum(np.sqrt(np.square(np.diff(xs)) + np.square(np.diff(ys))))
        fitness = np.exp(self.DNA_size * 2 / total_distance)
        return fitness, total_distance

    def select(self, fitness):
        idx = np.random.choice(np.arange(self.pop_size), size=self.pop_size, replace=True, p=fitness / fitness.sum())
        return self.pop[idx]

    def crossover(self, parent, pop):
        if np.random.rand() < self.cross_rate:
            i_ = np.random.randint(0, self.pop_size, size=1)                        # select another individual from pop
            cross_points = np.random.randint(0, 2, self.DNA_size).astype(np.bool)   # choose crossover points
            keep_city = parent[~cross_points]                                       # find the city number
            swap_city = pop[i_, np.isin(pop[i_].ravel(), keep_city, invert=True)]
            parent[:] = np.concatenate((keep_city, swap_city))
        return parent

    def mutate(self, child):
        for point in range(self.DNA_size):
            if np.random.rand() < self.mutate_rate:
                swap_point = np.random.randint(0, self.DNA_size)
                swapA, swapB = child[point], child[swap_point]
                child[point], child[swap_point] = swapB, swapA
        return child

    def evolve(self, fitness):
        pop = self.select(fitness)
        pop_copy = pop.copy()
        for parent in pop:  # for every parent
            child = self.crossover(parent, pop_copy)
            child = self.mutate(child)
            parent[:] = child
        self.pop = pop


class TravelSalesPerson(object):
    def __init__(self, n_cities):
        self.city_position = np.random.rand(n_cities, 2)
        plt.ion()

    def plotting(self, lx, ly, total_d):
        plt.cla()
        plt.scatter(self.city_position[:, 0].T, self.city_position[:, 1].T, s=100, c=‘k‘)
        plt.plot(lx.T, ly.T, ‘r-‘)
        plt.text(-0.05, -0.05, "Total distance=%.2f" % total_d, fontdict={‘size‘: 20, ‘color‘: ‘red‘})
        plt.xlim((-0.1, 1.1))
        plt.ylim((-0.1, 1.1))
        plt.pause(0.01)


ga = GA(DNA_size=N_CITIES, cross_rate=CROSS_RATE, mutation_rate=MUTATE_RATE, pop_size=POP_SIZE)

env = TravelSalesPerson(N_CITIES)
for generation in range(N_GENERATIONS):
    lx, ly = ga.translateDNA(ga.pop, env.city_position)
    fitness, total_distance = ga.get_fitness(lx, ly)
    ga.evolve(fitness)
    best_idx = np.argmax(fitness)
    print(‘Gen:‘, generation, ‘| best fit: %.2f‘ % fitness[best_idx],)

    env.plotting(lx[best_idx], ly[best_idx], total_distance[best_idx])

plt.ioff()
plt.show()

参考链接：莫烦PYTHON-旅行商问题（Travel Sales Problem）