nsga2算法求解vrp问题的python代码，并绘图

NSGA-II（Non-dominated Sorting Genetic Algorithm II）是一个常用的多目标优化算法，可以用于解决VRP问题。

以下是使用Python实现NSGA-II算法求解VRP问题的示例代码：

import numpy as np
import random
import matplotlib.pyplot as plt

# 参数设置
POP_SIZE = 50  # 种群大小
MUT_PROB = 0.1  # 变异概率
CROSS_PROB = 0.8  # 交叉概率
MAX_GEN = 100  # 最大迭代次数
N = 20  # 客户数
K = 5  # 车辆数
Q = 1  # 车辆容量
L = 100  # 路径长度限制
ALPHA = 1  # 距离惩罚系数

# 生成随机客户坐标和需求量
x = np.random.rand(N+1) * L
y = np.random.rand(N+1) * L
d = np.random.randint(1, Q+1, N+1)

# 计算客户之间的距离
dist = np.zeros((N+1, N+1))
for i in range(N+1):
    for j in range(i+1, N+1):
        dist[i][j] = dist[j][i] = np.sqrt((x[i]-x[j])**2 + (y[i]-y[j])**2)

# NSGA-II算法实现
class Individual:
    def __init__(self):
        self.x = [0] * (N+1)
        self.f = [0] * 2

    def evaluate(self):
        # 计算每辆车的路径长度和
        total_dist = [0] * K
        load = [0] * K
        for i in range(1, N+1):
            k = self.x[i]
            total_dist[k] += dist[self.x[i-1]][i] + ALPHA * max(load[k]+d[i]-Q, 0)
            load[k] += d[i]
        self.f[0] = max(total_dist)
        self.f[1] = sum(total_dist)

    def mutate(self):
        for i in range(1, N+1):
            if random.random() < MUT_PROB:
                self.x[i] = random.randint(0, K-1)

    def crossover(self, other):
        if random.random() < CROSS_PROB:
            c = random.randint(1, N-1)
            for i in range(c, N+1):
                self.x[i], other.x[i] = other.x[i], self.x[i]

class Population:
    def __init__(self):
        self.individuals = [Individual() for _ in range(POP_SIZE)]

    def evaluate(self):
        for individual in self.individuals:
            individual.evaluate()

    def select(self):
        fronts = [[]]
        S = [[] for _ in range(POP_SIZE)]
        n = [0] * POP_SIZE
        rank = [0] * POP_SIZE
        for p in range(POP_SIZE):
            for q in range(POP_SIZE):
                if self.individuals[p].f[0] > self.individuals[q].f[0] and self.individuals[p].f[1] > self.individuals[q].f[1]:
                    S[p].append(q)
                elif self.individuals[p].f[0] < self.individuals[q].f[0] and self.individuals[p].f[1] < self.individuals[q].f[1]:
                    n[p] += 1
            if n[p] == 0:
                rank[p] = 0
                fronts[0].append(p)
        i = 0
        while fronts[i]:
            Q = []
            for p in fronts[i]:
                for q in S[p]:
                    n[q] -= 1
                    if n[q] == 0:
                        rank[q] = i + 1
                        Q.append(q)
            i += 1
            fronts.append(Q)
        del fronts[-1]
        selected = []
        for f in fronts:
            if len(selected) + len(f) <= POP_SIZE:
                selected += f
            else:
                c = len(selected) + len(f) - POP_SIZE
                dists = []
                for p in f:
                    dist = 0
                    for q in selected:
                        dist += np.sqrt((self.individuals[p].f[0]-self.individuals[q].f[0])**2 + (self.individuals[p].f[1]-self.individuals[q].f[1])**2)
                    dists.append(dist)
                for p in np.argsort(dists)[:c]:
                    selected.append(f[p])
                break
        return [self.individuals[i] for i in selected]

    def evolve(self):
        self.evaluate()
        offspring = []
        for _ in range(POP_SIZE):
            p1, p2 = random.sample(self.select(), 2)
            child = Individual()
            child.x = p1.x.copy()
            child.crossover(p2)
            child.mutate()
            offspring.append(child)
        self.individuals += offspring
        self.evaluate()
        fronts = [[]]
        for p in range(POP_SIZE*2):
            for q in range(POP_SIZE*2):
                if self.individuals[p].f[0] > self.individuals[q].f[0] and self.individuals[p].f[1] > self.individuals[q].f[1]:
                    self.individuals[p].dom += 1
                elif self.individuals[p].f[0] < self.individuals[q].f[0] and self.individuals[p].f[1] < self.individuals[q].f[1]:
                    self.individuals[p].dom -= 1
            if self.individuals[p].dom == 0:
                self.individuals[p].rank = 0
                fronts[0].append(p)
        i = 0
        while fronts[i]:
            Q = []
            for p in fronts[i]:
                for q in range(POP_SIZE*2):
                    if self.individuals[p].dom == 1 and self.individuals[q].dom == -1:
                        self.individuals[q].dom = 0
                        self.individuals[q].rank = i + 1
                        Q.append(q)
            i += 1
            fronts.append(Q)
        del fronts[-1]
        self.individuals = [self.individuals[i] for f in fronts for i in f]
        while len(self.individuals) > POP_SIZE:
            del self.individuals[-1]

    def show(self):
        plt.scatter([ind.f[0] for ind in self.individuals], [ind.f[1] for ind in self.individuals])
        plt.xlabel('Max Distance')
        plt.ylabel('Total Distance')
        plt.show()

# 初始化种群
pop = Population()
for ind in pop.individuals:
    ind.x = [random.randint(0, K-1) for _ in range(N+1)]

# 进化
for gen in range(MAX_GEN):
    pop.evolve()

# 显示结果
pop.show()

上面的代码中使用了Matplotlib库来绘制种群的Pareto前沿线。在此之前需要先执行以下代码安装Matplotlib：

!pip install matplotlib

执行完毕后，可以调用NSGA-II算法求解VRP问题，并绘制出结果。其中，每个点代表一个解，横坐标为最大路径长度，纵坐标为总路径长度。Pareto前沿线上的点都是非支配解，即没有其他解能同时达到比它更好的两个目标值。

注意：由于NSGA-II算法使用了随机性，每次运行的结果可能会有所不同。