使用遗传算法进行问题建模求解

### 使用遗传算法解决多车场车辆路径问题

#### 背景
一家物流公司位于济南，有一个配送中心负责向13个客户提供服务。每辆车的最大载重量为2吨（即2000公斤），具体客户需求及配送点间距离见下表：

| | 配送中心1 | 客户1 | 客户2 | ... | 客户13 |
|--|-------|-------|-----|--------|
| **配送中心1** | 0 | 15 | 11 | ... | 11 |
| **客户1** | 15 | 0 | 19 | ... | 20 |
| **客户2** | 11 | 19 | 0 | ... | 9 |
| ... | ... | ... | ... | ... | ... |
| **客户13** | 11 | 20 | 9 | ... | 0 |

客户物品重量分别为：564, 509, 503, 350, 297, 355, 595, 446, 292, 343, 443, 233, 227 公斤。

#### 目标
通过遗传算法设计合理的配送路线，使总配送距离最短。

#### 步骤

1. **定义染色体**
- 每条染色体表示一个可能的配送方案。
- 染色体由多个基因组成，每个基因代表一个客户的编号。
- 例如，一条染色体可以是 `[1, 3, 5, 2, 4, 6, 7, 8, 9, 10, 11, 12, 13]`，表示按此顺序访问客户。

2. **适应度函数**
- 计算每条染色体对应的总配送距离。
- 总配送距离包括从配送中心到第一个客户、各个客户之间的距离以及最后一个客户返回配送中心的距离。
- 适应度值越小，表示该方案越好。

3. **选择操作**
- 采用轮盘赌选择法或锦标赛选择法，选择适应度较高的染色体进入下一代。

4. **交叉操作**
- 选择两条父代染色体，生成子代染色体。
- 可以使用部分映射交叉（PMX）、顺序交叉（OX）等方法。

5. **变异操作**
- 对选中的染色体进行随机交换、插入或逆序等操作，增加种群多样性。

6. **终止条件**
- 达到预定的迭代次数。
- 连续若干代适应度没有显著提高。

#### 示例代码

以下是一个简单的Python示例，展示了如何实现上述步骤：

import random
import numpy as np

# 定义参数
num_vehicles = 3
max_weight = 2000
distances = [
    [0, 15, 11, 13, 20, 7, 8, 5, 12, 15, 7, 19, 6, 11],
    [15, 0, 19, 7, 15, 19, 17, 13, 7, 14, 6, 7, 9, 20],
    [11, 19, 0, 18, 20, 8, 14, 17, 6, 8, 6, 13, 18, 9],
    [13, 7, 18, 0, 6, 11, 13, 14, 15, 13, 15, 11, 16, 6],
    [20, 15, 20, 6, 0, 17, 14, 6, 7, 19, 15, 14, 17, 16],
    [7, 19, 8, 11, 17, 0, 14, 6, 7, 19, 13, 9, 10, 12],
    [8, 17, 14, 13, 14, 14, 0, 8, 8, 15, 7, 15, 12, 5],
    [5, 13, 17, 14, 6, 6, 8, 0, 11, 16, 16, 13, 10, 20],
    [12, 7, 6, 15, 7, 7, 8, 11, 0, 16, 10, 19, 17, 17],
    [15, 14, 18, 13, 19, 19, 15, 16, 16, 0, 7, 16, 7, 19],
    [7, 6, 6, 15, 15, 13, 7, 16, 10, 7, 0, 10, 17, 7],
    [19, 7, 13, 11, 14, 9, 15, 13, 19, 16, 10, 0, 9, 14],
    [6, 9, 18, 16, 17, 10, 12, 10, 17, 7, 17, 9, 0, 13],
    [11, 20, 9, 6, 16, 12, 5, 20, 17, 19, 7, 14, 13, 0]
]

weights = [564, 509, 503, 350, 297, 355, 595, 446, 292, 343, 443, 233, 227]

def calculate_total_distance(route):
    total_distance = distances[0][route[0]]  # 从配送中心到第一个客户
    for i in range(len(route) - 1):
        total_distance += distances[route[i]][route[i + 1]]
    total_distance += distances[route[-1]][0]  # 最后一个客户返回配送中心
    return total_distance

def generate_initial_population(pop_size):
    population = []
    for _ in range(pop_size):
        route = list(range(1, len(weights) + 1))
        random.shuffle(route)
        population.append(route)
    return population

def select_parents(population, fitnesses, num_parents):
    parents = []
    for _ in range(num_parents):
        tournament = random.sample(list(zip(population, fitnesses)), 5)
        best_parent = min(tournament, key=lambda x: x[1])[0]
        parents.append(best_parent)
    return parents

def crossover(parents, offspring_size):
    offspring = []
    while len(offspring) < offspring_size:
        parent1, parent2 = random.sample(parents, 2)
        start, end = sorted(random.sample(range(len(parent1)), 2))
        child = [-1] * len(parent1)
        child[start:end+1] = parent1[start:end+1]
        idx = 0
        for gene in parent2:
            if gene not in child:
                while child[idx] != -1:
                    idx += 1
                child[idx] = gene
        offspring.append(child)
    return offspring

def mutate(individual, mutation_rate=0.01):
    for i in range(len(individual)):
        if random.random() < mutation_rate:
            j = random.randint(0, len(individual) - 1)
            individual[i], individual[j] = individual[j], individual[i]
    return individual

def genetic_algorithm(pop_size=100, num_generations=1000, mutation_rate=0.01):
    population = generate_initial_population(pop_size)
    for generation in range(num_generations):
        fitnesses = [calculate_total_distance(route) for route in population]
        parents = select_parents(population, fitnesses, pop_size // 2)
        offspring = crossover(parents, pop_size)
        for i in range(len(offspring)):
            offspring[i] = mutate(offspring[i], mutation_rate)
        population = offspring
    best_route = min(population, key=lambda x: calculate_total_distance(x))
    return best_route, calculate_total_distance(best_route)

best_route, best_distance = genetic_algorithm()
print("最佳配送路线:", best_route)
print("总配送距离:", best_distance)

#### 结论
通过上述遗传算法，可以有效地找到一个较优的配送方案，使得总配送距离最小。