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TSP和VRP是在运输领域中常见的两个重要问题。这两个问题在不同的场景中都需要求解最优的路径或路线，以降低运输成本、优化资源利用效率。
MIP（混合整数规模模型）是一种常用的数学建模方法，可以有效地解决这些问题。对于小规模的TSP、VRP问题，可以使用MIP数学建模方法来快速求解。
本文介绍通过pulp来建立MIP模型，来求解TSP和VRP问题。
安装pulp
pip install pulp
TSP问题
1.导入包
###导入需要的包
import itertools
import numpy as np
import pandas as pd
from scipy.spatial import distance_matrix
import matplotlib
import matplotlib.pylab as plt
import seaborn as sns
import pulp
import warnings
warnings.filterwarnings("ignore")
2.生成模拟数据
# 生成TSP的样例数据
np.random.seed(0)#随机种子，固定数据
n_customer = 9
n_point = n_customer + 1
df = pd.DataFrame({
'x': np.random.randint(0, 100, n_point),
'y': np.random.randint(0, 100, n_point),
})
df.iloc[0]['x'] = 0
df.iloc[0]['y'] = 0
df
3.生成距离矩阵
# 生成距离矩阵
distances = pd.DataFrame(distance_matrix(df[['x', 'y']].values, df[['x', 'y']].values), index=df.index, columns=df.index).values
fig, ax = plt.subplots(figsize=(8, 7))
sns.heatmap(distances, ax=ax, cmap='Blues', annot=True, fmt='.0f', cbar=True, cbar_kws={"shrink": .3}, linewidths=.1)
plt.title('distance matrix')
plt.show()
4.查看网点分布图
# 查看网点分布图
plt.figure(figsize=(5, 5))
# draw problem state
for i, row in df.iterrows():
if i == 0:
plt.scatter(row['x'], row['y'], c='r')
plt.text(row['x'] + 1, row['y'] + 1, 'depot')
else:
plt.scatter(row['x'], row['y'], c='black')
plt.text(row['x'] + 1, row['y'] + 1, f'{i}')
plt.xlim([-10, 110])
plt.ylim([-10, 110])
plt.title('points: id')
plt.show()
5.建模及求解
%%time
# 建立模型
problem = pulp.LpProblem('tsp_mip', pulp.LpMinimize)
# 设置变量
x = pulp.LpVariable.dicts('x', ((i, j) for i in range(n_point) for j in range(n_point)), lowBound=0, upBound=1, cat='Binary')
# 定义每个网点的名称1-9
u = pulp.LpVariable.dicts('u', (i for i in range(n_point)), lowBound=1, upBound=n_point, cat='Integer')
# 设置目标函数
problem += pulp.lpSum(distances[j] * x[i, j] for i in range(n_point) for j in range(n_point))
# 设置约束
for i in range(n_point):
problem += x[i, i] == 0
for i in range(n_point):
problem += pulp.lpSum(x[i, j] for j in range(n_point)) == 1
problem += pulp.lpSum(x[j, i] for j in range(n_point)) == 1
# 消除子回路
for i in range(n_point):
for j in range(n_point):
if i != j and (i != 0 and j != 0):
problem += u - u[j] <= n_point * (1 - x[i, j]) - 1
# 求解
status = problem.solve()
# 输出结果
status, pulp.LpStatus[status], pulp.value(problem.objective)
6.画最优线路图
# 画图
plt.figure(figsize=(5, 5))
for i, row in df.iterrows():
if i == 0:
plt.scatter(row['x'], row['y'], c='r')
plt.text(row['x'] + 1, row['y'] + 1, 'depot')
else:
plt.scatter(row['x'], row['y'], c='black')
plt.text(row['x'] + 1, row['y'] + 1, f'{i}')
plt.xlim([-10, 110])
plt.ylim([-10, 110])
plt.title('points: id')
# 画线路
routes = [(i, j) for i in range(n_point) for j in range(n_point) if pulp.value(x[i, j]) == 1]
arrowprops = dict(arrowstyle='->', connectionstyle='arc3', edgecolor='blue')
for i, j in routes:
plt.annotate('', xy=[df.iloc[j]['x'], df.iloc[j]['y']], xytext=[df.iloc['x'], df.iloc['y']], arrowprops=arrowprops)
plt.show()
VRP问题
1.生成模拟数据
与TSP不同的是，生成了每个点的需求量、车辆的最大装载。
# 生成模拟数据
np.random.seed(0)
n_customer = 9
n_point = n_customer + 1
vehicle_capacity = 8
df = pd.DataFrame({
'x': np.random.randint(0, 100, n_point),
'y': np.random.randint(0, 100, n_point),
'demand': np.random.randint(1, 5, n_point),
})
df.iloc[0]['x'] = 0
df.iloc[0]['y'] = 0
df.iloc[0]['demand'] = 0
df
2.生成距离矩阵
与TSP问题生成方式相同。
3.画网点定位图
画图
plt.figure(figsize=(5, 5))
for i, row in df.iterrows():
if i == 0:
plt.scatter(row['x'], row['y'], c='r')
plt.text(row['x'] + 1, row['y'] + 1, 'depot')
else:
plt.scatter(row['x'], row['y'], c='black')
demand = row['demand']
plt.text(row['x'] + 1, row['y'] + 1, f'{i}({demand})')
plt.xlim([-10, 110])
plt.ylim([-10, 110])
plt.title('points: id(demand)')
plt.show()
4.建模和求解
%%time
demands = df['demand'].values
# 建立模型
problem = pulp.LpProblem('cvrp_mip', pulp.LpMinimize)
# 设定变量
x = pulp.LpVariable.dicts('x', ((i, j) for i in range(n_point) for j in range(n_point)), lowBound=0, upBound=1, cat='Binary')
n_vehicle = pulp.LpVariable('n_vehicle', lowBound=0, upBound=100, cat='Integer')
# 设定目标函数
problem += pulp.lpSum([distances[j] * x[i, j] for i in range(n_point) for j in range(n_point)])
# 设置约束
for i in range(n_point):
problem += x[i, i] == 0
for i in range(1, n_point):
problem += pulp.lpSum(x[j, i] for j in range(n_point)) == 1
problem += pulp.lpSum(x[i, j] for j in range(n_point)) == 1
problem += pulp.lpSum(x[i, 0] for i in range(n_point)) == n_vehicle
problem += pulp.lpSum(x[0, i] for i in range(n_point)) == n_vehicle
# 消除子环路
subtours = []
for length in range(2, n_point):
subtours += itertools.combinations(range(1, n_point), length)
for st in subtours:
demand = np.sum([demands for s in st])
arcs = [x[i, j] for i, j in itertools.permutations(st, 2)]
problem += pulp.lpSum(arcs) <= np.max([0, len(st) - np.ceil(demand / vehicle_capacity)])
# 模型求解
status = problem.solve()
# 输出结果
status, pulp.LpStatus[status], pulp.value(problem.objective)
5.查看使用了几辆车
pulp.value(n_vehicle)
6.画最终的线路图
# 画图
plt.figure(figsize=(5, 5))
# 画点
for i, row in df.iterrows():
if i == 0:
plt.scatter(row['x'], row['y'], c='r')
plt.text(row['x'] + 1, row['y'] + 1, 'depot')
else:
plt.scatter(row['x'], row['y'], c='black')
demand = row['demand']
plt.text(row['x'] + 1, row['y'] + 1, f'{i}({demand})')
plt.xlim([-10, 110])
plt.ylim([-10, 110])
plt.title('points: id(demand)')
# 画线路
cmap = matplotlib.cm.get_cmap('Dark2')
routes = [(i, j) for i in range(n_point) for j in range(n_point) if pulp.value(x[i, j]) == 1]
for v in range(int(pulp.value(n_vehicle))):
# 定义每一个车辆
vehicle_route = [routes[v]]
while vehicle_route[-1][1] != 0:
for p in routes:
if p[0] == vehicle_route[-1][1]:
vehicle_route.append(p)
break
# 不同颜色，画每一个车辆的线路
arrowprops = dict(arrowstyle='->', connectionstyle='arc3', edgecolor=cmap(v))
for i, j in vehicle_route:
plt.annotate('', xy=[df.iloc[j]['x'], df.iloc[j]['y']], xytext=[df.iloc['x'], df.iloc['y']], arrowprops=arrowprops)
plt.show()
本文详细介绍了使用pulp通过建立MIP模型来解决TSP、VRP问题的步骤，代码均直接提供，感兴趣的朋友可直接复制本文代码运行。
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