Python中使用线性规划和网络单纯形法解决最大流问题

# 导入模块
import pulp

# 创建问题
problem = pulp.LpProblem("Max_Flow", pulp.LpMaximize)

# 创建决策变量
x_a_b = pulp.LpVariable('x_a_b', lowBound=0, cat='Integer')  # a -> b的流量
x_a_c = pulp.LpVariable('x_a_c', lowBound=0, cat='Integer')  # a -> c的流量
x_b_c = pulp.LpVariable('x_b_c', lowBound=0, cat='Integer')  # b -> c的流量
x_b_d = pulp.LpVariable('x_b_d', lowBound=0, cat='Integer')  # b -> d的流量
x_c_d = pulp.LpVariable('x_c_d', lowBound=0, cat='Integer')  # c -> d的流量

# 添加约束条件
problem += x_a_b + x_a_c <= 14  # 限制a的出流
problem += x_b_d <= 25  # 限制d的入流
problem += x_a_b <= 20  # 限制a -> b的流量
problem += x_a_c <= 10  # 限制a -> c的流量
problem += x_b_c <= 15  # 限制b -> c的流量
problem += x_c_d <= 30  # 限制c -> d的流量

# 添加目标函数
problem += x_a_b + x_a_c + x_b_c + x_b_d + x_c_d

# 求解问题
problem.solve()

# 打印结果
print("最大流为:", pulp.value(problem.objective))
print("决策变量结果:")
for v in problem.variables():
    print(v.name, "=", v.varValue)

# 导入模块
import networkx as nx
from scipy.optimize import linprog

# 创建图
G = nx.DiGraph()
G.add_node('a')
G.add_node('b')
G.add_node('c')
G.add_node('d')
G.add_edges_from([('a', 'b', {'capacity': 20}), ('a', 'c', {'capacity': 10}),
                  ('b', 'c', {'capacity': 15}), ('b', 'd', {'capacity': 25}),
                  ('c', 'd', {'capacity': 30})])

# 构建约束矩阵和目标向量
n = len(G.nodes())
A = []
b = []
c = []
s = 0  # 源点
t = n - 1  # 汇点
for i in range(n):
    a = [0] * n
    for j in G[i]:
        a[j] = G.get_edge_data(i, j)['capacity']
    A.append(a)
    if i == s:
        b.append(-1)
        c.append(0)
    elif i == t:
        b.append(1)
        c.append(0)
    else:
        b.append(0)
        c.append(1)
A = A[:-1]
b = b[:-1]

# 进行线性规划求解
res = linprog(c, A_eq=A, b_eq=b)

# 打印结果
print("最大流为:", int(round(res.fun)))