示例网络，箭头上的数字代表弧允许通过的最大流量

增加两个虚拟节点，节点0和节点7

import gurobipy as gp
from gurobipy import GRB

# Input the information of the network
dict_capacity = {(0, 1): 99999,
                 (1, 2): 70, (1, 3): 100, (1, 4): 90,
                 (2, 6): 80,
                 (3, 4): 40, (3, 5): 70,
                 (4, 5): 40, (4, 6): 100,
                 (5, 6): 90,
                 (6, 7): 99999
                 }

arc, capacity = gp.multidict(dict_capacity)

# Construct the model
m = gp.Model("maxflow")

# Decision variables
X = m.addVars(arc, name='X')

# Add constraints
for i, j in arc:
    # The volume of flow cannot exceed the capacity of the arc
    m.addConstr(X[i, j] <= capacity[i, j])
    if i == 0 or j == 7:
        continue
    else:
        # Flow balance constraint
        m.addConstr(X.sum(i, '*') == X.sum('*', i))

# Define the objective function
m.setObjective(X.sum(1, '*'), sense=GRB.MAXIMIZE)

# Solve the model
m.optimize()

# Output the results
print('*' * 60)
print("The objective value is：", m.objVal)
print('The flow in each arc is:')
for i, j in arc:
    print("Node %d --> Node %d: %d" % (i, j, X[i, j].x))

控制台输出结果

Using license file C:\Users\86158\gurobi.lic
Gurobi Optimizer version 9.1.1 build v9.1.1rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 20 rows, 11 columns and 42 nonzeros
Model fingerprint: 0x43a077e0
Coefficient statistics:
  Matrix range     [1e+00, 1e+00]
  Objective range  [1e+00, 1e+00]
  Bounds range     [0e+00, 0e+00]
  RHS range        [4e+01, 1e+05]
Presolve removed 17 rows and 5 columns
Presolve time: 0.00s
Presolved: 3 rows, 6 columns, 9 nonzeros

Iteration    Objective       Primal Inf.    Dual Inf.      Time
       0    2.6000000e+02   3.125000e+01   0.000000e+00      0s
       3    2.6000000e+02   0.000000e+00   0.000000e+00      0s

Solved in 3 iterations and 0.00 seconds
Optimal objective  2.600000000e+02
************************************************************
The objective value is： 260.0
The flow in each arc is:
Node 0 --> Node 1: 260
Node 1 --> Node 2: 70
Node 1 --> Node 3: 100
Node 1 --> Node 4: 90
Node 2 --> Node 6: 70
Node 3 --> Node 4: 40
Node 3 --> Node 5: 60
Node 4 --> Node 5: 30
Node 4 --> Node 6: 100
Node 5 --> Node 6: 90
Node 6 --> Node 7: 0

Process finished with exit code 0