Copyright 2025 Google LLC.

Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

max_flow_taha

<table align="left"> <td> <a href="https://colab.research.google.com/github/google/or-tools/blob/main/examples/notebook/contrib/max_flow_taha.ipynb">Run in Google Colab</a> </td> <td> <a href="https://github.com/google/or-tools/blob/main/examples/contrib/max_flow_taha.py">View source on GitHub</a> </td> </table>

First, you must install ortools package in this colab.

python

%pip install ortools

Max flow problem in Google CP Solver.

From Taha 'Introduction to Operations Research', Example 6.4-2

Translated from the AMPL code at http://taha.ineg.uark.edu/maxflo.txt

Compare with the following model:

MiniZinc: http://www.hakank.org/minizinc/max_flow_taha.mzn

This model was created by Hakan Kjellerstrand ([email protected]) Also see my other Google CP Solver models: http://www.hakank.org/google_or_tools/

python

from ortools.constraint_solver import pywrapcp


def main():

  # Create the solver.
  solver = pywrapcp.Solver('Max flow problem, Taha')

  #
  # data
  #
  n = 5
  start = 0
  end = n - 1

  nodes = list(range(n))

  # cost matrix
  c = [[0, 20, 30, 10, 0], [0, 0, 40, 0, 30], [0, 0, 0, 10, 20],
       [0, 0, 5, 0, 20], [0, 0, 0, 0, 0]]

  #
  # declare variables
  #
  x = {}
  for i in nodes:
    for j in nodes:
      x[i, j] = solver.IntVar(0, c[i][j], 'x[%i,%i]' % (i, j))

  x_flat = [x[i, j] for i in nodes for j in nodes]
  out_flow = [solver.IntVar(0, 10000, 'out_flow[%i]' % i) for i in nodes]
  in_flow = [solver.IntVar(0, 10000, 'in_flow[%i]' % i) for i in nodes]

  total = solver.IntVar(0, 10000, 'z')

  #
  # constraints
  #
  cost_sum = solver.Sum([x[start, j] for j in nodes if c[start][j] > 0])
  solver.Add(total == cost_sum)

  for i in nodes:
    in_flow_sum = solver.Sum([x[j, i] for j in nodes if c[j][i] > 0])
    solver.Add(in_flow[i] == in_flow_sum)

    out_flow_sum = solver.Sum([x[i, j] for j in nodes if c[i][j] > 0])
    solver.Add(out_flow[i] == out_flow_sum)

  # in_flow == out_flow
  for i in nodes:
    if i != start and i != end:
      solver.Add(out_flow[i] - in_flow[i] == 0)

  s1 = [x[i, start] for i in nodes if c[i][start] > 0]
  if len(s1) > 0:
    solver.Add(solver.Sum([x[i, start] for i in nodes if c[i][start] > 0] == 0))

  s2 = [x[end, j] for j in nodes if c[end][j] > 0]
  if len(s2) > 0:
    solver.Add(solver.Sum([x[end, j] for j in nodes if c[end][j] > 0]) == 0)

  # objective: maximize total cost
  objective = solver.Maximize(total, 1)

  #
  # solution and search
  #
  db = solver.Phase(x_flat, solver.INT_VAR_DEFAULT, solver.ASSIGN_MAX_VALUE)

  solver.NewSearch(db, [objective])
  num_solutions = 0
  while solver.NextSolution():
    num_solutions += 1
    print('total:', total.Value())
    print('in_flow:', [in_flow[i].Value() for i in nodes])
    print('out_flow:', [out_flow[i].Value() for i in nodes])
    for i in nodes:
      for j in nodes:
        print('%2i' % x[i, j].Value(), end=' ')
      print()
    print()

  print('num_solutions:', num_solutions)
  print('failures:', solver.Failures())
  print('branches:', solver.Branches())
  print('WallTime:', solver.WallTime(), 'ms')


main()