netflow.py


#!/usr/bin/env python3.7

# Copyright 2020, Gurobi Optimization, LLC

# Solve a multi-commodity flow problem.  Two products ('Pencils' and 'Pens')
# are produced in 2 cities ('Detroit' and 'Denver') and must be sent to
# warehouses in 3 cities ('Boston', 'New York', and 'Seattle') to
# satisfy demand ('inflow[h,i]').
#
# Flows on the transportation network must respect arc capacity constraints
# ('capacity[i,j]'). The objective is to minimize the sum of the arc
# transportation costs ('cost[i,j]').

import gurobipy as gp
from gurobipy import GRB

# Base data
commodities = ['Pencils', 'Pens']
nodes = ['Detroit', 'Denver', 'Boston', 'New York', 'Seattle']

arcs, capacity = gp.multidict({
    ('Detroit', 'Boston'):   100,
    ('Detroit', 'New York'):  80,
    ('Detroit', 'Seattle'):  120,
    ('Denver',  'Boston'):   120,
    ('Denver',  'New York'): 120,
    ('Denver',  'Seattle'):  120})

# Cost for triplets commodity-source-destination
cost = {
    ('Pencils', 'Detroit', 'Boston'):   10,
    ('Pencils', 'Detroit', 'New York'): 20,
    ('Pencils', 'Detroit', 'Seattle'):  60,
    ('Pencils', 'Denver',  'Boston'):   40,
    ('Pencils', 'Denver',  'New York'): 40,
    ('Pencils', 'Denver',  'Seattle'):  30,
    ('Pens',    'Detroit', 'Boston'):   20,
    ('Pens',    'Detroit', 'New York'): 20,
    ('Pens',    'Detroit', 'Seattle'):  80,
    ('Pens',    'Denver',  'Boston'):   60,
    ('Pens',    'Denver',  'New York'): 70,
    ('Pens',    'Denver',  'Seattle'):  30}

# Demand for pairs of commodity-city
inflow = {
    ('Pencils', 'Detroit'):   50,
    ('Pencils', 'Denver'):    60,
    ('Pencils', 'Boston'):   -50,
    ('Pencils', 'New York'): -50,
    ('Pencils', 'Seattle'):  -10,
    ('Pens',    'Detroit'):   60,
    ('Pens',    'Denver'):    40,
    ('Pens',    'Boston'):   -40,
    ('Pens',    'New York'): -30,
    ('Pens',    'Seattle'):  -30}

# Create optimization model
m = gp.Model('netflow')

# Create variables
flow = m.addVars(commodities, arcs, obj=cost, name="flow")

# Arc-capacity constraints
m.addConstrs(
    (flow.sum('*', i, j) <= capacity[i, j] for i, j in arcs), "cap")

# Equivalent version using Python looping
# for i, j in arcs:
#   m.addConstr(sum(flow[h, i, j] for h in commodities) <= capacity[i, j],
#               "cap[%s, %s]" % (i, j))


# Flow-conservation constraints
m.addConstrs(
    (flow.sum(h, '*', j) + inflow[h, j] == flow.sum(h, j, '*')
        for h in commodities for j in nodes), "node")

# Alternate version:
# m.addConstrs(
#   (gp.quicksum(flow[h, i, j] for i, j in arcs.select('*', j)) + inflow[h, j] ==
#     gp.quicksum(flow[h, j, k] for j, k in arcs.select(j, '*'))
#     for h in commodities for j in nodes), "node")

# Compute optimal solution
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    solution = m.getAttr('x', flow)
    for h in commodities:
        print('\nOptimal flows for %s:' % h)
        for i, j in arcs:
            if solution[h, i, j] > 0:
                print('%s -> %s: %g' % (i, j, solution[h, i, j]))

Try Gurobi for Free

Choose the evaluation license that fits you best, and start working with our Expert Team for technical guidance and support.

Evaluation License
Get a free, full-featured license of the Gurobi Optimizer to experience the performance, support, benchmarking and tuning services we provide as part of our product offering.
Academic License
Gurobi supports the teaching and use of optimization within academic institutions. We offer free, full-featured copies of Gurobi for use in class, and for research.
Cloud Trial

Request free trial hours, so you can see how quickly and easily a model can be solved on the cloud.

Search