example example

Our example solves a multi-commodity flow model on a small network. In the example, two commodities (Pencils and Pens) are produced in two cities (Detroit and Denver), and must be shipped to warehouses in three cities (Boston, New York, and Seattle) to satisfy given demand. Each arc in the transportation network has a cost associated with it, and a total capacity.

This is the complete source code for our example (also available in <installdir>/examples/python/

from gurobipy import *

# Model data

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

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

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 }

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 = Model('netflow')

# Create variables
flow = {}
for h in commodities:
    for i,j in arcs:
        flow[h,i,j] = m.addVar(ub=capacity[i,j], obj=cost[h,i,j],
                               name='flow_%s_%s_%s' % (h, i, j))

# Arc capacity constraints
for i,j in arcs:
    m.addConstr(quicksum(flow[h,i,j] for h in commodities) <= capacity[i,j],
                'cap_%s_%s' % (i, j))

# Flow conservation constraints
for h in commodities:
    for j in nodes:
          quicksum(flow[h,i,j] for i,j in'*',j)) +
              inflow[h,j] ==
          quicksum(flow[h,j,k] for j,k in,'*')),
                   'node_%s_%s' % (h, j))

# Compute optimal solution

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