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### netflow.py

#!/usr/bin/env python3.7

# Copyright 2021, 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
(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
(flow.sum(h, '*', j) + inflow[h, j] == flow.sum(h, j, '*')
for h in commodities for j in nodes), "node")

# Alternate version:
#   (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]))


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