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netflow.py
#!/usr/bin/env python3.7
# Copyright 2023, 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 supply/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}
# Supply (> 0) and demand (< 0) 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]))
