netflow.py


#!/usr/bin/env python3.11

# 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(f"\nOptimal flows for {h}:")
        for i, j in arcs:
            if solution[h, i, j] > 0:
                print(f"{i} -> {j}: {solution[h, i, j]:g}")

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