Vehicle Routing Problem (VRP / CVRP) Solver

VRP variants supported

CVRP

Capacitated VRP — each vehicle has a weight/volume limit

VRPTW

VRP with Time Windows — customers must be visited within a time range

MDVRP

Multi-depot VRP — vehicles start and end at different depots

VRPPD

VRP with Pickups & Deliveries — paired pickup/drop locations

Basic CVRP example

python

import nerox
import numpy as np

client = nerox.Client()

n_customers = 50
n_vehicles = 5
vehicle_capacity = 100

# Random demand and distance matrix
rng = np.random.default_rng(0)
demands = rng.integers(5, 25, size=n_customers + 1)  # +1 for depot
demands[0] = 0   # depot has no demand
coords = rng.random((n_customers + 1, 2)) * 100
dist = np.linalg.norm(coords[:, None] - coords[None, :], axis=-1)

job = client.optimize.cvrp(
    distance_matrix=dist,
    demands=demands,
    vehicle_capacity=vehicle_capacity,
    n_vehicles=n_vehicles,
    depot=0,              # index of depot node
    solver="gpu",
)

result = job.wait()
print(f"Total distance: {result.objective:.1f}")
for i, route in enumerate(result.routes):
    print(f"Vehicle {i}: depot → {' → '.join(str(c) for c in route)} → depot")

Time window constraints

python

# Each customer has an [earliest, latest] arrival window
time_windows = [
    (0, 0),       # depot: always open
    (8, 12),      # customer 1: must arrive between 8:00–12:00
    (10, 14),     # customer 2
    # ...
]

job = client.optimize.cvrp(
    distance_matrix=dist,
    demands=demands,
    vehicle_capacity=vehicle_capacity,
    n_vehicles=n_vehicles,
    depot=0,
    time_windows=time_windows,
    service_times=[0] + [15] * n_customers,  # 15-min service per stop
    solver="gpu",
)

How NEROX encodes VRP as QUBO

Vehicle routing is internally reformulated as a QUBO by introducing binary variables x[v, c, t] = 1 if vehicle v visits customer c at position t in its route. Capacity, ordering, and depot-return constraints are encoded as quadratic penalty terms. The objective minimizes total edge cost weighted by the route assignment variables.

This formulation is handled automatically — you supply the problem parameters, and NEROX builds the QUBO, solves it on GPU, and decodes the result back into routes.

Real-world scale

NEROX handles fleet dispatch problems with up to 500 customers and 50 vehicles in a single API call. For larger instances, use the Hybrid Solver which decomposes the route assignment into geographic clusters before optimizing each cluster's tour independently.