# Vehicle Routing Problem OR-tools - Distance between solutions / vehicles

I have an alternative problem which I reckon can be solved using OR-tools. For this problem, apart from the distance between nodes within a solution, I also need the distance between the n-th node of each solution.

For instance, my sample below yields:

• v1: 2 -> 1 -> 2
• v2: 4 -> 12 -> 11 -> 15 -> 3 -> 4
• v3: 6 -> 16 -> 10 -> 9 -> 5 -> 6
• v4: 8 -> 0 -> 7 -> 13 -> 14 -> 8

Then, i would need the distance between all v[n] for each n, and 0 if nothing is planned on that index for one of the vehicles. For the first n, this would be

• distance from 2-4 i.e. distance from v1[n] to v2[n]
• distance from 2-6
• distance from 2-8
• distance from 4-6
• distance from 4-8
• distance from 6-8

These would be added to the total distance travelled within a solution. The overall objective is to minimze the total distance.

Is something like that feasible and possible? Below is the standard OR-tools sample adjusted for start & end points per vehicle

from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcp

def create_data_model():
"""Stores the data for the problem."""
data = {}
data["distance_matrix"] = [
# fmt: off
[0, 548, 776, 696, 582, 274, 502, 194, 308, 194, 536, 502, 388, 354, 468, 776, 662],
[548, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674, 1016, 868, 1210],
[776, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164, 1130, 788, 1552, 754],
[696, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822, 1164, 560, 1358],
[582, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708, 1050, 674, 1244],
[274, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628, 514, 1050, 708],
[502, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856, 514, 1278, 480],
[194, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320, 662, 742, 856],
[308, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662, 320, 1084, 514],
[194, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388, 274, 810, 468],
[536, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764, 730, 388, 1152, 354],
[502, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114, 308, 650, 274, 844],
[388, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194, 536, 388, 730],
[354, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0, 342, 422, 536],
[468, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536, 342, 0, 764, 194],
[776, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274, 388, 422, 764, 0, 798],
[662, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730, 536, 194, 798, 0],
# fmt: on
]
data["num_vehicles"] = 4

# custom start and end index, always return back
data["starts"] = [2,4,6,8]
data["ends"] = [2,4,6,8]
return data

def print_solution(data, manager, routing, solution):
"""Prints solution on console."""
print(f"Objective: {solution.ObjectiveValue()}")
max_route_distance = 0
for vehicle_id in range(data["num_vehicles"]):
index = routing.Start(vehicle_id)
plan_output = f"Route for vehicle {vehicle_id}:\n"
route_distance = 0
while not routing.IsEnd(index):
plan_output += f" {manager.IndexToNode(index)} -> "
previous_index = index
index = solution.Value(routing.NextVar(index))
route_distance += routing.GetArcCostForVehicle(
previous_index, index, vehicle_id
)
plan_output += f"{manager.IndexToNode(index)}\n"
plan_output += f"Distance of the route: {route_distance}m\n"
print(plan_output)
max_route_distance = max(route_distance, max_route_distance)
print(f"Maximum of the route distances: {max_route_distance}m")

def main():
"""Entry point of the program."""
# Instantiate the data problem.
data = create_data_model()

# Create the routing index manager.
manager = pywrapcp.RoutingIndexManager(
len(data["distance_matrix"]), data["num_vehicles"], data["starts"], data['ends']
)

# Create Routing Model.
routing = pywrapcp.RoutingModel(manager)

# Create and register a transit callback.
def distance_callback(from_index, to_index):
"""Returns the distance between the two nodes."""
# Convert from routing variable Index to distance matrix NodeIndex.
from_node = manager.IndexToNode(from_index)
to_node = manager.IndexToNode(to_index)
return data["distance_matrix"][from_node][to_node]

transit_callback_index = routing.RegisterTransitCallback(distance_callback)

# Define cost of each arc.
routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index)

dimension_name = "Distance"
transit_callback_index,
0,  # no slack
3000,  # vehicle maximum travel distance
True,  # start cumul to zero
dimension_name,
)
distance_dimension = routing.GetDimensionOrDie(dimension_name)
distance_dimension.SetGlobalSpanCostCoefficient(100)

# Setting first solution heuristic.
search_parameters = pywrapcp.DefaultRoutingSearchParameters()
search_parameters.first_solution_strategy = (
routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC
)

# Solve the problem.
solution = routing.SolveWithParameters(search_parameters)

# Print solution on console.
if solution:
print_solution(data, manager, routing, solution)
else:
print("No solution found !")

if __name__ == "__

main__":
main()

• I'm sceptical. While the routing-solver basically allows everything to model in regards to feasibility through it's cp-solver, there are only a few objective-related patterns one can use (adding funcs is virtually impossible). Furthermore, the solver core is based on the arc-flow / next-var formulation (as i would call it) which basically has no information about absolute positioning within some tour. Deducing the index needs accumulation of fake counting-dimensions. And even then... consumption and further processing is hard. Therefore i don't think there is a general-purpose "nice" solution. Commented Jan 12 at 23:19