In diesem Abschnitt wird erläutert, wie Sie mit Einschränkungen umgehen, für die es keine durchführbare Lösung gibt. Wenn Sie beispielsweise eine VRP mit Kapazitätseinschränkungen erhalten, bei der die Gesamtnachfrage an allen Standorten die Gesamtkapazität der Fahrzeuge überschreitet, ist keine Lösung möglich. In solchen Fällen müssen die Fahrzeuge Besuche an einigen Standorten unterbrechen. Das Problem ist die Entscheidung, welche Besuche verworfen werden sollen.
Zur Lösung dieses Problems gelten an allen Standorten neue Kosten, sogenannte Strafen. Wird ein Besuch für einen Standort verworfen, wird die Strafe der Gesamtstrecke abgezogen. Der Matherechner findet dann eine Route, die die Gesamtstrecke plus die Summe der Strafen für alle verworfenen Standorte minimiert.
Nehmen wir als Beispiel die einfache VRP mit Kapazitätseinschränkungen aus der folgenden Grafik, in der die Zahlen neben den drei Standorten (außer dem Depot) Nachfragen darstellen.
Angenommen, es gibt nur ein Fahrzeug mit 50 Sitzplätzen. Es können nicht alle drei Standorte (A, B und C) besucht werden, da die Gesamtnachfrage 60 beträgt. Zur Lösung des Problems weisen Sie jedem Standort eine große Strafe, z. B. 100, zu. Nachdem festgestellt wurde, dass das Problem nicht umsetzbar ist, löscht der Rechner Standort B und gibt die folgende Route zurück: Depot -> A -> C -> Depot
Dies ist die kürzeste Route, die zwei der drei Orte besucht (Entfernung 55).
Größen für Strafen
Im obigen Beispiel haben wir Strafen ausgewählt, die größer sind als die Summe aller Entfernungen zwischen Standorten (mit Ausnahme des Depots). Nach dem Löschen eines Standorts, um das Problem umsetzbar zu machen, werden nach dem Löschen keine weiteren Standorte mehr entfernt, da dies zu einer weiteren Reduzierung der Reisestrecke führen würde.
Wenn Sie so viele Lieferungen wie möglich durchführen möchten, ist dies eine zufriedenstellende Lösung für das Problem.
Wenn Sie nicht so viele Zustellungen wie möglich vornehmen müssen, empfiehlt es sich, kleinere Strafen zuzuweisen. In diesem Fall kann der Löser mehr Standorte abgeben, als erforderlich sind, um das Problem möglich zu machen. Das ist z. B. dann der Fall, wenn zusätzliche Kosten über die grundlegenden Reisekosten hinaus für den Besuch bestimmter Orte anfallen.
Beispiel
Als Nächstes zeigen wir ein größeres Beispiel für eine VRP, die durch Strafen gelöst werden kann. Das Beispiel ähnelt dem vorherigen CVRP-Beispiel, aber dieses Mal haben wir einige der Anforderungen erhöht, was dazu geführt hat, dass einige Fahrzeuge die Website gelöscht haben.
Unten sehen Sie eine Grafik mit den Standorten und neuen Anforderungen.
Beispiel mit OR-Tools lösen
In den folgenden Abschnitten wird erläutert, wie Sie das Beispiel mit OR-Tools lösen.
Daten erstellen
Die Daten für dieses Beispiel umfassen die Daten aus dem vorherigen VRP-Beispiel. Außerdem werden die folgenden Anforderungen und Kapazitäten hinzugefügt:
Python
data["demands"] = [0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8]
data["vehicle_capacities"] = [15, 15, 15, 15]
C++
const std::vector<int64_t> demands{
0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8,
};
const std::vector<int64_t> vehicle_capacities{15, 15, 15, 15};
Java
public final long[] demands = {0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8};
public final long[] vehicleCapacities = {15, 15, 15, 15};
C#
public long[] Demands = { 0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8 };
public long[] VehicleCapacities = { 15, 15, 15, 15 };
Kapazitätsbeschränkungen und Strafen hinzufügen
Mit dem folgenden Code werden der Callback für die Nachfrage und die Kapazitätseinschränkungen sowie Strafen mit der Methode AddDisjunction hinzugefügt.
Python
def demand_callback(from_index):
"""Returns the demand of the node."""
# Convert from routing variable Index to demands NodeIndex.
from_node = manager.IndexToNode(from_index)
return data["demands"][from_node]
demand_callback_index = routing.RegisterUnaryTransitCallback(demand_callback)
routing.AddDimensionWithVehicleCapacity(
demand_callback_index,
0, # null capacity slack
data["vehicle_capacities"], # vehicle maximum capacities
True, # start cumul to zero
"Capacity",
)
# Allow to drop nodes.
penalty = 1000
for node in range(1, len(data["distance_matrix"])):
routing.AddDisjunction([manager.NodeToIndex(node)], penalty)
C++
const int demand_callback_index = routing.RegisterUnaryTransitCallback(
[&data, &manager](const int64_t from_index) -> int64_t {
// Convert from routing variable Index to demand NodeIndex.
const int from_node = manager.IndexToNode(from_index).value();
return data.demands[from_node];
});
routing.AddDimensionWithVehicleCapacity(
demand_callback_index, // transit callback index
int64_t{0}, // null capacity slack
data.vehicle_capacities, // vehicle maximum capacities
true, // start cumul to zero
"Capacity");
// Allow to drop nodes.
int64_t penalty{1000};
for (int i = 1; i < data.distance_matrix.size(); ++i) {
routing.AddDisjunction(
{manager.NodeToIndex(RoutingIndexManager::NodeIndex(i))}, penalty);
}
Java
final int demandCallbackIndex = routing.registerUnaryTransitCallback((long fromIndex) -> {
// Convert from routing variable Index to user NodeIndex.
int fromNode = manager.indexToNode(fromIndex);
return data.demands[fromNode];
});
routing.addDimensionWithVehicleCapacity(demandCallbackIndex, 0, // null capacity slack
data.vehicleCapacities, // vehicle maximum capacities
true, // start cumul to zero
"Capacity");
// Allow to drop nodes.
long penalty = 1000;
for (int i = 1; i < data.distanceMatrix.length; ++i) {
routing.addDisjunction(new long[] {manager.nodeToIndex(i)}, penalty);
}
C#
int demandCallbackIndex = routing.RegisterUnaryTransitCallback((long fromIndex) =>
{
// Convert from routing variable Index to
// demand NodeIndex.
var fromNode =
manager.IndexToNode(fromIndex);
return data.Demands[fromNode];
});
routing.AddDimensionWithVehicleCapacity(demandCallbackIndex, 0, // null capacity slack
data.VehicleCapacities, // vehicle maximum capacities
true, // start cumul to zero
"Capacity");
// Allow to drop nodes.
long penalty = 1000;
for (int i = 1; i < data.DistanceMatrix.GetLength(0); ++i)
{
routing.AddDisjunction(new long[] { manager.NodeToIndex(i) }, penalty);
}
In diesem Kontext ist eine Disjunktion einfach eine Variable, mit der der Rechner entscheidet, ob ein bestimmter Standort in die Lösung aufgenommen wird. In diesem Beispiel werden für jeden Standort dieselben Strafen hinzugefügt. Im Allgemeinen können Sie jedoch für verschiedene Standorte unterschiedliche Strafen festlegen.
Lösungsdrucker hinzufügen
Der unten dargestellte Lösungsdrucker ähnelt dem im CVRP-Beispiel, zeigt jedoch auch die verworfenen Standorte an.
Python
def print_solution(data, manager, routing, assignment):
"""Prints assignment on console."""
print(f"Objective: {assignment.ObjectiveValue()}")
# Display dropped nodes.
dropped_nodes = "Dropped nodes:"
for node in range(routing.Size()):
if routing.IsStart(node) or routing.IsEnd(node):
continue
if assignment.Value(routing.NextVar(node)) == node:
dropped_nodes += f" {manager.IndexToNode(node)}"
print(dropped_nodes)
# Display routes
total_distance = 0
total_load = 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
route_load = 0
while not routing.IsEnd(index):
node_index = manager.IndexToNode(index)
route_load += data["demands"][node_index]
plan_output += f" {node_index} Load({route_load}) -> "
previous_index = index
index = assignment.Value(routing.NextVar(index))
route_distance += routing.GetArcCostForVehicle(
previous_index, index, vehicle_id
)
plan_output += f" {manager.IndexToNode(index)} Load({route_load})\n"
plan_output += f"Distance of the route: {route_distance}m\n"
plan_output += f"Load of the route: {route_load}\n"
print(plan_output)
total_distance += route_distance
total_load += route_load
print(f"Total Distance of all routes: {total_distance}m")
print(f"Total Load of all routes: {total_load}")
C++
//! @brief Print the solution.
//! @param[in] data Data of the problem.
//! @param[in] manager Index manager used.
//! @param[in] routing Routing solver used.
//! @param[in] solution Solution found by the solver.
void PrintSolution(const DataModel& data, const RoutingIndexManager& manager,
const RoutingModel& routing, const Assignment& solution) {
// Display dropped nodes.
std::ostringstream dropped_nodes;
for (int64_t node = 0; node < routing.Size(); ++node) {
if (routing.IsStart(node) || routing.IsEnd(node)) continue;
if (solution.Value(routing.NextVar(node)) == node) {
dropped_nodes << " " << manager.IndexToNode(node).value();
}
}
LOG(INFO) << "Dropped nodes:" << dropped_nodes.str();
// Display routes
int64_t total_distance{0};
int64_t total_load{0};
for (int vehicle_id = 0; vehicle_id < data.num_vehicles; ++vehicle_id) {
int64_t index = routing.Start(vehicle_id);
LOG(INFO) << "Route for Vehicle " << vehicle_id << ":";
int64_t route_distance{0};
int64_t route_load{0};
std::ostringstream route;
while (!routing.IsEnd(index)) {
const int node_index = manager.IndexToNode(index).value();
route_load += data.demands[node_index];
route << node_index << " Load(" << route_load << ") -> ";
const int64_t previous_index = index;
index = solution.Value(routing.NextVar(index));
route_distance += routing.GetArcCostForVehicle(previous_index, index,
int64_t{vehicle_id});
}
LOG(INFO) << route.str() << manager.IndexToNode(index).value();
LOG(INFO) << "Distance of the route: " << route_distance << "m";
LOG(INFO) << "Load of the route: " << route_load;
total_distance += route_distance;
total_load += route_load;
}
LOG(INFO) << "Total distance of all routes: " << total_distance << "m";
LOG(INFO) << "Total load of all routes: " << total_load;
LOG(INFO) << "";
LOG(INFO) << "Advanced usage:";
LOG(INFO) << "Problem solved in " << routing.solver()->wall_time() << "ms";
}
Java
/// @brief Print the solution.
static void printSolution(
DataModel data, RoutingModel routing, RoutingIndexManager manager, Assignment solution) {
// Solution cost.
logger.info("Objective: " + solution.objectiveValue());
// Inspect solution.
// Display dropped nodes.
String droppedNodes = "Dropped nodes:";
for (int node = 0; node < routing.size(); ++node) {
if (routing.isStart(node) || routing.isEnd(node)) {
continue;
}
if (solution.value(routing.nextVar(node)) == node) {
droppedNodes += " " + manager.indexToNode(node);
}
}
logger.info(droppedNodes);
// Display routes
long totalDistance = 0;
long totalLoad = 0;
for (int i = 0; i < data.vehicleNumber; ++i) {
long index = routing.start(i);
logger.info("Route for Vehicle " + i + ":");
long routeDistance = 0;
long routeLoad = 0;
String route = "";
while (!routing.isEnd(index)) {
long nodeIndex = manager.indexToNode(index);
routeLoad += data.demands[(int) nodeIndex];
route += nodeIndex + " Load(" + routeLoad + ") -> ";
long previousIndex = index;
index = solution.value(routing.nextVar(index));
routeDistance += routing.getArcCostForVehicle(previousIndex, index, i);
}
route += manager.indexToNode(routing.end(i));
logger.info(route);
logger.info("Distance of the route: " + routeDistance + "m");
totalDistance += routeDistance;
totalLoad += routeLoad;
}
logger.info("Total Distance of all routes: " + totalDistance + "m");
logger.info("Total Load of all routes: " + totalLoad);
}
C#
/// <summary>
/// Print the solution.
/// </summary>
static void PrintSolution(in DataModel data, in RoutingModel routing, in RoutingIndexManager manager,
in Assignment solution)
{
Console.WriteLine($"Objective {solution.ObjectiveValue()}:");
// Inspect solution.
// Display dropped nodes.
string droppedNodes = "Dropped nodes:";
for (int index = 0; index < routing.Size(); ++index)
{
if (routing.IsStart(index) || routing.IsEnd(index))
{
continue;
}
if (solution.Value(routing.NextVar(index)) == index)
{
droppedNodes += " " + manager.IndexToNode(index);
}
}
Console.WriteLine("{0}", droppedNodes);
// Inspect solution.
long totalDistance = 0;
long totalLoad = 0;
for (int i = 0; i < data.VehicleNumber; ++i)
{
Console.WriteLine("Route for Vehicle {0}:", i);
long routeDistance = 0;
long routeLoad = 0;
var index = routing.Start(i);
while (routing.IsEnd(index) == false)
{
long nodeIndex = manager.IndexToNode(index);
routeLoad += data.Demands[nodeIndex];
Console.Write("{0} Load({1}) -> ", nodeIndex, routeLoad);
var previousIndex = index;
index = solution.Value(routing.NextVar(index));
routeDistance += routing.GetArcCostForVehicle(previousIndex, index, 0);
}
Console.WriteLine("{0}", manager.IndexToNode((int)index));
Console.WriteLine("Distance of the route: {0}m", routeDistance);
totalDistance += routeDistance;
totalLoad += routeLoad;
}
Console.WriteLine("Total Distance of all routes: {0}m", totalDistance);
Console.WriteLine("Total Load of all routes: {0}m", totalLoad);
}
Ausführen des Programms
Wenn Sie das Programm ausführen, wird die folgende Ausgabe zurückgegeben. Der Matherechner löscht die Positionen 6 und 15.
Objective: 7936 Dropped nodes: 6 15 Route for vehicle 0: 0 Load(0) -> 9 Load(1) -> 14 Load(7) -> 16 Load(15) -> 0 Load(15) Distance of the route: 1324m Load of the route: 15 Route for vehicle 1: 0 Load(0) -> 12 Load(2) -> 11 Load(3) -> 4 Load(9) -> 3 Load(12) -> 1 Load(13) -> 0 Load(13) Distance of the route: 1872m Load of the route: 13 Route for vehicle 2: 0 Load(0) -> 7 Load(8) -> 13 Load(14) -> 0 Load(14) Distance of the route: 868m Load of the route: 14 Route for vehicle 3: 0 Load(0) -> 8 Load(8) -> 10 Load(10) -> 2 Load(11) -> 5 Load(14) -> 0 Load(14) Distance of the route: 1872m Load of the route: 14 Total Distance of all routes: 5936m Total Load of all routes: 56
Hier ist ein Diagramm der Routen.
Programme abschließen
Hier sind die vollständigen Programme.
Python
"""Capacited Vehicles Routing Problem (CVRP)."""
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["demands"] = [0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8]
data["vehicle_capacities"] = [15, 15, 15, 15]
data["num_vehicles"] = 4
data["depot"] = 0
return data
def print_solution(data, manager, routing, assignment):
"""Prints assignment on console."""
print(f"Objective: {assignment.ObjectiveValue()}")
# Display dropped nodes.
dropped_nodes = "Dropped nodes:"
for node in range(routing.Size()):
if routing.IsStart(node) or routing.IsEnd(node):
continue
if assignment.Value(routing.NextVar(node)) == node:
dropped_nodes += f" {manager.IndexToNode(node)}"
print(dropped_nodes)
# Display routes
total_distance = 0
total_load = 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
route_load = 0
while not routing.IsEnd(index):
node_index = manager.IndexToNode(index)
route_load += data["demands"][node_index]
plan_output += f" {node_index} Load({route_load}) -> "
previous_index = index
index = assignment.Value(routing.NextVar(index))
route_distance += routing.GetArcCostForVehicle(
previous_index, index, vehicle_id
)
plan_output += f" {manager.IndexToNode(index)} Load({route_load})\n"
plan_output += f"Distance of the route: {route_distance}m\n"
plan_output += f"Load of the route: {route_load}\n"
print(plan_output)
total_distance += route_distance
total_load += route_load
print(f"Total Distance of all routes: {total_distance}m")
print(f"Total Load of all routes: {total_load}")
def main():
"""Solve the CVRP problem."""
# Instantiate the data problem.
data = create_data_model()
# Create the routing index manager.
manager = pywrapcp.RoutingIndexManager(
len(data["distance_matrix"]), data["num_vehicles"], data["depot"]
)
# 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)
# Add Capacity constraint.
def demand_callback(from_index):
"""Returns the demand of the node."""
# Convert from routing variable Index to demands NodeIndex.
from_node = manager.IndexToNode(from_index)
return data["demands"][from_node]
demand_callback_index = routing.RegisterUnaryTransitCallback(demand_callback)
routing.AddDimensionWithVehicleCapacity(
demand_callback_index,
0, # null capacity slack
data["vehicle_capacities"], # vehicle maximum capacities
True, # start cumul to zero
"Capacity",
)
# Allow to drop nodes.
penalty = 1000
for node in range(1, len(data["distance_matrix"])):
routing.AddDisjunction([manager.NodeToIndex(node)], penalty)
# Setting first solution heuristic.
search_parameters = pywrapcp.DefaultRoutingSearchParameters()
search_parameters.first_solution_strategy = (
routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC
)
search_parameters.local_search_metaheuristic = (
routing_enums_pb2.LocalSearchMetaheuristic.GUIDED_LOCAL_SEARCH
)
search_parameters.time_limit.FromSeconds(1)
# Solve the problem.
assignment = routing.SolveWithParameters(search_parameters)
# Print solution on console.
if assignment:
print_solution(data, manager, routing, assignment)
if __name__ == "__main__":
main()
C++
#include <cstdint>
#include <sstream>
#include <vector>
#include "google/protobuf/duration.pb.h"
#include "ortools/constraint_solver/routing.h"
#include "ortools/constraint_solver/routing_enums.pb.h"
#include "ortools/constraint_solver/routing_index_manager.h"
#include "ortools/constraint_solver/routing_parameters.h"
namespace operations_research {
struct DataModel {
const std::vector<std::vector<int64_t>> distance_matrix{
{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},
};
const std::vector<int64_t> demands{
0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8,
};
const std::vector<int64_t> vehicle_capacities{15, 15, 15, 15};
const int num_vehicles = 4;
const RoutingIndexManager::NodeIndex depot{0};
};
//! @brief Print the solution.
//! @param[in] data Data of the problem.
//! @param[in] manager Index manager used.
//! @param[in] routing Routing solver used.
//! @param[in] solution Solution found by the solver.
void PrintSolution(const DataModel& data, const RoutingIndexManager& manager,
const RoutingModel& routing, const Assignment& solution) {
// Display dropped nodes.
std::ostringstream dropped_nodes;
for (int64_t node = 0; node < routing.Size(); ++node) {
if (routing.IsStart(node) || routing.IsEnd(node)) continue;
if (solution.Value(routing.NextVar(node)) == node) {
dropped_nodes << " " << manager.IndexToNode(node).value();
}
}
LOG(INFO) << "Dropped nodes:" << dropped_nodes.str();
// Display routes
int64_t total_distance{0};
int64_t total_load{0};
for (int vehicle_id = 0; vehicle_id < data.num_vehicles; ++vehicle_id) {
int64_t index = routing.Start(vehicle_id);
LOG(INFO) << "Route for Vehicle " << vehicle_id << ":";
int64_t route_distance{0};
int64_t route_load{0};
std::ostringstream route;
while (!routing.IsEnd(index)) {
const int node_index = manager.IndexToNode(index).value();
route_load += data.demands[node_index];
route << node_index << " Load(" << route_load << ") -> ";
const int64_t previous_index = index;
index = solution.Value(routing.NextVar(index));
route_distance += routing.GetArcCostForVehicle(previous_index, index,
int64_t{vehicle_id});
}
LOG(INFO) << route.str() << manager.IndexToNode(index).value();
LOG(INFO) << "Distance of the route: " << route_distance << "m";
LOG(INFO) << "Load of the route: " << route_load;
total_distance += route_distance;
total_load += route_load;
}
LOG(INFO) << "Total distance of all routes: " << total_distance << "m";
LOG(INFO) << "Total load of all routes: " << total_load;
LOG(INFO) << "";
LOG(INFO) << "Advanced usage:";
LOG(INFO) << "Problem solved in " << routing.solver()->wall_time() << "ms";
}
void VrpDropNodes() {
// Instantiate the data problem.
DataModel data;
// Create Routing Index Manager
RoutingIndexManager manager(data.distance_matrix.size(), data.num_vehicles,
data.depot);
// Create Routing Model.
RoutingModel routing(manager);
// Create and register a transit callback.
const int transit_callback_index = routing.RegisterTransitCallback(
[&data, &manager](const int64_t from_index,
const int64_t to_index) -> int64_t {
// Convert from routing variable Index to distance matrix NodeIndex.
const int from_node = manager.IndexToNode(from_index).value();
const int to_node = manager.IndexToNode(to_index).value();
return data.distance_matrix[from_node][to_node];
});
// Define cost of each arc.
routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index);
// Add Capacity constraint.
const int demand_callback_index = routing.RegisterUnaryTransitCallback(
[&data, &manager](const int64_t from_index) -> int64_t {
// Convert from routing variable Index to demand NodeIndex.
const int from_node = manager.IndexToNode(from_index).value();
return data.demands[from_node];
});
routing.AddDimensionWithVehicleCapacity(
demand_callback_index, // transit callback index
int64_t{0}, // null capacity slack
data.vehicle_capacities, // vehicle maximum capacities
true, // start cumul to zero
"Capacity");
// Allow to drop nodes.
int64_t penalty{1000};
for (int i = 1; i < data.distance_matrix.size(); ++i) {
routing.AddDisjunction(
{manager.NodeToIndex(RoutingIndexManager::NodeIndex(i))}, penalty);
}
// Setting first solution heuristic.
RoutingSearchParameters search_parameters = DefaultRoutingSearchParameters();
search_parameters.set_first_solution_strategy(
FirstSolutionStrategy::PATH_CHEAPEST_ARC);
search_parameters.set_local_search_metaheuristic(
LocalSearchMetaheuristic::GUIDED_LOCAL_SEARCH);
search_parameters.mutable_time_limit()->set_seconds(1);
// Solve the problem.
const Assignment* solution = routing.SolveWithParameters(search_parameters);
// Print solution on console.
PrintSolution(data, manager, routing, *solution);
}
} // namespace operations_research
int main(int /*argc*/, char* /*argv*/[]) {
operations_research::VrpDropNodes();
return EXIT_SUCCESS;
}
Java
package com.google.ortools.constraintsolver.samples;
import com.google.ortools.Loader;
import com.google.ortools.constraintsolver.Assignment;
import com.google.ortools.constraintsolver.FirstSolutionStrategy;
import com.google.ortools.constraintsolver.LocalSearchMetaheuristic;
import com.google.ortools.constraintsolver.RoutingIndexManager;
import com.google.ortools.constraintsolver.RoutingModel;
import com.google.ortools.constraintsolver.RoutingSearchParameters;
import com.google.ortools.constraintsolver.main;
import com.google.protobuf.Duration;
import java.util.logging.Logger;
/** Minimal VRP.*/
public class VrpDropNodes {
private static final Logger logger = Logger.getLogger(VrpDropNodes.class.getName());
static class DataModel {
public final long[][] distanceMatrix = {
{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},
};
public final long[] demands = {0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8};
public final long[] vehicleCapacities = {15, 15, 15, 15};
public final int vehicleNumber = 4;
public final int depot = 0;
}
/// @brief Print the solution.
static void printSolution(
DataModel data, RoutingModel routing, RoutingIndexManager manager, Assignment solution) {
// Solution cost.
logger.info("Objective: " + solution.objectiveValue());
// Inspect solution.
// Display dropped nodes.
String droppedNodes = "Dropped nodes:";
for (int node = 0; node < routing.size(); ++node) {
if (routing.isStart(node) || routing.isEnd(node)) {
continue;
}
if (solution.value(routing.nextVar(node)) == node) {
droppedNodes += " " + manager.indexToNode(node);
}
}
logger.info(droppedNodes);
// Display routes
long totalDistance = 0;
long totalLoad = 0;
for (int i = 0; i < data.vehicleNumber; ++i) {
long index = routing.start(i);
logger.info("Route for Vehicle " + i + ":");
long routeDistance = 0;
long routeLoad = 0;
String route = "";
while (!routing.isEnd(index)) {
long nodeIndex = manager.indexToNode(index);
routeLoad += data.demands[(int) nodeIndex];
route += nodeIndex + " Load(" + routeLoad + ") -> ";
long previousIndex = index;
index = solution.value(routing.nextVar(index));
routeDistance += routing.getArcCostForVehicle(previousIndex, index, i);
}
route += manager.indexToNode(routing.end(i));
logger.info(route);
logger.info("Distance of the route: " + routeDistance + "m");
totalDistance += routeDistance;
totalLoad += routeLoad;
}
logger.info("Total Distance of all routes: " + totalDistance + "m");
logger.info("Total Load of all routes: " + totalLoad);
}
public static void main(String[] args) throws Exception {
Loader.loadNativeLibraries();
// Instantiate the data problem.
final DataModel data = new DataModel();
// Create Routing Index Manager
RoutingIndexManager manager =
new RoutingIndexManager(data.distanceMatrix.length, data.vehicleNumber, data.depot);
// Create Routing Model.
RoutingModel routing = new RoutingModel(manager);
// Create and register a transit callback.
final int transitCallbackIndex =
routing.registerTransitCallback((long fromIndex, long toIndex) -> {
// Convert from routing variable Index to user NodeIndex.
int fromNode = manager.indexToNode(fromIndex);
int toNode = manager.indexToNode(toIndex);
return data.distanceMatrix[fromNode][toNode];
});
// Define cost of each arc.
routing.setArcCostEvaluatorOfAllVehicles(transitCallbackIndex);
// Add Capacity constraint.
final int demandCallbackIndex = routing.registerUnaryTransitCallback((long fromIndex) -> {
// Convert from routing variable Index to user NodeIndex.
int fromNode = manager.indexToNode(fromIndex);
return data.demands[fromNode];
});
routing.addDimensionWithVehicleCapacity(demandCallbackIndex, 0, // null capacity slack
data.vehicleCapacities, // vehicle maximum capacities
true, // start cumul to zero
"Capacity");
// Allow to drop nodes.
long penalty = 1000;
for (int i = 1; i < data.distanceMatrix.length; ++i) {
routing.addDisjunction(new long[] {manager.nodeToIndex(i)}, penalty);
}
// Setting first solution heuristic.
RoutingSearchParameters searchParameters =
main.defaultRoutingSearchParameters()
.toBuilder()
.setFirstSolutionStrategy(FirstSolutionStrategy.Value.PATH_CHEAPEST_ARC)
.setLocalSearchMetaheuristic(LocalSearchMetaheuristic.Value.GUIDED_LOCAL_SEARCH)
.setTimeLimit(Duration.newBuilder().setSeconds(1).build())
.build();
// Solve the problem.
Assignment solution = routing.solveWithParameters(searchParameters);
// Print solution on console.
printSolution(data, routing, manager, solution);
}
}
C#
using System;
using System.Collections.Generic;
using Google.OrTools.ConstraintSolver;
using Google.Protobuf.WellKnownTypes; // Duration
/// <summary>
/// Minimal Vrp with drop nodes.
/// </summary>
public class VrpDropNodes
{
class DataModel
{
public long[,] DistanceMatrix = {
{ 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 }
};
public long[] Demands = { 0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8 };
public long[] VehicleCapacities = { 15, 15, 15, 15 };
public int VehicleNumber = 4;
public int Depot = 0;
};
/// <summary>
/// Print the solution.
/// </summary>
static void PrintSolution(in DataModel data, in RoutingModel routing, in RoutingIndexManager manager,
in Assignment solution)
{
Console.WriteLine($"Objective {solution.ObjectiveValue()}:");
// Inspect solution.
// Display dropped nodes.
string droppedNodes = "Dropped nodes:";
for (int index = 0; index < routing.Size(); ++index)
{
if (routing.IsStart(index) || routing.IsEnd(index))
{
continue;
}
if (solution.Value(routing.NextVar(index)) == index)
{
droppedNodes += " " + manager.IndexToNode(index);
}
}
Console.WriteLine("{0}", droppedNodes);
// Inspect solution.
long totalDistance = 0;
long totalLoad = 0;
for (int i = 0; i < data.VehicleNumber; ++i)
{
Console.WriteLine("Route for Vehicle {0}:", i);
long routeDistance = 0;
long routeLoad = 0;
var index = routing.Start(i);
while (routing.IsEnd(index) == false)
{
long nodeIndex = manager.IndexToNode(index);
routeLoad += data.Demands[nodeIndex];
Console.Write("{0} Load({1}) -> ", nodeIndex, routeLoad);
var previousIndex = index;
index = solution.Value(routing.NextVar(index));
routeDistance += routing.GetArcCostForVehicle(previousIndex, index, 0);
}
Console.WriteLine("{0}", manager.IndexToNode((int)index));
Console.WriteLine("Distance of the route: {0}m", routeDistance);
totalDistance += routeDistance;
totalLoad += routeLoad;
}
Console.WriteLine("Total Distance of all routes: {0}m", totalDistance);
Console.WriteLine("Total Load of all routes: {0}m", totalLoad);
}
public static void Main(String[] args)
{
// Instantiate the data problem.
DataModel data = new DataModel();
// Create Routing Index Manager
RoutingIndexManager manager =
new RoutingIndexManager(data.DistanceMatrix.GetLength(0), data.VehicleNumber, data.Depot);
// Create Routing Model.
RoutingModel routing = new RoutingModel(manager);
// Create and register a transit callback.
int transitCallbackIndex = routing.RegisterTransitCallback((long fromIndex, long toIndex) =>
{
// Convert from routing variable Index to
// distance matrix NodeIndex.
var fromNode = manager.IndexToNode(fromIndex);
var toNode = manager.IndexToNode(toIndex);
return data.DistanceMatrix[fromNode, toNode];
});
// Define cost of each arc.
routing.SetArcCostEvaluatorOfAllVehicles(transitCallbackIndex);
// Add Capacity constraint.
int demandCallbackIndex = routing.RegisterUnaryTransitCallback((long fromIndex) =>
{
// Convert from routing variable Index to
// demand NodeIndex.
var fromNode =
manager.IndexToNode(fromIndex);
return data.Demands[fromNode];
});
routing.AddDimensionWithVehicleCapacity(demandCallbackIndex, 0, // null capacity slack
data.VehicleCapacities, // vehicle maximum capacities
true, // start cumul to zero
"Capacity");
// Allow to drop nodes.
long penalty = 1000;
for (int i = 1; i < data.DistanceMatrix.GetLength(0); ++i)
{
routing.AddDisjunction(new long[] { manager.NodeToIndex(i) }, penalty);
}
// Setting first solution heuristic.
RoutingSearchParameters searchParameters =
operations_research_constraint_solver.DefaultRoutingSearchParameters();
searchParameters.FirstSolutionStrategy = FirstSolutionStrategy.Types.Value.PathCheapestArc;
searchParameters.LocalSearchMetaheuristic = LocalSearchMetaheuristic.Types.Value.GuidedLocalSearch;
searchParameters.TimeLimit = new Duration { Seconds = 1 };
// Solve the problem.
Assignment solution = routing.SolveWithParameters(searchParameters);
// Print solution on console.
PrintSolution(data, routing, manager, solution);
}
}