Pénalités et visites abandonnées

Dans cette section, nous expliquons comment gérer des problèmes de routage qui ne sont pas réalisables. en raison de contraintes. Par exemple, si vous recevez VRP avec contraintes de capacité où la demande totale en tout site dépasse la capacité totale aucune solution n'est possible. Dans ce cas, les véhicules ne peuvent plus visiter certains lieux. Le problème est comment identifier les visites à abandonner.

Pour résoudre le problème, nous introduisons de nouveaux coûts, appelés pénalités partout. Chaque fois qu'une visite dans un établissement est abandonnée, la pénalité est ajouté à la distance totale parcourue. Le résolveur trouve ensuite un itinéraire réduit la distance totale plus la somme des pénalités pour tous les éléments supprimés emplacements.

Prenons l'exemple d'un VRP simple avec des contraintes de capacité imposées par le graphique ci-dessous, qui affiche les chiffres à côté des trois lieux (autres que le de dépôt) sont des demandes.

Supposons qu'il n'y ait qu'un seul véhicule d'une capacité de 50 personnes. Il ne peut pas accéder aux trois emplacements, A, B et C, car la demande totale est de 60. Pour résoudre le problème, vous attribuez une pénalité élevée (100) à chaque site. Après découvrant que le problème est impossible, le résolveur abandonne l'emplacement B et renvoie l'itinéraire suivant: Depot -> A -> C -> Depot

Il s'agit de l'itinéraire le plus court permettant de visiter deux des trois lieux est 55).

Tailles de pénalité

Dans l'exemple ci-dessus, nous avons choisi des pénalités plus élevées que la somme les distances entre les lieux (à l'exception du dépôt). Par conséquent, après avoir abandonné un endroit pour rendre le problème réalisable, le résolveur n'abandonne d'adresses supplémentaires, car la pénalité liée à cette action de réduction de la distance parcourue.

En supposant que vous vouliez faire le plus de livraisons possible, cela donne une une solution satisfaisante au problème.

Si vous n'avez pas besoin d'effectuer autant de livraisons que possible, vous voudrez peut-être attribuer des pénalités plus faibles, auquel cas le résolveur risque de supprimer plus d'emplacements que est nécessaire pour rendre le problème réalisable. Par exemple, vous pouvez le faire si des frais supplémentaires, au-delà des frais de déplacement de base, s'appliquent à la visite de certains emplacements.

Exemple

Nous présenterons ensuite un exemple plus large de VRP pouvant être résolu à l'aide de pénalités. Cet exemple est semblable à l'exemple précédent exemple de CVRP, mais cette fois, nous avons augmenté ce qui a entraîné une baisse du nombre de visites chez certains véhicules.

Un graphique des emplacements et des nouvelles demandes est présenté ci-dessous.

Résoudre l'exemple avec OR-Tools

Les sections suivantes expliquent comment résoudre l'exemple avec OR-Tools.

Créer les données

Dans cet exemple, les données sont celles de la exemple de VRP, et ajoute les demandes suivantes : et capacités:

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 };

Ajouter les contraintes et les pénalités de capacité

Le code suivant ajoute les contraintes de rappel de demande et de capacité, et ajoute de pénalités à l'aide AddDisjunction .

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);
        }

Dans ce contexte, une disjonction est simplement une variable que le résolveur utilise pour décider d'inclure ou non un emplacement donné dans la solution. Dans cet exemple, ajoute la même pénalité pour chaque site, mais en général, vous pouvez ajouter des sanctions différentes selon les sites.

Ajouter l'imprimante de la solution

L'imprimante de la solution, illustrée ci-dessous, est semblable à celle de la section Exemple de CVRP, mais affiche également zones géographiques supprimées.

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);
    }

Exécution du programme

Lorsque vous exécutez le programme, il renvoie la sortie suivante, illustrée ci-dessous. Notez que le résolveur abandonne les lieux 6 et 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

Voici un schéma des routes.

Terminer les programmes

Voici les programmes complets.

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);
    }
}