Häufige Routingaufgaben

In den folgenden Abschnitten wird erläutert, wie Sie einige häufige Aufgaben bei der Routenplanung.

Suchlimits

Die Behebung von Routenproblemen für viele Standorte kann lange dauern. Für ist es sinnvoll, ein Suchlimit festzulegen, das die nach einer bestimmten Zeitspanne oder Anzahl zurückgegebener Lösungen suchen.

Zeitlimits

Die folgenden Beispiele zeigen, wie Sie ein Zeitlimit von 30 Sekunden für eine Suche festlegen.

Python

search_parameters = pywrapcp.DefaultRoutingSearchParameters()
search_parameters.time_limit.seconds = 30

C++

RoutingSearchParameters searchParameters = DefaultRoutingSearchParameters();
searchParameters.mutable_time_limit()->set_seconds(30);

Java

Fügen Sie den folgenden „Import“ am Anfang des Programms hinzu: 
import com.google.protobuf.Duration;
 Legen Sie dann die Suchparameter so fest: 
RoutingSearchParameters searchParameters =
        main.defaultRoutingSearchParameters()
            .toBuilder()
            .setTimeLimit(Duration.newBuilder().setSeconds(30).build())
            .build();

C#

Fügen Sie am Anfang des Programms die folgende Zeile ein:

using Google.Protobuf.WellKnownTypes; // Duration
 Legen Sie dann die Suchparameter so fest: 
RoutingSearchParameters searchParameters =
  operations_research_constraint_solver.DefaultRoutingSearchParameters();
searchParameters.TimeLimit = new Duration { Seconds = 10 };

Weitere Informationen finden Sie unter Ändern der Suchstrategie für ein in dem ein Zeitlimit festgelegt wird.

Lösungslimits

Die folgenden Beispiele zeigen, wie Sie für eine Suche ein Lösungslimit von 100 festlegen.

Python

search_parameters = pywrapcp.DefaultRoutingSearchParameters()
search_parameters.solution_limit = 100

C++

RoutingSearchParameters searchParameters = DefaultRoutingSearchParameters();
searchParameters.set_solution_limit(100);

Java

RoutingSearchParameters searchParameters =
        main.defaultRoutingSearchParameters()
            .toBuilder()
            .setSolutionLimit(100)
            .build();

C#

RoutingSearchParameters searchParameters =
  operations_research_constraint_solver.DefaultRoutingSearchParameters();
searchParameters.SolutionLimit(100);

Bei einigen Problemen möchten Sie vielleicht eine Reihe von Anfangsrouten für ein VRP, anstatt eine erste Lösung finden zu lassen, z. B. Sie haben bereits eine gute Lösung für ein Problem gefunden und möchten diese als zur Lösung eines modifizierten Problems.

So erstellen Sie die ersten Routen:

  1. Definieren Sie ein Array, das die Ausgangsrouten enthält.
  2. Erstellen Sie die erste Lösung mit der Methode ReadAssignmentFromRoutes.

Mit dem folgenden Code werden die Anfangsrouten in den Daten definiert.

Python

    data["initial_routes"] = [
        # fmt: off
      [8, 16, 14, 13, 12, 11],
      [3, 4, 9, 10],
      [15, 1],
      [7, 5, 2, 6],
        # fmt: on
    ]

C++

  const std::vector<std::vector<int64_t>> initial_routes{
      {8, 16, 14, 13, 12, 11},
      {3, 4, 9, 10},
      {15, 1},
      {7, 5, 2, 6},
  };

Java

    public final long[][] initialRoutes = {
        {8, 16, 14, 13, 12, 11},
        {3, 4, 9, 10},
        {15, 1},
        {7, 5, 2, 6},
    };

C#

        public long[][] InitialRoutes = {
            new long[] { 8, 16, 14, 13, 12, 11 },
            new long[] { 3, 4, 9, 10 },
            new long[] { 15, 1 },
            new long[] { 7, 5, 2, 6 },
        };

Mit dem folgenden Code wird die erste Lösung aus den Routen erstellt. führt eine Suche von der ursprünglichen Lösung aus.

Das Programm zeigt sowohl die ursprüngliche Lösung als auch die bei der Suche gefundene Lösung an.

Python

    initial_solution = routing.ReadAssignmentFromRoutes(data["initial_routes"], True)
    print("Initial solution:")
    print_solution(data, manager, routing, initial_solution)

C++

  const Assignment* initial_solution =
      routing.ReadAssignmentFromRoutes(data.initial_routes, true);
  // Print initial solution on console.
  LOG(INFO) << "Initial solution: ";
  PrintSolution(data, manager, routing, *initial_solution);

Java

    Assignment initialSolution = routing.readAssignmentFromRoutes(data.initialRoutes, true);
    logger.info("Initial solution:");
    printSolution(data, routing, manager, initialSolution);

C#

        Assignment initialSolution = routing.ReadAssignmentFromRoutes(data.InitialRoutes, true);
        // Print initial solution on console.
        Console.WriteLine("Initial solution:");
        PrintSolution(data, routing, manager, initialSolution);

Wenn Sie diesen Code zum vorherigen VRP-Programm an und führen das Programm aus, zeigt die folgende Ausgabe an:

Initial solution:
Route for vehicle 0:
 0 ->  8 ->  16 ->  14 ->  13 ->  12 ->  11 -> 0
Distance of the route: 2168m

Route for vehicle 1:
 0 ->  3 ->  4 ->  9 ->  10 -> 0
Distance of the route: 2464m

Route for vehicle 2:
 0 ->  15 ->  1 -> 0
Distance of the route: 2192m

Route for vehicle 3:
 0 ->  7 ->  5 ->  2 ->  6 -> 0
Distance of the route: 1780m

Maximum of the route distances: 2464m

Solution after search:

Route for vehicle 0:
 0 ->  9 ->  10 ->  16 ->  14 -> 0
Distance of the route: 1552m

Route for vehicle 1:
 0 ->  12 ->  11 ->  15 ->  13 -> 0
Distance of the route: 1552

Route for vehicle 2:
 0 ->  3 ->  4 ->  1 ->  7 -> 0
Distance of the route: 1552

Route for vehicle 3:
 0 ->  5 ->  2 ->  6 ->  8 -> 0
Distance of the route: 1552

Maximum of the route distances: 1552

Hier sind die vollständigen Programme, die Anfangsrouten festlegen.

Python

"""Vehicles Routing Problem (VRP)."""

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["initial_routes"] = [
        # fmt: off
      [8, 16, 14, 13, 12, 11],
      [3, 4, 9, 10],
      [15, 1],
      [7, 5, 2, 6],
        # fmt: on
    ]
    data["num_vehicles"] = 4
    data["depot"] = 0
    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():
    """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 Distance constraint.
    dimension_name = "Distance"
    routing.AddDimension(
        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)

    # Close model with the custom search parameters.
    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(5)
    # When an initial solution is given for search, the model will be closed with
    # the default search parameters unless it is explicitly closed with the custom
    # search parameters.
    routing.CloseModelWithParameters(search_parameters)

    # Get initial solution from routes after closing the model.
    initial_solution = routing.ReadAssignmentFromRoutes(data["initial_routes"], True)
    print("Initial solution:")
    print_solution(data, manager, routing, initial_solution)

    # Solve the problem.
    solution = routing.SolveFromAssignmentWithParameters(
        initial_solution, search_parameters
    )

    # Print solution on console.
    if solution:
        print("Solution after search:")
        print_solution(data, manager, routing, solution)


if __name__ == "__main__":
    main()

C++

#include <algorithm>
#include <cstdint>
#include <cstdlib>
#include <sstream>
#include <vector>

#include "google/protobuf/duration.pb.h"
#include "ortools/base/logging.h"
#include "ortools/constraint_solver/constraint_solver.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<std::vector<int64_t>> initial_routes{
      {8, 16, 14, 13, 12, 11},
      {3, 4, 9, 10},
      {15, 1},
      {7, 5, 2, 6},
  };
  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) {
  LOG(INFO) << "Objective: " << solution.ObjectiveValue();
  int64_t max_route_distance{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};
    std::stringstream route;
    while (!routing.IsEnd(index)) {
      route << manager.IndexToNode(index).value() << " -> ";
      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";
    max_route_distance = std::max(route_distance, max_route_distance);
  }
  LOG(INFO) << "Maximum of the route distances: " << max_route_distance << "m";
  LOG(INFO) << "";
  LOG(INFO) << "Problem solved in " << routing.solver()->wall_time() << "ms";
}

void VrpInitialRoutes() {
  // 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 Distance constraint.
  routing.AddDimension(transit_callback_index, 0, 3000,
                       true,  // start cumul to zero
                       "Distance");
  routing.GetMutableDimension("Distance")->SetGlobalSpanCostCoefficient(100);

  // Close model with the custom search parameters
  RoutingSearchParameters searchParameters = DefaultRoutingSearchParameters();
  searchParameters.set_first_solution_strategy(
      FirstSolutionStrategy::PATH_CHEAPEST_ARC);
  searchParameters.set_local_search_metaheuristic(
      LocalSearchMetaheuristic::GUIDED_LOCAL_SEARCH);
  searchParameters.mutable_time_limit()->set_seconds(5);
  // When an initial solution is given for search, the model will be closed with
  // the default search parameters unless it is explicitly closed with the
  // custom search parameters.
  routing.CloseModelWithParameters(searchParameters);

  // Get initial solution from routes after closing the model.
  const Assignment* initial_solution =
      routing.ReadAssignmentFromRoutes(data.initial_routes, true);
  // Print initial solution on console.
  LOG(INFO) << "Initial solution: ";
  PrintSolution(data, manager, routing, *initial_solution);

  // Solve from initial solution.
  const Assignment* solution = routing.SolveFromAssignmentWithParameters(
      initial_solution, searchParameters);

  // Print solution on console.
  LOG(INFO) << "";
  LOG(INFO) << "Solution from search: ";
  PrintSolution(data, manager, routing, *solution);
}
}  // namespace operations_research

int main(int /*argc*/, char* /*argv*/[]) {
  operations_research::VrpInitialRoutes();
  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.RoutingDimension;
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 java.util.logging.Logger;

/** Minimal VRP. */
public class VrpInitialRoutes {
  private static final Logger logger = Logger.getLogger(VrpInitialRoutes.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[][] initialRoutes = {
        {8, 16, 14, 13, 12, 11},
        {3, 4, 9, 10},
        {15, 1},
        {7, 5, 2, 6},
    };
    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.
    long maxRouteDistance = 0;
    for (int i = 0; i < data.vehicleNumber; ++i) {
      long index = routing.start(i);
      logger.info("Route for Vehicle " + i + ":");
      long routeDistance = 0;
      String route = "";
      while (!routing.isEnd(index)) {
        route += manager.indexToNode(index) + " -> ";
        long previousIndex = index;
        index = solution.value(routing.nextVar(index));
        routeDistance += routing.getArcCostForVehicle(previousIndex, index, i);
      }
      logger.info(route + manager.indexToNode(index));
      logger.info("Distance of the route: " + routeDistance + "m");
      maxRouteDistance = Math.max(routeDistance, maxRouteDistance);
    }
    logger.info("Maximum of the route distances: " + maxRouteDistance + "m");
  }

  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 Distance constraint.
    routing.addDimension(transitCallbackIndex, 0, 3000,
        true, // start cumul to zero
        "Distance");
    RoutingDimension distanceDimension = routing.getMutableDimension("Distance");
    distanceDimension.setGlobalSpanCostCoefficient(100);

    Assignment initialSolution = routing.readAssignmentFromRoutes(data.initialRoutes, true);
    logger.info("Initial solution:");
    printSolution(data, routing, manager, initialSolution);

    // Setting first solution heuristic.
    RoutingSearchParameters searchParameters = main.defaultRoutingSearchParameters();

    // Solve the problem.
    Assignment solution = routing.solveFromAssignmentWithParameters(
        initialSolution, searchParameters);

    // Print solution on console.
    logger.info("Solution after search:");
    printSolution(data, routing, manager, solution);
  }
}

C#

// Copyright 2010-2024 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

using System;
using System.Collections.Generic;
using Google.OrTools.ConstraintSolver;

/// <summary>
///   VRP with initial routes.
/// </summary>
public class InitialRoutes
{
    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[][] InitialRoutes = {
            new long[] { 8, 16, 14, 13, 12, 11 },
            new long[] { 3, 4, 9, 10 },
            new long[] { 15, 1 },
            new long[] { 7, 5, 2, 6 },
        };
        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.
        long maxRouteDistance = 0;
        for (int i = 0; i < data.VehicleNumber; ++i)
        {
            Console.WriteLine("Route for Vehicle {0}:", i);
            long routeDistance = 0;
            var index = routing.Start(i);
            while (routing.IsEnd(index) == false)
            {
                Console.Write("{0} -> ", manager.IndexToNode((int)index));
                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}", routeDistance);
            maxRouteDistance = Math.Max(routeDistance, maxRouteDistance);
        }
        Console.WriteLine("Maximum distance of the routes: {0}", maxRouteDistance);
    }

    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 Distance constraint.
        routing.AddDimension(transitCallbackIndex, 0, 3000,
                             true, // start cumul to zero
                             "Distance");
        RoutingDimension distanceDimension = routing.GetMutableDimension("Distance");
        distanceDimension.SetGlobalSpanCostCoefficient(100);

        // Get initial solution from routes.
        Assignment initialSolution = routing.ReadAssignmentFromRoutes(data.InitialRoutes, true);
        // Print initial solution on console.
        Console.WriteLine("Initial solution:");
        PrintSolution(data, routing, manager, initialSolution);

        // Setting first solution heuristic.
        RoutingSearchParameters searchParameters =
            operations_research_constraint_solver.DefaultRoutingSearchParameters();

        // Solve the problem.
        Assignment solution = routing.SolveFromAssignmentWithParameters(
            initialSolution, searchParameters);

        // Print solution on console.
        Console.WriteLine("Solution after search:");
        PrintSolution(data, routing, manager, solution);
    }
}

Start- und Zielpositionen für Routen festlegen

Bisher sind wir davon ausgegangen, dass alle Fahrzeuge an einem bestimmten Ort starten und enden. im Depot. Sie können auch unterschiedliche Start- und Zielstandorte für jedes Fahrzeug in der das Problem zu lösen. Dazu übergeben Sie zwei Vektoren, die die Indizes für Start und Ende enthalten. Standorte, als Eingaben für die RoutingModel in der Funktion "Main". So erstellen Sie die Start- und Endvektoren Datenabschnitt des Programms:

Python

    data["starts"] = [1, 2, 15, 16]
    data["ends"] = [0, 0, 0, 0]

C++

  const std::vector<RoutingIndexManager::NodeIndex> starts{
      RoutingIndexManager::NodeIndex{1},
      RoutingIndexManager::NodeIndex{2},
      RoutingIndexManager::NodeIndex{15},
      RoutingIndexManager::NodeIndex{16},
  };
  const std::vector<RoutingIndexManager::NodeIndex> ends{
      RoutingIndexManager::NodeIndex{0},
      RoutingIndexManager::NodeIndex{0},
      RoutingIndexManager::NodeIndex{0},
      RoutingIndexManager::NodeIndex{0},
  };

Java

    public final int[] starts = {1, 2, 15, 16};
    public final int[] ends = {0, 0, 0, 0};

C#

        public int[] Starts = { 1, 2, 15, 16 };
        public int[] Ends = { 0, 0, 0, 0 };

Hier sind die vollständigen Programme, die die Start- und Endpositionen auf diese Weise festlegen.

Python

"""Simple Vehicles Routing Problem."""

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
    data["starts"] = [1, 2, 15, 16]
    data["ends"] = [0, 0, 0, 0]
    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)

    # Add Distance constraint.
    dimension_name = "Distance"
    routing.AddDimension(
        transit_callback_index,
        0,  # no slack
        2000,  # 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)


if __name__ == "__main__":
    main()

C++

#include <algorithm>
#include <cstdint>
#include <sstream>
#include <vector>

#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 int num_vehicles = 4;
  const std::vector<RoutingIndexManager::NodeIndex> starts{
      RoutingIndexManager::NodeIndex{1},
      RoutingIndexManager::NodeIndex{2},
      RoutingIndexManager::NodeIndex{15},
      RoutingIndexManager::NodeIndex{16},
  };
  const std::vector<RoutingIndexManager::NodeIndex> ends{
      RoutingIndexManager::NodeIndex{0},
      RoutingIndexManager::NodeIndex{0},
      RoutingIndexManager::NodeIndex{0},
      RoutingIndexManager::NodeIndex{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) {
  int64_t max_route_distance{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};
    std::stringstream route;
    while (!routing.IsEnd(index)) {
      route << manager.IndexToNode(index).value() << " -> ";
      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";
    max_route_distance = std::max(route_distance, max_route_distance);
  }
  LOG(INFO) << "Maximum of the route distances: " << max_route_distance << "m";
  LOG(INFO) << "";
  LOG(INFO) << "Problem solved in " << routing.solver()->wall_time() << "ms";
}

void VrpStartsEnds() {
  // Instantiate the data problem.
  DataModel data;

  // Create Routing Index Manager
  RoutingIndexManager manager(data.distance_matrix.size(), data.num_vehicles,
                              data.starts, data.ends);

  // 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 Distance constraint.
  routing.AddDimension(transit_callback_index, 0, 2000,
                       /*fix_start_cumul_to_zero=*/true, "Distance");
  routing.GetMutableDimension("Distance")->SetGlobalSpanCostCoefficient(100);

  // Setting first solution heuristic.
  RoutingSearchParameters searchParameters = DefaultRoutingSearchParameters();
  searchParameters.set_first_solution_strategy(
      FirstSolutionStrategy::PATH_CHEAPEST_ARC);

  // Solve the problem.
  const Assignment* solution = routing.SolveWithParameters(searchParameters);

  // Print solution on console.
  PrintSolution(data, manager, routing, *solution);
}
}  // namespace operations_research

int main(int /*argc*/, char* /*argv*/[]) {
  operations_research::VrpStartsEnds();
  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.RoutingDimension;
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 java.util.logging.Logger;

/** Minimal VRP.*/
public class VrpStartsEnds {
  private static final Logger logger = Logger.getLogger(VrpStartsEnds.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 int vehicleNumber = 4;
    public final int[] starts = {1, 2, 15, 16};
    public final int[] ends = {0, 0, 0, 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.
    long maxRouteDistance = 0;
    for (int i = 0; i < data.vehicleNumber; ++i) {
      long index = routing.start(i);
      logger.info("Route for Vehicle " + i + ":");
      long routeDistance = 0;
      String route = "";
      while (!routing.isEnd(index)) {
        route += manager.indexToNode(index) + " -> ";
        long previousIndex = index;
        index = solution.value(routing.nextVar(index));
        routeDistance += routing.getArcCostForVehicle(previousIndex, index, i);
      }
      logger.info(route + manager.indexToNode(index));
      logger.info("Distance of the route: " + routeDistance + "m");
      maxRouteDistance = Math.max(routeDistance, maxRouteDistance);
    }
    logger.info("Maximum of the route distances: " + maxRouteDistance + "m");
  }

  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.starts, data.ends);

    // 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 Distance constraint.
    routing.addDimension(transitCallbackIndex, 0, 2000,
        true, // start cumul to zero
        "Distance");
    RoutingDimension distanceDimension = routing.getMutableDimension("Distance");
    distanceDimension.setGlobalSpanCostCoefficient(100);

    // Setting first solution heuristic.
    RoutingSearchParameters searchParameters =
        main.defaultRoutingSearchParameters()
            .toBuilder()
            .setFirstSolutionStrategy(FirstSolutionStrategy.Value.PATH_CHEAPEST_ARC)
            .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;

/// <summary>
///   Minimal TSP using distance matrix.
/// </summary>
public class VrpStartsEnds
{
    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 int VehicleNumber = 4;
        public int[] Starts = { 1, 2, 15, 16 };
        public int[] Ends = { 0, 0, 0, 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.
        long maxRouteDistance = 0;
        for (int i = 0; i < data.VehicleNumber; ++i)
        {
            Console.WriteLine("Route for Vehicle {0}:", i);
            long routeDistance = 0;
            var index = routing.Start(i);
            while (routing.IsEnd(index) == false)
            {
                Console.Write("{0} -> ", manager.IndexToNode((int)index));
                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);
            maxRouteDistance = Math.Max(routeDistance, maxRouteDistance);
        }
        Console.WriteLine("Maximum distance of the routes: {0}m", maxRouteDistance);
    }

    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.Starts, data.Ends);

        // 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 Distance constraint.
        routing.AddDimension(transitCallbackIndex, 0, 2000,
                             true, // start cumul to zero
                             "Distance");
        RoutingDimension distanceDimension = routing.GetMutableDimension("Distance");
        distanceDimension.SetGlobalSpanCostCoefficient(100);

        // Setting first solution heuristic.
        RoutingSearchParameters searchParameters =
            operations_research_constraint_solver.DefaultRoutingSearchParameters();
        searchParameters.FirstSolutionStrategy = FirstSolutionStrategy.Types.Value.PathCheapestArc;

        // Solve the problem.
        Assignment solution = routing.SolveWithParameters(searchParameters);

        // Print solution on console.
        PrintSolution(data, routing, manager, solution);
    }
}

Wenn Sie das Programm ausführen, erhalten Sie die folgende Ausgabe, in der die Routen beginnen. und enden an den angegebenen Orten:

Route for vehicle 0:
 1 -> 4 -> 3 -> 7 -> 0
Distance of the route: 1004m

Route for vehicle 1:
 2 -> 6 -> 8 -> 5 -> 0
Distance of the route: 936m

Route for vehicle 2:
 15 -> 11 -> 12 -> 13 -> 0
Distance of the route: 936m

Route for vehicle 3:
 16 -> 14 -> 10 -> 9 -> 0
Distance of the route: 1118m

Total distance of all routes: 3994m

Die Gesamtstrecke ist kürzer als im vorherigen Beispiel, weil die Fahrzeuge müssen nicht im Lagerhaus beginnen oder enden.

Beliebige Start- und Endorte zulassen

Bei anderen Versionen der Fahrzeugroute dürfen Fahrzeuge und an beliebigen Orten enden. Um das Problem auf diese Weise zu lösen, ändern Sie einfach der Distanzmatrix, sodass die Entfernung vom Depot zu einem beliebigen anderen Standort 0 ist, indem Sie die erste Zeile und Spalte der Matrix so einstellen, dass sie nur Nullen enthält. Dadurch wird das Depot in einen Dummy-Standort verwandelt, der keinen Einfluss auf die optimalen Routen hat.

Hier ist ein Beispiel, in dem die Distanzmatrix aus der VRP-Beispiel wurde geändert, um den Abstand vom Depot zu allen anderen Knoten 0.

data['distance_matrix'] = [
        [
            0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
        ],
        [
            0, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674,
            1016, 868, 1210
        ],
        [
            0, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164,
            1130, 788, 1552, 754
        ],
        [
            0, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822,
            1164, 560, 1358
        ],
        [
            0, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708,
            1050, 674, 1244
        ],
        [
            0, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628,
            514, 1050, 708
        ],
        [
            0, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856,
            514, 1278, 480
        ],
        [
            0, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320,
            662, 742, 856
        ],
        [
            0, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662,
            0, 1084, 514
        ],
        [
            0, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388,
            0, 810, 468
        ],
        [
            0, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764,
            730, 388, 1152, 354
        ],
        [
            0, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114,
            308, 650, 274, 844
        ],
        [
            0, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194,
            536, 388, 730
        ],
        [
            0, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0,
            342, 422, 536
        ],
        [
            0, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536,
            342, 0, 764, 194
        ],
        [
            0, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274,
            388, 422, 764, 0, 798
        ],
        [
            0, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730,
            536, 194, 798, 0
        ],
    ]

Wenn Sie das VRP-Programm mit die modifizierte Distance Matrix (und den Lösungsdrucker so ändern, dass das Depot weggelassen wird) enthält, zeigt das Programm die folgenden Routen an:

Route for vehicle 0:
 5  -> 8 -> 6 -> 2
Distance of the route: 662m

Route for vehicle 1:
 7  -> 1 -> 4 -> 3
Distance of the route: 662m

Route for vehicle 2:
 16  -> 14 -> 13 -> 15
Distance of the route: 958m

Route for vehicle 3:
 10  -> 9 -> 12 -> 11
Distance of the route: 878m
Maximum of the route distances: 958m