تمت ترجمة هذه الصفحة بواسطة Cloud Translation API‏.

مهام التوجيه الشائعة

توضح الأقسام التالية كيفية القيام ببعض المهام الشائعة المتعلقة بحل مشكلات توجيه المركبات.

حدود البحث

قد يستغرق حلّ مشاكل توجيه المركبات في العديد من المواقع الجغرافية وقتًا طويلاً. بالنسبة مثل هذه المشكلات، يُستحسن تعيين حد للبحث، ينهي البحث بعد فترة زمنية محددة أو عدد من الحلول التي تم عرضها.

الحدود الزمنية

توضِّح الأمثلة أدناه كيفية تحديد مهلة زمنية تبلغ 30 ثانية لإجراء بحث.

Python

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

C++‎

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

لغة Java

أضِف عملية الاستيراد التالية في بداية البرنامج:

import com.google.protobuf.Duration;

وبعد ذلك، عيِّن معلمات البحث على النحو التالي:

RoutingSearchParameters searchParameters =
        main.defaultRoutingSearchParameters()
            .toBuilder()
            .setTimeLimit(Duration.newBuilder().setSeconds(30).build())
            .build();

#C

أضِف السطر التالي في بداية البرنامج:

using Google.Protobuf.WellKnownTypes; // Duration

وبعد ذلك، عيِّن معلمات البحث على النحو التالي:

RoutingSearchParameters searchParameters =
  operations_research_constraint_solver.DefaultRoutingSearchParameters();
searchParameters.TimeLimit = new Duration { Seconds = 10 };

عرض تغيير استراتيجية البحث مثال يحدد حدًا زمنيًا.

الحدود القصوى للحلول

توضِّح الأمثلة أدناه كيفية ضبط حدّ أقصى يبلغ 100 حل في عملية بحث.

Python

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

C++‎

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

لغة Java

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

#C

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

تعيين المسارات الأولية للبحث

بالنسبة لبعض المشكلات، قد ترغب في تحديد مجموعة من المسارات الأولية لـ VRP، بدلاً من السماح للأداة بإيجاد حل أولي — على سبيل المثال، إذا فقد وجدت بالفعل حلاً جيدًا لمشكلة ما، وتريد استخدامه لحل مشكلة معدلة.

لإنشاء المسارات الأولية، اتبع الخطوات التالية:

حدد صفيفًا يحتوي على المسارات الأولية.
يمكنك إنشاء الحل الأولي باستخدام الطريقة ReadAssignmentFromRoutes.

يحدد الكود التالي المسارات الأولية في البيانات.

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},
  };

لغة 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 },
        };

يُنشئ الكود التالي الحل الأولي من المسارات ثم يجري بحثًا بدءًا من الحل الأولي.

يعرض البرنامج كلاً من الحل الأولي والحل الذي تم العثور عليه من خلال البحث.

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);

لغة 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);

عند إضافة هذا الرمز إلى السابق برنامج VRP وتشغيل البرنامج، يعرض الناتج التالي:

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

إليك البرامج الكاملة التي تحدد المسارات الأولية.

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;
}

لغة 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);
    }
}

تحديد مواقع البدء والانتهاء للمسارات

إلى الآن، افترضنا أن جميع المركبات تبدأ وتنتهي في موقع واحد، المستودع. يمكنك أيضًا تحديد مواقع بدء وانتهاء مختلفة لكل مركبة في المشكلة. للقيام بذلك، مرر متجهين، يحتويان على فهارس البداية والنهاية والمواقع، كمدخلات إلى RoutingModel في الدالة الرئيسية. إليك كيفية إنشاء متجهات البداية والنهاية في قسم البيانات في البرنامج:

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},
  };

لغة 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 };

إليك البرامج الكاملة التي تحدد موقعَي البدء والانتهاء بهذه الطريقة.

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;
}

لغة 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);
    }
}

عند تشغيل البرنامج، تحصل على الناتج التالي الذي تبدأ فيه المسارات. وينتهي في المواقع المحددة:

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

المسافة الإجمالية أقصر مما كانت عليه في المثال السابق لأن المركبات لا يلزم أن تبدأ أو تنتهي عند المستودع.

السماح بمواقع البدء والانتهاء العشوائية

في الإصدارات الأخرى من مشكلة توجيه المركبات، يُسمح ببدء تشغيل المركبات وتنتهي في مواقع عشوائية ولإعداد المشكلة بهذه الطريقة، ما عليك سوى تعديل بمصفوفة المسافة بحيث تكون المسافة من المستودع إلى أي موقع آخر 0، عن طريق تعيين الصف الأول والعمود الأول للمصفوفة على جميع الأصفار. وهذا يحول المستودع إلى موقع وهمي ليس له أي تأثير على المسارات المثلى.

وفي ما يلي مثال تكون فيه مصفوفة المسافة من تم تعديل مثال VRP لجعل المسافة من المستودع إلى جميع العُقد الأخرى 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
        ],
    ]

عند تشغيل برنامج VRP باستخدام مصفوفة المسافة المعدلة (وتعديل طابعة الحل لحذف المستودع) ، يعرض البرنامج المسارات التالية:

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