עונשים וביקורים בירידה

בקטע הזה נסביר איך לטפל בבעיות מסלול שלא ניתן ליישם בגלל אילוצים. לדוגמה, אם נותנים לך VRP עם מגבלות קיבולת שבה הביקוש הכולל בכל המיקומים חורג מהקיבולת הכוללת של אין פתרון אפשרי. במקרים כאלה, אסור לכלי הרכב לבקר במיקומים מסוימים. הבעיה היא איך להחליט אילו ביקורים ירדו.

כדי לפתור את הבעיה, אנחנו מוסיפים עלויות חדשות – שנקראות קנסות – בכל המיקומים. בכל פעם שביקור במיקום מסוים מבוטל, העונש יפוג מתווסף למרחק הכולל של הנסיעה. לאחר מכן, הפותר מוצא מסלול מזעור המרחק הכולל וגם את סכום העונשים של כל נפילה מיקומים.

לדוגמה, נבחן את מודל ה-VRP הפשוט עם מגבלות הקיבולת שניתנו על ידי בתרשים הבא, שבו המספרים לצד שלושת המיקומים (מלבד הם ביקוש.

נניח שיש רק רכב אחד עם קיבולת של 50. הוא לא יכול לבקר בכל שלושת הסוגים המיקומים, א', ב' ו-ג', כי הביקוש הכולל הוא 60. כדי לפתור את הבעיה, אם מקצים עונש גדול על כל מיקום, למשל 100. אחרי גילוי שהבעיה אינה מעשית, הפותר מסיר את מיקום ב' מחזירה את המסלול הבא: Depot -> A -> C -> Depot

זהו המסלול הקצר ביותר שבו ניתן להגיע לשניים מתוך שלושת המיקומים (המרחק הוא 55).

עונשים בגין עונשים

בדוגמה שלמעלה, בחרנו עונשים שגדולים מהסכום של כל הסנקציות מרחקים בין מיקומים (לא כולל התחנה). כתוצאה מכך, לאחר שחרור מיקום אחד שנועד להפוך את הבעיה לאפשרית, הפתרון לא משחרר אף אחד מיקומים נוספים, מפני שהעונש בגין פעולה זו עלול להימשך במרחק הנסיעה.

בהנחה שאתם רוצים לבצע כמה שיותר משלוחים, פתרון משביע רצון לבעיה.

אם אין לך צורך לבצע כמה שיותר משלוחים, אולי כדאי להקצות עונשים קטנים יותר, ובמקרה כזה הפתרון עשוי להוריד יותר מיקומים מאשר יש צורך כדי לאפשר את הבעיה. לדוגמה, אפשר לעשות את זה אם חלות עלויות נוספות, מעל עלות הנסיעה הבסיסית, על ביקור במקומות מסוימים מיקומים.

דוגמה

בשלב הבא, נציג דוגמה גדולה יותר של מצב VRP שאפשר לפתור באמצעות עונשים. הדוגמה דומה לדוגמה הקודמת דוגמה ל-CVRP, אבל הפעם הגדלנו כמה מהביקוש, ומאלץ כלי רכב מסוימים להפחית את הביקורים.

בהמשך מוצג תרשים של המיקומים והדרישות החדשות.

פתרון הדוגמה באמצעות כלי OR

בקטעים הבאים מוסבר איך לפתור את הדוגמה באמצעות כלי OR.

יצירת הנתונים

הנתונים בדוגמה הזאת כוללים את הנתונים מהדוגמה הקודמת דוגמה ל-VRP, ומוסיף את הדרישות הבאות וקיבולת:

Python

    data["demands"] = [0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8]
    data["vehicle_capacities"] = [15, 15, 15, 15]

C++‎

  const std::vector<int64_t> demands{
      0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8,
  };
  const std::vector<int64_t> vehicle_capacities{15, 15, 15, 15};

Java

    public final long[] demands = {0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8};
    public final long[] vehicleCapacities = {15, 15, 15, 15};

C#‎

        public long[] Demands = { 0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8 };
        public long[] VehicleCapacities = { 15, 15, 15, 15 };

הוספת מגבלות ועונשים על קיבולת

הקוד הבא מוסיף את הקריאה החוזרת (callback) ומגבלות הקיבולת של הביקוש, ומוסיף לשימוש AddDisjunction .

Python

    def demand_callback(from_index):
        """Returns the demand of the node."""
        # Convert from routing variable Index to demands NodeIndex.
        from_node = manager.IndexToNode(from_index)
        return data["demands"][from_node]

    demand_callback_index = routing.RegisterUnaryTransitCallback(demand_callback)
    routing.AddDimensionWithVehicleCapacity(
        demand_callback_index,
        0,  # null capacity slack
        data["vehicle_capacities"],  # vehicle maximum capacities
        True,  # start cumul to zero
        "Capacity",
    )
    # Allow to drop nodes.
    penalty = 1000
    for node in range(1, len(data["distance_matrix"])):
        routing.AddDisjunction([manager.NodeToIndex(node)], penalty)

C++‎

  const int demand_callback_index = routing.RegisterUnaryTransitCallback(
      [&data, &manager](const int64_t from_index) -> int64_t {
        // Convert from routing variable Index to demand NodeIndex.
        const int from_node = manager.IndexToNode(from_index).value();
        return data.demands[from_node];
      });
  routing.AddDimensionWithVehicleCapacity(
      demand_callback_index,    // transit callback index
      int64_t{0},               // null capacity slack
      data.vehicle_capacities,  // vehicle maximum capacities
      true,                     // start cumul to zero
      "Capacity");
  // Allow to drop nodes.
  int64_t penalty{1000};
  for (int i = 1; i < data.distance_matrix.size(); ++i) {
    routing.AddDisjunction(
        {manager.NodeToIndex(RoutingIndexManager::NodeIndex(i))}, penalty);
  }

Java

    final int demandCallbackIndex = routing.registerUnaryTransitCallback((long fromIndex) -> {
      // Convert from routing variable Index to user NodeIndex.
      int fromNode = manager.indexToNode(fromIndex);
      return data.demands[fromNode];
    });
    routing.addDimensionWithVehicleCapacity(demandCallbackIndex, 0, // null capacity slack
        data.vehicleCapacities, // vehicle maximum capacities
        true, // start cumul to zero
        "Capacity");
    // Allow to drop nodes.
    long penalty = 1000;
    for (int i = 1; i < data.distanceMatrix.length; ++i) {
      routing.addDisjunction(new long[] {manager.nodeToIndex(i)}, penalty);
    }

C#‎

        int demandCallbackIndex = routing.RegisterUnaryTransitCallback((long fromIndex) =>
                                                                       {
                                                                           // Convert from routing variable Index to
                                                                           // demand NodeIndex.
                                                                           var fromNode =
                                                                               manager.IndexToNode(fromIndex);
                                                                           return data.Demands[fromNode];
                                                                       });
        routing.AddDimensionWithVehicleCapacity(demandCallbackIndex, 0, // null capacity slack
                                                data.VehicleCapacities, // vehicle maximum capacities
                                                true,                   // start cumul to zero
                                                "Capacity");
        // Allow to drop nodes.
        long penalty = 1000;
        for (int i = 1; i < data.DistanceMatrix.GetLength(0); ++i)
        {
            routing.AddDisjunction(new long[] { manager.NodeToIndex(i) }, penalty);
        }

בהקשר הזה, הבחנה היא פשוט משתנה שהפותר משתמש בו כדי להחליט אם לכלול מיקום נתון בפתרון. במשפט הזה, השיטה מוסיפה אותו קנס לכל מיקום, אבל באופן כללי אפשר להוסיף סנקציות שונות על מיקומים שונים.

הוספה של מדפסת הפתרונות

המדפסת של הפתרון, שמוצגת בהמשך, דומה למדפסת שמופיעה דוגמה ל-CVRP, וגם מציגה את מיקומים מסוימים ירדו.

Python

def print_solution(data, manager, routing, assignment):
    """Prints assignment on console."""
    print(f"Objective: {assignment.ObjectiveValue()}")
    # Display dropped nodes.
    dropped_nodes = "Dropped nodes:"
    for node in range(routing.Size()):
        if routing.IsStart(node) or routing.IsEnd(node):
            continue
        if assignment.Value(routing.NextVar(node)) == node:
            dropped_nodes += f" {manager.IndexToNode(node)}"
    print(dropped_nodes)
    # Display routes
    total_distance = 0
    total_load = 0
    for vehicle_id in range(data["num_vehicles"]):
        index = routing.Start(vehicle_id)
        plan_output = f"Route for vehicle {vehicle_id}:\n"
        route_distance = 0
        route_load = 0
        while not routing.IsEnd(index):
            node_index = manager.IndexToNode(index)
            route_load += data["demands"][node_index]
            plan_output += f" {node_index} Load({route_load}) -> "
            previous_index = index
            index = assignment.Value(routing.NextVar(index))
            route_distance += routing.GetArcCostForVehicle(
                previous_index, index, vehicle_id
            )
        plan_output += f" {manager.IndexToNode(index)} Load({route_load})\n"
        plan_output += f"Distance of the route: {route_distance}m\n"
        plan_output += f"Load of the route: {route_load}\n"
        print(plan_output)
        total_distance += route_distance
        total_load += route_load
    print(f"Total Distance of all routes: {total_distance}m")
    print(f"Total Load of all routes: {total_load}")

C++‎

//! @brief Print the solution.
//! @param[in] data Data of the problem.
//! @param[in] manager Index manager used.
//! @param[in] routing Routing solver used.
//! @param[in] solution Solution found by the solver.
void PrintSolution(const DataModel& data, const RoutingIndexManager& manager,
                   const RoutingModel& routing, const Assignment& solution) {
  // Display dropped nodes.
  std::ostringstream dropped_nodes;
  for (int64_t node = 0; node < routing.Size(); ++node) {
    if (routing.IsStart(node) || routing.IsEnd(node)) continue;
    if (solution.Value(routing.NextVar(node)) == node) {
      dropped_nodes << " " << manager.IndexToNode(node).value();
    }
  }
  LOG(INFO) << "Dropped nodes:" << dropped_nodes.str();
  // Display routes
  int64_t total_distance{0};
  int64_t total_load{0};
  for (int vehicle_id = 0; vehicle_id < data.num_vehicles; ++vehicle_id) {
    int64_t index = routing.Start(vehicle_id);
    LOG(INFO) << "Route for Vehicle " << vehicle_id << ":";
    int64_t route_distance{0};
    int64_t route_load{0};
    std::ostringstream route;
    while (!routing.IsEnd(index)) {
      const int node_index = manager.IndexToNode(index).value();
      route_load += data.demands[node_index];
      route << node_index << " Load(" << route_load << ") -> ";
      const int64_t previous_index = index;
      index = solution.Value(routing.NextVar(index));
      route_distance += routing.GetArcCostForVehicle(previous_index, index,
                                                     int64_t{vehicle_id});
    }
    LOG(INFO) << route.str() << manager.IndexToNode(index).value();
    LOG(INFO) << "Distance of the route: " << route_distance << "m";
    LOG(INFO) << "Load of the route: " << route_load;
    total_distance += route_distance;
    total_load += route_load;
  }
  LOG(INFO) << "Total distance of all routes: " << total_distance << "m";
  LOG(INFO) << "Total load of all routes: " << total_load;
  LOG(INFO) << "";
  LOG(INFO) << "Advanced usage:";
  LOG(INFO) << "Problem solved in " << routing.solver()->wall_time() << "ms";
}

Java

  /// @brief Print the solution.
  static void printSolution(
      DataModel data, RoutingModel routing, RoutingIndexManager manager, Assignment solution) {
    // Solution cost.
    logger.info("Objective: " + solution.objectiveValue());
    // Inspect solution.
    // Display dropped nodes.
    String droppedNodes = "Dropped nodes:";
    for (int node = 0; node < routing.size(); ++node) {
      if (routing.isStart(node) || routing.isEnd(node)) {
        continue;
      }
      if (solution.value(routing.nextVar(node)) == node) {
        droppedNodes += " " + manager.indexToNode(node);
      }
    }
    logger.info(droppedNodes);
    // Display routes
    long totalDistance = 0;
    long totalLoad = 0;
    for (int i = 0; i < data.vehicleNumber; ++i) {
      long index = routing.start(i);
      logger.info("Route for Vehicle " + i + ":");
      long routeDistance = 0;
      long routeLoad = 0;
      String route = "";
      while (!routing.isEnd(index)) {
        long nodeIndex = manager.indexToNode(index);
        routeLoad += data.demands[(int) nodeIndex];
        route += nodeIndex + " Load(" + routeLoad + ") -> ";
        long previousIndex = index;
        index = solution.value(routing.nextVar(index));
        routeDistance += routing.getArcCostForVehicle(previousIndex, index, i);
      }
      route += manager.indexToNode(routing.end(i));
      logger.info(route);
      logger.info("Distance of the route: " + routeDistance + "m");
      totalDistance += routeDistance;
      totalLoad += routeLoad;
    }
    logger.info("Total Distance of all routes: " + totalDistance + "m");
    logger.info("Total Load of all routes: " + totalLoad);
  }

C#‎

    /// <summary>
    ///   Print the solution.
    /// </summary>
    static void PrintSolution(in DataModel data, in RoutingModel routing, in RoutingIndexManager manager,
                              in Assignment solution)
    {
        Console.WriteLine($"Objective {solution.ObjectiveValue()}:");

        // Inspect solution.
        // Display dropped nodes.
        string droppedNodes = "Dropped nodes:";
        for (int index = 0; index < routing.Size(); ++index)
        {
            if (routing.IsStart(index) || routing.IsEnd(index))
            {
                continue;
            }
            if (solution.Value(routing.NextVar(index)) == index)
            {
                droppedNodes += " " + manager.IndexToNode(index);
            }
        }
        Console.WriteLine("{0}", droppedNodes);
        // Inspect solution.
        long totalDistance = 0;
        long totalLoad = 0;
        for (int i = 0; i < data.VehicleNumber; ++i)
        {
            Console.WriteLine("Route for Vehicle {0}:", i);
            long routeDistance = 0;
            long routeLoad = 0;
            var index = routing.Start(i);
            while (routing.IsEnd(index) == false)
            {
                long nodeIndex = manager.IndexToNode(index);
                routeLoad += data.Demands[nodeIndex];
                Console.Write("{0} Load({1}) -> ", nodeIndex, routeLoad);
                var previousIndex = index;
                index = solution.Value(routing.NextVar(index));
                routeDistance += routing.GetArcCostForVehicle(previousIndex, index, 0);
            }
            Console.WriteLine("{0}", manager.IndexToNode((int)index));
            Console.WriteLine("Distance of the route: {0}m", routeDistance);
            totalDistance += routeDistance;
            totalLoad += routeLoad;
        }
        Console.WriteLine("Total Distance of all routes: {0}m", totalDistance);
        Console.WriteLine("Total Load of all routes: {0}m", totalLoad);
    }

הפעלה של התוכנית

כשמריצים את התוכנית, הפלט הבא אמור להתקבל: שימו לב הפותר מוריד את מיקומים 6 ו-15.

Objective: 7936
Dropped nodes: 6 15
Route for vehicle 0:
 0 Load(0) ->  9 Load(1) ->  14 Load(7) ->  16 Load(15) ->  0 Load(15)
Distance of the route: 1324m
Load of the route: 15

Route for vehicle 1:
 0 Load(0) ->  12 Load(2) ->  11 Load(3) ->  4 Load(9) ->  3 Load(12) ->  1 Load(13) ->  0 Load(13)
Distance of the route: 1872m
Load of the route: 13

Route for vehicle 2:
 0 Load(0) ->  7 Load(8) ->  13 Load(14) ->  0 Load(14)
Distance of the route: 868m
Load of the route: 14

Route for vehicle 3:
 0 Load(0) ->  8 Load(8) ->  10 Load(10) ->  2 Load(11) ->  5 Load(14) ->  0 Load(14)
Distance of the route: 1872m
Load of the route: 14

Total Distance of all routes: 5936m
Total Load of all routes: 56

הנה תרשים של הנתיבים.

השלמת תוכניות

אלה התוכניות המלאות.

Python

"""Capacited Vehicles Routing Problem (CVRP)."""

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


def create_data_model():
    """Stores the data for the problem."""
    data = {}
    data["distance_matrix"] = [
        # fmt: off
      [0, 548, 776, 696, 582, 274, 502, 194, 308, 194, 536, 502, 388, 354, 468, 776, 662],
      [548, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674, 1016, 868, 1210],
      [776, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164, 1130, 788, 1552, 754],
      [696, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822, 1164, 560, 1358],
      [582, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708, 1050, 674, 1244],
      [274, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628, 514, 1050, 708],
      [502, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856, 514, 1278, 480],
      [194, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320, 662, 742, 856],
      [308, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662, 320, 1084, 514],
      [194, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388, 274, 810, 468],
      [536, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764, 730, 388, 1152, 354],
      [502, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114, 308, 650, 274, 844],
      [388, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194, 536, 388, 730],
      [354, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0, 342, 422, 536],
      [468, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536, 342, 0, 764, 194],
      [776, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274, 388, 422, 764, 0, 798],
      [662, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730, 536, 194, 798, 0],
        # fmt: on
    ]
    data["demands"] = [0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8]
    data["vehicle_capacities"] = [15, 15, 15, 15]
    data["num_vehicles"] = 4
    data["depot"] = 0
    return data


def print_solution(data, manager, routing, assignment):
    """Prints assignment on console."""
    print(f"Objective: {assignment.ObjectiveValue()}")
    # Display dropped nodes.
    dropped_nodes = "Dropped nodes:"
    for node in range(routing.Size()):
        if routing.IsStart(node) or routing.IsEnd(node):
            continue
        if assignment.Value(routing.NextVar(node)) == node:
            dropped_nodes += f" {manager.IndexToNode(node)}"
    print(dropped_nodes)
    # Display routes
    total_distance = 0
    total_load = 0
    for vehicle_id in range(data["num_vehicles"]):
        index = routing.Start(vehicle_id)
        plan_output = f"Route for vehicle {vehicle_id}:\n"
        route_distance = 0
        route_load = 0
        while not routing.IsEnd(index):
            node_index = manager.IndexToNode(index)
            route_load += data["demands"][node_index]
            plan_output += f" {node_index} Load({route_load}) -> "
            previous_index = index
            index = assignment.Value(routing.NextVar(index))
            route_distance += routing.GetArcCostForVehicle(
                previous_index, index, vehicle_id
            )
        plan_output += f" {manager.IndexToNode(index)} Load({route_load})\n"
        plan_output += f"Distance of the route: {route_distance}m\n"
        plan_output += f"Load of the route: {route_load}\n"
        print(plan_output)
        total_distance += route_distance
        total_load += route_load
    print(f"Total Distance of all routes: {total_distance}m")
    print(f"Total Load of all routes: {total_load}")


def main():
    """Solve the CVRP problem."""
    # Instantiate the data problem.
    data = create_data_model()

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

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

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

    transit_callback_index = routing.RegisterTransitCallback(distance_callback)

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

    # Add Capacity constraint.
    def demand_callback(from_index):
        """Returns the demand of the node."""
        # Convert from routing variable Index to demands NodeIndex.
        from_node = manager.IndexToNode(from_index)
        return data["demands"][from_node]

    demand_callback_index = routing.RegisterUnaryTransitCallback(demand_callback)
    routing.AddDimensionWithVehicleCapacity(
        demand_callback_index,
        0,  # null capacity slack
        data["vehicle_capacities"],  # vehicle maximum capacities
        True,  # start cumul to zero
        "Capacity",
    )
    # Allow to drop nodes.
    penalty = 1000
    for node in range(1, len(data["distance_matrix"])):
        routing.AddDisjunction([manager.NodeToIndex(node)], penalty)

    # Setting first solution heuristic.
    search_parameters = pywrapcp.DefaultRoutingSearchParameters()
    search_parameters.first_solution_strategy = (
        routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC
    )
    search_parameters.local_search_metaheuristic = (
        routing_enums_pb2.LocalSearchMetaheuristic.GUIDED_LOCAL_SEARCH
    )
    search_parameters.time_limit.FromSeconds(1)

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

    # Print solution on console.
    if assignment:
        print_solution(data, manager, routing, assignment)


if __name__ == "__main__":
    main()

C++‎

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

#include "google/protobuf/duration.pb.h"
#include "ortools/constraint_solver/routing.h"
#include "ortools/constraint_solver/routing_enums.pb.h"
#include "ortools/constraint_solver/routing_index_manager.h"
#include "ortools/constraint_solver/routing_parameters.h"

namespace operations_research {
struct DataModel {
  const std::vector<std::vector<int64_t>> distance_matrix{
      {0, 548, 776, 696, 582, 274, 502, 194, 308, 194, 536, 502, 388, 354, 468,
       776, 662},
      {548, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674,
       1016, 868, 1210},
      {776, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164, 1130,
       788, 1552, 754},
      {696, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822,
       1164, 560, 1358},
      {582, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708,
       1050, 674, 1244},
      {274, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628, 514,
       1050, 708},
      {502, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856, 514,
       1278, 480},
      {194, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320, 662,
       742, 856},
      {308, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662, 320,
       1084, 514},
      {194, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388, 274,
       810, 468},
      {536, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764, 730,
       388, 1152, 354},
      {502, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114, 308,
       650, 274, 844},
      {388, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194, 536,
       388, 730},
      {354, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0, 342,
       422, 536},
      {468, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536, 342,
       0, 764, 194},
      {776, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274, 388,
       422, 764, 0, 798},
      {662, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730, 536,
       194, 798, 0},
  };
  const std::vector<int64_t> demands{
      0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8,
  };
  const std::vector<int64_t> vehicle_capacities{15, 15, 15, 15};
  const int num_vehicles = 4;
  const RoutingIndexManager::NodeIndex depot{0};
};

//! @brief Print the solution.
//! @param[in] data Data of the problem.
//! @param[in] manager Index manager used.
//! @param[in] routing Routing solver used.
//! @param[in] solution Solution found by the solver.
void PrintSolution(const DataModel& data, const RoutingIndexManager& manager,
                   const RoutingModel& routing, const Assignment& solution) {
  // Display dropped nodes.
  std::ostringstream dropped_nodes;
  for (int64_t node = 0; node < routing.Size(); ++node) {
    if (routing.IsStart(node) || routing.IsEnd(node)) continue;
    if (solution.Value(routing.NextVar(node)) == node) {
      dropped_nodes << " " << manager.IndexToNode(node).value();
    }
  }
  LOG(INFO) << "Dropped nodes:" << dropped_nodes.str();
  // Display routes
  int64_t total_distance{0};
  int64_t total_load{0};
  for (int vehicle_id = 0; vehicle_id < data.num_vehicles; ++vehicle_id) {
    int64_t index = routing.Start(vehicle_id);
    LOG(INFO) << "Route for Vehicle " << vehicle_id << ":";
    int64_t route_distance{0};
    int64_t route_load{0};
    std::ostringstream route;
    while (!routing.IsEnd(index)) {
      const int node_index = manager.IndexToNode(index).value();
      route_load += data.demands[node_index];
      route << node_index << " Load(" << route_load << ") -> ";
      const int64_t previous_index = index;
      index = solution.Value(routing.NextVar(index));
      route_distance += routing.GetArcCostForVehicle(previous_index, index,
                                                     int64_t{vehicle_id});
    }
    LOG(INFO) << route.str() << manager.IndexToNode(index).value();
    LOG(INFO) << "Distance of the route: " << route_distance << "m";
    LOG(INFO) << "Load of the route: " << route_load;
    total_distance += route_distance;
    total_load += route_load;
  }
  LOG(INFO) << "Total distance of all routes: " << total_distance << "m";
  LOG(INFO) << "Total load of all routes: " << total_load;
  LOG(INFO) << "";
  LOG(INFO) << "Advanced usage:";
  LOG(INFO) << "Problem solved in " << routing.solver()->wall_time() << "ms";
}

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

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

  // Create Routing Model.
  RoutingModel routing(manager);

  // Create and register a transit callback.
  const int transit_callback_index = routing.RegisterTransitCallback(
      [&data, &manager](const int64_t from_index,
                        const int64_t to_index) -> int64_t {
        // Convert from routing variable Index to distance matrix NodeIndex.
        const int from_node = manager.IndexToNode(from_index).value();
        const int to_node = manager.IndexToNode(to_index).value();
        return data.distance_matrix[from_node][to_node];
      });

  // Define cost of each arc.
  routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index);

  // Add Capacity constraint.
  const int demand_callback_index = routing.RegisterUnaryTransitCallback(
      [&data, &manager](const int64_t from_index) -> int64_t {
        // Convert from routing variable Index to demand NodeIndex.
        const int from_node = manager.IndexToNode(from_index).value();
        return data.demands[from_node];
      });
  routing.AddDimensionWithVehicleCapacity(
      demand_callback_index,    // transit callback index
      int64_t{0},               // null capacity slack
      data.vehicle_capacities,  // vehicle maximum capacities
      true,                     // start cumul to zero
      "Capacity");
  // Allow to drop nodes.
  int64_t penalty{1000};
  for (int i = 1; i < data.distance_matrix.size(); ++i) {
    routing.AddDisjunction(
        {manager.NodeToIndex(RoutingIndexManager::NodeIndex(i))}, penalty);
  }

  // Setting first solution heuristic.
  RoutingSearchParameters search_parameters = DefaultRoutingSearchParameters();
  search_parameters.set_first_solution_strategy(
      FirstSolutionStrategy::PATH_CHEAPEST_ARC);
  search_parameters.set_local_search_metaheuristic(
      LocalSearchMetaheuristic::GUIDED_LOCAL_SEARCH);
  search_parameters.mutable_time_limit()->set_seconds(1);

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

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

int main(int /*argc*/, char* /*argv*/[]) {
  operations_research::VrpDropNodes();
  return EXIT_SUCCESS;
}

Java

package com.google.ortools.constraintsolver.samples;
import com.google.ortools.Loader;
import com.google.ortools.constraintsolver.Assignment;
import com.google.ortools.constraintsolver.FirstSolutionStrategy;
import com.google.ortools.constraintsolver.LocalSearchMetaheuristic;
import com.google.ortools.constraintsolver.RoutingIndexManager;
import com.google.ortools.constraintsolver.RoutingModel;
import com.google.ortools.constraintsolver.RoutingSearchParameters;
import com.google.ortools.constraintsolver.main;
import com.google.protobuf.Duration;
import java.util.logging.Logger;

/** Minimal VRP.*/
public class VrpDropNodes {
  private static final Logger logger = Logger.getLogger(VrpDropNodes.class.getName());

  static class DataModel {
    public final long[][] distanceMatrix = {
        {0, 548, 776, 696, 582, 274, 502, 194, 308, 194, 536, 502, 388, 354, 468, 776, 662},
        {548, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674, 1016, 868, 1210},
        {776, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164, 1130, 788, 1552, 754},
        {696, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822, 1164, 560, 1358},
        {582, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708, 1050, 674, 1244},
        {274, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628, 514, 1050, 708},
        {502, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856, 514, 1278, 480},
        {194, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320, 662, 742, 856},
        {308, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662, 320, 1084, 514},
        {194, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388, 274, 810, 468},
        {536, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764, 730, 388, 1152, 354},
        {502, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114, 308, 650, 274, 844},
        {388, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194, 536, 388, 730},
        {354, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0, 342, 422, 536},
        {468, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536, 342, 0, 764, 194},
        {776, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274, 388, 422, 764, 0, 798},
        {662, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730, 536, 194, 798, 0},
    };
    public final long[] demands = {0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8};
    public final long[] vehicleCapacities = {15, 15, 15, 15};
    public final int vehicleNumber = 4;
    public final int depot = 0;
  }

  /// @brief Print the solution.
  static void printSolution(
      DataModel data, RoutingModel routing, RoutingIndexManager manager, Assignment solution) {
    // Solution cost.
    logger.info("Objective: " + solution.objectiveValue());
    // Inspect solution.
    // Display dropped nodes.
    String droppedNodes = "Dropped nodes:";
    for (int node = 0; node < routing.size(); ++node) {
      if (routing.isStart(node) || routing.isEnd(node)) {
        continue;
      }
      if (solution.value(routing.nextVar(node)) == node) {
        droppedNodes += " " + manager.indexToNode(node);
      }
    }
    logger.info(droppedNodes);
    // Display routes
    long totalDistance = 0;
    long totalLoad = 0;
    for (int i = 0; i < data.vehicleNumber; ++i) {
      long index = routing.start(i);
      logger.info("Route for Vehicle " + i + ":");
      long routeDistance = 0;
      long routeLoad = 0;
      String route = "";
      while (!routing.isEnd(index)) {
        long nodeIndex = manager.indexToNode(index);
        routeLoad += data.demands[(int) nodeIndex];
        route += nodeIndex + " Load(" + routeLoad + ") -> ";
        long previousIndex = index;
        index = solution.value(routing.nextVar(index));
        routeDistance += routing.getArcCostForVehicle(previousIndex, index, i);
      }
      route += manager.indexToNode(routing.end(i));
      logger.info(route);
      logger.info("Distance of the route: " + routeDistance + "m");
      totalDistance += routeDistance;
      totalLoad += routeLoad;
    }
    logger.info("Total Distance of all routes: " + totalDistance + "m");
    logger.info("Total Load of all routes: " + totalLoad);
  }

  public static void main(String[] args) throws Exception {
    Loader.loadNativeLibraries();
    // Instantiate the data problem.
    final DataModel data = new DataModel();

    // Create Routing Index Manager
    RoutingIndexManager manager =
        new RoutingIndexManager(data.distanceMatrix.length, data.vehicleNumber, data.depot);

    // Create Routing Model.
    RoutingModel routing = new RoutingModel(manager);

    // Create and register a transit callback.
    final int transitCallbackIndex =
        routing.registerTransitCallback((long fromIndex, long toIndex) -> {
          // Convert from routing variable Index to user NodeIndex.
          int fromNode = manager.indexToNode(fromIndex);
          int toNode = manager.indexToNode(toIndex);
          return data.distanceMatrix[fromNode][toNode];
        });

    // Define cost of each arc.
    routing.setArcCostEvaluatorOfAllVehicles(transitCallbackIndex);

    // Add Capacity constraint.
    final int demandCallbackIndex = routing.registerUnaryTransitCallback((long fromIndex) -> {
      // Convert from routing variable Index to user NodeIndex.
      int fromNode = manager.indexToNode(fromIndex);
      return data.demands[fromNode];
    });
    routing.addDimensionWithVehicleCapacity(demandCallbackIndex, 0, // null capacity slack
        data.vehicleCapacities, // vehicle maximum capacities
        true, // start cumul to zero
        "Capacity");
    // Allow to drop nodes.
    long penalty = 1000;
    for (int i = 1; i < data.distanceMatrix.length; ++i) {
      routing.addDisjunction(new long[] {manager.nodeToIndex(i)}, penalty);
    }

    // Setting first solution heuristic.
    RoutingSearchParameters searchParameters =
        main.defaultRoutingSearchParameters()
            .toBuilder()
            .setFirstSolutionStrategy(FirstSolutionStrategy.Value.PATH_CHEAPEST_ARC)
            .setLocalSearchMetaheuristic(LocalSearchMetaheuristic.Value.GUIDED_LOCAL_SEARCH)
            .setTimeLimit(Duration.newBuilder().setSeconds(1).build())
            .build();

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

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

C#‎

using System;
using System.Collections.Generic;
using Google.OrTools.ConstraintSolver;
using Google.Protobuf.WellKnownTypes; // Duration

/// <summary>
///   Minimal Vrp with drop nodes.
/// </summary>
public class VrpDropNodes
{
    class DataModel
    {
        public long[,] DistanceMatrix = {
            { 0, 548, 776, 696, 582, 274, 502, 194, 308, 194, 536, 502, 388, 354, 468, 776, 662 },
            { 548, 0, 684, 308, 194, 502, 730, 354, 696, 742, 1084, 594, 480, 674, 1016, 868, 1210 },
            { 776, 684, 0, 992, 878, 502, 274, 810, 468, 742, 400, 1278, 1164, 1130, 788, 1552, 754 },
            { 696, 308, 992, 0, 114, 650, 878, 502, 844, 890, 1232, 514, 628, 822, 1164, 560, 1358 },
            { 582, 194, 878, 114, 0, 536, 764, 388, 730, 776, 1118, 400, 514, 708, 1050, 674, 1244 },
            { 274, 502, 502, 650, 536, 0, 228, 308, 194, 240, 582, 776, 662, 628, 514, 1050, 708 },
            { 502, 730, 274, 878, 764, 228, 0, 536, 194, 468, 354, 1004, 890, 856, 514, 1278, 480 },
            { 194, 354, 810, 502, 388, 308, 536, 0, 342, 388, 730, 468, 354, 320, 662, 742, 856 },
            { 308, 696, 468, 844, 730, 194, 194, 342, 0, 274, 388, 810, 696, 662, 320, 1084, 514 },
            { 194, 742, 742, 890, 776, 240, 468, 388, 274, 0, 342, 536, 422, 388, 274, 810, 468 },
            { 536, 1084, 400, 1232, 1118, 582, 354, 730, 388, 342, 0, 878, 764, 730, 388, 1152, 354 },
            { 502, 594, 1278, 514, 400, 776, 1004, 468, 810, 536, 878, 0, 114, 308, 650, 274, 844 },
            { 388, 480, 1164, 628, 514, 662, 890, 354, 696, 422, 764, 114, 0, 194, 536, 388, 730 },
            { 354, 674, 1130, 822, 708, 628, 856, 320, 662, 388, 730, 308, 194, 0, 342, 422, 536 },
            { 468, 1016, 788, 1164, 1050, 514, 514, 662, 320, 274, 388, 650, 536, 342, 0, 764, 194 },
            { 776, 868, 1552, 560, 674, 1050, 1278, 742, 1084, 810, 1152, 274, 388, 422, 764, 0, 798 },
            { 662, 1210, 754, 1358, 1244, 708, 480, 856, 514, 468, 354, 844, 730, 536, 194, 798, 0 }
        };
        public long[] Demands = { 0, 1, 1, 3, 6, 3, 6, 8, 8, 1, 2, 1, 2, 6, 6, 8, 8 };
        public long[] VehicleCapacities = { 15, 15, 15, 15 };
        public int VehicleNumber = 4;
        public int Depot = 0;
    };

    /// <summary>
    ///   Print the solution.
    /// </summary>
    static void PrintSolution(in DataModel data, in RoutingModel routing, in RoutingIndexManager manager,
                              in Assignment solution)
    {
        Console.WriteLine($"Objective {solution.ObjectiveValue()}:");

        // Inspect solution.
        // Display dropped nodes.
        string droppedNodes = "Dropped nodes:";
        for (int index = 0; index < routing.Size(); ++index)
        {
            if (routing.IsStart(index) || routing.IsEnd(index))
            {
                continue;
            }
            if (solution.Value(routing.NextVar(index)) == index)
            {
                droppedNodes += " " + manager.IndexToNode(index);
            }
        }
        Console.WriteLine("{0}", droppedNodes);
        // Inspect solution.
        long totalDistance = 0;
        long totalLoad = 0;
        for (int i = 0; i < data.VehicleNumber; ++i)
        {
            Console.WriteLine("Route for Vehicle {0}:", i);
            long routeDistance = 0;
            long routeLoad = 0;
            var index = routing.Start(i);
            while (routing.IsEnd(index) == false)
            {
                long nodeIndex = manager.IndexToNode(index);
                routeLoad += data.Demands[nodeIndex];
                Console.Write("{0} Load({1}) -> ", nodeIndex, routeLoad);
                var previousIndex = index;
                index = solution.Value(routing.NextVar(index));
                routeDistance += routing.GetArcCostForVehicle(previousIndex, index, 0);
            }
            Console.WriteLine("{0}", manager.IndexToNode((int)index));
            Console.WriteLine("Distance of the route: {0}m", routeDistance);
            totalDistance += routeDistance;
            totalLoad += routeLoad;
        }
        Console.WriteLine("Total Distance of all routes: {0}m", totalDistance);
        Console.WriteLine("Total Load of all routes: {0}m", totalLoad);
    }

    public static void Main(String[] args)
    {
        // Instantiate the data problem.
        DataModel data = new DataModel();

        // Create Routing Index Manager
        RoutingIndexManager manager =
            new RoutingIndexManager(data.DistanceMatrix.GetLength(0), data.VehicleNumber, data.Depot);

        // Create Routing Model.
        RoutingModel routing = new RoutingModel(manager);

        // Create and register a transit callback.
        int transitCallbackIndex = routing.RegisterTransitCallback((long fromIndex, long toIndex) =>
                                                                   {
                                                                       // Convert from routing variable Index to
                                                                       // distance matrix NodeIndex.
                                                                       var fromNode = manager.IndexToNode(fromIndex);
                                                                       var toNode = manager.IndexToNode(toIndex);
                                                                       return data.DistanceMatrix[fromNode, toNode];
                                                                   });

        // Define cost of each arc.
        routing.SetArcCostEvaluatorOfAllVehicles(transitCallbackIndex);

        // Add Capacity constraint.
        int demandCallbackIndex = routing.RegisterUnaryTransitCallback((long fromIndex) =>
                                                                       {
                                                                           // Convert from routing variable Index to
                                                                           // demand NodeIndex.
                                                                           var fromNode =
                                                                               manager.IndexToNode(fromIndex);
                                                                           return data.Demands[fromNode];
                                                                       });
        routing.AddDimensionWithVehicleCapacity(demandCallbackIndex, 0, // null capacity slack
                                                data.VehicleCapacities, // vehicle maximum capacities
                                                true,                   // start cumul to zero
                                                "Capacity");
        // Allow to drop nodes.
        long penalty = 1000;
        for (int i = 1; i < data.DistanceMatrix.GetLength(0); ++i)
        {
            routing.AddDisjunction(new long[] { manager.NodeToIndex(i) }, penalty);
        }

        // Setting first solution heuristic.
        RoutingSearchParameters searchParameters =
            operations_research_constraint_solver.DefaultRoutingSearchParameters();
        searchParameters.FirstSolutionStrategy = FirstSolutionStrategy.Types.Value.PathCheapestArc;
        searchParameters.LocalSearchMetaheuristic = LocalSearchMetaheuristic.Types.Value.GuidedLocalSearch;
        searchParameters.TimeLimit = new Duration { Seconds = 1 };

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

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