ในส่วนนี้ เราจะอธิบายวิธีจัดการการกําหนดเส้นทางที่ไม่มีวิธีแก้ไขที่เป็นไปได้เนื่องจากข้อจํากัด ตัวอย่างเช่น หากคุณได้รับ VRP ที่มีข้อจํากัดด้านความจุ ซึ่งความต้องการโดยรวมในทุกตําแหน่งมีจํานวนมากกว่าความจุรวมของยานพาหนะ ก็ไม่อาจแก้ปัญหาได้ ในกรณีดังกล่าว ยานพาหนะดังกล่าวจะต้องลดการเดินทางไปยังสถานที่บางแห่ง โจทย์คือวิธีตัดสินใจว่าจะมีการเข้าชมใด
เราขอแนะนําค่าใช้จ่ายใหม่ที่เรียกว่าบทลงโทษในสถานที่ตั้งดังกล่าวเพื่อแก้ปัญหา เมื่อใดก็ตามที่การเดินทางไปยังสถานที่ดังกล่าวถูกลงโทษ จะถูกเพิ่มให้กับระยะทางรวมในการเดินทาง จากนั้นเครื่องมือแก้โจทย์คณิตจะค้นหาเส้นทางที่ลดระยะทางรวมบวกกับบทลงโทษหากสถานที่ตกลงมาทั้งหมด
ตัวอย่างเช่น พิจารณา VRP แบบง่ายที่มีข้อจํากัดความจุที่กําหนดโดยกราฟด้านล่าง ซึ่งตัวเลขถัดจากตําแหน่ง 3 ตําแหน่ง (นอกเหนือจากสถานี) เป็นความต้องการ
สมมติว่ามียานพาหนะเพียง 1 คันที่มีความจุ 50 คน แต่จะเข้าชมสถานที่ตั้งทั้ง 3 แห่งไม่ได้ ซึ่งก็คือ A, B และ C เพราะความต้องการทั้งหมดคือ 60 รายการ หากต้องการแก้ปัญหา
คุณกําหนดบทลงโทษครั้งใหญ่ เช่น 100 ครั้ง ให้กับสถานที่แต่ละแห่ง หลังจากที่ตรวจพบว่าปัญหาเกิดจากการแก้โจทย์ไม่ได้ เครื่องมือแก้โจทย์วางตําแหน่ง B แล้วแสดงผลเส้นทางต่อไปนี้ Depot -> A -> C -> Depot
นี่คือเส้นทางที่สั้นที่สุดที่เข้าชม 2 จาก 3 สถานที่ (ระยะทางคือ 55)
บทลงโทษ
ในตัวอย่างข้างต้น เราเลือกบทลงโทษที่มีขนาดใหญ่กว่าผลรวมของระยะทางระหว่างสถานที่ (ยกเว้นสถานี) ผลที่ได้คือ การดําเนินการแก้ปัญหาลดลงทําไม่ได้หลังจากนั้นก็จะทําให้สถานที่ไม่ลดลงเช่นเดียวกัน เนื่องจากบทลงโทษเมื่อทําเช่นนั้นจะทําให้มีจํานวนการเดินทางที่ลดลง
สมมติว่าคุณต้องการให้นําส่งข้อมูลให้มากที่สุดเท่าที่จะทําได้ ให้ใช้วิธีแก้โจทย์ที่น่าพอใจ
หากคุณไม่ต้องการนําส่งให้มากที่สุดเท่าที่จะทําได้ คุณอาจต้องกําหนดบทลงโทษขนาดเล็ก ซึ่งในกรณีนี้เครื่องมือแก้โจทย์คณิตอาจลดตําแหน่งมากกว่าจํานวนที่จําเป็นเพื่อให้ปัญหาเกิดขึ้นได้ เช่น คุณอาจทําเช่นนี้ได้ในกรณีที่มีค่าใช้จ่ายเพิ่มเติม การเดินทางไปยังสถานที่บางแห่ง และค่าใช้จ่ายการเดินทางเพิ่มเติม
ตัวอย่าง
ถัดไป เราขอนําเสนอตัวอย่างขนาดใหญ่ขึ้นของ VRP ที่สามารถแก้ไขได้โดยใช้บทลงโทษ ตัวอย่างนี้คล้ายกับตัวอย่าง CVRP ก่อนหน้า แต่ครั้งนี้เราได้เพิ่มความต้องการบางอย่าง ซึ่งเป็นการบังคับให้ยานพาหนะบางส่วนลดลงการเข้าชม
กราฟของสถานที่ตั้งและความต้องการใหม่จะแสดงอยู่ด้านล่าง
การแก้ตัวอย่างด้วยเครื่องมือ OR
ส่วนต่อไปนี้จะอธิบายวิธีแก้ตัวอย่างด้วยเครื่องมือ OR
สร้างข้อมูล
ข้อมูลของตัวอย่างนี้มีข้อมูลในตัวอย่าง VRP ก่อนหน้า รวมทั้งเพิ่มความต้องการและความจุต่อไปนี้
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 };
เพิ่มข้อจํากัดด้านความจุและบทลงโทษ
โค้ดต่อไปนี้จะเพิ่มข้อจํากัดของการเรียกกลับและขีดจํากัดความจุ และเพิ่มบทลงโทษโดยใช้เมธอด 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);
}
ในบริบทนี้ การประกาศเป็นตัวแปรที่เครื่องมือแก้โจทย์ใช้ตัดสินใจว่าจะรวมสถานที่นั้นๆ ไว้ในโซลูชันหรือไม่ ในตัวอย่างนี้ บทลงโทษมีบทลงโทษเหมือนกันในแต่ละสถานที่ แต่โดยทั่วไป คุณเพิ่มบทลงโทษที่แตกต่างกันให้กับสถานที่แต่ละแห่งได้
เพิ่มเครื่องพิมพ์โซลูชัน
เครื่องพิมพ์โซลูชันที่แสดงอยู่ด้านล่างคล้ายกับเครื่องพิมพ์ในตัวอย่าง CVRP แต่ก็แสดงตําแหน่งที่ทิ้งด้วย
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);
}
การเรียกใช้โปรแกรม
เมื่อเรียกใช้โปรแกรม ระบบจะแสดงผลด้านล่างนี้ โปรดทราบว่าเครื่องมือแก้โจทย์วางตําแหน่ง 6 และ 15 ไว้
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
นี่คือแผนภาพเส้นทาง
สิ้นสุดโปรแกรม
นี่คือโปรแกรมทั้งหมด
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);
}
}