OR-Tools:6-路由问题（Routing）车辆路线，旅行商问题TSP

OR-Tools 解决的问题类型：

Linear optimization

Constraint optimization

Mixed-integer optimization

Bin packing

Network flows

Assignment

Scheduling

Routing

路由问题：

优化重要的解决问题是车辆规划方面，路线问题，成本问题

车辆路线，其目标是为访问一组地点的车队找到最佳路线。通常，"最佳"是指总距离或成本最少的路线。下面是一些路由问题的示例：

包裹递送公司希望为司机分配送货路线。
一家有线电视公司希望为技术人员分配线路，让他们拨打住宅服务电话。
一家拼车公司希望为司机分配接送乘客的路线。

最著名的路线问题是旅行推销员问题（TSP）：为需要拜访不同地点的客户并返回起点的销售人员找到最短的路线。TSP 可以由图形表示，其中节点对应于位置，而边（或弧）表示位置之间的直接移动。例如，下图显示了一个 TSP，只有四个位置，标记为 A、B、C 和 D。任何两个位置之间的距离由连接它们的边缘旁边的数字给出。

最短路径是 ACDBA，其总距离为 35 × 30 × 15 × 10 × 90。

在上面的示例中，只有六条路由。当地址很多时候路由数量激增。搜素空间很大，与成本的平衡都是需要优化的问题

TSP 的更通用版本是车辆路线问题（VRP），其中有多辆车。在大多数情况下，VRP 具有约束性：例如，车辆可能具有可携带的最大重量或数量，或者可能需要驾驶员在客户要求的指定时间窗口内访问位置。

关于TSP问题可以多参考一个有趣的网站http://www.math.uwaterloo.ca/tsp/index.html

解决的问题列表：

旅行推销员问题TSP，典型的路线问题，其中只有一辆车。
车辆路线问题，TSP 与多辆车的概括。
具有容量限制的 VRP，其中车辆具有最大容量，可携带的物品。
带时间窗的 VRP，车辆必须在指定的时间间隔内访问位置。
具有资源限制的 VRP，如空间或人员在仓库（路线的起点）装卸车辆。
VRP与下降的访问，其中车辆不需要访问所有地点，但必须支付每次下降访问的罚款。

eg1:一个人旅行从0出发后再回到0点。cost就是距离

"""Simple travelling salesman problem between cities."""from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}data['distance_matrix'] = [[0, 2451, 713, 1018, 1631, 1374, 2408, 213, 2571, 875, 1420, 2145, 1972],[2451, 0, 1745, 1524, 831, 1240, 959, 2596, 403, 1589, 1374, 357, 579],[713, 1745, 0, 355, 920, 803, 1737, 851, 1858, 262, 940, 1453, 1260],[1018, 1524, 355, 0, 700, 862, 1395, 1123, 1584, 466, 1056, 1280, 987],[1631, 831, 920, 700, 0, 663, 1021, 1769, 949, 796, 879, 586, 371],[1374, 1240, 803, 862, 663, 0, 1681, 1551, 1765, 547, 225, 887, 999],[2408, 959, 1737, 1395, 1021, 1681, 0, 2493, 678, 1724, 1891, 1114, 701],[213, 2596, 851, 1123, 1769, 1551, 2493, 0, 2699, 1038, 1605, 2300, 2099],[2571, 403, 1858, 1584, 949, 1765, 678, 2699, 0, 1744, 1645, 653, 600],[875, 1589, 262, 466, 796, 547, 1724, 1038, 1744, 0, 679, 1272, 1162],[1420, 1374, 940, 1056, 879, 225, 1891, 1605, 1645, 679, 0, 1017, 1200],[2145, 357, 1453, 1280, 586, 887, 1114, 2300, 653, 1272, 1017, 0, 504],[1972, 579, 1260, 987, 371, 999, 701, 2099, 600, 1162, 1200, 504, 0],]  # yapf: disabledata['num_vehicles'] = 1data['depot'] = 0return datadef print_solution(manager, routing, solution):"""Prints solution on console."""print('Objective: {} miles'.format(solution.ObjectiveValue()))index = routing.Start(0)plan_output = 'Route for vehicle 0:\n'route_distance = 0while not routing.IsEnd(index):plan_output += ' {} ->'.format(manager.IndexToNode(index))previous_index = indexindex = solution.Value(routing.NextVar(index))route_distance += routing.GetArcCostForVehicle(previous_index, index, 0)plan_output += ' {}\n'.format(manager.IndexToNode(index))print(plan_output)plan_output += 'Route distance: {}miles\n'.format(route_distance)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['depot'])# Create Routing Model.routing = pywrapcp.RoutingModel(manager)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)# 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(manager, routing, solution)if __name__ == '__main__':main()

Objective: 7293 miles
Route for vehicle 0:0 -> 7 -> 2 -> 3 -> 4 -> 12 -> 6 -> 8 -> 1 -> 11 -> 10 -> 5 -> 9 -> 0

eg2:在电路板上钻孔,找到钻头在板上的最短路线，以便钻出所有必需的孔。问题的数据由平面中的 280 个点组成，如图中的散点图所示

计算数据中任何两个点之间的欧几里德距离，并存储在数组中。由于路由解算器在整数上工作，因此函数将计算的距离四舍五（以整数）

# Euclidean distancedistances[from_counter][to_counter] = (int(math.hypot((from_node[0] - to_node[0]),(from_node[1] - to_node[1]))))

"""Simple travelling salesman problem on a circuit board."""from __future__ import print_function
import math
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}# Locations in block unitsdata['locations'] = [(288, 149), (288, 129), (270, 133), (256, 141), (256, 157), (246, 157),(236, 169), (228, 169), (228, 161), (220, 169), (212, 169), (204, 169),(196, 169), (188, 169), (196, 161), (188, 145), (172, 145), (164, 145),(156, 145), (148, 145), (140, 145), (148, 169), (164, 169), (172, 169),(156, 169), (140, 169), (132, 169), (124, 169), (116, 161), (104, 153),(104, 161), (104, 169), (90, 165), (80, 157), (64, 157), (64, 165),(56, 169), (56, 161), (56, 153), (56, 145), (56, 137), (56, 129),(56, 121), (40, 121), (40, 129), (40, 137), (40, 145), (40, 153),(40, 161), (40, 169), (32, 169), (32, 161), (32, 153), (32, 145),(32, 137), (32, 129), (32, 121), (32, 113), (40, 113), (56, 113),(56, 105), (48, 99), (40, 99), (32, 97), (32, 89), (24, 89),(16, 97), (16, 109), (8, 109), (8, 97), (8, 89), (8, 81),(8, 73), (8, 65), (8, 57), (16, 57), (8, 49), (8, 41),(24, 45), (32, 41), (32, 49), (32, 57), (32, 65), (32, 73),(32, 81), (40, 83), (40, 73), (40, 63), (40, 51), (44, 43),(44, 35), (44, 27), (32, 25), (24, 25), (16, 25), (16, 17),(24, 17), (32, 17), (44, 11), (56, 9), (56, 17), (56, 25),(56, 33), (56, 41), (64, 41), (72, 41), (72, 49), (56, 49),(48, 51), (56, 57), (56, 65), (48, 63), (48, 73), (56, 73),(56, 81), (48, 83), (56, 89), (56, 97), (104, 97), (104, 105),(104, 113), (104, 121), (104, 129), (104, 137), (104, 145), (116, 145),(124, 145), (132, 145), (132, 137), (140, 137), (148, 137), (156, 137),(164, 137), (172, 125), (172, 117), (172, 109), (172, 101), (172, 93),(172, 85), (180, 85), (180, 77), (180, 69), (180, 61), (180, 53),(172, 53), (172, 61), (172, 69), (172, 77), (164, 81), (148, 85),(124, 85), (124, 93), (124, 109), (124, 125), (124, 117), (124, 101),(104, 89), (104, 81), (104, 73), (104, 65), (104, 49), (104, 41),(104, 33), (104, 25), (104, 17), (92, 9), (80, 9), (72, 9),(64, 21), (72, 25), (80, 25), (80, 25), (80, 41), (88, 49),(104, 57), (124, 69), (124, 77), (132, 81), (140, 65), (132, 61),(124, 61), (124, 53), (124, 45), (124, 37), (124, 29), (132, 21),(124, 21), (120, 9), (128, 9), (136, 9), (148, 9), (162, 9),(156, 25), (172, 21), (180, 21), (180, 29), (172, 29), (172, 37),(172, 45), (180, 45), (180, 37), (188, 41), (196, 49), (204, 57),(212, 65), (220, 73), (228, 69), (228, 77), (236, 77), (236, 69),(236, 61), (228, 61), (228, 53), (236, 53), (236, 45), (228, 45),(228, 37), (236, 37), (236, 29), (228, 29), (228, 21), (236, 21),(252, 21), (260, 29), (260, 37), (260, 45), (260, 53), (260, 61),(260, 69), (260, 77), (276, 77), (276, 69), (276, 61), (276, 53),(284, 53), (284, 61), (284, 69), (284, 77), (284, 85), (284, 93),(284, 101), (288, 109), (280, 109), (276, 101), (276, 93), (276, 85),(268, 97), (260, 109), (252, 101), (260, 93), (260, 85), (236, 85),(228, 85), (228, 93), (236, 93), (236, 101), (228, 101), (228, 109),(228, 117), (228, 125), (220, 125), (212, 117), (204, 109), (196, 101),(188, 93), (180, 93), (180, 101), (180, 109), (180, 117), (180, 125),(196, 145), (204, 145), (212, 145), (220, 145), (228, 145), (236, 145),(246, 141), (252, 125), (260, 129), (280, 133)]  # yapf: disabledata['num_vehicles'] = 1data['depot'] = 0return datadef compute_euclidean_distance_matrix(locations):"""Creates callback to return distance between points."""distances = {}for from_counter, from_node in enumerate(locations):distances[from_counter] = {}for to_counter, to_node in enumerate(locations):if from_counter == to_counter:distances[from_counter][to_counter] = 0else:# Euclidean distancedistances[from_counter][to_counter] = (int(math.hypot((from_node[0] - to_node[0]),(from_node[1] - to_node[1]))))return distancesdef print_solution(manager, routing, solution):"""Prints solution on console."""print('Objective: {}'.format(solution.ObjectiveValue()))index = routing.Start(0)plan_output = 'Route:\n'route_distance = 0while not routing.IsEnd(index):plan_output += ' {} ->'.format(manager.IndexToNode(index))previous_index = indexindex = solution.Value(routing.NextVar(index))route_distance += routing.GetArcCostForVehicle(previous_index, index, 0)plan_output += ' {}\n'.format(manager.IndexToNode(index))print(plan_output)plan_output += 'Objective: {}m\n'.format(route_distance)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['locations']),data['num_vehicles'], data['depot'])# Create Routing Model.routing = pywrapcp.RoutingModel(manager)distance_matrix = compute_euclidean_distance_matrix(data['locations'])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 distance_matrix[from_node][to_node]transit_callback_index = routing.RegisterTransitCallback(distance_callback)# Define cost of each arc.routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index)# 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(manager, routing, solution)if __name__ == '__main__':main()

最好的算法可以常规地解决具有数万个节点的 TSP 实例,目前最好的记录VLSI应用的85,900 nodes. 对于具有数百万个节点的某些实例，已找到解决方案，其性能保证在最佳游览的 1% 以内。

由于路由解算器在整数上工作，因此如果距离矩阵具有非整数条目，则必须将距离舍入到整数。如果某些距离较小，则舍入可能会影响解。

为了避免舍入的任何问题，您可以缩放距离矩阵：将矩阵的所有条目乘以大数，例如 100。这会将任何路由的长度乘以 100，但它不会更改解决方案。优点是，现在舍入矩阵条目时，舍入量（最多 0.5）与距离相比非常小，因此不会影响解决方案。但是在打印输出时候注意除以缩放因子。

对于车辆路线问题VPR（Vehicle Routing Problem）这个就不是TSP，而是多个sales.

目标：尽量减少所有车辆中最长的单路线的长度。

但是限制条件也有不同的情况

容量限制：车辆需要在他们访问的每个地点取件，但具有最大的承载能力。
时间限制：必须在特定时间窗口内访问每个位置。

例子，一家公司需要拜访由相同矩形街区的城市中的客户。城市图如下所示，公司位置以黑色标记，地点以蓝色标记。

"""Vehicles Routing Problem (VRP)."""from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}data['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],]data['num_vehicles'] = 4data['depot'] = 0return datadef print_solution(data, manager, routing, solution):"""Prints solution on console."""max_route_distance = 0for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)route_distance = 0while not routing.IsEnd(index):plan_output += ' {} -> '.format(manager.IndexToNode(index))previous_index = indexindex = solution.Value(routing.NextVar(index))route_distance += routing.GetArcCostForVehicle(previous_index, index, vehicle_id)plan_output += '{}\n'.format(manager.IndexToNode(index))plan_output += 'Distance of the route: {}m\n'.format(route_distance)print(plan_output)max_route_distance = max(route_distance, max_route_distance)print('Maximum of the route distances: {}m'.format(max_route_distance))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 slack3000,  # vehicle maximum travel distanceTrue,  # start cumul to zerodimension_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()

Route for vehicle 0:0 ->  8 ->  6 ->  2 ->  5 -> 0
Distance of the route: 1552mRoute for vehicle 1:0 ->  7 ->  1 ->  4 ->  3 -> 0
Distance of the route: 1552mRoute for vehicle 2:0 ->  9 ->  10 ->  16 ->  14 -> 0
Distance of the route: 1552mRoute for vehicle 3:0 ->  12 ->  11 ->  15 ->  13 -> 0
Distance of the route: 1552mMaximum of the route distances: 1552m

使用谷歌距离矩阵 API创建距离矩阵

为地址或纬度和经度定义的任何位置集创建距离矩阵

要使用 API，您需要 API 密钥

距离矩阵 API 是一种服务，它根据起点和终点之间的推荐路径，为起点和目的地矩阵提供行驶距离和时间。若要使用距离矩阵 API，必须首先启用 API 并获取正确的身份验证凭据。

Google 地图平台是一组从 Google 云控制台（也称为云控制台）管理的 API 和 SDK。要开始使用 Google 地图平台，您需要：

创建帐单帐户
创建项目
启用一个或多个 API 或 SDK
获取、添加和限制 API 密钥

完成这些过程的选项取决于您是云控制台的新用户还是当前用户。所以说要有bill，收费的话。我们目前先不玩，后面有需要再弄这个。省掉先。

python 获取距离矩阵的例子

from __future__ import division
from __future__ import print_function
import requests
import json
import urllibdef create_data():"""Creates the data."""data = {}data['API_key'] = 'YOUR_API_KEY'data['addresses'] = ['3610+Hacks+Cross+Rd+Memphis+TN', # depot'1921+Elvis+Presley+Blvd+Memphis+TN','149+Union+Avenue+Memphis+TN','1034+Audubon+Drive+Memphis+TN','1532+Madison+Ave+Memphis+TN','706+Union+Ave+Memphis+TN','3641+Central+Ave+Memphis+TN','926+E+McLemore+Ave+Memphis+TN','4339+Park+Ave+Memphis+TN','600+Goodwyn+St+Memphis+TN','2000+North+Pkwy+Memphis+TN','262+Danny+Thomas+Pl+Memphis+TN','125+N+Front+St+Memphis+TN','5959+Park+Ave+Memphis+TN','814+Scott+St+Memphis+TN','1005+Tillman+St+Memphis+TN']return datadef create_distance_matrix(data):addresses = data["addresses"]API_key = data["API_key"]# Distance Matrix API only accepts 100 elements per request, so get rows in multiple requests.max_elements = 100num_addresses = len(addresses) # 16 in this example.# Maximum number of rows that can be computed per request (6 in this example).max_rows = max_elements // num_addresses# num_addresses = q * max_rows + r (q = 2 and r = 4 in this example).q, r = divmod(num_addresses, max_rows)dest_addresses = addressesdistance_matrix = []# Send q requests, returning max_rows rows per request.for i in range(q):origin_addresses = addresses[i * max_rows: (i + 1) * max_rows]response = send_request(origin_addresses, dest_addresses, API_key)distance_matrix += build_distance_matrix(response)# Get the remaining remaining r rows, if necessary.if r > 0:origin_addresses = addresses[q * max_rows: q * max_rows + r]response = send_request(origin_addresses, dest_addresses, API_key)distance_matrix += build_distance_matrix(response)return distance_matrixdef send_request(origin_addresses, dest_addresses, API_key):""" Build and send request for the given origin and destination addresses."""def build_address_str(addresses):# Build a pipe-separated string of addressesaddress_str = ''for i in range(len(addresses) - 1):address_str += addresses[i] + '|'address_str += addresses[-1]return address_strrequest = 'https://maps.googleapis.com/maps/api/distancematrix/json?units=imperial'origin_address_str = build_address_str(origin_addresses)dest_address_str = build_address_str(dest_addresses)request = request + '&origins=' + origin_address_str + '&destinations=' + \dest_address_str + '&key=' + API_keyjsonResult = urllib.urlopen(request).read()response = json.loads(jsonResult)return responsedef build_distance_matrix(response):distance_matrix = []for row in response['rows']:row_list = [row['elements'][j]['distance']['value'] for j in range(len(row['elements']))]distance_matrix.append(row_list)return distance_matrix########
# Main #
########
def main():"""Entry point of the program"""# Create the data.data = create_data()addresses = data['addresses']API_key = data['API_key']distance_matrix = create_distance_matrix(data)print(distance_matrix)
if __name__ == '__main__':main()

容量有限车辆路线问题（CVRP）是一种 VRP，其中承载能力有限的车辆需要在不同地点拾取或运送物品。物料的数量（如重量或体积）和车辆具有可携带的最大容量。问题是以最少的成本取件或交付物品，同时不超过车辆的容量。

问题是找到一个路线的分配到总距离最短的车辆，使车辆携带的总量永远不会超过其容量。每辆车的最大容量为 15。

"""Capacited Vehicles Routing Problem (CVRP)."""from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}data['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],]data['demands'] = [0, 1, 1, 2, 4, 2, 4, 8, 8, 1, 2, 1, 2, 4, 4, 8, 8]data['vehicle_capacities'] = [15, 15, 15, 15]data['num_vehicles'] = 4data['depot'] = 0return datadef print_solution(data, manager, routing, solution):"""Prints solution on console."""total_distance = 0total_load = 0for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)route_distance = 0route_load = 0while not routing.IsEnd(index):node_index = manager.IndexToNode(index)route_load += data['demands'][node_index]plan_output += ' {0} Load({1}) -> '.format(node_index, route_load)previous_index = indexindex = solution.Value(routing.NextVar(index))route_distance += routing.GetArcCostForVehicle(previous_index, index, vehicle_id)plan_output += ' {0} Load({1})\n'.format(manager.IndexToNode(index),route_load)plan_output += 'Distance of the route: {}m\n'.format(route_distance)plan_output += 'Load of the route: {}\n'.format(route_load)print(plan_output)total_distance += route_distancetotal_load += route_loadprint('Total distance of all routes: {}m'.format(total_distance))print('Total load of all routes: {}'.format(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 slackdata['vehicle_capacities'],  # vehicle maximum capacitiesTrue,  # start cumul to zero'Capacity')# 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()

Route for vehicle 0:0 Load(0) ->  1 Load(1) ->  4 Load(5) ->  3 Load(7) ->  15 Load(15) ->  0 Load(15)
Distance of the route: 2192m
Load of the route: 15Route for vehicle 1:0 Load(0) ->  14 Load(4) ->  16 Load(12) ->  10 Load(14) ->  2 Load(15) ->  0 Load(15)
Distance of the route: 2192m
Load of the route: 15Route for vehicle 2:0 Load(0) ->  7 Load(8) ->  13 Load(12) ->  12 Load(14) ->  11 Load(15) ->  0 Load(15)
Distance of the route: 1324m
Load of the route: 15Route for vehicle 3:0 Load(0) ->  9 Load(1) ->  8 Load(9) ->  6 Load(13) ->  5 Load(15) ->  0 Load(15)
Distance of the route: 1164m
Load of the route: 15Total distance of all routes: 6872m
Total load of all routes: 60

注意当运行大型搜素的时候最好设置搜素的时间和事件条目。防止无止境的搜素。方法请参考约束编程那篇。

例3：

每辆车在不同位置拾取物品，然后将它们丢弃到其他位置。问题是为车辆分配路线以拾取和交付所有物品，同时最大限度地缩短最长路线的长度。

取件和交货位置的定义

data['pickups_deliveries'] = [[1, 6],[2, 10],[4, 3],[5, 9],[7, 8],[15, 11],[13, 12],[16, 14],]

"""Simple Pickup Delivery Problem (PDP)."""from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}data['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],]data['pickups_deliveries'] = [[1, 6],[2, 10],[4, 3],[5, 9],[7, 8],[15, 11],[13, 12],[16, 14],]data['num_vehicles'] = 4data['depot'] = 0return datadef print_solution(data, manager, routing, solution):"""Prints solution on console."""total_distance = 0for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)route_distance = 0while not routing.IsEnd(index):plan_output += ' {} -> '.format(manager.IndexToNode(index))previous_index = indexindex = solution.Value(routing.NextVar(index))route_distance += routing.GetArcCostForVehicle(previous_index, index, vehicle_id)plan_output += '{}\n'.format(manager.IndexToNode(index))plan_output += 'Distance of the route: {}m\n'.format(route_distance)print(plan_output)total_distance += route_distanceprint('Total Distance of all routes: {}m'.format(total_distance))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['depot'])# Create Routing Model.routing = pywrapcp.RoutingModel(manager)# Define cost of each arc.def distance_callback(from_index, to_index):"""Returns the manhattan 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)routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index)# Add Distance constraint.dimension_name = 'Distance'routing.AddDimension(transit_callback_index,0,  # no slack3000,  # vehicle maximum travel distanceTrue,  # start cumul to zerodimension_name)distance_dimension = routing.GetDimensionOrDie(dimension_name)distance_dimension.SetGlobalSpanCostCoefficient(100)# Define Transportation Requests.for request in data['pickups_deliveries']:pickup_index = manager.NodeToIndex(request[0])delivery_index = manager.NodeToIndex(request[1])routing.AddPickupAndDelivery(pickup_index, delivery_index)routing.solver().Add(routing.VehicleVar(pickup_index) == routing.VehicleVar(delivery_index))routing.solver().Add(distance_dimension.CumulVar(pickup_index) <=distance_dimension.CumulVar(delivery_index))# Setting first solution heuristic.search_parameters = pywrapcp.DefaultRoutingSearchParameters()search_parameters.first_solution_strategy = (routing_enums_pb2.FirstSolutionStrategy.PARALLEL_CHEAPEST_INSERTION)# Solve the problem.solution = routing.SolveWithParameters(search_parameters)# Print solution on console.if solution:print_solution(data, manager, routing, solution)if __name__ == '__main__':main()

Route for vehicle 0:0 ->  13 ->  15 ->  11 ->  12 -> 0
Distance of the route: 1552mRoute for vehicle 1:0 ->  5 ->  2 ->  10 ->  16 ->  14 ->  9 -> 0
Distance of the route: 2192mRoute for vehicle 2:0 ->  4 ->  3 -> 0
Distance of the route: 1392mRoute for vehicle 3:0 ->  7 ->  1 ->  6 ->  8 -> 0
Distance of the route: 1780mTotal Distance of all routes: 6916m

许多车辆路线问题涉及计划访问仅在特定时间窗口可用的客户。这些问题称为车辆路线问题与时间窗（VRPTW）

由于问题涉及时间窗口，因此数据包括时间矩阵，其中包含位置之间的行驶时间（而不是前面示例中的距离矩阵）。

下图显示了要以蓝色访问的位置和黑色仓库。时间窗口显示在每个位置的上方。

目标是最大限度地减少车辆的总行驶时间。time_matrix是指走的步长代价，例如0-7需要2步，0到8需要3步

    data['time_matrix'] = [[0, 6, 9, 8, 7, 3, 6, 2, 3, 2, 6, 6, 4, 4, 5, 9, 7],[6, 0, 8, 3, 2, 6, 8, 4, 8, 8, 13, 7, 5, 8, 12, 10, 14],[9, 8, 0, 11, 10, 6, 3, 9, 5, 8, 4, 15, 14, 13, 9, 18, 9],[8, 3, 11, 0, 1, 7, 10, 6, 10, 10, 14, 6, 7, 9, 14, 6, 16],[7, 2, 10, 1, 0, 6, 9, 4, 8, 9, 13, 4, 6, 8, 12, 8, 14],[3, 6, 6, 7, 6, 0, 2, 3, 2, 2, 7, 9, 7, 7, 6, 12, 8],[6, 8, 3, 10, 9, 2, 0, 6, 2, 5, 4, 12, 10, 10, 6, 15, 5],[2, 4, 9, 6, 4, 3, 6, 0, 4, 4, 8, 5, 4, 3, 7, 8, 10],[3, 8, 5, 10, 8, 2, 2, 4, 0, 3, 4, 9, 8, 7, 3, 13, 6],[2, 8, 8, 10, 9, 2, 5, 4, 3, 0, 4, 6, 5, 4, 3, 9, 5],[6, 13, 4, 14, 13, 7, 4, 8, 4, 4, 0, 10, 9, 8, 4, 13, 4],[6, 7, 15, 6, 4, 9, 12, 5, 9, 6, 10, 0, 1, 3, 7, 3, 10],[4, 5, 14, 7, 6, 7, 10, 4, 8, 5, 9, 1, 0, 2, 6, 4, 8],[4, 8, 13, 9, 8, 7, 10, 3, 7, 4, 8, 3, 2, 0, 4, 5, 6],[5, 12, 9, 14, 12, 6, 6, 7, 3, 3, 4, 7, 6, 4, 0, 9, 2],[9, 10, 18, 6, 8, 12, 15, 8, 13, 9, 13, 3, 4, 5, 9, 0, 9],[7, 14, 9, 16, 14, 8, 5, 10, 6, 5, 4, 10, 8, 6, 2, 9, 0],]data['time_windows'] = [(0, 5),  # depot(7, 12),  # 1(10, 15),  # 2(16, 18),  # 3(10, 13),  # 4(0, 5),  # 5(5, 10),  # 6(0, 4),  # 7(5, 10),  # 8(0, 3),  # 9(10, 16),  # 10(10, 15),  # 11(0, 5),  # 12(5, 10),  # 13(7, 8),  # 14(10, 15),  # 15(11, 15),  # 16]

每辆车路线上总时间的总时间上限：30

"""Vehicles Routing Problem (VRP) with Time Windows."""from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}data['time_matrix'] = [[0, 6, 9, 8, 7, 3, 6, 2, 3, 2, 6, 6, 4, 4, 5, 9, 7],[6, 0, 8, 3, 2, 6, 8, 4, 8, 8, 13, 7, 5, 8, 12, 10, 14],[9, 8, 0, 11, 10, 6, 3, 9, 5, 8, 4, 15, 14, 13, 9, 18, 9],[8, 3, 11, 0, 1, 7, 10, 6, 10, 10, 14, 6, 7, 9, 14, 6, 16],[7, 2, 10, 1, 0, 6, 9, 4, 8, 9, 13, 4, 6, 8, 12, 8, 14],[3, 6, 6, 7, 6, 0, 2, 3, 2, 2, 7, 9, 7, 7, 6, 12, 8],[6, 8, 3, 10, 9, 2, 0, 6, 2, 5, 4, 12, 10, 10, 6, 15, 5],[2, 4, 9, 6, 4, 3, 6, 0, 4, 4, 8, 5, 4, 3, 7, 8, 10],[3, 8, 5, 10, 8, 2, 2, 4, 0, 3, 4, 9, 8, 7, 3, 13, 6],[2, 8, 8, 10, 9, 2, 5, 4, 3, 0, 4, 6, 5, 4, 3, 9, 5],[6, 13, 4, 14, 13, 7, 4, 8, 4, 4, 0, 10, 9, 8, 4, 13, 4],[6, 7, 15, 6, 4, 9, 12, 5, 9, 6, 10, 0, 1, 3, 7, 3, 10],[4, 5, 14, 7, 6, 7, 10, 4, 8, 5, 9, 1, 0, 2, 6, 4, 8],[4, 8, 13, 9, 8, 7, 10, 3, 7, 4, 8, 3, 2, 0, 4, 5, 6],[5, 12, 9, 14, 12, 6, 6, 7, 3, 3, 4, 7, 6, 4, 0, 9, 2],[9, 10, 18, 6, 8, 12, 15, 8, 13, 9, 13, 3, 4, 5, 9, 0, 9],[7, 14, 9, 16, 14, 8, 5, 10, 6, 5, 4, 10, 8, 6, 2, 9, 0],]data['time_windows'] = [(0, 5),  # depot(7, 12),  # 1(10, 15),  # 2(16, 18),  # 3(10, 13),  # 4(0, 5),  # 5(5, 10),  # 6(0, 4),  # 7(5, 10),  # 8(0, 3),  # 9(10, 16),  # 10(10, 15),  # 11(0, 5),  # 12(5, 10),  # 13(7, 8),  # 14(10, 15),  # 15(11, 15),  # 16]data['num_vehicles'] = 4data['depot'] = 0return datadef print_solution(data, manager, routing, solution):"""Prints solution on console."""time_dimension = routing.GetDimensionOrDie('Time')total_time = 0for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)while not routing.IsEnd(index):time_var = time_dimension.CumulVar(index)plan_output += '{0} Time({1},{2}) -> '.format(manager.IndexToNode(index), solution.Min(time_var),solution.Max(time_var))index = solution.Value(routing.NextVar(index))time_var = time_dimension.CumulVar(index)plan_output += '{0} Time({1},{2})\n'.format(manager.IndexToNode(index),solution.Min(time_var),solution.Max(time_var))plan_output += 'Time of the route: {}min\n'.format(solution.Min(time_var))print(plan_output)total_time += solution.Min(time_var)print('Total time of all routes: {}min'.format(total_time))def main():"""Solve the VRP with time windows."""# Instantiate the data problem.data = create_data_model()# Create the routing index manager.manager = pywrapcp.RoutingIndexManager(len(data['time_matrix']),data['num_vehicles'], data['depot'])# Create Routing Model.routing = pywrapcp.RoutingModel(manager)# Create and register a transit callback.def time_callback(from_index, to_index):"""Returns the travel time between the two nodes."""# Convert from routing variable Index to time matrix NodeIndex.from_node = manager.IndexToNode(from_index)to_node = manager.IndexToNode(to_index)return data['time_matrix'][from_node][to_node]transit_callback_index = routing.RegisterTransitCallback(time_callback)# Define cost of each arc.routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index)# Add Time Windows constraint.time = 'Time'routing.AddDimension(transit_callback_index,30,  # allow waiting time30,  # maximum time per vehicleFalse,  # Don't force start cumul to zero.time)time_dimension = routing.GetDimensionOrDie(time)# Add time window constraints for each location except depot.for location_idx, time_window in enumerate(data['time_windows']):if location_idx == 0:continueindex = manager.NodeToIndex(location_idx)time_dimension.CumulVar(index).SetRange(time_window[0], time_window[1])# Add time window constraints for each vehicle start node.for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)time_dimension.CumulVar(index).SetRange(data['time_windows'][0][0],data['time_windows'][0][1])# Instantiate route start and end times to produce feasible times.for i in range(data['num_vehicles']):routing.AddVariableMinimizedByFinalizer(time_dimension.CumulVar(routing.Start(i)))routing.AddVariableMinimizedByFinalizer(time_dimension.CumulVar(routing.End(i)))# 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()

Route for vehicle 0:
0 Time(0,0) -> 9 Time(2,3) -> 14 Time(7,8) -> 16 Time(11,11) -> 0 Time(18,18)
Time of the route: 18minRoute for vehicle 1:
0 Time(0,0) -> 7 Time(2,4) -> 1 Time(7,11) -> 4 Time(10,13) -> 3 Time(16,16) -> 0 Time(24,24)
Time of the route: 24minRoute for vehicle 2:
0 Time(0,0) -> 12 Time(4,4) -> 13 Time(6,6) -> 15 Time(11,11) -> 11 Time(14,14) -> 0 Time(20,20)
Time of the route: 20minRoute for vehicle 3:
0 Time(0,0) -> 5 Time(3,3) -> 8 Time(5,5) -> 6 Time(7,7) -> 2 Time(10,10) -> 10 Time(14,14) -> 0 Time(20,20)
Time of the route: 20minTotal time of all routes: 82min

行驶时间的问题的松弛变量：车辆必须在 I 位置等待 25 分钟才能出发;换句话说，位置 i 的松弛是 25 。假设车辆在一个步骤中从位置 i 转到位置 j，则松弛与这些变量相关

slack(i) = cumul(j) - cumul(i) - transit(i, j)

资源受限和时间受限

所有车辆需要加载之前离开仓库，并在返回时卸载。由于只有两个可用的装卸码头，因此最多可以同时装载或卸载两辆车。因此，有些车辆必须等待其他车辆被装载，延迟离开仓库。

    data['vehicle_load_time'] = 5data['vehicle_unload_time'] = 5data['depot_capacity'] = 2

"""Vehicles Routing Problem (VRP) with Resource Constraints."""from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}data['time_matrix'] = [[0, 6, 9, 8, 7, 3, 6, 2, 3, 2, 6, 6, 4, 4, 5, 9, 7],[6, 0, 8, 3, 2, 6, 8, 4, 8, 8, 13, 7, 5, 8, 12, 10, 14],[9, 8, 0, 11, 10, 6, 3, 9, 5, 8, 4, 15, 14, 13, 9, 18, 9],[8, 3, 11, 0, 1, 7, 10, 6, 10, 10, 14, 6, 7, 9, 14, 6, 16],[7, 2, 10, 1, 0, 6, 9, 4, 8, 9, 13, 4, 6, 8, 12, 8, 14],[3, 6, 6, 7, 6, 0, 2, 3, 2, 2, 7, 9, 7, 7, 6, 12, 8],[6, 8, 3, 10, 9, 2, 0, 6, 2, 5, 4, 12, 10, 10, 6, 15, 5],[2, 4, 9, 6, 4, 3, 6, 0, 4, 4, 8, 5, 4, 3, 7, 8, 10],[3, 8, 5, 10, 8, 2, 2, 4, 0, 3, 4, 9, 8, 7, 3, 13, 6],[2, 8, 8, 10, 9, 2, 5, 4, 3, 0, 4, 6, 5, 4, 3, 9, 5],[6, 13, 4, 14, 13, 7, 4, 8, 4, 4, 0, 10, 9, 8, 4, 13, 4],[6, 7, 15, 6, 4, 9, 12, 5, 9, 6, 10, 0, 1, 3, 7, 3, 10],[4, 5, 14, 7, 6, 7, 10, 4, 8, 5, 9, 1, 0, 2, 6, 4, 8],[4, 8, 13, 9, 8, 7, 10, 3, 7, 4, 8, 3, 2, 0, 4, 5, 6],[5, 12, 9, 14, 12, 6, 6, 7, 3, 3, 4, 7, 6, 4, 0, 9, 2],[9, 10, 18, 6, 8, 12, 15, 8, 13, 9, 13, 3, 4, 5, 9, 0, 9],[7, 14, 9, 16, 14, 8, 5, 10, 6, 5, 4, 10, 8, 6, 2, 9, 0],]data['time_windows'] = [(0, 5),  # depot(7, 12),  # 1(10, 15),  # 2(5, 14),  # 3(5, 13),  # 4(0, 5),  # 5(5, 10),  # 6(0, 10),  # 7(5, 10),  # 8(0, 5),  # 9(10, 16),  # 10(10, 15),  # 11(0, 5),  # 12(5, 10),  # 13(7, 12),  # 14(10, 15),  # 15(5, 15),  # 16]data['num_vehicles'] = 4data['vehicle_load_time'] = 5data['vehicle_unload_time'] = 5data['depot_capacity'] = 2data['depot'] = 0return datadef print_solution(data, manager, routing, solution):"""Prints solution on console."""time_dimension = routing.GetDimensionOrDie('Time')total_time = 0for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)while not routing.IsEnd(index):time_var = time_dimension.CumulVar(index)plan_output += '{0} Time({1},{2}) -> '.format(manager.IndexToNode(index), solution.Min(time_var),solution.Max(time_var))index = solution.Value(routing.NextVar(index))time_var = time_dimension.CumulVar(index)plan_output += '{0} Time({1},{2})\n'.format(manager.IndexToNode(index),solution.Min(time_var),solution.Max(time_var))plan_output += 'Time of the route: {}min\n'.format(solution.Min(time_var))print(plan_output)total_time += solution.Min(time_var)print('Total time of all routes: {}min'.format(total_time))def main():"""Solve the VRP with time windows."""# Instantiate the data problem.data = create_data_model()# Create the routing index manager.manager = pywrapcp.RoutingIndexManager(len(data['time_matrix']),data['num_vehicles'], data['depot'])# Create Routing Model.routing = pywrapcp.RoutingModel(manager)# Create and register a transit callback.def time_callback(from_index, to_index):"""Returns the travel time between the two nodes."""# Convert from routing variable Index to time matrix NodeIndex.from_node = manager.IndexToNode(from_index)to_node = manager.IndexToNode(to_index)return data['time_matrix'][from_node][to_node]transit_callback_index = routing.RegisterTransitCallback(time_callback)# Define cost of each arc.routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index)# Add Time Windows constraint.time = 'Time'routing.AddDimension(transit_callback_index,60,  # allow waiting time60,  # maximum time per vehicleFalse,  # Don't force start cumul to zero.time)time_dimension = routing.GetDimensionOrDie(time)# Add time window constraints for each location except depot.for location_idx, time_window in enumerate(data['time_windows']):if location_idx == 0:continueindex = manager.NodeToIndex(location_idx)time_dimension.CumulVar(index).SetRange(time_window[0], time_window[1])# Add time window constraints for each vehicle start node.for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)time_dimension.CumulVar(index).SetRange(data['time_windows'][0][0],data['time_windows'][0][1])# Add resource constraints at the depot.solver = routing.solver()intervals = []for i in range(data['num_vehicles']):# Add time windows at start of routesintervals.append(solver.FixedDurationIntervalVar(time_dimension.CumulVar(routing.Start(i)),data['vehicle_load_time'], 'depot_interval'))# Add time windows at end of routes.intervals.append(solver.FixedDurationIntervalVar(time_dimension.CumulVar(routing.End(i)),data['vehicle_unload_time'], 'depot_interval'))depot_usage = [1 for i in range(len(intervals))]solver.Add(solver.Cumulative(intervals, depot_usage, data['depot_capacity'],'depot'))# Instantiate route start and end times to produce feasible times.for i in range(data['num_vehicles']):routing.AddVariableMinimizedByFinalizer(time_dimension.CumulVar(routing.Start(i)))routing.AddVariableMinimizedByFinalizer(time_dimension.CumulVar(routing.End(i)))# 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)else:print('No solution found !')if __name__ == '__main__':main()

Route for vehicle 0:
0 Time(5,5) -> 8 Time(8,8) -> 14 Time(11,11) -> 16 Time(13,13) -> 0 Time(20,20)
Time of the route: 20minRoute for vehicle 1:
0 Time(0,0) -> 12 Time(4,4) -> 13 Time(6,6) -> 15 Time(11,11) -> 11 Time(14,14) -> 0 Time(20,20)
Time of the route: 20minRoute for vehicle 2:
0 Time(5,5) -> 7 Time(7,7) -> 1 Time(11,11) -> 4 Time(13,13) -> 3 Time(14,14) -> 0 Time(25,25)
Time of the route: 25minRoute for vehicle 3:
0 Time(0,0) -> 9 Time(2,3) -> 5 Time(4,5) -> 6 Time(6,9) -> 2 Time(10,12) -> 10 Time(14,16) -> 0 Time(25,25)
Time of the route: 25minTotal time of all routes: 90min

处理由于约束而没有可行解决方案的路由问题，所有位置的总需求超过车辆的总容量，则无法找到任何解决方案。在这种情况下，车辆必须放弃访问某些地点。问题是如何决定放弃哪些访问。在所有地点引入了新的成本（称为罚款）。每当放弃对位置的访问时，将处罚添加到行驶的总距离中。然后，解算器将找到一条路由，该路径将总距离加上所有掉落位置的处罚之和降至最低。在删除一个位置以使问题变得可行后，解算器不会丢弃任何其他位置，因为这样做的处罚将超过任何进一步缩短的旅行距离。我们增加了一些需求，迫使一些车辆放弃访问。

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]

"""Capacited Vehicles Routing Problem (CVRP)."""from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}data['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],]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'] = 4data['depot'] = 0return datadef print_solution(data, manager, routing, assignment):"""Prints assignment on console."""# Display dropped nodes.dropped_nodes = 'Dropped nodes:'for node in range(routing.Size()):if routing.IsStart(node) or routing.IsEnd(node):continueif assignment.Value(routing.NextVar(node)) == node:dropped_nodes += ' {}'.format(manager.IndexToNode(node))print(dropped_nodes)# Display routestotal_distance = 0total_load = 0for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)route_distance = 0route_load = 0while not routing.IsEnd(index):node_index = manager.IndexToNode(index)route_load += data['demands'][node_index]plan_output += ' {0} Load({1}) -> '.format(node_index, route_load)previous_index = indexindex = assignment.Value(routing.NextVar(index))route_distance += routing.GetArcCostForVehicle(previous_index, index, vehicle_id)plan_output += ' {0} Load({1})\n'.format(manager.IndexToNode(index),route_load)plan_output += 'Distance of the route: {}m\n'.format(route_distance)plan_output += 'Load of the route: {}\n'.format(route_load)print(plan_output)total_distance += route_distancetotal_load += route_loadprint('Total Distance of all routes: {}m'.format(total_distance))print('Total Load of all routes: {}'.format(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 slackdata['vehicle_capacities'],  # vehicle maximum capacitiesTrue,  # start cumul to zero'Capacity')# Allow to drop nodes.penalty = 1000for 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)# Solve the problem.assignment = routing.SolveWithParameters(search_parameters)# Print solution on console.if assignment:print_solution(data, manager, routing, assignment)if __name__ == '__main__':main()

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: 15Route 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: 13Route for vehicle 2:0 Load(0) ->  7 Load(8) ->  13 Load(14) ->  0 Load(14)
Distance of the route: 868m
Load of the route: 14Route 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: 14Total Distance of all routes: 5936m
Total Load of all routes: 56

对于某些问题，您可能希望为 VRP 指定一组初始路由，而不是让解算器找到初始解决方案，例如，如果您已经找到了问题的解决方案，并且希望使用它作为解决修改问题的起点。

    data['initial_routes'] = [[8, 16, 14, 13, 12, 11],[3, 4, 9, 10],[15, 1],[7, 5, 2, 6],]

"""Vehicles Routing Problem (VRP)."""from __future__ import print_function
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}data['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],]data['initial_routes'] = [[8, 16, 14, 13, 12, 11],[3, 4, 9, 10],[15, 1],[7, 5, 2, 6],]data['num_vehicles'] = 4data['depot'] = 0return datadef print_solution(data, manager, routing, solution):"""Prints solution on console."""max_route_distance = 0for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)route_distance = 0while not routing.IsEnd(index):plan_output += ' {} -> '.format(manager.IndexToNode(index))previous_index = indexindex = solution.Value(routing.NextVar(index))route_distance += routing.GetArcCostForVehicle(previous_index, index, vehicle_id)plan_output += '{}\n'.format(manager.IndexToNode(index))plan_output += 'Distance of the route: {}m\n'.format(route_distance)print(plan_output)max_route_distance = max(route_distance, max_route_distance)print('Maximum of the route distances: {}m'.format(max_route_distance))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 slack3000,  # vehicle maximum travel distanceTrue,  # start cumul to zerodimension_name)distance_dimension = routing.GetDimensionOrDie(dimension_name)distance_dimension.SetGlobalSpanCostCoefficient(100)initial_solution = routing.ReadAssignmentFromRoutes(data['initial_routes'],True)print('Initial solution:')print_solution(data, manager, routing, initial_solution)# Set default search parameters.search_parameters = pywrapcp.DefaultRoutingSearchParameters()# 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()

设置可能不同的开始和结束位置。

在程序的数据部分创建开始和结束向量：

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

"""Simple Vehicles Routing Problem."""from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcpdef create_data_model():"""Stores the data for the problem."""data = {}data['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],]data['num_vehicles'] = 4data['starts'] = [1, 2, 15, 16]data['ends'] = [0, 0, 0, 0]return datadef print_solution(data, manager, routing, solution):"""Prints solution on console."""max_route_distance = 0for vehicle_id in range(data['num_vehicles']):index = routing.Start(vehicle_id)plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)route_distance = 0while not routing.IsEnd(index):plan_output += ' {} -> '.format(manager.IndexToNode(index))previous_index = indexindex = solution.Value(routing.NextVar(index))route_distance += routing.GetArcCostForVehicle(previous_index, index, vehicle_id)plan_output += '{}\n'.format(manager.IndexToNode(index))plan_output += 'Distance of the route: {}m\n'.format(route_distance)print(plan_output)max_route_distance = max(route_distance, max_route_distance)print('Maximum of the route distances: {}m'.format(max_route_distance))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 slack2000,  # vehicle maximum travel distanceTrue,  # start cumul to zerodimension_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()

Route for vehicle 0:1 ->  4 ->  3 ->  7 -> 0
Distance of the route: 1004mRoute for vehicle 1:2 ->  6 ->  8 ->  5 -> 0
Distance of the route: 936mRoute for vehicle 2:15 ->  11 ->  12 ->  13 -> 0
Distance of the route: 936mRoute for vehicle 3:16 ->  14 ->  10 ->  9 -> 0
Distance of the route: 1118mMaximum of the route distances: 1118m

允许车辆在任意位置起动和结束。若要这样设置问题，只需修改距离矩阵，即可将矩阵的第一行和列设置为具有所有零，使从仓库到任何其他位置的距离为 0。这会将仓库变成对最佳路径没有任何影响虚拟位置。

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],]

Route for vehicle 0:1 ->  7 ->  13 ->  12 -> 0
Distance of the route: 868mRoute for vehicle 1:2 ->  6 ->  8 ->  5 -> 0
Distance of the route: 662mRoute for vehicle 2:15 ->  11 ->  4 ->  3 -> 0
Distance of the route: 788mRoute for vehicle 3:16 ->  10 ->  9 ->  14 -> 0
Distance of the route: 696mMaximum of the route distances: 868m

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