1.VRP回顾
车辆路径问题是旅行商问题的推广。在VRP中，目标是为向不同地点交付货物或服务的车队找到最优路线集。VRP最初是由Dantzig和Ramser在1959年提出的。
与TSP类似，VRP也可以用分配给边缘的距离的图来表示。
如果您试图找到一组总距离最小、对车辆没有附加约束的路线，那么最优解决方案是将所有位置分配给一辆车，其余位置空闲。在这种情况下，问题归结为TSP。
一个更有趣的问题是最小化所有车辆的最长路线距离的长度，或最长旅行时间的路线的消耗时间。VRPs还可以对车辆有额外的限制—例如，对每辆车访问的地点数量或每条路线的长度的下限和上限。
1.1最小化最长的单一路径
在下面的例子中，一个公司需要访问由相同矩形块组成的城市中的一些客户位置。下图是该城市的示意图，公司位置用黑色标示，参观地点用蓝色标示。
"""
[(456, 320), # location 0
(228, 0), # location 1
(912, 0), # location 2
(0, 80), # location 3
(114, 80), # location 4
(570, 160), # location 5
(798, 160), # location 6
(342, 240), # location 7
(684, 240), # location 8
(570, 400), # location 9
(912, 400), # location 10
(114, 480), # location 11
(228, 480), # location 12
(342, 560), # location 13
(684, 560), # location 14
(0, 640), # location 15
(798, 640)] # location 16
"""
在这里我们根据上面的坐标算出不同位置之间的距离，距离函数使用曼哈顿距离:|x1 - x2| + |y1 - y2|。
# vrp_global_span
# [START import]
from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcp
# [END import]
# [START data_model]
def 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'] = 4
data['depot'] = 0
return data
# [END data_model]
# [START solution_printer]
def print_solution(data, manager, routing, solution):
"""Prints solution on console."""
max_route_distance = 0
for vehicle_id in range(data['num_vehicles']):
index = routing.Start(vehicle_id)
plan_output = 'Route for vehicle {}:\n'.format(vehicle_id)
route_distance = 0
while not routing.IsEnd(index):
plan_output += ' {} -> '.format(manager.IndexToNode(index))
previous_index = index
index = 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))
# [END solution_printer]
def main():
"""Solve the CVRP problem."""
# Instantiate the data problem.
# [START data]
data = create_data_model()
# [END data]
# Create the routing index manager.
# [START index_manager]
manager = pywrapcp.RoutingIndexManager(len(data['distance_matrix']),
data['num_vehicles'], data['depot'])
# [END index_manager]
# Create Routing Model.
# [START routing_model]
routing = pywrapcp.RoutingModel(manager)
# [END routing_model]
# Create and register a transit callback.
# [START 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)
# [END transit_callback]
# Define cost of each arc.
# [START arc_cost]
routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index)
# [END arc_cost]
# Add Distance constraint.
# [START distance_constraint]
dimension_name = 'Distance'
routing.AddDimension(
transit_callback_index,
0, # no slack
3000, # vehicle maximum travel distance
True, # start cumul to zero
dimension_name)
distance_dimension = routing.GetDimensionOrDie(dimension_name)
distance_dimension.SetGlobalSpanCostCoefficient(100)
# [END distance_constraint]
# Setting first solution heuristic.
# [START parameters]
search_parameters = pywrapcp.DefaultRoutingSearchParameters()
search_parameters.first_solution_strategy = (
routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC)
# [END parameters]
# Solve the problem.
# [START solve]
solution = routing.SolveWithParameters(search_parameters)
# [END solve]
# Print solution on console.
# [START print_solution]
if solution:
print_solution(data, manager, routing, solution)
# [END print_solution]
if __name__ == '__main__':
main()
# [END program]
"""
Route for vehicle 0:
0 -> 8 -> 6 -> 2 -> 5 -> 0
Distance of the route: 1552m
Route for vehicle 1:
0 -> 7 -> 1 -> 4 -> 3 -> 0
Distance of the route: 1552m
Route for vehicle 2:
0 -> 9 -> 10 -> 16 -> 14 -> 0
Distance of the route: 1552m
Route for vehicle 3:
0 -> 12 -> 11 -> 15 -> 13 -> 0
Distance of the route: 1552m
Maximum of the route distances: 1552m
"""
注意函数添加距离维度(Add a distance dimension)，要解决这个VRP，需要创建一个距离维度，它计算每辆车沿其路线行驶的累计距离。然后，将成本设置为每条路线总距离的最大值。
若要添加距离维度，请使用求解器的AddDimension方法。下面的代码为距离创建一个维度，并设置累计距离成本。关于距离信息可以参考vrp_dimension_details.
def add_distance_dimension(routing, distance_callback):
"""Add Global Span constraint"""
distance = 'Distance'
maximum_distance = 3000 # 每辆车能形式的最大距离
routing.AddDimension(
distance_callback,
0, # null slack
maximum_distance,
True, # 从累积到零，意思应该和“走了这么久还剩多少汽油”差不多吧
distance)
distance_dimension = routing.GetDimensionOrDie(distance)
# Try to minimize the max distance among vehicles.
# 尽量减少车辆之间的最大距离。
distance_dimension.SetGlobalSpanCostCoefficient(100)
下面是求解的结果：
2.CVRP问题
容量约束的车辆路径问题是一个对车辆能力有附加约束的车辆路径问题。在CVRP中，每个位置都有一个需求，例如重量或体积，对应于在那里取走或交付的项目。每次一辆车访问一个地点，它所承载的总数量会根据该地点的需求增加(取货)或减少(送货)。此外，每辆车在任何时候都有最大的承载能力。
CVRP可以用一个图来表示，该图具有分配给边缘的距离和分配给节点的需求。
如下图所示，在每个位置都有一个与要取的物品相对应的需求。此外，每辆车的最大容量为15。这个问题和VRP中是差不多的，只不过增加了车辆容量约束限制条件而已。
from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcp
def main():
"""Entry point of the program."""
# Instantiate the data problem.
num_locations = 5
num_vehicles = 1
depot = 0
# Create the routing index manager.
manager = pywrapcp.RoutingIndexManager(num_locations,num_vehicles,depot)
# Create Routing Model.
routing = pywrapcp.RoutingModel(manager)
# Create and register a transit callback.
def distance_callback(from_index, to_index):
"""Returns the absolute difference between the two nodes."""
# Convert from routing variable Index to user NodeIndex.
from_node = int(manager.IndexToNode(from_index))
to_node = int(manager.IndexToNode(to_index))
return abs(to_node - from_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) # pylint: disable=no-member
# Solve the problem.
assignment = routing.SolveWithParameters(search_parameters)
# Print solution on console.
print('Objective: {}'.format(assignment.ObjectiveValue()))
index = routing.Start(0)
plan_output = 'Route for vehicle 0:\n'
route_distance = 0
while not routing.IsEnd(index):
plan_output += '{} -> '.format(manager.IndexToNode(index))
previous_index = index
index = assignment.Value(routing.NextVar(index))
route_distance += routing.GetArcCostForVehicle(previous_index, index, 0)
plan_output += '{}\n'.format(manager.IndexToNode(index))
plan_output += 'Distance of the route: {}m\n'.format(route_distance)
print(plan_output)
if __name__ == '__main__':
main()
"""
Objective: 8
Route for vehicle 0:
0 -> 1 -> 2 -> 3 -> 4 -> 0
Distance of the route: 8m
"""
3.VRPTW
许多车辆路径问题都涉及到安排送货时间，这些问题的一个自然约束是，客户只能在指定的时间窗口内接收交付或服务调用，这些问题称为时间窗车辆路径问题(VRPTW)。
下图以蓝色表示客户地点，黑色表示仓库。时间窗口(以分钟为单位)显示在每个位置的上方。
# [START import]
from __future__ import print_function
from ortools.constraint_solver import routing_enums_pb2
from ortools.constraint_solver import pywrapcp
# [END import]
# [START data_model]
def 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'] = 4
data['depot'] = 0
return data
# [END data_model]
# [START solution_printer]
def print_solution(data, manager, routing, solution):
"""Prints solution on console."""
time_dimension = routing.GetDimensionOrDie('Time')
total_time = 0
for 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))
# [END solution_printer]
def main():
"""Solve the VRP with time windows."""
# Instantiate the data problem.
# [START data]
data = create_data_model()
# [END data]
# Create the routing index manager.
# [START index_manager]
manager = pywrapcp.RoutingIndexManager(len(data['time_matrix']),
data['num_vehicles'], data['depot'])
# [END index_manager]
# Create Routing Model.
# [START routing_model]
routing = pywrapcp.RoutingModel(manager)
# [END routing_model]
# Create and register a transit callback.
# [START 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)
# [END transit_callback]
# Define cost of each arc.
# [START arc_cost]
routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index)
# [END arc_cost]
# Add Time Windows constraint.
# [START time_windows_constraint]
time = 'Time'
routing.AddDimension(
transit_callback_index,
30, # allow waiting time
30, # maximum time per vehicle
False, # 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:
continue
index = 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])
# [END time_windows_constraint]
# Instantiate route start and end times to produce feasible times.
# [START depot_start_end_times]
for i in range(data['num_vehicles']):
routing.AddVariableMinimizedByFinalizer(
time_dimension.CumulVar(routing.Start(i)))
routing.AddVariableMinimizedByFinalizer(
time_dimension.CumulVar(routing.End(i)))
# [END depot_start_end_times]
# Setting first solution heuristic.
# [START parameters]
search_parameters = pywrapcp.DefaultRoutingSearchParameters()
search_parameters.first_solution_strategy = (
routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC)
# [END parameters]
# Solve the problem.
# [START solve]
solution = routing.SolveWithParameters(search_parameters)
# [END solve]
# Print solution on console.
# [START print_solution]
if solution:
print_solution(data, manager, routing, solution)
# [END print_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: 18min
Route 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: 24min
Route 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: 20min
Route 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: 20min
Total time of all routes: 82min
"""
