1. 问题模型

VRP问题是车辆路径问题的缩写。问题是：有N辆车，都从原点出发，每辆车访问一些点后回到原点，要求所有的点都要被访问到，求最短的车辆行驶距离或最少需要的车辆数或最小化最长行驶距离。
常见的限制要求包括：车辆容量限制、时间窗限制、点访问顺序要求等。

2. RoutingModel的一些说明

对于VRP问题，我们可以使用ortools的RoutingModel模型。首先为车添加用于进行优化的dimension：

方法	输入	说明
AddDimension	evaluator, slack_max, capacity, fix_start_cumul_to_zero, name	Creates a dimension where the transit variable is constrained to be equal to evaluator(i, next(i)); ‘slack_max’ is the upper bound of the slack variable and ‘capacity’ is the upper bound of the cumul variables. ‘name’ is the name used to reference the dimension; this name is used to get cumul and transit variables from the routing model. Returns false if a dimension with the same name has already been created (and doesn’t create the new dimension). Takes ownership of the callback ‘evaluator’.

然后使用GetDimensionOrDie方法获取dimension，并设置优化目标。注意在VRP问题中，路径上给点赋的index和点实际的index不一样，需要使用IndexToNode方法进行转换才能得到实际的index。

3. 例子

先看一个简单的例子：距离用曼哈顿距离，目标函数是最小化各车辆行驶距离的差别。可以对dimension使用SetGlobalSpanCostCoefficient方法可以获得目标函数。global_span_cost = coefficient * (Max(dimension end value) - Min(dimension start value)).

代码如下：

from __future__ import print_function
from ortools.constraint_solver import pywrapcp
from ortools.constraint_solver import routing_enums_pb2###########################
# Problem Data Definition #
###########################
def create_data_model():"""Stores the data for the problem"""data = {}# Locations in block units_locations = \[(4, 4), # depot(2, 0), (8, 0), # locations to visit(0, 1), (1, 1),(5, 2), (7, 2),(3, 3), (6, 3),(5, 5), (8, 5),(1, 6), (2, 6),(3, 7), (6, 7),(0, 8), (7, 8)]# Multiply coordinates in block units by the dimensions of an average city block, 114m x 80m,# to get location coordinates.data["locations"] = [(l[0] * 114, l[1] * 80) for l in _locations]data["num_locations"] = len(data["locations"])data["num_vehicles"] = 4data["depot"] = 0return data
#######################
# Problem Constraints #
#######################
def manhattan_distance(position_1, position_2):"""Computes the Manhattan distance between two points"""return (abs(position_1[0] - position_2[0]) + abs(position_1[1] - position_2[1]))
def create_distance_callback(data):"""Creates callback to return distance between points."""_distances = {}for from_node in range(data["num_locations"]):_distances[from_node] = {}for to_node in range(data["num_locations"]):if from_node == to_node:_distances[from_node][to_node] = 0else:_distances[from_node][to_node] = (manhattan_distance(data["locations"][from_node],data["locations"][to_node]))def distance_callback(from_node, to_node):"""Returns the manhattan distance between the two nodes"""return _distances[from_node][to_node]return distance_callback
def add_distance_dimension(routing, distance_callback):"""Add Global Span constraint"""distance = 'Distance'maximum_distance = 3000  # Maximum distance per vehicle.routing.AddDimension(distance_callback,0,  # null slackmaximum_distance,True,  # start cumul to zerodistance)distance_dimension = routing.GetDimensionOrDie(distance)# Try to minimize the max distance among vehicles.distance_dimension.SetGlobalSpanCostCoefficient(100)
###########
# Printer #
###########
def print_solution(data, routing, assignment):"""Print routes 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)distance = 0while not routing.IsEnd(index):plan_output += ' {} ->'.format(routing.IndexToNode(index))previous_index = indexindex = assignment.Value(routing.NextVar(index))distance += routing.GetArcCostForVehicle(previous_index, index, vehicle_id)plan_output += ' {}\n'.format(routing.IndexToNode(index))plan_output += 'Distance of route: {}m\n'.format(distance)print(plan_output)total_distance += distanceprint('Total distance of all routes: {}m'.format(total_distance))
########
# Main #
########
def main():"""Entry point of the program"""# Instantiate the data problem.data = create_data_model()# Create Routing Modelrouting = pywrapcp.RoutingModel(data["num_locations"],data["num_vehicles"],data["depot"])# Define weight of each edgedistance_callback = create_distance_callback(data)routing.SetArcCostEvaluatorOfAllVehicles(distance_callback)add_distance_dimension(routing, distance_callback)# Setting first solution heuristic (cheapest addition).search_parameters = pywrapcp.RoutingModel.DefaultSearchParameters()search_parameters.first_solution_strategy = (routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC) # pylint: disable=no-member# Solve the problem.assignment = routing.SolveWithParameters(search_parameters)if assignment:print_solution(data, routing, assignment)
if __name__ == '__main__':main()

输出为：

结果如下图：

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运筹系列16：routing模型之VRP问题

1. 问题模型

2. RoutingModel的一些说明

3. 例子

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