Up to now, we have been looking at routing problems with constraints that apply during vehicle travel. Next, we present a VRPTW that also has constraints at the depot: all vehicles need to be loaded before departing the depot and unloaded upon return. Since there are only two available loading docks, at most two vehicles can be loaded or unloaded at the same time. As a result, some vehicles must wait for others to be loaded, delaying their departure from the depot. The problem is to find optimal vehicle routes for the VRPTW that also meet the loading and unloading constraints at the depot.
VRPTW example with resource constraints
The diagram below shows a VRPTW with resource constraints.
Solving the example with OR-Tools
The following sections show how to solve the VRPTW with resource constraints using OR-Tools. Some of the code for the example is the same as in the previous VRPTW example, so we'll just describe the parts that are new.
Create the data
The following code creates the data for the example.
Python
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
(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"] = 4
data["vehicle_load_time"] = 5
data["vehicle_unload_time"] = 5
data["depot_capacity"] = 2
data["depot"] = 0
return data
C++
struct DataModel {
const std::vector<std::vector<int64_t>> 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},
};
const std::vector<std::pair<int64_t, int64_t>> 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
};
const int num_vehicles = 4;
const int vehicle_load_time = 5;
const int vehicle_unload_time = 5;
const int depot_capacity = 2;
const RoutingIndexManager::NodeIndex depot{0};
};
Java
static class DataModel {
public final long[][] timeMatrix = {
{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},
};
public final long[][] timeWindows = {
{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
};
public final int vehicleNumber = 4;
public final int vehicleLoadTime = 5;
public final int vehicleUnloadTime = 5;
public final int depotCapacity = 2;
public final int depot = 0;
}
C#
class DataModel
{
public long[,] TimeMatrix = {
{ 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 },
};
public long[,] TimeWindows = {
{ 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
};
public int VehicleNumber = 4;
public int VehicleLoadTime = 5;
public int VehicleUnloadTime = 5;
public int DepotCapacity = 2;
public int Depot = 0;
};
The data includes the following:
time_matrix: An array of travel times between locations.time_windows: An array of time windows for requested visits to the locations.vehicle_load_time: The time required to load a vehicle.vehicle_unload_time: The time required to unload a vehicle.depot_capacity: The maximum number of vehicles that can load or unload at the same time.
Add time windows for loading and unloading
The following code adds time windows for loading and unloading the vehicles at
the depot.
These windows, created by the method FixedDurationIntervalVar, are
variable time windows, meaning they don't have fixed start and end times
(unlike the time windows at the locations). The widths of the windows are
specified by vehicle_load_time and vehicle_unload_time, which happen to be
the same in this example.
Python
solver = routing.solver()
intervals = []
for i in range(data["num_vehicles"]):
# Add time windows at start of routes
intervals.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",
)
)
C++
Solver* solver = routing.solver();
std::vector<IntervalVar*> intervals;
for (int i = 0; i < data.num_vehicles; ++i) {
// Add load duration at start of routes
intervals.push_back(solver->MakeFixedDurationIntervalVar(
time_dimension.CumulVar(routing.Start(i)), data.vehicle_load_time,
"depot_interval"));
// Add unload duration at end of routes.
intervals.push_back(solver->MakeFixedDurationIntervalVar(
time_dimension.CumulVar(routing.End(i)), data.vehicle_unload_time,
"depot_interval"));
}
Java
Solver solver = routing.solver();
IntervalVar[] intervals = new IntervalVar[data.vehicleNumber * 2];
for (int i = 0; i < data.vehicleNumber; ++i) {
// Add load duration at start of routes
intervals[2 * i] = solver.makeFixedDurationIntervalVar(
timeDimension.cumulVar(routing.start(i)), data.vehicleLoadTime, "depot_interval");
// Add unload duration at end of routes.
intervals[2 * i + 1] = solver.makeFixedDurationIntervalVar(
timeDimension.cumulVar(routing.end(i)), data.vehicleUnloadTime, "depot_interval");
}
C#
Solver solver = routing.solver();
IntervalVar[] intervals = new IntervalVar[data.VehicleNumber * 2];
for (int i = 0; i < data.VehicleNumber; ++i)
{
// Add load duration at start of routes
intervals[2 * i] = solver.MakeFixedDurationIntervalVar(timeDimension.CumulVar(routing.Start(i)),
data.VehicleLoadTime, "depot_interval");
// Add unload duration at end of routes.
intervals[2 * i + 1] = solver.MakeFixedDurationIntervalVar(timeDimension.CumulVar(routing.End(i)),
data.VehicleUnloadTime, "depot_interval");
}
Add resource constraints at the depot
The following code creates the constraint that at most two vehicles can be loaded or unloaded at the same time.
Python
depot_usage = [1 for _ in range(len(intervals))]
solver.Add(
solver.Cumulative(intervals, depot_usage, data["depot_capacity"], "depot")
)
C++
std::vector<int64_t> depot_usage(intervals.size(), 1);
solver->AddConstraint(solver->MakeCumulative(intervals, depot_usage,
data.depot_capacity, "depot"));
Java
long[] depotUsage = new long[intervals.length];
Arrays.fill(depotUsage, 1);
solver.addConstraint(solver.makeCumulative(intervals, depotUsage, data.depotCapacity, "depot"));
C#
long[] depot_usage = Enumerable.Repeat<long>(1, intervals.Length).ToArray();
solver.Add(solver.MakeCumulative(intervals, depot_usage, data.DepotCapacity, "depot"));
depot_capacity is the maximum number of vehicles that can be loaded or
unloaded at the same time, which is 2 in this example.
depot_usage is a vector containing the relative amounts of space required by
each vehicle during loading (or unloading). In this example, we assume that all
vehicles require the same amount of space, so depot_usage contains all ones.
This means that the maximum number of vehicles that can be loaded at the same
time is 2.
Running the program
The following shows the output of the program.
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: 20min Route 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: 20min Route 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: 25min Route 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: 25min Total time of all routes: 90min
See the previous VRPTW example for an explanation of the output.
Note that vehicles 1 and 3 depart the depot at time 0. Vehicles 0 and 2, which
must wait for the others to be loaded, depart at time 5, the value of
vehicle_load_time.
The diagram below shows the solution.
Complete programs
The complete programs for the capacitated vehicle routing problem with resource constraints are shown below.
Python
"""Vehicles Routing Problem (VRP) with Resource Constraints."""
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["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"] = 4
data["vehicle_load_time"] = 5
data["vehicle_unload_time"] = 5
data["depot_capacity"] = 2
data["depot"] = 0
return data
def print_solution(data, manager, routing, solution):
"""Prints solution on console."""
print(f"Objective: {solution.ObjectiveValue()}")
time_dimension = routing.GetDimensionOrDie("Time")
total_time = 0
for vehicle_id in range(data["num_vehicles"]):
index = routing.Start(vehicle_id)
plan_output = f"Route for vehicle {vehicle_id}:\n"
while not routing.IsEnd(index):
time_var = time_dimension.CumulVar(index)
plan_output += (
f"{manager.IndexToNode(index)}"
f" Time({solution.Min(time_var)}, {solution.Max(time_var)})"
" -> "
)
index = solution.Value(routing.NextVar(index))
time_var = time_dimension.CumulVar(index)
plan_output += (
f"{manager.IndexToNode(index)}"
f" Time({solution.Min(time_var)},{solution.Max(time_var)})\n"
)
plan_output += f"Time of the route: {solution.Min(time_var)}min\n"
print(plan_output)
total_time += solution.Min(time_var)
print(f"Total time of all routes: {total_time}min")
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 time
60, # 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]
)
# Add resource constraints at the depot.
solver = routing.solver()
intervals = []
for i in range(data["num_vehicles"]):
# Add time windows at start of routes
intervals.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 _ 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()
C++
#include <cstdint>
#include <sstream>
#include <string>
#include <utility>
#include <vector>
#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>> 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},
};
const std::vector<std::pair<int64_t, int64_t>> 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
};
const int num_vehicles = 4;
const int vehicle_load_time = 5;
const int vehicle_unload_time = 5;
const int depot_capacity = 2;
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) {
const RoutingDimension& time_dimension = routing.GetDimensionOrDie("Time");
int64_t total_time{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 << ":";
std::ostringstream route;
while (!routing.IsEnd(index)) {
auto time_var = time_dimension.CumulVar(index);
route << manager.IndexToNode(index).value() << " Time("
<< solution.Min(time_var) << ", " << solution.Max(time_var)
<< ") -> ";
index = solution.Value(routing.NextVar(index));
}
auto time_var = time_dimension.CumulVar(index);
LOG(INFO) << route.str() << manager.IndexToNode(index).value() << " Time("
<< solution.Min(time_var) << ", " << solution.Max(time_var)
<< ")";
LOG(INFO) << "Time of the route: " << solution.Min(time_var) << "min";
total_time += solution.Min(time_var);
}
LOG(INFO) << "Total time of all routes: " << total_time << "min";
LOG(INFO) << "";
LOG(INFO) << "Advanced usage:";
LOG(INFO) << "Problem solved in " << routing.solver()->wall_time() << "ms";
}
void VrpTimeWindows() {
// Instantiate the data problem.
DataModel data;
// Create Routing Index Manager
RoutingIndexManager manager(data.time_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 time matrix NodeIndex.
const int from_node = manager.IndexToNode(from_index).value();
const int to_node = manager.IndexToNode(to_index).value();
return data.time_matrix[from_node][to_node];
});
// Define cost of each arc.
routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index);
// Add Time constraint.
const std::string time = "Time";
routing.AddDimension(transit_callback_index, // transit callback index
int64_t{30}, // allow waiting time
int64_t{30}, // maximum time per vehicle
false, // Don't force start cumul to zero
time);
const RoutingDimension& time_dimension = routing.GetDimensionOrDie(time);
// Add time window constraints for each location except depot.
for (int i = 1; i < data.time_windows.size(); ++i) {
const int64_t index =
manager.NodeToIndex(RoutingIndexManager::NodeIndex(i));
time_dimension.CumulVar(index)->SetRange(data.time_windows[i].first,
data.time_windows[i].second);
}
// Add time window constraints for each vehicle start node.
for (int i = 0; i < data.num_vehicles; ++i) {
const int64_t index = routing.Start(i);
time_dimension.CumulVar(index)->SetRange(data.time_windows[0].first,
data.time_windows[0].second);
}
// Add resource constraints at the depot.
Solver* solver = routing.solver();
std::vector<IntervalVar*> intervals;
for (int i = 0; i < data.num_vehicles; ++i) {
// Add load duration at start of routes
intervals.push_back(solver->MakeFixedDurationIntervalVar(
time_dimension.CumulVar(routing.Start(i)), data.vehicle_load_time,
"depot_interval"));
// Add unload duration at end of routes.
intervals.push_back(solver->MakeFixedDurationIntervalVar(
time_dimension.CumulVar(routing.End(i)), data.vehicle_unload_time,
"depot_interval"));
}
std::vector<int64_t> depot_usage(intervals.size(), 1);
solver->AddConstraint(solver->MakeCumulative(intervals, depot_usage,
data.depot_capacity, "depot"));
// Instantiate route start and end times to produce feasible times.
for (int i = 0; i < data.num_vehicles; ++i) {
routing.AddVariableMinimizedByFinalizer(
time_dimension.CumulVar(routing.Start(i)));
routing.AddVariableMinimizedByFinalizer(
time_dimension.CumulVar(routing.End(i)));
}
// Setting first solution heuristic.
RoutingSearchParameters searchParameters = DefaultRoutingSearchParameters();
searchParameters.set_first_solution_strategy(
FirstSolutionStrategy::PATH_CHEAPEST_ARC);
// Solve the problem.
const Assignment* solution = routing.SolveWithParameters(searchParameters);
// Print solution on console.
PrintSolution(data, manager, routing, *solution);
}
} // namespace operations_research
int main(int /*argc*/, char* /*argv*/[]) {
operations_research::VrpTimeWindows();
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.IntVar;
import com.google.ortools.constraintsolver.IntervalVar;
import com.google.ortools.constraintsolver.RoutingDimension;
import com.google.ortools.constraintsolver.RoutingIndexManager;
import com.google.ortools.constraintsolver.RoutingModel;
import com.google.ortools.constraintsolver.RoutingSearchParameters;
import com.google.ortools.constraintsolver.Solver;
import com.google.ortools.constraintsolver.main;
import java.util.Arrays;
import java.util.logging.Logger;
/** Minimal VRP with Resource Constraints.*/
public class VrpResources {
private static final Logger logger = Logger.getLogger(VrpResources.class.getName());
static class DataModel {
public final long[][] timeMatrix = {
{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},
};
public final long[][] timeWindows = {
{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
};
public final int vehicleNumber = 4;
public final int vehicleLoadTime = 5;
public final int vehicleUnloadTime = 5;
public final int depotCapacity = 2;
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.
RoutingDimension timeDimension = routing.getMutableDimension("Time");
long totalTime = 0;
for (int i = 0; i < data.vehicleNumber; ++i) {
long index = routing.start(i);
logger.info("Route for Vehicle " + i + ":");
String route = "";
while (!routing.isEnd(index)) {
IntVar timeVar = timeDimension.cumulVar(index);
route += manager.indexToNode(index) + " Time(" + solution.min(timeVar) + ","
+ solution.max(timeVar) + ") -> ";
index = solution.value(routing.nextVar(index));
}
IntVar timeVar = timeDimension.cumulVar(index);
route += manager.indexToNode(index) + " Time(" + solution.min(timeVar) + ","
+ solution.max(timeVar) + ")";
logger.info(route);
logger.info("Time of the route: " + solution.min(timeVar) + "min");
totalTime += solution.min(timeVar);
}
logger.info("Total time of all routes: " + totalTime + "min");
}
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.timeMatrix.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.timeMatrix[fromNode][toNode];
});
// Define cost of each arc.
routing.setArcCostEvaluatorOfAllVehicles(transitCallbackIndex);
// Add Time constraint.
routing.addDimension(transitCallbackIndex, // transit callback
30, // allow waiting time
30, // vehicle maximum capacities
false, // start cumul to zero
"Time");
RoutingDimension timeDimension = routing.getMutableDimension("Time");
// Add time window constraints for each location except depot.
for (int i = 1; i < data.timeWindows.length; ++i) {
long index = manager.nodeToIndex(i);
timeDimension.cumulVar(index).setRange(data.timeWindows[i][0], data.timeWindows[i][1]);
}
// Add time window constraints for each vehicle start node.
for (int i = 0; i < data.vehicleNumber; ++i) {
long index = routing.start(i);
timeDimension.cumulVar(index).setRange(data.timeWindows[0][0], data.timeWindows[0][1]);
}
// Add resource constraints at the depot.
Solver solver = routing.solver();
IntervalVar[] intervals = new IntervalVar[data.vehicleNumber * 2];
for (int i = 0; i < data.vehicleNumber; ++i) {
// Add load duration at start of routes
intervals[2 * i] = solver.makeFixedDurationIntervalVar(
timeDimension.cumulVar(routing.start(i)), data.vehicleLoadTime, "depot_interval");
// Add unload duration at end of routes.
intervals[2 * i + 1] = solver.makeFixedDurationIntervalVar(
timeDimension.cumulVar(routing.end(i)), data.vehicleUnloadTime, "depot_interval");
}
long[] depotUsage = new long[intervals.length];
Arrays.fill(depotUsage, 1);
solver.addConstraint(solver.makeCumulative(intervals, depotUsage, data.depotCapacity, "depot"));
// Instantiate route start and end times to produce feasible times.
for (int i = 0; i < data.vehicleNumber; ++i) {
routing.addVariableMinimizedByFinalizer(timeDimension.cumulVar(routing.start(i)));
routing.addVariableMinimizedByFinalizer(timeDimension.cumulVar(routing.end(i)));
}
// Setting first solution heuristic.
RoutingSearchParameters searchParameters =
main.defaultRoutingSearchParameters()
.toBuilder()
.setFirstSolutionStrategy(FirstSolutionStrategy.Value.PATH_CHEAPEST_ARC)
.build();
// Solve the problem.
Assignment solution = routing.solveWithParameters(searchParameters);
// Print solution on console.
printSolution(data, routing, manager, solution);
}
}
C#
using System;
using System.Linq;
using System.Collections.Generic;
using Google.OrTools.ConstraintSolver;
/// <summary>
/// Vehicles Routing Problem (VRP) with Resource Constraints.
/// </summary>
public class VrpResources
{
class DataModel
{
public long[,] TimeMatrix = {
{ 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 },
};
public long[,] TimeWindows = {
{ 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
};
public int VehicleNumber = 4;
public int VehicleLoadTime = 5;
public int VehicleUnloadTime = 5;
public int DepotCapacity = 2;
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.
RoutingDimension timeDimension = routing.GetMutableDimension("Time");
long totalTime = 0;
for (int i = 0; i < data.VehicleNumber; ++i)
{
Console.WriteLine("Route for Vehicle {0}:", i);
var index = routing.Start(i);
while (routing.IsEnd(index) == false)
{
var timeVar = timeDimension.CumulVar(index);
Console.Write("{0} Time({1},{2}) -> ", manager.IndexToNode(index), solution.Min(timeVar),
solution.Max(timeVar));
index = solution.Value(routing.NextVar(index));
}
var endTimeVar = timeDimension.CumulVar(index);
Console.WriteLine("{0} Time({1},{2})", manager.IndexToNode(index), solution.Min(endTimeVar),
solution.Max(endTimeVar));
Console.WriteLine("Time of the route: {0}min", solution.Min(endTimeVar));
totalTime += solution.Min(endTimeVar);
}
Console.WriteLine("Total time of all routes: {0}min", totalTime);
}
public static void Main(String[] args)
{
// Instantiate the data problem.
DataModel data = new DataModel();
// Create Routing Index Manager
RoutingIndexManager manager =
new RoutingIndexManager(data.TimeMatrix.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.TimeMatrix[fromNode, toNode];
});
// Define cost of each arc.
routing.SetArcCostEvaluatorOfAllVehicles(transitCallbackIndex);
// Add Distance constraint.
routing.AddDimension(transitCallbackIndex, // transit callback
30, // allow waiting time
30, // vehicle maximum capacities
false, // start cumul to zero
"Time");
RoutingDimension timeDimension = routing.GetMutableDimension("Time");
// Add time window constraints for each location except depot.
for (int i = 1; i < data.TimeWindows.GetLength(0); ++i)
{
long index = manager.NodeToIndex(i);
timeDimension.CumulVar(index).SetRange(data.TimeWindows[i, 0], data.TimeWindows[i, 1]);
}
// Add time window constraints for each vehicle start node.
for (int i = 0; i < data.VehicleNumber; ++i)
{
long index = routing.Start(i);
timeDimension.CumulVar(index).SetRange(data.TimeWindows[0, 0], data.TimeWindows[0, 1]);
}
// Add resource constraints at the depot.
Solver solver = routing.solver();
IntervalVar[] intervals = new IntervalVar[data.VehicleNumber * 2];
for (int i = 0; i < data.VehicleNumber; ++i)
{
// Add load duration at start of routes
intervals[2 * i] = solver.MakeFixedDurationIntervalVar(timeDimension.CumulVar(routing.Start(i)),
data.VehicleLoadTime, "depot_interval");
// Add unload duration at end of routes.
intervals[2 * i + 1] = solver.MakeFixedDurationIntervalVar(timeDimension.CumulVar(routing.End(i)),
data.VehicleUnloadTime, "depot_interval");
}
long[] depot_usage = Enumerable.Repeat<long>(1, intervals.Length).ToArray();
solver.Add(solver.MakeCumulative(intervals, depot_usage, data.DepotCapacity, "depot"));
// Instantiate route start and end times to produce feasible times.
for (int i = 0; i < data.VehicleNumber; ++i)
{
routing.AddVariableMinimizedByFinalizer(timeDimension.CumulVar(routing.Start(i)));
routing.AddVariableMinimizedByFinalizer(timeDimension.CumulVar(routing.End(i)));
}
// Setting first solution heuristic.
RoutingSearchParameters searchParameters =
operations_research_constraint_solver.DefaultRoutingSearchParameters();
searchParameters.FirstSolutionStrategy = FirstSolutionStrategy.Types.Value.PathCheapestArc;
// Solve the problem.
Assignment solution = routing.SolveWithParameters(searchParameters);
// Print solution on console.
PrintSolution(data, routing, manager, solution);
}
}