Cho đến nay, chúng tôi đã xem xét các vấn đề định tuyến cùng với những hạn chế áp dụng trong khi xe di chuyển. Tiếp theo, chúng tôi giới thiệu một VRPTW cũng có những hạn chế tại kho: tất cả xe đều phải được chất lên trước khi khởi hành từ kho và dỡ hàng khi quay về. Vì chỉ có hai bến tàu, nên có thể có tối đa hai xe tải hoặc huỷ tải cùng một lúc. Do đó, một số xe phải đợi các thiết bị khác cần được chất, làm trì hoãn thời gian khởi hành của họ khỏi kho. Vấn đề là tìm các tuyến đường xe tối ưu cho VRPTW cũng đáp ứng tải trọng và dỡ hàng tại kho.
Ví dụ về VRPTW với các hạn chế về tài nguyên
Sơ đồ dưới đây biểu thị một VRPTW với các giới hạn về tài nguyên.
Giải ví dụ bằng OR-Tools
Các phần sau đây trình bày cách giải quyết VRPTW bằng các hạn chế về tài nguyên bằng cách sử dụng OR-Tools. Một số mã cho ví dụ này giống như trong trước Ví dụ về VRPTW, vì vậy, chúng ta sẽ chỉ mô tả các phần mới.
Tạo dữ liệu
Đoạn mã sau đây sẽ tạo dữ liệu cho ví dụ.
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;
};
Dữ liệu này bao gồm:
time_matrix: Mảng thời gian di chuyển giữa các vị trí.time_windows: Một dãy các khoảng thời gian cho những lượt ghé thăm các vị trí được yêu cầu.vehicle_load_time: Thời gian cần thiết để tải một chiếc xe.vehicle_unload_time: Thời gian cần thiết để dỡ một xe.depot_capacity: Số lượng phương tiện tối đa có thể chất hàng hoặc dỡ hàng tại .
Thêm khoảng thời gian tải và huỷ tải
Đoạn mã sau đây thêm khoảng thời gian bốc hàng và dỡ hàng cho xe tại
kho hàng.
Các cửa sổ này do phương thức FixedDurationIntervalVar tạo
khoảng thời gian có thể thay đổi, nghĩa là các khoảng thời gian này không có thời gian bắt đầu và thời gian kết thúc cố định
(không như cửa sổ thời gian tại các vị trí). Chiều rộng của cửa sổ là
do vehicle_load_time và vehicle_unload_time chỉ định, là
tương tự trong ví dụ này.
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");
}
Thêm các điều kiện ràng buộc về tài nguyên tại kho chứa
Đoạn mã sau đây tạo ra một quy tắc ràng buộc rằng có thể có tối đa 2 xe tải hoặc huỷ tải cùng một lúc.
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 là số lượng phương tiện tối đa có thể chở được hoặc
bị huỷ tải cùng một lúc, trong ví dụ này là 2.
depot_usage là một vectơ chứa lượng không gian tương đối cần thiết cho
từng chiếc xe trong quá trình bốc hàng (hoặc dỡ hàng). Trong ví dụ này, chúng tôi giả định rằng tất cả
các xe cần cùng một dung lượng, vì vậy depot_usage chứa tất cả các xe.
Điều này có nghĩa là số lượng phương tiện tối đa có thể chở được cùng một lúc
thời gian là 2.
Chạy chương trình
Sau đây là kết quả của chương trình.
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
Xem ví dụ về VRPTW trước để xem nội dung giải thích về kết quả.
Lưu ý rằng xe 1 và 3 xuất phát từ kho tại thời điểm 0. Xe 0 và 2, trong đó
phải đợi các quảng cáo khác được tải, khởi hành vào lúc 5, giá trị của
vehicle_load_time.
Sơ đồ dưới đây minh hoạ giải pháp.
Hoàn tất chương trình
Chương trình hoàn chỉnh cho vấn đề định tuyến xe có điện dung với tài nguyên các quy tắc ràng buộc được trình bày bên dưới.
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);
}
}