Bei vielen Problemen mit der Routenplanung werden Besuche von Kunden geplant, die nur zu bestimmten Zeiten verfügbar sind.
Diese Probleme werden als Fahrzeugroutenprobleme mit Zeitfenstern bezeichnet.
VRPTW-Beispiel
Auf dieser Seite zeigen wir anhand eines Beispiels, wie ein VRPTW gelöst werden kann. Da das Problem Zeitfenster umfasst, enthalten die Daten eine Zeitmatrix, die die Fahrtzeiten zwischen Standorten enthält (und nicht wie in den vorherigen Beispielen eine Entfernungsmatrix).
Das folgende Diagramm zeigt die zu besuchenden Standorte in Blau und das Lager in Schwarz. Die Zeitfenster werden über jedem Ort angezeigt. Weitere Informationen zur Definition der Standorte finden Sie unter Standortkoordinaten im VRP-Abschnitt.
Ziel ist es, die Gesamtreisezeit der Fahrzeuge zu minimieren.
Das VRPTW-Beispiel mit ODER-Tools lösen
In den folgenden Abschnitten wird beschrieben, wie Sie das VRPTW-Beispiel mit ODER-Tools lösen.
Daten erstellen
Mit der folgenden Funktion werden die Daten für das Problem erstellt.
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
(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
Die Daten setzen sich so zusammen:
-
data['time_matrix']: ein Array mit Reisezeiten zwischen Standorten. Beachten Sie, dass sich dies von den vorherigen Beispielen unterscheidet, in denen eine Distanzmatrix verwendet wird. Wenn alle Fahrzeuge mit der gleichen Geschwindigkeit fahren, erhalten Sie die gleiche Lösung, wenn Sie eine Distanz- oder eine Zeitmatrix verwenden, da die Reisedistanzen ein konstantes Vielfaches der Fahrtzeiten sind. -
data['time_windows']: ein Array mit Zeitfenstern für die Standorte, die du dir als angeforderte Zeiten für einen Besuch vorstellen kannst. Die Fahrzeuge müssen einen Ort innerhalb des angegebenen Zeitfensters aufsuchen. -
data['num_vehicles']: Die Anzahl der Fahrzeuge im Fuhrpark. -
data['depot']: Index des Depots.
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
{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
};
const int num_vehicles = 4;
const RoutingIndexManager::NodeIndex depot{0};
};
Die Daten setzen sich so zusammen:
-
time_matrix: ein Array mit Reisezeiten zwischen Standorten. Beachten Sie, dass sich dies von den vorherigen Beispielen unterscheidet, in denen eine Distanzmatrix verwendet wird. Wenn alle Fahrzeuge mit der gleichen Geschwindigkeit fahren, erhalten Sie die gleiche Lösung, wenn Sie eine Distanz- oder eine Zeitmatrix verwenden, da die Reisedistanzen ein konstantes Vielfaches der Fahrtzeiten sind. -
time_windows: ein Array mit Zeitfenstern für die Standorte, die du dir als angeforderte Zeiten für einen Besuch vorstellen kannst. Die Fahrzeuge müssen einen Ort innerhalb des angegebenen Zeitfensters aufsuchen. -
num_vehicles: Die Anzahl der Fahrzeuge im Fuhrpark. -
depot: Index des Depots.
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
{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
};
public final int vehicleNumber = 4;
public final int depot = 0;
}
Die Daten setzen sich so zusammen:
-
timeMatrix: ein Array mit Reisezeiten zwischen Standorten. Beachten Sie, dass sich dies von den vorherigen Beispielen unterscheidet, in denen eine Distanzmatrix verwendet wird. Wenn alle Fahrzeuge mit der gleichen Geschwindigkeit fahren, erhalten Sie die gleiche Lösung, wenn Sie eine Distanz- oder eine Zeitmatrix verwenden, da die Reisedistanzen ein konstantes Vielfaches der Fahrtzeiten sind. -
timeWindows: ein Array mit Zeitfenstern für die Standorte, die du dir als angeforderte Zeiten für einen Besuch vorstellen kannst. Die Fahrzeuge müssen einen Ort innerhalb des angegebenen Zeitfensters aufsuchen. -
vehicleNumber: Die Anzahl der Fahrzeuge im Fuhrpark. -
depot: Index des Depots.
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
{ 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
};
public int VehicleNumber = 4;
public int Depot = 0;
};
Die Daten setzen sich so zusammen:
-
TimeMatrix: ein Array mit Reisezeiten zwischen Standorten. Beachten Sie, dass sich dies von den vorherigen Beispielen unterscheidet, in denen eine Distanzmatrix verwendet wird. Wenn alle Fahrzeuge mit der gleichen Geschwindigkeit fahren, erhalten Sie die gleiche Lösung, wenn Sie eine Distanz- oder eine Zeitmatrix verwenden, da die Reisedistanzen ein konstantes Vielfaches der Fahrtzeiten sind. -
TimeWindows: ein Array mit Zeitfenstern für die Standorte, die du dir als angeforderte Zeiten für einen Besuch vorstellen kannst. Die Fahrzeuge müssen einen Ort innerhalb des angegebenen Zeitfensters aufsuchen. -
VehicleNumber: Die Anzahl der Fahrzeuge im Fuhrpark. -
Depot: Index des Depots.
Zeitlicher Rückruf
Die folgende Funktion erstellt den Zeit-Callback und übergibt ihn an den Matherechner. Außerdem werden die Bogenkosten, die die Fahrtkosten definieren, als die Fahrtzeiten zwischen Standorten festgelegt.
Python
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)
routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index)
C++
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];
});
routing.SetArcCostEvaluatorOfAllVehicles(transit_callback_index);
Java
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];
});
routing.setArcCostEvaluatorOfAllVehicles(transitCallbackIndex);
C#
int transitCallbackIndex = routing.RegisterTransitCallback((long fromIndex, long toIndex) =>
{
// Convert from routing variable Index to time
// matrix NodeIndex.
var fromNode = manager.IndexToNode(fromIndex);
var toNode = manager.IndexToNode(toIndex);
return data.TimeMatrix[fromNode, toNode];
});
routing.SetArcCostEvaluatorOfAllVehicles(transitCallbackIndex);
Zeitfenstereinschränkungen hinzufügen
Mit dem folgenden Code werden Zeitfensterbeschränkungen für alle Standorte hinzugefügt.
Python
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 == data["depot"]:
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.
depot_idx = data["depot"]
for vehicle_id in range(data["num_vehicles"]):
index = routing.Start(vehicle_id)
time_dimension.CumulVar(index).SetRange(
data["time_windows"][depot_idx][0], data["time_windows"][depot_idx][1]
)
for i in range(data["num_vehicles"]):
routing.AddVariableMinimizedByFinalizer(
time_dimension.CumulVar(routing.Start(i))
)
routing.AddVariableMinimizedByFinalizer(time_dimension.CumulVar(routing.End(i)))
C++
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);
}
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)));
}
Java
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]);
}
for (int i = 0; i < data.vehicleNumber; ++i) {
routing.addVariableMinimizedByFinalizer(timeDimension.cumulVar(routing.start(i)));
routing.addVariableMinimizedByFinalizer(timeDimension.cumulVar(routing.end(i)));
}
C#
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]);
}
for (int i = 0; i < data.VehicleNumber; ++i)
{
routing.AddVariableMinimizedByFinalizer(timeDimension.CumulVar(routing.Start(i)));
routing.AddVariableMinimizedByFinalizer(timeDimension.CumulVar(routing.End(i)));
}
Der Code erstellt eine Dimension für die Fahrzeit der Fahrzeuge, ähnlich den Dimensionen für Entfernungen oder Anforderungen in den vorherigen Beispielen. Dimensionen erfassen die Mengen, die sich über die Route eines Fahrzeugs ansammeln. Im Code oben ist time_dimension.CumulVar(index) die kumulierte Fahrtzeit, wenn ein Fahrzeug mit der angegebenen index am Ort ankommt.
Die Dimension wird mit der Methode AddDimension erstellt, die die folgenden Argumente hat:
- Der Index für den Fahrtzeit-Callback:
transit_callback_index - Obergrenze für Slack (die Wartezeiten an den Standorten):
30. Im CVRP-Beispiel war dieser Wert auf 0 gesetzt. Das VRPTW muss jedoch aufgrund der Zeitfenstereinschränkungen positive Wartezeiten zulassen. - Obergrenze für die Gesamtzeit auf der Route jedes Fahrzeugs:
30 - Eine boolesche Variable, die angibt, ob die kumulative Variable zu Beginn der Route jedes Fahrzeugs auf null gesetzt ist.
- Der Name der Dimension.
Als Nächstes werden die Linien
Python
for location_idx, time_window in enumerate(data["time_windows"]):
if location_idx == data["depot"]:
continue
index = manager.NodeToIndex(location_idx)
time_dimension.CumulVar(index).SetRange(time_window[0], time_window[1])
C++
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);
}
Java
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]);
}
C#
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]);
}
verlangen, dass ein Fahrzeug einen Standort innerhalb des entsprechenden Zeitfensters anfahren muss.
Suchparameter festlegen
Mit dem folgenden Code werden die Standardsuchparameter und eine heuristische Methode zum Auffinden der ersten Lösung festgelegt:
Python
search_parameters = pywrapcp.DefaultRoutingSearchParameters()
search_parameters.first_solution_strategy = (
routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC
)
C++
RoutingSearchParameters searchParameters = DefaultRoutingSearchParameters();
searchParameters.set_first_solution_strategy(
FirstSolutionStrategy::PATH_CHEAPEST_ARC);
Java
RoutingSearchParameters searchParameters =
main.defaultRoutingSearchParameters()
.toBuilder()
.setFirstSolutionStrategy(FirstSolutionStrategy.Value.PATH_CHEAPEST_ARC)
.build();
C#
RoutingSearchParameters searchParameters =
operations_research_constraint_solver.DefaultRoutingSearchParameters();
searchParameters.FirstSolutionStrategy = FirstSolutionStrategy.Types.Value.PathCheapestArc;
Lösungsdrucker hinzufügen
Die Funktion, mit der die Lösung angezeigt wird, ist unten zu sehen.
Python
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")
C++
//! @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";
}
Java
/// @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");
}
C#
/// <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);
}
Die Lösung zeigt die Fahrzeugrouten und das Lösungsfenster an jedem Standort an, wie im nächsten Abschnitt erläutert.
Lösungszeitfenster
Das Lösungsfenster an einem Standort ist das Zeitintervall, in dem ein Fahrzeug eintreffen muss, um im Zeitplan zu bleiben.Das Lösungsfenster ist im Einschränkungs-Zeitfenster am Standort enthalten und ist normalerweise kleiner als dieses.
In der obigen Lösungsdruckerfunktion wird das Lösungsfenster von (assignment.Min(time_var), assignment.Max(time_var) zurückgegeben, wobei time_var = time_dimension.CumulVar(index) die kumulative Fahrtzeit des Fahrzeugs am Standort ist.
Wenn der Mindest- und Höchstwert von time_var gleich ist, ist das Lösungsfenster ein einzelner Zeitpunkt. Das bedeutet, dass das Fahrzeug vom Standort abfahren muss, sobald es eintrifft. Wenn der Mindestwert jedoch unter dem Höchstwert liegt, kann das Fahrzeug warten, bevor es abfährt.
Im Abschnitt Programm ausführen werden die Lösungsfenster für dieses Beispiel beschrieben.
Lösungen anbieten
Die Hauptfunktion in diesem Beispiel ähnelt der für das TSP-Beispiel.
Python
solution = routing.SolveWithParameters(search_parameters)
C++
const Assignment* solution = routing.SolveWithParameters(searchParameters);
Java
Assignment solution = routing.solveWithParameters(searchParameters);
C#
Assignment solution = routing.SolveWithParameters(searchParameters);
Programm durchführen
Wenn Sie das Programm ausführen, wird die folgende Ausgabe angezeigt:
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
Für jeden Standort auf einer Route ist Time(a,b) das Lösungsfenster: Das Fahrzeug, das den Standort besucht, muss dies in diesem Zeitintervall tun, um im Fahrplan zu bleiben.
Sehen Sie sich als Beispiel den folgenden Teil der Route für Fahrzeug 0 an.
0 Time(0,0) -> 9 Time(2,3) -> 14 Time(7,8)
An Standort 9 ist das Lösungsfenster Time(2,3). Das bedeutet, dass das Fahrzeug zwischen den Zeiten 2 und 3 ankommen muss. Beachten Sie, dass das Lösungsfenster an dieser Stelle im Einschränkungszeitfenster (0, 3) enthalten ist, das in den Problemdaten angegeben ist. Das Lösungsfenster beginnt zu Zeitpunkt 2, da es 2 Zeiteinheiten (den Eintrag 0, 9 in der Zeitmatrix) benötigt, um vom Lager zu Standort 9 zu gelangen.
Warum kann das Fahrzeug zwischen 2 und 3 an Ort 9 fahren? Der Grund dafür ist, dass die Fahrtzeit von Ort 9 zu Ort 14 3 ist und das Fahrzeug vor Ort 14 vor Ort 14 abfährt, was zu früh für den Besuch ist. Das Fahrzeug muss also irgendwo warten, z.B. kann der Fahrer beschließen, an Position 9 bis zum Zeitpunkt 3 zu warten, ohne die Fertigstellung der Route zu verzögern.
Sie haben vielleicht bemerkt, dass einige Lösungszeitfenster (z.B. an den Standorten 9 und 14) unterschiedliche Start- und Endzeiten haben, während andere (z.B. auf Routen 2 und 3) dieselbe Start- und Endzeit haben. Im ersten Fall können die Fahrzeuge bis zum Ende des Fensters warten, bevor sie abfahren, während sie im zweiten Fall sofort bei Ankunft abfahren müssen.
Lösungsfenster in einer Liste oder einem Array speichern
Der TSP-Abschnitt zeigt, wie die Routen in einer Lösung in einer Liste oder einem Array gespeichert werden. Bei einem VRPTW können Sie auch die Lösungsfenster speichern. Mit den folgenden Funktionen werden die Lösungsfenster in einer Liste (Python) oder einem Array (C++) gespeichert.
Python
def get_cumul_data(solution, routing, dimension):
"""Get cumulative data from a dimension and store it in an array."""
# Returns an array cumul_data whose i,j entry contains the minimum and
# maximum of CumulVar for the dimension at the jth node on route :
# - cumul_data[i][j][0] is the minimum.
# - cumul_data[i][j][1] is the maximum.
cumul_data = []
for route_nbr in range(routing.vehicles()):
route_data = []
index = routing.Start(route_nbr)
dim_var = dimension.CumulVar(index)
route_data.append([solution.Min(dim_var), solution.Max(dim_var)])
while not routing.IsEnd(index):
index = solution.Value(routing.NextVar(index))
dim_var = dimension.CumulVar(index)
route_data.append([solution.Min(dim_var), solution.Max(dim_var)])
cumul_data.append(route_data)
return cumul_data
C++
std::vector<std::vector<std::pair<int64_t, int64_t>>> GetCumulData(
const Assignment& solution, const RoutingModel& routing,
const RoutingDimension& dimension) {
// Returns an array cumul_data, whose i, j entry is a pair containing
// the minimum and maximum of CumulVar for the dimension.:
// - cumul_data[i][j].first is the minimum.
// - cumul_data[i][j].second is the maximum.
std::vector<std::vector<std::pair<int64_t, int64_t>>> cumul_data(
routing.vehicles());
for (int vehicle_id = 0; vehicle_id < routing.vehicles(); ++vehicle_id) {
int64_t index = routing.Start(vehicle_id);
IntVar* dim_var = dimension.CumulVar(index);
cumul_data[vehicle_id].emplace_back(solution.Min(dim_var),
solution.Max(dim_var));
while (!routing.IsEnd(index)) {
index = solution.Value(routing.NextVar(index));
IntVar* dim_var = dimension.CumulVar(index);
cumul_data[vehicle_id].emplace_back(solution.Min(dim_var),
solution.Max(dim_var));
}
}
return cumul_data;
}
Die Funktionen speichern die Mindest- und Höchstwerte der kumulativen Daten für jede Dimension (nicht nur für die Zeit).
Im aktuellen Beispiel sind diese Werte die Unter- und Obergrenze des Lösungsfensters. Die an die Funktion übergebene Dimension ist time_dimension.
Die folgenden Funktionen geben die Lösung aus den Routen und den kumulativen Daten aus.
Python
def print_solution(routes, cumul_data):
"""Print the solution."""
total_time = 0
route_str = ""
for i, route in enumerate(routes):
route_str += "Route " + str(i) + ":\n"
start_time = cumul_data[i][0][0]
end_time = cumul_data[i][0][1]
route_str += (
" "
+ str(route[0])
+ " Time("
+ str(start_time)
+ ", "
+ str(end_time)
+ ")"
)
for j in range(1, len(route)):
start_time = cumul_data[i][j][0]
end_time = cumul_data[i][j][1]
route_str += (
" -> "
+ str(route[j])
+ " Time("
+ str(start_time)
+ ", "
+ str(end_time)
+ ")"
)
route_str += f"\n Route time: {start_time}min\n\n"
total_time += cumul_data[i][len(route) - 1][0]
route_str += f"Total time: {total_time}min"
print(route_str)
C++
void PrintSolution(
const std::vector<std::vector<int>> routes,
std::vector<std::vector<std::pair<int64_t, int64_t>>> cumul_data) {
int64_t total_time{0};
std::ostringstream route;
for (int vehicle_id = 0; vehicle_id < routes.size(); ++vehicle_id) {
route << "\nRoute " << vehicle_id << ": \n";
route << " " << routes[vehicle_id][0] << " Time("
<< cumul_data[vehicle_id][0].first << ", "
<< cumul_data[vehicle_id][0].second << ") ";
for (int j = 1; j < routes[vehicle_id].size(); ++j) {
route << "-> " << routes[vehicle_id][j] << " Time("
<< cumul_data[vehicle_id][j].first << ", "
<< cumul_data[vehicle_id][j].second << ") ";
}
route << "\n Route time: "
<< cumul_data[vehicle_id][routes[vehicle_id].size() - 1].first
<< " minutes\n";
total_time += cumul_data[vehicle_id][routes[vehicle_id].size() - 1].first;
}
route << "\nTotal travel time: " << total_time << " minutes";
LOG(INFO) << route.str();
}
Abgeschlossene Programme
Die vollständigen Programme für das Problem mit der Routenplanung mit Zeitfenstern sind unten aufgeführt.
Python
"""Vehicles Routing Problem (VRP) with Time Windows."""
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
(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
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,
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 == data["depot"]:
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.
depot_idx = data["depot"]
for vehicle_id in range(data["num_vehicles"]):
index = routing.Start(vehicle_id)
time_dimension.CumulVar(index).SetRange(
data["time_windows"][depot_idx][0], data["time_windows"][depot_idx][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()
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
{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
};
const int num_vehicles = 4;
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);
}
// 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.RoutingDimension;
import com.google.ortools.constraintsolver.RoutingIndexManager;
import com.google.ortools.constraintsolver.RoutingModel;
import com.google.ortools.constraintsolver.RoutingSearchParameters;
import com.google.ortools.constraintsolver.main;
import java.util.logging.Logger;
/** VRPTW. */
public class VrpTimeWindows {
private static final Logger logger = Logger.getLogger(VrpTimeWindows.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
{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
};
public final int vehicleNumber = 4;
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]);
}
// 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);
}
}
// [END_program_part1]
C#
using System;
using System.Collections.Generic;
using Google.OrTools.ConstraintSolver;
/// <summary>
/// Vehicles Routing Problem (VRP) with Time Windows.
/// </summary>
public class VrpTimeWindows
{
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
{ 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
};
public int VehicleNumber = 4;
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 time
// 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 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.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]);
}
// 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);
}
}