Organisationen, deren Mitarbeiter mehrere Schichten arbeiten, müssen für jede tägliche Schicht ausreichend Arbeitskräfte einplanen. In der Regel gelten für die Zeitpläne Einschränkungen wie z. B. „Kein Mitarbeiter sollte zwei Schichten hintereinander arbeiten“. Es kann schwierig sein, einen Zeitplan zu finden, der alle Einschränkungen erfüllt.
In den folgenden Abschnitten werden zwei Beispiele für Probleme bei der Planung von Mitarbeitern dargestellt. Außerdem wird gezeigt, wie du diese mit dem CP-SAT-Rechner lösen kannst.
Ein komplexeres Beispiel finden Sie in diesem Programm zur Schichtplanung auf GitHub.
Ein Problem mit der Terminplanung in der Pflegefachkraft
Im nächsten Beispiel muss eine Krankenhausvorgesetzte einen Zeitplan für vier Pflegekräfte über einen Zeitraum von drei Tagen erstellen, wobei die folgenden Bedingungen gelten:
- Jeder Tag ist in drei 8-Stunden-Schichten unterteilt.
- Jede Schicht wird täglich einer einzigen Krankenschwester zugewiesen und keine Pflegefachkraft arbeitet mehr als eine Schicht.
- Jede Pflegefachkraft wird während der dreitägigen Periode mindestens zwei Schichten zugewiesen.
In den folgenden Abschnitten wird eine Lösung für das Problem mit der Terminplanung der Pflegekräfte vorgestellt.
Bibliotheken importieren
Mit dem folgenden Code wird die erforderliche Bibliothek importiert.
Python
from ortools.sat.python import cp_model
C++
#include <stdlib.h> #include <atomic> #include <map> #include <numeric> #include <string> #include <tuple> #include <vector> #include "absl/strings/str_format.h" #include "ortools/base/logging.h" #include "ortools/sat/cp_model.h" #include "ortools/sat/cp_model.pb.h" #include "ortools/sat/cp_model_solver.h" #include "ortools/sat/model.h" #include "ortools/sat/sat_parameters.pb.h" #include "ortools/util/time_limit.h"
Java
import com.google.ortools.Loader; import com.google.ortools.sat.CpModel; import com.google.ortools.sat.CpSolver; import com.google.ortools.sat.CpSolverSolutionCallback; import com.google.ortools.sat.CpSolverStatus; import com.google.ortools.sat.LinearExpr; import com.google.ortools.sat.LinearExprBuilder; import com.google.ortools.sat.Literal; import java.util.ArrayList; import java.util.List; import java.util.stream.IntStream;
C#
using System; using System.Collections.Generic; using System.IO; using System.Linq; using Google.OrTools.Sat;
Daten für das Beispiel
Mit dem folgenden Code werden die Daten für das Beispiel erstellt.
Python
num_nurses = 4 num_shifts = 3 num_days = 3 all_nurses = range(num_nurses) all_shifts = range(num_shifts) all_days = range(num_days)
C++
const int num_nurses = 4; const int num_shifts = 3; const int num_days = 3; std::vector<int> all_nurses(num_nurses); std::iota(all_nurses.begin(), all_nurses.end(), 0); std::vector<int> all_shifts(num_shifts); std::iota(all_shifts.begin(), all_shifts.end(), 0); std::vector<int> all_days(num_days); std::iota(all_days.begin(), all_days.end(), 0);
Java
final int numNurses = 4; final int numDays = 3; final int numShifts = 3; final int[] allNurses = IntStream.range(0, numNurses).toArray(); final int[] allDays = IntStream.range(0, numDays).toArray(); final int[] allShifts = IntStream.range(0, numShifts).toArray();
C#
const int numNurses = 4; const int numDays = 3; const int numShifts = 3; int[] allNurses = Enumerable.Range(0, numNurses).ToArray(); int[] allDays = Enumerable.Range(0, numDays).ToArray(); int[] allShifts = Enumerable.Range(0, numShifts).ToArray();
Modell erstellen
Mit dem folgenden Code wird das Modell erstellt.
Python
model = cp_model.CpModel()
C++
CpModelBuilder cp_model;
Java
CpModel model = new CpModel();
C#
CpModel model = new CpModel(); model.Model.Variables.Capacity = numNurses * numDays * numShifts;
Variablen erstellen
Mit dem folgenden Code wird ein Array mit Variablen erstellt.
Python
shifts = {}
for n in all_nurses:
for d in all_days:
for s in all_shifts:
shifts[(n, d, s)] = model.new_bool_var(f"shift_n{n}_d{d}_s{s}")
C++
std::map<std::tuple<int, int, int>, BoolVar> shifts;
for (int n : all_nurses) {
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
shifts[key] = cp_model.NewBoolVar().WithName(
absl::StrFormat("shift_n%dd%ds%d", n, d, s));
}
}
}
Java
Literal[][][] shifts = new Literal[numNurses][numDays][numShifts];
for (int n : allNurses) {
for (int d : allDays) {
for (int s : allShifts) {
shifts[n][d][s] = model.newBoolVar("shifts_n" + n + "d" + d + "s" + s);
}
}
}
C#
Dictionary<(int, int, int), BoolVar> shifts =
new Dictionary<(int, int, int), BoolVar>(numNurses * numDays * numShifts);
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
shifts.Add((n, d, s), model.NewBoolVar($"shifts_n{n}d{d}s{s}"));
}
}
}
Das Array definiert Zuweisungen für Schichten an das Pflegepersonal so: shifts[(n, d, s)] ist gleich 1, wenn die Schicht n am Tag d der Krankenschwester n zugewiesen ist, ansonsten ist 0.
Krankenschwestern den Schichten zuweisen
Als Nächstes zeigen wir, wie Krankenschwestern unter Berücksichtigung der folgenden Einschränkungen Schichten zugewiesen werden:
- Jede Schicht wird einer einzigen Pflegekraft pro Tag zugewiesen.
- Jede Krankenschwester arbeitet höchstens eine Schicht pro Tag.
Hier ist der Code, mit dem die erste Bedingung erstellt wird.
Python
for d in all_days:
for s in all_shifts:
model.add_exactly_one(shifts[(n, d, s)] for n in all_nurses)
C++
for (int d : all_days) {
for (int s : all_shifts) {
std::vector<BoolVar> nurses;
for (int n : all_nurses) {
auto key = std::make_tuple(n, d, s);
nurses.push_back(shifts[key]);
}
cp_model.AddExactlyOne(nurses);
}
}
Java
for (int d : allDays) {
for (int s : allShifts) {
List<Literal> nurses = new ArrayList<>();
for (int n : allNurses) {
nurses.add(shifts[n][d][s]);
}
model.addExactlyOne(nurses);
}
}
C#
List<ILiteral> literals = new List<ILiteral>();
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
foreach (int n in allNurses)
{
literals.Add(shifts[(n, d, s)]);
}
model.AddExactlyOne(literals);
literals.Clear();
}
}
Die letzte Zeile besagt, dass für jede Schicht die Summe der Krankenschwestern, die dieser Schicht zugewiesen sind, 1 beträgt.
Als Nächstes folgt der Code, der erfordert, dass jede Pflegefachkraft höchstens eine Schicht pro Tag bearbeitet.
Python
for n in all_nurses:
for d in all_days:
model.add_at_most_one(shifts[(n, d, s)] for s in all_shifts)
C++
for (int n : all_nurses) {
for (int d : all_days) {
std::vector<BoolVar> work;
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
work.push_back(shifts[key]);
}
cp_model.AddAtMostOne(work);
}
}
Java
for (int n : allNurses) {
for (int d : allDays) {
List<Literal> work = new ArrayList<>();
for (int s : allShifts) {
work.add(shifts[n][d][s]);
}
model.addAtMostOne(work);
}
}
C#
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
literals.Add(shifts[(n, d, s)]);
}
model.AddAtMostOne(literals);
literals.Clear();
}
}
Für jede Pflegefachkraft beträgt die Summe der dieser Krankenschwester zugewiesenen Schichten höchstens 1 ("höchstens", weil eine Krankenschwester möglicherweise den Tag frei hat).
Schichten gleichmäßig zuweisen
Als Nächstes zeigen wir, wie den Pflegekräften Schichten so gleichmäßig wie möglich zugewiesen werden. Da es innerhalb von drei Tagen neun Schichten gibt, können wir jedem der vier Pflegekräfte zwei Schichten zuweisen. Danach verbleibt eine Schicht, die jeder beliebigen Pflegekraft zugewiesen werden kann.
Der folgende Code sorgt dafür, dass jede Pflegekraft mindestens zwei Schichten innerhalb der drei Tage arbeitet.
Python
# Try to distribute the shifts evenly, so that each nurse works
# min_shifts_per_nurse shifts. If this is not possible, because the total
# number of shifts is not divisible by the number of nurses, some nurses will
# be assigned one more shift.
min_shifts_per_nurse = (num_shifts * num_days) // num_nurses
if num_shifts * num_days % num_nurses == 0:
max_shifts_per_nurse = min_shifts_per_nurse
else:
max_shifts_per_nurse = min_shifts_per_nurse + 1
for n in all_nurses:
shifts_worked = []
for d in all_days:
for s in all_shifts:
shifts_worked.append(shifts[(n, d, s)])
model.add(min_shifts_per_nurse <= sum(shifts_worked))
model.add(sum(shifts_worked) <= max_shifts_per_nurse)
C++
// Try to distribute the shifts evenly, so that each nurse works
// min_shifts_per_nurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int min_shifts_per_nurse = (num_shifts * num_days) / num_nurses;
int max_shifts_per_nurse;
if ((num_shifts * num_days) % num_nurses == 0) {
max_shifts_per_nurse = min_shifts_per_nurse;
} else {
max_shifts_per_nurse = min_shifts_per_nurse + 1;
}
for (int n : all_nurses) {
std::vector<BoolVar> shifts_worked;
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
shifts_worked.push_back(shifts[key]);
}
}
cp_model.AddLessOrEqual(min_shifts_per_nurse,
LinearExpr::Sum(shifts_worked));
cp_model.AddLessOrEqual(LinearExpr::Sum(shifts_worked),
max_shifts_per_nurse);
}
Java
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0) {
maxShiftsPerNurse = minShiftsPerNurse;
} else {
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
for (int n : allNurses) {
LinearExprBuilder shiftsWorked = LinearExpr.newBuilder();
for (int d : allDays) {
for (int s : allShifts) {
shiftsWorked.add(shifts[n][d][s]);
}
}
model.addLinearConstraint(shiftsWorked, minShiftsPerNurse, maxShiftsPerNurse);
}
C#
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0)
{
maxShiftsPerNurse = minShiftsPerNurse;
}
else
{
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
List<IntVar> shiftsWorked = new List<IntVar>();
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
shiftsWorked.Add(shifts[(n, d, s)]);
}
}
model.AddLinearConstraint(LinearExpr.Sum(shiftsWorked), minShiftsPerNurse, maxShiftsPerNurse);
shiftsWorked.Clear();
}
Da es im Zeitplanzeitraum insgesamt num_shifts * num_days Energy Shifts gibt, können Sie mindestens (num_shifts * num_days) // num_nurses zuweisen
Schichten für die einzelnen Pflegekräfte, aber einige Schichten bleiben möglicherweise übrig. (Hier ist // der Python-Ganzzahloperator, der den Mindestpreis des üblichen Quotienten zurückgibt.)
Für die angegebenen Werte von num_nurses = 4, num_shifts = 3 und num_days = 3 hat der Ausdruck min_shifts_per_nurse den Wert (3 * 3 // 4) = 2, sodass Sie jeder Pflegefachkraft mindestens zwei Schichten zuweisen können. Dies wird durch die Einschränkung (hier in Python)
model.add(min_shifts_per_nurse <= sum(num_shifts_worked))angegeben.
Da es innerhalb von drei Tagen insgesamt neun Schichten gibt, verbleibt eine Schicht, nachdem jeder Pflegekraft zwei Schichten zugewiesen wurden. Die zusätzliche Schicht kann jeder Krankenschwester zugewiesen werden.
Die letzte Zeile (hier in Python)
model.add(sum(num_shifts_worked) <= max_shifts_per_nurse)
dass keiner Pflegekraft mehr als eine zusätzliche Schicht zugewiesen wird.
Die Einschränkung ist in diesem Fall nicht erforderlich, da es nur eine zusätzliche Verschiebung gibt. Für unterschiedliche Parameterwerte kann es jedoch mehrere zusätzliche Verschiebungen geben, in denen die Einschränkung erforderlich ist.
Solver-Parameter aktualisieren
Bei einem Modell ohne Optimierung kann die Suche für alle Lösungen aktiviert werden.
Python
solver = cp_model.CpSolver() solver.parameters.linearization_level = 0 # Enumerate all solutions. solver.parameters.enumerate_all_solutions = True
C++
Model model; SatParameters parameters; parameters.set_linearization_level(0); // Enumerate all solutions. parameters.set_enumerate_all_solutions(true); model.Add(NewSatParameters(parameters));
Java
CpSolver solver = new CpSolver(); solver.getParameters().setLinearizationLevel(0); // Tell the solver to enumerate all solutions. solver.getParameters().setEnumerateAllSolutions(true);
C#
CpSolver solver = new CpSolver(); // Tell the solver to enumerate all solutions. solver.StringParameters += "linearization_level:0 " + "enumerate_all_solutions:true ";
Lösungsrückruf registrieren
Sie müssen einen Callback auf dem Solver registrieren, der bei jeder Lösung aufgerufen wird.
Python
class NursesPartialSolutionPrinter(cp_model.CpSolverSolutionCallback):
"""Print intermediate solutions."""
def __init__(self, shifts, num_nurses, num_days, num_shifts, limit):
cp_model.CpSolverSolutionCallback.__init__(self)
self._shifts = shifts
self._num_nurses = num_nurses
self._num_days = num_days
self._num_shifts = num_shifts
self._solution_count = 0
self._solution_limit = limit
def on_solution_callback(self):
self._solution_count += 1
print(f"Solution {self._solution_count}")
for d in range(self._num_days):
print(f"Day {d}")
for n in range(self._num_nurses):
is_working = False
for s in range(self._num_shifts):
if self.value(self._shifts[(n, d, s)]):
is_working = True
print(f" Nurse {n} works shift {s}")
if not is_working:
print(f" Nurse {n} does not work")
if self._solution_count >= self._solution_limit:
print(f"Stop search after {self._solution_limit} solutions")
self.stop_search()
def solutionCount(self):
return self._solution_count
# Display the first five solutions.
solution_limit = 5
solution_printer = NursesPartialSolutionPrinter(
shifts, num_nurses, num_days, num_shifts, solution_limit
)
C++
// Create an atomic Boolean that will be periodically checked by the limit.
std::atomic<bool> stopped(false);
model.GetOrCreate<TimeLimit>()->RegisterExternalBooleanAsLimit(&stopped);
const int kSolutionLimit = 5;
int num_solutions = 0;
model.Add(NewFeasibleSolutionObserver([&](const CpSolverResponse& r) {
LOG(INFO) << "Solution " << num_solutions;
for (int d : all_days) {
LOG(INFO) << "Day " << std::to_string(d);
for (int n : all_nurses) {
bool is_working = false;
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
if (SolutionIntegerValue(r, shifts[key])) {
is_working = true;
LOG(INFO) << " Nurse " << std::to_string(n) << " works shift "
<< std::to_string(s);
}
}
if (!is_working) {
LOG(INFO) << " Nurse " << std::to_string(n) << " does not work";
}
}
}
num_solutions++;
if (num_solutions >= kSolutionLimit) {
stopped = true;
LOG(INFO) << "Stop search after " << kSolutionLimit << " solutions.";
}
}));
Java
final int solutionLimit = 5;
class VarArraySolutionPrinterWithLimit extends CpSolverSolutionCallback {
public VarArraySolutionPrinterWithLimit(
int[] allNurses, int[] allDays, int[] allShifts, Literal[][][] shifts, int limit) {
solutionCount = 0;
this.allNurses = allNurses;
this.allDays = allDays;
this.allShifts = allShifts;
this.shifts = shifts;
solutionLimit = limit;
}
@Override
public void onSolutionCallback() {
System.out.printf("Solution #%d:%n", solutionCount);
for (int d : allDays) {
System.out.printf("Day %d%n", d);
for (int n : allNurses) {
boolean isWorking = false;
for (int s : allShifts) {
if (booleanValue(shifts[n][d][s])) {
isWorking = true;
System.out.printf(" Nurse %d work shift %d%n", n, s);
}
}
if (!isWorking) {
System.out.printf(" Nurse %d does not work%n", n);
}
}
}
solutionCount++;
if (solutionCount >= solutionLimit) {
System.out.printf("Stop search after %d solutions%n", solutionLimit);
stopSearch();
}
}
public int getSolutionCount() {
return solutionCount;
}
private int solutionCount;
private final int[] allNurses;
private final int[] allDays;
private final int[] allShifts;
private final Literal[][][] shifts;
private final int solutionLimit;
}
VarArraySolutionPrinterWithLimit cb =
new VarArraySolutionPrinterWithLimit(allNurses, allDays, allShifts, shifts, solutionLimit);
C#
Definieren Sie zuerst die SolutionPrinter-Klasse.
public class SolutionPrinter : CpSolverSolutionCallback
{
public SolutionPrinter(int[] allNurses, int[] allDays, int[] allShifts,
Dictionary<(int, int, int), BoolVar> shifts, int limit)
{
solutionCount_ = 0;
allNurses_ = allNurses;
allDays_ = allDays;
allShifts_ = allShifts;
shifts_ = shifts;
solutionLimit_ = limit;
}
public override void OnSolutionCallback()
{
Console.WriteLine($"Solution #{solutionCount_}:");
foreach (int d in allDays_)
{
Console.WriteLine($"Day {d}");
foreach (int n in allNurses_)
{
bool isWorking = false;
foreach (int s in allShifts_)
{
if (Value(shifts_[(n, d, s)]) == 1L)
{
isWorking = true;
Console.WriteLine($" Nurse {n} work shift {s}");
}
}
if (!isWorking)
{
Console.WriteLine($" Nurse {d} does not work");
}
}
}
solutionCount_++;
if (solutionCount_ >= solutionLimit_)
{
Console.WriteLine($"Stop search after {solutionLimit_} solutions");
StopSearch();
}
}
public int SolutionCount()
{
return solutionCount_;
}
private int solutionCount_;
private int[] allNurses_;
private int[] allDays_;
private int[] allShifts_;
private Dictionary<(int, int, int), BoolVar> shifts_;
private int solutionLimit_;
}
Instanziieren Sie sie dann mit:
const int solutionLimit = 5; SolutionPrinter cb = new SolutionPrinter(allNurses, allDays, allShifts, shifts, solutionLimit);
Lösen aufrufen
Mit dem folgenden Code wird der Solver aufgerufen und die ersten fünf Lösungen angezeigt.
Python
solver.solve(model, solution_printer)
C++
const CpSolverResponse response = SolveCpModel(cp_model.Build(), &model);
Java
CpSolverStatus status = solver.solve(model, cb);
System.out.println("Status: " + status);
System.out.println(cb.getSolutionCount() + " solutions found.");
C#
CpSolverStatus status = solver.Solve(model, cb);
Console.WriteLine($"Solve status: {status}");
Lösungen
Hier sind die ersten fünf Lösungen.
Solution 0
Day 0
Nurse 0 does not work
Nurse 1 works shift 0
Nurse 2 works shift 1
Nurse 3 works shift 2
Day 1
Nurse 0 works shift 2
Nurse 1 does not work
Nurse 2 works shift 1
Nurse 3 works shift 0
Day 2
Nurse 0 works shift 2
Nurse 1 works shift 1
Nurse 2 works shift 0
Nurse 3 does not work
Solution 1
Day 0
Nurse 0 works shift 0
Nurse 1 does not work
Nurse 2 works shift 1
Nurse 3 works shift 2
Day 1
Nurse 0 does not work
Nurse 1 works shift 2
Nurse 2 works shift 1
Nurse 3 works shift 0
Day 2
Nurse 0 works shift 2
Nurse 1 works shift 1
Nurse 2 works shift 0
Nurse 3 does not work
Solution 2
Day 0 Nurse 0 works shift 0
Nurse 1 does not work
Nurse 2 works shift 1
Nurse 3 works shift 2
Day 1
Nurse 0 works shift 1
Nurse 1 works shift 2
Nurse 2 does not work
Nurse 3 works shift 0
Day 2
Nurse 0 works shift 2
Nurse 1 works shift 1
Nurse 2 works shift 0
Nurse 3 does not work
Solution 3
Day 0 Nurse 0 does not work
Nurse 1 works shift 0
Nurse 2 works shift 1
Nurse 3 works shift 2
Day 1
Nurse 0 works shift 1
Nurse 1 works shift 2
Nurse 2 does not work
Nurse 3 works shift 0
Day 2
Nurse 0 works shift 2
Nurse 1 works shift 1
Nurse 2 works shift 0
Nurse 3 does not work
Solution 4
Day 0
Nurse 0 does not work
Nurse 1 works shift 0
Nurse 2 works shift 1
Nurse 3 works shift 2
Day 1
Nurse 0 works shift 2
Nurse 1 works shift 1
Nurse 2 does not work
Nurse 3 works shift 0
Day 2
Nurse 0 works shift 2
Nurse 1 works shift 1
Nurse 2 works shift 0
Nurse 3 does not work
Statistics
- conflicts : 5
- branches : 142
- wall time : 0.002484 s
- solutions found: 5
Die Gesamtzahl der Lösungen beträgt 5.184. Das folgende Zählargument erklärt die Gründe.
Erstens gibt es vier Möglichkeiten für die Pflegefachkraft, die eine zusätzliche Schicht arbeitet. Nachdem diese Krankenschwester ausgewählt wurde, gibt es 3 Schichten, denen die Pflegekraft an jedem der drei Tage zugewiesen werden kann. Die Anzahl der Möglichkeiten, der Pflegekraft eine zusätzliche Schicht zuzuweisen, beträgt also 4 · 33 = 108. Nachdem diese Pflegekraft zugewiesen wurde, gibt es an jedem Tag zwei verbleibende, nicht zugewiesene Schichten.
Von den übrigen drei Pflegekräften arbeitet eine an Tag 0 und 1, eine an Tag 0 und 2 und eine an Tag 1 und 2. Es gibt drei! = 6 Möglichkeiten, die Pflegekräfte diesen Tagen zuzuweisen, wie im folgenden Diagramm dargestellt. (Die drei Pflegekräfte sind mit den Bezeichnungen A, B und C beschriftet, und wir haben sie noch nicht den Schichten zugewiesen.)
Day 0 Day 1 Day 2
A B A C B C
A B B C A C
A C A B B C
A C B C A B
B C A B A C
B C A C A B
Für jede Zeile im obigen Diagramm gibt es 23 = 8 mögliche Möglichkeiten, die verbleibenden Schichten den Pflegekräften zuzuweisen (zwei Auswahlmöglichkeiten pro Tag). Die Gesamtzahl der möglichen Zuweisungen beträgt also 108·6·8 = 5.184.
Gesamtes Programm
Hier ist das gesamte Programm für das Problem mit der Terminplanung für das Pflegepersonal.
Python
"""Example of a simple nurse scheduling problem."""
from ortools.sat.python import cp_model
def main() -> None:
# Data.
num_nurses = 4
num_shifts = 3
num_days = 3
all_nurses = range(num_nurses)
all_shifts = range(num_shifts)
all_days = range(num_days)
# Creates the model.
model = cp_model.CpModel()
# Creates shift variables.
# shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
shifts = {}
for n in all_nurses:
for d in all_days:
for s in all_shifts:
shifts[(n, d, s)] = model.new_bool_var(f"shift_n{n}_d{d}_s{s}")
# Each shift is assigned to exactly one nurse in the schedule period.
for d in all_days:
for s in all_shifts:
model.add_exactly_one(shifts[(n, d, s)] for n in all_nurses)
# Each nurse works at most one shift per day.
for n in all_nurses:
for d in all_days:
model.add_at_most_one(shifts[(n, d, s)] for s in all_shifts)
# Try to distribute the shifts evenly, so that each nurse works
# min_shifts_per_nurse shifts. If this is not possible, because the total
# number of shifts is not divisible by the number of nurses, some nurses will
# be assigned one more shift.
min_shifts_per_nurse = (num_shifts * num_days) // num_nurses
if num_shifts * num_days % num_nurses == 0:
max_shifts_per_nurse = min_shifts_per_nurse
else:
max_shifts_per_nurse = min_shifts_per_nurse + 1
for n in all_nurses:
shifts_worked = []
for d in all_days:
for s in all_shifts:
shifts_worked.append(shifts[(n, d, s)])
model.add(min_shifts_per_nurse <= sum(shifts_worked))
model.add(sum(shifts_worked) <= max_shifts_per_nurse)
# Creates the solver and solve.
solver = cp_model.CpSolver()
solver.parameters.linearization_level = 0
# Enumerate all solutions.
solver.parameters.enumerate_all_solutions = True
class NursesPartialSolutionPrinter(cp_model.CpSolverSolutionCallback):
"""Print intermediate solutions."""
def __init__(self, shifts, num_nurses, num_days, num_shifts, limit):
cp_model.CpSolverSolutionCallback.__init__(self)
self._shifts = shifts
self._num_nurses = num_nurses
self._num_days = num_days
self._num_shifts = num_shifts
self._solution_count = 0
self._solution_limit = limit
def on_solution_callback(self):
self._solution_count += 1
print(f"Solution {self._solution_count}")
for d in range(self._num_days):
print(f"Day {d}")
for n in range(self._num_nurses):
is_working = False
for s in range(self._num_shifts):
if self.value(self._shifts[(n, d, s)]):
is_working = True
print(f" Nurse {n} works shift {s}")
if not is_working:
print(f" Nurse {n} does not work")
if self._solution_count >= self._solution_limit:
print(f"Stop search after {self._solution_limit} solutions")
self.stop_search()
def solutionCount(self):
return self._solution_count
# Display the first five solutions.
solution_limit = 5
solution_printer = NursesPartialSolutionPrinter(
shifts, num_nurses, num_days, num_shifts, solution_limit
)
solver.solve(model, solution_printer)
# Statistics.
print("\nStatistics")
print(f" - conflicts : {solver.num_conflicts}")
print(f" - branches : {solver.num_branches}")
print(f" - wall time : {solver.wall_time} s")
print(f" - solutions found: {solution_printer.solutionCount()}")
if __name__ == "__main__":
main()
C++
// Example of a simple nurse scheduling problem.
#include <stdlib.h>
#include <atomic>
#include <map>
#include <numeric>
#include <string>
#include <tuple>
#include <vector>
#include "absl/strings/str_format.h"
#include "ortools/base/logging.h"
#include "ortools/sat/cp_model.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_solver.h"
#include "ortools/sat/model.h"
#include "ortools/sat/sat_parameters.pb.h"
#include "ortools/util/time_limit.h"
namespace operations_research {
namespace sat {
void NurseSat() {
const int num_nurses = 4;
const int num_shifts = 3;
const int num_days = 3;
std::vector<int> all_nurses(num_nurses);
std::iota(all_nurses.begin(), all_nurses.end(), 0);
std::vector<int> all_shifts(num_shifts);
std::iota(all_shifts.begin(), all_shifts.end(), 0);
std::vector<int> all_days(num_days);
std::iota(all_days.begin(), all_days.end(), 0);
// Creates the model.
CpModelBuilder cp_model;
// Creates shift variables.
// shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
std::map<std::tuple<int, int, int>, BoolVar> shifts;
for (int n : all_nurses) {
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
shifts[key] = cp_model.NewBoolVar().WithName(
absl::StrFormat("shift_n%dd%ds%d", n, d, s));
}
}
}
// Each shift is assigned to exactly one nurse in the schedule period.
for (int d : all_days) {
for (int s : all_shifts) {
std::vector<BoolVar> nurses;
for (int n : all_nurses) {
auto key = std::make_tuple(n, d, s);
nurses.push_back(shifts[key]);
}
cp_model.AddExactlyOne(nurses);
}
}
// Each nurse works at most one shift per day.
for (int n : all_nurses) {
for (int d : all_days) {
std::vector<BoolVar> work;
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
work.push_back(shifts[key]);
}
cp_model.AddAtMostOne(work);
}
}
// Try to distribute the shifts evenly, so that each nurse works
// min_shifts_per_nurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int min_shifts_per_nurse = (num_shifts * num_days) / num_nurses;
int max_shifts_per_nurse;
if ((num_shifts * num_days) % num_nurses == 0) {
max_shifts_per_nurse = min_shifts_per_nurse;
} else {
max_shifts_per_nurse = min_shifts_per_nurse + 1;
}
for (int n : all_nurses) {
std::vector<BoolVar> shifts_worked;
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
shifts_worked.push_back(shifts[key]);
}
}
cp_model.AddLessOrEqual(min_shifts_per_nurse,
LinearExpr::Sum(shifts_worked));
cp_model.AddLessOrEqual(LinearExpr::Sum(shifts_worked),
max_shifts_per_nurse);
}
Model model;
SatParameters parameters;
parameters.set_linearization_level(0);
// Enumerate all solutions.
parameters.set_enumerate_all_solutions(true);
model.Add(NewSatParameters(parameters));
// Display the first five solutions.
// Create an atomic Boolean that will be periodically checked by the limit.
std::atomic<bool> stopped(false);
model.GetOrCreate<TimeLimit>()->RegisterExternalBooleanAsLimit(&stopped);
const int kSolutionLimit = 5;
int num_solutions = 0;
model.Add(NewFeasibleSolutionObserver([&](const CpSolverResponse& r) {
LOG(INFO) << "Solution " << num_solutions;
for (int d : all_days) {
LOG(INFO) << "Day " << std::to_string(d);
for (int n : all_nurses) {
bool is_working = false;
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
if (SolutionIntegerValue(r, shifts[key])) {
is_working = true;
LOG(INFO) << " Nurse " << std::to_string(n) << " works shift "
<< std::to_string(s);
}
}
if (!is_working) {
LOG(INFO) << " Nurse " << std::to_string(n) << " does not work";
}
}
}
num_solutions++;
if (num_solutions >= kSolutionLimit) {
stopped = true;
LOG(INFO) << "Stop search after " << kSolutionLimit << " solutions.";
}
}));
const CpSolverResponse response = SolveCpModel(cp_model.Build(), &model);
// Statistics.
LOG(INFO) << "Statistics";
LOG(INFO) << CpSolverResponseStats(response);
LOG(INFO) << "solutions found : " << std::to_string(num_solutions);
}
} // namespace sat
} // namespace operations_research
int main() {
operations_research::sat::NurseSat();
return EXIT_SUCCESS;
}
Java
package com.google.ortools.sat.samples;
import com.google.ortools.Loader;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverSolutionCallback;
import com.google.ortools.sat.CpSolverStatus;
import com.google.ortools.sat.LinearExpr;
import com.google.ortools.sat.LinearExprBuilder;
import com.google.ortools.sat.Literal;
import java.util.ArrayList;
import java.util.List;
import java.util.stream.IntStream;
/** Nurses problem. */
public class NursesSat {
public static void main(String[] args) {
Loader.loadNativeLibraries();
final int numNurses = 4;
final int numDays = 3;
final int numShifts = 3;
final int[] allNurses = IntStream.range(0, numNurses).toArray();
final int[] allDays = IntStream.range(0, numDays).toArray();
final int[] allShifts = IntStream.range(0, numShifts).toArray();
// Creates the model.
CpModel model = new CpModel();
// Creates shift variables.
// shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
Literal[][][] shifts = new Literal[numNurses][numDays][numShifts];
for (int n : allNurses) {
for (int d : allDays) {
for (int s : allShifts) {
shifts[n][d][s] = model.newBoolVar("shifts_n" + n + "d" + d + "s" + s);
}
}
}
// Each shift is assigned to exactly one nurse in the schedule period.
for (int d : allDays) {
for (int s : allShifts) {
List<Literal> nurses = new ArrayList<>();
for (int n : allNurses) {
nurses.add(shifts[n][d][s]);
}
model.addExactlyOne(nurses);
}
}
// Each nurse works at most one shift per day.
for (int n : allNurses) {
for (int d : allDays) {
List<Literal> work = new ArrayList<>();
for (int s : allShifts) {
work.add(shifts[n][d][s]);
}
model.addAtMostOne(work);
}
}
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0) {
maxShiftsPerNurse = minShiftsPerNurse;
} else {
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
for (int n : allNurses) {
LinearExprBuilder shiftsWorked = LinearExpr.newBuilder();
for (int d : allDays) {
for (int s : allShifts) {
shiftsWorked.add(shifts[n][d][s]);
}
}
model.addLinearConstraint(shiftsWorked, minShiftsPerNurse, maxShiftsPerNurse);
}
CpSolver solver = new CpSolver();
solver.getParameters().setLinearizationLevel(0);
// Tell the solver to enumerate all solutions.
solver.getParameters().setEnumerateAllSolutions(true);
// Display the first five solutions.
final int solutionLimit = 5;
class VarArraySolutionPrinterWithLimit extends CpSolverSolutionCallback {
public VarArraySolutionPrinterWithLimit(
int[] allNurses, int[] allDays, int[] allShifts, Literal[][][] shifts, int limit) {
solutionCount = 0;
this.allNurses = allNurses;
this.allDays = allDays;
this.allShifts = allShifts;
this.shifts = shifts;
solutionLimit = limit;
}
@Override
public void onSolutionCallback() {
System.out.printf("Solution #%d:%n", solutionCount);
for (int d : allDays) {
System.out.printf("Day %d%n", d);
for (int n : allNurses) {
boolean isWorking = false;
for (int s : allShifts) {
if (booleanValue(shifts[n][d][s])) {
isWorking = true;
System.out.printf(" Nurse %d work shift %d%n", n, s);
}
}
if (!isWorking) {
System.out.printf(" Nurse %d does not work%n", n);
}
}
}
solutionCount++;
if (solutionCount >= solutionLimit) {
System.out.printf("Stop search after %d solutions%n", solutionLimit);
stopSearch();
}
}
public int getSolutionCount() {
return solutionCount;
}
private int solutionCount;
private final int[] allNurses;
private final int[] allDays;
private final int[] allShifts;
private final Literal[][][] shifts;
private final int solutionLimit;
}
VarArraySolutionPrinterWithLimit cb =
new VarArraySolutionPrinterWithLimit(allNurses, allDays, allShifts, shifts, solutionLimit);
// Creates a solver and solves the model.
CpSolverStatus status = solver.solve(model, cb);
System.out.println("Status: " + status);
System.out.println(cb.getSolutionCount() + " solutions found.");
// Statistics.
System.out.println("Statistics");
System.out.printf(" conflicts: %d%n", solver.numConflicts());
System.out.printf(" branches : %d%n", solver.numBranches());
System.out.printf(" wall time: %f s%n", solver.wallTime());
}
private NursesSat() {}
}
C#
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using Google.OrTools.Sat;
public class NursesSat
{
public class SolutionPrinter : CpSolverSolutionCallback
{
public SolutionPrinter(int[] allNurses, int[] allDays, int[] allShifts,
Dictionary<(int, int, int), BoolVar> shifts, int limit)
{
solutionCount_ = 0;
allNurses_ = allNurses;
allDays_ = allDays;
allShifts_ = allShifts;
shifts_ = shifts;
solutionLimit_ = limit;
}
public override void OnSolutionCallback()
{
Console.WriteLine($"Solution #{solutionCount_}:");
foreach (int d in allDays_)
{
Console.WriteLine($"Day {d}");
foreach (int n in allNurses_)
{
bool isWorking = false;
foreach (int s in allShifts_)
{
if (Value(shifts_[(n, d, s)]) == 1L)
{
isWorking = true;
Console.WriteLine($" Nurse {n} work shift {s}");
}
}
if (!isWorking)
{
Console.WriteLine($" Nurse {d} does not work");
}
}
}
solutionCount_++;
if (solutionCount_ >= solutionLimit_)
{
Console.WriteLine($"Stop search after {solutionLimit_} solutions");
StopSearch();
}
}
public int SolutionCount()
{
return solutionCount_;
}
private int solutionCount_;
private int[] allNurses_;
private int[] allDays_;
private int[] allShifts_;
private Dictionary<(int, int, int), BoolVar> shifts_;
private int solutionLimit_;
}
public static void Main(String[] args)
{
const int numNurses = 4;
const int numDays = 3;
const int numShifts = 3;
int[] allNurses = Enumerable.Range(0, numNurses).ToArray();
int[] allDays = Enumerable.Range(0, numDays).ToArray();
int[] allShifts = Enumerable.Range(0, numShifts).ToArray();
// Creates the model.
CpModel model = new CpModel();
model.Model.Variables.Capacity = numNurses * numDays * numShifts;
// Creates shift variables.
// shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
Dictionary<(int, int, int), BoolVar> shifts =
new Dictionary<(int, int, int), BoolVar>(numNurses * numDays * numShifts);
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
shifts.Add((n, d, s), model.NewBoolVar($"shifts_n{n}d{d}s{s}"));
}
}
}
// Each shift is assigned to exactly one nurse in the schedule period.
List<ILiteral> literals = new List<ILiteral>();
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
foreach (int n in allNurses)
{
literals.Add(shifts[(n, d, s)]);
}
model.AddExactlyOne(literals);
literals.Clear();
}
}
// Each nurse works at most one shift per day.
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
literals.Add(shifts[(n, d, s)]);
}
model.AddAtMostOne(literals);
literals.Clear();
}
}
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0)
{
maxShiftsPerNurse = minShiftsPerNurse;
}
else
{
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
List<IntVar> shiftsWorked = new List<IntVar>();
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
shiftsWorked.Add(shifts[(n, d, s)]);
}
}
model.AddLinearConstraint(LinearExpr.Sum(shiftsWorked), minShiftsPerNurse, maxShiftsPerNurse);
shiftsWorked.Clear();
}
CpSolver solver = new CpSolver();
// Tell the solver to enumerate all solutions.
solver.StringParameters += "linearization_level:0 " + "enumerate_all_solutions:true ";
// Display the first five solutions.
const int solutionLimit = 5;
SolutionPrinter cb = new SolutionPrinter(allNurses, allDays, allShifts, shifts, solutionLimit);
// Solve
CpSolverStatus status = solver.Solve(model, cb);
Console.WriteLine($"Solve status: {status}");
Console.WriteLine("Statistics");
Console.WriteLine($" conflicts: {solver.NumConflicts()}");
Console.WriteLine($" branches : {solver.NumBranches()}");
Console.WriteLine($" wall time: {solver.WallTime()}s");
}
}
Mit Schichtanfragen planen
In diesem Abschnitt nehmen wir das vorherige Beispiel und fügen Pflegeanfragen für bestimmte Schichten hinzu. Wir suchen dann nach einem Zeitplan, mit dem die Anzahl der erfüllten Anfragen maximiert wird. Bei den meisten Planungsproblemen empfiehlt es sich, eine Zielfunktion zu optimieren, da es normalerweise nicht zweckmäßig ist, alle möglichen Zeitpläne auszudrucken.
Dieses Beispiel hat die gleichen Einschränkungen wie das vorherige Beispiel.
Bibliotheken importieren
Mit dem folgenden Code wird die erforderliche Bibliothek importiert.
Python
from ortools.sat.python import cp_model
C++
#include <stdlib.h> #include <cstdint> #include <map> #include <numeric> #include <string> #include <tuple> #include <vector> #include "absl/strings/str_format.h" #include "ortools/base/logging.h" #include "ortools/sat/cp_model.h" #include "ortools/sat/cp_model.pb.h" #include "ortools/sat/cp_model_solver.h"
Java
import com.google.ortools.Loader; import com.google.ortools.sat.CpModel; import com.google.ortools.sat.CpSolver; import com.google.ortools.sat.CpSolverStatus; import com.google.ortools.sat.LinearExpr; import com.google.ortools.sat.LinearExprBuilder; import com.google.ortools.sat.Literal; import java.util.ArrayList; import java.util.List; import java.util.stream.IntStream;
C#
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.Sat;
Daten für das Beispiel
Die Daten für dieses Beispiel werden im Anschluss angezeigt.
Python
num_nurses = 5
num_shifts = 3
num_days = 7
all_nurses = range(num_nurses)
all_shifts = range(num_shifts)
all_days = range(num_days)
shift_requests = [
[[0, 0, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 1]],
[[0, 0, 0], [0, 0, 0], [0, 1, 0], [0, 1, 0], [1, 0, 0], [0, 0, 0], [0, 0, 1]],
[[0, 1, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0], [0, 1, 0], [0, 0, 0]],
[[0, 0, 1], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0]],
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0]],
]
C++
const int num_nurses = 5;
const int num_days = 7;
const int num_shifts = 3;
std::vector<int> all_nurses(num_nurses);
std::iota(all_nurses.begin(), all_nurses.end(), 0);
std::vector<int> all_days(num_days);
std::iota(all_days.begin(), all_days.end(), 0);
std::vector<int> all_shifts(num_shifts);
std::iota(all_shifts.begin(), all_shifts.end(), 0);
std::vector<std::vector<std::vector<int64_t>>> shift_requests = {
{
{0, 0, 1},
{0, 0, 0},
{0, 0, 0},
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 1},
},
{
{0, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 1, 0},
{1, 0, 0},
{0, 0, 0},
{0, 0, 1},
},
{
{0, 1, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
{
{0, 0, 1},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
},
{
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
};
Java
final int numNurses = 5;
final int numDays = 7;
final int numShifts = 3;
final int[] allNurses = IntStream.range(0, numNurses).toArray();
final int[] allDays = IntStream.range(0, numDays).toArray();
final int[] allShifts = IntStream.range(0, numShifts).toArray();
final int[][][] shiftRequests = new int[][][] {
{
{0, 0, 1},
{0, 0, 0},
{0, 0, 0},
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 1},
},
{
{0, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 1, 0},
{1, 0, 0},
{0, 0, 0},
{0, 0, 1},
},
{
{0, 1, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
{
{0, 0, 1},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
},
{
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
};
C#
const int numNurses = 5;
const int numDays = 7;
const int numShifts = 3;
int[] allNurses = Enumerable.Range(0, numNurses).ToArray();
int[] allDays = Enumerable.Range(0, numDays).ToArray();
int[] allShifts = Enumerable.Range(0, numShifts).ToArray();
int[,,] shiftRequests = new int[,,] {
{
{ 0, 0, 1 },
{ 0, 0, 0 },
{ 0, 0, 0 },
{ 0, 0, 0 },
{ 0, 0, 1 },
{ 0, 1, 0 },
{ 0, 0, 1 },
},
{
{ 0, 0, 0 },
{ 0, 0, 0 },
{ 0, 1, 0 },
{ 0, 1, 0 },
{ 1, 0, 0 },
{ 0, 0, 0 },
{ 0, 0, 1 },
},
{
{ 0, 1, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 0, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
},
{
{ 0, 0, 1 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 0, 0 },
},
{
{ 0, 0, 0 },
{ 0, 0, 1 },
{ 0, 1, 0 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
},
};
Modell erstellen
Mit dem folgenden Code wird das Modell erstellt.
Python
model = cp_model.CpModel()
C++
CpModelBuilder cp_model;
Java
CpModel model = new CpModel();
C#
CpModel model = new CpModel();
Variablen erstellen
Der folgende Code enthält ein Array mit Variablen für das Problem.
Zusätzlich zu den Variablen aus dem vorherigen Beispiel enthalten die Daten auch eine Reihe von Dreifachen, die den drei Schichten pro Tag entsprechen. Jedes Element des Triple ist 0 oder 1 und gibt an, ob eine Verschiebung angefordert wurde. Zum Beispiel zeigt das Dreifach [0, 0, 1] an der fünften Position von Zeile 1 an, dass Krankenschwester 1 an Tag 5 die 3. Schicht anfordert.
Python
shifts = {}
for n in all_nurses:
for d in all_days:
for s in all_shifts:
shifts[(n, d, s)] = model.new_bool_var(f"shift_n{n}_d{d}_s{s}")
C++
std::map<std::tuple<int, int, int>, BoolVar> shifts;
for (int n : all_nurses) {
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
shifts[key] = cp_model.NewBoolVar().WithName(
absl::StrFormat("shift_n%dd%ds%d", n, d, s));
}
}
}
Java
Literal[][][] shifts = new Literal[numNurses][numDays][numShifts];
for (int n : allNurses) {
for (int d : allDays) {
for (int s : allShifts) {
shifts[n][d][s] = model.newBoolVar("shifts_n" + n + "d" + d + "s" + s);
}
}
}
C#
Dictionary<Tuple<int, int, int>, IntVar> shifts = new Dictionary<Tuple<int, int, int>, IntVar>();
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
shifts.Add(Tuple.Create(n, d, s), model.NewBoolVar($"shifts_n{n}d{d}s{s}"));
}
}
}
Einschränkungen erstellen
Mit dem folgenden Code werden die Einschränkungen für das Problem erstellt.
Python
for d in all_days:
for s in all_shifts:
model.add_exactly_one(shifts[(n, d, s)] for n in all_nurses)
C++
for (int d : all_days) {
for (int s : all_shifts) {
std::vector<BoolVar> nurses;
for (int n : all_nurses) {
auto key = std::make_tuple(n, d, s);
nurses.push_back(shifts[key]);
}
cp_model.AddExactlyOne(nurses);
}
}
Java
for (int d : allDays) {
for (int s : allShifts) {
List<Literal> nurses = new ArrayList<>();
for (int n : allNurses) {
nurses.add(shifts[n][d][s]);
}
model.addExactlyOne(nurses);
}
}
C#
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
IntVar[] x = new IntVar[numNurses];
foreach (int n in allNurses)
{
var key = Tuple.Create(n, d, s);
x[n] = shifts[key];
}
model.Add(LinearExpr.Sum(x) == 1);
}
}
Python
for n in all_nurses:
for d in all_days:
model.add_at_most_one(shifts[(n, d, s)] for s in all_shifts)
C++
for (int n : all_nurses) {
for (int d : all_days) {
std::vector<BoolVar> work;
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
work.push_back(shifts[key]);
}
cp_model.AddAtMostOne(work);
}
}
Java
for (int n : allNurses) {
for (int d : allDays) {
List<Literal> work = new ArrayList<>();
for (int s : allShifts) {
work.add(shifts[n][d][s]);
}
model.addAtMostOne(work);
}
}
C#
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
IntVar[] x = new IntVar[numShifts];
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
x[s] = shifts[key];
}
model.Add(LinearExpr.Sum(x) <= 1);
}
}
Python
# Try to distribute the shifts evenly, so that each nurse works
# min_shifts_per_nurse shifts. If this is not possible, because the total
# number of shifts is not divisible by the number of nurses, some nurses will
# be assigned one more shift.
min_shifts_per_nurse = (num_shifts * num_days) // num_nurses
if num_shifts * num_days % num_nurses == 0:
max_shifts_per_nurse = min_shifts_per_nurse
else:
max_shifts_per_nurse = min_shifts_per_nurse + 1
for n in all_nurses:
num_shifts_worked = 0
for d in all_days:
for s in all_shifts:
num_shifts_worked += shifts[(n, d, s)]
model.add(min_shifts_per_nurse <= num_shifts_worked)
model.add(num_shifts_worked <= max_shifts_per_nurse)
C++
// Try to distribute the shifts evenly, so that each nurse works
// min_shifts_per_nurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int min_shifts_per_nurse = (num_shifts * num_days) / num_nurses;
int max_shifts_per_nurse;
if ((num_shifts * num_days) % num_nurses == 0) {
max_shifts_per_nurse = min_shifts_per_nurse;
} else {
max_shifts_per_nurse = min_shifts_per_nurse + 1;
}
for (int n : all_nurses) {
LinearExpr num_worked_shifts;
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
num_worked_shifts += shifts[key];
}
}
cp_model.AddLessOrEqual(min_shifts_per_nurse, num_worked_shifts);
cp_model.AddLessOrEqual(num_worked_shifts, max_shifts_per_nurse);
}
Java
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0) {
maxShiftsPerNurse = minShiftsPerNurse;
} else {
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
for (int n : allNurses) {
LinearExprBuilder numShiftsWorked = LinearExpr.newBuilder();
for (int d : allDays) {
for (int s : allShifts) {
numShiftsWorked.add(shifts[n][d][s]);
}
}
model.addLinearConstraint(numShiftsWorked, minShiftsPerNurse, maxShiftsPerNurse);
}
C#
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0)
{
maxShiftsPerNurse = minShiftsPerNurse;
}
else
{
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
foreach (int n in allNurses)
{
IntVar[] numShiftsWorked = new IntVar[numDays * numShifts];
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
numShiftsWorked[d * numShifts + s] = shifts[key];
}
}
model.AddLinearConstraint(LinearExpr.Sum(numShiftsWorked), minShiftsPerNurse, maxShiftsPerNurse);
}
Ziel für das Beispiel
Wir möchten die folgende Zielfunktion optimieren.
Python
model.maximize(
sum(
shift_requests[n][d][s] * shifts[(n, d, s)]
for n in all_nurses
for d in all_days
for s in all_shifts
)
)
C++
LinearExpr objective_expr;
for (int n : all_nurses) {
for (int d : all_days) {
for (int s : all_shifts) {
if (shift_requests[n][d][s] == 1) {
auto key = std::make_tuple(n, d, s);
objective_expr += shifts[key] * shift_requests[n][d][s];
}
}
}
}
cp_model.Maximize(objective_expr);
Java
LinearExprBuilder obj = LinearExpr.newBuilder();
for (int n : allNurses) {
for (int d : allDays) {
for (int s : allShifts) {
obj.addTerm(shifts[n][d][s], shiftRequests[n][d][s]);
}
}
}
model.maximize(obj);
C#
IntVar[] flatShifts = new IntVar[numNurses * numDays * numShifts];
int[] flatShiftRequests = new int[numNurses * numDays * numShifts];
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
flatShifts[n * numDays * numShifts + d * numShifts + s] = shifts[key];
flatShiftRequests[n * numDays * numShifts + d * numShifts + s] = shiftRequests[n, d, s];
}
}
}
model.Maximize(LinearExpr.WeightedSum(flatShifts, flatShiftRequests));
Da shift_requests[n][d][s] * shifts[(n, d, s) 1 ist, wenn die Schicht s der Pflegekraft n am d. Tag zugewiesen wird und diese Schicht diese Schicht angefordert hat (und andernfalls 0), ist das Ziel die Anzahl der Zuweisungen, die einer Anfrage entsprechen.
Lösen aufrufen
Mit dem folgenden Code wird der Solver aufgerufen.
Python
solver = cp_model.CpSolver() status = solver.solve(model)
C++
const CpSolverResponse response = Solve(cp_model.Build());
Java
CpSolver solver = new CpSolver(); CpSolverStatus status = solver.solve(model);
C#
CpSolver solver = new CpSolver();
CpSolverStatus status = solver.Solve(model);
Console.WriteLine($"Solve status: {status}");
Ergebnisse anzeigen
Der folgende Code zeigt die folgende Ausgabe an, die einen optimalen Zeitplan enthält (aber möglicherweise nicht der einzige). Die Ausgabe zeigt, welche Schichtzuweisungen angefordert und wie viele Anfragen erfüllt wurden.
Python
if status == cp_model.OPTIMAL:
print("Solution:")
for d in all_days:
print("Day", d)
for n in all_nurses:
for s in all_shifts:
if solver.value(shifts[(n, d, s)]) == 1:
if shift_requests[n][d][s] == 1:
print("Nurse", n, "works shift", s, "(requested).")
else:
print("Nurse", n, "works shift", s, "(not requested).")
print()
print(
f"Number of shift requests met = {solver.objective_value}",
f"(out of {num_nurses * min_shifts_per_nurse})",
)
else:
print("No optimal solution found !")
C++
if (response.status() == CpSolverStatus::OPTIMAL) {
LOG(INFO) << "Solution:";
for (int d : all_days) {
LOG(INFO) << "Day " << std::to_string(d);
for (int n : all_nurses) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
if (SolutionIntegerValue(response, shifts[key]) == 1) {
if (shift_requests[n][d][s] == 1) {
LOG(INFO) << " Nurse " << std::to_string(n) << " works shift "
<< std::to_string(s) << " (requested).";
} else {
LOG(INFO) << " Nurse " << std::to_string(n) << " works shift "
<< std::to_string(s) << " (not requested).";
}
}
}
}
LOG(INFO) << "";
}
LOG(INFO) << "Number of shift requests met = " << response.objective_value()
<< " (out of " << num_nurses * min_shifts_per_nurse << ")";
} else {
LOG(INFO) << "No optimal solution found !";
}
Java
if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) {
System.out.printf("Solution:%n");
for (int d : allDays) {
System.out.printf("Day %d%n", d);
for (int n : allNurses) {
for (int s : allShifts) {
if (solver.booleanValue(shifts[n][d][s])) {
if (shiftRequests[n][d][s] == 1) {
System.out.printf(" Nurse %d works shift %d (requested).%n", n, s);
} else {
System.out.printf(" Nurse %d works shift %d (not requested).%n", n, s);
}
}
}
}
}
System.out.printf("Number of shift requests met = %f (out of %d)%n", solver.objectiveValue(),
numNurses * minShiftsPerNurse);
} else {
System.out.printf("No optimal solution found !");
}
C#
if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible)
{
Console.WriteLine("Solution:");
foreach (int d in allDays)
{
Console.WriteLine($"Day {d}");
foreach (int n in allNurses)
{
bool isWorking = false;
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
if (solver.Value(shifts[key]) == 1L)
{
if (shiftRequests[n, d, s] == 1)
{
Console.WriteLine($" Nurse {n} work shift {s} (requested).");
}
else
{
Console.WriteLine($" Nurse {n} work shift {s} (not requested).");
}
}
}
}
}
Console.WriteLine(
$"Number of shift requests met = {solver.ObjectiveValue} (out of {numNurses * minShiftsPerNurse}).");
}
else
{
Console.WriteLine("No solution found.");
}
Wenn Sie das Programm ausführen, wird die folgende Ausgabe angezeigt:
Day 0
Nurse 1 works shift 0 (not requested).
Nurse 2 works shift 1 (requested).
Nurse 3 works shift 2 (requested).
Day 1
Nurse 0 works shift 0 (not requested).
Nurse 2 works shift 1 (requested).
Nurse 4 works shift 2 (requested).
Day 2
Nurse 1 works shift 2 (not requested).
Nurse 3 works shift 0 (requested).
Nurse 4 works shift 1 (requested).
Day 3
Nurse 2 works shift 0 (requested).
Nurse 3 works shift 1 (requested).
Nurse 4 works shift 2 (not requested).
Day 4
Nurse 0 works shift 2 (requested).
Nurse 1 works shift 0 (requested).
Nurse 4 works shift 1 (not requested).
Day 5
Nurse 0 works shift 2 (not requested).
Nurse 2 works shift 1 (requested).
Nurse 3 works shift 0 (requested).
Day 6
Nurse 0 works shift 1 (not requested).
Nurse 1 works shift 2 (requested).
Nurse 4 works shift 0 (not requested).
Statistics
- Number of shift requests met = 13 (out of 20 )
- wall time : 0.003571 s
Gesamtes Programm
Hier ist das gesamte Programm zur Terminplanung mit Schichtanfragen.
Python
"""Nurse scheduling problem with shift requests."""
from ortools.sat.python import cp_model
def main() -> None:
# This program tries to find an optimal assignment of nurses to shifts
# (3 shifts per day, for 7 days), subject to some constraints (see below).
# Each nurse can request to be assigned to specific shifts.
# The optimal assignment maximizes the number of fulfilled shift requests.
num_nurses = 5
num_shifts = 3
num_days = 7
all_nurses = range(num_nurses)
all_shifts = range(num_shifts)
all_days = range(num_days)
shift_requests = [
[[0, 0, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 1]],
[[0, 0, 0], [0, 0, 0], [0, 1, 0], [0, 1, 0], [1, 0, 0], [0, 0, 0], [0, 0, 1]],
[[0, 1, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0], [0, 1, 0], [0, 0, 0]],
[[0, 0, 1], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0]],
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0]],
]
# Creates the model.
model = cp_model.CpModel()
# Creates shift variables.
# shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
shifts = {}
for n in all_nurses:
for d in all_days:
for s in all_shifts:
shifts[(n, d, s)] = model.new_bool_var(f"shift_n{n}_d{d}_s{s}")
# Each shift is assigned to exactly one nurse in .
for d in all_days:
for s in all_shifts:
model.add_exactly_one(shifts[(n, d, s)] for n in all_nurses)
# Each nurse works at most one shift per day.
for n in all_nurses:
for d in all_days:
model.add_at_most_one(shifts[(n, d, s)] for s in all_shifts)
# Try to distribute the shifts evenly, so that each nurse works
# min_shifts_per_nurse shifts. If this is not possible, because the total
# number of shifts is not divisible by the number of nurses, some nurses will
# be assigned one more shift.
min_shifts_per_nurse = (num_shifts * num_days) // num_nurses
if num_shifts * num_days % num_nurses == 0:
max_shifts_per_nurse = min_shifts_per_nurse
else:
max_shifts_per_nurse = min_shifts_per_nurse + 1
for n in all_nurses:
num_shifts_worked = 0
for d in all_days:
for s in all_shifts:
num_shifts_worked += shifts[(n, d, s)]
model.add(min_shifts_per_nurse <= num_shifts_worked)
model.add(num_shifts_worked <= max_shifts_per_nurse)
model.maximize(
sum(
shift_requests[n][d][s] * shifts[(n, d, s)]
for n in all_nurses
for d in all_days
for s in all_shifts
)
)
# Creates the solver and solve.
solver = cp_model.CpSolver()
status = solver.solve(model)
if status == cp_model.OPTIMAL:
print("Solution:")
for d in all_days:
print("Day", d)
for n in all_nurses:
for s in all_shifts:
if solver.value(shifts[(n, d, s)]) == 1:
if shift_requests[n][d][s] == 1:
print("Nurse", n, "works shift", s, "(requested).")
else:
print("Nurse", n, "works shift", s, "(not requested).")
print()
print(
f"Number of shift requests met = {solver.objective_value}",
f"(out of {num_nurses * min_shifts_per_nurse})",
)
else:
print("No optimal solution found !")
# Statistics.
print("\nStatistics")
print(f" - conflicts: {solver.num_conflicts}")
print(f" - branches : {solver.num_branches}")
print(f" - wall time: {solver.wall_time}s")
if __name__ == "__main__":
main()
C++
// Nurse scheduling problem with shift requests.
#include <stdlib.h>
#include <cstdint>
#include <map>
#include <numeric>
#include <string>
#include <tuple>
#include <vector>
#include "absl/strings/str_format.h"
#include "ortools/base/logging.h"
#include "ortools/sat/cp_model.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_solver.h"
namespace operations_research {
namespace sat {
void ScheduleRequestsSat() {
const int num_nurses = 5;
const int num_days = 7;
const int num_shifts = 3;
std::vector<int> all_nurses(num_nurses);
std::iota(all_nurses.begin(), all_nurses.end(), 0);
std::vector<int> all_days(num_days);
std::iota(all_days.begin(), all_days.end(), 0);
std::vector<int> all_shifts(num_shifts);
std::iota(all_shifts.begin(), all_shifts.end(), 0);
std::vector<std::vector<std::vector<int64_t>>> shift_requests = {
{
{0, 0, 1},
{0, 0, 0},
{0, 0, 0},
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 1},
},
{
{0, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 1, 0},
{1, 0, 0},
{0, 0, 0},
{0, 0, 1},
},
{
{0, 1, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
{
{0, 0, 1},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
},
{
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
};
// Creates the model.
CpModelBuilder cp_model;
// Creates shift variables.
// shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
std::map<std::tuple<int, int, int>, BoolVar> shifts;
for (int n : all_nurses) {
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
shifts[key] = cp_model.NewBoolVar().WithName(
absl::StrFormat("shift_n%dd%ds%d", n, d, s));
}
}
}
// Each shift is assigned to exactly one nurse in the schedule period.
for (int d : all_days) {
for (int s : all_shifts) {
std::vector<BoolVar> nurses;
for (int n : all_nurses) {
auto key = std::make_tuple(n, d, s);
nurses.push_back(shifts[key]);
}
cp_model.AddExactlyOne(nurses);
}
}
// Each nurse works at most one shift per day.
for (int n : all_nurses) {
for (int d : all_days) {
std::vector<BoolVar> work;
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
work.push_back(shifts[key]);
}
cp_model.AddAtMostOne(work);
}
}
// Try to distribute the shifts evenly, so that each nurse works
// min_shifts_per_nurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int min_shifts_per_nurse = (num_shifts * num_days) / num_nurses;
int max_shifts_per_nurse;
if ((num_shifts * num_days) % num_nurses == 0) {
max_shifts_per_nurse = min_shifts_per_nurse;
} else {
max_shifts_per_nurse = min_shifts_per_nurse + 1;
}
for (int n : all_nurses) {
LinearExpr num_worked_shifts;
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
num_worked_shifts += shifts[key];
}
}
cp_model.AddLessOrEqual(min_shifts_per_nurse, num_worked_shifts);
cp_model.AddLessOrEqual(num_worked_shifts, max_shifts_per_nurse);
}
LinearExpr objective_expr;
for (int n : all_nurses) {
for (int d : all_days) {
for (int s : all_shifts) {
if (shift_requests[n][d][s] == 1) {
auto key = std::make_tuple(n, d, s);
objective_expr += shifts[key] * shift_requests[n][d][s];
}
}
}
}
cp_model.Maximize(objective_expr);
const CpSolverResponse response = Solve(cp_model.Build());
if (response.status() == CpSolverStatus::OPTIMAL) {
LOG(INFO) << "Solution:";
for (int d : all_days) {
LOG(INFO) << "Day " << std::to_string(d);
for (int n : all_nurses) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
if (SolutionIntegerValue(response, shifts[key]) == 1) {
if (shift_requests[n][d][s] == 1) {
LOG(INFO) << " Nurse " << std::to_string(n) << " works shift "
<< std::to_string(s) << " (requested).";
} else {
LOG(INFO) << " Nurse " << std::to_string(n) << " works shift "
<< std::to_string(s) << " (not requested).";
}
}
}
}
LOG(INFO) << "";
}
LOG(INFO) << "Number of shift requests met = " << response.objective_value()
<< " (out of " << num_nurses * min_shifts_per_nurse << ")";
} else {
LOG(INFO) << "No optimal solution found !";
}
// Statistics.
LOG(INFO) << "Statistics";
LOG(INFO) << CpSolverResponseStats(response);
}
} // namespace sat
} // namespace operations_research
int main() {
operations_research::sat::ScheduleRequestsSat();
return EXIT_SUCCESS;
}
Java
package com.google.ortools.sat.samples;
import com.google.ortools.Loader;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverStatus;
import com.google.ortools.sat.LinearExpr;
import com.google.ortools.sat.LinearExprBuilder;
import com.google.ortools.sat.Literal;
import java.util.ArrayList;
import java.util.List;
import java.util.stream.IntStream;
/** Nurses problem with schedule requests. */
public class ScheduleRequestsSat {
public static void main(String[] args) {
Loader.loadNativeLibraries();
final int numNurses = 5;
final int numDays = 7;
final int numShifts = 3;
final int[] allNurses = IntStream.range(0, numNurses).toArray();
final int[] allDays = IntStream.range(0, numDays).toArray();
final int[] allShifts = IntStream.range(0, numShifts).toArray();
final int[][][] shiftRequests = new int[][][] {
{
{0, 0, 1},
{0, 0, 0},
{0, 0, 0},
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 1},
},
{
{0, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 1, 0},
{1, 0, 0},
{0, 0, 0},
{0, 0, 1},
},
{
{0, 1, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
{
{0, 0, 1},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
},
{
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
};
// Creates the model.
CpModel model = new CpModel();
// Creates shift variables.
// shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
Literal[][][] shifts = new Literal[numNurses][numDays][numShifts];
for (int n : allNurses) {
for (int d : allDays) {
for (int s : allShifts) {
shifts[n][d][s] = model.newBoolVar("shifts_n" + n + "d" + d + "s" + s);
}
}
}
// Each shift is assigned to exactly one nurse in the schedule period.
for (int d : allDays) {
for (int s : allShifts) {
List<Literal> nurses = new ArrayList<>();
for (int n : allNurses) {
nurses.add(shifts[n][d][s]);
}
model.addExactlyOne(nurses);
}
}
// Each nurse works at most one shift per day.
for (int n : allNurses) {
for (int d : allDays) {
List<Literal> work = new ArrayList<>();
for (int s : allShifts) {
work.add(shifts[n][d][s]);
}
model.addAtMostOne(work);
}
}
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0) {
maxShiftsPerNurse = minShiftsPerNurse;
} else {
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
for (int n : allNurses) {
LinearExprBuilder numShiftsWorked = LinearExpr.newBuilder();
for (int d : allDays) {
for (int s : allShifts) {
numShiftsWorked.add(shifts[n][d][s]);
}
}
model.addLinearConstraint(numShiftsWorked, minShiftsPerNurse, maxShiftsPerNurse);
}
LinearExprBuilder obj = LinearExpr.newBuilder();
for (int n : allNurses) {
for (int d : allDays) {
for (int s : allShifts) {
obj.addTerm(shifts[n][d][s], shiftRequests[n][d][s]);
}
}
}
model.maximize(obj);
// Creates a solver and solves the model.
CpSolver solver = new CpSolver();
CpSolverStatus status = solver.solve(model);
if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) {
System.out.printf("Solution:%n");
for (int d : allDays) {
System.out.printf("Day %d%n", d);
for (int n : allNurses) {
for (int s : allShifts) {
if (solver.booleanValue(shifts[n][d][s])) {
if (shiftRequests[n][d][s] == 1) {
System.out.printf(" Nurse %d works shift %d (requested).%n", n, s);
} else {
System.out.printf(" Nurse %d works shift %d (not requested).%n", n, s);
}
}
}
}
}
System.out.printf("Number of shift requests met = %f (out of %d)%n", solver.objectiveValue(),
numNurses * minShiftsPerNurse);
} else {
System.out.printf("No optimal solution found !");
}
// Statistics.
System.out.println("Statistics");
System.out.printf(" conflicts: %d%n", solver.numConflicts());
System.out.printf(" branches : %d%n", solver.numBranches());
System.out.printf(" wall time: %f s%n", solver.wallTime());
}
private ScheduleRequestsSat() {}
}
C#
using System;
using System.Collections.Generic;
using System.Linq;
using Google.OrTools.Sat;
public class ScheduleRequestsSat
{
public static void Main(String[] args)
{
const int numNurses = 5;
const int numDays = 7;
const int numShifts = 3;
int[] allNurses = Enumerable.Range(0, numNurses).ToArray();
int[] allDays = Enumerable.Range(0, numDays).ToArray();
int[] allShifts = Enumerable.Range(0, numShifts).ToArray();
int[,,] shiftRequests = new int[,,] {
{
{ 0, 0, 1 },
{ 0, 0, 0 },
{ 0, 0, 0 },
{ 0, 0, 0 },
{ 0, 0, 1 },
{ 0, 1, 0 },
{ 0, 0, 1 },
},
{
{ 0, 0, 0 },
{ 0, 0, 0 },
{ 0, 1, 0 },
{ 0, 1, 0 },
{ 1, 0, 0 },
{ 0, 0, 0 },
{ 0, 0, 1 },
},
{
{ 0, 1, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 0, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
},
{
{ 0, 0, 1 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 0, 0 },
},
{
{ 0, 0, 0 },
{ 0, 0, 1 },
{ 0, 1, 0 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
},
};
// Creates the model.
CpModel model = new CpModel();
// Creates shift variables.
// shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
Dictionary<Tuple<int, int, int>, IntVar> shifts = new Dictionary<Tuple<int, int, int>, IntVar>();
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
shifts.Add(Tuple.Create(n, d, s), model.NewBoolVar($"shifts_n{n}d{d}s{s}"));
}
}
}
// Each shift is assigned to exactly one nurse in the schedule period.
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
IntVar[] x = new IntVar[numNurses];
foreach (int n in allNurses)
{
var key = Tuple.Create(n, d, s);
x[n] = shifts[key];
}
model.Add(LinearExpr.Sum(x) == 1);
}
}
// Each nurse works at most one shift per day.
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
IntVar[] x = new IntVar[numShifts];
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
x[s] = shifts[key];
}
model.Add(LinearExpr.Sum(x) <= 1);
}
}
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0)
{
maxShiftsPerNurse = minShiftsPerNurse;
}
else
{
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
foreach (int n in allNurses)
{
IntVar[] numShiftsWorked = new IntVar[numDays * numShifts];
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
numShiftsWorked[d * numShifts + s] = shifts[key];
}
}
model.AddLinearConstraint(LinearExpr.Sum(numShiftsWorked), minShiftsPerNurse, maxShiftsPerNurse);
}
IntVar[] flatShifts = new IntVar[numNurses * numDays * numShifts];
int[] flatShiftRequests = new int[numNurses * numDays * numShifts];
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
flatShifts[n * numDays * numShifts + d * numShifts + s] = shifts[key];
flatShiftRequests[n * numDays * numShifts + d * numShifts + s] = shiftRequests[n, d, s];
}
}
}
model.Maximize(LinearExpr.WeightedSum(flatShifts, flatShiftRequests));
// Solve
CpSolver solver = new CpSolver();
CpSolverStatus status = solver.Solve(model);
Console.WriteLine($"Solve status: {status}");
if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible)
{
Console.WriteLine("Solution:");
foreach (int d in allDays)
{
Console.WriteLine($"Day {d}");
foreach (int n in allNurses)
{
bool isWorking = false;
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
if (solver.Value(shifts[key]) == 1L)
{
if (shiftRequests[n, d, s] == 1)
{
Console.WriteLine($" Nurse {n} work shift {s} (requested).");
}
else
{
Console.WriteLine($" Nurse {n} work shift {s} (not requested).");
}
}
}
}
}
Console.WriteLine(
$"Number of shift requests met = {solver.ObjectiveValue} (out of {numNurses * minShiftsPerNurse}).");
}
else
{
Console.WriteLine("No solution found.");
}
Console.WriteLine("Statistics");
Console.WriteLine($" conflicts: {solver.NumConflicts()}");
Console.WriteLine($" branches : {solver.NumBranches()}");
Console.WriteLine($" wall time: {solver.WallTime()}s");
}
}