Organisationen, deren Mitarbeiter in mehreren Schichten arbeiten, müssen genügend Zeit Arbeiter:innen in jeder Schicht. Üblicherweise haben die Zeitpläne Einschränkungen, zum Beispiel: „Kein Mitarbeiter sollte zwei Schichten hintereinander arbeiten“. Die Suche nach einem Zeitplan, alle Einschränkungen erfüllt, kann rechenintensiv sein.
In den folgenden Abschnitten werden zwei Beispiele für Probleme bei der Zeitplanung von Mitarbeitenden vorgestellt. zeigen, wie sie mit dem CP-SAT-Löser gelöst werden können.
Ein komplexeres Beispiel finden Sie hier: Schichtplanungsprogramm auf GitHub.
Ein Problem mit der Zeitplanung für Pflegekräfte
Im nächsten Beispiel muss eine Krankenhausleitung einen Zeitplan für vier Krankenschwestern über einen Zeitraum von drei Tagen, sofern die folgenden Bedingungen erfüllt sind:
- Jeder Tag ist in drei 8-Stunden-Schichten unterteilt.
- Jeden Tag wird jede Schicht einer einzelnen Pflegekraft zugewiesen und keine Pflegekraft 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 Zeitplanung von Pflegekräften 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 von 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 die Arbeitsverhältnisse der Schichten an die Krankenschwestern wie folgt:
shifts[(n, d, s)] ist 1, wenn Pflegepersonal n an Tag d die Schicht n zugewiesen wird, und 0
sonst.
Schichten Krankenschwestern zuweisen
Als Nächstes zeigen wir, wie wir Krankenschwestern den Schichten zuweisen, die den folgenden Einschränkungen unterliegen:
- Jede Schicht wird täglich einer einzelnen Pflegekraft zugewiesen.
- Jede Pflegekraft arbeitet höchstens eine Schicht pro Tag.
Mit diesem Code wird die erste Bedingung 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#
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();
}
}
In der letzten Zeile steht, dass in jeder Schicht die Summe der Shift 1 ist.
Als Nächstes ist der Code, der erfordert, dass jede Krankenschwester höchstens eine Schicht pro Schicht arbeitet Tag.
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 Pflegekraft ist die Summe der Schichten, die ihr zugewiesen sind, höchstens 1 („höchstens“ weil eine Krankenschwester am Tag frei ist.
Schichten gleichmäßig zuweisen
Als Nächstes zeigen wir, wie sich die Schichten so gleichmäßig wie möglich den Pflegekräften zuteilen. Da es innerhalb des Drei-Tage-Zeitraums neun Schichten gibt, können wir zwei Schichten vier Krankenschwestern. Danach folgt eine Schicht, die jeder Pflegefachkraft zugewiesen werden kann.
Der folgende Code stellt sicher, dass jede Pflegekraft mindestens zwei Schichten im für einen Zeitraum von drei Tagen.
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 geplanten Zeitraum insgesamt num_shifts * num_days Energy Shifts gibt, gehen Sie so vor:
kann mindestens (num_shifts * num_days) // num_nurses zuweisen
Schichten für jede Pflegekraft, aber einige Schichten bleiben übrig. (Hier ist // die Python-
Ganzzahldivisionsoperator, der den Mindestbetrag des gewöhnlichen 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 Pflegekraft mindestens zwei Schichten zuweisen. Dies wird durch die
Einschränkung (hier in Python)
model.add(min_shifts_per_nurse <= sum(shifts_worked))
Da es innerhalb des Drei-Tage-Zeitraums insgesamt neun Schichten gibt, verbleibende Schicht, nachdem jeder Pflegekraft zwei Schichten zugewiesen wurden. Die zusätzliche Schicht kann einer Pflegefachkraft zugewiesen.
Die letzte Zeile (hier in Python)
model.add(sum(shifts_worked) <= max_shifts_per_nurse)
stellt sicher, dass keiner Pflegekraft mehr als eine zusätzliche Schicht zugewiesen wird.
Die Einschränkung ist in diesem Fall nicht erforderlich, die Umschalttaste gedrückt. Für unterschiedliche Parameterwerte können jedoch zusätzliche Verschiebungen auftreten, In diesem Fall ist die Einschränkung erforderlich.
Solver-Parameter aktualisieren
In einem Nicht-Optimierungsmodell können Sie die Suche für alle Lösungen aktivieren.
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 ";
Lösungs-Callback registrieren
Du musst im Matherechner einen Callback registrieren, der bei jedem Aufruf aufgerufen wird. Lösung.
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 Klasse SolutionPrinter.
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 es dann mit:
const int solutionLimit = 5; SolutionPrinter cb = new SolutionPrinter(allNurses, allDays, allShifts, shifts, solutionLimit);
Solver aufrufen
Mit dem folgenden Code wird der Rechner aufgerufen und die ersten fünf Lösungen werden 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, warum.
Zunächst gibt es 4 Optionen für die eine Krankenschwester, die eine zusätzliche Schicht ausführt. Nach der Wahl dieser Krankenschwester gibt es 3 Schichten, denen die Pflegekraft zugewiesen werden kann. 3 Tage gegliedert sein. Die Anzahl der Möglichkeiten, die Pflegefachkraft zuzuweisen, zusätzliche Verschiebung: 4 · 33 = 108. Nachdem diese Pflegekraft zugewiesen wurde, gibt es an jedem Tag zwei nicht zugewiesene Schichten.
Von den drei Krankenschwestern, eine, die Tage 0 und 1, die andere an den Tagen 0 und 2, und einer der Tage 1 und 2. Es gibt drei! = 6 Möglichkeiten, die Pflegekräfte diesen Tagen zuzuweisen, wie in der Diagramm unten. (Die drei Krankenschwestern sind mit A, B und C beschriftet, und wir haben 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 übrigen Schichten den Pflegekräften zuweisen (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 Zeitplanung für Pflegekräfte.
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");
}
}
Terminplanung mit Schichtanfragen
In diesem Abschnitt nehmen wir das vorherige Beispiel und fügen bei bestimmten Schichten. Wir suchen dann nach einem Zeitplan, der die Anzahl der erfüllten Anfragen maximiert. Bei den meisten Planungsproblemen ist es am besten, eine Zielfunktion zu optimieren, da sie in der Regel nicht sinnvoll, alle möglichen Zeitpläne zu drucken.
Dieses Beispiel hat dieselben Einschränkungen wie das vorherige Beispiel.
Bibliotheken importieren
Mit dem folgenden Code wird die erforderliche Bibliothek importiert.
Python
from typing import Union 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
Im Folgenden wird ein Array von Variablen für das Problem codiert.
Zusätzlich zu den Variablen aus dem vorherigen Beispiel enthalten die Daten auch einen des Dreifachens entsprechend den drei Schichten pro Tag. Jedes Element des Das Dreieck ist 0 oder 1 und gibt an, ob eine Verschiebung angefordert wurde. Beispiel: Der Parameter Das Dreifache [0, 0, 1] in der fünften Position von Zeile 1 zeigt an, dass Krankenschwester 1 die 3. Schicht an Tag 5.
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
Der folgende Code erstellt die Einschränkungen für das Problem.
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: Union[cp_model.LinearExpr, int] = 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 zugewiesen ist
an die Pflegefachkraft n am d. Tag und diese Schicht hat diese Schicht angefordert (und sonst 0),
Ziel ist die Verschiebung der Anzahl
der Zuweisungen, die einer Anfrage entsprechen.
Solver aufrufen
Mit dem folgenden Code wird der Rechner 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, die eine optimale (wenn auch nicht der einzige). Die Ausgabe zeigt, welche Verschiebung die angefragten Aufgaben und die Anzahl der erfüllten Anfragen.
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 für die Terminplanung mit Schichtanfragen.
Python
"""Nurse scheduling problem with shift requests."""
from typing import Union
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: Union[cp_model.LinearExpr, int] = 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");
}
}