Organizacje, których pracownicy pracują na wiele zmian, muszą zaplanować odpowiednią na każdą zmianę dzienną. Harmonogramy zazwyczaj mają ograniczenia, np. „żaden z pracowników nie powinien pracować na dwie zmiany z rzędu”. Ustalenie harmonogramu, spełnia wszystkie ograniczenia, może być trudne pod względem obliczeniowym.
W poniższych sekcjach przedstawione są 2 przykłady problemów związanych z planowaniem przez pracowników oraz pokazać, jak je rozwiązać za pomocą narzędzia CP-SAT.
Bardziej zaawansowany przykład znajdziesz tutaj program zmiany harmonogramu w GitHubie.
Problem z planowaniem pracy pielęgniarki
W następnym przykładzie kierownik szpitala musi utworzyć harmonogram dla czterech osób. pielęgniarek w ciągu 3 dni z uwzględnieniem następujących warunków:
- Każdy dzień jest podzielony na 3 8-godzinne zmiany.
- Codziennie każda zmiana jest przypisywana do jednej pielęgniarki i żadna pielęgniarka nie pracuje niż jedną zmianę.
- Każda pielęgniarka jest przypisana na co najmniej 2 zmiany w ciągu 3 dni.
Poniższe sekcje zawierają rozwiązanie problemu planowania pielęgniarki.
Zaimportuj biblioteki
Poniższy kod importuje wymaganą bibliotekę.
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;
Dane do przykładu
Ten kod tworzy dane dla przykładu.
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();
Tworzenie modelu
Ten model tworzy model.
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;
Tworzenie zmiennych
Ten kod tworzy tablicę zmiennych.
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}"));
}
}
}
Tablica definiuje przypisania przerw na pielęgniarki w ten sposób:
shifts[(n, d, s)] równa się 1, jeśli zmiana S jest przypisana do pielęgniarki N w dniu d, a koniec 0 ma wartość 0
w przeciwnym razie.
Przypisz pielęgniarki do zmian
Następnie pokazujemy, jak przypisać pielęgniarki na zmiany z uwzględnieniem tych ograniczeń:
- Każda zmiana jest przypisana do 1 pielęgniarki na dzień.
- Każda pielęgniarka pracuje najwyżej na 1 zmianie dziennie.
Oto kod, który tworzy pierwszy warunek.
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();
}
}
W ostatnim wierszu znajduje się informacja, że na każdej zmianie jest to suma liczby pielęgniarek przypisanych do danej Shift wynosi 1.
Oto kod, który wymaga, aby każda pielęgniarka pracuje maksymalnie na 1 zmianie dzień.
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();
}
}
Suma zmian przypisanych do każdej pielęgniarki wynosi maksymalnie 1 („maksymalnie” ponieważ pielęgniarka może mieć dzień wolny).
Równomierne przypisywanie zmian
Następnie pokazujemy, jak najbardziej równomiernie przypisywać zmiany pracy pielęgniarkom. Ponieważ w okresie 3 dni obowiązuje dziewięć zmian, możemy przypisać 2 zmiany każdej z czterech pielęgniarek. Po tej dacie zostanie jeszcze jedna zmiana, którą można przypisać dowolnej pielęgniarce.
Poniższy kod daje pewność, że każda pielęgniarka pracuje na co najmniej dwóch zmianach trzy dni.
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();
}
Ponieważ w okresie harmonogramu jest łączna liczba num_shifts * num_days zmian, musisz
może przypisać co najmniej (num_shifts * num_days) // num_nurses
na pielęgniarkę, ale pewne zmiany mogą zostać pominięte. (tutaj // to Python
operator dzielenia liczby całkowitej, który zwraca wartość minimalną ilorazu zwykłego).
Dla danych wartości num_nurses = 4, num_shifts = 3 i num_days = 3:
wyrażenie min_shifts_per_nurse ma wartość (3 * 3 // 4) = 2, więc
przypisać do każdej z pielęgniarek co najmniej 2 zmiany. Jest ona określona przez
ograniczenie (tutaj w Pythonie)
model.add(min_shifts_per_nurse <= sum(shifts_worked))
Ponieważ w okresie 3 dni było łącznie dziewięć zmian, oznacza to jedną pozostałej zmiany po przypisaniu 2 zmian do każdej z pielęgniarek. Dodatkowa zmiana może być przypisaną do dowolnej pielęgniarki.
Ostatni wiersz (w tym miejscu w Pythonie)
model.add(sum(shifts_worked) <= max_shifts_per_nurse)
gwarantuje, że żadna pielęgniarka nie otrzyma więcej niż jednej dodatkowej zmiany.
Ograniczenie nie jest w tym przypadku konieczne, ponieważ istnieje tylko jeden dodatkowy Shift. Jednak w przypadku różnych wartości parametrów może wystąpić kilka dodatkowych przesunięć, W takim przypadku ograniczenie jest konieczne.
Aktualizowanie parametrów rozwiązania
W modelu bez optymalizacji możesz włączyć wyszukiwanie wszystkich rozwiązań.
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 ";
Rejestrowanie wywołania zwrotnego rozwiązań
Musisz zarejestrować wywołanie zwrotne dla rozwiązania, które będzie wywoływane za każdym razem rozwiązanie.
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#
Najpierw zdefiniuj klasę 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_;
}
Następnie utwórz jej instancję, używając:
const int solutionLimit = 5; SolutionPrinter cb = new SolutionPrinter(allNurses, allDays, allShifts, shifts, solutionLimit);
Wywołaj rozwiązanie
Następujący kod wywołuje rozwiązanie i wyświetla 5 pierwszych rozwiązań.
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}");
Rozwiązania
Oto pierwsze 5 rozwiązań.
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
Łączna liczba rozwiązań to 5184. Poniższy argument liczenia wyjaśnia, dlaczego.
Dostępne są 4 opcje dla pielęgniarki, która pracuje na dodatkowej zmianie. Po wybraniu pielęgniarki są 3 zmiany, do których może być przypisana każdego z tych 3 dni, więc istnieje wiele sposobów przydzielenia pielęgniarce dodatkowa zmiana to 4 · 33 = 108. Po przypisaniu tej pielęgniarki każdego dnia pozostały 2 nieprzypisane zmiany.
Z pozostałych 3 pielęgniarek 1 dzień pracy 0 i 1 dzień pracy oraz 1 dzień pracy 0 i 2. i drugi dzień roboczy i drugi. Są 3. = 6 sposobów przypisywania pielęgniarek do danego dnia, tak jak w tabeli który znajduje się poniżej. (Te trzy pielęgniarki mają oznaczenia A, B i C. Nie przypisał je do zmian).
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
W każdym wierszu na powyższym diagramie istnieje 23 = 8 sposobów przypisz pozostałe zmiany do pielęgniarek (dwie możliwości wyboru każdego dnia). Łączna liczba możliwych przypisań to 108·6·8 = 5184.
Cały program
Oto cały program dotyczący planowania pracy pielęgniarek.
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");
}
}
Planowanie z prośbami o zmiany
W tej sekcji posłużmy się poprzednim przykładem i dodamy prośby pielęgniarek o określonych zmian. Następnie szukamy harmonogramu, który zmaksymalizuje liczbę spełnionych żądań. W przypadku większości problemów z planowaniem najlepiej jest zoptymalizować funkcję celu, ponieważ wydrukowanie wszystkich możliwych harmonogramów jest zwykle niepraktyczne.
Ten przykład ma te same ograniczenia co poprzedni przykład.
Zaimportuj biblioteki
Poniższy kod importuje wymaganą bibliotekę.
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;
Dane do przykładu
Dane z tego przykładu zostaną wyświetlone poniżej.
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 },
},
};
Tworzenie modelu
Ten model tworzy model.
Python
model = cp_model.CpModel()
C++
CpModelBuilder cp_model;
Java
CpModel model = new CpModel();
C#
CpModel model = new CpModel();
Tworzenie zmiennych
Poniższy kod to tablica zmiennych do rozwiązania zadania.
Oprócz zmiennych z poprzedniego przykładu dane zawierają również zbiór trójek, odpowiadających 3 zmianom w ciągu dnia. Każdy element argumentu wartość potrójna to 0 lub 1, co wskazuje, czy zażądano przesunięcia. Na przykład parametr potrójna wartość [0, 0, 1] na piątej pozycji w wierszu 1 wskazuje, że pielęgniarka 1 prosi o podanie przesunięcie o 3 w dniu 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}"));
}
}
}
Tworzenie ograniczeń
Ograniczenia tego problemu tworzy poniższy kod.
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);
}
Cel przykładu
Chcemy zoptymalizować poniższą funkcję celu.
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));
Ponieważ shift_requests[n][d][s] * shifts[(n, d, s) ma wartość 1, jeśli przypisano przesunięcie s
do pielęgniarki n dnia d oraz ta pielęgniarka poprosiła o zmianę (i 0 w innym przypadku)
cel to zmiana liczby przypisań, które spełniają określone żądanie.
Wywołaj rozwiązanie
Następujący kod wywołuje rozwiązanie.
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}");
Wyświetl wyniki
Poniższy kod wyświetla poniższe dane wyjściowe, które zawierają optymalną wartość (choć może nie jedyny). W wynikach widać, które przesunięcie poproszono o przypisanie oraz liczbę spełnionych próśb.
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.");
}
Po uruchomieniu programu wyświetlą się następujące dane wyjściowe:
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
Cały program
Oto cały program do planowania z prośbami o zmianę.
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");
}
}