Những tổ chức có nhân viên làm việc nhiều ca làm việc cần bố trí đủ nhân viên cho mỗi ca làm việc hằng ngày. Thông thường, lịch biểu sẽ có các điều kiện ràng buộc, chẳng hạn như "không có nhân viên nào nên làm hai ca liên tiếp". Việc tìm lịch biểu đáp ứng tất cả các quy tắc ràng buộc có thể khó tính toán.
Các phần sau đây trình bày 2 ví dụ về các vấn đề liên quan đến việc lập lịch biểu của nhân viên và trình bày cách giải bằng trình giải CP-SAT.
Để biết ví dụ phức tạp hơn, hãy xem chương trình lên lịch thay đổi này trên GitHub.
Vấn đề về lịch trình của một y tá
Trong ví dụ tiếp theo, một giám sát bệnh viện cần tạo lịch biểu cho 4 y tá trong khoảng thời gian 3 ngày, tuỳ theo các điều kiện sau:
- Mỗi ngày được chia thành 3 ca làm việc, mỗi ca làm việc 8 tiếng.
- Mỗi ngày, mỗi ca được chỉ định cho một y tá duy nhất và không có y tá nào làm việc nhiều hơn một ca.
- Mỗi y tá được phân công ít nhất 2 ca làm việc trong thời gian 3 ngày.
Các phần sau đây trình bày giải pháp cho vấn đề lên lịch của y tá.
Nhập thư viện
Mã sau đây nhập thư viện bắt buộc.
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;
Dữ liệu cho ví dụ
Đoạn mã sau đây sẽ tạo dữ liệu cho ví dụ.
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();
Tạo mô hình
Đoạn mã sau đây sẽ tạo mô hình.
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;
Tạo các biến
Đoạn mã sau đây tạo một mảng biến.
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}"));
}
}
}
Mảng này xác định các phép chỉ định ca làm việc cho y tá như sau:
shifts[(n, d, s)] bằng 1 nếu ca làm việc s được chỉ định cho điều dưỡng viên n vào ngày d và 0
nếu không.
Phân công y tá vào ca làm việc
Tiếp theo, chúng tôi sẽ trình bày cách chỉ định y tá làm ca theo những điều kiện ràng buộc sau:
- Mỗi ca làm việc sẽ được chỉ định cho một y tá duy nhất mỗi ngày.
- Mỗi y tá làm việc tối đa một ca mỗi ngày.
Dưới đây là mã tạo ra điều kiện đầu tiên.
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();
}
}
Dòng cuối cùng cho biết rằng đối với mỗi ca làm việc, tổng số y tá được chỉ định cho ca đó là 1.
Tiếp theo, đây là mã yêu cầu mỗi y tá làm việc tối đa một ca mỗi ngày.
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();
}
}
Đối với mỗi y tá, tổng số ca làm việc được chỉ định cho y tá đó tối đa là 1 ("tối đa" vì y tá có thể được nghỉ ngày).
Chỉ định ca làm việc đồng đều
Tiếp theo, chúng tôi sẽ tìm hiểu cách phân công ca làm việc cho các điều dưỡng viên một cách đồng đều nhất có thể. Vì có 9 ca làm việc trong khoảng thời gian 3 ngày, nên chúng ta có thể chỉ định 2 ca làm việc cho mỗi y tá. Sau thời gian này, sẽ còn một ca làm việc và có thể được chỉ định cho bất kỳ y tá nào.
Đoạn mã sau đây đảm bảo rằng mỗi y tá làm việc ít nhất 2 ca trong khoảng thời gian 3 ngày.
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();
}
Vì có tổng cộng num_shifts * num_days lượt chuyển trong khoảng thời gian theo lịch, nên bạn có thể chỉ định ít nhất (num_shifts * num_days) // num_nurses
chuyển cho từng y tá, nhưng một số ca làm việc có thể bị thừa. (Ở đây, // là toán tử chia số nguyên Python, trả về giá trị sàn của thương số thông thường.)
Đối với các giá trị đã cho của num_nurses = 4, num_shifts = 3 và num_days = 3, biểu thức min_shifts_per_nurse có giá trị (3 * 3 // 4) = 2, vì vậy, bạn
có thể chỉ định ít nhất hai ca làm việc cho mỗi y tá. Thuộc tính này được chỉ định theo quy tắc ràng buộc (ở đây trong Python)
model.add(min_shifts_per_nurse <= sum(num_shifts_worked))
Vì có tổng cộng 9 ca làm việc trong khoảng thời gian 3 ngày, nên sẽ còn 1 ca làm việc sau khi chỉ định 2 ca làm việc cho mỗi y tá. Bạn có thể chỉ định thêm ca làm việc cho bất kỳ y tá nào.
Dòng cuối cùng (ở đây bằng Python)
model.add(sum(num_shifts_worked) <= max_shifts_per_nurse)
đảm bảo rằng không có y tá nào được chỉ định nhiều hơn một ca làm việc.
Bạn không cần thiết lập quy tắc ràng buộc trong trường hợp này vì chỉ có thêm một lượt chuyển đổi. Tuy nhiên, đối với các giá trị tham số khác nhau, có thể sẽ có thêm vài thay đổi, trong trường hợp đó, bạn cần phải có quy tắc ràng buộc.
Cập nhật tham số của trình giải
Trong mô hình không tối ưu hoá, bạn có thể bật tính năng tìm kiếm tất cả các giải pháp.
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 ";
Đăng ký Lệnh gọi lại Giải pháp
Bạn cần đăng ký một lệnh gọi lại trên trình giải quyết mà hệ thống sẽ gọi ở mỗi giải pháp.
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#
Trước tiên, hãy xác định lớp 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_;
}
Sau đó, tạo thực thể bằng cách sử dụng:
const int solutionLimit = 5; SolutionPrinter cb = new SolutionPrinter(allNurses, allDays, allShifts, shifts, solutionLimit);
Gọi trình giải
Mã sau đây gọi trình giải toán và hiển thị 5 giải pháp đầu tiên.
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}");
Giải pháp
Sau đây là 5 giải pháp đầu tiên.
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
Tổng số giải pháp là 5184. Đối số tính sau đây giải thích lý do.
Thứ nhất, có 4 lựa chọn cho một y tá làm việc thêm ca. Sau khi đã chọn y tá đó, có thể chỉ định 3 ca cho y tá trong mỗi 3 ngày, vậy nên số cách có thể để chỉ định y tá với ca làm việc thêm là 4 · 33 = 108. Sau khi chỉ định điều dưỡng viên này, mỗi ngày còn có 2 ca làm việc chưa được chỉ định.
Trong số ba y tá còn lại, một y tá làm việc vào ngày 0 và 1, một y tá làm việc vào ngày 0 và 2, và một y tá làm việc vào ngày 1 và 2. Có 3! = 6 cách chỉ định y tá cho những ngày này, như minh hoạ trong sơ đồ dưới đây. (Ba y tá được gắn nhãn A, B và C và chúng tôi chưa chỉ định họ vào ca làm việc.)
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
Đối với mỗi hàng trong sơ đồ trên, có 23 = 8 cách có thể dùng để chỉ định số ca làm việc còn lại cho các y tá (hai lựa chọn mỗi ngày). Vậy tổng số bài tập có thể có là 108·6·8 = 5184.
Toàn bộ chương trình
Đây là toàn bộ chương trình cho vấn đề lên lịch của y tá.
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");
}
}
Lên lịch với yêu cầu ca làm việc
Trong phần này, chúng tôi lấy ví dụ trước và thêm yêu cầu của y tá cho các ca làm việc cụ thể. Sau đó, chúng tôi tìm kiếm một lịch biểu tối đa hóa số lượng yêu cầu được đáp ứng. Đối với hầu hết các vấn đề về việc lập lịch, tốt nhất bạn nên tối ưu hoá hàm mục tiêu vì việc in tất cả lịch biểu có thể không hữu ích.
Ví dụ này có các quy tắc ràng buộc tương tự như ví dụ trước.
Nhập thư viện
Mã sau đây nhập thư viện bắt buộc.
Python
from ortools.sat.python import cp_model
C++
#include <stdlib.h> #include <cstdint> #include <map> #include <numeric> #include <string> #include <tuple> #include <vector> #include "absl/strings/str_format.h" #include "ortools/base/logging.h" #include "ortools/sat/cp_model.h" #include "ortools/sat/cp_model.pb.h" #include "ortools/sat/cp_model_solver.h"
Java
import com.google.ortools.Loader; import com.google.ortools.sat.CpModel; import com.google.ortools.sat.CpSolver; import com.google.ortools.sat.CpSolverStatus; import com.google.ortools.sat.LinearExpr; import com.google.ortools.sat.LinearExprBuilder; import com.google.ortools.sat.Literal; import java.util.ArrayList; import java.util.List; import java.util.stream.IntStream;
C#
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.Sat;
Dữ liệu cho ví dụ
Dữ liệu của ví dụ này sẽ được hiển thị sau đó.
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 },
},
};
Tạo mô hình
Đoạn mã sau đây sẽ tạo mô hình.
Python
model = cp_model.CpModel()
C++
CpModelBuilder cp_model;
Java
CpModel model = new CpModel();
C#
CpModel model = new CpModel();
Tạo các biến
Đoạn mã sau đây cung cấp một mảng các biến cho bài toán.
Ngoài các biến trong ví dụ trước, dữ liệu cũng chứa một tập hợp bộ ba, tương ứng với ba lượt chuyển mỗi ngày. Mỗi phần tử của bộ ba là 0 hoặc 1, cho biết liệu có yêu cầu chuyển hay không. Ví dụ: bộ ba [0, 0, 1] ở vị trí thứ năm của hàng 1 cho biết rằng y tá 1 yêu cầu chuyển 3 vào ngày 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}"));
}
}
}
Tạo các điều kiện ràng buộc
Đoạn mã sau đây tạo các điều kiện ràng buộc cho bài toán này.
Python
for d in all_days:
for s in all_shifts:
model.add_exactly_one(shifts[(n, d, s)] for n in all_nurses)
C++
for (int d : all_days) {
for (int s : all_shifts) {
std::vector<BoolVar> nurses;
for (int n : all_nurses) {
auto key = std::make_tuple(n, d, s);
nurses.push_back(shifts[key]);
}
cp_model.AddExactlyOne(nurses);
}
}
Java
for (int d : allDays) {
for (int s : allShifts) {
List<Literal> nurses = new ArrayList<>();
for (int n : allNurses) {
nurses.add(shifts[n][d][s]);
}
model.addExactlyOne(nurses);
}
}
C#
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
IntVar[] x = new IntVar[numNurses];
foreach (int n in allNurses)
{
var key = Tuple.Create(n, d, s);
x[n] = shifts[key];
}
model.Add(LinearExpr.Sum(x) == 1);
}
}
Python
for n in all_nurses:
for d in all_days:
model.add_at_most_one(shifts[(n, d, s)] for s in all_shifts)
C++
for (int n : all_nurses) {
for (int d : all_days) {
std::vector<BoolVar> work;
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
work.push_back(shifts[key]);
}
cp_model.AddAtMostOne(work);
}
}
Java
for (int n : allNurses) {
for (int d : allDays) {
List<Literal> work = new ArrayList<>();
for (int s : allShifts) {
work.add(shifts[n][d][s]);
}
model.addAtMostOne(work);
}
}
C#
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
IntVar[] x = new IntVar[numShifts];
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
x[s] = shifts[key];
}
model.Add(LinearExpr.Sum(x) <= 1);
}
}
Python
# Try to distribute the shifts evenly, so that each nurse works
# min_shifts_per_nurse shifts. If this is not possible, because the total
# number of shifts is not divisible by the number of nurses, some nurses will
# be assigned one more shift.
min_shifts_per_nurse = (num_shifts * num_days) // num_nurses
if num_shifts * num_days % num_nurses == 0:
max_shifts_per_nurse = min_shifts_per_nurse
else:
max_shifts_per_nurse = min_shifts_per_nurse + 1
for n in all_nurses:
num_shifts_worked = 0
for d in all_days:
for s in all_shifts:
num_shifts_worked += shifts[(n, d, s)]
model.add(min_shifts_per_nurse <= num_shifts_worked)
model.add(num_shifts_worked <= max_shifts_per_nurse)
C++
// Try to distribute the shifts evenly, so that each nurse works
// min_shifts_per_nurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int min_shifts_per_nurse = (num_shifts * num_days) / num_nurses;
int max_shifts_per_nurse;
if ((num_shifts * num_days) % num_nurses == 0) {
max_shifts_per_nurse = min_shifts_per_nurse;
} else {
max_shifts_per_nurse = min_shifts_per_nurse + 1;
}
for (int n : all_nurses) {
LinearExpr num_worked_shifts;
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
num_worked_shifts += shifts[key];
}
}
cp_model.AddLessOrEqual(min_shifts_per_nurse, num_worked_shifts);
cp_model.AddLessOrEqual(num_worked_shifts, max_shifts_per_nurse);
}
Java
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0) {
maxShiftsPerNurse = minShiftsPerNurse;
} else {
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
for (int n : allNurses) {
LinearExprBuilder numShiftsWorked = LinearExpr.newBuilder();
for (int d : allDays) {
for (int s : allShifts) {
numShiftsWorked.add(shifts[n][d][s]);
}
}
model.addLinearConstraint(numShiftsWorked, minShiftsPerNurse, maxShiftsPerNurse);
}
C#
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0)
{
maxShiftsPerNurse = minShiftsPerNurse;
}
else
{
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
foreach (int n in allNurses)
{
IntVar[] numShiftsWorked = new IntVar[numDays * numShifts];
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
numShiftsWorked[d * numShifts + s] = shifts[key];
}
}
model.AddLinearConstraint(LinearExpr.Sum(numShiftsWorked), minShiftsPerNurse, maxShiftsPerNurse);
}
Mục tiêu của ví dụ này
Chúng ta muốn tối ưu hoá hàm mục tiêu sau.
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));
Vì shift_requests[n][d][s] * shifts[(n, d, s) là 1 nếu ca làm việc s được chỉ định cho y tá n vào ngày d và y tá đó yêu cầu ca thay đổi đó (và 0 nếu không), mục tiêu là mức thay đổi số lượng bài tập đáp ứng yêu cầu.
Gọi trình giải
Mã sau đây gọi trình giải toán.
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}");
Hiện kết quả
Mã sau đây hiển thị kết quả sau, có chứa lịch biểu tối ưu (mặc dù có lẽ không phải là lịch duy nhất). Kết quả đầu ra cho biết sự chỉ định ca làm việc nào được yêu cầu và số lượng yêu cầu được đáp ứng.
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.");
}
Khi chạy chương trình, bạn sẽ thấy kết quả sau:
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
Toàn bộ chương trình
Đây là toàn bộ chương trình để lên lịch với yêu cầu ca làm việc.
Python
"""Nurse scheduling problem with shift requests."""
from ortools.sat.python import cp_model
def main() -> None:
# This program tries to find an optimal assignment of nurses to shifts
# (3 shifts per day, for 7 days), subject to some constraints (see below).
# Each nurse can request to be assigned to specific shifts.
# The optimal assignment maximizes the number of fulfilled shift requests.
num_nurses = 5
num_shifts = 3
num_days = 7
all_nurses = range(num_nurses)
all_shifts = range(num_shifts)
all_days = range(num_days)
shift_requests = [
[[0, 0, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 1]],
[[0, 0, 0], [0, 0, 0], [0, 1, 0], [0, 1, 0], [1, 0, 0], [0, 0, 0], [0, 0, 1]],
[[0, 1, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0], [0, 1, 0], [0, 0, 0]],
[[0, 0, 1], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 0, 0]],
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 0]],
]
# Creates the model.
model = cp_model.CpModel()
# Creates shift variables.
# shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
shifts = {}
for n in all_nurses:
for d in all_days:
for s in all_shifts:
shifts[(n, d, s)] = model.new_bool_var(f"shift_n{n}_d{d}_s{s}")
# Each shift is assigned to exactly one nurse in .
for d in all_days:
for s in all_shifts:
model.add_exactly_one(shifts[(n, d, s)] for n in all_nurses)
# Each nurse works at most one shift per day.
for n in all_nurses:
for d in all_days:
model.add_at_most_one(shifts[(n, d, s)] for s in all_shifts)
# Try to distribute the shifts evenly, so that each nurse works
# min_shifts_per_nurse shifts. If this is not possible, because the total
# number of shifts is not divisible by the number of nurses, some nurses will
# be assigned one more shift.
min_shifts_per_nurse = (num_shifts * num_days) // num_nurses
if num_shifts * num_days % num_nurses == 0:
max_shifts_per_nurse = min_shifts_per_nurse
else:
max_shifts_per_nurse = min_shifts_per_nurse + 1
for n in all_nurses:
num_shifts_worked = 0
for d in all_days:
for s in all_shifts:
num_shifts_worked += shifts[(n, d, s)]
model.add(min_shifts_per_nurse <= num_shifts_worked)
model.add(num_shifts_worked <= max_shifts_per_nurse)
model.maximize(
sum(
shift_requests[n][d][s] * shifts[(n, d, s)]
for n in all_nurses
for d in all_days
for s in all_shifts
)
)
# Creates the solver and solve.
solver = cp_model.CpSolver()
status = solver.solve(model)
if status == cp_model.OPTIMAL:
print("Solution:")
for d in all_days:
print("Day", d)
for n in all_nurses:
for s in all_shifts:
if solver.value(shifts[(n, d, s)]) == 1:
if shift_requests[n][d][s] == 1:
print("Nurse", n, "works shift", s, "(requested).")
else:
print("Nurse", n, "works shift", s, "(not requested).")
print()
print(
f"Number of shift requests met = {solver.objective_value}",
f"(out of {num_nurses * min_shifts_per_nurse})",
)
else:
print("No optimal solution found !")
# Statistics.
print("\nStatistics")
print(f" - conflicts: {solver.num_conflicts}")
print(f" - branches : {solver.num_branches}")
print(f" - wall time: {solver.wall_time}s")
if __name__ == "__main__":
main()
C++
// Nurse scheduling problem with shift requests.
#include <stdlib.h>
#include <cstdint>
#include <map>
#include <numeric>
#include <string>
#include <tuple>
#include <vector>
#include "absl/strings/str_format.h"
#include "ortools/base/logging.h"
#include "ortools/sat/cp_model.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_solver.h"
namespace operations_research {
namespace sat {
void ScheduleRequestsSat() {
const int num_nurses = 5;
const int num_days = 7;
const int num_shifts = 3;
std::vector<int> all_nurses(num_nurses);
std::iota(all_nurses.begin(), all_nurses.end(), 0);
std::vector<int> all_days(num_days);
std::iota(all_days.begin(), all_days.end(), 0);
std::vector<int> all_shifts(num_shifts);
std::iota(all_shifts.begin(), all_shifts.end(), 0);
std::vector<std::vector<std::vector<int64_t>>> shift_requests = {
{
{0, 0, 1},
{0, 0, 0},
{0, 0, 0},
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 1},
},
{
{0, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 1, 0},
{1, 0, 0},
{0, 0, 0},
{0, 0, 1},
},
{
{0, 1, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
{
{0, 0, 1},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
},
{
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
};
// Creates the model.
CpModelBuilder cp_model;
// Creates shift variables.
// shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
std::map<std::tuple<int, int, int>, BoolVar> shifts;
for (int n : all_nurses) {
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
shifts[key] = cp_model.NewBoolVar().WithName(
absl::StrFormat("shift_n%dd%ds%d", n, d, s));
}
}
}
// Each shift is assigned to exactly one nurse in the schedule period.
for (int d : all_days) {
for (int s : all_shifts) {
std::vector<BoolVar> nurses;
for (int n : all_nurses) {
auto key = std::make_tuple(n, d, s);
nurses.push_back(shifts[key]);
}
cp_model.AddExactlyOne(nurses);
}
}
// Each nurse works at most one shift per day.
for (int n : all_nurses) {
for (int d : all_days) {
std::vector<BoolVar> work;
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
work.push_back(shifts[key]);
}
cp_model.AddAtMostOne(work);
}
}
// Try to distribute the shifts evenly, so that each nurse works
// min_shifts_per_nurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int min_shifts_per_nurse = (num_shifts * num_days) / num_nurses;
int max_shifts_per_nurse;
if ((num_shifts * num_days) % num_nurses == 0) {
max_shifts_per_nurse = min_shifts_per_nurse;
} else {
max_shifts_per_nurse = min_shifts_per_nurse + 1;
}
for (int n : all_nurses) {
LinearExpr num_worked_shifts;
for (int d : all_days) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
num_worked_shifts += shifts[key];
}
}
cp_model.AddLessOrEqual(min_shifts_per_nurse, num_worked_shifts);
cp_model.AddLessOrEqual(num_worked_shifts, max_shifts_per_nurse);
}
LinearExpr objective_expr;
for (int n : all_nurses) {
for (int d : all_days) {
for (int s : all_shifts) {
if (shift_requests[n][d][s] == 1) {
auto key = std::make_tuple(n, d, s);
objective_expr += shifts[key] * shift_requests[n][d][s];
}
}
}
}
cp_model.Maximize(objective_expr);
const CpSolverResponse response = Solve(cp_model.Build());
if (response.status() == CpSolverStatus::OPTIMAL) {
LOG(INFO) << "Solution:";
for (int d : all_days) {
LOG(INFO) << "Day " << std::to_string(d);
for (int n : all_nurses) {
for (int s : all_shifts) {
auto key = std::make_tuple(n, d, s);
if (SolutionIntegerValue(response, shifts[key]) == 1) {
if (shift_requests[n][d][s] == 1) {
LOG(INFO) << " Nurse " << std::to_string(n) << " works shift "
<< std::to_string(s) << " (requested).";
} else {
LOG(INFO) << " Nurse " << std::to_string(n) << " works shift "
<< std::to_string(s) << " (not requested).";
}
}
}
}
LOG(INFO) << "";
}
LOG(INFO) << "Number of shift requests met = " << response.objective_value()
<< " (out of " << num_nurses * min_shifts_per_nurse << ")";
} else {
LOG(INFO) << "No optimal solution found !";
}
// Statistics.
LOG(INFO) << "Statistics";
LOG(INFO) << CpSolverResponseStats(response);
}
} // namespace sat
} // namespace operations_research
int main() {
operations_research::sat::ScheduleRequestsSat();
return EXIT_SUCCESS;
}
Java
package com.google.ortools.sat.samples;
import com.google.ortools.Loader;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverStatus;
import com.google.ortools.sat.LinearExpr;
import com.google.ortools.sat.LinearExprBuilder;
import com.google.ortools.sat.Literal;
import java.util.ArrayList;
import java.util.List;
import java.util.stream.IntStream;
/** Nurses problem with schedule requests. */
public class ScheduleRequestsSat {
public static void main(String[] args) {
Loader.loadNativeLibraries();
final int numNurses = 5;
final int numDays = 7;
final int numShifts = 3;
final int[] allNurses = IntStream.range(0, numNurses).toArray();
final int[] allDays = IntStream.range(0, numDays).toArray();
final int[] allShifts = IntStream.range(0, numShifts).toArray();
final int[][][] shiftRequests = new int[][][] {
{
{0, 0, 1},
{0, 0, 0},
{0, 0, 0},
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 1},
},
{
{0, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 1, 0},
{1, 0, 0},
{0, 0, 0},
{0, 0, 1},
},
{
{0, 1, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
{
{0, 0, 1},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 0, 0},
},
{
{0, 0, 0},
{0, 0, 1},
{0, 1, 0},
{0, 0, 0},
{1, 0, 0},
{0, 1, 0},
{0, 0, 0},
},
};
// Creates the model.
CpModel model = new CpModel();
// Creates shift variables.
// shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
Literal[][][] shifts = new Literal[numNurses][numDays][numShifts];
for (int n : allNurses) {
for (int d : allDays) {
for (int s : allShifts) {
shifts[n][d][s] = model.newBoolVar("shifts_n" + n + "d" + d + "s" + s);
}
}
}
// Each shift is assigned to exactly one nurse in the schedule period.
for (int d : allDays) {
for (int s : allShifts) {
List<Literal> nurses = new ArrayList<>();
for (int n : allNurses) {
nurses.add(shifts[n][d][s]);
}
model.addExactlyOne(nurses);
}
}
// Each nurse works at most one shift per day.
for (int n : allNurses) {
for (int d : allDays) {
List<Literal> work = new ArrayList<>();
for (int s : allShifts) {
work.add(shifts[n][d][s]);
}
model.addAtMostOne(work);
}
}
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0) {
maxShiftsPerNurse = minShiftsPerNurse;
} else {
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
for (int n : allNurses) {
LinearExprBuilder numShiftsWorked = LinearExpr.newBuilder();
for (int d : allDays) {
for (int s : allShifts) {
numShiftsWorked.add(shifts[n][d][s]);
}
}
model.addLinearConstraint(numShiftsWorked, minShiftsPerNurse, maxShiftsPerNurse);
}
LinearExprBuilder obj = LinearExpr.newBuilder();
for (int n : allNurses) {
for (int d : allDays) {
for (int s : allShifts) {
obj.addTerm(shifts[n][d][s], shiftRequests[n][d][s]);
}
}
}
model.maximize(obj);
// Creates a solver and solves the model.
CpSolver solver = new CpSolver();
CpSolverStatus status = solver.solve(model);
if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) {
System.out.printf("Solution:%n");
for (int d : allDays) {
System.out.printf("Day %d%n", d);
for (int n : allNurses) {
for (int s : allShifts) {
if (solver.booleanValue(shifts[n][d][s])) {
if (shiftRequests[n][d][s] == 1) {
System.out.printf(" Nurse %d works shift %d (requested).%n", n, s);
} else {
System.out.printf(" Nurse %d works shift %d (not requested).%n", n, s);
}
}
}
}
}
System.out.printf("Number of shift requests met = %f (out of %d)%n", solver.objectiveValue(),
numNurses * minShiftsPerNurse);
} else {
System.out.printf("No optimal solution found !");
}
// Statistics.
System.out.println("Statistics");
System.out.printf(" conflicts: %d%n", solver.numConflicts());
System.out.printf(" branches : %d%n", solver.numBranches());
System.out.printf(" wall time: %f s%n", solver.wallTime());
}
private ScheduleRequestsSat() {}
}
C#
using System;
using System.Collections.Generic;
using System.Linq;
using Google.OrTools.Sat;
public class ScheduleRequestsSat
{
public static void Main(String[] args)
{
const int numNurses = 5;
const int numDays = 7;
const int numShifts = 3;
int[] allNurses = Enumerable.Range(0, numNurses).ToArray();
int[] allDays = Enumerable.Range(0, numDays).ToArray();
int[] allShifts = Enumerable.Range(0, numShifts).ToArray();
int[,,] shiftRequests = new int[,,] {
{
{ 0, 0, 1 },
{ 0, 0, 0 },
{ 0, 0, 0 },
{ 0, 0, 0 },
{ 0, 0, 1 },
{ 0, 1, 0 },
{ 0, 0, 1 },
},
{
{ 0, 0, 0 },
{ 0, 0, 0 },
{ 0, 1, 0 },
{ 0, 1, 0 },
{ 1, 0, 0 },
{ 0, 0, 0 },
{ 0, 0, 1 },
},
{
{ 0, 1, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 0, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
},
{
{ 0, 0, 1 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 0, 0 },
},
{
{ 0, 0, 0 },
{ 0, 0, 1 },
{ 0, 1, 0 },
{ 0, 0, 0 },
{ 1, 0, 0 },
{ 0, 1, 0 },
{ 0, 0, 0 },
},
};
// Creates the model.
CpModel model = new CpModel();
// Creates shift variables.
// shifts[(n, d, s)]: nurse 'n' works shift 's' on day 'd'.
Dictionary<Tuple<int, int, int>, IntVar> shifts = new Dictionary<Tuple<int, int, int>, IntVar>();
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
shifts.Add(Tuple.Create(n, d, s), model.NewBoolVar($"shifts_n{n}d{d}s{s}"));
}
}
}
// Each shift is assigned to exactly one nurse in the schedule period.
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
IntVar[] x = new IntVar[numNurses];
foreach (int n in allNurses)
{
var key = Tuple.Create(n, d, s);
x[n] = shifts[key];
}
model.Add(LinearExpr.Sum(x) == 1);
}
}
// Each nurse works at most one shift per day.
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
IntVar[] x = new IntVar[numShifts];
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
x[s] = shifts[key];
}
model.Add(LinearExpr.Sum(x) <= 1);
}
}
// Try to distribute the shifts evenly, so that each nurse works
// minShiftsPerNurse shifts. If this is not possible, because the total
// number of shifts is not divisible by the number of nurses, some nurses will
// be assigned one more shift.
int minShiftsPerNurse = (numShifts * numDays) / numNurses;
int maxShiftsPerNurse;
if ((numShifts * numDays) % numNurses == 0)
{
maxShiftsPerNurse = minShiftsPerNurse;
}
else
{
maxShiftsPerNurse = minShiftsPerNurse + 1;
}
foreach (int n in allNurses)
{
IntVar[] numShiftsWorked = new IntVar[numDays * numShifts];
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
numShiftsWorked[d * numShifts + s] = shifts[key];
}
}
model.AddLinearConstraint(LinearExpr.Sum(numShiftsWorked), minShiftsPerNurse, maxShiftsPerNurse);
}
IntVar[] flatShifts = new IntVar[numNurses * numDays * numShifts];
int[] flatShiftRequests = new int[numNurses * numDays * numShifts];
foreach (int n in allNurses)
{
foreach (int d in allDays)
{
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
flatShifts[n * numDays * numShifts + d * numShifts + s] = shifts[key];
flatShiftRequests[n * numDays * numShifts + d * numShifts + s] = shiftRequests[n, d, s];
}
}
}
model.Maximize(LinearExpr.WeightedSum(flatShifts, flatShiftRequests));
// Solve
CpSolver solver = new CpSolver();
CpSolverStatus status = solver.Solve(model);
Console.WriteLine($"Solve status: {status}");
if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible)
{
Console.WriteLine("Solution:");
foreach (int d in allDays)
{
Console.WriteLine($"Day {d}");
foreach (int n in allNurses)
{
bool isWorking = false;
foreach (int s in allShifts)
{
var key = Tuple.Create(n, d, s);
if (solver.Value(shifts[key]) == 1L)
{
if (shiftRequests[n, d, s] == 1)
{
Console.WriteLine($" Nurse {n} work shift {s} (requested).");
}
else
{
Console.WriteLine($" Nurse {n} work shift {s} (not requested).");
}
}
}
}
}
Console.WriteLine(
$"Number of shift requests met = {solver.ObjectiveValue} (out of {numNurses * minShiftsPerNurse}).");
}
else
{
Console.WriteLine("No solution found.");
}
Console.WriteLine("Statistics");
Console.WriteLine($" conflicts: {solver.NumConflicts()}");
Console.WriteLine($" branches : {solver.NumBranches()}");
Console.WriteLine($" wall time: {solver.WallTime()}s");
}
}