يصف هذا القسم مشكلة مهمة دراسية لم يتم السماح فيها سوى يمكن تعيين مجموعات من العمال للمهام. في المثال، يوجد اثني عشر العاملين، المرقم من 0 إلى 11. المجموعات المسموح بها هي مجموعات من أزواج العمّال التالية.
group1 = [[2, 3], # Subgroups of workers 0 - 3 [1, 3], [1, 2], [0, 1], [0, 2]]group2 = [[6, 7], # Subgroups of workers 4 - 7 [5, 7], [5, 6], [4, 5], [4, 7]]
group3 = [[10, 11], # Subgroups of workers 8 - 11 [9, 11], [9, 10], [8, 10], [8, 11]]
يمكن أن تكون المجموعة المسموح بها أي مجموعة من ثلاثة أزواج من العاملين، أي زوج واحد من
لكل من المجموعة 1 والمجموعة 2 والمجموعة 3.
على سبيل المثال، يؤدي الجمع بين [2, 3] و[6, 7] و[10, 11] إلى تغيير القيم المسموح بها،
المجموعة [2, 3, 6, 7, 10, 11].
ولأن كل مجموعة من المجموعات الثلاث تحتوي على خمسة عناصر، فإن العدد الإجمالي المسموح به
المجموعات 5 * 5 * 5 = 125.
لاحظ أن مجموعة من العاملين يمكن أن تكون حلاً للمشكلة إذا كانت تنتمي إلى لأي مجموعة من المجموعات المسموح بها. أو بعبارةٍ أخرى، تتكون المجموعة الممكنة من والنقاط التي تتم تلبية أي من القيود بها. هذا مثال على مشكلة غير محدبة. على النقيض من ذلك، يعرض مثال MIP، الموضح سابقًا، يمثل مشكلة محدبة: لكي تكون النقطة ممكنة، يتم التعامل مع جميع القيود إلى الرضا.
بالنسبة إلى المسائل غير المحدبة مثل هذه، عادة ما تكون أداة حل CP-SAT أسرع من أداة حل MIP. تعرض الأقسام التالية حلولاً للمشكلة باستخدام أداة حلّ CP-SAT. وأداة حل MIP، ومقارنة أوقات الحل لكلتيهما.
حل CP-SAT
أولاً، سنصف حلاً للمشكلة باستخدام أداة حل CP-SAT.
استيراد المكتبات
يستورد الرمز التالي المكتبة المطلوبة.
Python
from ortools.sat.python import cp_model
C++
#include <stdlib.h> #include <cstdint> #include <numeric> #include <vector> #include "absl/strings/str_format.h" #include "absl/types/span.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.IntVar; 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;
تعريف البيانات
ينشئ الرمز التالي البيانات للبرنامج.
Python
costs = [
[90, 76, 75, 70, 50, 74],
[35, 85, 55, 65, 48, 101],
[125, 95, 90, 105, 59, 120],
[45, 110, 95, 115, 104, 83],
[60, 105, 80, 75, 59, 62],
[45, 65, 110, 95, 47, 31],
[38, 51, 107, 41, 69, 99],
[47, 85, 57, 71, 92, 77],
[39, 63, 97, 49, 118, 56],
[47, 101, 71, 60, 88, 109],
[17, 39, 103, 64, 61, 92],
[101, 45, 83, 59, 92, 27],
]
num_workers = len(costs)
num_tasks = len(costs[0])C++
const std::vector<std::vector<int>> costs = {{ {{90, 76, 75, 70, 50, 74}}, {{35, 85, 55, 65, 48, 101}}, {{125, 95, 90, 105, 59, 120}}, {{45, 110, 95, 115, 104, 83}}, {{60, 105, 80, 75, 59, 62}}, {{45, 65, 110, 95, 47, 31}}, {{38, 51, 107, 41, 69, 99}}, {{47, 85, 57, 71, 92, 77}}, {{39, 63, 97, 49, 118, 56}}, {{47, 101, 71, 60, 88, 109}}, {{17, 39, 103, 64, 61, 92}}, {{101, 45, 83, 59, 92, 27}}, }}; const int num_workers = static_cast<int>(costs.size()); std::vector<int> all_workers(num_workers); std::iota(all_workers.begin(), all_workers.end(), 0); const int num_tasks = static_cast<int>(costs[0].size()); std::vector<int> all_tasks(num_tasks); std::iota(all_tasks.begin(), all_tasks.end(), 0);
Java
int[][] costs = {
{90, 76, 75, 70, 50, 74},
{35, 85, 55, 65, 48, 101},
{125, 95, 90, 105, 59, 120},
{45, 110, 95, 115, 104, 83},
{60, 105, 80, 75, 59, 62},
{45, 65, 110, 95, 47, 31},
{38, 51, 107, 41, 69, 99},
{47, 85, 57, 71, 92, 77},
{39, 63, 97, 49, 118, 56},
{47, 101, 71, 60, 88, 109},
{17, 39, 103, 64, 61, 92},
{101, 45, 83, 59, 92, 27},
};
final int numWorkers = costs.length;
final int numTasks = costs[0].length;
final int[] allWorkers = IntStream.range(0, numWorkers).toArray();
final int[] allTasks = IntStream.range(0, numTasks).toArray();#C
int[,] costs = {
{ 90, 76, 75, 70, 50, 74 }, { 35, 85, 55, 65, 48, 101 }, { 125, 95, 90, 105, 59, 120 },
{ 45, 110, 95, 115, 104, 83 }, { 60, 105, 80, 75, 59, 62 }, { 45, 65, 110, 95, 47, 31 },
{ 38, 51, 107, 41, 69, 99 }, { 47, 85, 57, 71, 92, 77 }, { 39, 63, 97, 49, 118, 56 },
{ 47, 101, 71, 60, 88, 109 }, { 17, 39, 103, 64, 61, 92 }, { 101, 45, 83, 59, 92, 27 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);
int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray();
int[] allTasks = Enumerable.Range(0, numTasks).ToArray();إنشاء المجموعات المسموح بها
لتحديد مجموعات العمال المسموح بها لأداة حل CP-SAT، يمكنك إنشاء النظام الثنائي
الصفائف التي تشير إلى العاملين الذين ينتمون إلى مجموعة ما. على سبيل المثال: group1
(العاملون 0 - 3)، يحدد المتجه الثنائي [0, 0, 1, 1] المجموعة التي تحتوي
العاملين 2 و3.
تُحدِّد الصفائف التالية مجموعات العمال المسموح بها.
Python
group1 = [
[0, 0, 1, 1], # Workers 2, 3
[0, 1, 0, 1], # Workers 1, 3
[0, 1, 1, 0], # Workers 1, 2
[1, 1, 0, 0], # Workers 0, 1
[1, 0, 1, 0], # Workers 0, 2
]
group2 = [
[0, 0, 1, 1], # Workers 6, 7
[0, 1, 0, 1], # Workers 5, 7
[0, 1, 1, 0], # Workers 5, 6
[1, 1, 0, 0], # Workers 4, 5
[1, 0, 0, 1], # Workers 4, 7
]
group3 = [
[0, 0, 1, 1], # Workers 10, 11
[0, 1, 0, 1], # Workers 9, 11
[0, 1, 1, 0], # Workers 9, 10
[1, 0, 1, 0], # Workers 8, 10
[1, 0, 0, 1], # Workers 8, 11
]C++
const std::vector<std::vector<int64_t>> group1 = {{ {{0, 0, 1, 1}}, // Workers 2, 3 {{0, 1, 0, 1}}, // Workers 1, 3 {{0, 1, 1, 0}}, // Workers 1, 2 {{1, 1, 0, 0}}, // Workers 0, 1 {{1, 0, 1, 0}}, // Workers 0, 2 }}; const std::vector<std::vector<int64_t>> group2 = {{ {{0, 0, 1, 1}}, // Workers 6, 7 {{0, 1, 0, 1}}, // Workers 5, 7 {{0, 1, 1, 0}}, // Workers 5, 6 {{1, 1, 0, 0}}, // Workers 4, 5 {{1, 0, 0, 1}}, // Workers 4, 7 }}; const std::vector<std::vector<int64_t>> group3 = {{ {{0, 0, 1, 1}}, // Workers 10, 11 {{0, 1, 0, 1}}, // Workers 9, 11 {{0, 1, 1, 0}}, // Workers 9, 10 {{1, 0, 1, 0}}, // Workers 8, 10 {{1, 0, 0, 1}}, // Workers 8, 11 }};
Java
int[][] group1 = {
{0, 0, 1, 1}, // Workers 2, 3
{0, 1, 0, 1}, // Workers 1, 3
{0, 1, 1, 0}, // Workers 1, 2
{1, 1, 0, 0}, // Workers 0, 1
{1, 0, 1, 0}, // Workers 0, 2
};
int[][] group2 = {
{0, 0, 1, 1}, // Workers 6, 7
{0, 1, 0, 1}, // Workers 5, 7
{0, 1, 1, 0}, // Workers 5, 6
{1, 1, 0, 0}, // Workers 4, 5
{1, 0, 0, 1}, // Workers 4, 7
};
int[][] group3 = {
{0, 0, 1, 1}, // Workers 10, 11
{0, 1, 0, 1}, // Workers 9, 11
{0, 1, 1, 0}, // Workers 9, 10
{1, 0, 1, 0}, // Workers 8, 10
{1, 0, 0, 1}, // Workers 8, 11
};#C
long[,] group1 = {
{ 0, 0, 1, 1 }, // Workers 2, 3
{ 0, 1, 0, 1 }, // Workers 1, 3
{ 0, 1, 1, 0 }, // Workers 1, 2
{ 1, 1, 0, 0 }, // Workers 0, 1
{ 1, 0, 1, 0 }, // Workers 0, 2
};
long[,] group2 = {
{ 0, 0, 1, 1 }, // Workers 6, 7
{ 0, 1, 0, 1 }, // Workers 5, 7
{ 0, 1, 1, 0 }, // Workers 5, 6
{ 1, 1, 0, 0 }, // Workers 4, 5
{ 1, 0, 0, 1 }, // Workers 4, 7
};
long[,] group3 = {
{ 0, 0, 1, 1 }, // Workers 10, 11
{ 0, 1, 0, 1 }, // Workers 9, 11
{ 0, 1, 1, 0 }, // Workers 9, 10
{ 1, 0, 1, 0 }, // Workers 8, 10
{ 1, 0, 0, 1 }, // Workers 8, 11
};بالنسبة لـ CP-SAT ليس من الضروري إنشاء كافة الـ 125 مجموعة من هذه المتجهات
بشكل متكرر. توفّر أداة حلّ CP-SAT الطريقة AllowedAssignments،
والتي تمكّنك من تحديد القيود للمجموعات المسموح بها بشكل منفصل
لكل مجموعة من مجموعات العاملين الثلاث (0 - 3، و4 - 7، و8 - 11).
إليك آلية العمل:
Python
# Create variables for each worker, indicating whether they work on some task. work = {} for worker in range(num_workers): work[worker] = model.new_bool_var(f"work[{worker}]") for worker in range(num_workers): for task in range(num_tasks): model.add(work[worker] == sum(x[worker, task] for task in range(num_tasks))) # Define the allowed groups of worders model.add_allowed_assignments([work[0], work[1], work[2], work[3]], group1) model.add_allowed_assignments([work[4], work[5], work[6], work[7]], group2) model.add_allowed_assignments([work[8], work[9], work[10], work[11]], group3)
C++
// Create variables for each worker, indicating whether they work on some // task. std::vector<IntVar> work(num_workers); for (int worker : all_workers) { work[worker] = IntVar( cp_model.NewBoolVar().WithName(absl::StrFormat("work[%d]", worker))); } for (int worker : all_workers) { LinearExpr task_sum; for (int task : all_tasks) { task_sum += x[worker][task]; } cp_model.AddEquality(work[worker], task_sum); } // Define the allowed groups of worders auto table1 = cp_model.AddAllowedAssignments({work[0], work[1], work[2], work[3]}); for (const auto& t : group1) { table1.AddTuple(t); } auto table2 = cp_model.AddAllowedAssignments({work[4], work[5], work[6], work[7]}); for (const auto& t : group2) { table2.AddTuple(t); } auto table3 = cp_model.AddAllowedAssignments({work[8], work[9], work[10], work[11]}); for (const auto& t : group3) { table3.AddTuple(t); }
Java
// Create variables for each worker, indicating whether they work on some task. IntVar[] work = new IntVar[numWorkers]; for (int worker : allWorkers) { work[worker] = model.newBoolVar("work[" + worker + "]"); } for (int worker : allWorkers) { LinearExprBuilder expr = LinearExpr.newBuilder(); for (int task : allTasks) { expr.add(x[worker][task]); } model.addEquality(work[worker], expr); } // Define the allowed groups of worders model.addAllowedAssignments(new IntVar[] {work[0], work[1], work[2], work[3]}) .addTuples(group1); model.addAllowedAssignments(new IntVar[] {work[4], work[5], work[6], work[7]}) .addTuples(group2); model.addAllowedAssignments(new IntVar[] {work[8], work[9], work[10], work[11]}) .addTuples(group3);
#C
// Create variables for each worker, indicating whether they work on some task. BoolVar[] work = new BoolVar[numWorkers]; foreach (int worker in allWorkers) { work[worker] = model.NewBoolVar($"work[{worker}]"); } foreach (int worker in allWorkers) { List<ILiteral> tasks = new List<ILiteral>(); foreach (int task in allTasks) { tasks.Add(x[worker, task]); } model.Add(work[worker] == LinearExpr.Sum(tasks)); } // Define the allowed groups of worders model.AddAllowedAssignments(new IntVar[] { work[0], work[1], work[2], work[3] }).AddTuples(group1); model.AddAllowedAssignments(new IntVar[] { work[4], work[5], work[6], work[7] }).AddTuples(group2); model.AddAllowedAssignments(new IntVar[] { work[8], work[9], work[10], work[11] }).AddTuples(group3);
المتغيرات work[i] هي 0-1 من المتغيرات التي تشير إلى حالة العمل أو
لكل عامل. أي أنّ work[i] تساوي 1 إذا تم تكليف موظف بمهمة ما،
0 في الحالات الأخرى. الخط
solver.Add(solver.AllowedAssignments([work[0], work[1], work[2], work[3]], group1))
القيد القائل بأن حالة العمل للعاملين 0 - 3 يجب أن تتطابق مع أحد
الأنماط في group1. يمكنك الاطّلاع على التفاصيل الكاملة للرمز في الخطوة التالية.
.
إنشاء النموذج
تُنشئ التعليمة البرمجية التالية النموذج.
Python
model = cp_model.CpModel()
C++
CpModelBuilder cp_model;
Java
CpModel model = new CpModel();
#C
CpModel model = new CpModel();
إنشاء المتغيّرات
تُنشئ التعليمة البرمجية التالية صفيفًا من المتغيرات للمشكلة.
Python
x = {} for worker in range(num_workers): for task in range(num_tasks): x[worker, task] = model.new_bool_var(f"x[{worker},{task}]")
C++
// x[i][j] is an array of Boolean variables. x[i][j] is true // if worker i is assigned to task j. std::vector<std::vector<BoolVar>> x(num_workers, std::vector<BoolVar>(num_tasks)); for (int worker : all_workers) { for (int task : all_tasks) { x[worker][task] = cp_model.NewBoolVar().WithName( absl::StrFormat("x[%d,%d]", worker, task)); } }
Java
Literal[][] x = new Literal[numWorkers][numTasks]; for (int worker : allWorkers) { for (int task : allTasks) { x[worker][task] = model.newBoolVar("x[" + worker + "," + task + "]"); } }
#C
BoolVar[,] x = new BoolVar[numWorkers, numTasks];
// Variables in a 1-dim array.
foreach (int worker in allWorkers)
{
foreach (int task in allTasks)
{
x[worker, task] = model.NewBoolVar($"x[{worker},{task}]");
}
}إضافة القيود
تضع التعليمة البرمجية التالية قيودًا للبرنامج.
Python
# Each worker is assigned to at most one task. for worker in range(num_workers): model.add_at_most_one(x[worker, task] for task in range(num_tasks)) # Each task is assigned to exactly one worker. for task in range(num_tasks): model.add_exactly_one(x[worker, task] for worker in range(num_workers))
C++
// Each worker is assigned to at most one task. for (int worker : all_workers) { cp_model.AddAtMostOne(x[worker]); } // Each task is assigned to exactly one worker. for (int task : all_tasks) { std::vector<BoolVar> tasks; for (int worker : all_workers) { tasks.push_back(x[worker][task]); } cp_model.AddExactlyOne(tasks); }
Java
// Each worker is assigned to at most one task. for (int worker : allWorkers) { List<Literal> tasks = new ArrayList<>(); for (int task : allTasks) { tasks.add(x[worker][task]); } model.addAtMostOne(tasks); } // Each task is assigned to exactly one worker. for (int task : allTasks) { List<Literal> workers = new ArrayList<>(); for (int worker : allWorkers) { workers.add(x[worker][task]); } model.addExactlyOne(workers); }
#C
// Each worker is assigned to at most one task. foreach (int worker in allWorkers) { List<ILiteral> tasks = new List<ILiteral>(); foreach (int task in allTasks) { tasks.Add(x[worker, task]); } model.AddAtMostOne(tasks); } // Each task is assigned to exactly one worker. foreach (int task in allTasks) { List<ILiteral> workers = new List<ILiteral>(); foreach (int worker in allWorkers) { workers.Add(x[worker, task]); } model.AddExactlyOne(workers); }
إنشاء الهدف
تُنشئ التعليمة البرمجية التالية الدالة الموضوعية.
Python
objective_terms = [] for worker in range(num_workers): for task in range(num_tasks): objective_terms.append(costs[worker][task] * x[worker, task]) model.minimize(sum(objective_terms))
C++
LinearExpr total_cost; for (int worker : all_workers) { for (int task : all_tasks) { total_cost += x[worker][task] * costs[worker][task]; } } cp_model.Minimize(total_cost);
Java
LinearExprBuilder obj = LinearExpr.newBuilder(); for (int worker : allWorkers) { for (int task : allTasks) { obj.addTerm(x[worker][task], costs[worker][task]); } } model.minimize(obj);
#C
LinearExprBuilder obj = LinearExpr.NewBuilder();
foreach (int worker in allWorkers)
{
foreach (int task in allTasks)
{
obj.AddTerm(x[worker, task], costs[worker, task]);
}
}
model.Minimize(obj);استدعاء أداة الحلّ
يستدعي الرمز التالي أداة الحلّ ويعرض النتائج.
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($"Solve status: {status}");
عرض النتائج
الآن، يمكننا طباعة الحل.
Python
if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE: print(f"Total cost = {solver.objective_value}\n") for worker in range(num_workers): for task in range(num_tasks): if solver.boolean_value(x[worker, task]): print( f"Worker {worker} assigned to task {task}." + f" Cost = {costs[worker][task]}" ) else: print("No solution found.")
C++
if (response.status() == CpSolverStatus::INFEASIBLE) { LOG(FATAL) << "No solution found."; } LOG(INFO) << "Total cost: " << response.objective_value(); LOG(INFO); for (int worker : all_workers) { for (int task : all_tasks) { if (SolutionBooleanValue(response, x[worker][task])) { LOG(INFO) << "Worker " << worker << " assigned to task " << task << ". Cost: " << costs[worker][task]; } } }
Java
// Check that the problem has a feasible solution. if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) { System.out.println("Total cost: " + solver.objectiveValue() + "\n"); for (int worker : allWorkers) { for (int task : allTasks) { if (solver.booleanValue(x[worker][task])) { System.out.println("Worker " + worker + " assigned to task " + task + ". Cost: " + costs[worker][task]); } } } } else { System.err.println("No solution found."); }
#C
// Check that the problem has a feasible solution. if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible) { Console.WriteLine($"Total cost: {solver.ObjectiveValue}\n"); foreach (int worker in allWorkers) { foreach (int task in allTasks) { if (solver.Value(x[worker, task]) > 0.5) { Console.WriteLine($"Worker {worker} assigned to task {task}. " + $"Cost: {costs[worker, task]}"); } } } } else { Console.WriteLine("No solution found."); }
في ما يلي نتيجة البرنامج.
Minimum cost = 239 Worker 0 assigned to task 4 Cost = 50 Worker 1 assigned to task 2 Cost = 55 Worker 5 assigned to task 5 Cost = 31 Worker 6 assigned to task 3 Cost = 41 Worker 10 assigned to task 0 Cost = 17 Worker 11 assigned to task 1 Cost = 45 Time = 0.0113 seconds
البرنامج بأكمله
إليك البرنامج بالكامل.
Python
"""Solves an assignment problem for given group of workers.""" from ortools.sat.python import cp_model def main() -> None: # Data costs = [ [90, 76, 75, 70, 50, 74], [35, 85, 55, 65, 48, 101], [125, 95, 90, 105, 59, 120], [45, 110, 95, 115, 104, 83], [60, 105, 80, 75, 59, 62], [45, 65, 110, 95, 47, 31], [38, 51, 107, 41, 69, 99], [47, 85, 57, 71, 92, 77], [39, 63, 97, 49, 118, 56], [47, 101, 71, 60, 88, 109], [17, 39, 103, 64, 61, 92], [101, 45, 83, 59, 92, 27], ] num_workers = len(costs) num_tasks = len(costs[0]) # Allowed groups of workers: group1 = [ [0, 0, 1, 1], # Workers 2, 3 [0, 1, 0, 1], # Workers 1, 3 [0, 1, 1, 0], # Workers 1, 2 [1, 1, 0, 0], # Workers 0, 1 [1, 0, 1, 0], # Workers 0, 2 ] group2 = [ [0, 0, 1, 1], # Workers 6, 7 [0, 1, 0, 1], # Workers 5, 7 [0, 1, 1, 0], # Workers 5, 6 [1, 1, 0, 0], # Workers 4, 5 [1, 0, 0, 1], # Workers 4, 7 ] group3 = [ [0, 0, 1, 1], # Workers 10, 11 [0, 1, 0, 1], # Workers 9, 11 [0, 1, 1, 0], # Workers 9, 10 [1, 0, 1, 0], # Workers 8, 10 [1, 0, 0, 1], # Workers 8, 11 ] # Model model = cp_model.CpModel() # Variables x = {} for worker in range(num_workers): for task in range(num_tasks): x[worker, task] = model.new_bool_var(f"x[{worker},{task}]") # Constraints # Each worker is assigned to at most one task. for worker in range(num_workers): model.add_at_most_one(x[worker, task] for task in range(num_tasks)) # Each task is assigned to exactly one worker. for task in range(num_tasks): model.add_exactly_one(x[worker, task] for worker in range(num_workers)) # Create variables for each worker, indicating whether they work on some task. work = {} for worker in range(num_workers): work[worker] = model.new_bool_var(f"work[{worker}]") for worker in range(num_workers): for task in range(num_tasks): model.add(work[worker] == sum(x[worker, task] for task in range(num_tasks))) # Define the allowed groups of worders model.add_allowed_assignments([work[0], work[1], work[2], work[3]], group1) model.add_allowed_assignments([work[4], work[5], work[6], work[7]], group2) model.add_allowed_assignments([work[8], work[9], work[10], work[11]], group3) # Objective objective_terms = [] for worker in range(num_workers): for task in range(num_tasks): objective_terms.append(costs[worker][task] * x[worker, task]) model.minimize(sum(objective_terms)) # Solve solver = cp_model.CpSolver() status = solver.solve(model) # Print solution. if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE: print(f"Total cost = {solver.objective_value}\n") for worker in range(num_workers): for task in range(num_tasks): if solver.boolean_value(x[worker, task]): print( f"Worker {worker} assigned to task {task}." + f" Cost = {costs[worker][task]}" ) else: print("No solution found.") if __name__ == "__main__": main()
C++
// Solve assignment problem for given group of workers. #include <stdlib.h> #include <cstdint> #include <numeric> #include <vector> #include "absl/strings/str_format.h" #include "absl/types/span.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 AssignmentGroups() { // Data const std::vector<std::vector<int>> costs = {{ {{90, 76, 75, 70, 50, 74}}, {{35, 85, 55, 65, 48, 101}}, {{125, 95, 90, 105, 59, 120}}, {{45, 110, 95, 115, 104, 83}}, {{60, 105, 80, 75, 59, 62}}, {{45, 65, 110, 95, 47, 31}}, {{38, 51, 107, 41, 69, 99}}, {{47, 85, 57, 71, 92, 77}}, {{39, 63, 97, 49, 118, 56}}, {{47, 101, 71, 60, 88, 109}}, {{17, 39, 103, 64, 61, 92}}, {{101, 45, 83, 59, 92, 27}}, }}; const int num_workers = static_cast<int>(costs.size()); std::vector<int> all_workers(num_workers); std::iota(all_workers.begin(), all_workers.end(), 0); const int num_tasks = static_cast<int>(costs[0].size()); std::vector<int> all_tasks(num_tasks); std::iota(all_tasks.begin(), all_tasks.end(), 0); // Allowed groups of workers: const std::vector<std::vector<int64_t>> group1 = {{ {{0, 0, 1, 1}}, // Workers 2, 3 {{0, 1, 0, 1}}, // Workers 1, 3 {{0, 1, 1, 0}}, // Workers 1, 2 {{1, 1, 0, 0}}, // Workers 0, 1 {{1, 0, 1, 0}}, // Workers 0, 2 }}; const std::vector<std::vector<int64_t>> group2 = {{ {{0, 0, 1, 1}}, // Workers 6, 7 {{0, 1, 0, 1}}, // Workers 5, 7 {{0, 1, 1, 0}}, // Workers 5, 6 {{1, 1, 0, 0}}, // Workers 4, 5 {{1, 0, 0, 1}}, // Workers 4, 7 }}; const std::vector<std::vector<int64_t>> group3 = {{ {{0, 0, 1, 1}}, // Workers 10, 11 {{0, 1, 0, 1}}, // Workers 9, 11 {{0, 1, 1, 0}}, // Workers 9, 10 {{1, 0, 1, 0}}, // Workers 8, 10 {{1, 0, 0, 1}}, // Workers 8, 11 }}; // Model CpModelBuilder cp_model; // Variables // x[i][j] is an array of Boolean variables. x[i][j] is true // if worker i is assigned to task j. std::vector<std::vector<BoolVar>> x(num_workers, std::vector<BoolVar>(num_tasks)); for (int worker : all_workers) { for (int task : all_tasks) { x[worker][task] = cp_model.NewBoolVar().WithName( absl::StrFormat("x[%d,%d]", worker, task)); } } // Constraints // Each worker is assigned to at most one task. for (int worker : all_workers) { cp_model.AddAtMostOne(x[worker]); } // Each task is assigned to exactly one worker. for (int task : all_tasks) { std::vector<BoolVar> tasks; for (int worker : all_workers) { tasks.push_back(x[worker][task]); } cp_model.AddExactlyOne(tasks); } // Create variables for each worker, indicating whether they work on some // task. std::vector<IntVar> work(num_workers); for (int worker : all_workers) { work[worker] = IntVar( cp_model.NewBoolVar().WithName(absl::StrFormat("work[%d]", worker))); } for (int worker : all_workers) { LinearExpr task_sum; for (int task : all_tasks) { task_sum += x[worker][task]; } cp_model.AddEquality(work[worker], task_sum); } // Define the allowed groups of worders auto table1 = cp_model.AddAllowedAssignments({work[0], work[1], work[2], work[3]}); for (const auto& t : group1) { table1.AddTuple(t); } auto table2 = cp_model.AddAllowedAssignments({work[4], work[5], work[6], work[7]}); for (const auto& t : group2) { table2.AddTuple(t); } auto table3 = cp_model.AddAllowedAssignments({work[8], work[9], work[10], work[11]}); for (const auto& t : group3) { table3.AddTuple(t); } // Objective LinearExpr total_cost; for (int worker : all_workers) { for (int task : all_tasks) { total_cost += x[worker][task] * costs[worker][task]; } } cp_model.Minimize(total_cost); // Solve const CpSolverResponse response = Solve(cp_model.Build()); // Print solution. if (response.status() == CpSolverStatus::INFEASIBLE) { LOG(FATAL) << "No solution found."; } LOG(INFO) << "Total cost: " << response.objective_value(); LOG(INFO); for (int worker : all_workers) { for (int task : all_tasks) { if (SolutionBooleanValue(response, x[worker][task])) { LOG(INFO) << "Worker " << worker << " assigned to task " << task << ". Cost: " << costs[worker][task]; } } } } } // namespace sat } // namespace operations_research int main(int argc, char** argv) { operations_research::sat::AssignmentGroups(); return EXIT_SUCCESS; }
Java
// CP-SAT example that solves an assignment problem. 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.IntVar; 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; /** Assignment problem. */ public class AssignmentGroupsSat { public static void main(String[] args) { Loader.loadNativeLibraries(); // Data int[][] costs = { {90, 76, 75, 70, 50, 74}, {35, 85, 55, 65, 48, 101}, {125, 95, 90, 105, 59, 120}, {45, 110, 95, 115, 104, 83}, {60, 105, 80, 75, 59, 62}, {45, 65, 110, 95, 47, 31}, {38, 51, 107, 41, 69, 99}, {47, 85, 57, 71, 92, 77}, {39, 63, 97, 49, 118, 56}, {47, 101, 71, 60, 88, 109}, {17, 39, 103, 64, 61, 92}, {101, 45, 83, 59, 92, 27}, }; final int numWorkers = costs.length; final int numTasks = costs[0].length; final int[] allWorkers = IntStream.range(0, numWorkers).toArray(); final int[] allTasks = IntStream.range(0, numTasks).toArray(); // Allowed groups of workers: int[][] group1 = { {0, 0, 1, 1}, // Workers 2, 3 {0, 1, 0, 1}, // Workers 1, 3 {0, 1, 1, 0}, // Workers 1, 2 {1, 1, 0, 0}, // Workers 0, 1 {1, 0, 1, 0}, // Workers 0, 2 }; int[][] group2 = { {0, 0, 1, 1}, // Workers 6, 7 {0, 1, 0, 1}, // Workers 5, 7 {0, 1, 1, 0}, // Workers 5, 6 {1, 1, 0, 0}, // Workers 4, 5 {1, 0, 0, 1}, // Workers 4, 7 }; int[][] group3 = { {0, 0, 1, 1}, // Workers 10, 11 {0, 1, 0, 1}, // Workers 9, 11 {0, 1, 1, 0}, // Workers 9, 10 {1, 0, 1, 0}, // Workers 8, 10 {1, 0, 0, 1}, // Workers 8, 11 }; // Model CpModel model = new CpModel(); // Variables Literal[][] x = new Literal[numWorkers][numTasks]; for (int worker : allWorkers) { for (int task : allTasks) { x[worker][task] = model.newBoolVar("x[" + worker + "," + task + "]"); } } // Constraints // Each worker is assigned to at most one task. for (int worker : allWorkers) { List<Literal> tasks = new ArrayList<>(); for (int task : allTasks) { tasks.add(x[worker][task]); } model.addAtMostOne(tasks); } // Each task is assigned to exactly one worker. for (int task : allTasks) { List<Literal> workers = new ArrayList<>(); for (int worker : allWorkers) { workers.add(x[worker][task]); } model.addExactlyOne(workers); } // Create variables for each worker, indicating whether they work on some task. IntVar[] work = new IntVar[numWorkers]; for (int worker : allWorkers) { work[worker] = model.newBoolVar("work[" + worker + "]"); } for (int worker : allWorkers) { LinearExprBuilder expr = LinearExpr.newBuilder(); for (int task : allTasks) { expr.add(x[worker][task]); } model.addEquality(work[worker], expr); } // Define the allowed groups of worders model.addAllowedAssignments(new IntVar[] {work[0], work[1], work[2], work[3]}) .addTuples(group1); model.addAllowedAssignments(new IntVar[] {work[4], work[5], work[6], work[7]}) .addTuples(group2); model.addAllowedAssignments(new IntVar[] {work[8], work[9], work[10], work[11]}) .addTuples(group3); // Objective LinearExprBuilder obj = LinearExpr.newBuilder(); for (int worker : allWorkers) { for (int task : allTasks) { obj.addTerm(x[worker][task], costs[worker][task]); } } model.minimize(obj); // Solve CpSolver solver = new CpSolver(); CpSolverStatus status = solver.solve(model); // Print solution. // Check that the problem has a feasible solution. if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) { System.out.println("Total cost: " + solver.objectiveValue() + "\n"); for (int worker : allWorkers) { for (int task : allTasks) { if (solver.booleanValue(x[worker][task])) { System.out.println("Worker " + worker + " assigned to task " + task + ". Cost: " + costs[worker][task]); } } } } else { System.err.println("No solution found."); } } private AssignmentGroupsSat() {} }
#C
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.Sat; public class AssignmentGroupsSat { public static void Main(String[] args) { // Data. int[,] costs = { { 90, 76, 75, 70, 50, 74 }, { 35, 85, 55, 65, 48, 101 }, { 125, 95, 90, 105, 59, 120 }, { 45, 110, 95, 115, 104, 83 }, { 60, 105, 80, 75, 59, 62 }, { 45, 65, 110, 95, 47, 31 }, { 38, 51, 107, 41, 69, 99 }, { 47, 85, 57, 71, 92, 77 }, { 39, 63, 97, 49, 118, 56 }, { 47, 101, 71, 60, 88, 109 }, { 17, 39, 103, 64, 61, 92 }, { 101, 45, 83, 59, 92, 27 }, }; int numWorkers = costs.GetLength(0); int numTasks = costs.GetLength(1); int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray(); int[] allTasks = Enumerable.Range(0, numTasks).ToArray(); // Allowed groups of workers: long[,] group1 = { { 0, 0, 1, 1 }, // Workers 2, 3 { 0, 1, 0, 1 }, // Workers 1, 3 { 0, 1, 1, 0 }, // Workers 1, 2 { 1, 1, 0, 0 }, // Workers 0, 1 { 1, 0, 1, 0 }, // Workers 0, 2 }; long[,] group2 = { { 0, 0, 1, 1 }, // Workers 6, 7 { 0, 1, 0, 1 }, // Workers 5, 7 { 0, 1, 1, 0 }, // Workers 5, 6 { 1, 1, 0, 0 }, // Workers 4, 5 { 1, 0, 0, 1 }, // Workers 4, 7 }; long[,] group3 = { { 0, 0, 1, 1 }, // Workers 10, 11 { 0, 1, 0, 1 }, // Workers 9, 11 { 0, 1, 1, 0 }, // Workers 9, 10 { 1, 0, 1, 0 }, // Workers 8, 10 { 1, 0, 0, 1 }, // Workers 8, 11 }; // Model. CpModel model = new CpModel(); // Variables. BoolVar[,] x = new BoolVar[numWorkers, numTasks]; // Variables in a 1-dim array. foreach (int worker in allWorkers) { foreach (int task in allTasks) { x[worker, task] = model.NewBoolVar($"x[{worker},{task}]"); } } // Constraints // Each worker is assigned to at most one task. foreach (int worker in allWorkers) { List<ILiteral> tasks = new List<ILiteral>(); foreach (int task in allTasks) { tasks.Add(x[worker, task]); } model.AddAtMostOne(tasks); } // Each task is assigned to exactly one worker. foreach (int task in allTasks) { List<ILiteral> workers = new List<ILiteral>(); foreach (int worker in allWorkers) { workers.Add(x[worker, task]); } model.AddExactlyOne(workers); } // Create variables for each worker, indicating whether they work on some task. BoolVar[] work = new BoolVar[numWorkers]; foreach (int worker in allWorkers) { work[worker] = model.NewBoolVar($"work[{worker}]"); } foreach (int worker in allWorkers) { List<ILiteral> tasks = new List<ILiteral>(); foreach (int task in allTasks) { tasks.Add(x[worker, task]); } model.Add(work[worker] == LinearExpr.Sum(tasks)); } // Define the allowed groups of worders model.AddAllowedAssignments(new IntVar[] { work[0], work[1], work[2], work[3] }).AddTuples(group1); model.AddAllowedAssignments(new IntVar[] { work[4], work[5], work[6], work[7] }).AddTuples(group2); model.AddAllowedAssignments(new IntVar[] { work[8], work[9], work[10], work[11] }).AddTuples(group3); // Objective LinearExprBuilder obj = LinearExpr.NewBuilder(); foreach (int worker in allWorkers) { foreach (int task in allTasks) { obj.AddTerm(x[worker, task], costs[worker, task]); } } model.Minimize(obj); // Solve CpSolver solver = new CpSolver(); CpSolverStatus status = solver.Solve(model); Console.WriteLine($"Solve status: {status}"); // Print solution. // Check that the problem has a feasible solution. if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible) { Console.WriteLine($"Total cost: {solver.ObjectiveValue}\n"); foreach (int worker in allWorkers) { foreach (int task in allTasks) { if (solver.Value(x[worker, task]) > 0.5) { Console.WriteLine($"Worker {worker} assigned to task {task}. " + $"Cost: {costs[worker, task]}"); } } } } else { Console.WriteLine("No solution found."); } Console.WriteLine("Statistics"); Console.WriteLine($" - conflicts : {solver.NumConflicts()}"); Console.WriteLine($" - branches : {solver.NumBranches()}"); Console.WriteLine($" - wall time : {solver.WallTime()}s"); } }
حل MIP
بعد ذلك، نُصِف حلاً للمشكلة باستخدام أداة حل MIP.
استيراد المكتبات
يستورد الرمز التالي المكتبة المطلوبة.
Python
from ortools.linear_solver import pywraplp
C++
#include <cstdint> #include <memory> #include <numeric> #include <utility> #include <vector> #include "absl/strings/str_format.h" #include "ortools/base/logging.h" #include "ortools/linear_solver/linear_solver.h"
Java
import com.google.ortools.Loader; import com.google.ortools.linearsolver.MPConstraint; import com.google.ortools.linearsolver.MPObjective; import com.google.ortools.linearsolver.MPSolver; import com.google.ortools.linearsolver.MPVariable; import java.util.stream.IntStream;
#C
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.LinearSolver;
تعريف البيانات
ينشئ الرمز التالي البيانات للبرنامج.
Python
costs = [
[90, 76, 75, 70, 50, 74],
[35, 85, 55, 65, 48, 101],
[125, 95, 90, 105, 59, 120],
[45, 110, 95, 115, 104, 83],
[60, 105, 80, 75, 59, 62],
[45, 65, 110, 95, 47, 31],
[38, 51, 107, 41, 69, 99],
[47, 85, 57, 71, 92, 77],
[39, 63, 97, 49, 118, 56],
[47, 101, 71, 60, 88, 109],
[17, 39, 103, 64, 61, 92],
[101, 45, 83, 59, 92, 27],
]
num_workers = len(costs)
num_tasks = len(costs[0])C++
const std::vector<std::vector<int64_t>> costs = {{ {{90, 76, 75, 70, 50, 74}}, {{35, 85, 55, 65, 48, 101}}, {{125, 95, 90, 105, 59, 120}}, {{45, 110, 95, 115, 104, 83}}, {{60, 105, 80, 75, 59, 62}}, {{45, 65, 110, 95, 47, 31}}, {{38, 51, 107, 41, 69, 99}}, {{47, 85, 57, 71, 92, 77}}, {{39, 63, 97, 49, 118, 56}}, {{47, 101, 71, 60, 88, 109}}, {{17, 39, 103, 64, 61, 92}}, {{101, 45, 83, 59, 92, 27}}, }}; const int num_workers = costs.size(); std::vector<int> all_workers(num_workers); std::iota(all_workers.begin(), all_workers.end(), 0); const int num_tasks = costs[0].size(); std::vector<int> all_tasks(num_tasks); std::iota(all_tasks.begin(), all_tasks.end(), 0);
Java
double[][] costs = {
{90, 76, 75, 70, 50, 74},
{35, 85, 55, 65, 48, 101},
{125, 95, 90, 105, 59, 120},
{45, 110, 95, 115, 104, 83},
{60, 105, 80, 75, 59, 62},
{45, 65, 110, 95, 47, 31},
{38, 51, 107, 41, 69, 99},
{47, 85, 57, 71, 92, 77},
{39, 63, 97, 49, 118, 56},
{47, 101, 71, 60, 88, 109},
{17, 39, 103, 64, 61, 92},
{101, 45, 83, 59, 92, 27},
};
int numWorkers = costs.length;
int numTasks = costs[0].length;
final int[] allWorkers = IntStream.range(0, numWorkers).toArray();
final int[] allTasks = IntStream.range(0, numTasks).toArray();#C
int[,] costs = {
{ 90, 76, 75, 70, 50, 74 }, { 35, 85, 55, 65, 48, 101 }, { 125, 95, 90, 105, 59, 120 },
{ 45, 110, 95, 115, 104, 83 }, { 60, 105, 80, 75, 59, 62 }, { 45, 65, 110, 95, 47, 31 },
{ 38, 51, 107, 41, 69, 99 }, { 47, 85, 57, 71, 92, 77 }, { 39, 63, 97, 49, 118, 56 },
{ 47, 101, 71, 60, 88, 109 }, { 17, 39, 103, 64, 61, 92 }, { 101, 45, 83, 59, 92, 27 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);
int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray();
int[] allTasks = Enumerable.Range(0, numTasks).ToArray();إنشاء المجموعات المسموح بها
ينشئ الرمز التالي المجموعات المسموح بها من خلال التكرار عبر المجموعات الثلاث من المجموعات الفرعية الموضحة أعلاه.
Python
group1 = [ # Subgroups of workers 0 - 3
[2, 3],
[1, 3],
[1, 2],
[0, 1],
[0, 2],
]
group2 = [ # Subgroups of workers 4 - 7
[6, 7],
[5, 7],
[5, 6],
[4, 5],
[4, 7],
]
group3 = [ # Subgroups of workers 8 - 11
[10, 11],
[9, 11],
[9, 10],
[8, 10],
[8, 11],
]C++
using WorkerIndex = int; using Binome = std::pair<WorkerIndex, WorkerIndex>; using AllowedBinomes = std::vector<Binome>; const AllowedBinomes group1 = {{ // group of worker 0-3 {2, 3}, {1, 3}, {1, 2}, {0, 1}, {0, 2}, }}; const AllowedBinomes group2 = {{ // group of worker 4-7 {6, 7}, {5, 7}, {5, 6}, {4, 5}, {4, 7}, }}; const AllowedBinomes group3 = {{ // group of worker 8-11 {10, 11}, {9, 11}, {9, 10}, {8, 10}, {8, 11}, }};
Java
int[][] group1 = {
// group of worker 0-3
{2, 3},
{1, 3},
{1, 2},
{0, 1},
{0, 2},
};
int[][] group2 = {
// group of worker 4-7
{6, 7},
{5, 7},
{5, 6},
{4, 5},
{4, 7},
};
int[][] group3 = {
// group of worker 8-11
{10, 11},
{9, 11},
{9, 10},
{8, 10},
{8, 11},
};#C
int[,] group1 = {
// group of worker 0-3
{ 2, 3 }, { 1, 3 }, { 1, 2 }, { 0, 1 }, { 0, 2 },
};
int[,] group2 = {
// group of worker 4-7
{ 6, 7 }, { 5, 7 }, { 5, 6 }, { 4, 5 }, { 4, 7 },
};
int[,] group3 = {
// group of worker 8-11
{ 10, 11 }, { 9, 11 }, { 9, 10 }, { 8, 10 }, { 8, 11 },
};تضمين أداة الحلّ
تُنشئ التعليمة البرمجية التالية أداة الحل.
Python
# Create the mip solver with the SCIP backend. solver = pywraplp.Solver.CreateSolver("SCIP") if not solver: return
C++
// Create the mip solver with the SCIP backend. std::unique_ptr<MPSolver> solver(MPSolver::CreateSolver("SCIP")); if (!solver) { LOG(WARNING) << "SCIP solver unavailable."; return; }
Java
// Create the linear solver with the SCIP backend. MPSolver solver = MPSolver.createSolver("SCIP"); if (solver == null) { System.out.println("Could not create solver SCIP"); return; }
#C
Solver solver = Solver.CreateSolver("SCIP"); if (solver is null) { return; }
إنشاء المتغيّرات
تُنشئ التعليمة البرمجية التالية صفيفًا من المتغيرات للمشكلة.
Python
# x[worker, task] is an array of 0-1 variables, which will be 1 # if the worker is assigned to the task. x = {} for worker in range(num_workers): for task in range(num_tasks): x[worker, task] = solver.BoolVar(f"x[{worker},{task}]")
C++
// x[i][j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. std::vector<std::vector<const MPVariable*>> x( num_workers, std::vector<const MPVariable*>(num_tasks)); for (int worker : all_workers) { for (int task : all_tasks) { x[worker][task] = solver->MakeBoolVar(absl::StrFormat("x[%d,%d]", worker, task)); } }
Java
// x[i][j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. MPVariable[][] x = new MPVariable[numWorkers][numTasks]; for (int worker : allWorkers) { for (int task : allTasks) { x[worker][task] = solver.makeBoolVar("x[" + worker + "," + task + "]"); } }
#C
// x[i, j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. Variable[,] x = new Variable[numWorkers, numTasks]; foreach (int worker in allWorkers) { foreach (int task in allTasks) { x[worker, task] = solver.MakeBoolVar($"x[{worker},{task}]"); } }
إضافة القيود
تضع التعليمة البرمجية التالية قيودًا للبرنامج.
Python
# The total size of the tasks each worker takes on is at most total_size_max. for worker in range(num_workers): solver.Add(solver.Sum([x[worker, task] for task in range(num_tasks)]) <= 1) # Each task is assigned to exactly one worker. for task in range(num_tasks): solver.Add(solver.Sum([x[worker, task] for worker in range(num_workers)]) == 1)
C++
// Each worker is assigned to at most one task. for (int worker : all_workers) { LinearExpr worker_sum; for (int task : all_tasks) { worker_sum += x[worker][task]; } solver->MakeRowConstraint(worker_sum <= 1.0); } // Each task is assigned to exactly one worker. for (int task : all_tasks) { LinearExpr task_sum; for (int worker : all_workers) { task_sum += x[worker][task]; } solver->MakeRowConstraint(task_sum == 1.0); }
Java
// Each worker is assigned to at most one task. for (int worker : allWorkers) { MPConstraint constraint = solver.makeConstraint(0, 1, ""); for (int task : allTasks) { constraint.setCoefficient(x[worker][task], 1); } } // Each task is assigned to exactly one worker. for (int task : allTasks) { MPConstraint constraint = solver.makeConstraint(1, 1, ""); for (int worker : allWorkers) { constraint.setCoefficient(x[worker][task], 1); } }
#C
// Each worker is assigned to at most one task. foreach (int worker in allWorkers) { Constraint constraint = solver.MakeConstraint(0, 1, ""); foreach (int task in allTasks) { constraint.SetCoefficient(x[worker, task], 1); } } // Each task is assigned to exactly one worker. foreach (int task in allTasks) { Constraint constraint = solver.MakeConstraint(1, 1, ""); foreach (int worker in allWorkers) { constraint.SetCoefficient(x[worker, task], 1); } }
إنشاء الهدف
تُنشئ التعليمة البرمجية التالية الدالة الموضوعية.
Python
objective_terms = [] for worker in range(num_workers): for task in range(num_tasks): objective_terms.append(costs[worker][task] * x[worker, task]) solver.Minimize(solver.Sum(objective_terms))
C++
MPObjective* const objective = solver->MutableObjective(); for (int worker : all_workers) { for (int task : all_tasks) { objective->SetCoefficient(x[worker][task], costs[worker][task]); } } objective->SetMinimization();
Java
MPObjective objective = solver.objective(); for (int worker : allWorkers) { for (int task : allTasks) { objective.setCoefficient(x[worker][task], costs[worker][task]); } } objective.setMinimization();
#C
Objective objective = solver.Objective();
foreach (int worker in allWorkers)
{
foreach (int task in allTasks)
{
objective.SetCoefficient(x[worker, task], costs[worker, task]);
}
}
objective.SetMinimization();استدعاء أداة الحلّ
يستدعي الرمز التالي أداة الحلّ ويعرض النتائج.
Python
print(f"Solving with {solver.SolverVersion()}")
status = solver.Solve()C++
const MPSolver::ResultStatus result_status = solver->Solve();
Java
MPSolver.ResultStatus resultStatus = solver.solve();
#C
Solver.ResultStatus resultStatus = solver.Solve();
عرض النتائج
الآن، يمكننا طباعة الحل.
Python
if status == pywraplp.Solver.OPTIMAL or status == pywraplp.Solver.FEASIBLE: print(f"Total cost = {solver.Objective().Value()}\n") for worker in range(num_workers): for task in range(num_tasks): if x[worker, task].solution_value() > 0.5: print( f"Worker {worker} assigned to task {task}." + f" Cost: {costs[worker][task]}" ) else: print("No solution found.")
C++
// Check that the problem has a feasible solution. if (result_status != MPSolver::OPTIMAL && result_status != MPSolver::FEASIBLE) { LOG(FATAL) << "No solution found."; } LOG(INFO) << "Total cost = " << objective->Value() << "\n\n"; for (int worker : all_workers) { for (int task : all_tasks) { // Test if x[i][j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker][task]->solution_value() > 0.5) { LOG(INFO) << "Worker " << worker << " assigned to task " << task << ". Cost: " << costs[worker][task]; } } }
Java
// Check that the problem has a feasible solution. if (resultStatus == MPSolver.ResultStatus.OPTIMAL || resultStatus == MPSolver.ResultStatus.FEASIBLE) { System.out.println("Total cost: " + objective.value() + "\n"); for (int worker : allWorkers) { for (int task : allTasks) { // Test if x[i][j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker][task].solutionValue() > 0.5) { System.out.println("Worker " + worker + " assigned to task " + task + ". Cost: " + costs[worker][task]); } } } } else { System.err.println("No solution found."); }
#C
// Check that the problem has a feasible solution. if (resultStatus == Solver.ResultStatus.OPTIMAL || resultStatus == Solver.ResultStatus.FEASIBLE) { Console.WriteLine($"Total cost: {solver.Objective().Value()}\n"); foreach (int worker in allWorkers) { foreach (int task in allTasks) { // Test if x[i, j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker, task].SolutionValue() > 0.5) { Console.WriteLine($"Worker {worker} assigned to task {task}. Cost: {costs[worker, task]}"); } } } } else { Console.WriteLine("No solution found."); }
في ما يلي نتيجة البرنامج:
Minimum cost = 239.0 Worker 0 assigned to task 4 Cost = 50 Worker 1 assigned to task 2 Cost = 55 Worker 5 assigned to task 5 Cost = 31 Worker 6 assigned to task 3 Cost = 41 Worker 10 assigned to task 0 Cost = 17 Worker 11 assigned to task 1 Cost = 45 Time = 0.3281 seconds
البرنامج بأكمله
إليك البرنامج بالكامل.
Python
"""Solve assignment problem for given group of workers.""" from ortools.linear_solver import pywraplp def main(): # Data costs = [ [90, 76, 75, 70, 50, 74], [35, 85, 55, 65, 48, 101], [125, 95, 90, 105, 59, 120], [45, 110, 95, 115, 104, 83], [60, 105, 80, 75, 59, 62], [45, 65, 110, 95, 47, 31], [38, 51, 107, 41, 69, 99], [47, 85, 57, 71, 92, 77], [39, 63, 97, 49, 118, 56], [47, 101, 71, 60, 88, 109], [17, 39, 103, 64, 61, 92], [101, 45, 83, 59, 92, 27], ] num_workers = len(costs) num_tasks = len(costs[0]) # Allowed groups of workers: group1 = [ # Subgroups of workers 0 - 3 [2, 3], [1, 3], [1, 2], [0, 1], [0, 2], ] group2 = [ # Subgroups of workers 4 - 7 [6, 7], [5, 7], [5, 6], [4, 5], [4, 7], ] group3 = [ # Subgroups of workers 8 - 11 [10, 11], [9, 11], [9, 10], [8, 10], [8, 11], ] # Solver. # Create the mip solver with the SCIP backend. solver = pywraplp.Solver.CreateSolver("SCIP") if not solver: return # Variables # x[worker, task] is an array of 0-1 variables, which will be 1 # if the worker is assigned to the task. x = {} for worker in range(num_workers): for task in range(num_tasks): x[worker, task] = solver.BoolVar(f"x[{worker},{task}]") # Constraints # The total size of the tasks each worker takes on is at most total_size_max. for worker in range(num_workers): solver.Add(solver.Sum([x[worker, task] for task in range(num_tasks)]) <= 1) # Each task is assigned to exactly one worker. for task in range(num_tasks): solver.Add(solver.Sum([x[worker, task] for worker in range(num_workers)]) == 1) # Create variables for each worker, indicating whether they work on some task. work = {} for worker in range(num_workers): work[worker] = solver.BoolVar(f"work[{worker}]") for worker in range(num_workers): solver.Add( work[worker] == solver.Sum([x[worker, task] for task in range(num_tasks)]) ) # Group1 constraint_g1 = solver.Constraint(1, 1) for index, _ in enumerate(group1): # a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] # p is True if a AND b, False otherwise constraint = solver.Constraint(0, 1) constraint.SetCoefficient(work[group1[index][0]], 1) constraint.SetCoefficient(work[group1[index][1]], 1) p = solver.BoolVar(f"g1_p{index}") constraint.SetCoefficient(p, -2) constraint_g1.SetCoefficient(p, 1) # Group2 constraint_g2 = solver.Constraint(1, 1) for index, _ in enumerate(group2): # a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] # p is True if a AND b, False otherwise constraint = solver.Constraint(0, 1) constraint.SetCoefficient(work[group2[index][0]], 1) constraint.SetCoefficient(work[group2[index][1]], 1) p = solver.BoolVar(f"g2_p{index}") constraint.SetCoefficient(p, -2) constraint_g2.SetCoefficient(p, 1) # Group3 constraint_g3 = solver.Constraint(1, 1) for index, _ in enumerate(group3): # a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] # p is True if a AND b, False otherwise constraint = solver.Constraint(0, 1) constraint.SetCoefficient(work[group3[index][0]], 1) constraint.SetCoefficient(work[group3[index][1]], 1) p = solver.BoolVar(f"g3_p{index}") constraint.SetCoefficient(p, -2) constraint_g3.SetCoefficient(p, 1) # Objective objective_terms = [] for worker in range(num_workers): for task in range(num_tasks): objective_terms.append(costs[worker][task] * x[worker, task]) solver.Minimize(solver.Sum(objective_terms)) # Solve print(f"Solving with {solver.SolverVersion()}") status = solver.Solve() # Print solution. if status == pywraplp.Solver.OPTIMAL or status == pywraplp.Solver.FEASIBLE: print(f"Total cost = {solver.Objective().Value()}\n") for worker in range(num_workers): for task in range(num_tasks): if x[worker, task].solution_value() > 0.5: print( f"Worker {worker} assigned to task {task}." + f" Cost: {costs[worker][task]}" ) else: print("No solution found.") if __name__ == "__main__": main()
C++
// Solve a simple assignment problem. #include <cstdint> #include <memory> #include <numeric> #include <utility> #include <vector> #include "absl/strings/str_format.h" #include "ortools/base/logging.h" #include "ortools/linear_solver/linear_solver.h" namespace operations_research { void AssignmentTeamsMip() { // Data const std::vector<std::vector<int64_t>> costs = {{ {{90, 76, 75, 70, 50, 74}}, {{35, 85, 55, 65, 48, 101}}, {{125, 95, 90, 105, 59, 120}}, {{45, 110, 95, 115, 104, 83}}, {{60, 105, 80, 75, 59, 62}}, {{45, 65, 110, 95, 47, 31}}, {{38, 51, 107, 41, 69, 99}}, {{47, 85, 57, 71, 92, 77}}, {{39, 63, 97, 49, 118, 56}}, {{47, 101, 71, 60, 88, 109}}, {{17, 39, 103, 64, 61, 92}}, {{101, 45, 83, 59, 92, 27}}, }}; const int num_workers = costs.size(); std::vector<int> all_workers(num_workers); std::iota(all_workers.begin(), all_workers.end(), 0); const int num_tasks = costs[0].size(); std::vector<int> all_tasks(num_tasks); std::iota(all_tasks.begin(), all_tasks.end(), 0); // Allowed groups of workers: using WorkerIndex = int; using Binome = std::pair<WorkerIndex, WorkerIndex>; using AllowedBinomes = std::vector<Binome>; const AllowedBinomes group1 = {{ // group of worker 0-3 {2, 3}, {1, 3}, {1, 2}, {0, 1}, {0, 2}, }}; const AllowedBinomes group2 = {{ // group of worker 4-7 {6, 7}, {5, 7}, {5, 6}, {4, 5}, {4, 7}, }}; const AllowedBinomes group3 = {{ // group of worker 8-11 {10, 11}, {9, 11}, {9, 10}, {8, 10}, {8, 11}, }}; // Solver // Create the mip solver with the SCIP backend. std::unique_ptr<MPSolver> solver(MPSolver::CreateSolver("SCIP")); if (!solver) { LOG(WARNING) << "SCIP solver unavailable."; return; } // Variables // x[i][j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. std::vector<std::vector<const MPVariable*>> x( num_workers, std::vector<const MPVariable*>(num_tasks)); for (int worker : all_workers) { for (int task : all_tasks) { x[worker][task] = solver->MakeBoolVar(absl::StrFormat("x[%d,%d]", worker, task)); } } // Constraints // Each worker is assigned to at most one task. for (int worker : all_workers) { LinearExpr worker_sum; for (int task : all_tasks) { worker_sum += x[worker][task]; } solver->MakeRowConstraint(worker_sum <= 1.0); } // Each task is assigned to exactly one worker. for (int task : all_tasks) { LinearExpr task_sum; for (int worker : all_workers) { task_sum += x[worker][task]; } solver->MakeRowConstraint(task_sum == 1.0); } // Create variables for each worker, indicating whether they work on some // task. std::vector<const MPVariable*> work(num_workers); for (int worker : all_workers) { work[worker] = solver->MakeBoolVar(absl::StrFormat("work[%d]", worker)); } for (int worker : all_workers) { LinearExpr task_sum; for (int task : all_tasks) { task_sum += x[worker][task]; } solver->MakeRowConstraint(work[worker] == task_sum); } // Group1 { MPConstraint* g1 = solver->MakeRowConstraint(1, 1); for (int i = 0; i < group1.size(); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is true if a AND b, false otherwise MPConstraint* tmp = solver->MakeRowConstraint(0, 1); tmp->SetCoefficient(work[group1[i].first], 1); tmp->SetCoefficient(work[group1[i].second], 1); MPVariable* p = solver->MakeBoolVar(absl::StrFormat("g1_p%d", i)); tmp->SetCoefficient(p, -2); g1->SetCoefficient(p, 1); } } // Group2 { MPConstraint* g2 = solver->MakeRowConstraint(1, 1); for (int i = 0; i < group2.size(); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is true if a AND b, false otherwise MPConstraint* tmp = solver->MakeRowConstraint(0, 1); tmp->SetCoefficient(work[group2[i].first], 1); tmp->SetCoefficient(work[group2[i].second], 1); MPVariable* p = solver->MakeBoolVar(absl::StrFormat("g2_p%d", i)); tmp->SetCoefficient(p, -2); g2->SetCoefficient(p, 1); } } // Group3 { MPConstraint* g3 = solver->MakeRowConstraint(1, 1); for (int i = 0; i < group3.size(); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is true if a AND b, false otherwise MPConstraint* tmp = solver->MakeRowConstraint(0, 1); tmp->SetCoefficient(work[group3[i].first], 1); tmp->SetCoefficient(work[group3[i].second], 1); MPVariable* p = solver->MakeBoolVar(absl::StrFormat("g3_p%d", i)); tmp->SetCoefficient(p, -2); g3->SetCoefficient(p, 1); } } // Objective. MPObjective* const objective = solver->MutableObjective(); for (int worker : all_workers) { for (int task : all_tasks) { objective->SetCoefficient(x[worker][task], costs[worker][task]); } } objective->SetMinimization(); // Solve const MPSolver::ResultStatus result_status = solver->Solve(); // Print solution. // Check that the problem has a feasible solution. if (result_status != MPSolver::OPTIMAL && result_status != MPSolver::FEASIBLE) { LOG(FATAL) << "No solution found."; } LOG(INFO) << "Total cost = " << objective->Value() << "\n\n"; for (int worker : all_workers) { for (int task : all_tasks) { // Test if x[i][j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker][task]->solution_value() > 0.5) { LOG(INFO) << "Worker " << worker << " assigned to task " << task << ". Cost: " << costs[worker][task]; } } } } } // namespace operations_research int main(int argc, char** argv) { operations_research::AssignmentTeamsMip(); return EXIT_SUCCESS; }
Java
package com.google.ortools.linearsolver.samples; import com.google.ortools.Loader; import com.google.ortools.linearsolver.MPConstraint; import com.google.ortools.linearsolver.MPObjective; import com.google.ortools.linearsolver.MPSolver; import com.google.ortools.linearsolver.MPVariable; import java.util.stream.IntStream; /** MIP example that solves an assignment problem. */ public class AssignmentGroupsMip { public static void main(String[] args) { Loader.loadNativeLibraries(); // Data double[][] costs = { {90, 76, 75, 70, 50, 74}, {35, 85, 55, 65, 48, 101}, {125, 95, 90, 105, 59, 120}, {45, 110, 95, 115, 104, 83}, {60, 105, 80, 75, 59, 62}, {45, 65, 110, 95, 47, 31}, {38, 51, 107, 41, 69, 99}, {47, 85, 57, 71, 92, 77}, {39, 63, 97, 49, 118, 56}, {47, 101, 71, 60, 88, 109}, {17, 39, 103, 64, 61, 92}, {101, 45, 83, 59, 92, 27}, }; int numWorkers = costs.length; int numTasks = costs[0].length; final int[] allWorkers = IntStream.range(0, numWorkers).toArray(); final int[] allTasks = IntStream.range(0, numTasks).toArray(); // Allowed groups of workers: int[][] group1 = { // group of worker 0-3 {2, 3}, {1, 3}, {1, 2}, {0, 1}, {0, 2}, }; int[][] group2 = { // group of worker 4-7 {6, 7}, {5, 7}, {5, 6}, {4, 5}, {4, 7}, }; int[][] group3 = { // group of worker 8-11 {10, 11}, {9, 11}, {9, 10}, {8, 10}, {8, 11}, }; // Solver // Create the linear solver with the SCIP backend. MPSolver solver = MPSolver.createSolver("SCIP"); if (solver == null) { System.out.println("Could not create solver SCIP"); return; } // Variables // x[i][j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. MPVariable[][] x = new MPVariable[numWorkers][numTasks]; for (int worker : allWorkers) { for (int task : allTasks) { x[worker][task] = solver.makeBoolVar("x[" + worker + "," + task + "]"); } } // Constraints // Each worker is assigned to at most one task. for (int worker : allWorkers) { MPConstraint constraint = solver.makeConstraint(0, 1, ""); for (int task : allTasks) { constraint.setCoefficient(x[worker][task], 1); } } // Each task is assigned to exactly one worker. for (int task : allTasks) { MPConstraint constraint = solver.makeConstraint(1, 1, ""); for (int worker : allWorkers) { constraint.setCoefficient(x[worker][task], 1); } } // Create variables for each worker, indicating whether they work on some task. MPVariable[] work = new MPVariable[numWorkers]; for (int worker : allWorkers) { work[worker] = solver.makeBoolVar("work[" + worker + "]"); } for (int worker : allWorkers) { // MPVariable[] vars = new MPVariable[numTasks]; MPConstraint constraint = solver.makeConstraint(0, 0, ""); for (int task : allTasks) { // vars[task] = x[worker][task]; constraint.setCoefficient(x[worker][task], 1); } // solver.addEquality(work[worker], LinearExpr.sum(vars)); constraint.setCoefficient(work[worker], -1); } // Group1 MPConstraint constraintG1 = solver.makeConstraint(1, 1, ""); for (int i = 0; i < group1.length; ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise MPConstraint constraint = solver.makeConstraint(0, 1, ""); constraint.setCoefficient(work[group1[i][0]], 1); constraint.setCoefficient(work[group1[i][1]], 1); MPVariable p = solver.makeBoolVar("g1_p" + i); constraint.setCoefficient(p, -2); constraintG1.setCoefficient(p, 1); } // Group2 MPConstraint constraintG2 = solver.makeConstraint(1, 1, ""); for (int i = 0; i < group2.length; ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise MPConstraint constraint = solver.makeConstraint(0, 1, ""); constraint.setCoefficient(work[group2[i][0]], 1); constraint.setCoefficient(work[group2[i][1]], 1); MPVariable p = solver.makeBoolVar("g2_p" + i); constraint.setCoefficient(p, -2); constraintG2.setCoefficient(p, 1); } // Group3 MPConstraint constraintG3 = solver.makeConstraint(1, 1, ""); for (int i = 0; i < group3.length; ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise MPConstraint constraint = solver.makeConstraint(0, 1, ""); constraint.setCoefficient(work[group3[i][0]], 1); constraint.setCoefficient(work[group3[i][1]], 1); MPVariable p = solver.makeBoolVar("g3_p" + i); constraint.setCoefficient(p, -2); constraintG3.setCoefficient(p, 1); } // Objective MPObjective objective = solver.objective(); for (int worker : allWorkers) { for (int task : allTasks) { objective.setCoefficient(x[worker][task], costs[worker][task]); } } objective.setMinimization(); // Solve MPSolver.ResultStatus resultStatus = solver.solve(); // Print solution. // Check that the problem has a feasible solution. if (resultStatus == MPSolver.ResultStatus.OPTIMAL || resultStatus == MPSolver.ResultStatus.FEASIBLE) { System.out.println("Total cost: " + objective.value() + "\n"); for (int worker : allWorkers) { for (int task : allTasks) { // Test if x[i][j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker][task].solutionValue() > 0.5) { System.out.println("Worker " + worker + " assigned to task " + task + ". Cost: " + costs[worker][task]); } } } } else { System.err.println("No solution found."); } } private AssignmentGroupsMip() {} }
#C
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.LinearSolver; public class AssignmentGroupsMip { static void Main() { // Data. int[,] costs = { { 90, 76, 75, 70, 50, 74 }, { 35, 85, 55, 65, 48, 101 }, { 125, 95, 90, 105, 59, 120 }, { 45, 110, 95, 115, 104, 83 }, { 60, 105, 80, 75, 59, 62 }, { 45, 65, 110, 95, 47, 31 }, { 38, 51, 107, 41, 69, 99 }, { 47, 85, 57, 71, 92, 77 }, { 39, 63, 97, 49, 118, 56 }, { 47, 101, 71, 60, 88, 109 }, { 17, 39, 103, 64, 61, 92 }, { 101, 45, 83, 59, 92, 27 }, }; int numWorkers = costs.GetLength(0); int numTasks = costs.GetLength(1); int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray(); int[] allTasks = Enumerable.Range(0, numTasks).ToArray(); // Allowed groups of workers: int[,] group1 = { // group of worker 0-3 { 2, 3 }, { 1, 3 }, { 1, 2 }, { 0, 1 }, { 0, 2 }, }; int[,] group2 = { // group of worker 4-7 { 6, 7 }, { 5, 7 }, { 5, 6 }, { 4, 5 }, { 4, 7 }, }; int[,] group3 = { // group of worker 8-11 { 10, 11 }, { 9, 11 }, { 9, 10 }, { 8, 10 }, { 8, 11 }, }; // Solver. Solver solver = Solver.CreateSolver("SCIP"); if (solver is null) { return; } // Variables. // x[i, j] is an array of 0-1 variables, which will be 1 // if worker i is assigned to task j. Variable[,] x = new Variable[numWorkers, numTasks]; foreach (int worker in allWorkers) { foreach (int task in allTasks) { x[worker, task] = solver.MakeBoolVar($"x[{worker},{task}]"); } } // Constraints // Each worker is assigned to at most one task. foreach (int worker in allWorkers) { Constraint constraint = solver.MakeConstraint(0, 1, ""); foreach (int task in allTasks) { constraint.SetCoefficient(x[worker, task], 1); } } // Each task is assigned to exactly one worker. foreach (int task in allTasks) { Constraint constraint = solver.MakeConstraint(1, 1, ""); foreach (int worker in allWorkers) { constraint.SetCoefficient(x[worker, task], 1); } } // Create variables for each worker, indicating whether they work on some task. Variable[] work = new Variable[numWorkers]; foreach (int worker in allWorkers) { work[worker] = solver.MakeBoolVar($"work[{worker}]"); } foreach (int worker in allWorkers) { Variable[] vars = new Variable[numTasks]; foreach (int task in allTasks) { vars[task] = x[worker, task]; } solver.Add(work[worker] == LinearExprArrayHelper.Sum(vars)); } // Group1 Constraint constraint_g1 = solver.MakeConstraint(1, 1, ""); for (int i = 0; i < group1.GetLength(0); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise Constraint constraint = solver.MakeConstraint(0, 1, ""); constraint.SetCoefficient(work[group1[i, 0]], 1); constraint.SetCoefficient(work[group1[i, 1]], 1); Variable p = solver.MakeBoolVar($"g1_p{i}"); constraint.SetCoefficient(p, -2); constraint_g1.SetCoefficient(p, 1); } // Group2 Constraint constraint_g2 = solver.MakeConstraint(1, 1, ""); for (int i = 0; i < group2.GetLength(0); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise Constraint constraint = solver.MakeConstraint(0, 1, ""); constraint.SetCoefficient(work[group2[i, 0]], 1); constraint.SetCoefficient(work[group2[i, 1]], 1); Variable p = solver.MakeBoolVar($"g2_p{i}"); constraint.SetCoefficient(p, -2); constraint_g2.SetCoefficient(p, 1); } // Group3 Constraint constraint_g3 = solver.MakeConstraint(1, 1, ""); for (int i = 0; i < group3.GetLength(0); ++i) { // a*b can be transformed into 0 <= a + b - 2*p <= 1 with p in [0,1] // p is True if a AND b, False otherwise Constraint constraint = solver.MakeConstraint(0, 1, ""); constraint.SetCoefficient(work[group3[i, 0]], 1); constraint.SetCoefficient(work[group3[i, 1]], 1); Variable p = solver.MakeBoolVar($"g3_p{i}"); constraint.SetCoefficient(p, -2); constraint_g3.SetCoefficient(p, 1); } // Objective Objective objective = solver.Objective(); foreach (int worker in allWorkers) { foreach (int task in allTasks) { objective.SetCoefficient(x[worker, task], costs[worker, task]); } } objective.SetMinimization(); // Solve Solver.ResultStatus resultStatus = solver.Solve(); // Print solution. // Check that the problem has a feasible solution. if (resultStatus == Solver.ResultStatus.OPTIMAL || resultStatus == Solver.ResultStatus.FEASIBLE) { Console.WriteLine($"Total cost: {solver.Objective().Value()}\n"); foreach (int worker in allWorkers) { foreach (int task in allTasks) { // Test if x[i, j] is 0 or 1 (with tolerance for floating point // arithmetic). if (x[worker, task].SolutionValue() > 0.5) { Console.WriteLine($"Worker {worker} assigned to task {task}. Cost: {costs[worker, task]}"); } } } } else { Console.WriteLine("No solution found."); } } }
أوقات الحلول
في ما يلي أوقات الحل للأداتَين:
- CP-SAT: 0.0113 ثانية
- MIP: 0.3281 ثانية
ويكون اختبار CP-SAT أسرع بكثير من MIP في هذه المشكلة.