تعيين المهام مع فرق من العاملين

هناك العديد من الإصدارات الخاصة بمشكلة المهمة، والتي تحتوي على القيود على العمال أو المهام. في المثال التالي، هناك ستة موظفين مقسمًا إلى فريقين، ويمكن لكل فريق أداء مهمتين على الأكثر.

تقدم الأقسام التالية برنامج بايثون يحل هذه المسألة باستخدام أداة حل CP-SAT أو MIP. للحصول على حل باستخدام أداة حل تدفق التكلفة الدنيا، راجع القسم المهام الدراسية مع الفِرق:

حل CP-SAT

دعونا نلقي نظرة أولاً على حل CP-SAT للمشكلة.

استيراد المكتبات

يستورد الرمز التالي المكتبة المطلوبة.

Python

from ortools.sat.python import cp_model

C++‎

#include <stdlib.h>

#include <numeric>
#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;

تعريف البيانات

ينشئ الرمز التالي البيانات للبرنامج.

Python

costs = [
    [90, 76, 75, 70],
    [35, 85, 55, 65],
    [125, 95, 90, 105],
    [45, 110, 95, 115],
    [60, 105, 80, 75],
    [45, 65, 110, 95],
]
num_workers = len(costs)
num_tasks = len(costs[0])

team1 = [0, 2, 4]
team2 = [1, 3, 5]
# Maximum total of tasks for any team
team_max = 2

C++‎

const std::vector<std::vector<int>> costs = {{
    {{90, 76, 75, 70}},
    {{35, 85, 55, 65}},
    {{125, 95, 90, 105}},
    {{45, 110, 95, 115}},
    {{60, 105, 80, 75}},
    {{45, 65, 110, 95}},
}};
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);

const std::vector<int> team1 = {{0, 2, 4}};
const std::vector<int> team2 = {{1, 3, 5}};
// Maximum total of tasks for any team
const int team_max = 2;

Java

int[][] costs = {
    {90, 76, 75, 70},
    {35, 85, 55, 65},
    {125, 95, 90, 105},
    {45, 110, 95, 115},
    {60, 105, 80, 75},
    {45, 65, 110, 95},
};
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();

final int[] team1 = {0, 2, 4};
final int[] team2 = {1, 3, 5};
// Maximum total of tasks for any team
final int teamMax = 2;

#C

int[,] costs = {
    { 90, 76, 75, 70 },   { 35, 85, 55, 65 },  { 125, 95, 90, 105 },
    { 45, 110, 95, 115 }, { 60, 105, 80, 75 }, { 45, 65, 110, 95 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);

int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray();
int[] allTasks = Enumerable.Range(0, numTasks).ToArray();

int[] team1 = { 0, 2, 4 };
int[] team2 = { 1, 3, 5 };
// Maximum total of tasks for any team
int teamMax = 2;

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];
// Variables in a 1-dim array.
for (int worker : allWorkers) {
  for (int task : allTasks) {
    x[worker][task] = model.newBoolVar("x[" + worker + "," + task + "]");
  }
}

#C

BoolVar[,] x = new BoolVar[numWorkers, numTasks];
foreach (int worker in allWorkers)
{
    foreach (int task in allTasks)
    {
        x[worker, task] = model.NewBoolVar($"x[{worker},{task}]");
    }
}

هناك متغير واحد لكل زوج من العامل والمهمة. لاحظ أن العمال ويتم ترقيم المهام بقيمة 0 - 5، بينما تم ترقيم المهام بـ 0 - 3، على عكس المثال الأصلي، والذي كان يلزم فيه ترقيم جميع العقد بشكل مختلف، على النحو المطلوب بالحد الأدنى للتكلفة أداة حل التدفق.

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))

# Each team takes at most two tasks.
team1_tasks = []
for worker in team1:
    for task in range(num_tasks):
        team1_tasks.append(x[worker, task])
model.add(sum(team1_tasks) <= team_max)

team2_tasks = []
for worker in team2:
    for task in range(num_tasks):
        team2_tasks.append(x[worker, task])
model.add(sum(team2_tasks) <= team_max)

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);
}

// Each team takes at most two tasks.
LinearExpr team1_tasks;
for (int worker : team1) {
  for (int task : all_tasks) {
    team1_tasks += x[worker][task];
  }
}
cp_model.AddLessOrEqual(team1_tasks, team_max);

LinearExpr team2_tasks;
for (int worker : team2) {
  for (int task : all_tasks) {
    team2_tasks += x[worker][task];
  }
}
cp_model.AddLessOrEqual(team2_tasks, team_max);

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);
}

// Each team takes at most two tasks.
LinearExprBuilder team1Tasks = LinearExpr.newBuilder();
for (int worker : team1) {
  for (int task : allTasks) {
    team1Tasks.add(x[worker][task]);
  }
}
model.addLessOrEqual(team1Tasks, teamMax);

LinearExprBuilder team2Tasks = LinearExpr.newBuilder();
for (int worker : team2) {
  for (int task : allTasks) {
    team2Tasks.add(x[worker][task]);
  }
}
model.addLessOrEqual(team2Tasks, teamMax);

#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);
}

// Each team takes at most two tasks.
List<IntVar> team1Tasks = new List<IntVar>();
foreach (int worker in team1)
{
    foreach (int task in allTasks)
    {
        team1Tasks.Add(x[worker, task]);
    }
}
model.Add(LinearExpr.Sum(team1Tasks.ToArray()) <= teamMax);

List<IntVar> team2Tasks = new List<IntVar>();
foreach (int worker in team2)
{
    foreach (int task in allTasks)
    {
        team2Tasks.Add(x[worker, task]);
    }
}
model.Add(LinearExpr.Sum(team2Tasks.ToArray()) <= teamMax);

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);

قيمة الدالة الموضوعية هي التكلفة الإجمالية على جميع المتغيرات التي يتم تعيين القيمة 1 بواسطة أداة الحلّ.

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.

Total cost:  250
Worker 0 assigned to task 2. Cost = 75
Worker 1 assigned to task 0. Cost = 35
Worker 4 assigned to task 3. Cost = 75
Worker 5 assigned to task 1. Cost = 65
Time =  6  milliseconds

البرنامج بأكمله

إليك البرنامج بالكامل.

Python

"""Solves a simple assignment problem."""
from ortools.sat.python import cp_model


def main() -> None:
    # Data
    costs = [
        [90, 76, 75, 70],
        [35, 85, 55, 65],
        [125, 95, 90, 105],
        [45, 110, 95, 115],
        [60, 105, 80, 75],
        [45, 65, 110, 95],
    ]
    num_workers = len(costs)
    num_tasks = len(costs[0])

    team1 = [0, 2, 4]
    team2 = [1, 3, 5]
    # Maximum total of tasks for any team
    team_max = 2

    # 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))

    # Each team takes at most two tasks.
    team1_tasks = []
    for worker in team1:
        for task in range(num_tasks):
            team1_tasks.append(x[worker, task])
    model.add(sum(team1_tasks) <= team_max)

    team2_tasks = []
    for worker in team2:
        for task in range(num_tasks):
            team2_tasks.append(x[worker, task])
    model.add(sum(team2_tasks) <= team_max)

    # 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 a simple assignment problem.
#include <stdlib.h>

#include <numeric>
#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 AssignmentTeamsSat() {
  // Data
  const std::vector<std::vector<int>> costs = {{
      {{90, 76, 75, 70}},
      {{35, 85, 55, 65}},
      {{125, 95, 90, 105}},
      {{45, 110, 95, 115}},
      {{60, 105, 80, 75}},
      {{45, 65, 110, 95}},
  }};
  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);

  const std::vector<int> team1 = {{0, 2, 4}};
  const std::vector<int> team2 = {{1, 3, 5}};
  // Maximum total of tasks for any team
  const int team_max = 2;

  // 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);
  }

  // Each team takes at most two tasks.
  LinearExpr team1_tasks;
  for (int worker : team1) {
    for (int task : all_tasks) {
      team1_tasks += x[worker][task];
    }
  }
  cp_model.AddLessOrEqual(team1_tasks, team_max);

  LinearExpr team2_tasks;
  for (int worker : team2) {
    for (int task : all_tasks) {
      team2_tasks += x[worker][task];
    }
  }
  cp_model.AddLessOrEqual(team2_tasks, team_max);

  // 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::AssignmentTeamsSat();
  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.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 AssignmentTeamsSat {
  public static void main(String[] args) {
    Loader.loadNativeLibraries();
    // Data
    int[][] costs = {
        {90, 76, 75, 70},
        {35, 85, 55, 65},
        {125, 95, 90, 105},
        {45, 110, 95, 115},
        {60, 105, 80, 75},
        {45, 65, 110, 95},
    };
    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();

    final int[] team1 = {0, 2, 4};
    final int[] team2 = {1, 3, 5};
    // Maximum total of tasks for any team
    final int teamMax = 2;

    // Model
    CpModel model = new CpModel();

    // Variables
    Literal[][] x = new Literal[numWorkers][numTasks];
    // Variables in a 1-dim array.
    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);
    }

    // Each team takes at most two tasks.
    LinearExprBuilder team1Tasks = LinearExpr.newBuilder();
    for (int worker : team1) {
      for (int task : allTasks) {
        team1Tasks.add(x[worker][task]);
      }
    }
    model.addLessOrEqual(team1Tasks, teamMax);

    LinearExprBuilder team2Tasks = LinearExpr.newBuilder();
    for (int worker : team2) {
      for (int task : allTasks) {
        team2Tasks.add(x[worker][task]);
      }
    }
    model.addLessOrEqual(team2Tasks, teamMax);

    // 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 AssignmentTeamsSat() {}
}

#C

using System;
using System.Collections.Generic;
using System.Linq;
using Google.OrTools.Sat;

public class AssignmentTeamsSat
{
    public static void Main(String[] args)
    {
        // Data.
        int[,] costs = {
            { 90, 76, 75, 70 },   { 35, 85, 55, 65 },  { 125, 95, 90, 105 },
            { 45, 110, 95, 115 }, { 60, 105, 80, 75 }, { 45, 65, 110, 95 },
        };
        int numWorkers = costs.GetLength(0);
        int numTasks = costs.GetLength(1);

        int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray();
        int[] allTasks = Enumerable.Range(0, numTasks).ToArray();

        int[] team1 = { 0, 2, 4 };
        int[] team2 = { 1, 3, 5 };
        // Maximum total of tasks for any team
        int teamMax = 2;

        // Model.
        CpModel model = new CpModel();

        // Variables.
        BoolVar[,] x = new BoolVar[numWorkers, numTasks];
        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);
        }

        // Each team takes at most two tasks.
        List<IntVar> team1Tasks = new List<IntVar>();
        foreach (int worker in team1)
        {
            foreach (int task in allTasks)
            {
                team1Tasks.Add(x[worker, task]);
            }
        }
        model.Add(LinearExpr.Sum(team1Tasks.ToArray()) <= teamMax);

        List<IntVar> team2Tasks = new List<IntVar>();
        foreach (int worker in team2)
        {
            foreach (int task in allTasks)
            {
                team2Tasks.Add(x[worker, task]);
            }
        }
        model.Add(LinearExpr.Sum(team2Tasks.ToArray()) <= teamMax);

        // 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، سنوضح حلاً لمشكلة التعيين هذه 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 <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],
    [35, 85, 55, 65],
    [125, 95, 90, 105],
    [45, 110, 95, 115],
    [60, 105, 80, 75],
    [45, 65, 110, 95],
]
num_workers = len(costs)
num_tasks = len(costs[0])

team1 = [0, 2, 4]
team2 = [1, 3, 5]
# Maximum total of tasks for any team
team_max = 2

C++‎

const std::vector<std::vector<int64_t>> costs = {{
    {{90, 76, 75, 70}},
    {{35, 85, 55, 65}},
    {{125, 95, 90, 105}},
    {{45, 110, 95, 115}},
    {{60, 105, 80, 75}},
    {{45, 65, 110, 95}},
}};
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);

const std::vector<int64_t> team1 = {{0, 2, 4}};
const std::vector<int64_t> team2 = {{1, 3, 5}};
// Maximum total of tasks for any team
const int team_max = 2;

Java

double[][] costs = {
    {90, 76, 75, 70},
    {35, 85, 55, 65},
    {125, 95, 90, 105},
    {45, 110, 95, 115},
    {60, 105, 80, 75},
    {45, 65, 110, 95},
};
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();

final int[] team1 = {0, 2, 4};
final int[] team2 = {1, 3, 5};
// Maximum total of tasks for any team
final int teamMax = 2;

#C

int[,] costs = {
    { 90, 76, 75, 70 },   { 35, 85, 55, 65 },  { 125, 95, 90, 105 },
    { 45, 110, 95, 115 }, { 60, 105, 80, 75 }, { 45, 65, 110, 95 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);

int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray();
int[] allTasks = Enumerable.Range(0, numTasks).ToArray();

int[] team1 = { 0, 2, 4 };
int[] team2 = { 1, 3, 5 };
// Maximum total of tasks for any team
int teamMax = 2;

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[i, j] is an array of 0-1 variables, which will be 1
# if worker i is assigned to task j.
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

# Each worker is assigned at most 1 task.
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)

# Each team takes at most two tasks.
team1_tasks = []
for worker in team1:
    for task in range(num_tasks):
        team1_tasks.append(x[worker, task])
solver.Add(solver.Sum(team1_tasks) <= team_max)

team2_tasks = []
for worker in team2:
    for task in range(num_tasks):
        team2_tasks.append(x[worker, task])
solver.Add(solver.Sum(team2_tasks) <= team_max)

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);
}

// Each team takes at most two tasks.
LinearExpr team1_tasks;
for (int worker : team1) {
  for (int task : all_tasks) {
    team1_tasks += x[worker][task];
  }
}
solver->MakeRowConstraint(team1_tasks <= team_max);

LinearExpr team2_tasks;
for (int worker : team2) {
  for (int task : all_tasks) {
    team2_tasks += x[worker][task];
  }
}
solver->MakeRowConstraint(team2_tasks <= team_max);

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);
  }
}

// Each team takes at most two tasks.
MPConstraint team1Tasks = solver.makeConstraint(0, teamMax, "");
for (int worker : team1) {
  for (int task : allTasks) {
    team1Tasks.setCoefficient(x[worker][task], 1);
  }
}

MPConstraint team2Tasks = solver.makeConstraint(0, teamMax, "");
for (int worker : team2) {
  for (int task : allTasks) {
    team2Tasks.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);
    }
}

// Each team takes at most two tasks.
Constraint team1Tasks = solver.MakeConstraint(0, teamMax, "");
foreach (int worker in team1)
{
    foreach (int task in allTasks)
    {
        team1Tasks.SetCoefficient(x[worker, task], 1);
    }
}

Constraint team2Tasks = solver.MakeConstraint(0, teamMax, "");
foreach (int worker in team2)
{
    foreach (int task in allTasks)
    {
        team2Tasks.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.")
print(f"Time = {solver.WallTime()} ms")

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 مخرجات البرنامج.

Minimum cost assignment:  250.0
Worker 0 assigned to task 2. Cost = 75
Worker 1 assigned to task 0. Cost = 35
Worker 4 assigned to task 3. Cost = 75
Worker 5 assigned to task 1. Cost = 65
Time =  6  milliseconds

البرنامج بأكمله

إليك البرنامج بالكامل.

Python

"""MIP example that solves an assignment problem."""
from ortools.linear_solver import pywraplp


def main():
    # Data
    costs = [
        [90, 76, 75, 70],
        [35, 85, 55, 65],
        [125, 95, 90, 105],
        [45, 110, 95, 115],
        [60, 105, 80, 75],
        [45, 65, 110, 95],
    ]
    num_workers = len(costs)
    num_tasks = len(costs[0])

    team1 = [0, 2, 4]
    team2 = [1, 3, 5]
    # Maximum total of tasks for any team
    team_max = 2

    # Solver
    # Create the mip solver with the SCIP backend.
    solver = pywraplp.Solver.CreateSolver("SCIP")
    if not solver:
        return

    # Variables
    # x[i, j] is an array of 0-1 variables, which will be 1
    # if worker i is assigned to task j.
    x = {}
    for worker in range(num_workers):
        for task in range(num_tasks):
            x[worker, task] = solver.BoolVar(f"x[{worker},{task}]")

    # Constraints
    # Each worker is assigned at most 1 task.
    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)

    # Each team takes at most two tasks.
    team1_tasks = []
    for worker in team1:
        for task in range(num_tasks):
            team1_tasks.append(x[worker, task])
    solver.Add(solver.Sum(team1_tasks) <= team_max)

    team2_tasks = []
    for worker in team2:
        for task in range(num_tasks):
            team2_tasks.append(x[worker, task])
    solver.Add(solver.Sum(team2_tasks) <= team_max)

    # 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.")
    print(f"Time = {solver.WallTime()} ms")


if __name__ == "__main__":
    main()

C++‎

// Solve a simple assignment problem.
#include <cstdint>
#include <memory>
#include <numeric>
#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}},
      {{35, 85, 55, 65}},
      {{125, 95, 90, 105}},
      {{45, 110, 95, 115}},
      {{60, 105, 80, 75}},
      {{45, 65, 110, 95}},
  }};
  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);

  const std::vector<int64_t> team1 = {{0, 2, 4}};
  const std::vector<int64_t> team2 = {{1, 3, 5}};
  // Maximum total of tasks for any team
  const int team_max = 2;

  // 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);
  }

  // Each team takes at most two tasks.
  LinearExpr team1_tasks;
  for (int worker : team1) {
    for (int task : all_tasks) {
      team1_tasks += x[worker][task];
    }
  }
  solver->MakeRowConstraint(team1_tasks <= team_max);

  LinearExpr team2_tasks;
  for (int worker : team2) {
    for (int task : all_tasks) {
      team2_tasks += x[worker][task];
    }
  }
  solver->MakeRowConstraint(team2_tasks <= team_max);

  // 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 AssignmentTeamsMip {
  public static void main(String[] args) {
    Loader.loadNativeLibraries();
    // Data
    double[][] costs = {
        {90, 76, 75, 70},
        {35, 85, 55, 65},
        {125, 95, 90, 105},
        {45, 110, 95, 115},
        {60, 105, 80, 75},
        {45, 65, 110, 95},
    };
    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();

    final int[] team1 = {0, 2, 4};
    final int[] team2 = {1, 3, 5};
    // Maximum total of tasks for any team
    final int teamMax = 2;

    // 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);
      }
    }

    // Each team takes at most two tasks.
    MPConstraint team1Tasks = solver.makeConstraint(0, teamMax, "");
    for (int worker : team1) {
      for (int task : allTasks) {
        team1Tasks.setCoefficient(x[worker][task], 1);
      }
    }

    MPConstraint team2Tasks = solver.makeConstraint(0, teamMax, "");
    for (int worker : team2) {
      for (int task : allTasks) {
        team2Tasks.setCoefficient(x[worker][task], 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 AssignmentTeamsMip() {}
}

#C

using System;
using System.Collections.Generic;
using System.Linq;
using Google.OrTools.LinearSolver;

public class AssignmentTeamsMip
{
    static void Main()
    {
        // Data.
        int[,] costs = {
            { 90, 76, 75, 70 },   { 35, 85, 55, 65 },  { 125, 95, 90, 105 },
            { 45, 110, 95, 115 }, { 60, 105, 80, 75 }, { 45, 65, 110, 95 },
        };
        int numWorkers = costs.GetLength(0);
        int numTasks = costs.GetLength(1);

        int[] allWorkers = Enumerable.Range(0, numWorkers).ToArray();
        int[] allTasks = Enumerable.Range(0, numTasks).ToArray();

        int[] team1 = { 0, 2, 4 };
        int[] team2 = { 1, 3, 5 };
        // Maximum total of tasks for any team
        int teamMax = 2;

        // 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);
            }
        }

        // Each team takes at most two tasks.
        Constraint team1Tasks = solver.MakeConstraint(0, teamMax, "");
        foreach (int worker in team1)
        {
            foreach (int task in allTasks)
            {
                team1Tasks.SetCoefficient(x[worker, task], 1);
            }
        }

        Constraint team2Tasks = solver.MakeConstraint(0, teamMax, "");
        foreach (int worker in team2)
        {
            foreach (int task in allTasks)
            {
                team2Tasks.SetCoefficient(x[worker, task], 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.");
        }
    }
}