Atribuição com grupos permitidos

Esta seção descreve um problema de atribuição em que apenas alguns tipos grupos de workers podem ser atribuídos às tarefas. No exemplo, há 12 trabalhadores, numerados de 0 a 11. Os grupos permitidos são combinações dos seguintes pares de workers.

  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]]

Um grupo permitido pode ser qualquer combinação de três pares de workers, um par de para cada grupo1, grupo2 e grupo3. Por exemplo, combinar [2, 3], [6, 7] e [10, 11] resulta no grupo [2, 3, 6, 7, 10, 11]. Como cada um dos três conjuntos contém cinco elementos, o número total de grupos é 5 * 5 * 5 = 125.

Observe que um grupo de workers pode ser uma solução para o problema se ele pertencer a qualquer um dos grupos permitidos. Em outras palavras, o conjunto viável consiste em pontos em que qualquer uma das restrições é satisfeita. Esse é um exemplo de um problema não convexo. Por outro lado, o Exemplo de MIP, descrito anteriormente, é um problema convexo: para que um ponto seja viável, todas as restrições precisam ser satisfeitos.

Para problemas não convexos como este, o solucionador CP-SAT costuma ser mais rápido um solucionador MIP. As seções a seguir apresentam soluções para o problema usando o solucionador CP-SAT e o solucionador MIP e comparar os tempos de solução dos dois.

Solução CP-SAT

Primeiro, vamos descrever uma solução para o problema usando o solucionador CP-SAT.

Importar as bibliotecas

O código a seguir importa a biblioteca necessária.

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;

Definir os dados

O código a seguir cria os dados para o programa.

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();

Criar os grupos permitidos

Para definir os grupos de workers permitidos para o solucionador CP-SAT, crie arquivos matrizes que indicam quais workers pertencem a um grupo. Por exemplo, para group1 (workers 0 - 3), o vetor binário [0, 0, 1, 1] especifica o grupo que contém workers 2 e 3.

Observação :dessa forma, especificar os grupos é um pouco diferente de como eles são. especificada na solução MIP, mas equivale à mesma coisa.

As matrizes a seguir definem os grupos permitidos de workers.

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
};

Para o CP-SAT, não é necessário criar todas as 125 combinações desses vetores em um loop. O solucionador CP-SAT fornece um método, AllowedAssignments, que permite especificar separadamente as restrições para os grupos permitidos. para cada um dos três conjuntos de workers (0 - 3, 4 - 7 e 8 - 11). Veja como funciona:

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);

As variáveis work[i] são de 0 a 1 que indicam o status do trabalho ou cada worker. Ou seja, work[i] é igual a 1 se o worker i for atribuído a uma tarefa. 0 caso contrário. A linha solver.Add(solver.AllowedAssignments([work[0], work[1], work[2], work[3]], group1)) define a restrição de que o status de trabalho dos workers 0 - 3 deve corresponder a um dos os padrões em group1. Confira os detalhes completos do código no nesta seção.

Criar o modelo

O código a seguir cria o modelo.

Python

model = cp_model.CpModel()

C++

CpModelBuilder cp_model;

Java

CpModel model = new CpModel();

C#

CpModel model = new CpModel();

Criar as variáveis

O código a seguir cria uma matriz de variáveis para o problema.

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}]");
    }
}

Adicionar as restrições

O código a seguir cria as restrições para o programa.

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);
}

Criar o objetivo

O código a seguir cria a função de objetivo.

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);

Invocar o solucionador

O código a seguir invoca o solucionador e exibe os resultados.

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}");

Mostrar os resultados

Agora, podemos mostrar a solução.

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.");
}

Esta é a saída do programa.

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

Todo o programa

Aqui está o programa completo.

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");
    }
}

Solução MIP

Em seguida, descrevemos uma solução para o problema usando o solucionador MIP.

Importar as bibliotecas

O código a seguir importa a biblioteca necessária.

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;

Definir os dados

O código a seguir cria os dados para o programa.

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();

Criar os grupos permitidos

O código a seguir cria os grupos permitidos percorrendo os três conjuntos dos subgrupos mostrados acima.

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 },
};

Declarar o solucionador

O código a seguir cria o solucionador.

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;
}

Criar as variáveis

O código a seguir cria uma matriz de variáveis para o problema.

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}]");
    }
}

Adicionar as restrições

O código a seguir cria as restrições para o programa.

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);
    }
}

Criar o objetivo

O código a seguir cria a função de objetivo.

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();

Invocar o solucionador

O código a seguir invoca o solucionador e exibe os resultados.

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();

Mostrar os resultados

Agora, podemos mostrar a solução.

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.");
}

Esta é a saída do programa:

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

Todo o programa

Aqui está o programa completo.

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.");
        }
    }
}

Tempos das soluções

Os tempos de solução dos dois solucionadores são os seguintes:

• CP-SAT: 0,0113 segundos
• MIP: 0,3281 segundo

O CP-SAT é significativamente mais rápido do que o MIP para esse problema.