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Zuweisung mit zulässigen Gruppen

In diesem Abschnitt wird ein Zuweisungsproblem beschrieben, bei dem nur bestimmte Gruppen von Workern können den Aufgaben zugewiesen werden. In diesem Beispiel sind zwölf Worker, nummeriert von 0–11. Die zulässigen Gruppen sind Kombinationen der folgenden Worker-Paare.

  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]]

Eine zulässige Gruppe kann eine beliebige Kombination aus drei Worker-Paaren sein, ein Paar aus „group1“, „group2“ und „group3“. Beispielsweise führt die Kombination von [2, 3], [6, 7] und [10, 11] zu den zulässigen Gruppe [2, 3, 6, 7, 10, 11]. Da jeder der drei Sätze fünf Elemente enthält, ist die Gesamtzahl der zulässigen Gruppen ist 5 * 5 * 5 = 125.

Beachten Sie, dass eine Gruppe von Workern eine Lösung für das Problem sein kann, wenn sie zu einer der zulässigen Gruppen. Mit anderen Worten, die mögliche Menge besteht aus Punkte, für die eine der Einschränkungen erfüllt ist. Dies ist ein Beispiel für ein nicht konvexes Problem. Im Gegensatz dazu enthält das MIP-Beispiel, das ist ein konvexes Problem: Damit ein Punkt realisierbar ist, müssen alle Einschränkungen zufrieden sein müssen.

Bei nicht konvexen Problemen wie diesem ist der CP-SAT-Löser in der Regel schneller als einen MIP-Resolver. In den folgenden Abschnitten werden Lösungen für das Problem mithilfe des CP-SAT-Lösers vorgestellt. und den MIP-Löser und vergleichen die Zeitangaben für die jeweilige Lösung.

CP-SAT-Lösung

Zuerst beschreiben wir eine Lösung für das Problem mithilfe des CP-SAT-Lösers.

Bibliotheken importieren

Mit dem folgenden Code wird die erforderliche Bibliothek importiert.

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;

Daten definieren

Mit dem folgenden Code werden die Daten für das Programm erstellt.

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();

Zulässige Gruppen erstellen

Zum Definieren der zulässigen Arbeitergruppen für den CP-SAT-Löser erstellen Sie eine Binärdatei Arrays, die anzeigen, welche Worker zu einer Gruppe gehören. Beispiel: Für group1 (Worker 0–3) gibt der binäre Vektor [0, 0, 1, 1] die Gruppe an, die Worker 2 und 3.

Die folgenden Arrays definieren die zulässigen Gruppen von Workern.

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
};

Für CP-SAT sind nicht alle 125 Kombinationen dieser Vektoren erforderlich. in einer Schleife. Der CP-SAT-Löser bietet die Methode AllowedAssignments, Hier können Sie die Einschränkungen für die zulässigen Gruppen separat festlegen. für jeden der drei Worker-Sätze (0–3, 4–7 und 8–11). So funktionierts:

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);

Die Variablen work[i] sind 0-1-Variablen, die den Arbeitsstatus oder für jeden Worker. Das heißt, work[i] ist 1, wenn Worker i einer Aufgabe zugewiesen ist, und Andernfalls 0. Die Linie solver.Add(solver.AllowedAssignments([work[0], work[1], work[2], work[3]], group1)) definiert die Einschränkung, dass der Arbeitsstatus der Worker 0–3 mit einem der folgenden Werte übereinstimmen muss: die Muster in group1. Die vollständigen Codedetails findest du in der nächsten .

Modell erstellen

Mit dem folgenden Code wird das Modell erstellt.

Python

model = cp_model.CpModel()

C++

CpModelBuilder cp_model;

Java

CpModel model = new CpModel();

C#

CpModel model = new CpModel();

Variablen erstellen

Mit dem folgenden Code wird ein Array von Variablen für das Problem erstellt.

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}]");
    }
}

Einschränkungen hinzufügen

Der folgende Code erstellt die Einschränkungen für das Programm.

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);
}

Ziel erstellen

Mit dem folgenden Code wird die Zielfunktion erstellt.

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);

Solver aufrufen

Mit dem folgenden Code wird der Löser aufgerufen und die Ergebnisse angezeigt.

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}");

Ergebnisse anzeigen

Jetzt können wir die Lösung drucken.

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.");
}

Hier ist die Ausgabe des Programms.

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

Das gesamte Programm

Hier ist das gesamte Programm.

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");
    }
}

MIP-Lösung

Als Nächstes beschreiben wir eine Lösung für dieses Problem mithilfe des MIP-lösers.

Bibliotheken importieren

Mit dem folgenden Code wird die erforderliche Bibliothek importiert.

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;

Daten definieren

Mit dem folgenden Code werden die Daten für das Programm erstellt.

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();

Zulässige Gruppen erstellen

Mit dem folgenden Code werden die zulässigen Gruppen erstellt, indem die drei Sets als Schleife durchlaufen werden. von Untergruppen (siehe oben).

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 },
};

Löser deklarieren

Mit dem folgenden Code wird der Löser erstellt.

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;
}

Variablen erstellen

Mit dem folgenden Code wird ein Array von Variablen für das Problem erstellt.

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}]");
    }
}

Einschränkungen hinzufügen

Der folgende Code erstellt die Einschränkungen für das Programm.

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);
    }
}

Ziel erstellen

Mit dem folgenden Code wird die Zielfunktion erstellt.

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();

Solver aufrufen

Mit dem folgenden Code wird der Löser aufgerufen und die Ergebnisse angezeigt.

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();

Ergebnisse anzeigen

Jetzt können wir die Lösung drucken.

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.");
}

Hier ist die Ausgabe des Programms:

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

Das gesamte Programm

Hier ist das gesamte Programm.

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.");
        }
    }
}

Lösungszeiten

Die Lösungszeiten für die beiden Matherechner sehen so aus:

CP-SAT: 0,0113 Sekunden
MIP: 0,3281 Sekunden

CP-SAT ist bei diesem Problem wesentlich schneller als MIP.