تکلیف با اندازه وظایف

این بخش یک مسئله تکلیف را توضیح می‌دهد که در آن هر کار اندازه‌ای دارد که نشان‌دهنده زمان یا تلاشی است که آن کار نیاز دارد. اندازه کل وظایف انجام شده توسط هر کارگر دارای یک حد ثابت است.

ما برنامه های پایتون را ارائه خواهیم کرد که این مشکل را با استفاده از حل کننده CP-SAT و حل کننده MIP حل می کنند.

راه حل CP-SAT

ابتدا، اجازه دهید نگاهی به راه حل CP-SAT برای مشکل بیندازیم.

کتابخانه ها را وارد کنید

کد زیر کتابخانه مورد نیاز را وارد می کند.

پایتون

from ortools.sat.python import cp_model

C++

#include <stdlib.h>

#include <cstdint>
#include <numeric>
#include <vector>

#include "absl/strings/str_format.h"
#include "ortools/base/logging.h"
#include "ortools/sat/cp_model.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_solver.h"

جاوا

import com.google.ortools.Loader;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverStatus;
import com.google.ortools.sat.LinearExpr;
import com.google.ortools.sat.LinearExprBuilder;
import com.google.ortools.sat.Literal;
import java.util.ArrayList;
import java.util.List;
import java.util.stream.IntStream;

سی شارپ

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

داده ها را تعریف کنید

کد زیر داده های برنامه را ایجاد می کند.

پایتون

costs = [
    [90, 76, 75, 70, 50, 74, 12, 68],
    [35, 85, 55, 65, 48, 101, 70, 83],
    [125, 95, 90, 105, 59, 120, 36, 73],
    [45, 110, 95, 115, 104, 83, 37, 71],
    [60, 105, 80, 75, 59, 62, 93, 88],
    [45, 65, 110, 95, 47, 31, 81, 34],
    [38, 51, 107, 41, 69, 99, 115, 48],
    [47, 85, 57, 71, 92, 77, 109, 36],
    [39, 63, 97, 49, 118, 56, 92, 61],
    [47, 101, 71, 60, 88, 109, 52, 90],
]
num_workers = len(costs)
num_tasks = len(costs[0])

task_sizes = [10, 7, 3, 12, 15, 4, 11, 5]
# Maximum total of task sizes for any worker
total_size_max = 15

C++

const std::vector<std::vector<int>> costs = {{
    {{90, 76, 75, 70, 50, 74, 12, 68}},
    {{35, 85, 55, 65, 48, 101, 70, 83}},
    {{125, 95, 90, 105, 59, 120, 36, 73}},
    {{45, 110, 95, 115, 104, 83, 37, 71}},
    {{60, 105, 80, 75, 59, 62, 93, 88}},
    {{45, 65, 110, 95, 47, 31, 81, 34}},
    {{38, 51, 107, 41, 69, 99, 115, 48}},
    {{47, 85, 57, 71, 92, 77, 109, 36}},
    {{39, 63, 97, 49, 118, 56, 92, 61}},
    {{47, 101, 71, 60, 88, 109, 52, 90}},
}};
const int num_workers = static_cast<int>(costs.size());
std::vector<int> all_workers(num_workers);
std::iota(all_workers.begin(), all_workers.end(), 0);

const int num_tasks = static_cast<int>(costs[0].size());
std::vector<int> all_tasks(num_tasks);
std::iota(all_tasks.begin(), all_tasks.end(), 0);

const std::vector<int64_t> task_sizes = {{10, 7, 3, 12, 15, 4, 11, 5}};
// Maximum total of task sizes for any worker
const int total_size_max = 15;

جاوا

int[][] costs = {
    {90, 76, 75, 70, 50, 74, 12, 68},
    {35, 85, 55, 65, 48, 101, 70, 83},
    {125, 95, 90, 105, 59, 120, 36, 73},
    {45, 110, 95, 115, 104, 83, 37, 71},
    {60, 105, 80, 75, 59, 62, 93, 88},
    {45, 65, 110, 95, 47, 31, 81, 34},
    {38, 51, 107, 41, 69, 99, 115, 48},
    {47, 85, 57, 71, 92, 77, 109, 36},
    {39, 63, 97, 49, 118, 56, 92, 61},
    {47, 101, 71, 60, 88, 109, 52, 90},
};
final int numWorkers = costs.length;
final int numTasks = costs[0].length;

final int[] allWorkers = IntStream.range(0, numWorkers).toArray();
final int[] allTasks = IntStream.range(0, numTasks).toArray();

final int[] taskSizes = {10, 7, 3, 12, 15, 4, 11, 5};
// Maximum total of task sizes for any worker
final int totalSizeMax = 15;

سی شارپ

int[,] costs = {
    { 90, 76, 75, 70, 50, 74, 12, 68 },    { 35, 85, 55, 65, 48, 101, 70, 83 },
    { 125, 95, 90, 105, 59, 120, 36, 73 }, { 45, 110, 95, 115, 104, 83, 37, 71 },
    { 60, 105, 80, 75, 59, 62, 93, 88 },   { 45, 65, 110, 95, 47, 31, 81, 34 },
    { 38, 51, 107, 41, 69, 99, 115, 48 },  { 47, 85, 57, 71, 92, 77, 109, 36 },
    { 39, 63, 97, 49, 118, 56, 92, 61 },   { 47, 101, 71, 60, 88, 109, 52, 90 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);

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

int[] taskSizes = { 10, 7, 3, 12, 15, 4, 11, 5 };
// Maximum total of task sizes for any worker
int totalSizeMax = 15;

مانند مثال های قبلی، ماتریس هزینه، هزینه کارگر i را برای انجام وظیفه j می دهد. بردار sizes اندازه هر کار را نشان می دهد. total_size_max حد بالایی در اندازه کل وظایف انجام شده توسط هر کارگر منفرد است.

مدل را ایجاد کنید

کد زیر مدل را ایجاد می کند.

پایتون

model = cp_model.CpModel()

C++

CpModelBuilder cp_model;

جاوا

CpModel model = new CpModel();

سی شارپ

CpModel model = new CpModel();

متغیرها را ایجاد کنید

کد زیر آرایه ای از متغیرها را برای مشکل ایجاد می کند.

پایتون

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));
  }
}

جاوا

Literal[][] x = new Literal[numWorkers][numTasks];
for (int worker : allWorkers) {
  for (int task : allTasks) {
    x[worker][task] = model.newBoolVar("x[" + worker + "," + task + "]");
  }
}

سی شارپ

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

محدودیت ها را اضافه کنید

کد زیر محدودیت هایی را برای برنامه ایجاد می کند.

پایتون

# Each worker is assigned to at most one task.
for worker in range(num_workers):
    model.add(
        sum(task_sizes[task] * x[worker, task] for task in range(num_tasks))
        <= total_size_max
    )

# 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) {
  LinearExpr task_sum;
  for (int task : all_tasks) {
    task_sum += x[worker][task] * task_sizes[task];
  }
  cp_model.AddLessOrEqual(task_sum, total_size_max);
}
// 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);
}

جاوا

// Each worker has a maximum capacity.
for (int worker : allWorkers) {
  LinearExprBuilder expr = LinearExpr.newBuilder();
  for (int task : allTasks) {
    expr.addTerm(x[worker][task], taskSizes[task]);
  }
  model.addLessOrEqual(expr, totalSizeMax);
}

// 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);
}

سی شارپ

// Each worker is assigned to at most max task size.
foreach (int worker in allWorkers)
{
    BoolVar[] vars = new BoolVar[numTasks];
    foreach (int task in allTasks)
    {
        vars[task] = x[worker, task];
    }
    model.Add(LinearExpr.WeightedSum(vars, taskSizes) <= totalSizeMax);
}

// 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);
}

هدف را ایجاد کنید

کد زیر تابع هدف را ایجاد می کند.

پایتون

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);

جاوا

LinearExprBuilder obj = LinearExpr.newBuilder();
for (int worker : allWorkers) {
  for (int task : allTasks) {
    obj.addTerm(x[worker][task], costs[worker][task]);
  }
}
model.minimize(obj);

سی شارپ

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);

حل کننده را فراخوانی کنید

کد زیر حل کننده را فراخوانی می کند.

پایتون

solver = cp_model.CpSolver()
status = solver.solve(model)

C++

const CpSolverResponse response = Solve(cp_model.Build());

جاوا

CpSolver solver = new CpSolver();
CpSolverStatus status = solver.solve(model);

سی شارپ

CpSolver solver = new CpSolver();
CpSolverStatus status = solver.Solve(model);
Console.WriteLine($"Solve status: {status}");

نمایش نتایج

اکنون می توانیم راه حل را چاپ کنیم.

پایتون

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];
    }
  }
}

جاوا

// 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.");
}

سی شارپ

// 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.");
}

اینم خروجی برنامه

Minimum cost: 326
Worker  0  assigned to task  6   Cost =  12
Worker  1  assigned to task  0   Cost =  35
Worker  1  assigned to task  2   Cost =  55
Worker  2  assigned to task  4   Cost =  59
Worker  5  assigned to task  5   Cost =  31
Worker  5  assigned to task  7   Cost =  34
Worker  6  assigned to task  1   Cost =  51
Worker  8  assigned to task  3   Cost =  49

Time =  2.2198 seconds

کل برنامه

در اینجا کل برنامه است.

پایتون

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


def main() -> None:
    # Data
    costs = [
        [90, 76, 75, 70, 50, 74, 12, 68],
        [35, 85, 55, 65, 48, 101, 70, 83],
        [125, 95, 90, 105, 59, 120, 36, 73],
        [45, 110, 95, 115, 104, 83, 37, 71],
        [60, 105, 80, 75, 59, 62, 93, 88],
        [45, 65, 110, 95, 47, 31, 81, 34],
        [38, 51, 107, 41, 69, 99, 115, 48],
        [47, 85, 57, 71, 92, 77, 109, 36],
        [39, 63, 97, 49, 118, 56, 92, 61],
        [47, 101, 71, 60, 88, 109, 52, 90],
    ]
    num_workers = len(costs)
    num_tasks = len(costs[0])

    task_sizes = [10, 7, 3, 12, 15, 4, 11, 5]
    # Maximum total of task sizes for any worker
    total_size_max = 15

    # 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(
            sum(task_sizes[task] * x[worker, task] for task in range(num_tasks))
            <= total_size_max
        )

    # 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))

    # 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.
#include <stdlib.h>

#include <cstdint>
#include <numeric>
#include <vector>

#include "absl/strings/str_format.h"
#include "ortools/base/logging.h"
#include "ortools/sat/cp_model.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_solver.h"

namespace operations_research {
namespace sat {

void AssignmentTaskSizes() {
  // Data
  const std::vector<std::vector<int>> costs = {{
      {{90, 76, 75, 70, 50, 74, 12, 68}},
      {{35, 85, 55, 65, 48, 101, 70, 83}},
      {{125, 95, 90, 105, 59, 120, 36, 73}},
      {{45, 110, 95, 115, 104, 83, 37, 71}},
      {{60, 105, 80, 75, 59, 62, 93, 88}},
      {{45, 65, 110, 95, 47, 31, 81, 34}},
      {{38, 51, 107, 41, 69, 99, 115, 48}},
      {{47, 85, 57, 71, 92, 77, 109, 36}},
      {{39, 63, 97, 49, 118, 56, 92, 61}},
      {{47, 101, 71, 60, 88, 109, 52, 90}},
  }};
  const int num_workers = static_cast<int>(costs.size());
  std::vector<int> all_workers(num_workers);
  std::iota(all_workers.begin(), all_workers.end(), 0);

  const int num_tasks = static_cast<int>(costs[0].size());
  std::vector<int> all_tasks(num_tasks);
  std::iota(all_tasks.begin(), all_tasks.end(), 0);

  const std::vector<int64_t> task_sizes = {{10, 7, 3, 12, 15, 4, 11, 5}};
  // Maximum total of task sizes for any worker
  const int total_size_max = 15;

  // 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) {
    LinearExpr task_sum;
    for (int task : all_tasks) {
      task_sum += x[worker][task] * task_sizes[task];
    }
    cp_model.AddLessOrEqual(task_sum, total_size_max);
  }
  // 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);
  }

  // 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::AssignmentTaskSizes();
  return EXIT_SUCCESS;
}

جاوا

// CP-SAT example that solves an assignment problem.
package com.google.ortools.sat.samples;
import com.google.ortools.Loader;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverStatus;
import com.google.ortools.sat.LinearExpr;
import com.google.ortools.sat.LinearExprBuilder;
import com.google.ortools.sat.Literal;
import java.util.ArrayList;
import java.util.List;
import java.util.stream.IntStream;

/** Assignment problem. */
public class AssignmentTaskSizesSat {
  public static void main(String[] args) {
    Loader.loadNativeLibraries();
    // Data
    int[][] costs = {
        {90, 76, 75, 70, 50, 74, 12, 68},
        {35, 85, 55, 65, 48, 101, 70, 83},
        {125, 95, 90, 105, 59, 120, 36, 73},
        {45, 110, 95, 115, 104, 83, 37, 71},
        {60, 105, 80, 75, 59, 62, 93, 88},
        {45, 65, 110, 95, 47, 31, 81, 34},
        {38, 51, 107, 41, 69, 99, 115, 48},
        {47, 85, 57, 71, 92, 77, 109, 36},
        {39, 63, 97, 49, 118, 56, 92, 61},
        {47, 101, 71, 60, 88, 109, 52, 90},
    };
    final int numWorkers = costs.length;
    final int numTasks = costs[0].length;

    final int[] allWorkers = IntStream.range(0, numWorkers).toArray();
    final int[] allTasks = IntStream.range(0, numTasks).toArray();

    final int[] taskSizes = {10, 7, 3, 12, 15, 4, 11, 5};
    // Maximum total of task sizes for any worker
    final int totalSizeMax = 15;

    // 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 has a maximum capacity.
    for (int worker : allWorkers) {
      LinearExprBuilder expr = LinearExpr.newBuilder();
      for (int task : allTasks) {
        expr.addTerm(x[worker][task], taskSizes[task]);
      }
      model.addLessOrEqual(expr, totalSizeMax);
    }

    // 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);
    }

    // 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 AssignmentTaskSizesSat() {}
}

سی شارپ

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

public class AssignmentTaskSizesSat
{
    public static void Main(String[] args)
    {
        // Data.
        int[,] costs = {
            { 90, 76, 75, 70, 50, 74, 12, 68 },    { 35, 85, 55, 65, 48, 101, 70, 83 },
            { 125, 95, 90, 105, 59, 120, 36, 73 }, { 45, 110, 95, 115, 104, 83, 37, 71 },
            { 60, 105, 80, 75, 59, 62, 93, 88 },   { 45, 65, 110, 95, 47, 31, 81, 34 },
            { 38, 51, 107, 41, 69, 99, 115, 48 },  { 47, 85, 57, 71, 92, 77, 109, 36 },
            { 39, 63, 97, 49, 118, 56, 92, 61 },   { 47, 101, 71, 60, 88, 109, 52, 90 },
        };
        int numWorkers = costs.GetLength(0);
        int numTasks = costs.GetLength(1);

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

        int[] taskSizes = { 10, 7, 3, 12, 15, 4, 11, 5 };
        // Maximum total of task sizes for any worker
        int totalSizeMax = 15;

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

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

        // Constraints
        // Each worker is assigned to at most max task size.
        foreach (int worker in allWorkers)
        {
            BoolVar[] vars = new BoolVar[numTasks];
            foreach (int task in allTasks)
            {
                vars[task] = x[worker, task];
            }
            model.Add(LinearExpr.WeightedSum(vars, taskSizes) <= totalSizeMax);
        }

        // 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);
        }

        // 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");
    }
}

راه حل MIP

در مرحله بعد، ما یک راه حل برای مشکل انتساب با استفاده از حل کننده MIP توضیح می دهیم.

کتابخانه ها را وارد کنید

کد زیر کتابخانه مورد نیاز را وارد می کند.

پایتون

from ortools.linear_solver import pywraplp

C++

#include <cstdint>
#include <memory>
#include <numeric>
#include <vector>

#include "absl/strings/str_format.h"
#include "ortools/base/logging.h"
#include "ortools/linear_solver/linear_solver.h"

جاوا

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;

سی شارپ

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

داده ها را تعریف کنید

کد زیر داده های برنامه را ایجاد می کند.

پایتون

costs = [
    [90, 76, 75, 70, 50, 74, 12, 68],
    [35, 85, 55, 65, 48, 101, 70, 83],
    [125, 95, 90, 105, 59, 120, 36, 73],
    [45, 110, 95, 115, 104, 83, 37, 71],
    [60, 105, 80, 75, 59, 62, 93, 88],
    [45, 65, 110, 95, 47, 31, 81, 34],
    [38, 51, 107, 41, 69, 99, 115, 48],
    [47, 85, 57, 71, 92, 77, 109, 36],
    [39, 63, 97, 49, 118, 56, 92, 61],
    [47, 101, 71, 60, 88, 109, 52, 90],
]
num_workers = len(costs)
num_tasks = len(costs[0])

task_sizes = [10, 7, 3, 12, 15, 4, 11, 5]
# Maximum total of task sizes for any worker
total_size_max = 15

C++

const std::vector<std::vector<int64_t>> costs = {{
    {{90, 76, 75, 70, 50, 74, 12, 68}},
    {{35, 85, 55, 65, 48, 101, 70, 83}},
    {{125, 95, 90, 105, 59, 120, 36, 73}},
    {{45, 110, 95, 115, 104, 83, 37, 71}},
    {{60, 105, 80, 75, 59, 62, 93, 88}},
    {{45, 65, 110, 95, 47, 31, 81, 34}},
    {{38, 51, 107, 41, 69, 99, 115, 48}},
    {{47, 85, 57, 71, 92, 77, 109, 36}},
    {{39, 63, 97, 49, 118, 56, 92, 61}},
    {{47, 101, 71, 60, 88, 109, 52, 90}},
}};
const int num_workers = costs.size();
std::vector<int> all_workers(num_workers);
std::iota(all_workers.begin(), all_workers.end(), 0);

const int num_tasks = costs[0].size();
std::vector<int> all_tasks(num_tasks);
std::iota(all_tasks.begin(), all_tasks.end(), 0);

const std::vector<int64_t> task_sizes = {{10, 7, 3, 12, 15, 4, 11, 5}};
// Maximum total of task sizes for any worker
const int total_size_max = 15;

جاوا

double[][] costs = {
    {90, 76, 75, 70, 50, 74, 12, 68},
    {35, 85, 55, 65, 48, 101, 70, 83},
    {125, 95, 90, 105, 59, 120, 36, 73},
    {45, 110, 95, 115, 104, 83, 37, 71},
    {60, 105, 80, 75, 59, 62, 93, 88},
    {45, 65, 110, 95, 47, 31, 81, 34},
    {38, 51, 107, 41, 69, 99, 115, 48},
    {47, 85, 57, 71, 92, 77, 109, 36},
    {39, 63, 97, 49, 118, 56, 92, 61},
    {47, 101, 71, 60, 88, 109, 52, 90},
};
int numWorkers = costs.length;
int numTasks = costs[0].length;

final int[] allWorkers = IntStream.range(0, numWorkers).toArray();
final int[] allTasks = IntStream.range(0, numTasks).toArray();

final int[] taskSizes = {10, 7, 3, 12, 15, 4, 11, 5};
// Maximum total of task sizes for any worker
final int totalSizeMax = 15;

سی شارپ

int[,] costs = {
    { 90, 76, 75, 70, 50, 74, 12, 68 },    { 35, 85, 55, 65, 48, 101, 70, 83 },
    { 125, 95, 90, 105, 59, 120, 36, 73 }, { 45, 110, 95, 115, 104, 83, 37, 71 },
    { 60, 105, 80, 75, 59, 62, 93, 88 },   { 45, 65, 110, 95, 47, 31, 81, 34 },
    { 38, 51, 107, 41, 69, 99, 115, 48 },  { 47, 85, 57, 71, 92, 77, 109, 36 },
    { 39, 63, 97, 49, 118, 56, 92, 61 },   { 47, 101, 71, 60, 88, 109, 52, 90 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);

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

int[] taskSizes = { 10, 7, 3, 12, 15, 4, 11, 5 };
// Maximum total of task sizes for any worker
int totalSizeMax = 15;

حل کننده را اعلام کنید

کد زیر حل کننده را ایجاد می کند.

پایتون

# 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;
}

جاوا

// 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;
}

سی شارپ

Solver solver = Solver.CreateSolver("SCIP");
if (solver is null)
{
    return;
}

متغیرها را ایجاد کنید

کد زیر آرایه ای از متغیرها را برای مشکل ایجاد می کند.

پایتون

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

C++

// x[i][j] is an array of 0-1 variables, which will be 1
// if worker i is assigned to task j.
std::vector<std::vector<const MPVariable*>> x(
    num_workers, std::vector<const MPVariable*>(num_tasks));
for (int worker : all_workers) {
  for (int task : all_tasks) {
    x[worker][task] =
        solver->MakeBoolVar(absl::StrFormat("x[%d,%d]", worker, task));
  }
}

جاوا

// 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 + "]");
  }
}

سی شارپ

// 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}]");
    }
}

محدودیت ها را اضافه کنید

کد زیر محدودیت هایی را برای برنامه ایجاد می کند.

پایتون

# 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(
            [task_sizes[task] * x[worker, task] for task in range(num_tasks)]
        )
        <= total_size_max
    )

# 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 += LinearExpr(x[worker][task]) * task_sizes[task];
  }
  solver->MakeRowConstraint(worker_sum <= total_size_max);
}
// 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);
}

جاوا

// Each worker is assigned to at most max task size.
for (int worker : allWorkers) {
  MPConstraint constraint = solver.makeConstraint(0, totalSizeMax, "");
  for (int task : allTasks) {
    constraint.setCoefficient(x[worker][task], taskSizes[task]);
  }
}
// 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);
  }
}

سی شارپ

// Each worker is assigned to at most max task size.
foreach (int worker in allWorkers)
{
    Constraint constraint = solver.MakeConstraint(0, totalSizeMax, "");
    foreach (int task in allTasks)
    {
        constraint.SetCoefficient(x[worker, task], taskSizes[task]);
    }
}
// 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);
    }
}

هدف را ایجاد کنید

کد زیر تابع هدف را ایجاد می کند.

پایتون

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();

جاوا

MPObjective objective = solver.objective();
for (int worker : allWorkers) {
  for (int task : allTasks) {
    objective.setCoefficient(x[worker][task], costs[worker][task]);
  }
}
objective.setMinimization();

سی شارپ

Objective objective = solver.Objective();
foreach (int worker in allWorkers)
{
    foreach (int task in allTasks)
    {
        objective.SetCoefficient(x[worker, task], costs[worker, task]);
    }
}
objective.SetMinimization();

حل کننده را فراخوانی کنید

کد زیر حل کننده را فراخوانی می کند و نتایج را نمایش می دهد.

پایتون

print(f"Solving with {solver.SolverVersion()}")
status = solver.Solve()

C++

const MPSolver::ResultStatus result_status = solver->Solve();

جاوا

MPSolver.ResultStatus resultStatus = solver.solve();

سی شارپ

Solver.ResultStatus resultStatus = solver.Solve();

نمایش نتایج

اکنون می توانیم راه حل را چاپ کنیم.

پایتون

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];
    }
  }
}

جاوا

// 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.");
}

سی شارپ

// 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.");
}

اینم خروجی برنامه

Minimum cost =  326.0

Worker 0  assigned to task 6   Cost =  12
Worker 1  assigned to task 0   Cost =  35
Worker 1  assigned to task 2   Cost =  55
Worker 4  assigned to task 4   Cost =  59
Worker 5  assigned to task 5   Cost =  31
Worker 5  assigned to task 7   Cost =  34
Worker 6  assigned to task 1   Cost =  51
Worker 8  assigned to task 3   Cost =  49

Time =  0.0167 seconds

کل برنامه

در اینجا کل برنامه است.

پایتون

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


def main():
    # Data
    costs = [
        [90, 76, 75, 70, 50, 74, 12, 68],
        [35, 85, 55, 65, 48, 101, 70, 83],
        [125, 95, 90, 105, 59, 120, 36, 73],
        [45, 110, 95, 115, 104, 83, 37, 71],
        [60, 105, 80, 75, 59, 62, 93, 88],
        [45, 65, 110, 95, 47, 31, 81, 34],
        [38, 51, 107, 41, 69, 99, 115, 48],
        [47, 85, 57, 71, 92, 77, 109, 36],
        [39, 63, 97, 49, 118, 56, 92, 61],
        [47, 101, 71, 60, 88, 109, 52, 90],
    ]
    num_workers = len(costs)
    num_tasks = len(costs[0])

    task_sizes = [10, 7, 3, 12, 15, 4, 11, 5]
    # Maximum total of task sizes for any worker
    total_size_max = 15

    # Solver
    # Create the mip solver with the SCIP backend.
    solver = pywraplp.Solver.CreateSolver("SCIP")

    if not solver:
        return

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

    # Constraints
    # 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(
                [task_sizes[task] * x[worker, task] for task in range(num_tasks)]
            )
            <= total_size_max
        )

    # 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)

    # 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 <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, 12, 68}},
      {{35, 85, 55, 65, 48, 101, 70, 83}},
      {{125, 95, 90, 105, 59, 120, 36, 73}},
      {{45, 110, 95, 115, 104, 83, 37, 71}},
      {{60, 105, 80, 75, 59, 62, 93, 88}},
      {{45, 65, 110, 95, 47, 31, 81, 34}},
      {{38, 51, 107, 41, 69, 99, 115, 48}},
      {{47, 85, 57, 71, 92, 77, 109, 36}},
      {{39, 63, 97, 49, 118, 56, 92, 61}},
      {{47, 101, 71, 60, 88, 109, 52, 90}},
  }};
  const int num_workers = costs.size();
  std::vector<int> all_workers(num_workers);
  std::iota(all_workers.begin(), all_workers.end(), 0);

  const int num_tasks = costs[0].size();
  std::vector<int> all_tasks(num_tasks);
  std::iota(all_tasks.begin(), all_tasks.end(), 0);

  const std::vector<int64_t> task_sizes = {{10, 7, 3, 12, 15, 4, 11, 5}};
  // Maximum total of task sizes for any worker
  const int total_size_max = 15;

  // 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 += LinearExpr(x[worker][task]) * task_sizes[task];
    }
    solver->MakeRowConstraint(worker_sum <= total_size_max);
  }
  // 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);
  }

  // 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;
}

جاوا

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 AssignmentTaskSizesMip {
  public static void main(String[] args) {
    Loader.loadNativeLibraries();
    // Data
    double[][] costs = {
        {90, 76, 75, 70, 50, 74, 12, 68},
        {35, 85, 55, 65, 48, 101, 70, 83},
        {125, 95, 90, 105, 59, 120, 36, 73},
        {45, 110, 95, 115, 104, 83, 37, 71},
        {60, 105, 80, 75, 59, 62, 93, 88},
        {45, 65, 110, 95, 47, 31, 81, 34},
        {38, 51, 107, 41, 69, 99, 115, 48},
        {47, 85, 57, 71, 92, 77, 109, 36},
        {39, 63, 97, 49, 118, 56, 92, 61},
        {47, 101, 71, 60, 88, 109, 52, 90},
    };
    int numWorkers = costs.length;
    int numTasks = costs[0].length;

    final int[] allWorkers = IntStream.range(0, numWorkers).toArray();
    final int[] allTasks = IntStream.range(0, numTasks).toArray();

    final int[] taskSizes = {10, 7, 3, 12, 15, 4, 11, 5};
    // Maximum total of task sizes for any worker
    final int totalSizeMax = 15;

    // 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 max task size.
    for (int worker : allWorkers) {
      MPConstraint constraint = solver.makeConstraint(0, totalSizeMax, "");
      for (int task : allTasks) {
        constraint.setCoefficient(x[worker][task], taskSizes[task]);
      }
    }
    // 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);
      }
    }

    // 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 AssignmentTaskSizesMip() {}
}

سی شارپ

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

public class AssignmentTaskSizesMip
{
    static void Main()
    {
        // Data.
        int[,] costs = {
            { 90, 76, 75, 70, 50, 74, 12, 68 },    { 35, 85, 55, 65, 48, 101, 70, 83 },
            { 125, 95, 90, 105, 59, 120, 36, 73 }, { 45, 110, 95, 115, 104, 83, 37, 71 },
            { 60, 105, 80, 75, 59, 62, 93, 88 },   { 45, 65, 110, 95, 47, 31, 81, 34 },
            { 38, 51, 107, 41, 69, 99, 115, 48 },  { 47, 85, 57, 71, 92, 77, 109, 36 },
            { 39, 63, 97, 49, 118, 56, 92, 61 },   { 47, 101, 71, 60, 88, 109, 52, 90 },
        };
        int numWorkers = costs.GetLength(0);
        int numTasks = costs.GetLength(1);

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

        int[] taskSizes = { 10, 7, 3, 12, 15, 4, 11, 5 };
        // Maximum total of task sizes for any worker
        int totalSizeMax = 15;

        // 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 max task size.
        foreach (int worker in allWorkers)
        {
            Constraint constraint = solver.MakeConstraint(0, totalSizeMax, "");
            foreach (int task in allTasks)
            {
                constraint.SetCoefficient(x[worker, task], taskSizes[task]);
            }
        }
        // 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);
            }
        }

        // 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.");
        }
    }
}