Phần này mô tả một vấn đề về bài tập, trong đó mỗi nhiệm vụ có một kích thước, thể hiện thời gian hoặc công sức của nhiệm vụ đó. Tổng kích thước của các tác vụ do mỗi worker thực hiện có một giới hạn cố định.
Chúng tôi sẽ trình bày các chương trình Python giải quyết vấn đề này bằng cách sử dụng trình giải CP-SAT và trình giải MIP.
Giải pháp CP-SAT
Trước tiên, hãy cùng tìm hiểu giải pháp CP-SAT cho vấn đề.
Nhập thư viện
Mã sau đây nhập thư viện bắt buộc.
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 "ortools/base/logging.h" #include "ortools/sat/cp_model.h" #include "ortools/sat/cp_model.pb.h" #include "ortools/sat/cp_model_solver.h"
Java
import com.google.ortools.Loader; import com.google.ortools.sat.CpModel; import com.google.ortools.sat.CpSolver; import com.google.ortools.sat.CpSolverStatus; import com.google.ortools.sat.LinearExpr; import com.google.ortools.sat.LinearExprBuilder; import com.google.ortools.sat.Literal; import java.util.ArrayList; import java.util.List; import java.util.stream.IntStream;
C#
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.Sat;
Xác định dữ liệu
Đoạn mã sau đây tạo dữ liệu cho chương trình.
Python
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;
Java
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;
C#
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;
Như trong các ví dụ trước, ma trận chi phí đưa ra chi phí để trình thực thi i thực hiện tác vụ j.
Vectơ sizes cung cấp kích thước của mỗi tác vụ.
total_size_max là giới hạn trên trên tổng kích thước của các tác vụ do một trình thực thi bất kỳ thực hiện.
Tạo mô hình
Mã sau đây sẽ tạo mô hình.
Python
model = cp_model.CpModel()
C++
CpModelBuilder cp_model;
Java
CpModel model = new CpModel();
C#
CpModel model = new CpModel();
Tạo các biến
Mã sau đây tạo một mảng biến cho bài toán.
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];
foreach (int worker in allWorkers)
{
foreach (int task in allTasks)
{
x[worker, task] = model.NewBoolVar($"x[{worker},{task}]");
}
}
Thêm các điều kiện ràng buộc
Mã sau đây tạo các điều kiện ràng buộc cho chương trình.
Python
# 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);
}
Java
// 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);
}
C#
// 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);
}
Tạo mục tiêu
Mã sau đây tạo hàm mục tiêu.
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);
Gọi trình giải
Mã sau đây gọi trình giải.
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}");
Hiện kết quả
Bây giờ, chúng ta có thể in giải pháp.
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.");
}
Đây là kết quả của chương trình.
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
Toàn bộ chương trình
Sau đây là toàn bộ chương trình.
Python
"""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;
}
Java
// CP-SAT example that solves an assignment problem.
package com.google.ortools.sat.samples;
import com.google.ortools.Loader;
import com.google.ortools.sat.CpModel;
import com.google.ortools.sat.CpSolver;
import com.google.ortools.sat.CpSolverStatus;
import com.google.ortools.sat.LinearExpr;
import com.google.ortools.sat.LinearExprBuilder;
import com.google.ortools.sat.Literal;
import java.util.ArrayList;
import java.util.List;
import java.util.stream.IntStream;
/** Assignment problem. */
public class 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() {}
}
C#
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");
}
}
Giải pháp MIP
Tiếp theo, chúng tôi sẽ mô tả cách giải quyết vấn đề về bài tập bằng cách sử dụng trình giải MIP.
Nhập thư viện
Mã sau đây nhập thư viện bắt buộc.
Python
from ortools.linear_solver import pywraplp
C++
#include <cstdint> #include <memory> #include <numeric> #include <vector> #include "absl/strings/str_format.h" #include "ortools/base/logging.h" #include "ortools/linear_solver/linear_solver.h"
Java
import com.google.ortools.Loader; import com.google.ortools.linearsolver.MPConstraint; import com.google.ortools.linearsolver.MPObjective; import com.google.ortools.linearsolver.MPSolver; import com.google.ortools.linearsolver.MPVariable; import java.util.stream.IntStream;
C#
using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.LinearSolver;
Xác định dữ liệu
Đoạn mã sau đây tạo dữ liệu cho chương trình.
Python
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;
Java
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;
C#
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;
Khai báo trình giải
Mã sau đây sẽ tạo trình giải.
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;
}
Tạo các biến
Mã sau đây tạo một mảng biến cho bài toán.
Python
# x[i, j] is an array of 0-1 variables, which will be 1
# if worker i is assigned to task j.
x = {}
for worker in range(num_workers):
for task in range(num_tasks):
x[worker, task] = solver.BoolVar(f"x[{worker},{task}]")
C++
// x[i][j] is an array of 0-1 variables, which will be 1
// if worker i is assigned to task j.
std::vector<std::vector<const MPVariable*>> x(
num_workers, std::vector<const MPVariable*>(num_tasks));
for (int worker : all_workers) {
for (int task : all_tasks) {
x[worker][task] =
solver->MakeBoolVar(absl::StrFormat("x[%d,%d]", worker, task));
}
}
Java
// x[i][j] is an array of 0-1 variables, which will be 1
// if worker i is assigned to task j.
MPVariable[][] x = new MPVariable[numWorkers][numTasks];
for (int worker : allWorkers) {
for (int task : allTasks) {
x[worker][task] = solver.makeBoolVar("x[" + worker + "," + task + "]");
}
}
C#
// x[i, j] is an array of 0-1 variables, which will be 1
// if worker i is assigned to task j.
Variable[,] x = new Variable[numWorkers, numTasks];
foreach (int worker in allWorkers)
{
foreach (int task in allTasks)
{
x[worker, task] = solver.MakeBoolVar($"x[{worker},{task}]");
}
}
Thêm các điều kiện ràng buộc
Mã sau đây tạo các điều kiện ràng buộc cho chương trình.
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(
[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);
}
Java
// 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);
}
}
C#
// 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);
}
}
Tạo mục tiêu
Mã sau đây tạo hàm mục tiêu.
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();
Gọi trình giải
Mã sau đây gọi trình phân giải và hiển thị kết quả.
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();
Hiện kết quả
Bây giờ, chúng ta có thể in giải pháp.
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.");
}
Đây là kết quả của chương trình.
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
Toàn bộ chương trình
Sau đây là toàn bộ chương trình.
Python
"""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;
}
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 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() {}
}
C#
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.");
}
}
}