এই বিভাগটি একটি উদাহরণ উপস্থাপন করে যা দেখায় কিভাবে MIP সল্ভার এবং CP-SAT সমাধানকারী উভয় ব্যবহার করে একটি অ্যাসাইনমেন্ট সমস্যা সমাধান করা যায়।
উদাহরণ
উদাহরণে পাঁচজন কর্মী (সংখ্যাযুক্ত 0-4) এবং চারটি কাজ (0-3 সংখ্যাযুক্ত) রয়েছে। উল্লেখ্য যে ওভারভিউতে উদাহরণের চেয়ে আরও একজন কর্মী আছে।
কাজের জন্য কর্মীদের নিয়োগের খরচ নিম্নলিখিত টেবিলে দেখানো হয়েছে।
| কর্মী | টাস্ক 0 | কার্যক্রম 1 | টাস্ক 2 | টাস্ক 3 |
|---|---|---|---|---|
| 0 | 90 | 80 | 75 | 70 |
| 1 | 35 | 85 | 55 | 65 |
| 2 | 125 | 95 | 90 | 95 |
| 3 | 45 | 110 | 95 | 115 |
| 4 | 50 | 100 | 90 | 100 |
সমস্যা হল প্রতিটি কর্মীকে সর্বাধিক একটি কাজের জন্য বরাদ্দ করা, যেখানে কোনও দুইজন কর্মী একই কাজ সম্পাদন করে না, মোট খরচ কমিয়ে দেয়। যেহেতু কাজের চেয়ে বেশি কর্মী রয়েছে, তাই একজন কর্মীকে একটি কাজ দেওয়া হবে না।
এমআইপি সমাধান
নিম্নলিখিত বিভাগগুলি বর্ণনা করে কিভাবে MPSolver র্যাপার ব্যবহার করে সমস্যার সমাধান করা যায়।
লাইব্রেরি আমদানি করুন
নিম্নলিখিত কোড প্রয়োজনীয় লাইব্রেরি আমদানি করে।
পাইথন
from ortools.linear_solver import pywraplp
সি++
#include <memory> #include <vector> #include "ortools/base/logging.h" #include "ortools/linear_solver/linear_solver.h"
জাভা
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;
সি#
using System; using Google.OrTools.LinearSolver;
ডেটা তৈরি করুন
নিম্নলিখিত কোডটি সমস্যার জন্য ডেটা তৈরি করে।
পাইথন
costs = [
[90, 80, 75, 70],
[35, 85, 55, 65],
[125, 95, 90, 95],
[45, 110, 95, 115],
[50, 100, 90, 100],
]
num_workers = len(costs)
num_tasks = len(costs[0])সি++
const std::vector<std::vector<double>> costs{
{90, 80, 75, 70}, {35, 85, 55, 65}, {125, 95, 90, 95},
{45, 110, 95, 115}, {50, 100, 90, 100},
};
const int num_workers = costs.size();
const int num_tasks = costs[0].size();জাভা
double[][] costs = {
{90, 80, 75, 70},
{35, 85, 55, 65},
{125, 95, 90, 95},
{45, 110, 95, 115},
{50, 100, 90, 100},
};
int numWorkers = costs.length;
int numTasks = costs[0].length;সি#
int[,] costs = {
{ 90, 80, 75, 70 }, { 35, 85, 55, 65 }, { 125, 95, 90, 95 }, { 45, 110, 95, 115 }, { 50, 100, 90, 100 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1); costs অ্যারেটি উপরে দেখানো কাজের জন্য কর্মীদের নিয়োগের জন্য খরচের টেবিলের সাথে মিলে যায়।
এমআইপি সমাধানকারী ঘোষণা করুন
নিম্নলিখিত কোড MIP সমাধানকারী ঘোষণা করে।
পাইথন
# Create the mip solver with the SCIP backend.
solver = pywraplp.Solver.CreateSolver("SCIP")
if not solver:
returnসি++
// Create the mip solver with the SCIP backend.
std::unique_ptr<MPSolver> solver(MPSolver::CreateSolver("SCIP"));
if (!solver) {
LOG(WARNING) << "SCIP solver unavailable.";
return;
}জাভা
// 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;
}সি#
Solver solver = Solver.CreateSolver("SCIP");
if (solver is null)
{
return;
}ভেরিয়েবল তৈরি করুন
নিম্নলিখিত কোডটি সমস্যার জন্য বাইনারি পূর্ণসংখ্যা ভেরিয়েবল তৈরি করে।
পাইথন
# x[i, j] is an array of 0-1 variables, which will be 1
# if worker i is assigned to task j.
x = {}
for i in range(num_workers):
for j in range(num_tasks):
x[i, j] = solver.IntVar(0, 1, "")সি++
// 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 i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
x[i][j] = solver->MakeIntVar(0, 1, "");
}
}জাভা
// 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 i = 0; i < numWorkers; ++i) {
for (int j = 0; j < numTasks; ++j) {
x[i][j] = solver.makeIntVar(0, 1, "");
}
}সি#
// 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];
for (int i = 0; i < numWorkers; ++i)
{
for (int j = 0; j < numTasks; ++j)
{
x[i, j] = solver.MakeIntVar(0, 1, $"worker_{i}_task_{j}");
}
}সীমাবদ্ধতা তৈরি করুন
নিম্নলিখিত কোডটি সমস্যার জন্য সীমাবদ্ধতা তৈরি করে।
পাইথন
# Each worker is assigned to at most 1 task.
for i in range(num_workers):
solver.Add(solver.Sum([x[i, j] for j in range(num_tasks)]) <= 1)
# Each task is assigned to exactly one worker.
for j in range(num_tasks):
solver.Add(solver.Sum([x[i, j] for i in range(num_workers)]) == 1)সি++
// Each worker is assigned to at most one task.
for (int i = 0; i < num_workers; ++i) {
LinearExpr worker_sum;
for (int j = 0; j < num_tasks; ++j) {
worker_sum += x[i][j];
}
solver->MakeRowConstraint(worker_sum <= 1.0);
}
// Each task is assigned to exactly one worker.
for (int j = 0; j < num_tasks; ++j) {
LinearExpr task_sum;
for (int i = 0; i < num_workers; ++i) {
task_sum += x[i][j];
}
solver->MakeRowConstraint(task_sum == 1.0);
}জাভা
// Each worker is assigned to at most one task.
for (int i = 0; i < numWorkers; ++i) {
MPConstraint constraint = solver.makeConstraint(0, 1, "");
for (int j = 0; j < numTasks; ++j) {
constraint.setCoefficient(x[i][j], 1);
}
}
// Each task is assigned to exactly one worker.
for (int j = 0; j < numTasks; ++j) {
MPConstraint constraint = solver.makeConstraint(1, 1, "");
for (int i = 0; i < numWorkers; ++i) {
constraint.setCoefficient(x[i][j], 1);
}
}সি#
// Each worker is assigned to at most one task.
for (int i = 0; i < numWorkers; ++i)
{
Constraint constraint = solver.MakeConstraint(0, 1, "");
for (int j = 0; j < numTasks; ++j)
{
constraint.SetCoefficient(x[i, j], 1);
}
}
// Each task is assigned to exactly one worker.
for (int j = 0; j < numTasks; ++j)
{
Constraint constraint = solver.MakeConstraint(1, 1, "");
for (int i = 0; i < numWorkers; ++i)
{
constraint.SetCoefficient(x[i, j], 1);
}
}উদ্দেশ্য ফাংশন তৈরি করুন
নিম্নলিখিত কোডটি সমস্যার জন্য উদ্দেশ্য ফাংশন তৈরি করে।
পাইথন
objective_terms = []
for i in range(num_workers):
for j in range(num_tasks):
objective_terms.append(costs[i][j] * x[i, j])
solver.Minimize(solver.Sum(objective_terms))সি++
MPObjective* const objective = solver->MutableObjective();
for (int i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
objective->SetCoefficient(x[i][j], costs[i][j]);
}
}
objective->SetMinimization();জাভা
MPObjective objective = solver.objective();
for (int i = 0; i < numWorkers; ++i) {
for (int j = 0; j < numTasks; ++j) {
objective.setCoefficient(x[i][j], costs[i][j]);
}
}
objective.setMinimization();সি#
Objective objective = solver.Objective();
for (int i = 0; i < numWorkers; ++i)
{
for (int j = 0; j < numTasks; ++j)
{
objective.SetCoefficient(x[i, j], costs[i, j]);
}
}
objective.SetMinimization();উদ্দেশ্য ফাংশনের মান হল সমস্ত ভেরিয়েবলের মোট খরচ যা সমাধানকারী দ্বারা 1 মান নির্ধারণ করা হয়েছে।
সমাধানকারীকে আহ্বান করুন
নিম্নলিখিত কোডটি সমাধানকারীকে আহ্বান করে।
পাইথন
print(f"Solving with {solver.SolverVersion()}")
status = solver.Solve()সি++
const MPSolver::ResultStatus result_status = solver->Solve();
জাভা
MPSolver.ResultStatus resultStatus = solver.solve();
সি#
Solver.ResultStatus resultStatus = solver.Solve();
সমাধান প্রিন্ট করুন
নিম্নলিখিত কোডটি সমস্যার সমাধান প্রিন্ট করে।
পাইথন
if status == pywraplp.Solver.OPTIMAL or status == pywraplp.Solver.FEASIBLE:
print(f"Total cost = {solver.Objective().Value()}\n")
for i in range(num_workers):
for j in range(num_tasks):
# Test if x[i,j] is 1 (with tolerance for floating point arithmetic).
if x[i, j].solution_value() > 0.5:
print(f"Worker {i} assigned to task {j}." + f" Cost: {costs[i][j]}")
else:
print("No solution found.")সি++
// 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 i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
// Test if x[i][j] is 0 or 1 (with tolerance for floating point
// arithmetic).
if (x[i][j]->solution_value() > 0.5) {
LOG(INFO) << "Worker " << i << " assigned to task " << j
<< ". Cost = " << costs[i][j];
}
}
}জাভা
// 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 i = 0; i < numWorkers; ++i) {
for (int j = 0; j < numTasks; ++j) {
// Test if x[i][j] is 0 or 1 (with tolerance for floating point
// arithmetic).
if (x[i][j].solutionValue() > 0.5) {
System.out.println(
"Worker " + i + " assigned to task " + j + ". Cost = " + costs[i][j]);
}
}
}
} else {
System.err.println("No solution found.");
}সি#
// 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");
for (int i = 0; i < numWorkers; ++i)
{
for (int j = 0; j < numTasks; ++j)
{
// Test if x[i, j] is 0 or 1 (with tolerance for floating point
// arithmetic).
if (x[i, j].SolutionValue() > 0.5)
{
Console.WriteLine($"Worker {i} assigned to task {j}. Cost: {costs[i, j]}");
}
}
}
}
else
{
Console.WriteLine("No solution found.");
}এখানে প্রোগ্রামের আউটপুট আছে।
Total cost = 265.0 Worker 0 assigned to task 3. Cost = 70 Worker 1 assigned to task 2. Cost = 55 Worker 2 assigned to task 1. Cost = 95 Worker 3 assigned to task 0. Cost = 45
সম্পূর্ণ প্রোগ্রাম
এখানে MIP সমাধানের জন্য সম্পূর্ণ প্রোগ্রাম আছে.
পাইথন
from ortools.linear_solver import pywraplp
def main():
# Data
costs = [
[90, 80, 75, 70],
[35, 85, 55, 65],
[125, 95, 90, 95],
[45, 110, 95, 115],
[50, 100, 90, 100],
]
num_workers = len(costs)
num_tasks = len(costs[0])
# 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 i in range(num_workers):
for j in range(num_tasks):
x[i, j] = solver.IntVar(0, 1, "")
# Constraints
# Each worker is assigned to at most 1 task.
for i in range(num_workers):
solver.Add(solver.Sum([x[i, j] for j in range(num_tasks)]) <= 1)
# Each task is assigned to exactly one worker.
for j in range(num_tasks):
solver.Add(solver.Sum([x[i, j] for i in range(num_workers)]) == 1)
# Objective
objective_terms = []
for i in range(num_workers):
for j in range(num_tasks):
objective_terms.append(costs[i][j] * x[i, j])
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 i in range(num_workers):
for j in range(num_tasks):
# Test if x[i,j] is 1 (with tolerance for floating point arithmetic).
if x[i, j].solution_value() > 0.5:
print(f"Worker {i} assigned to task {j}." + f" Cost: {costs[i][j]}")
else:
print("No solution found.")
if __name__ == "__main__":
main()সি++
#include <memory>
#include <vector>
#include "ortools/base/logging.h"
#include "ortools/linear_solver/linear_solver.h"
namespace operations_research {
void AssignmentMip() {
// Data
const std::vector<std::vector<double>> costs{
{90, 80, 75, 70}, {35, 85, 55, 65}, {125, 95, 90, 95},
{45, 110, 95, 115}, {50, 100, 90, 100},
};
const int num_workers = costs.size();
const int num_tasks = costs[0].size();
// 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 i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
x[i][j] = solver->MakeIntVar(0, 1, "");
}
}
// Constraints
// Each worker is assigned to at most one task.
for (int i = 0; i < num_workers; ++i) {
LinearExpr worker_sum;
for (int j = 0; j < num_tasks; ++j) {
worker_sum += x[i][j];
}
solver->MakeRowConstraint(worker_sum <= 1.0);
}
// Each task is assigned to exactly one worker.
for (int j = 0; j < num_tasks; ++j) {
LinearExpr task_sum;
for (int i = 0; i < num_workers; ++i) {
task_sum += x[i][j];
}
solver->MakeRowConstraint(task_sum == 1.0);
}
// Objective.
MPObjective* const objective = solver->MutableObjective();
for (int i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
objective->SetCoefficient(x[i][j], costs[i][j]);
}
}
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 i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
// Test if x[i][j] is 0 or 1 (with tolerance for floating point
// arithmetic).
if (x[i][j]->solution_value() > 0.5) {
LOG(INFO) << "Worker " << i << " assigned to task " << j
<< ". Cost = " << costs[i][j];
}
}
}
}
} // namespace operations_research
int main(int argc, char** argv) {
operations_research::AssignmentMip();
return EXIT_SUCCESS;
}জাভা
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;
/** MIP example that solves an assignment problem. */
public class AssignmentMip {
public static void main(String[] args) {
Loader.loadNativeLibraries();
// Data
double[][] costs = {
{90, 80, 75, 70},
{35, 85, 55, 65},
{125, 95, 90, 95},
{45, 110, 95, 115},
{50, 100, 90, 100},
};
int numWorkers = costs.length;
int numTasks = costs[0].length;
// 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 i = 0; i < numWorkers; ++i) {
for (int j = 0; j < numTasks; ++j) {
x[i][j] = solver.makeIntVar(0, 1, "");
}
}
// Constraints
// Each worker is assigned to at most one task.
for (int i = 0; i < numWorkers; ++i) {
MPConstraint constraint = solver.makeConstraint(0, 1, "");
for (int j = 0; j < numTasks; ++j) {
constraint.setCoefficient(x[i][j], 1);
}
}
// Each task is assigned to exactly one worker.
for (int j = 0; j < numTasks; ++j) {
MPConstraint constraint = solver.makeConstraint(1, 1, "");
for (int i = 0; i < numWorkers; ++i) {
constraint.setCoefficient(x[i][j], 1);
}
}
// Objective
MPObjective objective = solver.objective();
for (int i = 0; i < numWorkers; ++i) {
for (int j = 0; j < numTasks; ++j) {
objective.setCoefficient(x[i][j], costs[i][j]);
}
}
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 i = 0; i < numWorkers; ++i) {
for (int j = 0; j < numTasks; ++j) {
// Test if x[i][j] is 0 or 1 (with tolerance for floating point
// arithmetic).
if (x[i][j].solutionValue() > 0.5) {
System.out.println(
"Worker " + i + " assigned to task " + j + ". Cost = " + costs[i][j]);
}
}
}
} else {
System.err.println("No solution found.");
}
}
private AssignmentMip() {}
}সি#
using System;
using Google.OrTools.LinearSolver;
public class AssignmentMip
{
static void Main()
{
// Data.
int[,] costs = {
{ 90, 80, 75, 70 }, { 35, 85, 55, 65 }, { 125, 95, 90, 95 }, { 45, 110, 95, 115 }, { 50, 100, 90, 100 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);
// 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];
for (int i = 0; i < numWorkers; ++i)
{
for (int j = 0; j < numTasks; ++j)
{
x[i, j] = solver.MakeIntVar(0, 1, $"worker_{i}_task_{j}");
}
}
// Constraints
// Each worker is assigned to at most one task.
for (int i = 0; i < numWorkers; ++i)
{
Constraint constraint = solver.MakeConstraint(0, 1, "");
for (int j = 0; j < numTasks; ++j)
{
constraint.SetCoefficient(x[i, j], 1);
}
}
// Each task is assigned to exactly one worker.
for (int j = 0; j < numTasks; ++j)
{
Constraint constraint = solver.MakeConstraint(1, 1, "");
for (int i = 0; i < numWorkers; ++i)
{
constraint.SetCoefficient(x[i, j], 1);
}
}
// Objective
Objective objective = solver.Objective();
for (int i = 0; i < numWorkers; ++i)
{
for (int j = 0; j < numTasks; ++j)
{
objective.SetCoefficient(x[i, j], costs[i, j]);
}
}
objective.SetMinimization();
// Solve
Solver.ResultStatus resultStatus = solver.Solve();
// Print solution.
// Check that the problem has a feasible solution.
if (resultStatus == Solver.ResultStatus.OPTIMAL || resultStatus == Solver.ResultStatus.FEASIBLE)
{
Console.WriteLine($"Total cost: {solver.Objective().Value()}\n");
for (int i = 0; i < numWorkers; ++i)
{
for (int j = 0; j < numTasks; ++j)
{
// Test if x[i, j] is 0 or 1 (with tolerance for floating point
// arithmetic).
if (x[i, j].SolutionValue() > 0.5)
{
Console.WriteLine($"Worker {i} assigned to task {j}. Cost: {costs[i, j]}");
}
}
}
}
else
{
Console.WriteLine("No solution found.");
}
}
}CP SAT সমাধান
নিম্নলিখিত বিভাগগুলি CP-SAT সমাধানকারী ব্যবহার করে কীভাবে সমস্যার সমাধান করতে হয় তা বর্ণনা করে।
লাইব্রেরি আমদানি করুন
নিম্নলিখিত কোড প্রয়োজনীয় লাইব্রেরি আমদানি করে।
পাইথন
import io import pandas as pd from ortools.sat.python import cp_model
সি++
#include <stdlib.h> #include <vector> #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"
জাভা
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;
সি#
using System; using System.Collections.Generic; using Google.OrTools.Sat;
মডেল ঘোষণা করুন
নিম্নলিখিত কোডটি CP-SAT মডেল ঘোষণা করে।
পাইথন
model = cp_model.CpModel()
সি++
CpModelBuilder cp_model;
জাভা
CpModel model = new CpModel();
সি#
CpModel model = new CpModel();
ডেটা তৈরি করুন
নিম্নলিখিত কোডটি সমস্যার জন্য ডেটা সেট আপ করে।
পাইথন
data_str = """
worker task cost
w1 t1 90
w1 t2 80
w1 t3 75
w1 t4 70
w2 t1 35
w2 t2 85
w2 t3 55
w2 t4 65
w3 t1 125
w3 t2 95
w3 t3 90
w3 t4 95
w4 t1 45
w4 t2 110
w4 t3 95
w4 t4 115
w5 t1 50
w5 t2 110
w5 t3 90
w5 t4 100
"""
data = pd.read_table(io.StringIO(data_str), sep=r"\s+")সি++
const std::vector<std::vector<int>> costs{
{90, 80, 75, 70}, {35, 85, 55, 65}, {125, 95, 90, 95},
{45, 110, 95, 115}, {50, 100, 90, 100},
};
const int num_workers = static_cast<int>(costs.size());
const int num_tasks = static_cast<int>(costs[0].size());জাভা
int[][] costs = {
{90, 80, 75, 70},
{35, 85, 55, 65},
{125, 95, 90, 95},
{45, 110, 95, 115},
{50, 100, 90, 100},
};
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();সি#
int[,] costs = {
{ 90, 80, 75, 70 }, { 35, 85, 55, 65 }, { 125, 95, 90, 95 }, { 45, 110, 95, 115 }, { 50, 100, 90, 100 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1); costs অ্যারেটি উপরে দেখানো কাজের জন্য কর্মীদের নিয়োগের জন্য খরচের টেবিলের সাথে মিলে যায়।
ভেরিয়েবল তৈরি করুন
নিম্নলিখিত কোডটি সমস্যার জন্য বাইনারি পূর্ণসংখ্যা ভেরিয়েবল তৈরি করে।
পাইথন
x = model.new_bool_var_series(name="x", index=data.index)
সি++
// 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 i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
x[i][j] = cp_model.NewBoolVar();
}
}জাভা
Literal[][] x = new Literal[numWorkers][numTasks];
for (int worker : allWorkers) {
for (int task : allTasks) {
x[worker][task] = model.newBoolVar("x[" + worker + "," + task + "]");
}
}সি#
BoolVar[,] x = new BoolVar[numWorkers, numTasks];
// Variables in a 1-dim array.
for (int worker = 0; worker < numWorkers; ++worker)
{
for (int task = 0; task < numTasks; ++task)
{
x[worker, task] = model.NewBoolVar($"worker_{worker}_task_{task}");
}
}সীমাবদ্ধতা তৈরি করুন
নিম্নলিখিত কোডটি সমস্যার জন্য সীমাবদ্ধতা তৈরি করে।
পাইথন
# Each worker is assigned to at most one task.
for unused_name, tasks in data.groupby("worker"):
model.add_at_most_one(x[tasks.index])
# Each task is assigned to exactly one worker.
for unused_name, workers in data.groupby("task"):
model.add_exactly_one(x[workers.index])সি++
// Each worker is assigned to at most one task.
for (int i = 0; i < num_workers; ++i) {
cp_model.AddAtMostOne(x[i]);
}
// Each task is assigned to exactly one worker.
for (int j = 0; j < num_tasks; ++j) {
std::vector<BoolVar> tasks;
for (int i = 0; i < num_workers; ++i) {
tasks.push_back(x[i][j]);
}
cp_model.AddExactlyOne(tasks);
}জাভা
// Each worker is assigned to at most one task.
for (int worker : allWorkers) {
List<Literal> tasks = new ArrayList<>();
for (int task : allTasks) {
tasks.add(x[worker][task]);
}
model.addAtMostOne(tasks);
}
// Each task is assigned to exactly one worker.
for (int task : allTasks) {
List<Literal> workers = new ArrayList<>();
for (int worker : allWorkers) {
workers.add(x[worker][task]);
}
model.addExactlyOne(workers);
}সি#
// Each worker is assigned to at most one task.
for (int worker = 0; worker < numWorkers; ++worker)
{
List<ILiteral> tasks = new List<ILiteral>();
for (int task = 0; task < numTasks; ++task)
{
tasks.Add(x[worker, task]);
}
model.AddAtMostOne(tasks);
}
// Each task is assigned to exactly one worker.
for (int task = 0; task < numTasks; ++task)
{
List<ILiteral> workers = new List<ILiteral>();
for (int worker = 0; worker < numWorkers; ++worker)
{
workers.Add(x[worker, task]);
}
model.AddExactlyOne(workers);
}উদ্দেশ্য ফাংশন তৈরি করুন
নিম্নলিখিত কোডটি সমস্যার জন্য উদ্দেশ্য ফাংশন তৈরি করে।
পাইথন
model.minimize(data.cost.dot(x))
সি++
LinearExpr total_cost;
for (int i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
total_cost += x[i][j] * costs[i][j];
}
}
cp_model.Minimize(total_cost);জাভা
LinearExprBuilder obj = LinearExpr.newBuilder();
for (int worker : allWorkers) {
for (int task : allTasks) {
obj.addTerm(x[worker][task], costs[worker][task]);
}
}
model.minimize(obj);সি#
LinearExprBuilder obj = LinearExpr.NewBuilder();
for (int worker = 0; worker < numWorkers; ++worker)
{
for (int task = 0; task < numTasks; ++task)
{
obj.AddTerm((IntVar)x[worker, task], costs[worker, task]);
}
}
model.Minimize(obj);উদ্দেশ্য ফাংশনের মান হল সমস্ত ভেরিয়েবলের মোট খরচ যা সমাধানকারী দ্বারা 1 মান নির্ধারণ করা হয়েছে।
সমাধানকারীকে আহ্বান করুন
নিম্নলিখিত কোডটি সমাধানকারীকে আহ্বান করে।
পাইথন
solver = cp_model.CpSolver() status = solver.solve(model)
সি++
const CpSolverResponse response = Solve(cp_model.Build());
জাভা
CpSolver solver = new CpSolver(); CpSolverStatus status = solver.solve(model);
সি#
CpSolver solver = new CpSolver();
CpSolverStatus status = solver.Solve(model);
Console.WriteLine($"Solve status: {status}");সমাধান প্রিন্ট করুন
নিম্নলিখিত কোডটি সমস্যার সমাধান প্রিন্ট করে।
পাইথন
if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE:
print(f"Total cost = {solver.objective_value}\n")
selected = data.loc[solver.boolean_values(x).loc[lambda x: x].index]
for unused_index, row in selected.iterrows():
print(f"{row.task} assigned to {row.worker} with a cost of {row.cost}")
elif status == cp_model.INFEASIBLE:
print("No solution found")
else:
print("Something is wrong, check the status and the log of the solve")সি++
if (response.status() == CpSolverStatus::INFEASIBLE) {
LOG(FATAL) << "No solution found.";
}
LOG(INFO) << "Total cost: " << response.objective_value();
LOG(INFO);
for (int i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
if (SolutionBooleanValue(response, x[i][j])) {
LOG(INFO) << "Task " << i << " assigned to worker " << j
<< ". Cost: " << costs[i][j];
}
}
}জাভা
// Check that the problem has a feasible solution.
if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) {
System.out.println("Total cost: " + solver.objectiveValue() + "\n");
for (int i = 0; i < numWorkers; ++i) {
for (int j = 0; j < numTasks; ++j) {
if (solver.booleanValue(x[i][j])) {
System.out.println(
"Worker " + i + " assigned to task " + j + ". Cost: " + costs[i][j]);
}
}
}
} else {
System.err.println("No solution found.");
}সি#
// Check that the problem has a feasible solution.
if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible)
{
Console.WriteLine($"Total cost: {solver.ObjectiveValue}\n");
for (int i = 0; i < numWorkers; ++i)
{
for (int j = 0; j < numTasks; ++j)
{
if (solver.Value(x[i, j]) > 0.5)
{
Console.WriteLine($"Worker {i} assigned to task {j}. Cost: {costs[i, j]}");
}
}
}
}
else
{
Console.WriteLine("No solution found.");
}এখানে প্রোগ্রামের আউটপুট আছে।
Total cost = 265 Worker 0 assigned to task 3 Cost = 70 Worker 1 assigned to task 2 Cost = 55 Worker 2 assigned to task 1 Cost = 95 Worker 3 assigned to task 0 Cost = 45
সম্পূর্ণ প্রোগ্রাম
এখানে CP-SAT সমাধানের জন্য সম্পূর্ণ প্রোগ্রাম রয়েছে।
পাইথন
import io
import pandas as pd
from ortools.sat.python import cp_model
def main() -> None:
# Data
data_str = """
worker task cost
w1 t1 90
w1 t2 80
w1 t3 75
w1 t4 70
w2 t1 35
w2 t2 85
w2 t3 55
w2 t4 65
w3 t1 125
w3 t2 95
w3 t3 90
w3 t4 95
w4 t1 45
w4 t2 110
w4 t3 95
w4 t4 115
w5 t1 50
w5 t2 110
w5 t3 90
w5 t4 100
"""
data = pd.read_table(io.StringIO(data_str), sep=r"\s+")
# Model
model = cp_model.CpModel()
# Variables
x = model.new_bool_var_series(name="x", index=data.index)
# Constraints
# Each worker is assigned to at most one task.
for unused_name, tasks in data.groupby("worker"):
model.add_at_most_one(x[tasks.index])
# Each task is assigned to exactly one worker.
for unused_name, workers in data.groupby("task"):
model.add_exactly_one(x[workers.index])
# Objective
model.minimize(data.cost.dot(x))
# 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")
selected = data.loc[solver.boolean_values(x).loc[lambda x: x].index]
for unused_index, row in selected.iterrows():
print(f"{row.task} assigned to {row.worker} with a cost of {row.cost}")
elif status == cp_model.INFEASIBLE:
print("No solution found")
else:
print("Something is wrong, check the status and the log of the solve")
if __name__ == "__main__":
main()সি++
#include <stdlib.h>
#include <vector>
#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 IntegerProgrammingExample() {
// Data
const std::vector<std::vector<int>> costs{
{90, 80, 75, 70}, {35, 85, 55, 65}, {125, 95, 90, 95},
{45, 110, 95, 115}, {50, 100, 90, 100},
};
const int num_workers = static_cast<int>(costs.size());
const int num_tasks = static_cast<int>(costs[0].size());
// 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 i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
x[i][j] = cp_model.NewBoolVar();
}
}
// Constraints
// Each worker is assigned to at most one task.
for (int i = 0; i < num_workers; ++i) {
cp_model.AddAtMostOne(x[i]);
}
// Each task is assigned to exactly one worker.
for (int j = 0; j < num_tasks; ++j) {
std::vector<BoolVar> tasks;
for (int i = 0; i < num_workers; ++i) {
tasks.push_back(x[i][j]);
}
cp_model.AddExactlyOne(tasks);
}
// Objective
LinearExpr total_cost;
for (int i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
total_cost += x[i][j] * costs[i][j];
}
}
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 i = 0; i < num_workers; ++i) {
for (int j = 0; j < num_tasks; ++j) {
if (SolutionBooleanValue(response, x[i][j])) {
LOG(INFO) << "Task " << i << " assigned to worker " << j
<< ". Cost: " << costs[i][j];
}
}
}
}
} // namespace sat
} // namespace operations_research
int main(int argc, char** argv) {
operations_research::sat::IntegerProgrammingExample();
return EXIT_SUCCESS;
}জাভা
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 AssignmentSat {
public static void main(String[] args) {
Loader.loadNativeLibraries();
// Data
int[][] costs = {
{90, 80, 75, 70},
{35, 85, 55, 65},
{125, 95, 90, 95},
{45, 110, 95, 115},
{50, 100, 90, 100},
};
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();
// Model
CpModel model = new CpModel();
// Variables
Literal[][] x = new Literal[numWorkers][numTasks];
for (int worker : allWorkers) {
for (int task : allTasks) {
x[worker][task] = model.newBoolVar("x[" + worker + "," + task + "]");
}
}
// Constraints
// Each worker is assigned to at most one task.
for (int worker : allWorkers) {
List<Literal> tasks = new ArrayList<>();
for (int task : allTasks) {
tasks.add(x[worker][task]);
}
model.addAtMostOne(tasks);
}
// Each task is assigned to exactly one worker.
for (int task : allTasks) {
List<Literal> workers = new ArrayList<>();
for (int worker : allWorkers) {
workers.add(x[worker][task]);
}
model.addExactlyOne(workers);
}
// 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 i = 0; i < numWorkers; ++i) {
for (int j = 0; j < numTasks; ++j) {
if (solver.booleanValue(x[i][j])) {
System.out.println(
"Worker " + i + " assigned to task " + j + ". Cost: " + costs[i][j]);
}
}
}
} else {
System.err.println("No solution found.");
}
}
private AssignmentSat() {}
}
সি#
using System;
using System.Collections.Generic;
using Google.OrTools.Sat;
public class AssignmentSat
{
public static void Main(String[] args)
{
// Data.
int[,] costs = {
{ 90, 80, 75, 70 }, { 35, 85, 55, 65 }, { 125, 95, 90, 95 }, { 45, 110, 95, 115 }, { 50, 100, 90, 100 },
};
int numWorkers = costs.GetLength(0);
int numTasks = costs.GetLength(1);
// Model.
CpModel model = new CpModel();
// Variables.
BoolVar[,] x = new BoolVar[numWorkers, numTasks];
// Variables in a 1-dim array.
for (int worker = 0; worker < numWorkers; ++worker)
{
for (int task = 0; task < numTasks; ++task)
{
x[worker, task] = model.NewBoolVar($"worker_{worker}_task_{task}");
}
}
// Constraints
// Each worker is assigned to at most one task.
for (int worker = 0; worker < numWorkers; ++worker)
{
List<ILiteral> tasks = new List<ILiteral>();
for (int task = 0; task < numTasks; ++task)
{
tasks.Add(x[worker, task]);
}
model.AddAtMostOne(tasks);
}
// Each task is assigned to exactly one worker.
for (int task = 0; task < numTasks; ++task)
{
List<ILiteral> workers = new List<ILiteral>();
for (int worker = 0; worker < numWorkers; ++worker)
{
workers.Add(x[worker, task]);
}
model.AddExactlyOne(workers);
}
// Objective
LinearExprBuilder obj = LinearExpr.NewBuilder();
for (int worker = 0; worker < numWorkers; ++worker)
{
for (int task = 0; task < numTasks; ++task)
{
obj.AddTerm((IntVar)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");
for (int i = 0; i < numWorkers; ++i)
{
for (int j = 0; j < numTasks; ++j)
{
if (solver.Value(x[i, j]) > 0.5)
{
Console.WriteLine($"Worker {i} assigned to task {j}. Cost: {costs[i, j]}");
}
}
}
}
else
{
Console.WriteLine("No solution found.");
}
Console.WriteLine("Statistics");
Console.WriteLine($" - conflicts : {solver.NumConflicts()}");
Console.WriteLine($" - branches : {solver.NumBranches()}");
Console.WriteLine($" - wall time : {solver.WallTime()}s");
}
}