La section précédente vous a montré comment trouver toutes les solutions à un problème de CP. Ensuite, nous allons vous montrer comment trouver une solution optimale. Par exemple, nous allons résoudre le problème d'optimisation suivant.
- Maximisez 2x + 2y + 3z en tenant compte des contraintes suivantes:
-
x + 7⁄2 y + 3⁄2 z ≤ 25 3x – 5y + 7z ≤ 45 5x + 2y - 6z ≤ 37 x, y, z ≥ 0 Entiers x, y, z
Afin d'augmenter la vitesse de calcul, le résolveur CP-SAT travaille sur des entiers. Cela signifie que toutes les contraintes et l'objectif doivent avoir des coefficients entiers. Dans l'exemple ci-dessus, la première contrainte ne remplit pas cette condition. Pour résoudre le problème, vous devez d'abord transformer la contrainte en la multipliant par un entier suffisamment grand pour convertir tous les coefficients en entiers. Ce processus est illustré dans la section Contraintes ci-dessous.
Solution utilisant le résolveur CP-SAT
Les sections suivantes présentent un programme Python qui résout le problème à l'aide du résolveur CP-SAT.
Importer les bibliothèques
Le code suivant importe la bibliothèque requise.
Python
from ortools.sat.python import cp_model
C++
#include <stdint.h> #include <stdlib.h> #include <algorithm> #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" #include "ortools/util/sorted_interval_list.h"
Java
import static java.util.Arrays.stream; import com.google.ortools.Loader; import com.google.ortools.sat.CpModel; import com.google.ortools.sat.CpSolver; import com.google.ortools.sat.CpSolverStatus; import com.google.ortools.sat.IntVar; import com.google.ortools.sat.LinearExpr;
C#
using System; using System.Linq; using Google.OrTools.Sat;
Déclarer le modèle
Le code suivant déclare le modèle posant problème.
Python
model = cp_model.CpModel()
C++
CpModelBuilder cp_model;
Java
CpModel model = new CpModel();
C#
CpModel model = new CpModel();
Créer les variables
Le code suivant crée les variables correspondant au problème.
Python
var_upper_bound = max(50, 45, 37) x = model.new_int_var(0, var_upper_bound, "x") y = model.new_int_var(0, var_upper_bound, "y") z = model.new_int_var(0, var_upper_bound, "z")
C++
int64_t var_upper_bound = std::max({50, 45, 37}); const Domain domain(0, var_upper_bound); const IntVar x = cp_model.NewIntVar(domain).WithName("x"); const IntVar y = cp_model.NewIntVar(domain).WithName("y"); const IntVar z = cp_model.NewIntVar(domain).WithName("z");
Java
int varUpperBound = stream(new int[] {50, 45, 37}).max().getAsInt(); IntVar x = model.newIntVar(0, varUpperBound, "x"); IntVar y = model.newIntVar(0, varUpperBound, "y"); IntVar z = model.newIntVar(0, varUpperBound, "z");
C#
int varUpperBound = new int[] { 50, 45, 37 }.Max(); IntVar x = model.NewIntVar(0, varUpperBound, "x"); IntVar y = model.NewIntVar(0, varUpperBound, "y"); IntVar z = model.NewIntVar(0, varUpperBound, "z");
Définir les contraintes
Depuis la première contrainte,
x + 7⁄2 y + 3⁄2 z | ≤ | 25 |
présente des coefficients non entiers, vous devez d'abord multiplier la contrainte entière par un entier suffisamment grand pour convertir les coefficients en entiers. Dans ce cas, vous pouvez multiplier par 2, ce qui donne la nouvelle contrainte.
2x + 7y + 3z | ≤ | 50 |
Cela ne modifie pas le problème, car la contrainte d'origine a exactement les mêmes solutions que la contrainte transformée.
Le code suivant définit les trois contraintes linéaires pour le problème:
Python
model.add(2 * x + 7 * y + 3 * z <= 50) model.add(3 * x - 5 * y + 7 * z <= 45) model.add(5 * x + 2 * y - 6 * z <= 37)
C++
cp_model.AddLessOrEqual(2 * x + 7 * y + 3 * z, 50); cp_model.AddLessOrEqual(3 * x - 5 * y + 7 * z, 45); cp_model.AddLessOrEqual(5 * x + 2 * y - 6 * z, 37);
Java
model.addLessOrEqual(LinearExpr.weightedSum(new IntVar[] {x, y, z}, new long[] {2, 7, 3}), 50); model.addLessOrEqual(LinearExpr.weightedSum(new IntVar[] {x, y, z}, new long[] {3, -5, 7}), 45); model.addLessOrEqual(LinearExpr.weightedSum(new IntVar[] {x, y, z}, new long[] {5, 2, -6}), 37);
C#
model.Add(2 * x + 7 * y + 3 * z <= 50); model.Add(3 * x - 5 * y + 7 * z <= 45); model.Add(5 * x + 2 * y - 6 * z <= 37);
Définir la fonction objectif
Le code suivant définit la fonction objectif du problème et la déclare comme un problème de maximisation:
Python
model.maximize(2 * x + 2 * y + 3 * z)
C++
cp_model.Maximize(2 * x + 2 * y + 3 * z);
Java
model.maximize(LinearExpr.weightedSum(new IntVar[] {x, y, z}, new long[] {2, 2, 3}));
C#
model.Maximize(2 * x + 2 * y + 3 * z);
Appeler le résolveur
Le code suivant appelle le résolveur.
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);
Afficher la solution
Le code suivant affiche les résultats.
Python
if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE: print(f"Maximum of objective function: {solver.objective_value}\n") print(f"x = {solver.value(x)}") print(f"y = {solver.value(y)}") print(f"z = {solver.value(z)}") else: print("No solution found.")
C++
if (response.status() == CpSolverStatus::OPTIMAL || response.status() == CpSolverStatus::FEASIBLE) { // Get the value of x in the solution. LOG(INFO) << "Maximum of objective function: " << response.objective_value(); LOG(INFO) << "x = " << SolutionIntegerValue(response, x); LOG(INFO) << "y = " << SolutionIntegerValue(response, y); LOG(INFO) << "z = " << SolutionIntegerValue(response, z); } else { LOG(INFO) << "No solution found."; }
Java
if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) { System.out.printf("Maximum of objective function: %f%n", solver.objectiveValue()); System.out.println("x = " + solver.value(x)); System.out.println("y = " + solver.value(y)); System.out.println("z = " + solver.value(z)); } else { System.out.println("No solution found."); }
C#
if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible) { Console.WriteLine($"Maximum of objective function: {solver.ObjectiveValue}"); Console.WriteLine("x = " + solver.Value(x)); Console.WriteLine("y = " + solver.Value(y)); Console.WriteLine("z = " + solver.Value(z)); } else { Console.WriteLine("No solution found."); }
La sortie est illustrée ci-dessous:
Maximum of objective function: 35 x value: 7 y value: 3 z value: 5
L'ensemble du programme
Le programme complet est présenté ci-dessous.
Python
"""Simple solve.""" from ortools.sat.python import cp_model def main() -> None: """Minimal CP-SAT example to showcase calling the solver.""" # Creates the model. model = cp_model.CpModel() # Creates the variables. var_upper_bound = max(50, 45, 37) x = model.new_int_var(0, var_upper_bound, "x") y = model.new_int_var(0, var_upper_bound, "y") z = model.new_int_var(0, var_upper_bound, "z") # Creates the constraints. model.add(2 * x + 7 * y + 3 * z <= 50) model.add(3 * x - 5 * y + 7 * z <= 45) model.add(5 * x + 2 * y - 6 * z <= 37) model.maximize(2 * x + 2 * y + 3 * z) # Creates a solver and solves the model. solver = cp_model.CpSolver() status = solver.solve(model) if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE: print(f"Maximum of objective function: {solver.objective_value}\n") print(f"x = {solver.value(x)}") print(f"y = {solver.value(y)}") print(f"z = {solver.value(z)}") else: print("No solution found.") # Statistics. print("\nStatistics") print(f" status : {solver.status_name(status)}") print(f" conflicts: {solver.num_conflicts}") print(f" branches : {solver.num_branches}") print(f" wall time: {solver.wall_time} s") if __name__ == "__main__": main()
C++
#include <stdint.h> #include <stdlib.h> #include <algorithm> #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" #include "ortools/util/sorted_interval_list.h" namespace operations_research { namespace sat { void CpSatExample() { CpModelBuilder cp_model; int64_t var_upper_bound = std::max({50, 45, 37}); const Domain domain(0, var_upper_bound); const IntVar x = cp_model.NewIntVar(domain).WithName("x"); const IntVar y = cp_model.NewIntVar(domain).WithName("y"); const IntVar z = cp_model.NewIntVar(domain).WithName("z"); cp_model.AddLessOrEqual(2 * x + 7 * y + 3 * z, 50); cp_model.AddLessOrEqual(3 * x - 5 * y + 7 * z, 45); cp_model.AddLessOrEqual(5 * x + 2 * y - 6 * z, 37); cp_model.Maximize(2 * x + 2 * y + 3 * z); // Solving part. const CpSolverResponse response = Solve(cp_model.Build()); if (response.status() == CpSolverStatus::OPTIMAL || response.status() == CpSolverStatus::FEASIBLE) { // Get the value of x in the solution. LOG(INFO) << "Maximum of objective function: " << response.objective_value(); LOG(INFO) << "x = " << SolutionIntegerValue(response, x); LOG(INFO) << "y = " << SolutionIntegerValue(response, y); LOG(INFO) << "z = " << SolutionIntegerValue(response, z); } else { LOG(INFO) << "No solution found."; } // Statistics. LOG(INFO) << "Statistics"; LOG(INFO) << CpSolverResponseStats(response); } } // namespace sat } // namespace operations_research int main() { operations_research::sat::CpSatExample(); return EXIT_SUCCESS; }
Java
package com.google.ortools.sat.samples; import static java.util.Arrays.stream; import com.google.ortools.Loader; import com.google.ortools.sat.CpModel; import com.google.ortools.sat.CpSolver; import com.google.ortools.sat.CpSolverStatus; import com.google.ortools.sat.IntVar; import com.google.ortools.sat.LinearExpr; /** Minimal CP-SAT example to showcase calling the solver. */ public final class CpSatExample { public static void main(String[] args) { Loader.loadNativeLibraries(); // Create the model. CpModel model = new CpModel(); // Create the variables. int varUpperBound = stream(new int[] {50, 45, 37}).max().getAsInt(); IntVar x = model.newIntVar(0, varUpperBound, "x"); IntVar y = model.newIntVar(0, varUpperBound, "y"); IntVar z = model.newIntVar(0, varUpperBound, "z"); // Create the constraints. model.addLessOrEqual(LinearExpr.weightedSum(new IntVar[] {x, y, z}, new long[] {2, 7, 3}), 50); model.addLessOrEqual(LinearExpr.weightedSum(new IntVar[] {x, y, z}, new long[] {3, -5, 7}), 45); model.addLessOrEqual(LinearExpr.weightedSum(new IntVar[] {x, y, z}, new long[] {5, 2, -6}), 37); model.maximize(LinearExpr.weightedSum(new IntVar[] {x, y, z}, new long[] {2, 2, 3})); // Create a solver and solve the model. CpSolver solver = new CpSolver(); CpSolverStatus status = solver.solve(model); if (status == CpSolverStatus.OPTIMAL || status == CpSolverStatus.FEASIBLE) { System.out.printf("Maximum of objective function: %f%n", solver.objectiveValue()); System.out.println("x = " + solver.value(x)); System.out.println("y = " + solver.value(y)); System.out.println("z = " + solver.value(z)); } else { System.out.println("No solution found."); } // Statistics. System.out.println("Statistics"); System.out.printf(" conflicts: %d%n", solver.numConflicts()); System.out.printf(" branches : %d%n", solver.numBranches()); System.out.printf(" wall time: %f s%n", solver.wallTime()); } private CpSatExample() {} }
C#
using System; using System.Linq; using Google.OrTools.Sat; public class CpSatExample { static void Main() { // Creates the model. CpModel model = new CpModel(); // Creates the variables. int varUpperBound = new int[] { 50, 45, 37 }.Max(); IntVar x = model.NewIntVar(0, varUpperBound, "x"); IntVar y = model.NewIntVar(0, varUpperBound, "y"); IntVar z = model.NewIntVar(0, varUpperBound, "z"); // Creates the constraints. model.Add(2 * x + 7 * y + 3 * z <= 50); model.Add(3 * x - 5 * y + 7 * z <= 45); model.Add(5 * x + 2 * y - 6 * z <= 37); model.Maximize(2 * x + 2 * y + 3 * z); // Creates a solver and solves the model. CpSolver solver = new CpSolver(); CpSolverStatus status = solver.Solve(model); if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible) { Console.WriteLine($"Maximum of objective function: {solver.ObjectiveValue}"); Console.WriteLine("x = " + solver.Value(x)); Console.WriteLine("y = " + solver.Value(y)); Console.WriteLine("z = " + solver.Value(z)); } 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"); } }