Questa sezione mostra come risolvere il problema degli zaini per più zaini utilizzando sia il risolutore MIP sia il risolutore CP-SAT. In questo caso, è comune fare riferimento ai container come bin, degli zaini.
Il prossimo esempio mostra come trovare il modo ottimale per imballare gli articoli in cinque bin.
Esempio
Come nell'esempio precedente, inizi con una raccolta di elementi con diversi pesi e valori. Il problema è pacchettizzare degli articoli in cinque bin, ognuno dei quali ha una capacità massima di 100, in modo che il valore pacchettizzato totale sia il massimo.
Le seguenti sezioni presentano sezioni di programmi che risolvono questo problema. Per i programmi completi, vedi Programmi completi.
soluzione MIP
Le sezioni seguenti descrivono come risolvere il problema utilizzando il comando Wrapper MPSolver.
Importa le librerie
Il codice seguente importa le librerie richieste.
Python
from ortools.linear_solver import pywraplp
C++
#include <iostream> #include <memory> #include <numeric> #include <vector> #include "absl/strings/str_format.h" #include "ortools/linear_solver/linear_expr.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;
crea i dati
Il seguente codice crea i dati per il problema.
Python
data = {} data["weights"] = [48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36] data["values"] = [10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25] assert len(data["weights"]) == len(data["values"]) data["num_items"] = len(data["weights"]) data["all_items"] = range(data["num_items"]) data["bin_capacities"] = [100, 100, 100, 100, 100] data["num_bins"] = len(data["bin_capacities"]) data["all_bins"] = range(data["num_bins"])
C++
const std::vector<int> weights = { {48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36}}; const std::vector<int> values = { {10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25}}; const int num_items = weights.size(); std::vector<int> all_items(num_items); std::iota(all_items.begin(), all_items.end(), 0); const std::vector<int> bin_capacities = {{100, 100, 100, 100, 100}}; const int num_bins = bin_capacities.size(); std::vector<int> all_bins(num_bins); std::iota(all_bins.begin(), all_bins.end(), 0);
Java
final double[] weights = {48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36};
final double[] values = {10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25};
final int numItems = weights.length;
final int[] allItems = IntStream.range(0, numItems).toArray();
final double[] binCapacities = {100, 100, 100, 100, 100};
final int numBins = binCapacities.length;
final int[] allBins = IntStream.range(0, numBins).toArray();C#
double[] Weights = { 48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36 };
double[] Values = { 10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25 };
int NumItems = Weights.Length;
int[] allItems = Enumerable.Range(0, NumItems).ToArray();
double[] BinCapacities = { 100, 100, 100, 100, 100 };
int NumBins = BinCapacities.Length;
int[] allBins = Enumerable.Range(0, NumBins).ToArray();I dati includono:
weights: un vettore contenente i pesi degli elementi.values: un vettore contenente i valori degli elementi.capacities: un vettore contenente le capacità delle fasce.
In questo esempio, tutti i bin hanno la stessa capacità, ma non è necessario che sia vero in generale.
Dichiara il risolutore MIP
Il seguente codice dichiara il risolutore MIP.
Python
solver = pywraplp.Solver.CreateSolver("SCIP") if solver is None: print("SCIP solver unavailable.") return
C++
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#
// Create the linear solver with the SCIP backend. Solver solver = Solver.CreateSolver("SCIP"); if (solver is null) { return; }
Crea le variabili
Il seguente codice crea le variabili per il problema.
Python
# x[i, b] = 1 if item i is packed in bin b. x = {} for i in data["all_items"]: for b in data["all_bins"]: x[i, b] = solver.BoolVar(f"x_{i}_{b}")
C++
// x[i][b] = 1 if item i is packed in bin b. std::vector<std::vector<const MPVariable*>> x( num_items, std::vector<const MPVariable*>(num_bins)); for (int i : all_items) { for (int b : all_bins) { x[i][b] = solver->MakeBoolVar(absl::StrFormat("x_%d_%d", i, b)); } }
Java
MPVariable[][] x = new MPVariable[numItems][numBins]; for (int i : allItems) { for (int b : allBins) { x[i][b] = solver.makeBoolVar("x_" + i + "_" + b); } }
C#
Variable[,] x = new Variable[NumItems, NumBins];
foreach (int i in allItems)
{
foreach (int b in allBins)
{
x[i, b] = solver.MakeBoolVar($"x_{i}_{b}");
}
}Ogni x[(i, j)] è una variabile 0-1, dove i è un elemento e j è un bin. Nella
la soluzione, x[(i, j)] è 1 se l'elemento i si trova nella fascia j e 0
negli altri casi.
Definisci i vincoli
Il seguente codice definisce i vincoli per il problema:
Python
# Each item is assigned to at most one bin. for i in data["all_items"]: solver.Add(sum(x[i, b] for b in data["all_bins"]) <= 1) # The amount packed in each bin cannot exceed its capacity. for b in data["all_bins"]: solver.Add( sum(x[i, b] * data["weights"][i] for i in data["all_items"]) <= data["bin_capacities"][b] )
C++
// Each item is assigned to at most one bin. for (int i : all_items) { LinearExpr sum; for (int b : all_bins) { sum += x[i][b]; } solver->MakeRowConstraint(sum <= 1.0); } // The amount packed in each bin cannot exceed its capacity. for (int b : all_bins) { LinearExpr bin_weight; for (int i : all_items) { bin_weight += LinearExpr(x[i][b]) * weights[i]; } solver->MakeRowConstraint(bin_weight <= bin_capacities[b]); }
Java
// Each item is assigned to at most one bin. for (int i : allItems) { MPConstraint constraint = solver.makeConstraint(0, 1, ""); for (int b : allBins) { constraint.setCoefficient(x[i][b], 1); } } // The amount packed in each bin cannot exceed its capacity. for (int b : allBins) { MPConstraint constraint = solver.makeConstraint(0, binCapacities[b], ""); for (int i : allItems) { constraint.setCoefficient(x[i][b], weights[i]); } }
C#
// Each item is assigned to at most one bin. foreach (int i in allItems) { Constraint constraint = solver.MakeConstraint(0, 1, ""); foreach (int b in allBins) { constraint.SetCoefficient(x[i, b], 1); } } // The amount packed in each bin cannot exceed its capacity. foreach (int b in allBins) { Constraint constraint = solver.MakeConstraint(0, BinCapacities[b], ""); foreach (int i in allItems) { constraint.SetCoefficient(x[i, b], Weights[i]); } }
I vincoli sono i seguenti:
- Ogni articolo può essere posizionato al massimo in un contenitore. Questo vincolo è impostato da
richiedere che la somma di
x[i, j]su tutti i binjsia minore o uguale a a 1. - Il peso totale imballato in ogni bin non può superare la sua capacità. Questo
Il vincolo viene impostato richiedendo la somma dei pesi degli elementi posizionati in una fascia
jsia inferiore o uguale alla capacità del contenitore.
Definisci l'obiettivo
Il seguente codice definisce la funzione obiettivo per il problema, ovvero valore totale degli articoli imballati.
Python
# Maximize total value of packed items. objective = solver.Objective() for i in data["all_items"]: for b in data["all_bins"]: objective.SetCoefficient(x[i, b], data["values"][i]) objective.SetMaximization()
C++
// Maximize total value of packed items. MPObjective* const objective = solver->MutableObjective(); LinearExpr objective_value; for (int i : all_items) { for (int b : all_bins) { objective_value += LinearExpr(x[i][b]) * values[i]; } } objective->MaximizeLinearExpr(objective_value);
Java
// Maximize total value of packed items. MPObjective objective = solver.objective(); for (int i : allItems) { for (int b : allBins) { objective.setCoefficient(x[i][b], values[i]); } } objective.setMaximization();
C#
Objective objective = solver.Objective(); foreach (int i in allItems) { foreach (int b in allBins) { objective.SetCoefficient(x[i, b], Values[i]); } } objective.SetMaximization();
Tieni presente che x[i, j] * data['values'][i] aggiunge il valore dell'elemento i al
se l'elemento si trova nella fascia j. Se i non è inserito in nessun bin, la relativa
non contribuisce all'obiettivo.
Richiama il risolutore
Il seguente codice richiama il risolutore.
Python
print(f"Solving with {solver.SolverVersion()}")
status = solver.Solve()C++
const MPSolver::ResultStatus result_status = solver->Solve();
Java
final MPSolver.ResultStatus status = solver.solve();
C#
Solver.ResultStatus resultStatus = solver.Solve();
Stampa la soluzione
Il seguente codice visualizza la soluzione al problema.
Python
if status == pywraplp.Solver.OPTIMAL: print(f"Total packed value: {objective.Value()}") total_weight = 0 for b in data["all_bins"]: print(f"Bin {b}") bin_weight = 0 bin_value = 0 for i in data["all_items"]: if x[i, b].solution_value() > 0: print( f"Item {i} weight: {data['weights'][i]} value:" f" {data['values'][i]}" ) bin_weight += data["weights"][i] bin_value += data["values"][i] print(f"Packed bin weight: {bin_weight}") print(f"Packed bin value: {bin_value}\n") total_weight += bin_weight print(f"Total packed weight: {total_weight}") else: print("The problem does not have an optimal solution.")
C++
if (result_status == MPSolver::OPTIMAL) { LOG(INFO) << "Total packed value: " << objective->Value(); double total_weight = 0.0; for (int b : all_bins) { LOG(INFO) << "Bin " << b; double bin_weight = 0.0; double bin_value = 0.0; for (int i : all_items) { if (x[i][b]->solution_value() > 0) { LOG(INFO) << "Item " << i << " weight: " << weights[i] << " value: " << values[i]; bin_weight += weights[i]; bin_value += values[i]; } } LOG(INFO) << "Packed bin weight: " << bin_weight; LOG(INFO) << "Packed bin value: " << bin_value; total_weight += bin_weight; } LOG(INFO) << "Total packed weight: " << total_weight; } else { LOG(INFO) << "The problem does not have an optimal solution."; }
Java
// Check that the problem has an optimal solution. if (status == MPSolver.ResultStatus.OPTIMAL) { System.out.println("Total packed value: " + objective.value()); double totalWeight = 0; for (int b : allBins) { double binWeight = 0; double binValue = 0; System.out.println("Bin " + b); for (int i : allItems) { if (x[i][b].solutionValue() == 1) { System.out.println("Item " + i + " weight: " + weights[i] + " value: " + values[i]); binWeight += weights[i]; binValue += values[i]; } } System.out.println("Packed bin weight: " + binWeight); System.out.println("Packed bin value: " + binValue); totalWeight += binWeight; } System.out.println("Total packed weight: " + totalWeight); } else { System.err.println("The problem does not have an optimal solution."); }
C#
// Check that the problem has an optimal solution. if (resultStatus == Solver.ResultStatus.OPTIMAL) { Console.WriteLine($"Total packed value: {solver.Objective().Value()}"); double TotalWeight = 0.0; foreach (int b in allBins) { double BinWeight = 0.0; double BinValue = 0.0; Console.WriteLine("Bin " + b); foreach (int i in allItems) { if (x[i, b].SolutionValue() == 1) { Console.WriteLine($"Item {i} weight: {Weights[i]} values: {Values[i]}"); BinWeight += Weights[i]; BinValue += Values[i]; } } Console.WriteLine("Packed bin weight: " + BinWeight); Console.WriteLine("Packed bin value: " + BinValue); TotalWeight += BinWeight; } Console.WriteLine("Total packed weight: " + TotalWeight); } else { Console.WriteLine("The problem does not have an optimal solution!"); }
Per ogni fascia, il codice visualizza gli elementi che si trovano nella fascia e i relativi il valore totale e la ponderazione. Il codice visualizza anche il valore totale complessivo peso degli articoli imballati.
Quando esegui il programma, viene visualizzato l'output seguente.
Total packed value: 395.0 Bin 0 Item 3 - weight: 36 value: 50 Item 13 - weight: 36 value: 30 Packed bin weight: 72 Packed bin value: 80 Bin 1 Item 5 - weight: 48 value: 30 Item 7 - weight: 42 value: 40 Packed bin weight: 90 Packed bin value: 70 Bin 2 Item 1 - weight: 30 value: 30 Item 10 - weight: 30 value: 45 Item 14 - weight: 36 value: 25 Packed bin weight: 96 Packed bin value: 100 Bin 3 Item 2 - weight: 42 value: 25 Item 12 - weight: 42 value: 20 Packed bin weight: 84 Packed bin value: 45 Bin 4 Item 4 - weight: 36 value: 35 Item 8 - weight: 36 value: 30 Item 9 - weight: 24 value: 35 Packed bin weight: 96 Packed bin value: 100 Total packed weight: 438
Completa i programmi
Di seguito sono riportati i programmi completi per più zaini.
Python
"""Solve a multiple knapsack problem using a MIP solver.""" from ortools.linear_solver import pywraplp def main(): data = {} data["weights"] = [48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36] data["values"] = [10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25] assert len(data["weights"]) == len(data["values"]) data["num_items"] = len(data["weights"]) data["all_items"] = range(data["num_items"]) data["bin_capacities"] = [100, 100, 100, 100, 100] data["num_bins"] = len(data["bin_capacities"]) data["all_bins"] = range(data["num_bins"]) # Create the mip solver with the SCIP backend. solver = pywraplp.Solver.CreateSolver("SCIP") if solver is None: print("SCIP solver unavailable.") return # Variables. # x[i, b] = 1 if item i is packed in bin b. x = {} for i in data["all_items"]: for b in data["all_bins"]: x[i, b] = solver.BoolVar(f"x_{i}_{b}") # Constraints. # Each item is assigned to at most one bin. for i in data["all_items"]: solver.Add(sum(x[i, b] for b in data["all_bins"]) <= 1) # The amount packed in each bin cannot exceed its capacity. for b in data["all_bins"]: solver.Add( sum(x[i, b] * data["weights"][i] for i in data["all_items"]) <= data["bin_capacities"][b] ) # Objective. # Maximize total value of packed items. objective = solver.Objective() for i in data["all_items"]: for b in data["all_bins"]: objective.SetCoefficient(x[i, b], data["values"][i]) objective.SetMaximization() print(f"Solving with {solver.SolverVersion()}") status = solver.Solve() if status == pywraplp.Solver.OPTIMAL: print(f"Total packed value: {objective.Value()}") total_weight = 0 for b in data["all_bins"]: print(f"Bin {b}") bin_weight = 0 bin_value = 0 for i in data["all_items"]: if x[i, b].solution_value() > 0: print( f"Item {i} weight: {data['weights'][i]} value:" f" {data['values'][i]}" ) bin_weight += data["weights"][i] bin_value += data["values"][i] print(f"Packed bin weight: {bin_weight}") print(f"Packed bin value: {bin_value}\n") total_weight += bin_weight print(f"Total packed weight: {total_weight}") else: print("The problem does not have an optimal solution.") if __name__ == "__main__": main()
C++
// Solve a multiple knapsack problem using a MIP solver. #include <iostream> #include <memory> #include <numeric> #include <vector> #include "absl/strings/str_format.h" #include "ortools/linear_solver/linear_expr.h" #include "ortools/linear_solver/linear_solver.h" namespace operations_research { void MultipleKnapsackMip() { const std::vector<int> weights = { {48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36}}; const std::vector<int> values = { {10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25}}; const int num_items = weights.size(); std::vector<int> all_items(num_items); std::iota(all_items.begin(), all_items.end(), 0); const std::vector<int> bin_capacities = {{100, 100, 100, 100, 100}}; const int num_bins = bin_capacities.size(); std::vector<int> all_bins(num_bins); std::iota(all_bins.begin(), all_bins.end(), 0); // 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][b] = 1 if item i is packed in bin b. std::vector<std::vector<const MPVariable*>> x( num_items, std::vector<const MPVariable*>(num_bins)); for (int i : all_items) { for (int b : all_bins) { x[i][b] = solver->MakeBoolVar(absl::StrFormat("x_%d_%d", i, b)); } } // Constraints. // Each item is assigned to at most one bin. for (int i : all_items) { LinearExpr sum; for (int b : all_bins) { sum += x[i][b]; } solver->MakeRowConstraint(sum <= 1.0); } // The amount packed in each bin cannot exceed its capacity. for (int b : all_bins) { LinearExpr bin_weight; for (int i : all_items) { bin_weight += LinearExpr(x[i][b]) * weights[i]; } solver->MakeRowConstraint(bin_weight <= bin_capacities[b]); } // Objective. // Maximize total value of packed items. MPObjective* const objective = solver->MutableObjective(); LinearExpr objective_value; for (int i : all_items) { for (int b : all_bins) { objective_value += LinearExpr(x[i][b]) * values[i]; } } objective->MaximizeLinearExpr(objective_value); const MPSolver::ResultStatus result_status = solver->Solve(); if (result_status == MPSolver::OPTIMAL) { LOG(INFO) << "Total packed value: " << objective->Value(); double total_weight = 0.0; for (int b : all_bins) { LOG(INFO) << "Bin " << b; double bin_weight = 0.0; double bin_value = 0.0; for (int i : all_items) { if (x[i][b]->solution_value() > 0) { LOG(INFO) << "Item " << i << " weight: " << weights[i] << " value: " << values[i]; bin_weight += weights[i]; bin_value += values[i]; } } LOG(INFO) << "Packed bin weight: " << bin_weight; LOG(INFO) << "Packed bin value: " << bin_value; total_weight += bin_weight; } LOG(INFO) << "Total packed weight: " << total_weight; } else { LOG(INFO) << "The problem does not have an optimal solution."; } } } // namespace operations_research int main(int argc, char** argv) { operations_research::MultipleKnapsackMip(); return EXIT_SUCCESS; }
Java
// Solve a multiple knapsack problem using a MIP solver. 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; /** Multiple knapsack problem. */ public class MultipleKnapsackMip { public static void main(String[] args) { Loader.loadNativeLibraries(); // Instantiate the data problem. final double[] weights = {48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36}; final double[] values = {10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25}; final int numItems = weights.length; final int[] allItems = IntStream.range(0, numItems).toArray(); final double[] binCapacities = {100, 100, 100, 100, 100}; final int numBins = binCapacities.length; final int[] allBins = IntStream.range(0, numBins).toArray(); // 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. MPVariable[][] x = new MPVariable[numItems][numBins]; for (int i : allItems) { for (int b : allBins) { x[i][b] = solver.makeBoolVar("x_" + i + "_" + b); } } // Constraints. // Each item is assigned to at most one bin. for (int i : allItems) { MPConstraint constraint = solver.makeConstraint(0, 1, ""); for (int b : allBins) { constraint.setCoefficient(x[i][b], 1); } } // The amount packed in each bin cannot exceed its capacity. for (int b : allBins) { MPConstraint constraint = solver.makeConstraint(0, binCapacities[b], ""); for (int i : allItems) { constraint.setCoefficient(x[i][b], weights[i]); } } // Objective. // Maximize total value of packed items. MPObjective objective = solver.objective(); for (int i : allItems) { for (int b : allBins) { objective.setCoefficient(x[i][b], values[i]); } } objective.setMaximization(); final MPSolver.ResultStatus status = solver.solve(); // Check that the problem has an optimal solution. if (status == MPSolver.ResultStatus.OPTIMAL) { System.out.println("Total packed value: " + objective.value()); double totalWeight = 0; for (int b : allBins) { double binWeight = 0; double binValue = 0; System.out.println("Bin " + b); for (int i : allItems) { if (x[i][b].solutionValue() == 1) { System.out.println("Item " + i + " weight: " + weights[i] + " value: " + values[i]); binWeight += weights[i]; binValue += values[i]; } } System.out.println("Packed bin weight: " + binWeight); System.out.println("Packed bin value: " + binValue); totalWeight += binWeight; } System.out.println("Total packed weight: " + totalWeight); } else { System.err.println("The problem does not have an optimal solution."); } } private MultipleKnapsackMip() {} }
C#
// Solve a multiple knapsack problem using a MIP solver. using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.LinearSolver; public class MultipleKnapsackMip { public static void Main() { // Instantiate the data problem. double[] Weights = { 48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36 }; double[] Values = { 10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25 }; int NumItems = Weights.Length; int[] allItems = Enumerable.Range(0, NumItems).ToArray(); double[] BinCapacities = { 100, 100, 100, 100, 100 }; int NumBins = BinCapacities.Length; int[] allBins = Enumerable.Range(0, NumBins).ToArray(); // Create the linear solver with the SCIP backend. Solver solver = Solver.CreateSolver("SCIP"); if (solver is null) { return; } // Variables. Variable[,] x = new Variable[NumItems, NumBins]; foreach (int i in allItems) { foreach (int b in allBins) { x[i, b] = solver.MakeBoolVar($"x_{i}_{b}"); } } // Constraints. // Each item is assigned to at most one bin. foreach (int i in allItems) { Constraint constraint = solver.MakeConstraint(0, 1, ""); foreach (int b in allBins) { constraint.SetCoefficient(x[i, b], 1); } } // The amount packed in each bin cannot exceed its capacity. foreach (int b in allBins) { Constraint constraint = solver.MakeConstraint(0, BinCapacities[b], ""); foreach (int i in allItems) { constraint.SetCoefficient(x[i, b], Weights[i]); } } // Objective. Objective objective = solver.Objective(); foreach (int i in allItems) { foreach (int b in allBins) { objective.SetCoefficient(x[i, b], Values[i]); } } objective.SetMaximization(); Solver.ResultStatus resultStatus = solver.Solve(); // Check that the problem has an optimal solution. if (resultStatus == Solver.ResultStatus.OPTIMAL) { Console.WriteLine($"Total packed value: {solver.Objective().Value()}"); double TotalWeight = 0.0; foreach (int b in allBins) { double BinWeight = 0.0; double BinValue = 0.0; Console.WriteLine("Bin " + b); foreach (int i in allItems) { if (x[i, b].SolutionValue() == 1) { Console.WriteLine($"Item {i} weight: {Weights[i]} values: {Values[i]}"); BinWeight += Weights[i]; BinValue += Values[i]; } } Console.WriteLine("Packed bin weight: " + BinWeight); Console.WriteLine("Packed bin value: " + BinValue); TotalWeight += BinWeight; } Console.WriteLine("Total packed weight: " + TotalWeight); } else { Console.WriteLine("The problem does not have an optimal solution!"); } } }
Soluzione CP SAT
Le seguenti sezioni descrivono come risolvere il problema utilizzando il risolutore CP-SAT.
Importa le librerie
Il codice seguente importa le librerie richieste.
Python
from ortools.sat.python import cp_model
C++
#include <stdlib.h> #include <map> #include <numeric> #include <tuple> #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; public class MultipleKnapsackSat { public static void Main(String[] args) { // Instantiate the data problem. int[] Weights = { 48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36 }; int[] Values = { 10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25 }; int NumItems = Weights.Length; int[] allItems = Enumerable.Range(0, NumItems).ToArray(); int[] BinCapacities = { 100, 100, 100, 100, 100 }; int NumBins = BinCapacities.Length; int[] allBins = Enumerable.Range(0, NumBins).ToArray(); // Model. CpModel model = new CpModel(); // Variables. ILiteral[,] x = new ILiteral[NumItems, NumBins]; foreach (int i in allItems) { foreach (int b in allBins) { x[i, b] = model.NewBoolVar($"x_{i}_{b}"); } } // Constraints. // Each item is assigned to at most one bin. foreach (int i in allItems) { List<ILiteral> literals = new List<ILiteral>(); foreach (int b in allBins) { literals.Add(x[i, b]); } model.AddAtMostOne(literals); } // The amount packed in each bin cannot exceed its capacity. foreach (int b in allBins) { List<ILiteral> items = new List<ILiteral>(); foreach (int i in allItems) { items.Add(x[i, b]); } model.Add(LinearExpr.WeightedSum(items, Weights) <= BinCapacities[b]); } // Objective. LinearExprBuilder obj = LinearExpr.NewBuilder(); foreach (int i in allItems) { foreach (int b in allBins) { obj.AddTerm(x[i, b], Values[i]); } } model.Maximize(obj); // Solve CpSolver solver = new CpSolver(); CpSolverStatus status = solver.Solve(model); // Print solution. // Check that the problem has a feasible solution. if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible) { Console.WriteLine($"Total packed value: {solver.ObjectiveValue}"); double TotalWeight = 0.0; foreach (int b in allBins) { double BinWeight = 0.0; double BinValue = 0.0; Console.WriteLine($"Bin {b}"); foreach (int i in allItems) { if (solver.BooleanValue(x[i, b])) { Console.WriteLine($"Item {i} weight: {Weights[i]} values: {Values[i]}"); BinWeight += Weights[i]; BinValue += Values[i]; } } Console.WriteLine("Packed bin weight: " + BinWeight); Console.WriteLine("Packed bin value: " + BinValue); TotalWeight += BinWeight; } Console.WriteLine("Total packed weight: " + TotalWeight); } 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"); } }
Dichiara il modello
Il seguente codice dichiara il modello CP-SAT.
Python
model = cp_model.CpModel()
C++
CpModelBuilder cp_model;
Java
CpModel model = new CpModel();
C#
CpModel model = new CpModel();
crea i dati
Il seguente codice configura i dati per il problema.
Python
data = {} data["weights"] = [48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36] data["values"] = [10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25] assert len(data["weights"]) == len(data["values"]) num_items = len(data["weights"]) all_items = range(num_items) data["bin_capacities"] = [100, 100, 100, 100, 100] num_bins = len(data["bin_capacities"]) all_bins = range(num_bins)
C++
const std::vector<int> weights = { {48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36}}; const std::vector<int> values = { {10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25}}; const int num_items = static_cast<int>(weights.size()); std::vector<int> all_items(num_items); std::iota(all_items.begin(), all_items.end(), 0); const std::vector<int> bin_capacities = {{100, 100, 100, 100, 100}}; const int num_bins = static_cast<int>(bin_capacities.size()); std::vector<int> all_bins(num_bins); std::iota(all_bins.begin(), all_bins.end(), 0);
Java
final int[] weights = {48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36};
final int[] values = {10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25};
final int numItems = weights.length;
final int[] allItems = IntStream.range(0, numItems).toArray();
final int[] binCapacities = {100, 100, 100, 100, 100};
final int numBins = binCapacities.length;
final int[] allBins = IntStream.range(0, numBins).toArray();C#
int[] Weights = { 48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36 };
int[] Values = { 10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25 };
int NumItems = Weights.Length;
int[] allItems = Enumerable.Range(0, NumItems).ToArray();
int[] BinCapacities = { 100, 100, 100, 100, 100 };
int NumBins = BinCapacities.Length;
int[] allBins = Enumerable.Range(0, NumBins).ToArray();L'array costs corrisponde alla tabella dei costi
per assegnare i worker alle attività, come mostrato sopra.
Crea le variabili
Il seguente codice crea variabili intere binarie per il problema.
Python
# x[i, b] = 1 if item i is packed in bin b. x = {} for i in all_items: for b in all_bins: x[i, b] = model.new_bool_var(f"x_{i}_{b}")
C++
// x[i, b] = 1 if item i is packed in bin b. std::map<std::tuple<int, int>, BoolVar> x; for (int i : all_items) { for (int b : all_bins) { auto key = std::make_tuple(i, b); x[key] = cp_model.NewBoolVar().WithName(absl::StrFormat("x_%d_%d", i, b)); } }
Java
Literal[][] x = new Literal[numItems][numBins]; for (int i : allItems) { for (int b : allBins) { x[i][b] = model.newBoolVar("x_" + i + "_" + b); } }
C#
ILiteral[,] x = new ILiteral[NumItems, NumBins];
foreach (int i in allItems)
{
foreach (int b in allBins)
{
x[i, b] = model.NewBoolVar($"x_{i}_{b}");
}
}crea i vincoli
Il seguente codice crea i vincoli per il problema.
Python
# Each item is assigned to at most one bin. for i in all_items: model.add_at_most_one(x[i, b] for b in all_bins) # The amount packed in each bin cannot exceed its capacity. for b in all_bins: model.add( sum(x[i, b] * data["weights"][i] for i in all_items) <= data["bin_capacities"][b] )
C++
// Each item is assigned to at most one bin. for (int i : all_items) { std::vector<BoolVar> copies; for (int b : all_bins) { copies.push_back(x[std::make_tuple(i, b)]); } cp_model.AddAtMostOne(copies); } // The amount packed in each bin cannot exceed its capacity. for (int b : all_bins) { LinearExpr bin_weight; for (int i : all_items) { bin_weight += x[std::make_tuple(i, b)] * weights[i]; } cp_model.AddLessOrEqual(bin_weight, bin_capacities[b]); }
Java
// Each item is assigned to at most one bin. for (int i : allItems) { List<Literal> bins = new ArrayList<>(); for (int b : allBins) { bins.add(x[i][b]); } model.addAtMostOne(bins); } // The amount packed in each bin cannot exceed its capacity. for (int b : allBins) { LinearExprBuilder load = LinearExpr.newBuilder(); for (int i : allItems) { load.addTerm(x[i][b], weights[i]); } model.addLessOrEqual(load, binCapacities[b]); }
C#
// Each item is assigned to at most one bin. foreach (int i in allItems) { List<ILiteral> literals = new List<ILiteral>(); foreach (int b in allBins) { literals.Add(x[i, b]); } model.AddAtMostOne(literals); } // The amount packed in each bin cannot exceed its capacity. foreach (int b in allBins) { List<ILiteral> items = new List<ILiteral>(); foreach (int i in allItems) { items.Add(x[i, b]); } model.Add(LinearExpr.WeightedSum(items, Weights) <= BinCapacities[b]); }
Crea la funzione obiettivo
Il seguente codice crea la funzione obiettivo per il problema.
Python
# maximize total value of packed items. objective = [] for i in all_items: for b in all_bins: objective.append(cp_model.LinearExpr.term(x[i, b], data["values"][i])) model.maximize(cp_model.LinearExpr.sum(objective))
C++
// Maximize total value of packed items. LinearExpr objective; for (int i : all_items) { for (int b : all_bins) { objective += x[std::make_tuple(i, b)] * values[i]; } } cp_model.Maximize(objective);
Java
// Maximize total value of packed items. LinearExprBuilder obj = LinearExpr.newBuilder(); for (int i : allItems) { for (int b : allBins) { obj.addTerm(x[i][b], values[i]); } } model.maximize(obj);
C#
LinearExprBuilder obj = LinearExpr.NewBuilder(); foreach (int i in allItems) { foreach (int b in allBins) { obj.AddTerm(x[i, b], Values[i]); } } model.Maximize(obj);
Il valore della funzione obiettivo è il costo totale di tutte le variabili assegnate e il valore 1 del risolutore.
Richiama il risolutore
Il seguente codice richiama il risolutore.
Python
solver = cp_model.CpSolver() status = solver.solve(model)
C++
const CpSolverResponse response = Solve(cp_model.Build());
Java
CpSolver solver = new CpSolver(); final CpSolverStatus status = solver.solve(model);
C#
CpSolver solver = new CpSolver(); CpSolverStatus status = solver.Solve(model);
Stampa la soluzione
Il seguente codice visualizza la soluzione al problema.
Python
if status == cp_model.OPTIMAL: print(f"Total packed value: {solver.objective_value}") total_weight = 0 for b in all_bins: print(f"Bin {b}") bin_weight = 0 bin_value = 0 for i in all_items: if solver.value(x[i, b]) > 0: print( f'Item:{i} weight:{data["weights"][i]} value:{data["values"][i]}' ) bin_weight += data["weights"][i] bin_value += data["values"][i] print(f"Packed bin weight: {bin_weight}") print(f"Packed bin value: {bin_value}\n") total_weight += bin_weight print(f"Total packed weight: {total_weight}") else: print("The problem does not have an optimal solution.")
C++
if (response.status() == CpSolverStatus::OPTIMAL || response.status() == CpSolverStatus::FEASIBLE) { LOG(INFO) << "Total packed value: " << response.objective_value(); double total_weight = 0.0; for (int b : all_bins) { LOG(INFO) << "Bin " << b; double bin_weight = 0.0; double bin_value = 0.0; for (int i : all_items) { auto key = std::make_tuple(i, b); if (SolutionIntegerValue(response, x[key]) > 0) { LOG(INFO) << "Item " << i << " weight: " << weights[i] << " value: " << values[i]; bin_weight += weights[i]; bin_value += values[i]; } } LOG(INFO) << "Packed bin weight: " << bin_weight; LOG(INFO) << "Packed bin value: " << bin_value; total_weight += bin_weight; } LOG(INFO) << "Total packed weight: " << total_weight; } else { LOG(INFO) << "The problem does not have an optimal solution."; }
Java
// Check that the problem has an optimal solution. if (status == CpSolverStatus.OPTIMAL) { System.out.println("Total packed value: " + solver.objectiveValue()); long totalWeight = 0; for (int b : allBins) { long binWeight = 0; long binValue = 0; System.out.println("Bin " + b); for (int i : allItems) { if (solver.booleanValue(x[i][b])) { System.out.println("Item " + i + " weight: " + weights[i] + " value: " + values[i]); binWeight += weights[i]; binValue += values[i]; } } System.out.println("Packed bin weight: " + binWeight); System.out.println("Packed bin value: " + binValue); totalWeight += binWeight; } System.out.println("Total packed weight: " + totalWeight); } else { System.err.println("The problem does not have an optimal solution."); }
C#
// Check that the problem has a feasible solution. if (status == CpSolverStatus.Optimal || status == CpSolverStatus.Feasible) { Console.WriteLine($"Total packed value: {solver.ObjectiveValue}"); double TotalWeight = 0.0; foreach (int b in allBins) { double BinWeight = 0.0; double BinValue = 0.0; Console.WriteLine($"Bin {b}"); foreach (int i in allItems) { if (solver.BooleanValue(x[i, b])) { Console.WriteLine($"Item {i} weight: {Weights[i]} values: {Values[i]}"); BinWeight += Weights[i]; BinValue += Values[i]; } } Console.WriteLine("Packed bin weight: " + BinWeight); Console.WriteLine("Packed bin value: " + BinValue); TotalWeight += BinWeight; } Console.WriteLine("Total packed weight: " + TotalWeight); } else { Console.WriteLine("No solution found."); }
Ecco l'output del programma.
Total packed value: 395.0 Bin 0 Item 3 - weight: 36 value: 50 Item 13 - weight: 36 value: 30 Packed bin weight: 72 Packed bin value: 80 Bin 1 Item 5 - weight: 48 value: 30 Item 7 - weight: 42 value: 40 Packed bin weight: 90 Packed bin value: 70 Bin 2 Item 1 - weight: 30 value: 30 Item 10 - weight: 30 value: 45 Item 14 - weight: 36 value: 25 Packed bin weight: 96 Packed bin value: 100 Bin 3 Item 2 - weight: 42 value: 25 Item 12 - weight: 42 value: 20 Packed bin weight: 84 Packed bin value: 45 Bin 4 Item 4 - weight: 36 value: 35 Item 8 - weight: 36 value: 30 Item 9 - weight: 24 value: 35 Packed bin weight: 96 Packed bin value: 100 Total packed weight: 438
Completa i programmi
Ecco i programmi completi per la soluzione CP-SAT.
Python
"""Solves a multiple knapsack problem using the CP-SAT solver.""" from ortools.sat.python import cp_model def main() -> None: data = {} data["weights"] = [48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36] data["values"] = [10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25] assert len(data["weights"]) == len(data["values"]) num_items = len(data["weights"]) all_items = range(num_items) data["bin_capacities"] = [100, 100, 100, 100, 100] num_bins = len(data["bin_capacities"]) all_bins = range(num_bins) model = cp_model.CpModel() # Variables. # x[i, b] = 1 if item i is packed in bin b. x = {} for i in all_items: for b in all_bins: x[i, b] = model.new_bool_var(f"x_{i}_{b}") # Constraints. # Each item is assigned to at most one bin. for i in all_items: model.add_at_most_one(x[i, b] for b in all_bins) # The amount packed in each bin cannot exceed its capacity. for b in all_bins: model.add( sum(x[i, b] * data["weights"][i] for i in all_items) <= data["bin_capacities"][b] ) # Objective. # maximize total value of packed items. objective = [] for i in all_items: for b in all_bins: objective.append(cp_model.LinearExpr.term(x[i, b], data["values"][i])) model.maximize(cp_model.LinearExpr.sum(objective)) solver = cp_model.CpSolver() status = solver.solve(model) if status == cp_model.OPTIMAL: print(f"Total packed value: {solver.objective_value}") total_weight = 0 for b in all_bins: print(f"Bin {b}") bin_weight = 0 bin_value = 0 for i in all_items: if solver.value(x[i, b]) > 0: print( f'Item:{i} weight:{data["weights"][i]} value:{data["values"][i]}' ) bin_weight += data["weights"][i] bin_value += data["values"][i] print(f"Packed bin weight: {bin_weight}") print(f"Packed bin value: {bin_value}\n") total_weight += bin_weight print(f"Total packed weight: {total_weight}") else: print("The problem does not have an optimal solution.") if __name__ == "__main__": main()
C++
// Solves a multiple knapsack problem using the CP-SAT solver. #include <stdlib.h> #include <map> #include <numeric> #include <tuple> #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 MultipleKnapsackSat() { const std::vector<int> weights = { {48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36}}; const std::vector<int> values = { {10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25}}; const int num_items = static_cast<int>(weights.size()); std::vector<int> all_items(num_items); std::iota(all_items.begin(), all_items.end(), 0); const std::vector<int> bin_capacities = {{100, 100, 100, 100, 100}}; const int num_bins = static_cast<int>(bin_capacities.size()); std::vector<int> all_bins(num_bins); std::iota(all_bins.begin(), all_bins.end(), 0); CpModelBuilder cp_model; // Variables. // x[i, b] = 1 if item i is packed in bin b. std::map<std::tuple<int, int>, BoolVar> x; for (int i : all_items) { for (int b : all_bins) { auto key = std::make_tuple(i, b); x[key] = cp_model.NewBoolVar().WithName(absl::StrFormat("x_%d_%d", i, b)); } } // Constraints. // Each item is assigned to at most one bin. for (int i : all_items) { std::vector<BoolVar> copies; for (int b : all_bins) { copies.push_back(x[std::make_tuple(i, b)]); } cp_model.AddAtMostOne(copies); } // The amount packed in each bin cannot exceed its capacity. for (int b : all_bins) { LinearExpr bin_weight; for (int i : all_items) { bin_weight += x[std::make_tuple(i, b)] * weights[i]; } cp_model.AddLessOrEqual(bin_weight, bin_capacities[b]); } // Objective. // Maximize total value of packed items. LinearExpr objective; for (int i : all_items) { for (int b : all_bins) { objective += x[std::make_tuple(i, b)] * values[i]; } } cp_model.Maximize(objective); const CpSolverResponse response = Solve(cp_model.Build()); if (response.status() == CpSolverStatus::OPTIMAL || response.status() == CpSolverStatus::FEASIBLE) { LOG(INFO) << "Total packed value: " << response.objective_value(); double total_weight = 0.0; for (int b : all_bins) { LOG(INFO) << "Bin " << b; double bin_weight = 0.0; double bin_value = 0.0; for (int i : all_items) { auto key = std::make_tuple(i, b); if (SolutionIntegerValue(response, x[key]) > 0) { LOG(INFO) << "Item " << i << " weight: " << weights[i] << " value: " << values[i]; bin_weight += weights[i]; bin_value += values[i]; } } LOG(INFO) << "Packed bin weight: " << bin_weight; LOG(INFO) << "Packed bin value: " << bin_value; total_weight += bin_weight; } LOG(INFO) << "Total packed weight: " << total_weight; } else { LOG(INFO) << "The problem does not have an optimal solution."; } // Statistics. LOG(INFO) << "Statistics"; LOG(INFO) << CpSolverResponseStats(response); } } // namespace sat } // namespace operations_research int main() { operations_research::sat::MultipleKnapsackSat(); return EXIT_SUCCESS; }
Java
// Solves a multiple knapsack problem using the CP-SAT solver. 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; /** Sample showing how to solve a multiple knapsack problem. */ public class MultipleKnapsackSat { public static void main(String[] args) { Loader.loadNativeLibraries(); // Instantiate the data problem. final int[] weights = {48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36}; final int[] values = {10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25}; final int numItems = weights.length; final int[] allItems = IntStream.range(0, numItems).toArray(); final int[] binCapacities = {100, 100, 100, 100, 100}; final int numBins = binCapacities.length; final int[] allBins = IntStream.range(0, numBins).toArray(); CpModel model = new CpModel(); // Variables. Literal[][] x = new Literal[numItems][numBins]; for (int i : allItems) { for (int b : allBins) { x[i][b] = model.newBoolVar("x_" + i + "_" + b); } } // Constraints. // Each item is assigned to at most one bin. for (int i : allItems) { List<Literal> bins = new ArrayList<>(); for (int b : allBins) { bins.add(x[i][b]); } model.addAtMostOne(bins); } // The amount packed in each bin cannot exceed its capacity. for (int b : allBins) { LinearExprBuilder load = LinearExpr.newBuilder(); for (int i : allItems) { load.addTerm(x[i][b], weights[i]); } model.addLessOrEqual(load, binCapacities[b]); } // Objective. // Maximize total value of packed items. LinearExprBuilder obj = LinearExpr.newBuilder(); for (int i : allItems) { for (int b : allBins) { obj.addTerm(x[i][b], values[i]); } } model.maximize(obj); CpSolver solver = new CpSolver(); final CpSolverStatus status = solver.solve(model); // Check that the problem has an optimal solution. if (status == CpSolverStatus.OPTIMAL) { System.out.println("Total packed value: " + solver.objectiveValue()); long totalWeight = 0; for (int b : allBins) { long binWeight = 0; long binValue = 0; System.out.println("Bin " + b); for (int i : allItems) { if (solver.booleanValue(x[i][b])) { System.out.println("Item " + i + " weight: " + weights[i] + " value: " + values[i]); binWeight += weights[i]; binValue += values[i]; } } System.out.println("Packed bin weight: " + binWeight); System.out.println("Packed bin value: " + binValue); totalWeight += binWeight; } System.out.println("Total packed weight: " + totalWeight); } else { System.err.println("The problem does not have an optimal solution."); } } private MultipleKnapsackSat() {} }
C#
// Solves a multiple knapsack problem using the CP-SAT solver. using System; using System.Collections.Generic; using System.Linq; using Google.OrTools.Sat; public class MultipleKnapsackSat { public static void Main(String[] args) { // Instantiate the data problem. int[] Weights = { 48, 30, 42, 36, 36, 48, 42, 42, 36, 24, 30, 30, 42, 36, 36 }; int[] Values = { 10, 30, 25, 50, 35, 30, 15, 40, 30, 35, 45, 10, 20, 30, 25 }; int NumItems = Weights.Length; int[] allItems = Enumerable.Range(0, NumItems).ToArray(); int[] BinCapacities = { 100, 100, 100, 100, 100 }; int NumBins = BinCapacities.Length; int[] allBins = Enumerable.Range(0, NumBins).ToArray(); // Model. CpModel model = new CpModel(); // Variables. ILiteral[,] x = new ILiteral[NumItems, NumBins]; foreach (int i in allItems) { foreach (int b in allBins) { x[i, b] = model.NewBoolVar($"x_{i}_{b}"); } } // Constraints. // Each item is assigned to at most one bin. foreach (int i in allItems) { List<ILiteral> literals = new List<ILiteral>(); foreach (int b in allBins) { literals.Add(x[i, b]); } model.AddAtMostOne(literals); } // The amount packed in each bin cannot exceed its capacity. foreach (int b in allBins) { List<ILiteral> items = new List<ILiteral>(); foreach (int i in allItems) { items.Add(x[i, b]); } model.Add(LinearExpr.WeightedSum(items, Weights) <= BinCapacities[b]); } // Objective. LinearExprBuilder obj = LinearExpr.NewBuilder(); foreach (int i in allItems) { foreach (int b in allBins) { obj.AddTerm(x[i, b], Values[i]); } } model.Maximize(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) { Console.WriteLine($"Total packed value: {solver.ObjectiveValue}"); double TotalWeight = 0.0; foreach (int b in allBins) { double BinWeight = 0.0; double BinValue = 0.0; Console.WriteLine($"Bin {b}"); foreach (int i in allItems) { if (solver.BooleanValue(x[i, b])) { Console.WriteLine($"Item {i} weight: {Weights[i]} values: {Values[i]}"); BinWeight += Weights[i]; BinValue += Values[i]; } } Console.WriteLine("Packed bin weight: " + BinWeight); Console.WriteLine("Packed bin value: " + BinValue); TotalWeight += BinWeight; } Console.WriteLine("Total packed weight: " + TotalWeight); } else { Console.WriteLine("No solution found."); } Console.WriteLine("Statistics"); Console.WriteLine($" conflicts: {solver.NumConflicts()}"); Console.WriteLine($" branches : {solver.NumBranches()}"); Console.WriteLine($" wall time: {solver.WallTime()}s"); } }