कई क्नैपैक की समस्या हल करना

इस सेक्शन में, एक से ज़्यादा नैपसैक की समस्या को हल करने का तरीका बताया गया है इसके लिए, हम MIP सॉल्वर और CP-SAT सॉल्वर, दोनों का इस्तेमाल करते हैं. इस मामले में, कंटेनर को बिन ही देखना आम बात है, से ज़्यादा.

अगला उदाहरण, किसी आइटम को पांच बिन में पैक करने का सबसे सही तरीका ढूंढने का तरीका बताता है.

उदाहरण

पिछले उदाहरण की तरह, इसमें अलग-अलग वज़न और वैल्यू के आइटम का कलेक्शन. समस्या है एक पैकेज में आइटम के सबसेट को पांच बिन में रखें, जिनमें से हर एक की क्षमता के लिए 100 से ज़्यादा बिन बनाएं, ताकि पैक की गई कुल वैल्यू ज़्यादा से ज़्यादा हो.

नीचे दिए गए सेक्शन में, इस समस्या को हल करने वाले प्रोग्राम के सेक्शन दिए गए हैं. सभी प्रोग्राम के बारे में जानने के लिए, सभी प्रोग्राम देखें.

एमआईपी सलूशन

नीचे दिए सेक्शन में बताया गया है कि MPSolver रैपर.

लाइब्रेरी इंपोर्ट करें

यहां दिया गया कोड ज़रूरी लाइब्रेरी इंपोर्ट करता है.

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;

डेटा बनाना

यह कोड समस्या के लिए डेटा बनाता है.

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();

डेटा में ये चीज़ें शामिल होती हैं:

• weights: आइटम का वज़न बताने वाला वेक्टर.
• values: एक वेक्टर, जिसमें आइटम की वैल्यू होती हैं.
• capacities: बिन की क्षमता वाला वेक्टर.

इस उदाहरण में, सभी बिन की कपैसिटी एक जैसी है. हालांकि, यह ज़रूरी नहीं है कि किया जा सकता है.

MIP सॉल्वर के बारे में बताएं

यहां दिया गया कोड, MIP सॉल्वर के बारे में बताता है.

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;
}

वैरिएबल बनाना

यह कोड समस्या के लिए वैरिएबल बनाता है.

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}");
    }
}

हर x[(i, j)] एक 0-1 वैरिएबल होता है. इसमें i एक आइटम है और j एक बिन है. तय सीमा में इस विकल्प को चुनने पर, अगर i आइटम को बिन j में रखा गया है, तो x[(i, j)] की वैल्यू 1 होगी. इसके अलावा, 0 होने पर नहीं तो.

सीमाएं तय करना

यह कोड समस्या की सीमाओं के बारे में बताता है:

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]);
    }
}

शर्तें नीचे दी गई हैं:

• हर आइटम को ज़्यादा से ज़्यादा एक बिन में रखा जा सकता है. यह कंस्ट्रेंट इसे सेट करता है: x[i, j] के योग को सभी बिन के j से कम या इसके बराबर होना ज़रूरी है पहला चरण.
• हर बिन में पैक किया गया कुल वज़न इसकी क्षमता से ज़्यादा नहीं हो सकता. यह कंस्ट्रेंट को बिन में रखे गए आइटम के वज़न के योग की ज़रूरत के हिसाब से सेट किया जाता है j की वैल्यू, बिन की कपैसिटी से कम या उसके बराबर होनी चाहिए.

मकसद तय करना

नीचे दिया गया कोड, सवाल के उस मकसद के फ़ंक्शन को तय करता है जो पैक किए गए आइटम की कुल कीमत.

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();

ध्यान दें कि x[i, j] * data['values'][i], आइटम i की वैल्यू को अगर आइटम को बिन j में रखा गया है, तो इसका मकसद. अगर i को किसी भी बिन में नहीं रखा गया है, तो वैल्यू से मकसद पूरा नहीं होता.

सॉल्वर आज़माएं

यह कोड, सॉल्वर को शुरू करता है.

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();

समाधान को प्रिंट करें

निम्न कोड समस्या का समाधान प्रिंट करता है.

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!");
}

हर बिन के लिए, कोड उस बिन में रखे गए आइटम और बिन के आइटम दिखाता है कुल वैल्यू और वज़न. कोड यह भी दिखाता है कि कुल वैल्यू पैक किए गए आइटम का वज़न.

प्रोग्राम को चलाने पर, यह नीचे दिया गया आउटपुट दिखाता है.

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

प्रोग्राम पूरे करें

एक से ज़्यादा नैपसैक के पूरे प्रोग्राम नीचे दिए गए हैं.

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!");
        }
    }
}

सीपी एसएटी सलूशन

इन सेक्शन में CP-SAT सॉल्वर का इस्तेमाल करके, समस्या को हल करने का तरीका बताया गया है.

लाइब्रेरी इंपोर्ट करें

यहां दिया गया कोड ज़रूरी लाइब्रेरी इंपोर्ट करता है.

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");
    }
}

मॉडल का एलान करना

यहां दिया गया कोड, CP-SAT मॉडल के बारे में बताता है.

Python

model = cp_model.CpModel()

C++

CpModelBuilder cp_model;

Java

CpModel model = new CpModel();

C#

CpModel model = new CpModel();

डेटा बनाना

यहां दिया गया कोड, समस्या के लिए डेटा सेट अप करता है.

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();

costs कलेक्शन, लागत की टेबल से मेल खाता है टास्क में कर्मचारियों को असाइन करने के लिए किया जा सकता है.

वैरिएबल बनाना

नीचे दिया गया कोड, इस सवाल के लिए बाइनरी इंटीजर वैरिएबल बनाता है.

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}");
    }
}

कंस्ट्रेंट बनाएं

यह कोड समस्या के लिए पाबंदियां बनाता है.

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]);
}

मकसद फ़ंक्शन बनाना

यह कोड समस्या के लिए मकसद फ़ंक्शन बनाता है.

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);

मकसद फ़ंक्शन की वैल्यू, असाइन किए गए सभी वैरिएबल की कुल लागत होती है वैल्यू 1 को सॉल्वर से हल करें.

सॉल्वर आज़माएं

यह कोड, सॉल्वर को शुरू करता है.

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);

समाधान को प्रिंट करें

निम्न कोड समस्या का समाधान प्रिंट करता है.

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.");
}

ये रहे प्रोग्राम का आउटपुट.

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

प्रोग्राम पूरे करें

यहां CP-SAT समाधान के लिए सभी प्रोग्राम दिए गए हैं.

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");
    }
}