يجب تجميع مجموعة من العناصر بقيم معيّنة ضمن مشكلة في حقيبة الظهر والأحجام (مثل الأوزان أو الأحجام)، في حاوية بحد أقصى للسعة . إذا تجاوز الحجم الإجمالي للعناصر السعة، لا يمكنك تعبئتها جميعًا. وفي هذه الحالة، تكمن المشكلة في اختيار مجموعة فرعية من عناصر الحد الأقصى لإجمالي التي ستتناسب مع الحاوية.
توضح الأقسام التالية كيفية حل مشكلة الحقيبة باستخدام أدوات OR.
مثال
في ما يلي شرح تصويري لمشكلة في حقيبة الظهر:
في الصورة المتحركة أعلاه، تم تعبئة 50 عنصر في سلة. ولكل عنصر قيمة
(الرقم الموجود على العنصر) والوزن (يتناسب تقريبًا مع مساحة
هذا العنصر).
تمّ الإعلان عن أنّ السلة بسعة 850، وهدفنا هو إيجاد المجموعة
السلع التي ستزيد القيمة الإجمالية إلى أقصى حد بدون تجاوز السعة.
تصف الأقسام التالية البرامج التي تحل مشكلة الحقيبة. للاطلاع على البرامج الكاملة، راجع البرامج المكتملة.
استيراد المكتبات
يستورد الرمز التالي المكتبات المطلوبة.
Python
from ortools.algorithms.python import knapsack_solver
C++
#include <algorithm> #include <cstdint> #include <iterator> #include <numeric> #include <sstream> #include <vector> #include "ortools/algorithms/knapsack_solver.h"
Java
import com.google.ortools.Loader; import com.google.ortools.algorithms.KnapsackSolver; import java.util.ArrayList;
#C
using System; using Google.OrTools.Algorithms;
إنشاء البيانات
يقوم التعليمة البرمجية أدناه بإنشاء البيانات للمشكلة.
Python
values = [
# fmt:off
360, 83, 59, 130, 431, 67, 230, 52, 93, 125, 670, 892, 600, 38, 48, 147,
78, 256, 63, 17, 120, 164, 432, 35, 92, 110, 22, 42, 50, 323, 514, 28,
87, 73, 78, 15, 26, 78, 210, 36, 85, 189, 274, 43, 33, 10, 19, 389, 276,
312
# fmt:on
]
weights = [
# fmt: off
[7, 0, 30, 22, 80, 94, 11, 81, 70, 64, 59, 18, 0, 36, 3, 8, 15, 42, 9, 0,
42, 47, 52, 32, 26, 48, 55, 6, 29, 84, 2, 4, 18, 56, 7, 29, 93, 44, 71,
3, 86, 66, 31, 65, 0, 79, 20, 65, 52, 13],
# fmt: on
]
capacities = [850]
C++
std::vector<int64_t> values = {
360, 83, 59, 130, 431, 67, 230, 52, 93, 125, 670, 892, 600,
38, 48, 147, 78, 256, 63, 17, 120, 164, 432, 35, 92, 110,
22, 42, 50, 323, 514, 28, 87, 73, 78, 15, 26, 78, 210,
36, 85, 189, 274, 43, 33, 10, 19, 389, 276, 312};
std::vector<std::vector<int64_t>> weights = {
{7, 0, 30, 22, 80, 94, 11, 81, 70, 64, 59, 18, 0, 36, 3, 8, 15,
42, 9, 0, 42, 47, 52, 32, 26, 48, 55, 6, 29, 84, 2, 4, 18, 56,
7, 29, 93, 44, 71, 3, 86, 66, 31, 65, 0, 79, 20, 65, 52, 13}};
std::vector<int64_t> capacities = {850};
Java
final long[] values = {360, 83, 59, 130, 431, 67, 230, 52, 93, 125, 670, 892, 600, 38, 48, 147,
78, 256, 63, 17, 120, 164, 432, 35, 92, 110, 22, 42, 50, 323, 514, 28, 87, 73, 78, 15, 26,
78, 210, 36, 85, 189, 274, 43, 33, 10, 19, 389, 276, 312};
final long[][] weights = {{7, 0, 30, 22, 80, 94, 11, 81, 70, 64, 59, 18, 0, 36, 3, 8, 15, 42, 9,
0, 42, 47, 52, 32, 26, 48, 55, 6, 29, 84, 2, 4, 18, 56, 7, 29, 93, 44, 71, 3, 86, 66, 31,
65, 0, 79, 20, 65, 52, 13}};
final long[] capacities = {850};
#C
long[] values = { 360, 83, 59, 130, 431, 67, 230, 52, 93, 125, 670, 892, 600, 38, 48, 147, 78,
256, 63, 17, 120, 164, 432, 35, 92, 110, 22, 42, 50, 323, 514, 28, 87, 73,
78, 15, 26, 78, 210, 36, 85, 189, 274, 43, 33, 10, 19, 389, 276, 312 };
long[,] weights = { { 7, 0, 30, 22, 80, 94, 11, 81, 70, 64, 59, 18, 0, 36, 3, 8, 15,
42, 9, 0, 42, 47, 52, 32, 26, 48, 55, 6, 29, 84, 2, 4, 18, 56,
7, 29, 93, 44, 71, 3, 86, 66, 31, 65, 0, 79, 20, 65, 52, 13 } };
long[] capacities = { 850 };
وتشمل البيانات ما يلي:
weights: متجه يحتوي على أوزان العناصر.values: متجه يحتوي على قيم العناصر.capacities: متجه يحتوي على إدخال واحد فقط، وهو سعة الكيس.
تضمين أداة الحلّ
تعلن التعليمة البرمجية التالية عن أداة حل knapsack، وهي أداة متخصصة للحل مشكلات الحقائب المحمولة.
Python
solver = knapsack_solver.KnapsackSolver(
knapsack_solver.SolverType.KNAPSACK_MULTIDIMENSION_BRANCH_AND_BOUND_SOLVER,
"KnapsackExample",
)
C++
KnapsackSolver solver(
KnapsackSolver::KNAPSACK_MULTIDIMENSION_BRANCH_AND_BOUND_SOLVER,
"KnapsackExample");
Java
KnapsackSolver solver = new KnapsackSolver(
KnapsackSolver.SolverType.KNAPSACK_MULTIDIMENSION_BRANCH_AND_BOUND_SOLVER, "test");
#C
KnapsackSolver solver = new KnapsackSolver(
KnapsackSolver.SolverType.KNAPSACK_MULTIDIMENSION_BRANCH_AND_BOUND_SOLVER, "KnapsackExample");
يوجِّه الخيار KNAPSACK_MULTIDIMENSION_BRANCH_AND_BOUND_SOLVER أداة الحلّ إلى
استخدام خوارزمية الفرع والربط لحل المسألة.
طلب أداة حلّ المشكلة
تستدعي التعليمة البرمجية التالية أداة الحل ويطبع الحل.
Python
solver.init(values, weights, capacities)
computed_value = solver.solve()
packed_items = []
packed_weights = []
total_weight = 0
print("Total value =", computed_value)
for i in range(len(values)):
if solver.best_solution_contains(i):
packed_items.append(i)
packed_weights.append(weights[0][i])
total_weight += weights[0][i]
print("Total weight:", total_weight)
print("Packed items:", packed_items)
print("Packed_weights:", packed_weights)
C++
solver.Init(values, weights, capacities);
int64_t computed_value = solver.Solve();
std::vector<int> packed_items;
for (std::size_t i = 0; i < values.size(); ++i) {
if (solver.BestSolutionContains(i)) packed_items.push_back(i);
}
std::ostringstream packed_items_ss;
std::copy(packed_items.begin(), packed_items.end() - 1,
std::ostream_iterator<int>(packed_items_ss, ", "));
packed_items_ss << packed_items.back();
std::vector<int64_t> packed_weights;
packed_weights.reserve(packed_items.size());
for (const auto& it : packed_items) {
packed_weights.push_back(weights[0][it]);
}
std::ostringstream packed_weights_ss;
std::copy(packed_weights.begin(), packed_weights.end() - 1,
std::ostream_iterator<int>(packed_weights_ss, ", "));
packed_weights_ss << packed_weights.back();
int64_t total_weights =
std::accumulate(packed_weights.begin(), packed_weights.end(), int64_t{0});
LOG(INFO) << "Total value: " << computed_value;
LOG(INFO) << "Packed items: {" << packed_items_ss.str() << "}";
LOG(INFO) << "Total weight: " << total_weights;
LOG(INFO) << "Packed weights: {" << packed_weights_ss.str() << "}";
Java
solver.init(values, weights, capacities);
final long computedValue = solver.solve();
ArrayList<Integer> packedItems = new ArrayList<>();
ArrayList<Long> packedWeights = new ArrayList<>();
int totalWeight = 0;
System.out.println("Total value = " + computedValue);
for (int i = 0; i < values.length; i++) {
if (solver.bestSolutionContains(i)) {
packedItems.add(i);
packedWeights.add(weights[0][i]);
totalWeight = (int) (totalWeight + weights[0][i]);
}
}
System.out.println("Total weight: " + totalWeight);
System.out.println("Packed items: " + packedItems);
System.out.println("Packed weights: " + packedWeights);
#C
solver.Init(values, weights, capacities);
long computedValue = solver.Solve();
Console.WriteLine("Optimal Value = " + computedValue);
يهيئ البرنامج أداة الحلّ أولاً، ثم يستدعيها عن طريق
computed_value = solver.Solve()
القيمة الإجمالية للحلّ الأمثل هي computed_value، وهي القيمة نفسها
كالوزن الإجمالي في هذه الحالة. يحصل البرنامج بعد ذلك على مؤشرات
العناصر المعبأة في الحل على النحو التالي:
packed_items = [x for x in range(0, len(weights[0]))
if solver.BestSolutionContains(x)]
بما أنّ "solver.BestSolutionContains(x)" تعرض "TRUE" في حال تضمين العنصر x.
في الحل، `packed_items` عبارة عن قائمة بالعناصر المعبأة المثلى.
وبالمثل، فإن `packed_weights` هي أوزان العناصر المعبأة.
### ناتج البرنامج
في ما يلي مخرجات البرنامج.
Total value = 7534
Total weight: 850
Packed items: [0, 1, 3, 4, 6, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 24, 27, 28, 29, 30, 31,
32, 34, 38, 39, 41, 42, 44, 47, 48, 49]
Packed_weights: [7, 0, 22, 80, 11, 59, 18, 0, 3, 8, 15, 42, 9, 0, 47, 52, 26, 6, 29, 84, 2, 4,
18, 7, 71, 3, 66, 31, 0, 65, 52, 13]
إكمال البرامج
فيما يلي البرامج الكاملة التي تحل مشكلة الحقيبة.
Python
from ortools.algorithms.python import knapsack_solver
def main():
# Create the solver.
solver = knapsack_solver.KnapsackSolver(
knapsack_solver.SolverType.KNAPSACK_MULTIDIMENSION_BRANCH_AND_BOUND_SOLVER,
"KnapsackExample",
)
values = [
# fmt:off
360, 83, 59, 130, 431, 67, 230, 52, 93, 125, 670, 892, 600, 38, 48, 147,
78, 256, 63, 17, 120, 164, 432, 35, 92, 110, 22, 42, 50, 323, 514, 28,
87, 73, 78, 15, 26, 78, 210, 36, 85, 189, 274, 43, 33, 10, 19, 389, 276,
312
# fmt:on
]
weights = [
# fmt: off
[7, 0, 30, 22, 80, 94, 11, 81, 70, 64, 59, 18, 0, 36, 3, 8, 15, 42, 9, 0,
42, 47, 52, 32, 26, 48, 55, 6, 29, 84, 2, 4, 18, 56, 7, 29, 93, 44, 71,
3, 86, 66, 31, 65, 0, 79, 20, 65, 52, 13],
# fmt: on
]
capacities = [850]
solver.init(values, weights, capacities)
computed_value = solver.solve()
packed_items = []
packed_weights = []
total_weight = 0
print("Total value =", computed_value)
for i in range(len(values)):
if solver.best_solution_contains(i):
packed_items.append(i)
packed_weights.append(weights[0][i])
total_weight += weights[0][i]
print("Total weight:", total_weight)
print("Packed items:", packed_items)
print("Packed_weights:", packed_weights)
if __name__ == "__main__":
main()
C++
#include <algorithm>
#include <cstdint>
#include <iterator>
#include <numeric>
#include <sstream>
#include <vector>
#include "ortools/algorithms/knapsack_solver.h"
namespace operations_research {
void RunKnapsackExample() {
// Instantiate the solver.
KnapsackSolver solver(
KnapsackSolver::KNAPSACK_MULTIDIMENSION_BRANCH_AND_BOUND_SOLVER,
"KnapsackExample");
std::vector<int64_t> values = {
360, 83, 59, 130, 431, 67, 230, 52, 93, 125, 670, 892, 600,
38, 48, 147, 78, 256, 63, 17, 120, 164, 432, 35, 92, 110,
22, 42, 50, 323, 514, 28, 87, 73, 78, 15, 26, 78, 210,
36, 85, 189, 274, 43, 33, 10, 19, 389, 276, 312};
std::vector<std::vector<int64_t>> weights = {
{7, 0, 30, 22, 80, 94, 11, 81, 70, 64, 59, 18, 0, 36, 3, 8, 15,
42, 9, 0, 42, 47, 52, 32, 26, 48, 55, 6, 29, 84, 2, 4, 18, 56,
7, 29, 93, 44, 71, 3, 86, 66, 31, 65, 0, 79, 20, 65, 52, 13}};
std::vector<int64_t> capacities = {850};
solver.Init(values, weights, capacities);
int64_t computed_value = solver.Solve();
// Print solution
std::vector<int> packed_items;
for (std::size_t i = 0; i < values.size(); ++i) {
if (solver.BestSolutionContains(i)) packed_items.push_back(i);
}
std::ostringstream packed_items_ss;
std::copy(packed_items.begin(), packed_items.end() - 1,
std::ostream_iterator<int>(packed_items_ss, ", "));
packed_items_ss << packed_items.back();
std::vector<int64_t> packed_weights;
packed_weights.reserve(packed_items.size());
for (const auto& it : packed_items) {
packed_weights.push_back(weights[0][it]);
}
std::ostringstream packed_weights_ss;
std::copy(packed_weights.begin(), packed_weights.end() - 1,
std::ostream_iterator<int>(packed_weights_ss, ", "));
packed_weights_ss << packed_weights.back();
int64_t total_weights =
std::accumulate(packed_weights.begin(), packed_weights.end(), int64_t{0});
LOG(INFO) << "Total value: " << computed_value;
LOG(INFO) << "Packed items: {" << packed_items_ss.str() << "}";
LOG(INFO) << "Total weight: " << total_weights;
LOG(INFO) << "Packed weights: {" << packed_weights_ss.str() << "}";
}
} // namespace operations_research
int main(int argc, char** argv) {
operations_research::RunKnapsackExample();
return EXIT_SUCCESS;
}
Java
package com.google.ortools.algorithms.samples;
import com.google.ortools.Loader;
import com.google.ortools.algorithms.KnapsackSolver;
import java.util.ArrayList;
/**
* Sample showing how to model using the knapsack solver.
*/
public class Knapsack {
private Knapsack() {}
private static void solve() {
KnapsackSolver solver = new KnapsackSolver(
KnapsackSolver.SolverType.KNAPSACK_MULTIDIMENSION_BRANCH_AND_BOUND_SOLVER, "test");
final long[] values = {360, 83, 59, 130, 431, 67, 230, 52, 93, 125, 670, 892, 600, 38, 48, 147,
78, 256, 63, 17, 120, 164, 432, 35, 92, 110, 22, 42, 50, 323, 514, 28, 87, 73, 78, 15, 26,
78, 210, 36, 85, 189, 274, 43, 33, 10, 19, 389, 276, 312};
final long[][] weights = {{7, 0, 30, 22, 80, 94, 11, 81, 70, 64, 59, 18, 0, 36, 3, 8, 15, 42, 9,
0, 42, 47, 52, 32, 26, 48, 55, 6, 29, 84, 2, 4, 18, 56, 7, 29, 93, 44, 71, 3, 86, 66, 31,
65, 0, 79, 20, 65, 52, 13}};
final long[] capacities = {850};
solver.init(values, weights, capacities);
final long computedValue = solver.solve();
ArrayList<Integer> packedItems = new ArrayList<>();
ArrayList<Long> packedWeights = new ArrayList<>();
int totalWeight = 0;
System.out.println("Total value = " + computedValue);
for (int i = 0; i < values.length; i++) {
if (solver.bestSolutionContains(i)) {
packedItems.add(i);
packedWeights.add(weights[0][i]);
totalWeight = (int) (totalWeight + weights[0][i]);
}
}
System.out.println("Total weight: " + totalWeight);
System.out.println("Packed items: " + packedItems);
System.out.println("Packed weights: " + packedWeights);
}
public static void main(String[] args) throws Exception {
Loader.loadNativeLibraries();
Knapsack.solve();
}
}
#C
using System;
using Google.OrTools.Algorithms;
public class Knapsack
{
static void Main()
{
KnapsackSolver solver = new KnapsackSolver(
KnapsackSolver.SolverType.KNAPSACK_MULTIDIMENSION_BRANCH_AND_BOUND_SOLVER, "KnapsackExample");
long[] values = { 360, 83, 59, 130, 431, 67, 230, 52, 93, 125, 670, 892, 600, 38, 48, 147, 78,
256, 63, 17, 120, 164, 432, 35, 92, 110, 22, 42, 50, 323, 514, 28, 87, 73,
78, 15, 26, 78, 210, 36, 85, 189, 274, 43, 33, 10, 19, 389, 276, 312 };
long[,] weights = { { 7, 0, 30, 22, 80, 94, 11, 81, 70, 64, 59, 18, 0, 36, 3, 8, 15,
42, 9, 0, 42, 47, 52, 32, 26, 48, 55, 6, 29, 84, 2, 4, 18, 56,
7, 29, 93, 44, 71, 3, 86, 66, 31, 65, 0, 79, 20, 65, 52, 13 } };
long[] capacities = { 850 };
solver.Init(values, weights, capacities);
long computedValue = solver.Solve();
Console.WriteLine("Optimal Value = " + computedValue);
}
}