Knapsack Problem Research Articles

Matheuristic fixed set search applied to the multidimensional knapsack problem and the knapsack problem with forfeit sets

In this paper, we present a solution method for the multidimensional knapsack problem (MKP) and the knapsack problem with forfeit sets (KPFS) using a population-based matheuristic approach. Specifically, the learning mechanism of the fixed set search (FSS) metaheuristic is combined with the use of integer programming for solving subproblems. This is achieved by introducing a new ground set of elements that can be used for both the MKP and the KPFS that aim to maximize the information provided by the fixed set. The method for creating fixed sets is also adjusted to enhance the diversity of generated solutions. Compared to state-of-the-art methods for the MKP and the KPFS, the proposed approach offers an implementation that can be easily extended to other variants of the knapsack problem. Computational experiments indicate that the matheuristic FSS is highly competitive to best-performing methods from the literature. The proposed approach is robust in the sense of having a good performance for a wide range of parameter values of the method.

Role of combinatorial optimization problems for the efficient decision making of a company- Case study-

All company seeks to improve their performance through achieving several objectives to stay in the labor market and increase competitiveness. The problem lies in the fact that it often relies on traditional methods for efficient decision-making, causing a waste of time and effort. The paper aims to find the best car service stations among several proposed stations by the Naftal Company, which seeks to achieve the two most important objectives; maximizing lubricants and tires revenues, and maximizing fuel and LPG revenues for the year 2019. The new applied case study was done with the help of the binary multi-objective knapsack problem, which is considered one of the most important problems in multi-objective combinatorial optimization problems. The problem was solved by the epsilon constraint method using MATLAB and CPLEX software. The study results confirmed that the proposed efficient solution set (Pareto optimal solutions) provides the decision maker with a comprehensive view of how to make their decision in a scientific, thoughtful, logical, and convincing manner that suits his situation and puts him at ease, away from trial and error or relying on self-experience.

An Accelerated Fixed-Point Algorithm Applied to Quadratic Convex Separable Knapsack Problems

In this article, we propose a root-finding algorithm for solving a quadratic convex separable knapsack problem, which is more straightforward than existing methods and competitive in practice. Besides, we also present an extension of the proposal, which improves its computational time, and then we incorporate the accelerated Anderson’s and Aitken’s fixed-point algorithms to obtain better results. The algorithm only performs function evaluations. We present partial convergence results of the algorithm. Moreover, we illustrate superior computational results in medium and large problems as well as the applicability of the algorithm with real-life applications to show their efficiency.

An adaptive binary quantum-behaved particle swarm optimization algorithm for the multidimensional knapsack problem

The multidimensional knapsack problem (MKP) is a classical combinatorial optimization problem with wide real-life applications. Binary quantum-behaved particle swarm optimization (BQPSO) algorithm is a popular heuristic algorithm used in binary optimization. While BQPSO exhibits strong global search capabilities, it is still prone to local optima due to particle aggregation. To address this issue, an adaptive BQPSO (ABQPSO) algorithm is proposed to solve the MKP efficiently. A hybrid encoding population initialization scheme is employed, leveraging specific knowledge of MKP to increase population diversity and improve search efficiency. Furthermore, ABQPSO uses a mapping strategy that converts continuous values into discrete values based on the average position of particles. An adaptive repair operator considering two pseudo-utility ratios is introduced to enable particles to explore different feasible regions, which dynamically adjusts current pseudo-utility ratios based on changes in the global best solution. A local search method is applied to guide particles towards convergence to the optimum. A local sparseness degree measurement and a diversity mechanism are utilized to avoid local optima. To evaluate the effectiveness of ABQPSO, it is compared against ten state-of-the-art algorithms using 168 MKP benchmark instances of varying scales. Experimental results reveal that ABQPSO outperforms the comparison algorithms, especially for large-scale problems, demonstrating better solution accuracy.

A survey of dynamic programming algorithms

Dynamic programming is an important algorithmic idea with widespread applications in computer science and other disciplines. With the continuous improvement of computing power and the increasing complexity of practical problems, dynamic programming algorithms have also received more and more attention. However, due to the limitations of the dynamic programming algorithm itself, it also brings challenges to the optimization of the algorithm and its application in different fields. This paper uses the methods of literature review and case analysis to systematically summarize and sort out the basic principles, complexity analysis, classic problems, and applications of dynamic programming algorithms. First of all, this paper discusses the basic concepts of splitting and optimal substructure characteristics of related problems in dynamic programming; secondly, it discusses the time complexity and space complexity of dynamic programming algorithms in detail; then, in the classic dynamic programming The case part focuses on the knapsack problem, including the 0-1 knapsack problem and multiple knapsack problems; finally, this paper discusses the wide application and optimization of dynamic programming algorithms in practice, covering natural language processing, bioinformatics and How to optimize the dynamic programming algorithm to improve the efficiency of problem solving and reduce resource consumption. To sum up, this paper fully demonstrates the basic principles and applications of dynamic programming algorithms, as well as optimization methods and development trends, and provides guidance and references for their academic and practical applications.

Research on solving 0-1 knapsack problem by dynamic programmin

The 0-1 Knapsack Problem poses the challenge of maximizing the total value of a subset of items, while respecting a capacity constraint in a knapsack. This paper explores the problems solution and aims to find an optimal solution. The Dynamic Programming approach, based on the principles of optimal substructure and overlapping subproblems, breaks down the problem into smaller, solvable subproblems. The article discusses the key features of Dynamic Programming, including its ability to handle non-linearities and provide a bottom-up approach for optimal substructure. A code implementation showcases the pseudocode, demonstrating the iterative calculation and efficient storage of results. Reviewing the research, the Dynamic Programming is a suitable and powerful tool as the benefits are summarized and listed into less computations, high efficiency and the optimal final result. By leveraging Dynamic Programming, the 0-1 Knapsack Problem can be solved optimally by finding the subset of items that maximizes the value within the capacity constraint.

Stability constraints in a 3D knapsack problem with non parallelepipedic items

This paper concerns stability issues that we encountered while solving an industrial pallet loading problem. The practical problem can be modeled as a three-dimensional knapsack with original constraints related to the stability of the pallet: when the placement of the boxes on the pallet is computed, the weight of the boxes and the placement sequence are not known.Despite this lack of information, the placement must remain stable throughout the construction of the pallet. Even more important, the major difference with classical placement problems is that the boxes are parallelepipedic rectangle with edge reduction (the upper surface may be smaller than the bottom face).In this paper, we review stability constraints used in the literature and the assumptions under which these constraints can be used and we detail their adequacy to the problem we consider. We then propose a new stability constraint which takes into account the practical features, which can be integrated into a placement algorithm and which is not too restrictive.Numerical experiments on benchmarks from the literature show that this constraint, added to a classical placement algorithm, obtains results which are similar to algorithms from the literature dedicated to a specific stability constraint for parallelepipedic boxes. By adjusting the benchmark to take into account the practical problem specificities, we show that this new stability constraint is perfectly suited to the industrial problem and obtains very good results compared to classical constraints. Therefore, it has been integrated in the placement software developed by the company.

Research on the pricing and replenishment decision-making problem of vegetable products based on neural networks and grey wolf algorithm

This article deals with the knapsack problem, where, under real-world conditions, the wholesale prices for the day are not known, and the sales of vegetable products exhibit seasonal patterns. Therefore, this article utilizes an LSTM neural network model to predict the daily wholesale prices. Subsequently, the top 60 projects with the highest profits are selected to create a histogram, and available projects are chosen from these 60.Initially, this article identifies five factors that impact profit and transforms them into positive indicators. Weight allocation is done using the entropy weight method and fuzzy comprehensive evaluation method. The TOPSIS method is employed to obtain composite score indicators for the 60 projects. A linear programming objective function is established, and the results are solved using the Grey Wolf Optimization (GWO) algorithm.

On the Use of Quality Diversity Algorithms for the Travelling Thief Problem

In real-world optimisation, it is common to face several sub-problems interacting and forming the main problem. There is an inter-dependency between the sub-problems, making it impossible to solve such a problem by focusing on only one component. The travelling thief problem (TTP) belongs to this category and is formed by the integration of the travelling salesperson problem (TSP) and the knapsack problem (KP). In this paper, we investigate the inter-dependency of the TSP and the KP by means of quality diversity (QD) approaches. QD algorithms provide a powerful tool not only to obtain high-quality solutions but also to illustrate the distribution of high-performing solutions in the behavioural space. We introduce a multi-dimensional archive of phenotypic elites (MAP-Elites) based evolutionary algorithm using well-known TSP and KP search operators, taking the TSP and KP score as the behavioural descriptor. MAP-Elites algorithms are QD-based techniques to explore high-performing solutions in a behavioural space. Afterwards, we conduct comprehensive experimental studies that show the usefulness of using the QD approach applied to the TTP. First, we provide insights regarding high-quality TTP solutions in the TSP/KP behavioural space. Afterwards, we show that better solutions for the TTP can be obtained by using our QD approach, and it can improve the best-known solution for a number of TTP instances used for benchmarking in the literature.

Chaotic Binarization Schemes for Solving Combinatorial Optimization Problems Using Continuous Metaheuristics

Chaotic maps are sources of randomness formed by a set of rules and chaotic variables. They have been incorporated into metaheuristics because they improve the balance of exploration and exploitation, and with this, they allow one to obtain better results. In the present work, chaotic maps are used to modify the behavior of the binarization rules that allow continuous metaheuristics to solve binary combinatorial optimization problems. In particular, seven different chaotic maps, three different binarization rules, and three continuous metaheuristics are used, which are the Sine Cosine Algorithm, Grey Wolf Optimizer, and Whale Optimization Algorithm. A classic combinatorial optimization problem is solved: the 0-1 Knapsack Problem. Experimental results indicate that chaotic maps have an impact on the binarization rule, leading to better results. Specifically, experiments incorporating the standard binarization rule and the complement binarization rule performed better than experiments incorporating the elitist binarization rule. The experiment with the best results was STD_TENT, which uses the standard binarization rule and the tent chaotic map.

Комбинаторные свойства задачи об ограниченном рюкзаке

The combinatorial properties of the set of solutions to the bounded knapsack problem are considered. As in the general case, this problem is an NP-complete combinatorial optimization problem and its exact solution requires the use of search algorithms with the decomposition of a set of feasible solutions. In this regard, the question of determining and evaluating the properties of the set of acceptable solutions to the problem is relevant. In this paper, formulas are obtained which allow to calculate the average value of the functional of a problem on the set of its feasible solutions and the power of this set through the number of solutions of subtasks of smaller dimension. The basic technique for obtaining results is the method of generating functions. We also consider the knapsack problem with arbitrary values of variables, in which the coefficients of the constraint vector and the objective function coincide. For this, the “continuity” of the set of solutions is assumed. Estimates of the values of the functional in this problem are found. The results may be of interest for the design of computational algorithms for finding and estimating the number of solutions and the value of the functional for optimal solutions. The expressions found can also be used in auxiliary procedures to evaluate the optimality of the solution in decomposition or heuristic algorithms for solving the knapsack problem.

Automated Design of Collaboration-Based Hybrid Metaheuristics.

Hybridization plays a prominent role in bolstering the performance of optimization algorithms (OAs), yet designing efficient hybrid OAs tailored to intricate optimization problems persists as a formidable task. This article introduces a novel top-down methodology for the automated design of hybrid OAs, treating algorithm design as a meta-optimization problem. A general design template for collaboration-based hybrid OAs is developed, integrating a multitude of hybridization strategies for the first time. Besides, a mathematical model is built to formulate the meta-optimization problem of algorithm design. To address the meta-optimization challenge, an improved multifactorial evolutionary algorithm is proposed to automatically design efficient hybrid metaheuristics in a multitasking environment for the given instances with diverse features. To verify the effectiveness of the proposed design methodology, it is applied to the CEC2017 benchmark functions and the binary knapsack problem. Numerical results have demonstrated the feasibility and effectiveness of the proposed methodology for both continuous and combinatorial optimization benchmarks.

Discrete Cuckoo Search for 0-1 knapsack problem

Discrete Cuckoo Search for 0-1 knapsack problem

Workload-Aware Scheduling of Real-Time Jobs in Cloud Computing to Minimize Energy Consumption

Cloud computing is a powerful paradigm that can provide high-quality computation services to customers. Because its energy consumption has a large effect on its service price, this study investigates how to minimize the energy consumption while achieving adequate response times for requested computations. Such a problem is formulated as a nonlinear integer program. By deriving a state transition equation, this problem is transformed into an unconventional 0-1 knapsack problem, and dynamic programming is then used to solve it. In addition to this solution, we develop an energy-efficient job accommodation scheme that can manage dynamic jobs with varying frequencies throughout a day. Unlike existing studies that abruptly switch off old virtual machines and create new ones for upcoming jobs, this scheme tries to accommodate them with current virtual machines, and new virtual machines are not created unless necessary. Conversely, when the workload declines, jobs on energy-inefficient servers are moved to other servers, such that some energy-inefficient servers can be switched off to save energy. This scheme adjusts the computing power adaptively and smoothly without lowering the system’s quality of service. Experimental results demonstrate that the proposed solution outperforms a particle swarm optimizer and two other heuristics in terms of accommodating jobs and saving energy consumption.

An Improved Shuffled Frog-Leaping Algorithm to Solving 0–1 Knapsack Problem

An Improved Shuffled Frog-Leaping Algorithm to Solving 0–1 Knapsack Problem

Supported nondominated points as a representation of the nondominated set: An empirical analysis

AbstractThe nondominated set of a multiple objective discrete optimization problem is known to contain unsupported nondominated points, which outnumber the supported ones and are more difficult to obtain. We treat supported nondominated points as a representation and analyse their quality using different metrics beyond their sheer numbers. Under different data generation schemes on multiobjective knapsack and assignment problems, we observe that supported nondominated points almost always provide a good representation of the entire nondominated set.

Optimal topology control for grounding protection sensor networks in shipboard electrical systems.

Intelligent monitoring of environmental information in ships is an important prerequisite for reducing and avoiding accidents and is of great importance for safety at sea. The use of a ship environmental monitoring system based on wireless sensor networks has the unique advantages of low cost, stability, and reliability, allowing real-time monitoring of many environmental factors in ships. This article proposes an optimal topology control to maximize collected data satisfaction and reduce energy consumption and transmission latency. First, the cluster head selection method is obtained by initializing the wireless sensor network with a fuzzy clustering algorithm, then the amount of collection data is obtained by a sub-gradient descent algorithm, and a sensor association scheme based on ant colony optimization is proposed to solve the knapsack problem. Finally, simulation results show that the proposed algorithm in this article can effectively reduce the energy consumption and transmission delay of the wireless sensor network while ensuring the monitoring of the grounding protection in the electrical system of the offshore ship.

Chance-Constrained Multiple-Choice Knapsack Problem: Model, Algorithms, and Applications.

The multiple-choice knapsack problem (MCKP) is a classic NP-hard combinatorial optimization problem. Motivated by several significant real-world applications, this work investigates a novel variant of MCKP called the chance-constrained MCKP (CCMCKP), where item weights are random variables. In particular, we focus on the practical scenario of CCMCKP, in which the probability distributions of random weights are unknown and only sample data is available. We first present the problem formulation of CCMCKP and then establish the two benchmark sets. The first set contains synthetic instances, while the second set is designed to simulate a real-world application scenario of a telecommunication company. To solve CCMCKP, we propose a data-driven adaptive local search (DDALS) algorithm. Compared to existing stochastic optimization and distributionally robust optimization methods, the main novelty of DDALS lies in its data-driven solution evaluation approach, which does not make any assumptions about the underlying distributions and is highly effective even when faced with a high intensity of the chance constraint and a limited amount of sample data. Experimental results demonstrate the superiority of DDALS over the baselines on both the benchmarks. Finally, DDALS can serve as the baseline for future research, and the benchmark sets are open-sourced to further promote research on this challenging problem.

Sensitivity analysis of the optimum solution to perturbations of the profit of an item in the multiple-choice knapsack problem

Sensitivity analysis of the optimum solution to perturbations of the profit of an item in the multiple-choice knapsack problem

Discrete Preference-Based Stepping Ahead Firefly Algorithm for Solving Multidimensional Knapsack Problems

Discrete Preference-Based Stepping Ahead Firefly Algorithm for Solving Multidimensional Knapsack Problems