Knapsack Problem Research Articles

Asymptotic Analysis of Optimal Vaccination Policies

Targeted vaccination policies can have a significant impact on the number of infections and deaths in an epidemic. However, optimising such policies is complicated, and the resultant solution may be difficult to explain to policy-makers and to the public. The key novelty of this paper is a derivation of the leading-order optimal vaccination policy under multi-group susceptible–infected–recovered dynamics in two different cases. Firstly, it considers the case of a small vulnerable subgroup in a population and shows that (in the asymptotic limit) it is optimal to vaccinate this group first, regardless of the properties of the other groups. Then, it considers the case of a small vaccine supply and transforms the optimal vaccination problem into a simple knapsack problem by linearising the final size equations. Both of these cases are then explored further through numerical examples, which show that these solutions are also directly useful for realistic parameter values. Moreover, the findings of this paper give some general principles for optimal vaccination policies which will help policy-makers and the public to understand the reasoning behind optimal vaccination programs in more generic cases.

Scenario-dominance to multi-stage stochastic lot-sizing and knapsack problems

This paper presents strong scenario dominance cuts for effectively solving the multi-stage stochastic mixed-integer programs (M-SMIPs), specifically focusing on the two most well-known M-SMIPs: stochastic capacitated multi-item lot-sizing (S-MCLSP) and the stochastic dynamic multi-dimensional knapsack (S-MKP) problems. Scenario dominance is characterized by a partial ordering of scenarios based on the pairwise comparisons of random variable realizations in a scenario tree of a stochastic program. In this paper, we study the implications of scenario-dominance relations and inferences obtained by solving scenario sub-problems to drive new strong cutting planes to solve S-MCLSP and S-MKP instances faster. Computational experiments demonstrate that our strong scenario dominance cuts can significantly reduce the solution time for such M-SMIP problems with an average of 0.06% deviation from the optimal solution. The results with up to 81 random variables for S-MKP show that strong dominance cuts improve the state-of-the-art solver solution of two hours by 0.13% in five minutes. The proposed framework can also be applied to other scenario-based optimization problems.

A method for rockfall risk quantification and optimal arrangement of protection structures along a road

The purpose of this study is to introduce a novel framework that aids in the installation of rockfall protection structures for the risk reduction of rockfall-related damage. In this study, we propose a computational method that performs risk analysis for a road and plans an optimal arrangement of rockfall protection structures. First, a rockfall trajectory simulation is performed using a three-dimensional numerical analysis method to obtain sufficient data for risk analysis. Then, the risk to motorized traffic during a certain period is quantified considering the relationship between rock block volume and rockfall frequency, the movement of rock blocks, the vulnerability of elements at risk, and some other factors. Finally, a knapsack problem that aims to minimize the total risk on a road while satisfying a budget constraint is solved. The proposed method is applied to an actual site and represents that it can derive a layout for protection structures to maximize the cost-benefit performance depending on the road traffic and budget conditions. The obtained results of the structural protection plan can be used to support road managers in decision making.

Discrete equilibrium optimizer combined with simulated annealing for feature selection

This paper proposes a binary adaptation of the recently proposed meta-heuristic, Equilibrium Optimizer (EO), called Discrete EO (DEO), to solve binary optimization problems. A U-shaped transfer function is used to map the continuous values of EO into the binary domain. To further improve the exploitation capability of DEO, Simulated Annealing (SA) is used as a local search procedure and the combination is named as DEOSA. The proposed DEOSA algorithm is applied to 18 well-known UCI datasets and compared with a wide range of algorithms. The results are statistically validated using Wilcoxon rank-sum test and Friedman test. In order to test the scalability and robustness of DEOSA, it is additionally tested over seven high-dimensional Microarray datasets and 25 binary Knapsack problems. The results evidently demonstrate the superiority and merits of DEOSA when solving binary optimization problems.

Wasserstein distributionally robust chance-constrained program with moment information

This paper studies a distributionally robust joint chance-constrained program with a hybrid ambiguity set including the Wasserstein metric, and moment and bounded support information of uncertain parameters. For the considered mathematical program, the random variables are located in a given support space, so a set of random constraints with a high threshold probability for all the distributions that are within a specified Wasserstein distance from an empirical distribution, and a series of moment constraints have to be simultaneously satisfied. We first demonstrate how to transform the distributionally robust joint chance-constrained program into an equivalent reformulation, and show that such a program with binary variables can be equivalently reformulated as a mixed 0–1 integer conic program. To reduce the computational complexity, we derive a relaxed approximation of the joint DRCCP-H using McCormick envelop relaxation, and introduce linear relaxed and conservative approximations by using norm-based inequalities when the Wasserstein metric uses the lp-norm with p≠1 and p≠∞. Finally, we apply this new scheme to address the multi-dimensional knapsack and surgery block allocation problems. The results show that the model with a hybrid ambiguity set yields less conservative solutions when encountering uncertainty over the model with an ambiguity set involving only the Wasserstein metric or moment information, verifying the merit of considering the hybrid ambiguity set, and that the linear approximations significantly reduce the computational time while maintaining high solution quality.

An Effective Hybrid Ant Colony Optimization for the Knapsack Problem Using Multi-Directional Search

An Effective Hybrid Ant Colony Optimization for the Knapsack Problem Using Multi-Directional Search

Incorporating travel time means and standard deviations into transportation network design problem: a hybrid method based on column generation and Lagrangian relaxation

ABSTRACT Travel times in real-world transportation networks are affected by many disruptions. When we conduct the network design optimization, the traffic condition and its resulting travel time variability should be taken into account. However, most of the previous network design optimizations adopted the lengths or expected travel times of links. Based on travel time means and standard deviations, we develop an arc-based model that is a nonlinear and concave integer program. By the Dantzig-Wolfe reformulation, we transform it into an equivalent column-based model that is an integer linear program with a large number of variables. Based on the column-based model, we develop a hybrid method based on column generation and Lagrangian relaxation. The restricted master problem can be settled by the linear programming solvers. The pricing subproblems incorporate independent reliable shortest path problems and a knapsack problem. In numerical experiments, the proposed method can generate feasible solutions with good integrality gaps.

Binary light spectrum optimizer for knapsack problems: An improved model

This paper presents a binary variant of a novel physics-based meta-heuristic optimization algorithm, namely Light spectrum optimizer (LSO), for tackling both the 0–1 knapsack (KP01) and multidimensional knapsack problems (MKP). Because of the continuous nature of the standard LSO that contradicts the knapsack problem's discrete nature, two various transfer functions: S-shaped and X-shaped, are used to convert the continuous values produced by LSO into discrete ones. Some binary solutions produced by the binary LSO (BLSO) may be infeasible, so an improvement-repair strategy is used to convert those solutions into feasible ones by making some improvements on them. Moreover, the classical LSO was modified in this study to propose a new binary variant, namely BMLSO, with better exploration and exploitation operators for overcoming the knapsack problems. Additionally, a novel method, which simulates the swarm intelligence behaviors and the simulated binary crossover (SBX) to accelerate the convergence speed with avoiding stuck into local minima, has been proposed for producing a new binary variant of MLSO known as BHLSO. To verify the performance of the proposed binary variants of LSO, 45 benchmark instances of KP01 and 30 benchmark instances of MKP used commonly in the literature have been used in our experiments. The experimental findings show the superiority of BHLSO for both KP01 and MKP compared with several well-known algorithms in terms of CPU time, convergence speed, and accuracy.

Combinatorial algorithms for solving the constrained knapsack problems with divisible item sizes and penalties

Combinatorial algorithms for solving the constrained knapsack problems with divisible item sizes and penalties

Comparative analysis of linear programming relaxations for the robust knapsack problem

Comparative analysis of linear programming relaxations for the robust knapsack problem

A Feasible Method for Solving an SDP Relaxation of the Quadratic Knapsack Problem

In this paper, we consider a semidefinite programming (SDP) relaxation of the quadratic knapsack problem. After applying low-rank factorization, we get a nonconvex problem, whose feasible region is an algebraic variety with certain good geometric properties, which we analyze. We derive a rank condition under which these two formulations are equivalent. This rank condition is much weaker than the classical rank condition if the coefficient matrix has certain special structures. We also prove that, under an appropriate rank condition, the nonconvex problem has no spurious local minima without assuming linearly independent constraint qualification. We design a feasible method that can escape from nonoptimal nonregular points. Numerical experiments are conducted to verify the high efficiency and robustness of our algorithm as compared with other solvers. In particular, our algorithm is able to solve a one-million-dimensional sparse SDP problem accurately in about 20 minutes on a modest computer. Funding: The research of K.-C. Toh is supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant call [Grant MOE-2019-T3-1-010].

Application network programming method – knapsack problem and it’s modification

New method of multi-extremal optimization has been developed - a network programming method, which is based on the possibility to represent a complex function as a superposition of simpler functions. The paper deals with network (dichotomous) programming method and its application for solving multi-extremal and discrete optimization problems in Project Management. The main goal of the paper is to develop a new approach to solving knapsack problem and to solve a number of optimization problems of project building management. The optimal structure of the dichotomous representation is determined. On the basis dichotomous programming method new algorithm to solve a optimization problem is proposed

Solving the 0/1 Knapsack Problem Using Metaheuristic and Neural Networks for the Virtual Machine Placement Process in Cloud Computing Environment

Virtual machine placement (VMP) is carried out during virtual machine migration to choose the best physical computer to host the virtual machines. It is a crucial task in cloud computing. It directly affects data center performance, resource utilization, and power consumption, and it can help cloud providers save money on data center maintenance. To optimize various characteristics that affect data centers, VMs, and their runs, numerous VMP strategies have been developed in the cloud computing environment. This paper aims to compare the accuracy and efficiency of nine distinct strategies for treating the VMP as a knapsack problem. In the numerical analysis, we test out various conditions to determine how well the system works. We first illustrate the rate of convergence for algorithms, then the rate of execution time growth for a given number of virtual machines, and lastly the rate of development of CPU usage rate supplied by the nine methods throughout the three analyzed conditions. The obtained results reveal that the neural network algorithm performs better than the other eight approaches. The model performed well, as shown by its ability to provide near‐optimal solutions to test cases.

Positional Knapsack Problem: NP-hardness and approximation scheme (Brief Announcement)

We present the Positional Knapsack Problem (PKP), show that it is NP-hard and admits a Fully Polynomial-Time Approximation Scheme (FPTAS). This problem is a variant of the classical Binary Knapsack Problem (KP) in which the contribution of an item to the objective function varies according to the position in which it is added. The change in the valuation adds new properties to the problem that do not hold for KP as PKP is not a generalization of KP. Our FPTAS is based on a dynamic programming algorithm and uses a recursive rounding approach, which is necessary since the objective function depends on each item's value and position.

An improved discreet Jaya optimisation algorithm with mutation operator and opposition-based learning to solve the 0-1 knapsack problem

An improved discreet Jaya optimisation algorithm with mutation operator and opposition-based learning to solve the 0-1 knapsack problem

An improved discreet Jaya optimization algorithm with mutation operator and opposition based learning to solve the 0-1 knapsack problem.

An improved discreet Jaya optimization algorithm with mutation operator and opposition based learning to solve the 0-1 knapsack problem.

Remanufacturing Process Optimization - Introducing Dynamic New Part Infill to Increase the Utilization Rate of Remanufactured Components for Multi-Variant Cores

Climate change, resource scarcity, and the current energy crisis demand fundamental change to overcome future problems. The Circular Economy represents a sustainable process transformation by closing value chain loops. Remanufacturing is one method that extends the life cycle of materials by returning worn-out products (cores) to the manufacturer to produce like-new remanufactured products. In detail, cores are disassembled into core components, which are inspected, sorted out, cleaned, remanufactured, and reassembled into a remanufactured product. The challenge is that multiple components in different conditions must be combined into one remanufactured product because only a certain quantity of core components can be used for remanufacturing due to quality criteria (yield rates). The cores’ uncertain quantity and different yield rates of core components limit the maximum amount of remanufactured products, and the component with the lowest quantity acts as the bottleneck. In other words, although the rest of the parts have been remanufactured through costly and time-consuming processes, they remain unused, and the output of newly remanufactured products is obstructed. Such inefficiencies must be reduced to increase efficiency and create a sustainable process. This paper proposes a problem formulation and a concept that optimizes the utilization of remanufactured components in remanufacturing processes by modifying the bills of material of remanufactured products. The concept implements a dynamic infill of new parts, applying a dynamic mixture of new and remanufactured components to increase the number of remanufactured products while following technical and process restrictions (e.g., production and material planning, quality requirements). The utilization problem is identified as a combinatorial optimization problem and translated into a general multidimensional knapsack problem. The resulting knapsack problem is optimized with a meta-heuristic to find a satisfying solution for the utilization of remanufactured components. Following the case study methodology, the concept is developed, tested, and validated with an automotive OEM.

Continuous linear knapsack problems revisited

In this paper, the continuous linear knapsack problem is considered. Some preliminary results are formulated and proved, and theorems concerning the optimal solution of the considered problem are stated and proved.

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

Modified Artificial Bee Colony Algorithm for Multiple-Choice Multidimensional Knapsack Problem

The multiple-choice multidimensional knapsack problem (MMKP) is a well-known NP-hard problem that has many real-time applications. However, owing to its complexity, finding computationally efficient solutions for the MMKP remains a challenging task. In this study, we propose a Modified Artificial Bee Colony algorithm (MABC) to solve the MMKP. The MABC employs surrogate relaxation, Hamming distance, and a tabu list to enhance the local search process and exploit neighborhood information. We evaluated the performance of the MABC on standard benchmark instances and compared it with several state-of-the-art algorithms, including RLS, ALMMKP, ACO, PEGF-PERC, TIKS-TIKS <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and D-QPSO. The experimental results reveal that MABC produces highly competitive solutions in terms of the best solutions found, achieving approximately 2% of the optimal solutions with trivial (milliseconds) CPU time. The Kruskal-Wallis test revealed that there was no statistically significant difference in the objective function values between the MABC algorithm and other state-of-the-art algorithms (H = 0.31506, p = 0.98882). However, for CPU efficiency, the test showed a statistically significant difference (H = 84.90850, p = 0), indicating that the MABC algorithm exhibited significantly better CPU efficiency (with shorter execution times) than the other algorithms did. Along with these findings, the ease of implementation of the algorithm and the small number of control parameters make our approach highly adaptive for large-scale real-time systems.