Multiple Knapsack Problem Research Articles

Solving Transport Infrastructure Investment Project Selection and Scheduling Using Genetic Algorithms

The development of transport infrastructure is crucial for economic growth, social connectivity, and sustainable development. Many countries have historically underinvested in transport infrastructure, necessitating more efficient strategic planning in the implementation of transport infrastructure investment projects. This article addresses the selection and scheduling of transport infrastructure projects, specifically within the context of utilizing pre-allocated funds within a multi-annual budget investment program. The current decision-making process relies heavily on expert judgment and lacks quantitative decision support methods. We propose a genetic algorithm as a decision-support tool, framing the problem as an NP-hard 0–1 multiple knapsack problem. The proposed genetic algorithm (GA) is unique for its matrix-encoded chromosomes, specially designed genetic operators, and a customized repair operator to address the large number of invalid chromosomes generated during the GA computation. In computational experiments, the proposed GA is compared to an exact solution and proves to be efficient in terms of quality of obtained solutions and computational time, with an average computational time of 108 s and the quality of obtained solutions typically ranging between 85% and 95% of the optimal solution. These results highlight the potential of the proposed GA to enhance strategic decision-making in transport infrastructure development.

Two-set inequalities for the binary knapsack polyhedra

This paper presents two-set inequalities, a class of valid inequalities for knapsack and multiple knapsack problems. Two-set inequalities are generated from two arbitrary sets of variables from a knapsack constraint. This class of cutting planes is not a traditional type of lifting since a valid inequality over a restricted space is not required to start. Furthermore, they cannot be derived using any existing lifting technique. The paper presents a quadratic algorithm to efficiently generate many two-set inequalities. Conditions for facet-defining two-set inequalities are also derived. Computational experiments tested these inequalities as pre-processing cuts versus CPLEX, a high-performance mathematical programming solver, at default settings. Overall, two-set inequalities reduced the time to solve some benchmark multiple knapsack instances to up to 80%. Computational results also showed the potential of this new class of cutting planes to solve computationally challenging binary integer programs.

Towards Collaborative Edge Intelligence: Blockchain-Based Data Valuation and Scheduling for Improved Quality of Service

Collaborative edge intelligence, a distributed computing paradigm, refers to a system where multiple edge devices work together to process data and perform distributed machine learning (DML) tasks locally. Decentralized Internet of Things (IoT) devices share knowledge and resources to improve the quality of service (QoS) of the system with reduced reliance on centralized cloud infrastructure. However, the paradigm is vulnerable to free-riding attacks, where some devices benefit from the collective intelligence without contributing their fair share, potentially disincentivizing collaboration and undermining the system’s effectiveness. Moreover, data collected from heterogeneous IoT devices may contain biased information that decreases the prediction accuracy of DML models. To address these challenges, we propose a novel incentive mechanism that relies on time-dependent blockchain records and multi-access edge computing (MEC). We formulate the QoS problem as an unbounded multiple knapsack problem at the network edge. Furthermore, a decentralized valuation protocol is introduced atop blockchain to incentivize contributors and disincentivize free-riders. To improve model prediction accuracy within latency requirements, a data scheduling algorithm is given based on a curriculum learning framework. Based on our computer simulations using heterogeneous datasets, we identify two critical factors for enhancing the QoS in collaborative edge intelligence systems: (1) mitigating the impact of information loss and free-riders via decentralized data valuation and (2) optimizing the marginal utility of individual data samples by adaptive data scheduling.

Improved approximation for two-dimensional vector multiple knapsack

We study the uniform 2-dimensional vector multiple knapsack (2VMK) problem, a natural variant of multiple knapsack arising in real-world applications such as virtual machine placement. The input for 2VMK is a set of items, each associated with a 2-dimensional weight vector and a positive profit, along with m 2-dimensional bins of uniform (unit) capacity in each dimension. The goal is to find an assignment of a subset of the items to the bins, such that the total weight of items assigned to a single bin is at most one in each dimension, and the total profit is maximized.Our main result is a (1−ln⁡22−ε)-approximation algorithm for 2VMK, for every fixed ε>0, thus improving the best known ratio of (1−1e−ε) which follows as a special case from a result of Fleischer et al. (2011) [6].Our algorithm relies on an adaptation of the Round&Approx framework of Bansal et al. (2010) [15], originally designed for set covering problems, to maximization problems. The algorithm uses randomized rounding of a configuration-LP solution to assign items to ≈m⋅ln⁡2≈0.693⋅m of the bins, followed by a reduction to the (1-dimensional) Multiple Knapsack problem for assigning items to the remaining bins.

A novel discrete differential evolution algorithm combining transfer function with modulo operation for solving the multiple knapsack problem

In this paper, an efficient method for solving multiple knapsack problem (MKP) using discrete differential evolution is proposed. Firstly, an integer programming model of MKP suitable for discrete evolutionary algorithm is established. Secondly, a new method for discretizing continuous evolutionary algorithm is proposed based on the combination of transfer function and modulo operation. Therefrom, a new discrete differential evolution algorithm (named TMDDE) is proposed. Thirdly, the algorithm GROA is proposed to eliminate infeasible solutions of MKP. On this basis, a new method for solving MKP using TMDDE is proposed. Finally, the performance of TMDDE using S-shaped, U-shaped, V-shaped, and Taper-shaped transfer functions combined with modulo operation is compared, respectively. It is pointed out that T3-TMDDE which used Taper-shaped transfer function T3 is the best. The comparison results of solving 30 MKP instances show that the performance of T3-TMDDE is better than five advanced evolutionary algorithms. It not only indicates that TMDDE is more competitive for solving MKP, but also demonstrates that the proposed discretization method is an effective method.

Design of daily load profiles in environmentally friendly commercial and industrial microgrids

The purpose of the study is to solve the problem of design of daily load profiles for optimal control of an environmentally friendly commercial and industrial microgrid (CIM) (power is generated only by renewable energy sources) connected to the power system by a power transmission line. This goal is achieved by adjusting the planned daily load profiles of consumers located on the territory of the CIM. The adjustment means shifting power consumption to another time of the day in question (power consumption is delayed). The problem of the delayed power allocation is represented as an optimization multiple knapsack problem that adapts to the problem-solving process. The proposed algorithm was tested on a 6-node system according to the scenario that involved adjustment of the load profile in the CIM to ensure that the power flow from the power system remains within specified limits. Compliance with the limits guarantees uninterrupted power supply from the power system, which is a fundamental requirement when developing load profiles. Experiments were carried out to evaluate the delivery of uninterrupted power supply to CIM consumers depending on the initial data. The findings indicate that the disruptions in power supply to CIM consumers are completely eliminated if the load can be divided to shift it to other hours of the day and when the load of 0.151 MW is disconnected in operating state 7. The total load should be divided into at least three parts. Disconnection of 0.151 MW is performed to prevent disconnection of the commercial and industrial microgrid from the power system, which would result in a power deficit of 4.652 MW.

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.

Adjustable Robust Optimization with Discrete Uncertainty

In this paper, we study adjustable robust optimization (ARO) problems with discrete uncertainty. Under a very general modeling framework, we show that such two-stage robust problems can be exactly reformulated as ARO problems with objective uncertainty only. This reformulation is valid with and without the fixed recourse assumption and is not limited to continuous wait-and-see decision variables unlike most of the existing literature. Additionally, we extend an enumerative algorithm akin to a branch-and-cut scheme for which we study the asymptotic convergence. We discuss how to apply the reformulation on two variants of well-known optimization problems, a facility location problem in which uncertainty may affect the capacity values and a multiple knapsack problem with uncertain weights, and we report extensive computational results demonstrating the effectiveness of the approach. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported by the Air Force Office of Scientific Research [Grants FA8655-20-1-7012, FA8655-20-1-7019].

Efficient Techniques for Residential Appliances Scheduling in Smart Homes for Energy Management Using Multiple Knapsack Problem

The evolution of the smart grid has enabled residential users to manage the ever-growing energy demand in an efficient manner. The smart grid plays an important role in managing this huge energy demand of residential households. A home energy management system enhances the efficiency of the energy infrastructure of smart homes and provides an opportunity for residential users to optimize their energy consumption. Smart homes contribute significantly to reducing electricity consumption costs by scheduling domestic appliances effectively. This residential appliance scheduling problem is the motivation to find an optimal appliance schedule for users that could balance the load profile of the home and helps in minimizing electricity cost (EC) and peak-to-average ratio (PAR). In this paper, we have focused on appliance scheduling on the consumer side. Two novel home energy management models are proposed using multiple scheduling options. The residential appliance scheduling problem is formulated using the multiple knapsack technique. Serial and parallel scheduling algorithms of home appliances namely MKSI (Multiple knapsacks with serial implementation) and MKPI (Multiple knapsacks with parallel implementation) are proposed to reduce electricity cost and PAR. Price-based demand response techniques are incorporated to shift appliances from peak hours to off-peak hours to optimize energy consumption. The proposed algorithms are tested on real-time datasets and evaluated based on time of use pricing tariff and critical peak pricing. The performance of both the algorithms is compared with the unscheduled scenario and existing algorithm. Simulations show that both proposed algorithms are efficient methods for home energy management to minimize PAR and electricity bills of consumers. The proposed MKSI algorithm achieves cost reduction of 20.26% and 42.53% for TOU and CPP, respectively as compared to the unscheduled scenario while PAR is reduced by 45.07% and 39.51% for TOU and CPP, respectively. The proposed MKPI algorithm achieves 22.33% and 46.36% cost reduction compared to the unscheduled case for TOU and CPP while the PAR ratio is reduced by 46.47% and 41.16% for TOU and CPP respectively.

Lagrangian matheuristics for the Quadratic Multiple Knapsack Problem

The Quadratic Multiple Knapsack Problem (QMKP) is a challenging combinatorial optimization problem combining the well-known Quadratic Knapsack Problem with the Multiple Knapsack Problem. After a long stream of research devoted to metaheuristic approaches for large-scale instances, only recently some authors started to study the mathematical properties of the QMKP and proposed exact solution methods for benchmark instances of smaller size. In this paper, we propose the first matheuristic approach for solving the QMKP approximately. The proposed method exploits the strength of a Lagrangian relaxation for the natural quadratic formulation of the QMKP to derive heuristic solutions. Postoptimization local search procedures are embedded in the final framework. Experimental studies show that the resulting deterministic matheuristic approach consistently delivers solutions of very good quality in short computing times. • First matheuristic for the Quadratic Multiple Knapsack Problem. • Competitive with currently best performing exact method. • Tighter Lagrangian bounds on medium-size instances.

Optimization model for combining drug vials in the preparation of doses for outpatient

Planning the use of chemotherapy drugs in outpatient treatment is a complex problem due to the variability of cancer, resulting in different chemotherapy protocols. This process involves factors such as the cyclical nature of treatment protocols and clinical resources. Within this context, an optimization model is needed to plan the use of chemotherapy drugs in the preparation of doses for patient treatment, considering the operational particularities of treatment centers. This study proposed a linear programming model based on the multiple knapsack problem, in order to optimize the combination of different vials of chemotherapy drugs, minimizing the total cost of treatment. The model, based on the daily schedule of patients, provides the best combination of drug vials and supports the preparation process of each dose prescribed by the doctor for each patient, respecting the treatment protocol and resource limitations. The model was implemented in the CPLEX 12.9.0 application, and the computational tests were performed with real data. The results demonstrated that the costs when applying the model were 5% lower when compared to the current manner in which the oncology pharmacy combines the drugs vials.

Efficiently Generating Bounded Solutions for Very Large Multiple Knapsack Assignment Problems

The Multiple Knapsack Assignment Problem (MKAP) is an interesting generalization of the Multiple Knapsack Problem which has logistical applications in transportation and shipping. In addition to trying to insert items into knapsacks in order to maximize the profit of the items in the knapsacks, the MKAP partitions the items into classes and only items from the same class can be inserted into a knapsack. In the literature, the Gurobi integer programming software has solved MKAPs with up to 1240 variables and 120 constraints in at most 20 minutes on a standard PC. In this article, using a standard PC and iteratively loosening the acceptable tolerance gap for 180 MKAPs with up to 20,100 variables and 1,120 constraints, we show that Gurobi can, on average, generate solutions that are guaranteed to be at most 0.17% from the optimums in 43 seconds. However, for very large MKAPs (over a million variables), Gurobi’s performance can be significantly improved when an initial feasible solution is provided. Specifically, using from the literature, a heuristic and 42 MKAP instances with over 6 million variables and nearly 90,000 constraints, Gurobi generated solutions guaranteed to be, on average, within 0.21% of the optimums in 10 minutes. This is a 99% reduction in the final solution bound (gap between the best Gurobi solution and the best upper bound) compared to the approach without initial solution inputs. Hence, a major objective of this article is to demonstrate for what size MKAP instances providing Gurobi with an initial heuristic solution significantly improves performance in terms of both execution time and solution quality.

Using general-purpose integer programming software to generate bounded solutions for the multiple knapsack problem: a guide for or practitioners

An NP-Hard combinatorial optimization problem that has significant industrial applications is the Multiple Knapsack Problem. If approximate solution approaches are used to solve the Multiple Knapsack Problem there are no guarantees on solution quality and exact solution approaches can be intricate and challenging to implement. This article demonstrates the iterative use of general-purpose integer programming software (Gurobi) to generate solutions for test problems that are available in the literature. Using the software package Gurobi on a standard PC, we generate in a relatively straightforward manner solutions to these problems in an average of less than a minute that are guaranteed to be within 0.16% of the optimum. This algorithm, called the Simple Sequential Increasing Tolerance (SSIT) algorithm, iteratively increases tolerances in Gurobi to generate a solution that is guaranteed to be close to the optimum in a short time. This solution strategy generates bounded solutions in a timely manner without requiring the coding of a problem-specific algorithm. This approach is attractive to management for solving industrial problems because it is both cost and time effective and guarantees the quality of the generated solutions. Finally, comparing SSIT results for 480 large multiple knapsack problem instances to results using published multiple knapsack problem algorithms demonstrates that SSIT outperforms these specialized algorithms.

Research on Resource Allocation Algorithm for Satellite Uplink to Provide Differentiated QoS Support

Broadband satellite systems are built to serve different user groups ranging from enterprise and military users to common personal users, with various kinds of services calling for dissimilar quality of service (QoS). Obviously, the satellite systems are expected to distinguish the specific type of an incoming service request as well as the importance of its owner in order to perform an efficient assignment of valuable resources. In this article, the uplink resource management challenge in a multifrequency-time-division multiple access satellite system is addressed where provisioning the discriminated QoS is of vital importance. Having taken both user priority and service type into account, a differentiated QoS support principle conforming to the digital video broadcast - second generation DVB interactive satellite system (DVB-RCS2) standard is put forward, based on which the uplink timeslot assignment problem is formulated.The NP-hardness of the problem is proved, and a differentiated QoS resource allocation scheme is designed, decoupling the problem into the coarse-level of resource allocation and fine-level of timeslot assignment. The former stage involves the constant rate assignment, bandwidth adjustment, QoS degradation and user scheduling, among which the subproblem of user scheduling is an integer nonlinear program. To tackle the issue, the problem is first innovatively converted to an integer linear program, and is then modeled as the multiple-choice multiple knapsack (MCMK) problem. Furthermore, an MCMK algorithm is presented to give the optimal solution to the user scheduling problem, after which the timeslot assignment is investigated in the second phase to generate the terminal burst time plan table for return channel satellite terminals.

Data-driven multi-layered intelligent energy management system for domestic decentralized power distribution systems

Energy management techniques have been immensely adopted in residential, industrial, and commercial sectors to meritoriously flatten the load profile of consumers. In this article, a cutting-edge intelligent energy management system is proposed to analytically monitor and regulate the load demand of a residential area to meritoriously address the load balancing and scheduling optimization problem with an aim to design an autonomous domestic complex, less reliant on the focal power grid. To do so, single- and multi- objective functions are defined for the presented double-layered framework. In the first layer, a day-ahead scheduling problem is evaluated to achieve the mutual revenue of electricity utilities and consumers, while in the second layer, a real-time coordination scheme for robust energy transactions among appliances, multiple microgrids, and the main power system is examined. Herein, a 0−1 multiple knapsack problem incorporated with the mixed-integer linear programming method is used to formulate the system mathematical model. To evaluate the intended system model, three heuristic algorithms (genetic algorithm, wind-driven optimization, and their hybrid) are modified. Furthermore, the Monte–Carlo simulations are performed to mitigate the impact of randomness and avoid pre-convergence. From the results of extensive simulations, it is corroborated that the proposed methodology is robust, reliable, and eco-social compared to prior schemes. This article reveals that the hybrid approach outperformed the other two strategies in terms of energy consumption, electricity cost, peak-to-average ratio, and user comfort level.

A Genetic algorithm combined with mathematical programming to solve Generalised Quadratic Multiple Knapsack Problem

A Genetic algorithm combined with mathematical programming to solve Generalised Quadratic Multiple Knapsack Problem

Exact and Heuristic Optimization Algorithms for Auto-Rack Loading Problem

The transportation of new vehicles over the rail network takes place in specially designed railcars / wagons called auto-racks. This paper introduces a new problem – Auto-Rack Loading Problem (ARLP), involving transportation of vehicles on auto-racks with the objective of maximizing the revenue from the loaded vehicles subject to constraints. Existing literature only addresses methods to model and optimize transportation of vehicles on trucks called the auto-carrier loading problem. Initially, a model is formulated as an integer programming problem which is proven to be NP-hard since a special case of the ARLP reduces to a 0-1 multiple knapsack problem. Consequently, a multi-phase heuristic procedure is developed to solve large instances of the problem in reasonable time. The performance of the integer programming model and the heuristic procedure are evaluated and compared with an upper bound for randomly generated instances involving auto-racks and vehicles designed for use in the Indian rail network.

Computation power maximization for mobile edge computing enabled dense network

High-density connection is among the natures of next-generation wireless communication systems. Meanwhile, various computation-intensive smart applications are becoming growing popularity with the technology boom. Thus, strong computation power will become a crucial requirement for wireless communication systems. Mobile edge computing provides a promising solution to this requirement by pushing a cloud-like computation capacity to the network edge. This study aims to maximize the computation power of a mobile edge computing enabled dense network. To this end, a computation bits maximization problem is formulated by jointly optimizing offloading decision and resource allocation. The problem is a mixed-variable nonlinear programming problem. First, by analyzing the feasibility of computation modes and constructing a user distribution strategy, the original problem is decoupled into two sub-problems, i.e., the assignment of the offloading users and resource allocation. The offloading users’ assignment is modeled as a restricted multiple knapsack problem, and a profit density is defined to maximize the overall profits of multiple knapsacks. Then, an improved differential evolution algorithm is developed to address the knapsack problem, in which mutation and repair operators are designed according to the profit density. Based on the optimal solution to the knapsack problem, the resource allocation sub-problem is solved by the corresponding calculation. Finally, extensive experiments are conducted to evaluate the performance of our scheme. Results show that: (1) our scheme provides superior computation power compared to that of benchmark schemes; (2) the performance gain of our scheme over benchmark schemes expands with growing connection density. Therefore, our scheme is an effective computation power optimization scheme.

QMKPy: A Python Testbed for the Quadratic Multiple Knapsack Problem

QMKPy provides a Python framework for modeling and solving the quadratic multiple knapsack problem (QMKP). It is primarily aimed at researchers who develop new solution algorithms for the QMKP. QMKPy therefore mostly functions as a testbed to quickly implement novel algorithms and compare their results with existing ones. However, the package also already includes implementations of established algorithms for those who only need to solve a QMKP as part of their research.

LP relaxation and dynamic programming enhancing VNS for the multiple knapsack problem with setup

AbstractWe consider the multiple knapsack problem (KP) with setup (MKPS), which is an extension of the KP with setup (KPS). We propose a new solving approach denoted by LP&DP‐VNS that combines linear programming (LP) relaxation and dynamic programming (DP) to enhance variable neighborhood search (VNS). The LP&DP‐VNS is tailored to the characteristics of the MKPS and reduced to LP&DP to solve the KPS. The approach is tested on 200 KPS and 360 MKPS benchmark instances. Computational experiments show the effectiveness of the LP&DP‐VNS, compared to integer programming (using CPLEX) and the best state‐of‐the‐art algorithms. It reaches 299/342 optimal solutions and 316/318 best‐known solutions and provides 127 new best solutions. An additional computational study shows that the LP&DP‐VNS scales up extremely well, solving optimally and near‐optimally very large instances with up 200 families and 300,000 items in a reasonable amount of time.