Multidimensional Knapsack Problem Research Articles

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

AbstractIn 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.

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.

On the Best Fitness Function for the WSN Lifetime Maximization: A Solution Based on a Modified Salp Swarm Algorithm for Centralized Clustering and Routing

Wireless sensor networks (WSNs) have become increasingly important in recent years due to their ability to monitor and collect data in various environments. However, WSNs are often limited by their energy resources, making energy efficiency a critical concern in designing WSNs. Clustering and multi-hop routing are effective methods for maximizing the lifetime of WSNs. The energy efficient clustering and routing problem in WSNs can be formulated as a multi-dimensional knapsack problem (d-kp). This formulation can generalize all clustering and routing protocols in regard to energy efficiency. In this paper, we investigate the hardness of the energy-efficient clustering and routing problem in WSNs based on the d-kp formulation and introduce a modified Salp Swarm Algorithm (SSA) as a method to solve this problem. SSA is a metaheuristic algorithm inspired by the behavior of salps in the ocean. Moreover, we prove that it is impossible to design a polynomial time fitness function to maximize a long-term metric such as FND (time of first node death), which is a commonly used metric for network lifetime. Our results have implications for the design of energy-efficient WSNs and call for the development of more efficient algorithms that can handle the complexity of this problem. furthermore, we evaluate our proposed algorithm using simulation and compare it to existing algorithms. Our results show that using SSA with a carefully designed fitness function based on the theoretical findings and a unique one-to-one routing protocol to mitigate the energy hole problem, outperforms many of the existing algorithms in terms of energy efficiency and network lifetime.

Verifying new instances of the multidemand multidimensional knapsack problem with instance space analysis

The optimization literature contains numerous studies defining a problem and showing a state-of-the-art implementation of a solution method. These studies may involve an empirical evaluation of the solution methods but assume the current collection of test instances is sufficient. This paper is concerned with instance generation methods for the multidemand multidimensional knapsack problem. We use instance space analysis to characterize the landscape of existing instances and validate the novelty of new instances. We discuss gaps present within the current instance space landscape and fill them with three new sets of instances. We find feasibility issues within instance generation methods and address them through a primal problem instance generator (PPIG). The instance generator is capable of producing feasible and diverse instances by directly controlling the problem features. PPIG contributes to the previous collections of instances and is validated through instance space analysis.

Discrete Data Rate Adaptation for Wireless Body Area Networks

eHealth services require continuous data streaming and a stable level of quality of service. However, wireless network connections can be characterized by variable bandwidths. This requires continuous adaptation of systems, including adapting the bit rates of data streamed by sensors. Assigning appropriate rates to the data represents a main task in congestion control. Most of the current methods look for proper sensor data rates within continuous domains. We examine the case when sensors can generate data streams with several different qualities (e.g., sampling rates, sampling accuracies, etc.). For this reason, the domain of the data rate values can be restricted to the discrete values representing the data rates of the possible quality variations. This paper examines the optimization of the utility of the delivered data under resource constraints by selecting an appropriate variation of the provided data from a discrete set. We provide a formal model for delivering data streams in WBANs and recommend an optimization algorithm to solve the problem. Our recommended solutions are related to the multiple-choice multidimensional knapsack problem. By comparing the proposed algorithms, we found that the greedy method closely approximates the optimum in a short running time.

BE-GWO: Binary extremum-based grey wolf optimizer for discrete optimization problems

Since most metaheuristic algorithms for continuous search space have been developed, a number of transfer functions have been proposed including S-shaped, V-shaped, linear, U-shaped, and X-shaped to convert the continuous solution to the binary one. However, most existing transfer functions do not provide exploration and exploitation required to solve complex discrete problems. Thus, in this study, an improved binary GWO named extremum-based GWO (BE-GWO) algorithm is introduced. The proposed algorithm proposes a new cosine transfer function (CTF) to convert the continuous GWO to the binary form and then introduces an extremum (Ex) search strategy to improve the efficiency of converted binary solutions. The performance of the BE-GWO was evaluated through solving two binary optimization problems, the feature selection and the 0-1 multidimensional knapsack problem (MKP). The results of feature selection problems were compared with several well-known binary metaheuristic algorithms such as BPSO, BGSA, BitABC, bALO, bGWO, BDA, BSSA, and BinABC. Moreover, the results were compared with four versions of the binary GWO, the binary PSO, and the binary ABC. In addition, the BE-GWO algorithm was evaluated to solve the 0-1 MKP with difficult and very difficult benchmark instances and the results were compared with several binary GWO variants. The results of two binary problems were statistically analyzed by the Friedman test. The experimental results showed that the proposed BE-GWO algorithm enhances the performance of binary GWO in terms of solution accuracy, convergence speed, exploration, and balancing between exploration and exploitation.

A Learning—Based Particle Swarm Optimizer for Solving Mathematical Combinatorial Problems

This paper presents a set of adaptive parameter control methods through reinforcement learning for the particle swarm algorithm. The aim is to adjust the algorithm’s parameters during the run, to provide the metaheuristics with the ability to learn and adapt dynamically to the problem and its context. The proposal integrates Q–Learning into the optimization algorithm for parameter control. The applied strategies include a shared Q–table, separate tables per parameter, and flexible state representation. The study was evaluated through various instances of the multidimensional knapsack problem belonging to the NP-hard class. It can be formulated as a mathematical combinatorial problem involving a set of items with multiple attributes or dimensions, aiming to maximize the total value or utility while respecting constraints on the total capacity or available resources. Experimental and statistical tests were carried out to compare the results obtained by each of these hybridizations, concluding that they can significantly improve the quality of the solutions found compared to the native version of the algorithm.

A recipe for an omnichannel warehouse storage system: Improving the storage efficiency by integrating block stacking and racking

Warehouses in multi-channel retailing are expected to possess storage capacity based on the purpose of operations. However, the inefficient use of storage capacity due to omnichannel retail restructuring, wherein mixed-storage items feature both online and offline, has prompted us to revisit the nature of existing storage systems (block stacking and racking). Most warehouses continue to improvise and undertake critical decision-making with regard to profiling balanced storage systems between block stacking and racking, including the efficient slotting of unit-load items across storage systems. This study aimed to establish a framework to systemically improve storage efficiency by achieving an optimum balance between block stacking and racking under inventory uncertainty – lot size and item dimensions. This study ingeniously proposes a two-stage stochastic programming model for robust optimisation during long-run profiling (redesign) and the multi-dimensional knapsack problem for achieving minimum space wastage during slotting (reshuffle). The study adopted a two-fold approach. First, the hypothetical directions for a balanced system, combining increases in buildings’ ceiling heights with its tendency to store low-volume, high-mix items, were examined. Second, the applicability of the two methods – redesign and reshuffle – was examined by considering a real-life case study and the life cycle of warehouse storage design. It was found that the reshuffling approach can resourcefully serve as a short-term solution for warehouses with imminent space issues. Redesigning also demonstrated a long-term space gain in floor space, thus minimising space wastage.

Hybrid Learning Moth Search Algorithm for Solving Multidimensional Knapsack Problems

The moth search algorithm (MS) is a relatively new metaheuristic optimization algorithm which mimics the phototaxis and Lévy flights of moths. Being an NP-hard problem, the 0–1 multidimensional knapsack problem (MKP) is a classical multi-constraint complicated combinatorial optimization problem with numerous applications. In this paper, we present a hybrid learning MS (HLMS) by incorporating two learning mechanisms, global-best harmony search (GHS) learning and Baldwinian learning for solving MKP. (1) GHS learning guides moth individuals to search for more valuable space and the potential dimensional learning uses the difference between two random dimensions to generate a large jump. (2) Baldwinian learning guides moth individuals to change the search space by making full use of the beneficial information of other individuals. Hence, GHS learning mainly provides global exploration and Baldwinian learning works for local exploitation. We demonstrate the competitiveness and effectiveness of the proposed HLMS by conducting extensive experiments on 87 benchmark instances. The experimental results show that the proposed HLMS has better or at least competitive performance against the original MS and some other state-of-the-art metaheuristic algorithms. In addition, the parameter sensitivity of Baldwinian learning is analyzed and two important components of HLMS are investigated to understand their impacts on the performance of the proposed algorithm.

A comparison of first-come-first-served and multidimensional heuristic approaches for asset allocation of floor cleaning machines

This paper presents a comparison of three binary heuristic algorithms - Particle Swarm Optimization (BPSO), Differential Evolution (BDE), and a Genetic Algorithm (GA) - under two asset allocation approaches to obtain optimal deployment combinations of assets for a cleaning company. When a company requires planning for the best asset allocation, the common practice is to rely on the experience of managers to estimate what combination of assets is appropriate to deploy to fulfill the different cleaning tasks that are carried out simultaneously in different locations. This work considers a given asset inventory of 168 machines composed of 29 different types of cleaners obtained online from three vendors. The comparison is made for three scenarios with 5, 10, and 15 tasks. Each task has a desired cleaning area and maximum cleaning time that are taken as constraints while minimizing the number of selected machines to be allocated for each task. The allocation approaches are inspired by the First-Come-First-Served (FCFS) approach and the Multidimensional Knapsack Problem (MKP). In total, 30 trials were computed for each scenario where the best, worst, average, and standard deviation were obtained. The results indicate that the combination of BDE and the FCFS approach provides the best solutions by having the lowest number of assets selected for each scenario. Additionally, it has the lowest standard deviation in cleaning area and time, while having a low execution time compared to the other approaches and heuristic algorithms.

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.

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.

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.

A novel binary multi-swarms fruit fly optimisation algorithm for the 0-1 multidimensional knapsack problem

A novel binary multi-swarms fruit fly optimisation algorithm for the 0-1 multidimensional knapsack problem

Application and assessment of divide-and-conquer-based heuristic algorithms for some integer optimization problems

In this paper three heuristic algorithms using the Divide-and-Conquer paradigm are developed and assessed for three integer optimizations problems: Multidimensional Knapsack Problem (d-KP), Bin Packing Problem (BPP) and Travelling Salesman Problem (TSP). For each case, the algorithm is introduced, together with the design of numerical experiments, in order to empirically establish its performance from both points of view: its computational time and its numerical accuracy.

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.

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.

Solving Multidimensional Knapsack Problem with Bayesian Multiploid Genetic Algorithm

Solving optimization problems is still a big challenge in the area of optimization algorithms. Many proposed algorithms in the literature don’t consider the relations between the variables of the nature of the problem. However, a recently published algorithm, called “Bayesian Multiploid Genetic Algorithm” exploits the relations between the variables and then solves the given problem. It also uses more than one genotype unlike the simple Genetic Algorithm (GA) and it acts like an implicit memory in order to remember the old but good solutions. In this work, the well-known Multidimensional Knapsack Problem (MKP) is solved by the Bayesian Multiploid Genetic Algorithm. And the results show that exploiting relations between the variables gets a huge advantage in solving the given problem.

Dynamic Multidimensional Knapsack Problem benchmark datasets

With increasing research on solving Dynamic Optimization Problems (DOPs), many metaheuristic algorithms and their adaptations have been proposed to solve them. However, from currently existing research results, it is hard to evaluate the algorithm performance in a repeatable way for combinatorial DOPs due to the fact that each research work has created its own version of a dynamic problem dataset using stochastic methods. Up to date, there are no combinatorial DOP benchmarks with replicable qualities. This work introduces a non-stochastic consistent Dynamic Multidimensional Knapsack Problem (Dynamic MKP) dataset generation method that is also extensible to solve the research replicability problem. Using this method, generated and published 1405 Dynamic MKP benchmark datasets using existing famous static MKP benchmark instances as the initial state. Then the datasets are quantitatively and qualitatively analyzed. Furthermore, 445 datasets have the optimal result found of each state using a linear solver. The optimal results and result scores are included with published datasets.

Revisiting surrogate relaxation for the multidimensional knapsack problem

The multidimensional knapsack problem (MKP) is a classic problem in combinatorial optimisation. Several authors have proposed to use surrogate relaxation to compute upper bounds for the MKP. In their papers, the surrogate dual is solved heuristically. We show that, using a modern dual simplex solver as a subroutine, one can solve the surrogate dual exactly in reasonable computing times. On the other hand, the resulting upper bound tends to be strong only for relatively small MKP instances.