Binary metaheuristic algorithms for 0–1 knapsack problems: Performance analysis, hybrid variants, and real-world application

This paper proposes a novel approach for the multidimensional knapsack problem (MDKP) using differential evolution. Firstly, the principle and the pseudo-code of binary differential evolution with hybrid encoding (HBDE) are presented. On the basis of the existing repair operator 2 (RO2), an improved repair operator 3 (RO3) for handling the infeasible solutions of MDKP is developed. Then, combine HBDE with RO3, an efficient algorithm (HBDE-RO3) for MDKP is proposed. Finally, the experiment results of the 138 well-known MDKP benchmarks show that RO3 is advantageous to deal with the infeasible solutions than RO2, and the proposed algorithm HBDE-RO3 has superior performance for solving MDKP than the state-of-the-art algorithms.

This paper presents a new binary optimization technique for solving the 0–1 knapsack problem. This algorithm is based on converting the continuous search space of the recently proposed quadratic interpolation optimization (QIO) into discrete search space using various V-shaped and S-shaped transfer functions; this algorithm is abbreviated as BQIO. To further improve its performance, it is effectively integrated with a uniform crossover operator and a swap operator to explore the discrete binary search space more effectively. This improved variant is called BIQIO. Both BQIO and BIQIO are assessed using 20 well-known knapsack instances and compared to four recently published metaheuristic algorithms to reveal their effectiveness. The comparison among algorithms is based on three performance metrics: the mean fitness value, Friedman mean rank and computational cost. The first two metrics are used to observe the accuracy of the results, while the last metric is employed to show the efficiency of each algorithm. The results of this comparison reveal the superiority of BIQIO over the classical BQIO and four rival optimizers.

The pigeon-inspired optimization algorithm is a category of a newly proposed swarm intelligence-based algorithm that belongs to the population-based solution technique. The MKP is a class of complex optimization problems that have many practical applications in the fields of engineering and sciences. Due to the practical applications of MKP, numerous algorithmic-based methods like local search and population-based search algorithms have been proposed to solve the MKP in the past few decades. This paper proposes a modified binary pigeon-inspired optimization algorithm named (Modified-BPIO) for the 0 - 1 multidimensional knapsack problem (MKP). The utilization of the binary pigeon-inspired optimization (BPIO) for solving the multidimensional knapsack problem came with huge success. However, it can be observed that the BPIO converges prematurely due to lost diversity during the search activities. Given the above, the crossover operator is integrated with the landmark component of the BPIO to improve the diversity of the solution space. The MKP benchmarks from the Operations Research (OR) library are utilized to test the performance of the proposed binary method. Experimentally, it is concluded that the proposed Modified-BPIO has a better performance when compared with the BPIO and existing state-of-the-arts that worked on the same MKP benchmarks.

Knapsack problems are among the most common problems in literature tackled with evolutionary algorithms (EA). Their major advantage lies in the fact that they are relatively simple to implement while they allow generalizations for a wide range of real world problems. The multi-dimensional knapsack problem (MKP), which belongs to the class of NP-complete combinatorial optimization problems, is one of the variations of the knapsack problem. The MKP has a wide range of real world applications such as cargo loading, selecting projects to fund, budget management, cutting stock, etc. The MKP has been studied quite extensively in the EA community. Due to the constrained nature of the problem, constraint handling techniques gain great importance in the performance of the proposed EA approaches. In this study, the applicability of a generational EA that uses a penalty-based constraint handling technique and a gene locus based, asymmetric, adaptive mutation scheme is explored for the MKP. The effects of the parameters of the explored approach is determined through tests. Further experiments, using large MKP instances from commonly used benchmarks available through the Internet are performed. Comparison tables are given for the performance of the explored approach and other good performing EAs found in literature for the MKP. Results show that performance improves greatly when compared with other penalty-based techniques, but the explored approach is still not the best performer among all. However, unlike the explored technique, the EAs using the other constraint handling techniques require a great amount of extra computational effort and need heuristic information specific to the optimization problem. Based on these observations, and the fact that the performance difference between the explored scheme and the better performers is not too high, research on improving the explored approach is still in progress.

The multi-dimensional knapsack problem (MDKP) is a well-known NP-hard problem in combinatorial optimization. As it has various real-life applications, the MDKP has been intensively studied in the literature. On the other hand, far too little attention has been paid to the multi-objective version of the MDKP. In this chapter, we consider the bi-objective multi-dimensional knapsack problem (BOMDKP). We propose a Binary Genetic Algorithm (BGA) with an external archive for the problem. Our proposed BGA algorithm also employs a binary local search. The non-dominated solution sets are obtained for various bi-objective benchmark instances with 100, 250, 500 and 750 items, by employing the proposed BGA. Then, the performance of the BGA is compared with other multi-objective algorithms from the literature, i.e., MOEA/D and MOFPA. Furthermore, it is observed that the Pareto-optimal solution set provided by Zitzler and Laumans for 500 items and 2 knapsacks includes 30 dominated solutions. Also, the Pareto-optimal solutions for the scenario with 750 items are not reported in Zitzler and Thiele [43]. Hence, the true Pareto-optimal solution sets are found for all benchmark problem instances using Improved Augmented Epsilon Constraint (AUGMECON2) method. The non-dominated solution sets of the BGA, MOEA/D and MOFPA are compared with the Pareto-optimal solution sets for all test instances. The computational results indicate that the proposed BGA is more effective to solve the BOMDKP than the best-performing algorithms from the literature.

While the classical knapsack problem has been the object to be solved by optimization algorithm proposals for many years, another version of this problem, discounted {0-1} knapsack problem, is gaining a lot of attention recently. The original knapsack problem requires selecting specific items from an item set to maximize the total benefit while ensuring that the total weight does not exceed the knapsack capacity. Meanwhile, discounted {0-1} knapsack problem has more stringent requirements in which items are divided into groups, and only up to one item from a particular group can be selected. This constraint, which does not exist in the original knapsack problem, makes discounted {0-1} knapsack problem even more challenging. In this paper, we propose a new algorithm based on salp swarm algorithm in the form of four different variants to resolve the discounted {0-1} knapsack problem. In addition, we also make use of an effective data modeling mechanism and a greedy repair operator that helps overcome local optima when finding the global optimal solution. Experimental and statistical results show that our algorithm is superior to currently available algorithms in terms of solution quality, convergence, and other statistical criteria.

A binary differential evolution algorithm learning from explored solutions

This paper presents a novel Simple Human Learning Optimization (SHLO) algorithm, which is inspired by human learning mechanisms. Three learning operators are developed to generate new solutions and search for the optima by mimicking the learning behaviors of human. The 0-1 knapsack problems are adopted as benchmark problems to validate the performance of SHLO, and the results are compared with those of binary particle swarm optimization (BPSO), modified binary differential evolution (MBDE), binary fruit fly optimization algorithm (bFOA) and adaptive binary harmony search algorithm (ABHS). The experimental results demonstrate that SHLO significantly outperforms BPSO, MBDE, bFOA and ABHS. Considering the ease of implementation and the excellence of global search ability, SHLO is a promising optimization tool.

In machine learning and data mining, feature selection (FS) is a traditional and complicated optimization problem. Since the run time increases exponentially, FS is treated as an NP-hard problem. The researcher’s effort to build a new FS solution was inspired by the ongoing need for an efficient FS framework and the success rates of swarming outcomes in different optimization scenarios. This paper presents two binary variants of a Hunger Games Search Optimization (HGSO) algorithm based on V- and S-shaped transfer functions within a wrapper FS model for choosing the best features from a large dataset. The proposed technique transforms the continuous HGSO into a binary variant using V- and S-shaped transfer functions (BHGSO-V and BHGSO-S). To validate the accuracy, 16 famous UCI datasets are considered and compared with different state-of-the-art metaheuristic binary algorithms. The findings demonstrate that BHGSO-V achieves better performance in terms of the selected number of features, classification accuracy, run time, and fitness values than other state-of-the-art algorithms. The results demonstrate that the BHGSO-V algorithm can reduce dimensionality and choose the most helpful features for classification problems. The proposed BHGSO-V achieves 95% average classification accuracy for most of the datasets, and run time is less than 5 sec. for low and medium dimensional datasets and less than 10 sec for high dimensional datasets.

A novel binary greater cane rat algorithm for feature selection

In this paper, we have proposed a novel algorithm based on Ant Colony Optimization (ACO) for finding near-optimal solutions for the Multi-dimensional Multi-choice Knapsack Problem (MMKP). MMKP is a discrete optimization problem, which is a variant of the classical 0-1 Knapsack Problem and is also an NP-hard problem. Due to its high computational complexity, exact solutions of MMKP are not suitable for most real-time decision-making applications e.g. QoS and Admission Control for Adaptive Multimedia Systems, Service Level Agreement (SLA) etc. Although ACO algorithms are known to have scalability and slow convergence issues, here we have augmented the traditional ACO algorithm with a unique random local search, which not only produces near-optimal solutions but also greatly enhances convergence speed. A comparative analysis with other state-of-the-art heuristic algorithms based on public MMKP dataset shows that, in all cases our approaches outperform others. We have also shown that our algorithms find near optimal (within 3% of the optimal value) solutions within milliseconds, which makes our approach very attractive for large scale real time systems.

Binary Particle Swarm Optimization (BPSO) has extended the capacity of the conventional particle swarm optimization (PSO) for optimizing the discrete combinational optimization problems. The transfer function of BPSO is key for its capacity to search solution. This paper discuss the weight setting of two variants called the S-Shaped and the V-Shaped transfer functions in BPSO. The experimental results indicate that the increasing weight setting is beneficial to the performance of BPSO with the S-shaped transfer function, while the decreasing weight setting is flavorful for that of BPSO with the V-shaped transfer function. The Multi dimension Knapsack Problem (MKP) are used in the experiment for testing the discussable conclusions. The experimental results illustrate that the transfer function can be adjusted to improve the performance of BPSO, and the recommended weight setting is obtained, accordingly.KeywordsBinary Particle Swarm OptimizationTransfer functionWeight settingKnapsack problem

Binary knapsack problem has received considerable attention in combinational optimization. Various meta-heuristic algorithms are dedicated to solve this problem in the literature. Recently, a binary flower pollination algorithm (BFPA) was proposed, which has been successfully applied to solve 0-1 knapsack problem. In this paper, two new time-varying transfer functions are proposed to improve the exploration and exploitation capability of the BFPA with the best solution and short computing time. Based on small, medium, and high-dimensional scales of the knapsack problem, the computational results reveal that the proposed time-varying transfer functions not only to find the best possible solutions but also to have less computational time. Compared to the standard transfer functions, the efficiency of the proposed time-varying transfer functions is superior, especially in the high-dimensional scales.

Knapsack problem is a classical optimization problem. It is widely used in resource allocation problems. We are aware of 0-1 knapsack and fractional knapsack problem. Here, we will discuss about a newer version, unified knapsack problem, which is a combination of above two versions. In 0-1 knapsack problem each item is either be taken or left behind but in fractional knapsack problem a fraction of any item can be taken, while in unified knapsack problem some items can be fractioned and some cannot.KeywordsResource allocationfractional knapsack problemunified knapsack problem0-1 knapsack problem

An adaptive simplified human learning optimization algorithm