Hybrid Genetic Algorithms to Solve the Multidimensional Knapsack Problem

In this paper, we presented a New Hybrid Binary Particle Swarm Optimization (NHBPSO). This hybridization consists at combining some principles of Particle Swarm Optimization (PSO) and Crossover Operation of the Genetic Algorithm (GA). The proposed algorithm is used to solving the NP-hard combinatorial optimization problem of Multidimensional Knapsack Problem (MKP). In the aim to access the efficiency and performance of our NHBPSO algorithm we have tested it on some benchmarks from OR-Library and we have compared our results with the obtained results by the standard binary Particle Swarm Optimization with penalty function technique (PSO-P) algorithm and the quantum version (QICSA) of the new metaheuristic Cuckoo Search. The experimental results show a good and promise solution quality obtained by the proposed algorithm which outperforms the PSO-P and QICSA algorithms.

A two-phase tabu-evolutionary algorithm for the 0–1 multidimensional knapsack problem

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.

Solving the bi-objective multi-dimensional knapsack problem exploiting the concept of core

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.

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.

The multidimensional knapsack problem (MKP) is an NP-hard combinatorial optimization problem whose solution consists of determining a subset of items of the maximum total profit that does not violate capacity constraints. Due to its hardness, large-scale MKP instances are usually a target for metaheuristics, a context in which effective feasibility maintenance strategies are crucial. In 1998, Chu and Beasley proposed an effective heuristic repair that is still relevant for recent metaheuristics. However, due to its deterministic nature, the diversity of solutions such heuristic provides is not sufficient for long runs. As a result, the search ceases to find new solutions after a while. This paper evaluates the use of efficiency groups to define a randomization strategy for the heuristic repair that increases the variability of the repaired solutions, without deteriorating quality and improves the overall results. We compared our randomized heuristic repair against the original one in 270 or-library instances, with improvements at the running time and solution quality found for many of them.

In recent years, with the rapid development of internet, digital data has been widely used in various science and engineering applications. But, this development has also led to several threats and attacks from the intruders. Most of the organizations are very much concerned about the Illegal copyright activities which have become a series issue. A number of copyright protection schemes have been developed in the literature. Watermarking is one of the best known methods for copyright protection. Earlier Bit Error Rate (BER) is been observed to provide good results for choosing the optimal feature region from the watermarked image. However it fails to produce more accurate image quality measures. This research work mainly focuses on a novel technique implemented based on the simulated attacking approach and the GA-based Multi Dimensional Knapsack Problem (MDKP) solving technique to select the most sufficient feature regions for robust digital image watermarking without affecting the image quality. A major limitation in existing methods is the security problems which lead to decrease in the integrity of digital watermarking scheme. With that problem, the definite optimization problems (variant problems) cannot be resolved using genetic algorithms. To focus on the above mentioned problems, the proposed work uses two approaches. Chaotic Iterative Process encryption scheme is used to solve the security problems for watermarking image. Next, the optimal feature points can be chosen from the watermarked image by Modified version of Peak Signal Noise Ratio (MPSNR). MPSNR is based on the enhanced quality measurement of which point opposes the predefined and undefined attacks. With those two approaches the PSO algorithm is employed as a search strategy for identifying an optimal feature subset. The experimental results illustrates that comparing with various well-known feature-based techniques, the proposed method shows better performance in robust and more secure digital watermarking.

The evolutionary metaheuristic called scatter search has been applied successfully to optimization problems for several years. In this paper, we apply the scatter search technique to the well-known 0–1 multidimensional knapsack problem. We propose a new relaxation-based diversification generator, which produces an initial population with elite solutions. The computational results obtained for a set of classic and correlated instances clearly show that (1) this generator can also be used as a heuristic for solving the multidimensional knapsack problem; (2) using the population produced by our generator as a starting point for the scatter search algorithm leads to better performance. We also enhance the scatter search algorithm by integrating memory and by using adapted intensification phases. Overall, the results are interesting and competitive compared to other population-based algorithms, such as genetic algorithms.

We develop and test a new approach for generating initial populations for the application of genetic algorithms (GA) to problems in combinatorial optimisation, specifically the multidimensional knapsack problem. We focus the empirical study of our approach on a set of two dimensional knapsack problems (2KP) used in a past study of 2KP algorithm performance. Our proposed approach for initial population generation focuses on generating populations that are stronger in terms of solution quality, solution diversity and in terms of solutions hovering near the border of feasible and infeasible solutions within the problem solution space. We report the results of a Monte Carlo experiment comparing our approach with the traditional initial population generation approach and report the results of computational tests involving 1120 2KP instances that cover a range of problem constraint characteristics. The collective of these computational results show that our proposed approach provides an initial population of sufficient quality and diversity to produce improved convergence to near optimal solutions which can equate to reduced computational burden in applications involving complex computations.

A binary moth search algorithm based on self-learning for multidimensional knapsack problems

In this paper, the problem of jointly allocating bits, subcarriers and power to maximize the total transmit bit rate in an OFDM based cognitive radio (CR) is studied. Moreover, different service requirements of SUs are also considered. Since "underlay" is adopted as the spectrum shared model, secondary users (SU) in the CR system are allowed to utilize any part of frequency band as long as the interference is under the threshold. We formulate this allocation problem as a multidimensional knapsack problem (MKP) which is NP hard. Considering the real time application, a fast genetic algorithm (FGA) is proposed to obtain a suboptimal solution in short time. For comparison, a reduced complexity (CR) algorithm based on greedy algorithm is also analyzed. Simulation results show that the novel algorithm outperforms RC algorithm and achieves a throughput close to the optimal solution.

[EN] In this research, a new model of computing is presented, within the framework of natural computing and computer models inspired by biology: Networks of Genetic Processors (NGP). This new model is based on the one hand, by the family of networks of bio-inspired processors, specifically Networks of Evolutionary Processors (NEP) and Networks of Splicing Processors (NSP), and on the other hand by Genetic Algorithms. We can define the new model as a network of biologically-inspired processors where operations used by processors are crossover and mutation. One of the major interests studying the NGP is the computational power of the operations of crossover and mutation acting together. The NEP is a complete model that uses operations of symbol mutation: insertion, substitution and deletion, the NSP is a complete model that uses splicing operations, but the NEP is no longer a complete model using only substitution operations, as it happens to the NSP if we restrict the context of its splicing rules to the empty context. The study of the new model presented here responds to what happens when we put together in a single model the substitution rules from the NEP (called mutation rules) and the splicing rules with empty context from the NSP (called crossover rules).\n\nWhen we work with networks of biologically-inspired processors there are two basic types of networks, accepting networks and generating networks. These types of networks are mainly used to work at a theoretical level or to solve decision problems. To work on a more practical level such as solving optimization problems, we propose a new type of network, Networks of Genetic processors as Parallel Genetic Algorithms, inspired by the Parallel Genetic Algorithms.\n\nIn this work we prove the computational completeness of our new model by showing that it is equivalent to the Turing machine. We prove the computational completeness of Parallel Genetic Algorithms by using this result and the similarity between the NGP and Parallel Genetic Algorithms. Moreover, we propose a characterization of the Chomsky hierarchy by using the NGP. Here, we simulate every grammar in the language classes of the Chomsky's hierarchy by using a NGP with an small number of processors required for each simulation. Hence, it gives an appreciable idea of the descriptional complexity of the different families of languages.\n\nFinally, in this work there is an experimental study of the behavior of the model for the resolution of some practical problems. First, we design and implement a simulator that allows the execution of networks of Genetic Processors in any of its three defined types: accepting networks, generating networks or as Parallel Genetic Algorithms. This allows us to test the model with different optimization problems. Then, we make a study to see if the new model could solve NP problems in polynomial time. We use the decision problem of Hamiltonian cycle in a graph. Finally, we test the simulator with two optimization problems showing a good computational behavior. The problems are the Multidimensional Knapsack problem and the Traveling Salesman problem.

We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different integer linear programming (ILP)-based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to the LP relaxation and the original problem and then introduce a new core concept for the multidimensional knapsack problem (MKP), which we study extensively. The empirical analysis is then used to develop new concepts for solving the MKP using ILP-based and memetic algorithms. Different collaborative combinations of the presented methods are discussed and evaluated. Further computational experiments with longer run times are also performed to compare the solutions of our approaches to the best-known solutions of another so-far leading approach for common MKP benchmark instances. The extensive computational experiments show the effectiveness of the proposed methods, which yield highly competitive results in significantly shorter run times than do previously described approaches.