The 0/1 Multidimensional Knapsack Problem (0/1 MKP) is an interesting NP-hard combinatorial optimization problem that can model a number of challenging applications in logistics, finance, telecommunications and other fields. In the 0/1 MKP, a set of items is given, each with a size and value, which has to be placed into a knapsack that has a certain number of dimensions having each a limited capacity. The goal is to find a subset of items leading to the maximum total profit while respecting the capacity constraints. Even though the 0/1 MKP is well studied in the literature, we can just find a little number of recent review papers on this problem. Furthermore, the existing reviews focus particularly on some specific issues. This paper aims to give a general and comprehensive survey of the considered problem so that it can be useful for both researchers and practitioners. Indeed, we first describe the 0/1 MKP and its relevant variants. Then, we present the detailed models of some important real-world applications of this problem. Moreover, an important collection of recently published heuristics and metaheuristics is categorized and briefly reviewed. These approaches are then quantitatively compared through some indicative statistics. Finally, some synthetic remarks and research directions are highlighted in the conclusion.

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Multiobjective evolutionary algorithms (MOEAs) Crossover operators Multiobjective combinatorial optimization (MOCO) Multiobjective 0/1 knapsack problem (MOKP)In this paper, we first propose a new recombination operator called the two-stage recombination and then we test its performance in the context of the multiobjective 0/1 knapsack problem (MOKP). The proposed recombination operator generates only one offspring solution from a selected pair of parents according to the following two stages. In the first stage, called genetic shared-information stage or similarity-preserving stage, the generated offspring inherits all parent similar genes (i.e., genes or decision variables having the same positions and the same values in both parents). In the second stage, called problem fitness-information stage, the parent non-similar genes (i.e., genes or decision variables having the same positions but different values regarding the two parents) are selected from one of the two parents using some fitness information. Initially, we propose two different approaches for the second stage: the general version and the restricted version. However, the application of the restricted version to the MOKP leads to an improved version which is more specific to this problem. The general and the MOKP-specific versions of the two-stage recombination are compared against three traditional crossovers using two well-known multiobjective evolutionary algorithms. Promising results are obtained. We also provide a comparison between the general version and the MOKP-specific version.

Purpose The purpose of this paper is to provide an improved genetic algorithm to solve 0/1 multidimensional knapsack problem (0/1 MKP), by proposing new selection and crossover operators that cooperate to explore the search space. Design/methodology/approach The authors first present a new sexual selection strategy that significantly improves the one proposed by (Varnamkhasti and Lee, 2012), while working in phenotype space. Then they propose two variants of the two-stage recombination operator of (Aghezzaf and Naimi, 2009), while they adapt the latter in the context of 0/1 MKP. The authors evaluate the efficiency of both proposed operators on a large set of 0/1 MKP benchmark instances. The obtained results are compared against that of conventional selection and crossover operators, in terms of solution quality and computing time. Findings The paper shows that the proposed selection respects the two major factors of any metaheuristic: exploration and exploitation aspects. Furthermore, the first variant of the two-stage recombination operator pushes the search space towards exploitation, while the second variant increases the genetic diversity. The paper then demonstrates that the improved genetic algorithm combining the two proposed operators is a competitive method for solving the 0/1 MKP. Practical implications Although only 0/1 MKP standard instances were tested in the empirical experiments in this paper, the improved genetic algorithm can be used as a powerful tool to solve many real-world applications of 0/1 MKP, as the latter models several industrial and investment issues. Moreover, the proposed selection and crossover operators can be incorporated into other bio-inspired algorithms to improve their performance. Furthermore, the two proposed operators can be adapted to solve other binary combinatorial optimization problems. Originality/value This research study provides an effective solution for a well-known non-deterministic polynomial-time (NP)-hard combinatorial optimization problem; that is 0/1 MKP, by tackling it with an improved genetic algorithm. The proposed evolutionary mechanism is based on two new genetic operators. The first proposed operator is a new and deeply different variant of the so-called sexual selection that has been rarely addressed in the literature. The second proposed operator is an adaptation of the two-stage recombination operator in the 0/1 MKP context. This adaptation results in two variants of the two-stage recombination operator that aim to improve the quality of encountered solutions, while taking advantage of the sexual selection criteria to prevent the classical issue of genetic algorithm that is premature convergence.