Abstract
Due to its NP-hard nature, it is still difficult to find an optimal solution for instances of the binary knapsack problem as small as 100 variables. In this paper, we developed a three-level hyper-heuristic framework to generate algorithms for the problem. From elementary components and multiple sets of problem instances, algorithms are generated. The best algorithms are selected to go through a second step process, where they are evaluated with problem instances that differ in size and difficulty. The problem instances are generated according to methods that are found in the literature. In all of the larger problem instances, the generated algorithms have less than 1 % error with respect to the optimal solution. Additionally, generated algorithms are efficient, taking on average fractions of a second to find a solution for any instance, with a standard deviation of 1 s. In terms of structure, hyper-heuristic algorithms are compact in size compared with those in the literature, allowing an in-depth analysis of their structure and their presentation to the scientific world.
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Abbreviations
- BKP:
-
Binary knapsack problem
- GP:
-
Genetic programming
- GPC++:
-
Genetic programming platform for evolving tree structures of code
- IKL:
-
In the knapsack list; data structure used by the evolved algorithms
- OKL:
-
Out of knapsack list; data structure used by the evolved algorithms
- UC:
-
Uncorrelated instance of the binary knapsack problem
- WC:
-
Weakly correlated instance of the binary knapsack problem
- SC:
-
Strongly correlated instance of the binary knapsack problem
- SS:
-
Subset sum instance of the binary knapsack problem
- FC:
-
Fitness cases i.e. problem instances of the binary knapsack problem used to evolve algorithms
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Acknowledgments
The authors would like to thank the Complex Engineering Systems Institute ICM: P-05-004-F, CONICYT: FBO16, DICYT: 61219-USACH, ECOS/CONICYT: C13E04, STICAMSUD: 13STIC-05.
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Parada, L., Herrera, C., Sepúlveda, M. et al. Evolution of new algorithms for the binary knapsack problem. Nat Comput 15, 181–193 (2016). https://doi.org/10.1007/s11047-015-9483-8
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DOI: https://doi.org/10.1007/s11047-015-9483-8