Abstract
This paper proposes a group theory-based optimization algorithm (GTOA) for knapsack problems, which draws algebraic group operations into the evolution process. The key parts of GTOA are that the feasible solution of the knapsack problem is considered as an element of the direct product of groups and that the evolution process is implemented by multiplication and inverse operations of the direct product of groups. Based on the algorithms for handling infeasible solutions, GTOA is used to solve knapsack problems such as the Set-union knapsack problem, the Discounted {0-1} knapsack problem, and the Bounded knapsack problem. GTOA is validated to be an efficient algorithm for solving knapsack problems. A comparison between GTOA and existing evolutionary algorithms such as genetic algorithm, binary particle swarm optimization, binary artificial bee colony, and their improved variations is conducted and the comparative results show that GTOA has a better performance than other algorithms. In addition, GTOA is not only an efficient algorithm for solving knapsack problems but is also the first paradigm that applies group theory to directly design an evolutionary algorithm.
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