Ternary selection based Hybrid Evolutionary Algorithm for maximizing water retention on magic squares

Water retention on magic square is the optimization version of the classical magic square problem that has recently attracted significant considerations from researchers. Different approaches have been used to handle this challenging combinatorial optimization problem. In this paper, we have proposed two methodologies. The first one is an improved constraint based tabu search approach that mitigates all the identified limitations of the state-of-the-art procedure. The second one is a hybrid evolutionary algorithm that comprises a tabu search and a genetic algorithm. In the proposed tabu search approach three new strategies have been presented such as an incremental calculation of the objective function, greediness reduction technique during variable selection and taking care of long-term cycling problem through a similarity checking method. The hybrid evolutionary algorithm incorporates our tabu search approach for mutation, a ternary selection method for parent selection including a one-point crossover to combine two magic squares. Experimental results show that our proposed algorithm performs better than the state-of-the-art approach with respect to solution quality and computation time. Moreover, our empirical study reveals that our proposed ternary selection method performs better than the traditional parent selection strategies. By using this new ternary selection method in our hybrid evolutionary algorithm we are able to improve our previous retention values for magic square of order 8. © 2022, The Author(s), under exclusive licence to Bharati Vidyapeeth's Institute of Computer Applications and Management.