Application of new hybrid harmony search algorithms based on cellular automata theory for solving magic square problems

Abstract

Magic square construction is a complex and hard permutation problem of recreational combinatorics with a long history. The complexity level enhances rapidly when the number of magic squares increases with the order of magic square. This paper proposes two hybrid metaheuristic algorithms, so-called cellular harmony search (CHS) and smallest-small-world cellular harmony search (SSWCHS) for solving magic square problems. The inspiration of the CHS is based on the cellular automata (CA) formation, while the SSWCHS is inspired by the structure of smallest-small-world network (SSWN) and CA using the concept of HS. Numerical optimization results obtained are compared with different optimizers in terms of statistical results and number of found feasible solutions. Computational results show that the proposed hybrid optimizers are computationally effective and highly efficient for tackling magic square problems.