Hybrid metaheuristics: integrating Chebyshev map and cuckoo search with tuna swarm optimization

Metaheuristic algorithms have been widely applied to complex optimization problems; however, many still suffer from premature convergence and entrapment in local optimal. The Tuna Swarm Optimization (TSO) algorithm, despite its effectiveness, also faces these limitations. To address this issue, two enhanced variants are proposed: Chebyshev Map-based TSO (TSO-CM) and Cuckoo Search (CS)–integrated TSO-CM (CSTSO-CM). TSO-CM improves population diversity through chaotic dynamics, while the integration of CS strengthens exploration capability, achieving a better balance between exploration and exploitation. The proposed algorithms were evaluated on 23 benchmark functions over 500 iterations with a population size of 30, and each experiment was repeated 30 times independently. Comparative results indicate that TSO-CM outperforms the standard TSO, while CSTSO-CM achieves the best or near-best solutions in the majority of the 23 test functions. Both the Wilcoxon signed-rank and Friedman tests confirm the statistically significant superiority of CSTSO-CM (p < 0.05). Furthermore, applications to gear train and welded beam design problems confirm the practical effectiveness of CSTSO-CM. These findings demonstrate that the proposed approaches successfully overcome the limitations of TSO and provide efficient solutions for challenging real-world optimization tasks.