**7. Conclusions**

PSO is a relatively new metaheuristic for global optimization of a multimodal objective function with continuous variables, and has been recognized a standard global optimizer. Although a wealth of efforts have been devoted to improve its convergence speed, solution quality, and algorithm stability, the performance of the existing PSOs are still unsatisfactory. For example, a premature convergence and the loss of diversity are two challenging issues to be addressed for existing PSOs. In this respect, a novel adaptive mutation operator is designed to ensure the diversity of particles in the optimization process, and a dynamic factor is proposed to ensure a good balance between exploration and exploitation searches. The numerical results on mathematical test problems and an engineering application prototype have validated the effectiveness of the proposed PSO algorithm. Consequently, the present work provides a feasible global optimizer for optimizations of multimodal functions with continuous variables.

In future study, we would really want to analyze the convergence problem using a hybrid optimization algorithm (PSO & ABC) and introducing novel formulations for the cognitive and social components, designing novel selection methods for the leader particle, and creating new equations for the personal best particle using the idea of neighborhood. At the same time, we may choose other case studies such as, solenoid problems, as well as using some novel shifted or rotated mathematical test functions.

**Author Contributions:** Conceptualization, R.A.K. and S.Y.; methodology, software and validation, R.A.K. formal analysis, investigation and resources, S.F.; writing—original draft preparation, R.A.K.; writing—review and editing, S.K.; visualization, K.; supervision, S.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not available.

**Informed Consent Statement:** All authors agree.

**Data Availability Statement:** The data that support the study's findings, such as numerical simulation, model, or code generated or used during the study, are available upon request from the journal and the corresponding author.

**Acknowledgments:** Thanks to the China Scholarship Council, Zhejiang University, Hangzhou, China (www.zju.edu.cn, accessed on 5 December 2021) and the University of Science & Technology Bannu, Pakistan (www.ustb.edu.pk, accessed on 5 December 2021) for providing us great research environment during this work.

**Conflicts of Interest:** The corresponding author declares that there is no contradiction on behalf of all authors.
