۱- Introduction

۲- Whale optimization algorithm

۳- Differential evolution

۴- The proposed approach

۵- Simulation results and discussion

۶- Conclusion

References

Abstract

The whale optimization algorithm (WOA) is a new bio-inspired meta-heuristic algorithm which is presented based on the social hunting behavior of humpback whales. WOA suffers premature convergence that causes it to trap in local optima. In order to overcome this limitation of WOA, in this paper WOA is hybridized with differential evolution (DE) which has good exploration ability for function optimization problems. The proposed method is named Improved WOA (IWOA). The proposed method, combines exploitation of WOA with exploration of DE and therefore provides a promising candidate solution. In addition, IWOA+ is presented in this paper which is an extended form of IWOA. IWOA+ utilizes re-initialization and adaptive parameter which controls the whole search process to obtain better solutions. IWOA and IWOA+ are validated on a set of 25 benchmark functions, and they are compared with PSO, DE, BBO, DE/BBO, PSO/GSA, SCA, MFO and WOA. Furthermore, the effects of dimensionality and population size on the performance of our proposed algorithms are studied. The results demonstrate that IWOA and IWOA+ outperform the other algorithms in terms of quality of the final solution and convergence rate.

Introduction

Nature-inspired algorithms are very popular for solving different problems in various fields such as engineering (Hadavandi, Mostafayi, & Soltani, 2018; Lucas, Nasiri-Gheidari, & Tootoonchian, 2010), bioinformatics (Das, Abraham, & Konar, 2008), economy (Hafezi, Shahrabi, & Hadavandi, 2015) and medicine (Lin, Mimori, & Chen, 2012), for they don’t require prior knowledge about the problems (Ghasemi, Ghavidel, Rahmani, Roosta, & Falah, 2014). Swarm intelligence (SI) is one of the main categories of nature-inspired algorithms which is developed by simulating the social behavior of some simple organisms. Some of the most popular swarm intelligence algorithms are Particle Swarm Optimization (PSO) (Clerc & Kennedy, 2002; Eberhart & Shi, 2004), Krill-Herd (KH) (Gandomi & Alavi, 2012), Moth-flame Optimization (Mirjalili, 2015), Cuckoo Search (CS) (Yang & Deb, 2009), Biogeography-Based Optimization (BBO) (Simon & Member, 2008) and Grey Wolf Optimization (GWO) (Mirjalili, Mirjalili, & Lewis, 2014). The Whale Optimization Algorithm (WOA) (Mirjalili & Lewis, 2016) is a new swarm intelligence algorithm developed by Mirjalili and Lewis and it is inspired from social behavior of humpback whales. WOA searches the global optimum through encircling prey, searching for prey and attacking the prey. The performance of WOA was tested on 29 optimization benchmark functions and some 6 structural design problems. Results indicates the effectiveness of WOA in comparison with some state-of-the-art natureinspired algorithms (Mirjalili & Lewis, 2016). Several studies of WOA are presented which can be divided into two categories, i.e., (1) improving the WOA’s performance and (2) Appling the WOA to solve some optimization problems. Some of WOA’s improvements are (Kaur & Arora, 2017; Ling, Zhou, & Luo, 2017; Sun, Wang, Chen, & Liu, 2018). Kaur and Arora proposed chaotic WOA (CWOA) (Kaur & Arora, 2017) in which the chaos theory is used to tuning the main parameters of WOA to enhance the convergence speed of it. The experimental results on 20 optimization benchmark functions demonstrated that CWOA can improve the performance of WOA. In Ling et al. (2017) Lévy flight trajectory-based WOA (LWOA) is introduced. The LWOA employed Lévy flight trajectory to increase the diversity of population. The results indicated that LWOA outperformed the original WOA. Sun et al. (2018) proposed modified WOA (MWOA) for solving largescale global optimization problems. In MWOA the Lévy flight strategy was employed to improve WOA’s exploration ability.