Table 1 A summary of improved WOA algorithms.

Hybrid other meta heuristics algorithms

Hybrid WOA and GWO (HWGO)

Korashy et al.³³

Implementing a leadership hierarchy of GWO to enhances the exploitative phase of the WOA

Combines WOA with artificial bee colony algorithm and firefly algorithm (WOA-AEFS)

Strumberger et al.³⁶

Trigger ABC’s exploration phase to improve WOA’s exploration phase, etc

Combines the features of whale optimization and lion optimization (WL-optimization)

Saxena et al.⁴⁰

replaces the exploitation phase of Incorporating a mutated Lion operator into WOA to hinder premature trapping of solutions in local optima

Random Walk or Flight Strategy

WOA based on Levy flight and memory (WOALFM)

Zong X et al.⁴¹

The diversity of solutions is expanded through the utilization of Lévy flight and memory strategy

Levy whale optimization algorithm (LWOA)

Chen Z et al.⁴²

Using the Lévy flight strategy instead of the encircling prey technique

A hybrid WOA with integrated symbiotic strategy (HWOAMS)

Li M et al.⁴³

Lévy flight is introduced into the exploration process 

Examining the components of the WOA in light of the evolutionary cores of Gaussian walk, CMA-ES, and evolution strategy(VCSWOA)

Hussien A G et al.⁴⁴

Applying Gaussian walks to augment population diversity

Chaos initialization strategy

Chaotic Whale optimization algorithm (CWOA)

Kaur et al.¹⁰

The inclusion of diverse chaotic maps in WOA allows for the optimization of the main parameter, boosting the harmony between exploration and exploitation

Improved logical whale optimization algorithm (ILWOA)

Si et al.⁴⁵

The initial population of whales is enhanced using an upgraded logistic chaotic mapping approach

A novel version of the WOA (ANWOA)

Elmogy et al.⁴⁶

The utilization of two discrete chaotic maps enables the selection of an optimal initial population, ultimately leading to global optimality