# An Improved Multi-Objective Grey Wolf Optimizer for Aerodynamic Optimization of Axial Cooling Fans
## 1 Introduction
Prominent algorithms in this category include the multi-objective gravitational search algorithm (MOGSA) [7], multi-objective atom search optimizer (MOASO) [8], multi-objective ray optimization algorithm (MOROA) [9], and multi-objective multi-verse optimizer (MOMOVO) [10]. MOGSA is inspired by one of nature's most fundamental forces: gravity. It combines Pareto optimality principles with operators derived from evolutionary algorithms. While gravitational-based algorithms are known for their efficiency, MOGSA tends to have a slightly higher execution time per iteration compared to conventional methods like particle swarm optimization [7]. MOASO leverages molecular forces to provide enhanced capabilities for tackling complex global optimization problems. MOROA employs [PERSON]'s law of refraction to find global solutions by altering the path of light as it travels through materials with different refractive indices. MOMVO adapts the concepts of the multi-verse optimizer to address multi-objective optimization problems. It integrates an archive mechanism to maintain non-dominated solutions and utilizes a leader selection strategy to guide the optimization process. The algorithm initializes a population of solutions evaluated across multiple objectives and employs Pareto dominance principles to classify solutions and update the archive with the best non-dominated solutions. During optimization, MOMVO dynamically adjusts solution positions by exchanging variables between population solutions and those in the archive, balancing exploration and exploitation of the search space. The algorithm's robustness and ability to maintain solution diversity are enhanced by using multiple operators to induce abrupt changes in solution variables. This makes MOMVO particularly effective for problems with up to three objectives, delivering competitive results in terms of both accuracy and execution time [10]. Swarm intelligence (SI) algorithms are renowned for their decentralized nature, shape-formation capabilities, and self-organization properties. These algorithms are inspired by the collective behaviors of social creatures such as bird flocking, animal herding, and ant foraging. Some of the most popular SI algorithms include the multi-objective particle swarm optimizer (MOPSO) [11], multi-objective cat swarm optimizer (MOCSO) [12], and multi-objective grey wolf optimizer (MOGWO) [13]. MOPSO extends the traditional particle swarm optimization algorithm to handle multi-objective optimization problems. It is a population-based algorithm with relatively simple implementation, incorporating an external archive (referred to as a \"repository\") and a geographically inspired strategy to preserve solution diversity [11]. MOCSO, proposed by [PERSON] and [PERSON] [12], extends the cat swarm optimization algorithm to multi-objective problems. By incorporating both an external archive and the Pareto dominance criterion, it ensures effective preservation and management of non-dominated solutions. MOGWO is a sophisticated algorithm designed specifically for multi-objective optimization by extending the grey wolf optimizer (GWO) [14; 15; 16; 17]. MOGWO incorporates two key components to enhance its functionality: an archive for storing non-dominated solutions and a leader selection mechanism. The archive preserves the best solutions found during the optimization process, balancing diversity and convergence. The leader selection mechanism involves choosing three leaders (\(\alpha\), \(\beta\), and \(\delta\) wolves) from the archive to guide the search agents. This strategy improves the navigation of the search space by effectively balancing the exploration and exploitation phases, resulting in a well-distributed Pareto front. The algorithm's effectiveness is due to its unique mechanisms for maintaining diversity and guiding the search process with the best non-dominated solutions.