# An Improved Multi-Objective Grey Wolf Optimizer for Aerodynamic Optimization of Axial Cooling Fans
## 1 Introduction  
Real-world optimization problems are pervasive across all scientific disciplines and engineering applications, often being naturally formulated as optimization problems. These challenges are characterized by their complexity, including nonlinearity, multiple conflicting objectives, discontinuities, high dimensionality, uncertainties, non-convex regions, etc. The primary aim of multi-objective optimization algorithms is to accurately approximate the true Pareto-optimal solutions while maintaining maximal diversity among these solutions [1]. This diversity is crucial as it provides decision-makers with a comprehensive set of design options. Unlike traditional methods that combine multiple objectives into asingle aggregate objective, contemporary multi-objective optimization techniques preserve the distinct multi-objective formulation. This approach allows for the thorough exploration of trade-offs between conflicting objectives, enabling the approximation of the entire Pareto-optimal front within a single optimization process. Nature-inspired algorithms are widely used for multi-objective optimization problems, and they can be broadly categorized into three types: (1) evolutionary algorithms, (2) swarm intelligence algorithms, and (3) physics-based algorithms [2]. Evolutionary algorithms simulate natural evolutionary processes through iterative optimization, where the best individuals are combined to form new generations. This iterative combination promotes population improvement over successive iterations. Among the most prominent evolutionary algorithms are the non-dominated sorting genetic algorithm II (NSGA II) [3], NSGA III [4; 5], and the multi-objective differential evolution algorithm (MODEA) [6]. NSGA II, a well-known evolutionary algorithm for multi-objective optimization, addresses the limitations of its predecessor, NSGA, by incorporating fast non-dominated sorting, an elitist mechanism, and a diversity-preserving operator. It begins with a randomly generated population, classifies individuals into different fronts based on dominance, and determines fitness using front rank and crowding distance. This ensures a diverse set of solutions. Through binary tournament selection, crossover, and mutation, offspring are generated and combined with the parent population. The combined population undergoes non-dominated sorting again, and the best individuals are selected to form the new population. A key feature of NSGA II is its ability to maintain a diverse set of Pareto-optimal solutions via the crowding distance mechanism, which favors individuals in less crowded regions, ensuring a well-distributed Pareto front. This iterative process continues until a termination criterion, typically a maximum number of generations, is met. NSGA II is renowned for its effectiveness in approximating the true Pareto front and maintaining solution diversity, making it a benchmark in multi-objective optimization. Building upon NSGA II, NSGA III introduces enhancements in diversity preservation and efficiency, enabling a more accurate approximation of the Pareto front [4]. However, its performance in high-dimensional objective spaces is somewhat limited in maintaining diversity. The multi-objective differential evolution algorithm (MODEA) modifies the selection scheme of traditional differential evolution to address multi-objective problems, adjusting domination criteria and improving offspring generation before performing domination checks. Despite these improvements, the practical application of differential evolution algorithms for multi-objective optimization, particularly in nutrition decision problems, still requires further enhancement [6]. Physics-based algorithms mimic the fundamental physical laws observed in nature, allowing individuals to communicate and navigate the search space using concepts such as gravitational force, inertia, light refraction, and molecular dynamics.