## 3 Establishment of Multi-Objective Optimization Algorithm
### Multi-Objective Beluga Optimization Algorithm (NSGA-III-LBWO)  
Original beluga whale optimization is suitable for solving single-objective optimization problems. When there are multiple optimization objectives (total pressure, efficiency, and sound pressure level), they are usually added in a normalized and weighted manner [25] to transform into a single-objective optimization problem. However, it is not easy to achieve optimal values for each optimization variable through this means. To address the aforementioned limitation, this study proposes a multi-objective beluga whale optimization algorithm (NSGA-III-LBWO), integrating strategies from both LBWO [26] and NSGA-III [27] (Table 3). NSGA-III is a multi-objective evolutionary algorithm that incorporates the Pareto dominance mechanism [28]. The evaluation of individuals is no longer based solely on objective value magnitudes; instead, solution quality is assessed through the dominance relationships defined by the Pareto dominance mechanism. In the population, an individual p is said to dominate another individual q if the condition specified in Equation (11) is satisfied. The Pareto front is defined as the set of objective function values corresponding to the Pareto-optimal solutions, wherein all solutions are mutually non-dominated and no external solution dominates any member of the set. \[\forall i\in(1,2,\ldots,\zeta),f_{i}(p)\leq f_{i}(q)\ \wedge\ \exists l\in(1,2,\ldots,\zeta),f_{l}(p)<f_{l}(q) \tag{11}\]  
where \(f_{i}(X)\) is the value of individual \(X\) on the \(i\)th objective function and \(\zeta\) is the number of objective functions. Compared to the single-objective optimization algorithm, NSGA-III based on non-dominated sorting enhances the exploration capability of the search space. Building upon the framework of NSGA-II, NSGA-III introduces a novel reference-point-based mecha \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Category** & **LBWO** & **NSGA-III** & **NSGA-III-LBWO** \\ \hline Core & Logistic chaotic initialization & Non-dominated sorting + & Hybrid of LBWO and NSGA-III, \\ Mechanism & + Levy flight search & Reference point mechanism & dynamically generates \\ \hline Advantages & 1. Strong global search & 1. High-dimensional multi-objective optimization & 1. Global–local balance \\
2. High population diversity & 2. Uniform solution distribution & 2. High-quality solutions (verified in engineering applications) \\ \hline Limitations & Single-objective, prone to local optima & Slow convergence, weak local search & Parameter sensitivity, moderate computational cost \\ \hline Summary & NSGA-III-LBWO integrates the global search capability of single-target LBWO and the diversity maintenance mechanism of multi-target NSGA-III, realizing the complementary advantages of both. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparative analysis of LBWO, NSGA-III, and NSGA-III-LBWO. nism for evaluating individual solutions. By generating a series of uniformly distributed reference points in space, individuals and reference points are associated, so that some individuals with clustered distribution are eliminated, ensuring that the Pareto front can also maintain uniform distribution characteristics, which can ensure the uniformity and accuracy of the calculation results when analyzing nonlinear problems such as centrifugal fan performance optimization. During the offspring generation phase in NSGA-III-LBWO, a portion of the initial population is derived from the current Pareto-optimal set, while the remainder is generated using the Logistic chaotic mapping defined in Equation (10). The positions of individuals are iteratively updated following the optimization rules of LBWO, with the final updated solutions being retained as the offspring individuals.