## 4 Developed Multi-Objective Optimization Method Based on [PERSON]  
Usually, multi-objective optimization is crucial for simultaneously minimizing multiple conflicting objectives while adhering to system constraints. The introduction of many scholarly works primarily focused on addressing single-objective optimization problems. Few researchers have attempted to tackle multi-objective problems, and in some cases, they could not conclusively prove the effectiveness of their proposed methods for both single- and multi-objective scenarios. In this study, we aim to solve both single- and multi-objective problems simultaneously and present the performance of our proposed HHO method. In essence, the multi-objective function can be defined as follows [35]:  
\[\text{Minimize, }J_{m}(\hat{x}):\sum_{m=1}^{N}\omega_{m}f_{m}(\hat{x})= \text{sum}\left(\left[ \begin{array}{c}\omega_{1}\\ \omega_{2}\\ \cdot\\ \omega_{m}\end{array} \right]\left[ \begin{array}{ccc}f_{1}(\hat{x})&f_{2}( \hat{x})&\dots&f_{m}(\hat{x})\end{array} \right]\right), \tag{20}\]  
subject to the following:  
\[g(\hat{x}) =0\] \[h(\hat{x}) \leq 0\]  
In the above equations, \(J\) represents the multi-objective function comprising m functions, where \(f\) denotes each individual objective function, and \(w\) is a scalar weight applied to each objective, ranging from 0 to 1. Importantly, the sum of all weights is one. To address the multi-objective function in this study, the no-preference weighted sum approach is employed. This approach assigns no particular significance to any of the objectives but employs a heuristic to obtain a single optimal solution. Drawing inspiration from the Harris hawk, a highly intelligent and skilled predator bird in nature known for its remarkable group hunting abilities when pursuing potential prey (such as rabbits), an analogy is made. In this analogy, the initial population is likened to a group of hawks, each attempting to chose the target rabbit (representing the solution to the optimization problem) from various directions using seven distinct hunting strategies or surprise pounces. Initially, the leading hawk tries to capture the prey. If it fails to secure the animal due to the prey's dynamic and evasive behavior, a switch in tactics occurs. Other members of the group (the hawks) then join in the pursuit to capture the escaped prey until it is seized. The key advantage of this collaborative tactic lies in the ability of the hawks to persistently chase the elusive rabbit, leading to its eventual capture through a combination of confusion and exhaustion. In the context of HHO, the candidate solutions are analogous to the [PERSON] hawks themselves, while the optimal or global solution represents the targeted prey [36; 37]. It is essential to note that the objective functions are designed to be either minimized or maximized, depending on the specific problem and how the objective function is defined. It is crucial to understand that solving a multi-objective optimization problem leads to a set of optimal solutions, whereas a single-objective optimization problem results in a single optimal solution. In the case of a bi-objective problem, each element in this set consists of a pair of values. From this standpoint, every element within this collection of optimal solutions is referred to as a Pareto-optimal solution, meaning it is not dominated by or inferior to any other solution in the set. The core principle behind the Pareto-optimal front hinges on the idea that it is impossible to enhance the value of one objective function without causing a detrimental impact on the values of other objective functions. Among the various available multi-objective optimization techniques, the epsilon-constraint method has proven to be exceptionally effective, particularly when compared to the weighted sum approach. In the weighted sum method, all objective functions receive weights and are combined into a single-objective function.