*3.1. Conventional Arithmetic Optimization Algorithm*

Arithmetic optimization algorithm (AOA) is a stochastic population-based metaheuristic optimization algorithm proposed by Abualigah et al. [48] in the year 2021. The algorithm is motivated by the distribution behavior of four key arithmetic operators in the field of mathematics, which includes addition, subtraction, multiplication, and division. In the area of science and engineering, there are complex, non-convex, and high dimension problems, which are difficult to solve using conventional gradient-based optimization algorithms. Metaheuristic is a high-level search algorithm that easily finds the optimal solution for diverse problems without getting stuck in local optimal solution. These algorithms first create a random solution in the search space and iteratively discovers the solution through different search strategies. The phenomenon of how these algorithms update its solution is defined by mathematical behavior of algorithms. Based on these mathematical-concepts, these algorithms are classified as evolutionary, swarm, physics-based, and human-based algorithms. Genetic algorithm (GA), particle swarm optimization (PSO), gravitational search algorithm (GSA), whale optimization algorithm (WOA), and grey wolf optimization (GWO) are some of the metaheuristic algorithms that have efficiently solved non-linear and high-computational engineering design problems. Exploration and exploitation are other unique characteristics that define the functionality of these algorithms. Exploration is defined as the global search capability of the algorithm, while exploitation is defined as the capability of algorithm to explore the nearby promising regions. The efficiency of a metaheuristic algorithm depends on how efficiently the algorithm maintains the balance between exploration and exploitation. AOA uses high and low dispersion nature of arithmetic operators to creates this balance. Multiplication and division operators have high

distributed values, therefore these operators are used in the exploration phase to discover the optimal solution in a diverse region of search space with the following equations:

$$\mathbf{x}\_{i,j}(\mathbb{C}\_{\text{Iter}} + 1) = \begin{cases} \begin{array}{c} \text{best} \left( \mathbf{x}\_{j} \right) \div (\text{MOP} + \boldsymbol{\varepsilon}) \times \left( \left( \mathbb{U}B\_{j} - \mathbb{L}B\_{j} \right) \times \boldsymbol{\mu} + \mathbb{L}B\_{j} \right), \ r\_{2} < 0.5\\ \text{best} \left( \mathbf{x}\_{j} \right) \times \text{MOP} \times \left( \left( \mathbb{U}B\_{j} - \mathbb{L}B\_{j} \right) \times \boldsymbol{\mu} + \mathbb{L}B\_{j} \right), \text{ otherwise} \end{array} \tag{20}$$

where *xi*,*<sup>j</sup>* represents the *j*th position of the *i*th solution, *best xj* is the *j*th position of the best obtained solution, *UB<sup>j</sup>* and *LB<sup>j</sup>* are the upper and the lower bound of the *j*th position, *ǫ* is a constant parameter, *µ* is the control parameter that regulates the search process, and *r*<sup>2</sup> is the random number in the range [0, 1]. The MOP is math optimizer probability and defined as:

$$MOP(\text{C}\_{\text{Iter}}) = 1 - \frac{\left(\text{C}\_{\text{Iter}}\right)^{1\_{\text{f}}}}{\left(M\_{\text{Iter}}\right)^{1\_{\text{f}}}} \tag{21}$$

where *C*Iter represents the current iteration, *M*Iter represents the maximum number of iterations, and *α* is the constant parameter.

Subtraction and multiplication operators have low distributed values, therefore, these operators easily find the optimal solution in the areas that were discovered in the exploration phase. These exploitation operators iteratively reach the solution with the following equations :

$$\mathbf{x}\_{i,j}(\mathbb{C}\_{\text{Iter}} + 1) = \begin{cases} \begin{array}{c} \text{best}(\mathbf{x}\_{j}) - (\text{MOP} + \boldsymbol{\varepsilon}) \times \left( \left( \mathbb{I} \mathbf{B}\_{j} - \mathbb{I} \mathbf{B}\_{j} \right) \times \boldsymbol{\mu} + \mathbb{I} \mathbf{B}\_{j} \right), \ r\_{3} < 0.5\\ \text{best}(\mathbf{x}\_{j}) + \text{MOP} \times \left( \left( \mathbb{I} \mathbf{B}\_{j} - \mathbb{I} \mathbf{B}\_{j} \right) \times \boldsymbol{\mu} + \mathbb{I} \mathbf{B}\_{j} \right), \text{ otherwise} \end{array} \tag{22}$$

where *r*<sup>3</sup> is the random number defined in range [0, 1].

The exploration and exploitation phases are balanced by Math Optimizer accelerated (*MOA*) function, which is defined as:

$$MOA(\mathbf{C\_{Iter}}) = Min + \mathbf{C\_{Iter}} \times \left(\frac{Max - Min}{M\_{\text{Iter}}}\right) \tag{23}$$

where *min* and *max* represent the minimum and the maximum value of the accelerated function. Exploration phase is executed when the value of *r*1, which is a random number in range [0, 1] is greater than MOA, otherwise the exploitation phase is executed.
