The specific behaviors of Arctic puffins in the air and on the water form the basis for the design of the APO algorithm. The APO algorithm initializes the population using the following formula:

${\stackrel{\to }{X}}_i^t=rand\ast \left(ub-lb\right)+lb,i=1,2,3,\cdots ,N$ (1)

where ${\stackrel{\to }{X}}_i^t$ represents the position of the ith Arctic Puffin; rand is a random number between 0 and 1; ub and lb represent the upper and lower bounds, respectively; N is the population size.

The Arctic puffins rely on unique flying and foraging strategies to cope with their challenging survival. In their daily lives, they must adapt flexibly between the ocean and the air, meet their nutritional needs, and adjust to different environments. During the aerial foraging phase, the Arctic Puffin employs two key strategies to address different situations, namely the aerial search strategy and the swooping predation strategy.

The aerial search strategy simulates the behavior of the Arctic Puffin searching for suitable foraging waters while in the air, utilizing Levy flight as its powerful wings to change position. When encountering predators such as seagulls, it employs a spiral flight strategy to evade the predators. The following is the position update formula:

${\stackrel{\to }{Y}}_i^{t+1}={\stackrel{\to }{X}}_i^t+\left({\stackrel{\to }{X}}_i^t-{\stackrel{\to }{X}}_r^t\right)\ast L\left(D\right)+R$ (2)

$R=round\left(0.5\ast \left(0.5+rand\right)\right)\ast \alpha$ (3)

$\alpha \text{~}Normal\left(0,1\right)$ (4)

where r is a random integer between 1 and N − 1, excluding i; ${\stackrel{\to }{X}}_i^t$ represents the current ith candidate solution in the population; ${\stackrel{\to }{X}}_r^t$ is a candidate solution randomly selected from the current population, with ${\stackrel{\to }{X}}_i^t\ne {\stackrel{\to }{X}}_r^t$; L(D) denotes a random number generated through Levy flight; D is the dimensionality; $\alpha$ is a random number following a standard normal distribution.

The swooping predation strategy simulates the behavior of the Arctic Puffin rapidly changing its flight direction to dive and feed upon spotting prey. To ensure their survival, they must ensure faster and more successful capture of their prey. To simulate this diving behavior, the algorithm introduces a velocity coefficient S to adjust the displacement of the Arctic Puffin during the dive process. The following is the position update formula:

${\stackrel{\to }{Z}}_i^{t+1}={\stackrel{\to }{Y}}_i^{t+1}\ast S$ (5)

$S=\tan \left(\left(rand-0.5\right)\ast \pi \right)$ (6)

To achieve the best results under various conditions, the algorithm combines the candidate positions generated by the two strategies, sorts them based on their fitness values, and selects the top N individuals with the best fitness values to form a new population. The equation describing this process is as follows:

${\stackrel{\to }{P}}_i^{t+1}={\stackrel{\to }{Y}}_i^{t+1}\cup {\stackrel{\to }{Z}}_i^{t+1}$ (7)

$new=sort\left({\stackrel{\to }{P}}_i^{t+1}\right)$ (8)

${\stackrel{\to }{X}}_i^{t+1}=new\left(1:N\right)$ (9)

The survival strategy of the Arctic puffin involves two crucial aspects: aerial flight and underwater foraging. The underwater foraging phase consists of three main strategies, each employed under specific circumstances to enhance predation efficiency. These three strategies are gathering foraging, intensifying search, and avoiding predators.

The gathering foraging strategy simulates the cooperative foraging behavior of Arctic puffins, and the following equation describes the location update:

${\stackrel{\to }{W}}_i^{t+1}\{\begin{array}{c}{\stackrel{\to }{X}}_{r_1}^t+F\ast L\left(D\right)\ast \left({\stackrel{\to }{X}}_{r_2}^t-{\stackrel{\to }{X}}_{r_3}^t\right)rand\ge 0.5\\ {\stackrel{\to }{X}}_{r_1}^t+F\ast \left({\stackrel{\to }{X}}_{r_2}^t-{\stackrel{\to }{X}}_{r_3}^t\right)rand<0.5\end{array}$ (10)

where F represents the cooperative factor, adjusting the predation behavior of Arctic puffins. In this paper, F = 0.5. The variables $r_1,r_2,r_3$ are random integers between 1 and N – 1 (excluding i), and ${\stackrel{\to }{X}}_{r_1}^t,{\stackrel{\to }{X}}_{r_2}^t,{\stackrel{\to }{X}}_{r_3}^t$ are candidate solutions randomly selected from the current population, and $r_1\ne r_2\ne r_{\text{3}}$, ${\stackrel{\to }{X}}_{r_2}^t\ne {\stackrel{\to }{X}}_{r_3}^t$.

In equation (10), when $rand<0.5$ is present, it represents the cooperative foraging behavior of the Arctic Puffin with other members, utilizing a cooperation factor F and engaging in random motion to explore the surrounding environment. When $rand\ge 0.5$ is present, it signifies a more complex food search strategy where the Arctic Puffin initially follows other members and, upon discovering a school of fish, quickly swims to join a more advantageous predatory group.

The intensified search strategy describes a situation where, as predation proceeds, the food resources in the current foraging area gradually become depleted. To continue meeting their needs, Arctic Puffins must change their underwater position to seek out new food sources. The position update equation for this phase is as follows:

${\stackrel{\to }{Y}}_i^{t+1}={\stackrel{\to }{W}}_i^{t+1}\ast \left(1+f\right)$ (11)

$f=0.1\ast \left(rand-1\right)\ast \frac{\left(T-t\right)}{T}$ (12)

where f is an adaptive factor used to adjust the position of the Arctic puffin in the water. T represents the total number of iterations, and t denotes the current iteration count.

The avoiding predator strategy is used to describe the behavior of an Arctic puffin when it spots a nearby predator, and here is the location update equation used for this strategy:

${\stackrel{\to }{Z}}_i^{t+1}\{\begin{array}{c}{\stackrel{\to }{X}}_{r_1}^t+F\ast L\left(D\right)\ast \left({\stackrel{\to }{X}}_{r_1}^t-{\stackrel{\to }{X}}_{r_2}^t\right)rand\ge 0.5\\ {\stackrel{\to }{X}}_{r_1}^t+\beta \ast \left({\stackrel{\to }{X}}_{r_1}^t-{\stackrel{\to }{X}}_{r_2}^t\right)rand<0.5\end{array}$ (13)

where β is a uniformly distributed number between 0 and 1.

To achieve the best results under various conditions, the algorithm merges the candidate positions generated by the three strategies, sorts them based on their fitness values, and selects the top N individuals with the superior fitness values to form a new population. The equation describing this process is as follows:

${\stackrel{\to }{P}}_i^{t+1}={\stackrel{\to }{W}}_i^{t+1}\cup {\stackrel{\to }{Y}}_i^{t+1}\cup {\stackrel{\to }{Z}}_i^{t+1}$ (14)

$new=sort\left({\stackrel{\to }{P}}_i^{t+1}\right)$ (15)

${\stackrel{\to }{X}}_i^{t+1}=new\left(1:N\right)$ (16)

The APO algorithm favors aerial foraging for global search in the early stages of iteration and underwater foraging for local exploitation in the later stages. This search mechanism is derived from the natural behavior of the Arctic Puffin, which initially searches for suitable foraging waters in the air and then primarily forages underwater. To achieve a smooth transition from global search to local exploitation, the algorithm incorporates a behavior conversion factor, denoted as B. Here is its definition:

$B=2\ast \text{log}\left(\frac{1}{rand}\right)\ast \left(1-\frac{t}{T}\right)$ (17)

The value of B can be dynamically adjusted as the iterations progress to accommodate the exploration needs at different stages. Additionally, a parameter C is introduced within the algorithm to determine the strategy to be executed at the current iteration stage by comparing the values of B and C. In the paper, the parameter C is set to 0.5.

The Elite Reverse Learning Strategy [16] significantly enhances the performance of optimization algorithms by introducing elite particles and a reverse learning mechanism. This strategy endows algorithms with stronger global search capabilities and higher solution accuracy when tackling complex optimization problems and has been widely applied to the improvement of various optimization algorithms.

The optimization performance of an algorithm is greatly influenced by the quality of the initial solutions; a high-quality initial population can accelerate the convergence of the algorithm and facilitate the discovery of the global optimum. However, the APO algorithm uses random initialization for population, which can lead to poor population diversity and slow convergence. To address this issue, this paper applies the Elite Reverse Learning Strategy to the population initialization phase of the algorithm to improve the quality of initial solutions and enhance global search capabilities.

The basic steps for initializing the population using the Elite Reverse Learning Strategy are as follows:

1) Randomly initialize the population S, and select the top N/2 individuals with better fitness values to form the elite population E;

2) Determine the reverse population OE of the elite population E;

3) Merge the populations S and OE to form a new population, and select N individuals with better fitness values to constitute the initial population.

The Tangent Search Algorithm (TSA) [17] , proposed in 2021, is a novel optimization algorithm that introduces a new step size based on a tangent function, denoted as $step\ast \tan \left(\theta \right)$, which is akin to the Levy flight function and is referred to as tangent flight. The search equation that combines global and local wandering for the tangent flight function is as follows:

$X^{t+1}=X^t+step\times \tan \left(\theta \right)$ (18)

In this paper, the tangent flight function is used to replace the levy flight function in the original APO algorithm, and Equation (2) becomes (19):

${\stackrel{\to }{Y}}_i^{t+1}={\stackrel{\to }{X}}_i^t+\left({\stackrel{\to }{X}}_i^t-{\stackrel{\to }{X}}_r^t\right)\ast T\left(D\right)+R$ (19)

Equation (10) becomes (20):

${\stackrel{\to }{W}}_i^{t+1}\{\begin{array}{c}{\stackrel{\to }{X}}_{r_1}^t+F\ast T\left(D\right)\ast \left({\stackrel{\to }{X}}_{r_2}^t-{\stackrel{\to }{X}}_{r_3}^t\right)rand\ge 0.5\\ {\stackrel{\to }{X}}_{r_1}^t+F\ast \left({\stackrel{\to }{X}}_{r_2}^t-{\stackrel{\to }{X}}_{r_3}^t\right)rand<0.5\end{array}$ (20)

Equation (13) becomes (21):

${\stackrel{\to }{Z}}_i^{t+1}\{\begin{array}{c}{\stackrel{\to }{X}}_{r_1}^t+F\ast T\left(D\right)\ast \left({\stackrel{\to }{X}}_{r_1}^t-{\stackrel{\to }{X}}_{r_2}^t\right)rand\ge 0.5\\ {\stackrel{\to }{X}}_{r_1}^t+\beta \ast \left({\stackrel{\to }{X}}_{r_1}^t-{\stackrel{\to }{X}}_{r_2}^t\right)rand<0.5\end{array}$ (21)

Optimizing the step size is also crucial for algorithm optimization; a large step size is conducive to exploration, while a small step size is beneficial for exploitation. Figure 1 and Figure 2 illustrate the random walks of 1000 Levy flights and tangent flights simulated using the Mantegna method [18] .

From the figures, it can be observed that the step sizes generated by Levy flight exhibit poor randomness and a narrow range, which can lead to the algorithm searching with too small a distance in the early iterations and too large a distance in the later iterations. This results in prolonged optimization iteration cycles and insufficient precision. In contrast, tangent flight has a higher probability of producing large steps and a lower probability of producing small steps. This indicates that tangent flight avoids the issues of search distances being either too large or too small. Therefore, tangent flight is more conducive to helping the algorithm escape from local optima and conduct extensive searches, thereby providing the Arctic Puffin with more opportunities for predation.

The reasons for choosing the tangent flight function over other strategies are:

Thus, replacing the Levy flight function in the original APO algorithm with the tangent flight function allows the improved algorithm to not only avoid insufficient local exploitation but also to further enhance the global search capability of the APO algorithm.

The t-distribution, also known as the student distribution, has a probability density function of:

$p\left(x\right)=\frac{\Gamma \left(\frac{m+1}{2}\right)}{\sqrt{m\pi }\Gamma \left(\frac{m}{2}\right)}{\left(1+\frac{x^2}{2}\right)}^{-\frac{m+1}{2}}$ (22)

In the formula, $\Gamma$ represents the gamma function, m is the degrees of freedom parameter, and x is the random variable. The degrees of freedom parameter m influence the shape of the curve. As m approaches infinity, the curve manifests as a Gaussian distribution $N\left(0,1\right)$. When m approaches 1, the curve resembles a Cauchy distribution $C\left(0,1\right)$. The Gaussian and Cauchy distributions are two boundary special cases of the t-distribution. The density function distributions of the three are illustrated in Figure 3 .

The introduction of Gaussian and Cauchy mutations in algorithms has been proven to effectively enhance their performance. Since the Gaussian and Cauchy distributions are two special forms of the t-distribution, incorporating t-distribution mutation into algorithms can combine the advantages of both Gaussian and Cauchy mutations. Therefore, an adaptive t-distribution mutation strategy as shown in Equation (24) is applied to the positions of individuals during the Arctic Puffin’s predator avoidance phase:

$Z_{new}=X_i+X_i\times t\left(A\right)$ (23)

${\stackrel{\to }{Z}}_i^{t+1}\text{=}\{\begin{array}{c}{\stackrel{\to }{Z}}_i^{t+1},rand<0.5\\ Z_{new},rand\ge 0.5\end{array}$ (24)

here, $Z_{new}$ denotes the new position after t-distribution mutation, and $t\left(A\right)$ is the t-distribution with the number of iterations of the APO algorithm as its degrees of freedom. In the early stages of iteration, when the number of iterations is small, the t-distribution mutation resembles Cauchy mutation, which helps maintain population diversity and enhances the algorithm’s global exploration capability. In the later stages of iteration, when the number of iterations is large, the t-distribution mutation resembles Gaussian mutation, which is conducive to fine and stable local exploration around the optimal solution, thereby improving convergence accuracy. The flowchart of ETAAPO algorithm as shown in Figure 4 .

The pseudo code of the ETAAPO is described as follows:

Time complexity is an important measure of an algorithm’s operational efficiency. The time complexity of an algorithm primarily depends on the population size (N), the maximum number of iterations (T), and the dimension (dim). In the APO algorithm, there is an outer loop that runs T times, and within each iteration, an inner loop performs $O\left(N+\dim \right)$ operations for each individual, plus an $O\left(N\right)$operation for updating the global optimum. Therefore, the overall time complexity of the APO algorithm is $O\left(T\ast N\ast \left(N+\dim \right)\right)$.

In the ETAAPO algorithm, the time complexity of the population initialization phase is $O\left(N+\dim \right)$. After initializing the population, the algorithm needs to find the global optimum within the current population, which requires traversing the entire population to determine the best fitness value, making the time complexity of this step $O\left(N\right)$. The core of the algorithm consists of T iterations, with each iteration including a traversal of N individuals, thus the time complexity for this part is $O\left(T\ast N\right)$. At the end of each iteration, the algorithm needs to update the global optimum. The time complexity of this step is $O\left(N\right)$ as it requires traversing the entire population to identify the best fitness value. Combining the above analysis, the overall time complexity of the ETAAPO algorithm is $O\left(T\ast N\ast \left(N+\dim \right)\right)$.

Therefore, the ETAAPO algorithm proposed in this paper does not increase the time complexity and remains consistent with the APO algorithm.

The simulation experimental environment for this study is as follows: the operating system is Windows 10, 64-bit operating system, the processor is AMD Ryzen 7 4800H with Radeon Graphics 2.90 GHz, the memory is 16.0GB, and the simulation software is MATLAB R2022a.

To thoroughly validate the performance of the ETAAPO algorithm, this study selects the Grey Wolf Optimizer (GWO) [19] , Whale Optimization Algorithm (WOA) [20] , Harris Hawks Optimization (HHO) [21] , and the newly published Rime optimization algorithm (RIME) [22] in 2024, which have broad application scopes and good performance in recent years, as comparative algorithms based on the CEC2021 benchmark functions. The ETAAPO algorithm is compared with the original Arctic Puffin Optimization (APO) algorithm and the aforementioned comparative algorithms.

The unimodal test benchmark functions within the test suite are used to assess the local exploitation capabilities of the algorithms, while the multimodal test functions are employed to evaluate their global search capabilities. The specific function details are presented in Table 1 .

The ETAAPO algorithm proposed in this paper was performance tested against APO, GWO, WOA, HHO, and RIME on the ten benchmark test functions listed in Table 1 . In MATLAB R2022a, each experiment was independently executed 30 times with a dimensionality setting of 20 and an iteration count of 500. The evaluation criteria were based on the optimal values, standard deviations, and average values of the results.

Table 2 compiles the test results for the six algorithms, with the best results highlighted in bold. Correspondingly, the average fitness convergence curves for each algorithm are depicted in Figure 5 .

The test results presented in Table 2 demonstrate that the ETAAPO algorithm achieved a 100% optimization effectiveness on functions F3 to F7. Although it did not directly find the optimal values when solving other functions, its optimization accuracy still surpassed that of other comparative algorithms. Compared to other comparative algorithms, the ETAAPO algorithm exhibited a standard deviation of 0 on functions F1, F3 to F7, indicating its strong robustness. Additionally, whether for unimodal or multimodal test functions, the ETAAPO outperformed the other five algorithms not only in terms of optimization accuracy but also in stability.

The performance of an algorithm can be intuitively demonstrated through its convergence curves, which showcase the convergence speed and stability of the algorithm. Figures 5(a)-(j) present a comparison of the convergence curves for six algorithms, including GWO, WOA, HHO, RIME, APO, and ETAAPO, when applied to the aforementioned ten benchmark functions under a 20-dimensional setting. Observing the convergence curves of the aforementioned algorithms, it can be seen that the convergence curve of ETAAPO declines faster than the other five algorithms, and its convergence accuracy is also the best among these six algorithms. This not only indicates that ETAAPO converges faster and has better global search capabilities than the other algorithms, but also that it is less likely to get trapped in local optima, balancing global search capabilities and local development capabilities.

When assessing the performance of an algorithm, it is insufficient to rely solely on the best values, standard deviations, and average values to measure the performance of the improved algorithm. Further statistical testing is required to demonstrate the effectiveness of the Arctic Puffin Optimization algorithm improved with a mixture of strategies and to prove its significant advantages over other existing algorithms. Therefore, this paper employs the Wilcoxon rank-sum test [23] at a significance level of p = 5% and with a dimension of 20.

Table 3 presents the results of the Wilcoxon rank-sum test comparing the optimal values obtained from 30 independent runs of the ETAAPO algorithm with those of the Whale Optimization Algorithm (WOA), Grey Wolf Optimizer (GWO), Harris Hawk Optimizer (HHO), Recursive Interdiction Model for Energy (RIME), and the original APO algorithm. The p-value indicates the test result, and h denotes the significance judgment outcome. When $p<0.05$, it signifies that the ETAAPO algorithm outperforms the compared algorithm; when $p>0.05$, it indicates that the ETAAPO algorithm performs worse than the compared algorithm; “NaN” implies inapplicability, meaning that a significance judgment cannot be made. From the statistical results in the table, it can be observed that for most of the 10 benchmark test functions, the p-values are less than 0.05, indicating a significant difference between the ETAAPO and the compared algorithms, with the ETAAPO demonstrating markedly superior performance.

To validate the avant-garde nature of the ETAAPO algorithm and its superiority in practical engineering applications, this study selects the gear reducer design problem [24] for comparison with several improved algorithms that have demonstrated significant enhancements in recent years. These include the Adaptive Spiral Flight Sparrow Search Algorithm (ASFSSA) [25] , the Grey Wolf Optimizer with enhanced convergence factors and proportional weights (CGWO) [26] , the Nonlinear Chaotic Harris Hawk Optimization (NCHHO) [27] , and the original APO algorithm.

This problem is a relatively classic engineering optimization design challenge, where the optimization objective is to identify a set of seven decision variables that satisfy a range of constraints, including gear bending stress, contact stress, shaft torsional deformation, and stress. The decision variables are as follows: gear width ( $x_1$), gear modulus ( $x_2$), number of teeth on the small gear ( $x_3$), length of bearing 1 ( $x_4$), length of bearing 2 ( $x_5$), diameter of shaft 1 ( $x_6$), and diameter of shaft 2 ( $x_7$), with the goal of minimizing the weight of the reducer. This problem is a mixed-integer programming problem, with variable ( $x_3$) being an integer and all other variables being continuous. The mathematical model of this problem is described as follows:

$\begin{array}{l}\min f\left(x\right)=0.7854x_1{x}_2^2\left(3.3333x_3^2+14.9334x_3-43.0934\right)-1.508\left(x_6^2+x_7^3\right)\\ +7.4777\left(x_6^3+x_7^3\right)+0.7854\left(x_4{x}_6^2+x_5{x}_7^2\right)\end{array}$

$\begin{array}{l}s.t.g_1\left(x\right)=\frac{27}{x_1{x}_2^2{x}_3}-1\le 0\\ g_2\left(x\right)=\frac{397.5}{x_1{x}_2^2{x}_3^2}-1\le 0\\ g_3\left(x\right)=\frac{1.93x_4^3}{x_2{x}_6^4{x}_3}-1\le 0\\ g_4\left(x\right)=\frac{1.93x_5^3}{x_2{x}_7^4{x}_3}-1\le 0\\ g_5\left(x\right)=\frac{{\left[{\left(745x_4/x_2{x}_3\right)}^2+16.9\times {10}^6\right]}^{\text{1/2}}}{\text{110}\text{.0}{x}_6^3}-1\le 0\end{array}$

$\begin{array}{l}{g}_6\left(x\right)=\frac{{\left[{\left(745x_5/x_2{x}_3\right)}^2+157.5\times {10}^6\right]}^{\text{1/2}}}{\text{85}\text{.0}{x}_7^3}-1\le \text{0}\\ g_7\left(x\right)=\frac{x_2{x}_3}{40}-1\le 0\\ g_8\left(x\right)=\frac{5x_2}{x_1}-1\le 0\\ g_9\left(x\right)=\frac{x_1}{12x_2}-1\le 0\\ g_{10}\left(x\right)=\frac{1.5x_6+1.9}{x_4}-1\le 0\\ g_{11}\left(x\right)=\frac{1.1x_7+1.9}{x_5}-1\le 0\end{array}$

$\begin{array}{l}\text{2}\text{.6}\le x_1\le 3.6\\ 0.7\le x_2\le 0.8\\ 17\le x_3\le 28\\ 7.3\le x_4,x_5\le 8.3\\ 2.9\le x_6\le 3.95.0\le x_7\le 5.5\end{array}$

The solution results are presented in Table 4 , which compares the outcomes of the ETAAPO algorithm with those of three other improved algorithms and the original algorithm. From the table, it can be observed that in solving the gear reducer design problem, the best values, mean values, and standard deviations of the ETAAPO are all lower than those of the other four improved algorithms. This indicates that the ETAAPO algorithm has higher solution accuracy and better stability in addressing such problems.