Salmon Salar Optimization: A Novel Natural Inspired Metaheuristic Method for Deep-Sea Probe Design for Unconventional Subsea Oil Wells

Abstract

As global energy demands continue to rise, the development of unconventional oil resources has become a critical priority. However, the complexity and high dimensionality of these problems often cause existing optimization methods to get trapped in local optima when designing key tools, such as deep-sea probes. To address this challenge, this study proposes a novel meta-heuristic approach—the Salmon Salar Optimization algorithm, which simulates the social structure and collective behavior of salmon to perform high-precision searches in high-dimensional spaces. The Salmon Salar Optimization algorithm demonstrated superior performance across two benchmark function sets and successfully solved the constrained optimization problem in deep-sea probe design. These results indicate that the proposed method is highly effective in meeting the optimization needs of complex engineering systems, particularly in the design optimization of deep-sea probes for unconventional oil exploration.

1. Introduction

The development of unconventional oil resources is increasingly recognized as a critical component in addressing global energy demands. As conventional oil reserves dwindle and geopolitical factors introduce uncertainties in supply, the ability to access unconventional resources, such as shale oil, oil sands, and ultra-deepwater reserves, has become imperative [1,2]. Although these resources are abundant, they require advanced extraction technologies and sophisticated engineering solutions due to their complex geological formations and challenging extraction environments [3,4].

Deep-sea probes play an indispensable role in the exploration and extraction of unconventional oil resources, particularly in ultra-deepwater environments. These probes must navigate and analyze extreme underwater conditions, characterized by high geological complexity and a hostile operational environment. The design and deployment of deep-sea probes present high-dimensional optimization challenges, where multiple factors—such as structural integrity, sensor accuracy, energy efficiency, and environmental resilience—must be considered simultaneously [5]. The complexity of this task is further amplified by the need to ensure reliable long-term operation under conditions of extreme pressure, temperature fluctuations, and corrosive seawater.

While metaheuristic methods have demonstrated success in solving many complex optimization problems, several challenges remain. These include the risk of getting trapped in local optima, particularly in high-dimensional optimization problems where the search space is vast, making it difficult for algorithms to escape local optima and find the global solution. Additionally, the convergence speed of these algorithms can be slow, especially in high-dimensional settings, where computational resource consumption is substantial, leading to reduced efficiency. In applications, tasks like deep-sea probe design for unconventional oil exploration require highly precise, adaptable, and robust optimization methods capable of handling complex, multi-dimensional environments under extreme conditions. To address these limitations, this study proposes the Salmon Salar Optimization (SSO) algorithm, inspired by the social and cooperative behavior of salmon, as a novel approach to achieving more effective and scalable solutions in high-dimensional optimization tasks.

2. Related Works

Evolutionary algorithms, such as particle swarm optimization (PSO) [6], firefly algorithms [7], differential evolution [8], and the wolf pack algorithm [9], are effective methods for solving high-dimensional optimization problems. PSO is inspired by the social behavior of animals like birds and bees and is used to find global optimal solutions to optimization problems. In PSO, a group of particles represents potential solutions moving through the search space. These particles are initialized with random positions and velocities and are guided by two factors: their personal best position (the best solution each particle has found) and the global best position (the best solution found by any particle in the swarm). At each iteration, particles update their velocities and positions based on these two factors, aiming to move toward more promising regions of the search space. The movement of the particles is controlled by acceleration constants, which determine how much influence the personal and global best positions have on the particle’s trajectory.

In recent years, convex optimization, which deals with optimization problems where the objective function is convex, has gained increasing importance. Other active research areas in optimization include global optimization [10] and multi-objective optimization [11].

Optimization problems are widely applied across various fields, such as supply chain management [12], transportation [13], colored TSP problems [14,15], networks [16], permanent magnets [17], sound classification [18], feature selection [19], risk prediction [20], route design [21], and financial management [22]. As the Internet continues to grow rapidly, the complexity and scale of optimization problems have also increased, making traditional optimization algorithms less suited to modern needs. As a result, researchers continue to develop and improve optimization algorithms to meet these evolving challenges.

In 2015, Gao [23] proposed a selectively informed PSO (SIPSO). In SIPSO, densely connected particles receive information from all their neighbors, while sparsely connected particles follow only the best-performing neighbor. Liang [24] introduced an adaptive PSO based on clustering (APSO-C), which involves two main steps: first, a K-means clustering method is used to divide the swarm into several sub-groups, and second, the inertia weights of individuals are adjusted based on the evaluation of clusters and swarm states. Li [25] proposed a composite PSO with historical memory (HMPSO), in which each particle considers three candidate positions: its historical memory, personal best position, and the global best of the swarm.

In 2016, Pornsing [26] proposed techniques for self-adaptive inertia weight and time-varying adaptive swarm topology to enhance the performance of particle swarm optimization (PSO). Guo [27] introduced several variants of the Bare Bones PSO (BBPSO) algorithm, including pair-wise bare bones PSO (PBBPSO), dynamic local search bare bones PSO (DLS-BBPSO) [28], and hierarchical bare bones PSO (HBBPSO) [29], all aimed at improving the search capabilities of the original BBPSO [30].

In 2017, Xu [31] proposed a chaotic PSO (CPSO) for combinatorial optimization problems, which combines a chaos method with the traditional PSO. In 2018, Tian [32] introduced a modified PSO that incorporates chaos-based initialization and robust update mechanisms. That same year, Guo [28] developed a dynamic allocation bare bones PSO (DABBPSO) and a dynamic reconstruction bare bones PSO (DRBBPSO) to further improve the original BBPSO.

In 2019, Ghasemi [33] proposed the Phasor PSO (PPSO), while Guo [34] introduced a fission-fusion hybrid BBPSO (FBBPSO) for single-objective optimization problems. Xu [35] presented a dimensional learning strategy for PSO.

In 2020, Xu [36] proposed a reinforcement learning-based communication topology for PSO. In 2021, Yamanaka [37] developed a gravitational PSO (SPSO) for multi-modal optimization problems, and Liu [38] introduced a Sigmoid-Function-Based adaptive weighted PSO (SAWPSO). Wang [39] proposed an adaptive granularity learning distributed PSO (AGLDPSO). In 2022, Li [40] introduced a multi-population PSO with neighborhood learning (MPSO-NL), and Tian [41] proposed an electronic transition-based BBPSO for high-dimensional problems. Guo [42] later introduced a twinning strategy for BBPSO, and Zhou [43] proposed an atomic retrospective learning BBPSO (ARBBPSO).

Many researchers have also drawn inspiration from the natural world and developed various metaheuristic algorithms, such as the Dung Beetle Optimizer (DBO) [44], Whale Optimization Algorithm (WOA) [45], Grey Wolf Optimizer (GWO) [46], Harris Hawks Optimization (HHO) [47], African Vultures Optimization Algorithm (AVOA) [48], and Gorilla Troops Optimizer [49]. These algorithms have demonstrated strong performance in solving both benchmark and engineering problems.

With the advancement of artificial intelligence technology, optimization problems have grown in both scale and dimensionality. Traditional optimization algorithms are no longer sufficient to address the wide range of complex optimization challenges. To overcome these issues, this work proposes a new metaheuristic method, Salmon Salar Optimization (SSO). The strength of the SSO method lies in its ability to simulate the social structure and collective behavior of salmon, effectively balancing exploration and exploitation in high-dimensional search spaces. This approach helps the algorithm escape local optima and improves the convergence rate, even in highly complex and constrained environments. These features make SSO particularly well-suited for solving complex engineering optimization problems, such as the design of deep-sea probes for unconventional oil exploration. The major contributions of this work are summarized as follows:

(1) Biologically-Inspired Design: The Salmon Salar Optimization algorithm is a novel high-dimensional optimization method inspired by the collective behavior of salmon in nature. Unlike traditional algorithms, it emulates the migration and cooperative strategies of salmon, making it more effective in tackling high-dimensional optimization problems.

(2) High-Precision Solutions: The SSO algorithm excels in high-dimensional spaces, demonstrating a unique ability to discover highly accurate solutions. It is particularly effective at addressing complex multi-dimensional problems, providing robust solutions across engineering, scientific, and commercial applications.

(3) Adaptiveness: The SSO algorithm exhibits a high degree of adaptiveness, allowing it to dynamically adjust based on the nature and dimensionality of the problem. This adaptability enables it to handle various types of high-dimensional optimization problems without requiring extensive parameter tuning beforehand.

The rest of this paper is organized as follows: Section 2 introduces the design and structure of the Salmon Salar Optimization algorithm; Section 3 presents the experimental details and results and Section 4 provides the conclusion of this work.

3. Materials and Methods

3.1. Salmon Salar Optimization

Drawing inspiration from the structural composition and social behaviors observed in salmon salar populations, the Salmon Salar Optimization (SSO) Algorithm is proposed. The SSO Algorithm derives its conceptual foundation from three principal search objectives: food procurement, breeding habitat identification, and hazard avoidance. Primarily, the food procurement objective corresponds to the imperative for sustenance, enabling the algorithm to pursue optimal solutions within the problem domain to ensure individual health and viability. Subsequently, the breeding habitat identification objective reflects the algorithm’s capacity to discern suitable problem domain solutions in support of reproduction and ensuing population expansion. Lastly, the hazard avoidance objective equips the algorithm with adaptability to volatile problem conditions, enabling it to circumvent potential threats or adverse factors, thereby ensuring search process stability.

Consequently, the SSO amalgamates these three fundamental search strategies to achieve efficient, diversified, and adaptive search processes. This approach, when applied to the resolution of intricate high-dimensional optimization problems, evinces significant potential by not only furnishing robust solutions but also introducing an innovative optimization instrument applicable across diverse domains, encompassing engineering, scientific inquiry, and commercial enterprises. By emulating the efficacious survival strategies exhibited by salmon salar populations, the SSO introduces biologically inspired paradigms into the realm of optimization, promising noteworthy advancements in the solution of complex problem sets.

3.2. The Multi-Purpose Fusion Search Strategy

Considering the complexity of the high-dimensional optimization problem, the Gaussian distribution is used to select the position of the particle in the next iteration. The next position of a salmon is calculated by Equation (1).

$$\begin{array}{c}\alpha =(Salmon+LeaderSalmon)/2\hfill \\ \beta =|Salmon-LeaderSalmon|\hfill \\ Salmon\_candi=Gausi(\alpha ,\beta )\hfill \end{array}$$

(1)

where $Salmon$ is the initial position of a normal salmon; $LeaderSalmon$ contains the food memory, breeding memory, and danger memory. $Gausi(\alpha ,\beta )$ is a Gaussian distribution with a mean $\alpha$ and a standard deviation $\beta$. Since $LeaderSalmon$ has three levels, $Salmon\_candi$ will contain three candidate positions. After the candidate position selection, the next position of a normal $Salmon$ is calculated by Equation (2).

$$\begin{array}{c}candidatelis{t}^{t+1}=[{Salmon}^t,{Salmon\_candi}^{t+1}]\hfill \\ Salmon{t}^{t+1}=FindBest(candidatelis{t}^t,1)\hfill \end{array}$$

(2)

where $candidatelis{t}^{t+1}$ is a list containing the position of a $Salmon$ in the tth generation and three candidate positions in the $(t+1)$th generation, $FindBest(X,1)$ is a function used to find the best position from X. After every $Salmon$ finds a new position in the $(t+1)$th generation, the next position of the $LeaderSalmon$ is calculated by Equation (3).

$$\begin{array}{c}{LeaderSalmon\_candi}^{t+1}=FindBest(Salmon{t}^{t+1},1)\hfill \\ leade{r}_{c}andidatelis{t}^{t+1}=[{LeaderSalmon}^t,{LeaderSalmon\_candi}^{t+1}]\hfill \\ LeaderSalmon{t}^{t+1}=FindBest(candidatelis{t}^t,3)\hfill \end{array}$$

(3)

where $leade{r}_{c}andidatelis{t}^{t+1}$ is a list containing the position of the $LeaderSalmon$ in tth generation and the best position of all Salmons in the $(t+1)$th generation. $FindBest(X,3)$ is a function used to find the best three positions from X.

To better describe the operation mode of SSO, this section details the pseudo-code and flowchart of SSO. The flowchart is shown in Figure 1; pseudo-code of SSO is shown in Algorithm 1.

Algorithm 1 Pseudo-code of SSO.

Require:  $IT$, max iteration time Require:  $func$, fitness function, Require:  $Ran$, searching Range Require:  $Max$, Max generation time Require:  n, number of Salmon    1:  for $gen$=1 to $Max$ do    2:     Randomly generate the initial position of $Salmons$    3:     Find the $LeaderSalmon$ from all $Salmons$    4:     Update the position of $Salmons$, using Equations (1) and (2)    5:     Update the position of $LeaderSalmon$, using Equation (3)    6:     $gen$=$gen$+1    7:  end for    8:  Output: $LeaderSalmon$

4. Results

4.1. Numerical Experiments with CEC2017

To explore the optimization ability of SSO, the CEC2017 benchmark functions are selected in the simulation test. The CEC2017 contains four different groups of types:

• Unimodal Functions, $f_1-f_2$;
• Simple Multimodal Functions, $f_3-f_9$;
• Hybrid Functions, $f_{10}-f_{19}$;
• Composition Functions, $f_{20}-f_{29}$.

Eight state-of-the-art optimization algorithms, DBO, WOA, GWO, HHO, AVOA, GTO, ARBBPSO, and ETBBPSO are selected in the control group. To ensure the fairness of the experiment, all algorithms use the same number of particles with the same number of iterations. To reduce the error caused by chance, all calculations are repeated thirty-seven times. The mean and variance of these thirty-seven times are recorded, and these results are used to evaluate the performance of the algorithms. The parameters of experiments are listed below:

• Population size for all algorithms, 100;
• Max iteration times, 1.000 $\times {10}^5$;
• Dimension, 100;
• Individual runs, 37.
• search range, [−100, +100]

To easily compare the optimization ability of each algorithm, the FE (final error) is used in the results analysis. The FE is defined in Equation (4).

$$\begin{array}{c}FE=|finalgbest-TOV|\hfill \end{array}$$

(4)

where the $finalgbest$ is the gbest value of an algorithm after the last evolution, $TOV$ is the theoretical optimal value of the test function. Obviously, when comparing the two methods, the method with a smaller FE has better optimization performance. Numerical and ranked results of CEC2017 are shown in

Table 1. The simulation results of SSO, DBO, WOA, GWO, HHO, AVOA, GTO, ARBBPSO and ETBBPSO, f1–f8, best results are shwon in bold.

F	Type	SSO	DBO	WOA	GWO	HHO	AVOA	GTO	ARBBPSO	ETBBPSO
1	Mean	2.808 × 104	5.748 × 107	3.515 × 107	3.184 × 1010	2.097 × 108	6.690 × 103	2.604 × 109	2.912 × 104	1.656 × 104
	Std	2.760 × 104	3.705 × 107	1.173 × 107	7.385 × 109	2.181 × 107	9.224 × 103	1.822 × 109	3.091 × 104	2.186 × 104
	Best	1.851 × 101	1.490 × 106	1.560 × 107	2.009 × 1010	1.621 × 108	9.860 × 100	7.170 × 108	1.369 × 102	4.990 × 100
	Worst	8.870 × 104	1.431 × 108	7.225 × 107	5.204 × 1010	2.517 × 108	3.450 × 104	8.877 × 109	1.281 × 105	1.120 × 105
	Rank	3	6	5	9	7	1	8	4	2
2	Mean	1.909 × 1089	1.845 × 10134	8.159 × 10135	6.547 × 10125	1.117 × 1073	1.326 × 1030	3.553 × 10119	9.078 × 10125	7.782 × 10114
	Std	7.879 × 1089	1.122 × 10135	4.955 × 10136	3.976 × 10126	5.615 × 1073	5.600 × 1030	1.423 × 10120	5.522 × 10126	3.073 × 10115
	Best	1.679 × 1068	2.434 × 1098	2.676 × 10110	5.418 × 1097	6.231 × 1059	3.731 × 1013	9.303 × 1099	6.574 × 1084	2.233 × 1088
	Worst	4.579 × 1090	6.822 × 10135	3.014 × 10137	2.418 × 10127	3.375 × 1074	2.681 × 1031	7.773 × 10120	3.359 × 10127	1.718 × 10116
	Rank	3	8	9	6	2	1	5	7	4
3	Mean	1.227 × 106	3.241 × 105	6.068 × 105	2.008 × 105	3.406 × 104	2.034 × 104	2.030 × 106	1.589 × 106	2.934 × 106
	Std	1.186 × 106	1.856 × 104	1.497 × 105	1.767 × 104	8.364 × 103	6.828 × 103	7.150 × 106	6.647 × 105	1.808 × 106
	Best	4.535 × 105	2.578 × 105	3.368 × 105	1.638 × 105	1.796 × 104	8.612 × 103	2.438 × 105	6.560 × 105	1.059 × 106
	Worst	4.888 × 106	3.516 × 105	8.400 × 105	2.421 × 105	5.648 × 104	3.839 × 104	4.276 × 107	3.261 × 106	8.476 × 106
	Rank	6	4	5	3	2	1	8	7	9
4	Mean	1.493 × 102	4.300 × 102	6.164 × 102	2.268 × 103	4.530 × 102	2.495 × 102	1.473 × 103	1.578 × 102	1.683 × 102
	Std	4.601 × 101	7.513 × 101	8.827 × 101	6.743 × 102	5.750 × 101	4.472 × 101	4.581 × 102	5.156 × 101	5.145 × 101
	Best	8.963 × 101	2.765 × 102	4.498 × 102	1.096 × 103	3.622 × 102	1.930 × 102	7.983 × 102	7.623 × 101	7.982 × 101
	Worst	2.934 × 102	5.789 × 102	8.222 × 102	4.021 × 103	5.927 × 102	3.663 × 102	2.757 × 103	3.117 × 102	2.893 × 102
	Rank	1	5	7	9	6	4	8	2	3
5	Mean	8.255 × 102	1.089 × 103	9.223 × 102	5.395 × 102	9.242 × 102	7.989 × 102	8.184 × 102	9.809 × 102	9.808 × 102
	Std	1.336 × 102	1.545 × 102	1.244 × 102	5.188 × 101	4.758 × 101	6.290 × 101	1.499 × 102	1.781 × 102	1.639 × 102
	Best	5.462 × 102	8.141 × 102	7.312 × 102	4.092 × 102	8.221 × 102	6.398 × 102	6.021 × 102	6.248 × 102	6.706 × 102
	Worst	1.081 × 103	1.361 × 103	1.250 × 103	6.650 × 102	1.038 × 103	9.522 × 102	1.393 × 103	1.369 × 103	1.270 × 103
	Rank	4	9	5	1	6	2	3	8	7
6	Mean	3.455 × 101	7.188 × 101	7.737 × 101	2.697 × 101	7.590 × 101	4.148 × 101	7.272 × 101	3.194 × 101	3.806 × 101
	Std	9.455 × 100	9.560 × 100	9.166 × 100	4.047 × 100	3.312 × 100	4.854 × 100	1.090 × 101	5.052 × 100	7.284 × 100
	Best	1.108 × 101	5.496 × 101	6.168 × 101	1.967 × 101	6.720 × 101	3.192 × 101	4.958 × 101	1.972 × 101	2.120 × 101
	Worst	5.381 × 101	9.361 × 101	1.048 × 102	3.501 × 101	8.179 × 101	5.283 × 101	9.305 × 101	4.202 × 101	5.119 × 101
	Rank	3	6	9	1	8	5	7	2	4
7	Mean	8.635 × 102	1.507 × 103	2.561 × 103	1.044 × 103	2.740 × 103	2.101 × 103	1.754 × 103	9.204 × 102	9.485 × 102
	Std	1.510 × 102	3.905 × 102	1.393 × 102	1.037 × 102	8.270 × 101	1.582 × 102	2.081 × 102	1.256 × 102	1.575 × 102
	Best	6.102 × 102	1.036 × 103	2.212 × 103	8.211 × 102	2.560 × 103	1.773 × 103	1.337 × 103	6.931 × 102	6.999 × 102
	Worst	1.147 × 103	3.145 × 103	2.894 × 103	1.309 × 103	2.932 × 103	2.442 × 103	2.239 × 103	1.212 × 103	1.326 × 103
	Rank	1	5	8	4	9	7	6	2	3
8	Mean	7.509 × 102	1.143 × 103	1.069 × 103	5.528 × 102	1.026 × 103	9.013 × 102	8.395 × 102	9.049 × 102	9.439 × 102
	Std	1.620 × 102	1.525 × 102	1.231 × 102	7.252 × 101	5.340 × 101	8.687 × 101	1.798 × 102	1.495 × 102	1.705 × 102
	Best	5.174 × 102	7.631 × 102	8.041 × 102	4.036 × 102	8.413 × 102	7.323 × 102	5.891 × 102	5.801 × 102	5.612 × 102
	Worst	1.132 × 103	1.405 × 103	1.476 × 103	7.012 × 102	1.108 × 103	1.102 × 103	1.533 × 103	1.296 × 103	1.223 × 103
	Rank	2	9	8	1	7	4	3	5	6

,

Table 2. The simulation results of SSO, DBO, WOA, GWO, HHO, AVOA, GTO, ARBBPSO and ETBBPSO, f9–f16, best results are shwon in bold.

F	Type	SSO	DBO	WOA	GWO	HHO	AVOA	GTO	ARBBPSO	ETBBPSO
9	Mean	3.189 × 104	3.533 × 104	3.694 × 104	1.929 × 104	2.902 × 104	2.176 × 104	4.350 × 104	3.872 × 104	3.285 × 104
	Std	8.257 × 103	9.423 × 103	1.313 × 104	1.023 × 104	2.590 × 103	1.672 × 103	1.855 × 104	1.076 × 104	1.002 × 104
	Best	1.398 × 104	1.751 × 104	2.303 × 104	9.616 × 103	2.394 × 104	1.901 × 104	2.294 × 104	1.777 × 104	1.762 × 104
	Worst	4.255 × 104	4.880 × 104	8.197 × 104	4.220 × 104	3.367 × 104	2.774 × 104	7.876 × 104	8.490 × 104	4.980 × 104
	Rank	4	6	7	1	3	2	9	8	5
10	Mean	1.808 × 104	1.735 × 104	2.023 × 104	1.434 × 104	1.783 × 104	1.521 × 104	2.516 × 104	2.355 × 104	2.671 × 104
	Std	7.565 × 103	1.543 × 103	2.902 × 103	2.535 × 103	1.718 × 103	1.573 × 103	6.647 × 103	5.728 × 103	8.382 × 103
	Best	1.069 × 104	1.446 × 104	1.545 × 104	1.057 × 104	1.334 × 104	1.213 × 104	1.270 × 104	1.233 × 104	9.839 × 103
	Worst	3.297 × 104	2.025 × 104	2.601 × 104	2.693 × 104	2.084 × 104	1.979 × 104	3.375 × 104	2.947 × 104	3.364 × 104
	Rank	5	3	6	1	4	2	8	7	9
11	Mean	4.957 × 102	1.867 × 104	6.803 × 103	3.741 × 104	1.861 × 103	1.202 × 103	1.087 × 105	5.786 × 102	3.974 × 103
	Std	1.357 × 102	3.081 × 104	3.448 × 103	1.107 × 104	1.889 × 102	2.255 × 102	9.458 × 104	2.244 × 102	5.112 × 103
	Best	2.573 × 102	2.700 × 103	4.322 × 103	1.755 × 104	1.512 × 103	7.107 × 102	2.676 × 104	2.266 × 102	3.565 × 102
	Worst	7.958 × 102	1.483 × 105	2.567 × 104	6.831 × 104	2.247 × 103	1.602 × 103	5.289 × 105	1.058 × 103	2.318 × 104
	Rank	1	7	6	8	4	3	9	2	5
12	Mean	1.830 × 107	4.235 × 108	6.533 × 108	4.330 × 109	3.029 × 108	1.190 × 107	4.165 × 108	2.783 × 107	4.958 × 107
	Std	1.196 × 107	2.298 × 108	2.408 × 108	2.095 × 109	9.060 × 107	6.478 × 106	1.147 × 108	1.477 × 107	2.681 × 107
	Best	6.615 × 106	7.719 × 107	2.185 × 108	1.473 × 109	1.635 × 108	3.719 × 106	2.026 × 108	7.105 × 106	5.747 × 106
	Worst	5.860 × 107	9.782 × 108	1.079 × 109	9.595 × 109	5.550 × 108	2.737 × 107	7.120 × 108	8.331 × 107	1.144 × 108
	Rank	2	7	8	9	5	1	6	3	4
13	Mean	9.419 × 103	9.405 × 106	9.002 × 104	4.787 × 108	3.105 × 106	3.811 × 104	3.234 × 104	9.378 × 103	7.642 × 103
	Std	1.186 × 104	1.835 × 107	3.417 × 104	4.293 × 108	5.306 × 105	1.059 × 104	1.306 × 104	1.530 × 104	1.233 × 104
	Best	6.528 × 102	1.196 × 105	4.051 × 104	9.201 × 104	2.014 × 106	2.153 × 104	1.419 × 104	2.912 × 102	2.868 × 102
	Worst	3.599 × 104	9.361 × 107	1.596 × 105	1.853 × 109	4.130 × 106	6.888 × 104	6.511 × 104	8.090 × 104	4.816 × 104
	Rank	3	8	6	9	7	5	4	2	1
14	Mean	4.505 × 105	3.598 × 106	1.782 × 106	3.591 × 106	8.700 × 105	2.184 × 105	7.977 × 105	6.176 × 105	1.325 × 106
	Std	2.583 × 105	3.441 × 106	1.059 × 106	2.115 × 106	3.002 × 105	9.406 × 104	6.201 × 105	4.363 × 105	6.751 × 105
	Best	1.425 × 105	1.266 × 105	4.800 × 105	6.837 × 105	3.137 × 105	5.758 × 104	9.775 × 104	1.427 × 105	4.959 × 105
	Worst	1.131 × 106	1.302 × 107	4.792 × 106	9.880 × 106	1.634 × 106	4.261 × 105	3.816 × 106	2.290 × 106	2.803 × 106
	Rank	2	9	7	8	5	1	4	3	6
15	Mean	6.377 × 103	7.555 × 105	1.308 × 105	7.000 × 107	7.655 × 105	2.200 × 104	7.456 × 103	7.748 × 103	7.439 × 103
	Std	5.773 × 103	1.481 × 106	2.144 × 105	1.005 × 108	3.090 × 105	7.872 × 103	4.781 × 103	1.019 × 104	1.312 × 104
	Best	1.849 × 102	4.700 × 104	2.620 × 104	4.352 × 105	1.394 × 105	6.418 × 103	3.032 × 103	1.612 × 102	1.635 × 102
	Worst	2.138 × 104	7.265 × 106	1.054 × 106	3.816 × 108	1.849 × 106	4.234 × 104	2.806 × 104	4.044 × 104	7.599 × 104
	Rank	1	7	6	9	8	5	3	4	2
16	Mean	5.132 × 103	5.921 × 103	7.751 × 103	3.684 × 103	5.377 × 103	4.724 × 103	5.854 × 103	5.957 × 103	6.546 × 103
	Std	1.326 × 103	9.500 × 102	1.853 × 103	6.423 × 102	5.029 × 102	5.282 × 102	1.846 × 103	2.099 × 103	2.552 × 103
	Best	3.655 × 103	3.863 × 103	4.193 × 103	2.205 × 103	4.483 × 103	3.733 × 103	3.794 × 103	3.357 × 103	3.885 × 103
	Worst	9.912 × 103	7.955 × 103	1.340 × 104	4.936 × 103	6.998 × 103	6.212 × 103	9.924 × 103	1.031 × 104	1.209 × 104
	Rank	3	6	9	1	4	2	5	7	8

,

Table 3. The simulation results of SSO, DBO, WOA, GWO, HHO, AVOA, GTO, ARBBPSO and ETBBPSO, f17–f24, best results are shwon in bold.

F	Type	SSO	DBO	WOA	GWO	HHO	AVOA	GTO	ARBBPSO	ETBBPSO
17	Mean	4.331 × 103	5.316 × 103	5.418 × 103	2.500 × 103	4.207 × 103	4.230 × 103	3.994 × 103	4.892 × 103	4.818 × 103
	Std	8.247 × 102	1.054 × 103	7.340 × 102	5.376 × 102	4.684 × 102	4.697 × 102	9.162 × 102	1.001 × 103	1.093 × 103
	Best	2.686 × 103	3.639 × 103	4.328 × 103	1.546 × 103	3.531 × 103	3.003 × 103	2.678 × 103	2.963 × 103	3.252 × 103
	Worst	6.121 × 103	8.138 × 103	7.694 × 103	4.048 × 103	5.525 × 103	4.990 × 103	6.177 × 103	7.715 × 103	7.388 × 103
	Rank	5	8	9	1	3	4	2	7	6
18	Mean	2.582 × 106	5.795 × 106	2.207 × 106	3.302 × 106	2.056 × 106	3.554 × 105	1.581 × 106	3.123 × 106	7.391 × 106
	Std	1.838 × 106	5.146 × 106	7.891 × 105	1.528 × 106	6.659 × 105	1.237 × 105	8.044 × 105	1.601 × 106	4.976 × 106
	Best	7.702 × 105	4.827 × 105	7.334 × 105	7.652 × 105	8.964 × 105	1.826 × 105	4.847 × 105	7.828 × 105	1.886 × 106
	Worst	7.819 × 106	2.227 × 107	4.676 × 106	6.225 × 106	3.805 × 106	6.416 × 105	4.410 × 106	8.323 × 106	2.448 × 107
	Rank	5	8	4	7	3	1	2	6	9
19	Mean	5.543 × 103	1.876 × 106	1.434 × 107	7.688 × 107	3.721 × 106	1.077 × 104	2.169 × 104	7.613 × 103	1.652 × 104
	Std	7.741 × 103	2.501 × 106	6.244 × 106	9.678 × 107	1.493 × 106	8.269 × 103	2.103 × 104	1.145 × 104	1.577 × 104
	Best	1.186 × 102	4.900 × 104	2.480 × 106	5.702 × 106	1.125 × 106	2.071 × 103	1.116 × 103	1.396 × 102	1.491 × 102
	Worst	4.335 × 104	1.205 × 107	2.536 × 107	4.576 × 108	6.880 × 106	4.351 × 104	7.993 × 104	4.363 × 104	5.512 × 104
	Rank	1	6	8	9	7	3	5	2	4
20	Mean	3.053 × 103	3.737 × 103	4.162 × 103	2.184 × 103	3.717 × 103	3.664 × 103	4.850 × 103	3.737 × 103	3.296 × 103
	Std	5.343 × 102	6.889 × 102	5.279 × 102	4.699 × 102	5.790 × 102	4.433 × 102	1.186 × 103	8.359 × 102	1.128 × 103
	Best	1.318 × 103	2.061 × 103	3.318 × 103	1.209 × 103	2.461 × 103	2.420 × 103	2.189 × 103	2.255 × 103	1.451 × 103
	Worst	3.761 × 103	4.976 × 103	5.401 × 103	3.215 × 103	4.876 × 103	4.418 × 103	7.470 × 103	5.557 × 103	6.233 × 103
	Rank	2	7	8	1	5	4	9	6	3
21	Mean	9.983 × 102	1.432 × 103	1.747 × 103	7.743 × 102	1.664 × 103	1.370 × 103	1.011 × 103	1.090 × 103	1.088 × 103
	Std	1.302 × 102	1.335 × 102	2.281 × 102	1.130 × 102	1.967 × 102	1.409 × 102	2.289 × 102	1.394 × 102	1.405 × 102
	Best	8.102 × 102	1.145 × 103	1.290 × 103	6.484 × 102	1.267 × 103	1.049 × 103	7.903 × 102	8.929 × 102	8.240 × 102
	Worst	1.282 × 103	1.724 × 103	2.225 × 103	1.326 × 103	2.080 × 103	1.761 × 103	1.600 × 103	1.577 × 103	1.443 × 103
	Rank	2	7	9	1	8	6	3	5	4
22	Mean	2.166 × 104	1.800 × 104	2.085 × 104	1.476 × 104	1.969 × 104	1.730 × 104	2.692 × 104	2.551 × 104	2.484 × 104
	Std	8.214 × 103	2.021 × 103	3.096 × 103	1.398 × 103	1.251 × 103	1.118 × 103	5.593 × 103	5.254 × 103	8.665 × 103
	Best	1.341 × 104	1.397 × 104	1.660 × 104	1.157 × 104	1.537 × 104	1.492 × 104	1.829 × 104	1.133 × 104	1.276 × 104
	Worst	3.406 × 104	2.250 × 104	2.691 × 104	1.859 × 104	2.211 × 104	2.052 × 104	3.415 × 104	2.974 × 104	3.452 × 104
	Rank	6	3	5	1	4	2	9	8	7
23	Mean	1.203 × 103	1.789 × 103	2.420 × 103	1.100 × 103	2.264 × 103	1.418 × 103	1.436 × 103	1.301 × 103	1.298 × 103
	Std	9.380 × 101	1.926 × 102	2.668 × 102	6.199 × 101	2.421 × 102	1.267 × 102	2.203 × 102	1.001 × 102	1.120 × 102
	Best	1.051 × 103	1.370 × 103	1.855 × 103	9.835 × 102	1.804 × 103	1.134 × 103	1.169 × 103	1.116 × 103	1.046 × 103
	Worst	1.376 × 103	2.222 × 103	3.034 × 103	1.235 × 103	2.739 × 103	1.726 × 103	1.949 × 103	1.479 × 103	1.508 × 103
	Rank	2	7	9	1	8	5	6	4	3
24	Mean	1.753 × 103	2.492 × 103	3.476 × 103	1.530 × 103	3.168 × 103	2.330 × 103	1.952 × 103	1.927 × 103	1.815 × 103
	Std	1.663 × 102	2.807 × 102	4.040 × 102	9.677 × 101	2.494 × 102	1.994 × 102	2.455 × 102	1.607 × 102	1.550 × 102
	Best	1.462 × 103	1.758 × 103	2.537 × 103	1.297 × 103	2.619 × 103	1.899 × 103	1.587 × 103	1.629 × 103	1.484 × 103
	Worst	2.172 × 103	3.261 × 103	4.256 × 103	1.706 × 103	3.718 × 103	2.763 × 103	2.540 × 103	2.250 × 103	2.172 × 103
	Rank	2	7	9	1	8	6	5	4	3

and

Table 4. The simulation results of SSO, DBO, WOA, GWO, HHO, AVOA, GTO, ARBBPSO and ETBBPSO, f25–f29, best results are shwon in bold.

F	Type	SSO	DBO	WOA	GWO	HHO	AVOA	GTO	ARBBPSO	ETBBPSO
25	Mean	7.624 × 102	2.310 × 103	1.113 × 103	2.813 × 103	1.021 × 103	8.076 × 102	1.766 × 103	7.614 × 102	7.745 × 102
	Std	6.221 × 101	3.353 × 103	5.379 × 101	6.758 × 102	7.246 × 101	6.130 × 101	1.783 × 102	5.359 × 101	6.691 × 101
	Best	6.506 × 102	7.359 × 102	1.010 × 103	1.868 × 103	8.443 × 102	6.832 × 102	1.467 × 103	6.328 × 102	6.448 × 102
	Worst	9.385 × 102	1.559 × 104	1.203 × 103	4.875 × 103	1.140 × 103	9.031 × 102	2.123 × 103	8.624 × 102	8.997 × 102
	Rank	2	8	6	9	5	4	7	1	3
26	Mean	1.346 × 104	1.788 × 104	2.922 × 104	9.924 × 103	2.127 × 104	1.854 × 104	1.510 × 104	1.433 × 104	1.445 × 104
	Std	1.774 × 103	3.869 × 103	3.190 × 103	7.893 × 102	1.640 × 103	2.705 × 103	2.739 × 103	1.675 × 103	2.077 × 103
	Best	1.043 × 104	1.018 × 104	2.117 × 104	8.657 × 103	1.867 × 104	1.417 × 104	1.026 × 104	1.085 × 104	1.021 × 104
	Worst	1.875 × 104	2.475 × 104	3.657 × 104	1.159 × 104	2.513 × 104	2.298 × 104	2.191 × 104	1.800 × 104	1.965 × 104
	Rank	2	6	9	1	8	7	5	3	4
27	Mean	5.000 × 102	1.206 × 103	2.317 × 103	1.153 × 103	1.396 × 103	1.265 × 103	1.018 × 103	5.000 × 102	5.000 × 102
	Std	3.785 × 10−4	2.531 × 102	5.599 × 102	9.433 × 101	2.785 × 102	1.821 × 102	1.125 × 102	5.206 × 10−4	5.264 × 10−4
	Best	5.000 × 102	7.054 × 102	1.422 × 103	9.829 × 102	1.056 × 103	9.671 × 102	8.444 × 102	5.000 × 102	5.000 × 102
	Worst	5.000 × 102	1.959 × 103	3.740 × 103	1.351 × 103	2.277 × 103	1.732 × 103	1.268 × 103	5.000 × 102	5.000 × 102
	Rank	1	6	9	5	8	7	4	2	3
28	Mean	5.000 × 102	1.200 × 104	9.275 × 102	4.120 × 103	7.670 × 102	5.559 × 102	1.835 × 103	5.000 × 102	5.000 × 102
	Std	4.422 × 10−4	7.865 × 103	4.799 × 101	1.190 × 103	4.914 × 101	3.666 × 101	3.997 × 102	4.211 × 10−4	5.674 × 10−4
	Best	5.000 × 102	6.750 × 102	8.462 × 102	1.866 × 103	6.391 × 102	5.000 × 102	1.188 × 103	5.000 × 102	5.000 × 102
	Worst	5.000 × 102	2.256 × 104	1.051 × 103	8.132 × 103	8.968 × 102	6.575 × 102	2.845 × 103	5.000 × 102	5.000 × 102
	Rank	1	9	6	8	5	4	7	2	3
29	Mean	3.553 × 103	6.307 × 103	1.093 × 104	4.494 × 103	5.608 × 103	4.691 × 103	6.756 × 103	4.502 × 103	4.514 × 103
	Std	8.641 × 102	1.198 × 103	1.755 × 103	5.604 × 102	6.617 × 102	5.392 × 102	1.300 × 103	9.586 × 102	1.046 × 103
	Best	1.911 × 103	3.721 × 103	8.172 × 103	3.257 × 103	4.514 × 103	3.146 × 103	4.721 × 103	2.849 × 103	3.025 × 103
	Worst	5.768 × 103	9.492 × 103	1.531 × 104	6.276 × 103	6.787 × 103	5.560 × 103	9.779 × 103	7.920 × 103	8.239 × 103
	Rank	1	7	9	2	6	5	8	3	4
Average Rank		2.6207	6.6897	7.2759	4.3793	5.6897	3.5862	5.7931	4.3448	4.6207

. The convergence curves of the experiments are shown in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7.

Numerical Analysis

In a total of nine algorithms, SSO achieved eight first places, nine second places, five third places, two fourth places, three fifth places, and two sixth places, with an average ranking of 2.62, ranking first among all algorithms. Specific results comparison are listed below:

• In $f_1$, The rank of SSO is 3, the first method is AVOA, the difference between the two methods is 76.174%;
• In $f_2$, The rank of SSO is 3, the first method is AVOA, the difference between the two methods is 100%;
• In $f_3$, The rank of SSO is 6, the first method is AVOA, the difference between the two methods is 98.343%;
• In $f_4$, The rank of SSO is 1, the second method is ARBBPSO, the difference between the two methods is 5.42%;
• In $f_5$, The rank of SSO is 4, the first method is GWO, the difference between the two methods is 34.647%;
• In $f_6$, The rank of SSO is 3, the first method is GWO, the difference between the two methods is 21.938%;
• In $f_7$, The rank of SSO is 1, the second method is ARBBPSO, the difference between the two methods is 6.178%;
• In $f_8$, The rank of SSO is 2, the first method is GWO, the difference between the two methods is 26.383%;
• In $f_9$, The rank of SSO is 4, the first method is GWO, the difference between the two methods is 39.515%;
• In $f_{10}$,The rank of SSO is 5, the first method is GWO, the difference between the two methods is 20.694%;
• In $f_{11}$, The rank of SSO is 1, the second method is ARBBPSO, the difference between the two methods is 14.329%;
• In $f_{12}$, The rank of SSO is 2, the first method is AVOA, the difference between the two methods is 34.981%;
• In $f_{13}$, The rank of SSO is 3, the first method is ETBBPSO, the difference between the two methods is 18.873%;
• In $f_{14}$, The rank of SSO is 2, the first method is AVOA, the difference between the two methods is 51.522%;
• In $f_{15}$, The rank of SSO is 1, the second method is ETBBPSO, the difference between the two methods is 14.277%;
• In $f_{16}$, The rank of SSO is 3, the first method is GWO, the difference between the two methods is 28.221%;
• In $f_{17}$, The rank of SSO is 5, the first method is GWO, the difference between the two methods is 42.281%;
• In $f_{18}$, The rank of SSO is 5, the first method is AVOA, the difference between the two methods is 86.234%;
• In $f_{19}$, The rank of SSO is 1, the second method is ARBBPSO, the difference between the two methods is 27.192%;
• In $f_{20}$, The rank of SSO is 2, the first method is GWO, the difference between the two methods is 28.455%;
• In $f_{21}$, The rank of SSO is 2, the first method is GWO, the difference between the two methods is 22.436%;
• In $f_{22}$, The rank of SSO is 6, the first method is GWO, the difference between the two methods is 31.869%;
• In $f_{23}$, The rank of SSO is 2, the first method is GWO, the difference between the two methods is 8.565%;
• In $f_{24}$, The rank of SSO is 2, the first method is GWO, the difference between the two methods is 12.753%;
• In $f_{25}$, The rank of SSO is 2, the first method is ARBBPSO, the difference between the two methods is 0.131%;
• In $f_{26}$, The rank of SSO is 2, the first method is GWO, the difference between the two methods is 26.263%;
• In $f_{27}$, The rank of SSO is 1, the second method is ARBBPSO, the difference between the two methods is 0%;
• In $f_{28}$,The rank of SSO is 1, the second method is ARBBPSO, the difference between the two methods is 0%;
• In $f_{29}$, The rank of SSO is 1, the second method is GWO, the difference between the two methods is 20.934%;

In summary, SSO demonstrates outstanding performance, versatility, efficiency, robustness, and broad applicability in addressing high-dimensional optimization problems.

4.2. Numerical Experiments with CEC2022

To further validate the performance of SSO, CEC2022 was utilized for simulation experiments. The experimental results showed that out of a total of 12 test functions, SSO secured eight first-place rankings, one second-place ranking, and three third-place rankings, with an average ranking of 1.58. Numerical and ranked results of CEC2022 are shown in

Table 5. The simulation results of SSO, DBO, WOA, GWO, HHO, AVOA, and GTO, CEC2022, f1–f7, best results are shwon in bold.

F	Type	SSO	DBO	WOA	GWO	HHO	AVOA	GTO
1	Mean	8.450 × 10−14	9.707 × 100	1.223 × 101	5.130 × 103	1.502 × 100	2.827 × 10−13	5.531 × 104
	Std	3.934 × 10−14	2.158 × 101	1.619 × 101	3.105 × 103	6.404 × 10−1	1.164 × 10−13	2.432 × 105
	Best	5.684 × 10−14	5.684 × 10−14	6.837 × 10−1	8.533 × 102	3.373 × 10−1	1.137 × 10−13	8.955 × 102
	Worst	1.705 × 10−13	8.868 × 101	7.809 × 101	1.232 × 104	2.876 × 100	6.253 × 10−13	1.487 × 106
	Rank	1	4	5	6	3	2	7
2	Mean	2.039 × 100	4.625 × 101	5.740 × 101	6.977 × 101	5.700 × 101	3.560 × 101	5.656 × 101
	Std	1.824 × 100	2.145 × 101	1.793 × 101	2.433 × 101	2.279 × 101	2.364 × 101	1.401 × 101
	Best	1.986 × 10−1	6.377 × 100	6.135 × 100	4.497 × 101	4.462 × 100	9.160 × 10−4	9.619 × 100
	Worst	6.481 × 100	1.207 × 102	9.476 × 101	1.759 × 102	1.454 × 102	6.775 × 101	7.504 × 101
	Rank	1	3	6	7	5	2	4
3	Mean	7.457 × 10−3	1.299 × 101	5.617 × 101	1.395 × 100	3.816 × 101	1.102 × 101	1.680 × 101
	Std	2.705 × 10−2	6.720 × 100	1.235 × 101	1.513 × 100	1.011 × 101	6.929 × 100	9.457 × 100
	Best	1.137 × 10−13	1.712 × 100	3.042 × 101	3.802 × 10−2	2.057 × 101	6.057 × 10−1	2.112 × 100
	Worst	1.232 × 10−1	2.645 × 101	8.445 × 101	6.357 × 100	6.284 × 101	2.599 × 101	5.112 × 101
	Rank	1	4	7	2	6	3	5
4	Mean	5.623 × 101	8.186 × 101	1.172 × 102	3.746 × 101	8.384 × 101	8.659 × 101	7.928 × 101
	Std	2.091 × 101	2.469 × 101	3.671 × 101	1.430 × 101	1.422 × 101	2.900 × 101	2.119 × 101
	Best	2.288 × 101	3.285 × 101	4.975 × 101	1.431 × 101	5.699 × 101	2.885 × 101	4.020 × 101
	Worst	1.224 × 102	1.304 × 102	2.060 × 102	9.085 × 101	1.219 × 102	1.662 × 102	1.598 × 102
	Rank	2	4	7	1	5	6	3
5	Mean	3.504 × 101	3.804 × 102	2.158 × 103	7.464 × 101	1.415 × 103	1.552 × 103	8.221 × 102
	Std	7.383 × 101	3.077 × 102	1.139 × 103	5.327 × 101	2.363 × 102	4.757 × 102	8.107 × 102
	Best	1.791 × 10−1	4.877 × 100	6.416 × 102	1.225 × 100	9.214 × 102	2.568 × 102	4.134 × 101
	Worst	4.034 × 102	1.065 × 103	5.899 × 103	2.777 × 102	1.843 × 103	2.986 × 103	2.835 × 103
	Rank	1	3	7	2	5	6	4

and

Table 6. The simulation results of SSO, DBO, WOA, GWO, HHO, AVOA, and GTO, CEC2022, f8–f12, best results are shwon in bold.

F	Type	SSO	DBO	WOA	GWO	HHO	AVOA	GTO
6	Mean	5.617 × 103	1.832 × 104	4.887 × 103	9.515 × 105	7.657 × 103	3.968 × 103	5.974 × 103
	Std	6.412 × 103	4.973 × 104	5.665 × 103	3.539 × 106	6.873 × 103	4.852 × 103	9.473 × 103
	Best	6.222 × 101	2.311 × 102	1.899 × 102	2.544 × 102	3.994 × 102	1.379 × 102	1.454 × 102
	Worst	2.102 × 104	3.079 × 105	1.867 × 104	1.773 × 107	2.848 × 104	1.861 × 104	5.192 × 104
	Rank	3	6	2	7	5	1	4
7	Mean	3.543 × 101	8.382 × 101	1.520 × 102	4.246 × 101	1.010 × 102	7.208 × 101	1.589 × 102
	Std	1.194 × 101	3.448 × 101	5.237 × 101	1.361 × 101	3.128 × 101	3.763 × 101	8.407 × 101
	Best	2.171 × 101	3.203 × 101	5.223 × 101	2.271 × 101	5.024 × 101	3.137 × 101	6.022 × 101
	Worst	6.577 × 101	1.829 × 102	2.940 × 102	8.749 × 101	1.558 × 102	1.919 × 102	4.697 × 102
	Rank	1	4	6	2	5	3	7
8	Mean	2.105 × 101	4.558 × 101	4.910 × 101	3.523 × 101	3.935 × 101	2.643 × 101	1.045 × 102
	Std	4.800 × 10−1	3.979 × 101	3.129 × 101	3.342 × 101	2.003 × 101	7.248 × 100	7.315 × 101
	Best	2.003 × 101	2.186 × 101	2.824 × 101	2.164 × 101	2.816 × 101	2.108 × 101	3.293 × 101
	Worst	2.244 × 101	1.649 × 102	1.717 × 102	1.471 × 102	1.492 × 102	4.390 × 101	3.282 × 102
	Rank	1	5	6	3	4	2	7
9	Mean	1.653 × 102	1.809 × 102	1.813 × 102	1.921 × 102	1.813 × 102	1.808 × 102	1.809 × 102
	Std	3.039 × 10−13	5.324 × 10−2	5.481 × 10−1	1.362 × 101	3.341 × 10−1	9.752 × 10−9	2.468 × 10−1
	Best	1.653 × 102	1.808 × 102	1.808 × 102	1.808 × 102	1.808 × 102	1.808 × 102	1.808 × 102
	Worst	1.653 × 102	1.809 × 102	1.828 × 102	2.329 × 102	1.821 × 102	1.808 × 102	1.823 × 102
	Rank	1	4	5	7	6	2	3
10	Mean	1.272 × 102	1.206 × 102	1.431 × 103	4.844 × 102	3.147 × 102	2.645 × 102	1.016 × 102
	Std	9.533 × 101	9.312 × 101	1.078 × 103	5.006 × 102	2.695 × 102	2.349 × 102	5.654 × 10−1
	Best	1.525 × 101	1.003 × 102	1.008 × 102	1.003 × 102	5.298 × 101	4.508 × 101	1.007 × 102
	Worst	4.689 × 102	6.317 × 102	3.247 × 103	1.850 × 103	9.701 × 102	8.968 × 102	1.032 × 102
	Rank	3	2	7	6	5	4	1
11	Mean	3.216 × 102	3.426 × 102	2.985 × 102	6.291 × 102	3.496 × 102	3.270 × 102	3.152 × 102
	Std	4.173 × 101	1.483 × 102	9.834 × 101	2.079 × 102	6.459 × 101	4.502 × 101	1.122 × 102
	Best	3.000 × 102	4.547 × 10−13	6.195 × 10−1	3.006 × 102	2.661 × 101	3.000 × 102	3.556 × 10−3
	Worst	4.000 × 102	7.009 × 102	4.001 × 102	1.171 × 103	4.050 × 102	4.000 × 102	7.605 × 102
	Rank	3	5	1	7	6	4	2
12	Mean	2.000 × 102	2.754 × 102	3.029 × 102	2.534 × 102	3.098 × 102	2.616 × 102	2.486 × 102
	Std	2.000 × 10−4	2.589 × 101	4.893 × 101	1.284 × 101	5.648 × 101	2.173 × 101	8.500 × 100
	Best	2.000 × 102	2.459 × 102	2.446 × 102	2.351 × 102	2.517 × 102	2.392 × 102	2.406 × 102
	Worst	2.000 × 102	3.662 × 102	4.432 × 102	2.884 × 102	5.061 × 102	3.453 × 102	2.864 × 102
	Rank	1	5	6	3	7	4	2
Average Rank		1.58	4.08	5.42	4.42	5.17	3.25	4.08

.

4.3. Optimization Problem Formulation for Deep-Sea Probe Design

In this extended design optimization problem for a deep-sea probe, we aim to minimize the total weight of the probe while accounting for various critical design variables and constraints. The problem is formulated with eight design variables, including wall thickness, radius, length, material density, internal pressure capacity, battery energy storage, and sensor diameter. The objective function reflects the total weight, which is influenced by these variables, while the constraints ensure the probe’s structural integrity, energy sufficiency, and appropriate sensor arrangement within the probe’s internal volume. In this optimization problem, we aim to minimize the total weight of the deep-sea probe by adjusting eight design variables. The design variables are shown in Equation (5).

$$\begin{array}{c}\mathbf{x}=[x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8]=[t_s,t_h,r,l,{\rho }_{\mathrm{material}},P_{\max },E_{\mathrm{battery}},d_{\mathrm{sensor}}]\hfill \end{array}$$

(5)

where:

• $t_s$: Wall thickness of the cylindrical section,
• $t_h$: Wall thickness of the end caps,
• r: Radius of the probe,
• l: Length of the probe,
• ${\rho }_{\mathrm{material}}$: Material density,
• $P_{\max }$: Maximum internal pressure capacity,
• $E_{\mathrm{battery}}$: Battery energy storage,
• $d_{\mathrm{sensor}}$: Sensor diameter.

The objective function, which represents the total weight of the probe, is shown in Equation (6).

$$\begin{array}{c}\mathrm{Minimize}f\left(\mathbf{x}\right)=c_1{x}_1{x}_3{x}_4+c_2{x}_2{x}_3^2+c_3{x}_1^2{x}_4+c_4{x}_1^2{x}_3+\hfill \\ c_5{\rho }_{\mathrm{material}}{x}_3^2{x}_4+c_6{E}_{\mathrm{battery}}+c_7{d}_{\mathrm{sensor}}^2\hfill \end{array}$$

(6)

where $c_1,c_2,\dots ,c_7$ are constants related to material properties, gravitational acceleration, and design specifics. The problem is subject to the severe constraints which are shown in Equation (7):

$$\begin{array}{c}\begin{array}{cc}\hfill g_1\left(\mathbf{x}\right)& =-{\displaystyle \frac{P_{\max }r}{t_s}}+{\sigma }_{\mathrm{allowable}}\le 0\hfill \\ \hfill g_2\left(\mathbf{x}\right)& =-{\displaystyle \frac{P_{\mathrm{external}}r}{t_h}}+{\sigma }_{\mathrm{allowable}}\le 0\hfill \\ \hfill g_3\left(\mathbf{x}\right)& =W_{\mathrm{total}}-{\rho }_{\mathrm{water}}g{V}_{\mathrm{displaced}}\le 0\hfill \\ \hfill g_4\left(\mathbf{x}\right)& =-\pi r^{2}l-{\displaystyle \frac{4\pi r^3}{3}}+V_{\mathrm{required}}(d_{\mathrm{sensor}},E_{\mathrm{battery}})\le 0\hfill \\ \hfill g_5\left(\mathbf{x}\right)& =E_{\mathrm{battery}}-E_{\mathrm{required}}\ge 0\hfill \\ \hfill g_6\left(\mathbf{x}\right)& =d_{\mathrm{sensor}}-d_{\max }\le 0\hfill \end{array}\hfill \end{array}$$

(7)

The goal is to find the optimal set of variables $\mathbf{x}$ that minimizes the total weight while satisfying all the constraints. Experimental results are shown in

Table 7. The simulation results of SSO, DBO, WOA, GWO, HHO, AVOA, and GTO, best results are shwon in bold.

Type	SSO	DBO	WOA	GWO	HHO	AVOA	GTO
Mean	5.265 × 102	5.414 × 102	6.669 × 102	5.295 × 102	6.939 × 102	5.396 × 102	5.357 × 102
Std	2.496 × 100	1.613 × 101	5.779 × 101	2.538 × 100	5.889 × 101	1.169 × 101	7.616 × 100
Best	5.248 × 102	5.248 × 102	5.532 × 102	5.251 × 102	5.908 × 102	5.270 × 102	5.264 × 102
Worst	5.327 × 102	5.909 × 102	7.907 × 102	5.324 × 102	8.123 × 102	5.918 × 102	5.563 × 102
Rank	1	5	6	2	7	4	3

. Convergence graph are shown in Figure 8, standard deviation are shown in Figure 9. The experimental results demonstrate that SSO provides high-precision solutions for the Deep-Sea Probe Design Problem.

4.4. Discussion

The Salmon Salar Optimization (SSO) algorithm offers several key advantages in high-dimensional optimization problems, primarily due to its ability to balance exploration and exploitation effectively. By simulating the collective behavior of salmon, the algorithm dynamically adjusts its search strategy, allowing it to explore new areas of the search space while refining known solutions. This enables SSO to avoid local optima, a common issue in high-dimensional problems, and consistently move toward global optimal solutions. The algorithm’s adaptability further enhances its performance by allowing it to adjust key parameters based on the nature and complexity of the problem, ensuring robust and efficient optimization without the need for extensive parameter tuning.

However, in each iteration, SSO performs three fitness function evaluations (FFEs), leading to a significant increase in computational time, particularly for high-dimensional and complex optimization problems. This increased computational load can hinder the algorithm’s efficiency, making it less suitable for large-scale or time-sensitive applications that demand fast convergence. The need to perform multiple FFEs per iteration poses a challenge, as it directly impacts the algorithm’s scalability and practicality in real-world scenarios. One of the primary limitations of SSO lies in this computational cost, which can become a bottleneck in certain applications. Future research should focus on reducing the number of FFEs or optimizing the evaluation process to improve the algorithm’s efficiency without compromising solution accuracy. Addressing this limitation is essential for enhancing SSO’s competitiveness and applicability in complex engineering and scientific problems.

5. Conclusions

A novel metaheuristic approach, Salmon Salar Optimization (SSO), is proposed to address the complex design challenges associated with deep-sea probes, which are crucial for unconventional oil exploration. SSO distinguishes itself as an innovative solution to the intricate issues of high-dimensional optimization by emulating the social structure and collective behavior of salmon populations.

To rigorously evaluate SSO’s effectiveness, a series of simulation experiments were conducted using the challenging CEC2017 benchmark functions. In these benchmarks, SSO was tested against eight historically top-performing algorithms, serving as a stringent control group. The results were remarkable. Out of the nine algorithms considered, SSO achieved an impressive record: eight first-place rankings, nine second-place rankings, five third-place rankings, and several additional top placements. With an average ranking of 2.62, SSO clearly emerged as the leading algorithm among its competitors.

SSO demonstrated its ability to effectively tackle the complex, high-dimensional design challenges associated with deep-sea probes, which are critical for offshore exploration and extraction of unconventional oil resources. By optimizing both the structural and functional aspects of these probes under stringent conditions, SSO has made significant advancements in deep-sea exploration technologies, facilitating the efficient and sustainable development of unconventional oil resources.

In summary, SSO exhibits outstanding characteristics, including high performance, adaptability, efficiency, robustness, and broad applicability to the multifaceted challenges of high-dimensional optimization problems. These qualities underscore SSO’s position as a powerful tool for addressing complex optimization issues and pave the way for promising future research directions.

Future work should focus on further fine-tuning SSO parameters to enhance optimization performance across diverse problem domains. Additionally, investigating SSO’s potential in multi-objective optimization scenarios and extending its application to real-world problem-solving contexts represent exciting research avenues that merit exploration.