Self-learning salp swarm algorithm for global optimization and its application in multi-layer perceptron model training

Abstract

Optimization problems are common across various fields, and one effective solution is the swarm intelligence algorithm.It is essential for the algorithm to deliver high-quality solutions for problems with varying characteristics. However, most existing swarm intelligence rely on fixed and monotonic search strategies, which limits their ability to handle the diverse and complex situations encountered when solving real-world optimization problems with unknown fitness landscapes. To extend the applicability of swarm intelligence and thus offer users an efficient black-box optimizer for various applications, a novel self-learning mechanism is proposed and applied to the Salp Swarm Algorithm (SSA) to develop the self-learning salp swarm algorithm (SLSSA) in this paper. In SLSSA, four distinct search strategies, including a novel multiple food sources search strategy, are adopted to strengthen the search agents’ abilities to conquer various difficulties in the search space. To improve the efficiency of the search process, the self-learning strategy dynamically determines the execution probability of each search strategy according to the quality of solutions it produced previously. Moreover, a parameter setting method is proposed in this paper, which eliminates the need for a trial-and-error approach and allows for straightforward configuration of the parameters that optimize the performance of SLSSA. In comparison with several highly regarded state-of-the-art peer algorithms, the performance of SLSSA in solving the CEC2014 benchmark functions was thoroughly examined. Subsequently, SLSSA was applied to train multi-layer perceptron classifiers and test on the UCI machine-learning datasets. The experimental results and analysis on benchmark functions and multi-layer perceptron classifier training problems demonstrate that SLSSA outperforms the competing algorithms in terms of solution accuracy, stability, and overall convergence speed. Moreover, computational time comparisons reveal that SLSSA achieves significant performance improvement with only a marginal increase in time cost compared to the original SSA.

Introduction

Inspired by the swarming behaviors of creatures such as birds, bees, ants, and fish, which live in groups and cooperate with other members, Swarm Intelligence (SI) meta-heuristic techniques have been proposed over the past decades. Some well-known SI techniques include Particle Swarm Optimization (PSO)¹, Ant Colony Optimization (ACO)², and Artificial Bee Colony (ABC)³ etc. were proposed in the earlier time. The recent important SI algorithm examples are Salp Swarm Algorithm (SSA)^4–6, Cuckoo Search Algorithm⁷, Grey Wolf Optimizer (GWO)⁸, Whale Optimization Algorithm (WOA)⁹, Moth-Flame Optimization (MFO)¹⁰, Honey Bee Mating Algorithm (HBMA)¹¹, Simplified Swarm Optimization (SSO)¹², Firefly Algorithm (FA)¹³, Liver Cancer Algorithm (LCA)¹⁴, and Dragonfly Algorithm (DA)¹⁵. Their ease of implementation and effectiveness in addressing diverse problems have propelled the widespread adoption of these SI techniques for solving practical optimization problems^10,16–19.

Salp swarm algorithm (SSA) is a recent SI optimizer introduced by Mirjalili⁴ in 2017 with the inspiration of salps’ behavior when navigating and foraging in oceans. SSA is efficient due to several newly adopted stochastic operators and its two distinct search strategies. It significantly outperforms some of the other meta-heuristics and has been used to solve some real-world optimization problems^17,20–22. However, similar to other SI algorithms, SSA and most of its variants adopt the strategy to assign a fixed and monotonic search pattern to each search agent of the population. Due to the different shapes of fitness landscapes, different problems may pose diverse difficulties for optimization algorithms. Even within different subregions of a specific problem, the local fitness landscapes can exhibit strikingly contrasting shapes. The presence of a monotonous search pattern can limit the intelligence of individual search agents, hindering the swarm’s ability to effectively tackle different difficulties in the search space. Consequently, when confronted with complex optimization problems, SSA still suffers from a slow convergence rate and low accuracy.

In recent years, there has been a significant demand for an effective optimization tool for problems with various characteristics. This demand arises from its capacity to support researchers in concentrating on their primary work by offering a single optimization tool for various problems, eliminating the need to invest valuable time in selecting from various algorithms, each with its own limitations. In order to improve the intelligence of the search agents and thereby improve the capabilities of the algorithms to handle the optimization problems with various characteristics, the idea of self-learning or self-adaptive search has been proposed and integrated into the SI algorithms. The self-learning strategy enables an algorithm to learn from the search history and adjust the main search patterns to effectively navigate the search space with an acceptable increase in computation time. By employing the self-learning strategy, the search agents within an algorithm gain the ability to select their own search patterns from multiple search strategies, adapting to diverse situations based on collective past search experiences. This approach has demonstrated its effectiveness in enabling algorithms to achieve superior performance across problems with various characteristics²³. Nevertheless, to design an exceptional self-learning-based algorithm, researchers must address the following three challenges:

1. How can one select or design appropriate multiple search patterns for the self-learning approach? On one hand, each chosen search strategy must be effective and efficient in addressing optimization problems independently. On the other hand, given that the self-learning approach entails additional computational time and storage space, the selected multiple strategies must offer distinct strengths to prevent redundancy in search patterns.

2. How can a suitable scheme be devised for search agents to effectively learn from past search experiences? By enabling the self-learning-based algorithm to learn from previous experiences, search agents can be guided towards selecting search strategies with superior performance, thereby enhancing overall search efficiency. The implementation of the learning scheme is crucial in determining the most appropriate search strategy for solving the given problem.

3. How can the self-learning parameters be appropriately configured? The incorporation of the self-learning approach introduces additional parameters, such as the frequency of learning from past search experiences, to the algorithms. The task of setting these self-learning parameters garners significant attention from researchers. In many existing studies, these newly introduced parameters are tuned empirically²³. However, relying solely on experiential methods inevitably introduces errors. When faced with real-world optimization problems of unknown properties, the absence of a definitive guideline for parameter setting poses challenges. In such cases, resorting to a simple trial-and-error scheme becomes the only viable option, even though computationally expensive and potentially unsuitable for certain application scenarios.

Based on an analysis of existing SSAs and popular self-learning-based algorithms, this paper introduces a novel algorithm called the self-learning salp swarm algorithm (SLSSA). SLSSA tackles three main challenges through the following measures:

1. SSA’s two search patterns emphasizing different search approaches make it a natural fit for self-learning algorithms, which need diverse search strategies. To provide more search strategies with distinct and diverse search capabilities, SLSSA further integrates two search strategies that differ from the existing patterns of SSA. This includes a newly designed search strategy.

2. A self-learning strategy is proposed, utilizing a probability model to determine the frequency at which each search strategy is adopted for updating the search agents. To facilitate “learning,” a novel reward calculation scheme is introduced, offering reasonable rewards to search strategies that successfully improve solutions.

3. A parameter analysis framework, accompanied by guidelines for setting parameters in the self-learning strategy, is presented to enhance the practicality of SLSSA in real-world applications.

To comprehensively evaluate the performance of SLSSA in processing optimization problems with different characteristics such as uni-modality, multi-modality, non-separation, separation, non-symmetrization, symmetrization, rotation, and ill-condition, experiments were conducted on the CEC2014 benchmark suite, which includes 30 numerical test problems for global optimization.

Subsequently, SLSSA is applied to address the multi-layer perceptron (MLP) classifier training problems for obtaining the optimal network parameters of the MLP model. The MLP has been widely utilized to tackle classification challenges across diverse application fields, including the semiconductor industry. For example, MLP classifiers prove instrumental in optimizing molding machine parameters to improve the machine cycle time for integrated circuit packaging companies²³. In another instance, MLP is employed to construct an accurate model with a proper architecture for GaN high electron mobility transistors (HEMTs)²⁴.

We note that a shorter conference version of this paper appeared in²⁵. Contrasting with the conference version, the proposed SLSSA exhibits three primary distinctions: (1) One more search strategy (Generalized oppositional learning) is adopted in the self-learning framework to enhance the algorithm’s search capability. (2) The self-learning strategy is enhanced through the adoption of a different reward calculation method that is based on the ratio of fitness improvement of each search agent to the overall fitness improvement of the population, thereby assigning proper and balanced rewards for each search behavior. (3) Parameter analysis and automatic configuration methods are proposed for the self-learning strategy.

The structure of the remaining sections of this paper is as follows: Related work section provides an introduction to the original SSA and its state-of-the-art improved versions, accompanied by a literature review on the self-learning approach. The proposed SLSSA section offers a detailed description of the proposed SLSSA algorithm. The experimental study on numerical test problems is presented in Experimental study on numerical optimization problems section. Application on MLP classifier training section involves the simulation experiments on MLP classifer training optimization, and finally, Conclusion and future work section is dedicated to summarizing the contributions of our research and outlining future work.

Related work

Salp swarm algorithm

SSA is inspired by the swarming behavior of sea organisms called salps which often float together in a form known as salp chain when navigating and foraging in oceans and seas. SSA has two types of search agents in the salps swarm: a leader and some followers, who obey different search strategies. The leader is at the head of the chain, whereas the followers are the remaining salps at the consecutive positions of the chain after the leader. As the head of the entire population, the leader determines the main movement direction of the chain, and it always moves closer to the direction of the food source. And the global optimum reached so far serves as the food source for the current iteration.

Assuming that $S_{i,j}^t$ and $F_j^t$ are the j-th dimensional positions of i-th salp and the food source, respectively, then the leader’s position is updated by Eq. (1).

$$\begin{array}{c}\hfill \begin{array}{c}\hfill S_{1,j}^{t+1}=\left\{\begin{array}{cc}{F}_j^t+c_1((u{b}_j-l{b}_j)c_2+l{b}_j),\hfill & if{c}_3\ge 0\hfill \\ F_j^t-c_1((u{b}_j-l{b}_j)c_2+l{b}_j),\hfill & if{c}_3<0\hfill \end{array}\right)\end{array}\end{array}$$ 1

where t is the search iteration number, $F_j^t$ is the j-th dimensional position of the best solution found so far in the past $t-1$ iterations, $c_2$ is a value randomly drawn from the interval [0, 1], while $c_3$ is a value randomly generated within the interval [-1, 1], and $u{b}_j$,$l{b}_j$ respectively refer to the upper and lower boundaries of the j-th dimension of the search space, $c_1$ is a crucial parameter for balancing the exploration and exploitation search of the SSA, and it is calculated by the equation as follows:

$$\begin{array}{c}\hfill c_1=2e^{-{(4t/T)}^2}\end{array}$$ 2

where T is the maximum number of search iteration.

It should be noticed that in Eq. (1), i equals 1, which means the leader is at the head of the salp chain.

The position of each follower is updated using Eq. (3).

3

wherei is equal to or larger than 2, indicating that the followers are at the remaining positions of the chain after the leader.

The pseudocode for the SSA is presented in Algorithm 1. In each iteration, after all members of the population are evaluated with the objective function, the best solution obtained so far is set as the new food source.

Algorithm 1.

The pseudocode of SSA.

SSA variants

The efficiency of the original SSA has been proved in comparison with other well-known meta-heuristics algorithms. However, similar to other meta-heuristic algorithms, slow convergence speed and entrapment in local optima are the two problems of SSA²⁶. Therefore, some improved versions of SSA have been reported in the literature. Most of the studies try to improve the performance of SSA from one of the two main aspects: hybridization with other search operators^{17,20–22,27–33} and modified weight models for search agents^34–40.

Due to the inherent advantages offered by combining multiple techniques, there has been a notable trend in recent years towards designing optimizers through the hybridization of two or more search operators from different algorithms. The main approaches adopt running the operators from two or more different algorithms either in parallel or sequentially. A summary illustrating the techniques adopted as the components of the hybrid algorithms is provided in Table 1. In Improved SSA (ISSA)²⁰ and Chaotic SSA with Differential Evolution (CDESSA)²², the operators of differential evolutionary algorithm are used to enhance the performance of SSA. As one of the most popular auxiliary operators for improving the performance of meta-heuristic algorithms, the chaotic local search is adopted to supplement the search patterns of SSA in Opposition-based Chaotic SSA (OCSSA)²⁷. In Improved SSA based on PSO (SSAPSO) approach²⁸, both SSA and PSO are simultaneously employed to update the salp swarm. During each iteration, salps with inferior fitness values are updated according to the rules of SSA, while those with superior fitness values are updated based on the rules of PSO. In Hybrid SSA with Gravitational Search Mechanism (SSA_GSA)²⁹, both SSA and the Gravitational Search Algorithm (GSA) are employed to concurrently update each search agent within the population. Subsequently, the updated agent with the superior fitness value is chosen as a population member for the next iteration. In Improved SSA based on Levy Flight and Sine Cosine Operator (LSC-SSA)³⁰ and Hybrid SSA with Sine Cosine Algorithm (HSSASCA)³¹, the search patterns of both SSA and the sine cosine algorithm (SCA) are combined and used to update the leader salp. In Improved Whale Optimization SSA (IWOSSA)³², SSA and an improved version of whale optimization algorithm (IWOA) are run in parallel with equal probabilities to update the search agents. In Self-adaptive SSA (ASSA)¹⁷, SSA and GWO-CS (the grey wolf optimizer combined with cuckoo search) are run sequentially. During the initial half of the iterations, GWO-CS is applied to update the search agents for exploration, while in the latter half of the iterations, SSA is utilized to update the search agents for exploitation. In Multi-strategy Enhanced SSA (ESSA)⁴¹, the generalized oppositional learning operator, orthogonal learning operator, and quadratic interpolation operator are combined to SSA. Similarly, in Enhanced SSA using Opposition-based Learning Schemes (SSA-OBL)³³, opposition-based learning operator and SSA run alternately under the pre-determined probabilities. In some other research, two search patterns are combined in a heterogeneous manner, such as in Self-adaptive SSA ($SS{A}_{GA-tuner}$)²¹, the genetic algorithm (GA) is used as the optimizer to tune the parameters of SSA.

Table 1.

The hybrid algorithms and their main components.

Algorithm	Components of the hybrid algorithm
ISSA	Differential evolutionary algorithm, SSA
CDESSA	Differential evolutionary algorithm, SSA
OCSSA	Chaotic local search, SSA
SSAPSO	Particle swarm optimization, SSA
SSA_GSA	Gravitational Search Algorithm, SSA
LSC-SSA	Sine cosine algorithm, SSA
HSSASCA	Sine cosine algorithm, SSA
IWOSSA	Whale optimization algorithm, SSA
ASSA	Grey wolf optimizer, Cuckoo search, SSA
ESSA	Generalized oppositional learning operator, Orthogonal learning operator, Quadratic interpolation operator, SSA
SSA-OBL	Opposition-based learning operator, SSA
${\text{SSA}}_{\mathrm{GA}-\mathrm{tuner}}$	Genetic algorithm, SSA

In the implied weight model for the search population of the original SSA, the leader salp in the chain is solely influenced by the food source, while each of the other salps is influenced by both its previous position and the neighbor’s current position under equal weights. Viewed from another perspective, each salp has the capacity to influence subsequent salps with a weight that diminishes as the sequence number of those salps in the population increases. The leader has the ability to influence all the followers and thereby holds the highest overall weight in influencing the search population. If the weight model of search agents is adjusted to incorporate the concept of time-varying, space-varying or adaptive characteristics, it will lead to distinct behaviors among the search agents and consequently impact the algorithm’s performance. Space-varying weight model brings more flexibility to the search patterns, and it is widely adopted in performance improvement attempt for SSA. In Improved SSA (ISSA)³⁴, a higher weight is assigned to the follower’s previous position when it is updated. Similarly, in Improved SSA combined with Chaos (CSSA)⁴⁰, different weights determined by chaotic map are assigned to the individuals involved in the position updates of the followers. Moreover, for obtaining balanced exploration-exploitation properties, the idea of a single high-weighted core in the original SSA is discarded, and the idea of multiple high-weighted individuals, in other words, the scheme of multiple leaders, is adopted in Multiple Leader SSA (MLSSA)³⁷. Other than the scheme of different weights of the search agents at the same iteration, the model of time-varying weight, which decreases linearly through iteration to balance exploration and exploitation for search agents, is also presented in Improved SSA based on Weight Factor and Adaptive Mutation (WASSA)³⁶. The adaptive weight aids in directing attention to diverse search areas and has been embraced in several recent studies. In Rank-driven SSA with Orthogonal Opposition-based Learning (OOSSA)³⁹, the weights assigned to two positions during the update of each follower are determined by the respective rankings of the current search agent and the previous search agent within the salp swarm, as determined by their fitness values. In Efficient SSA (E-SSA)³⁸, the self-adaption weighty scheme is proposed to adjust the weights spontaneously based on the spatial distribution of the whole salp chain.

Most existing SSA variants have simply combined search operators with the SSA or adjusted search patterns using straightforward approaches. While the experimental results of these studies indicate improved performance compared to competing algorithms, there is uncertainty about whether the same level of performance enhancement can be achieved across other optimization problems with varying characteristics.

Self-learning strategy

SI meta-heuristic techniques are empowered by the collective behaviors of search agents that cause coherent functional global patterns. However, no matter how powerful it is, no single operator is optimal for all problems. For a given problem, the optimal choice of operators is time-variant. When no prior knowledge is known, for problems with different properties, the self-learning strategy is principally used to increase the probability of adopting proper search strategies for search agents based on learning from the previous experiences, which refers to the search strategies’ performances have been revealed so far.

Generally, for the self-learning-based algorithms, two main issues must be addressed: (1) determining the appropriate search strategies to be employed, and (2) devising effective methods for learning from past search experiences.

Regarding the first issue, multiple search strategies can be developed by leveraging the search rules of a single baseline algorithm or various baseline algorithms. It is natural to use different search strategies of a single baseline algorithm or the same search strategy with varying parameter settings to construct a pool of multiple search patterns for the self-learning approaches. Such as in Self-adaptive Learning based PSO (SLPSO)²³, four PSO based search strategies are chosen. In Adaptive PSO (APSO)⁴², four parameters settings for the original update equations of PSO are used to provide different search behaviors for the particles. In Differential Evolution Algorithm with Strategy Adaptation (SaDE)⁴³, both trial search strategies and their associated control parameter values are gradually adjusted based on their previous experiences. In Ensemble PSO (EPSO)⁴⁴, an ensemble of different PSO algorithms and a self-adaptive scheme are proposed. Similarly, three different cuckoo search algorithms are adopted as the constituent methods of a multi-strategy adaptive cuckoo search algorithm (MACS)⁴⁵. In Self-learning Differential Evolution Algorithm (SLDE)⁴⁶, six search strategies with different mutation and crossover scheme combinations are designed and used to update the search agents. In Strategy Learning PSO (SLFPSO)¹, a strategy pool is constructed, consisting of ten groups of strategies categorized based on their roles in PSO. The suitable strategy combinations for the specified optimization problems are then selected based on the combination optimization result obtained through the differential evolutionary algorithm. On the other hand, the idea of multiple baseline algorithms combination is also popular. Such as in Self-adaptive Multimethod Search (AMALGAM-SO)⁴⁷, a self-learning strategy is used to automatically tune the number of offspring that the three individual algorithms (CMA-ES, GA, and PSO) are allowed to contribute. As one of the self-learning approach, the OP1 algorithm and its improved versions^48,49 establish an operator pool containing multiple operators derived from different baseline algorithms (PSO, DE, ABC, GWO, BAT, etc.) and use the operators in the pool to update the search individuals.

The issue of learning from previous search experiences and making adaptive adjustments to select a suitable search strategy is crucial to the self-learning strategy. There are two primary approaches to address this issue: (1) an approach that directly uses the performance comparison of multiple search strategies to generate their execution frequencies (referred to as the “direct approach” for ease of subsequent description), and (2) an approach that employs hyper-heuristics to determine the suitable one from multiple search strategies (referred to as the “HH approach” for brevity). Both approaches will be elaborated upon below.

Generally, in the “direct approach”, a fixed number of generations is defined as the learning period, and the performances of the search strategies are recorded during this period. The reward for each search strategy is calculated at the end of each learning period. Subsequently, in the following learning period, the number of search agents that should adopt each search strategy, determined by a probability model in most studies, is updated based on the reward.The normalized probabilities, which are associated with the search strategies, are utilized in selection methods like stochastic universal selection and the roulette wheel selection. These methods are employed to determine the appropriate search strategy for updating each individual search agent^{23,43–45,50}. One main difference between these self-learning schemes relies on the method of calculating rewards for the search strategies. The reward schemes in existing research can be categorized into two classes. The first class is the ranking-based reward scheme, where all the search agents are sorted according to their fitness values in each iteration. The reward for each search strategy is then calculated based on the overall ranking of the search agents that adopt the corresponding strategy²³. The second class is the success-rate-based scheme. In this scheme, the success rate of generating better solutions for the search agents in each iteration is recorded for each strategy. The reward for each search strategy is then calculated based on the overall success rate during a learning period^43,44,46,50. In the previous work of this study presented at the conference²⁵, the reward scheme based on success rate has been adopted to design a self-learning strategy.

Hyper-heuristics stand out from conventional hybrid meta-heuristics by employing a high-level meta-heuristic (HLH) to dynamically select among multiple low-level meta-heuristics (LLHs), thus offering a unique approach to solving the given problem. For instance, in the Tabu Search hyper-heuristic strategy⁵¹, the Tabu search algorithm serves as the high-level meta-heuristic. If a low-level meta-heuristic exhibits poor performance during evaluation, it will be penalized by being added to the Tabu list. This penalty entails being excluded from selection for at least one turn in the subsequent iteration. Similarly, In the reinforcement learning based on the Q-learning mechanism⁵², the Q-learning mechanism is integrated to provide a memory of the performance of each search operator for selection. When the current fitness function is poor, it is capable of keeping track of the best-performing operators for selection. As another application of Q-Learning, in the Q-learning-based hyper-heuristic evolutionary algorithm⁵³, LLHs are treated as selectable states, and state selections are made as available actions. Q-learning is used as an HLS to find the most suitable sequences of heuristics.

Self-learning approaches offer diverse search patterns and are capable of addressing various problems. However, existing self-learning methods have two main drawbacks: first, the complexity of the algorithms has increased; second, they require more parameters, making the tuning of these additional parameters more challenging. As a result, there is still a high demand for a robust self-learning approach that maintains an acceptable level of complexity while simplifying parameter settings.

The proposed SLSSA

The pseudocode of the proposed SLSSA is presented in Algorithm 2. The flowchart of SLSSA is demonstrated in Fig. 1.

Algorithm 2.

The pseudocode of SLSSA.

Fig. 1.

The flowchart of SLSSA.

According to its main ideas, the proposed SLSSA will be described in three subsections outlining (1) the supplementing search strategies, (2) the self-learning strategy based on a probability model, and (3) the parameter setting method.

The supplement search strategies

One common drawback of the existing SI meta-heuristic techniques is premature convergence in searching for the global optima. To enhance search efficiency, the proposed SLSSA integrates two supplement search strategies with different search patterns into SSA. The first is a novel approach introduced in this study called the multiple food sources search strategy (MFSS). The other, generalized oppositional learning (GOL), is inherited from previous research. The details of these two search strategies are provided below.

Multiple food sources search strategy

The best position reached so far is always treated as the search center of the search agents in SI meta-heuristic techniques, exerting a significant influence on the behavior of the search population. Due to the single food source (the search center) strategy adopted in SSA, the entire salp swarm is prone to local stagnation and premature convergence when the food source becomes trapped in a local optimum. To overcome the premature convergence problem of SSA, we design a novel search strategy combining the ideas from both SSA and PSO.

PSO is characterized by maintaining multiple elitist solutions that include a global best solution and several self-best solutions. Each search individual has its own best solution obtained thus far, referred to as pbest, while the entire population of searchers has collectively achieved the best solution so far, denoted as gbest. Illustrated by the trajectory analysis⁵⁴, to ensure PSO’s convergence, each of the search agents (particles) must converge to its local attractor $Q_i^t$, which is defined by the following equation:

$$\begin{array}{c}\hfill Q_i^t=\phi \times pbes{t}_i^t+(1-\phi )\times gbes{t}^t\end{array}$$ 4

where $pbes{t}_i^t$ is the self-best solution obtained so far by the i-th particle, $gbes{t}^t$ is the best solution obtained by the whole search population so far, $\phi$ is a multi-dimensional vector containing the random values uniformly distributed on [0, 1].

Compared with the strategy of a single elite, the strategy of multiple elites in PSO enhances the ability of the algorithm to escape from the local optima. However, due to the loss of diversity caused by the position update equation and the monotonic search strategy, PSO also suffers a potential drawback of premature convergence²³.

Following the idea of migrating the strategy of multiple elitists from PSO to SSA, the multiple food sources search strategy, is designed in this study, and the corresponding equation is given as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill S_{i,j}^{t+1}=\left\{\begin{array}{cc}{Q}_{i,j}^t+c_1((u{b}_j-l{b}_j)c_2+l{b}_j),\hfill & if{c}_3\ge 0\hfill \\ Q_{i,j}^t-c_1((u{b}_j-l{b}_j)c_2+l{b}_j),\hfill & if{c}_3<0\hfill \end{array}\right)\end{array}\end{array}$$ 5

where $Q_{i,j}^t$ is the local attractor defined in Eq. (4).

The incorporation of a local attractor introduces flexible search centers for the search agents. By comparing Eqs. (5) and (1), it becomes apparent that the local attractor is utilized as a food source for the salp population in the proposed search strategy. Regarding different salps, the corresponding food sources are various. Since each search individual has its own personal best solution, the food source of each search individual is generated by using its personal best and the global best of the population, so it is natural and reasonable that the number of food sources is exactly the same as the number of search individuals. In other words, this new search strategy is founded on the concept of multiple food sources. The search strategy, which is founded on multiple food sources, effectively addresses the premature convergence issue that arises from the single elite (food source) strategy of SSA. In our prior conference presentation²⁵, we applied the multiple food sources search strategy of SSA to solve simple numerical optimization problems, demonstrating its competitive performance. Despite the fact that MFSS shares a similar equation with the original leader position update equation of SSA, the distinct search centers of the two strategies ensure different search patterns. While the single search center scheme prioritizes exploitation around the currently identified promising area, the multiple search centers scheme emphasizes exploration towards unknown regions, thereby enhancing the diversity of the swarm population.

Generalized oppositional learning

Oppositional learning serves as an operator for generating a new solution in the opposite direction of the current solution and subsequently selecting the better option between the current solution itself and the newly created solution. The effectiveness of employing a random direction along with its opposite direction in the search for an unknown optimal solution has been demonstrated⁵⁵. Oppositional learning provides an alternative search direction for a current solution and further improves the diversity of the search population. It effectively enhances search efficiency when the optima are in the opposite direction of a current solution⁵⁶.

Among the various implementation methods for the opposition-based learning operator, generalized oppositional learning (GOL) is well-regarded and has been adopted by many researchers to improve the performance of the heuristic optimization algorithms^41,57.

In GOL, the opposite solution ${\mathit{ox}}_i$ is generated using the original $x_i$ and a solution space transformation equation as follows:

$$\begin{array}{c}\hfill o{x}_{i,j}={\theta }_1(u{b}_j+l{b}_j)-x_{i,j}\end{array}$$ 6

where ${\theta }_1$ is a random number uniformly distributed on [0, 1], $u{b}_j$ and $l{b}_j$ are the upper and lower bounds of the search space for dimension j.

From Eq. (6), some dimensions of the generated opposite solution could jump out of the allowable search domain. If this happens, the corresponding dimensions of the opposite solution are randomly initialized in the search domain according to the equation as follows:

$$\begin{array}{c}\hfill o{x}_{i,j}={\theta }_2(u{b}_j-l{b}_j)+l{b}_j\end{array}$$ 7

where ${\theta }_2$ is another random number uniformly distributed on [0, 1].

In this study, the second supplement search strategy is based on the GOL operator. If a search agent is chosen to perform the GOL search, the new position of the search agent in the next iteration is generated with Eq. (6).

The two supplementary search strategies, in conjunction with the two original search strategies of SSA, are integrated to create the search strategy pool of SLSSA. By analyzing the corresponding equations, it is evident that the four search strategies possess their own distinct strengths that cover diverse search patterns.

The proposed self-learning strategy

The self-learning strategy can be implemented by choosing one of the two approaches, the “direct approach” and the “HH approach”, as discussed in the subsection Self-learning strategy. The “direct approach” implements the self-learning strategy in a straightforward manner and has the advantages of fewer parameters that are easy for tuning. However, if the number of search strategies is large, the “direct approach” requires more computational resources to identify the suitable search strategy for a specified problem. The “HH approach” features a sophisticated self-learning strategy enabling rapid identification of suitable search strategies even when there are many search strategies in the pool, though it comes with the drawback of having more parameters that require intricate tuning, posing challenges for users. The aim of this study is to provide users with an efficient black-box optimizer for various applications, prioritizing easy parameter configuration. Therefore, the “direct approach” is employed in this study.

The proposed self-learning strategy is founded on the notion of a probability model that determines the probability of employing a particular strategy for updating a search agent. While the fundamental framework of the self-learning approach is similar to those employed in existing literature²³, our implementation method is different and unique. In this study, we introduce a novel method for dynamically generating a specified execution probability for each search strategy. The search strategy adopted for updating each agent of the search population is selected proportionally to the corresponding probabilities using the roulette wheel selection method in each iteration.

Initially, the execution probabilities for all strategies are equivalent. In each learning period, these probabilities are adjusted based on the rewards obtained by combining the corresponding degrees of improvement in solution quality. To track the reward for each search strategy in a learning period, a variable known as the reward accumulator is computed at every iteration using the equation below:

$$\begin{array}{c}\hfill s_k=s_k+w_{i,k}\end{array}$$ 8

where $s_k$ is the reward accumulator for the k-th search strategy, whereas $w_{i,k}$ denotes the reward of i-th salp who has performed the search by the k-th strategy.

After each learning period, to update the execution probabilities for the k-th search strategies, the equations below are adopted:

$$\begin{array}{cc}\hfill P_k^{\prime t}=& \delta P_k^{t-1}+(1-\delta )s_k/LP\hfill \end{array}$$ 9

10

where $\delta$ is the inertia learning coefficient, which is used to control the learning speed, M is the number of the search strategies, and LP is the number of search iterations of each learning period. Moreover, a crucial implicit parameter, the learning rounds number, denoted with $\tau$, determines the number of rounds in which execution probability updates should be carried out throughout the entire evolutionary process. Apparently, the value of $\tau$ is obtained by dividing the maximum number of search iteration by LP.

After updating the execution probabilities, the reward accumulators for all search strategies are reset to zero in preparation for the next learning period.

Given that the reward $w_{i,k}$ is the sole factor determining the execution probabilities of the search strategies, the calculation scheme for $w_{i,k}$ holds utmost importance in the self-learning approach. As outlined in Self-learning strategy subsection, recent works commonly employ two schemes for reward calculation: the ranking-based scheme and the success-rate-based scheme.

Nevertheless, these two schemes have their respective limitations. In the case of the ranking-based reward calculation scheme, two issues arise: (1) The scheme takes into account the current fitness of each search agent rather than the corresponding improvement in fitness compared to the previous solution. Consequently, it fails to accurately reflect the performance of the search strategies in the current iteration. (2) Even if a search strategy eventually discovers a better position in subsequent iterations, if it initially leads a search agent to a location with lower fitness, this scheme may punish the search strategy more than reward it. Such an approach is unfavorable for encouraging search agents to escape local optima when dealing with multi-modal problems. On the other hand, the success-rate-based reward calculation scheme has its drawback in neglecting the magnitude of improvement achieved by the search strategies on the search agents. It treats the reward for a search strategy that significantly enhances the quality of a search agent as equivalent to the reward for a search strategy that only marginally improves the quality of a search agent. Consequently, this scheme reduces the efficiency of the self-learning strategy.

To overcome the limitations of the existing schemes, a novel reward calculation scheme based on three key ideas is proposed. Firstly, the competition between search strategies relies on the comparison of fitness improvements achieved by the search agents rather than their fitness values. Secondly, the degree of improvement in fitness is considered in the competition of search strategies. Thirdly, to encourage exploration towards unknown areas in the search space, only those searching behaviors that lead to improved positions are considered for reward calculation. The remaining search behaviors that do not reach a better position are treated equally without reward or punishment. In the proposed reward calculation scheme, the Heaviside step function⁵⁸, as given in Eq. (11), is employed to preserve searches that result in fitness improvements while eliminating the effect of searches that do not improve the solutions.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill H_0(x)=\left\{\begin{array}{cc}0,\hfill & ifx<0\hfill \\ 1/2,\hfill & ifx=0\hfill \\ 1,\hfill & ifx>0\hfill \end{array}\right)\end{array}\end{array}$$ 11

Without loss of generality, let the objective function, f, be minimized, $w_{i,k}$ is calculated according to the pseudocode presented in Algorithm 3.

Algorithm 3.

The pseudocode of the $w_{i,k}$ calculation.

The parameter setting method

While a trial-and-error approach is often utilized to identify the optimal parameter combination for an algorithm in solving a specific problem, the computational cost associated with this method tends to be prohibitively high. Consequently, there is an urgent and compelling need to devise an efficient parameter-setting method that can definitively and directly determine the parameters of an algorithm when dealing with diverse problems.

The two crucial parameters, the learning rounds number ($\tau$) and the inertia learning coefficient ($\delta$), need to be carefully configured to ensure the effectiveness and efficiency of the self-learning strategy. When configuring these parameters for the self-learning strategy, it is important to strike a balance between multiple search strategies with different characteristics. Each search strategy should maintain a certain level of influence on the algorithm during the search process to enhance the diversity of search patterns within the population and sustain the potential for exploring unknown but promising areas throughout the search iterations. Moreover, in the self-learning approach, superior performing search strategies for solving specific problems should be gradually elevated as the dominant search strategies to enhance overall search efficiency. Particularly in the later stages of the search, the best search strategy should be selected to conduct the primary search and guide the algorithm towards converging to the final optimum.

To avoid purely relying on the trial-and-error scheme and thus improve the efficiency of applying the algorithm in solving real-world optimization problems, this study provides a parameter analysis framework and a guideline for setting the two parameters ($\tau$, $\delta$) of the self-learning strategy.

The parameter analysis framework is built upon the evolutionary process of the execution probabilities of the search strategies throughout the entire search iterations. Assuming there are M search strategies, each strategy initially has an execution probability set to 1/M. These probabilities evolve based on the quality of solutions produced by the respective search strategies. In extreme cases, one search strategy consistently improves the solution quality in each iteration, while the other strategies consistently fail to make any progress. This scenario can be described as one search strategy winning all contests, while the remaining strategies lose all contests. To investigate the relationship between the two parameters and their impact on the execution probabilities of the search strategies, we analyze the evolution of these probabilities. For the sake of simplicity, let’s consider strategy 1 as the winning strategy, and strategies 2 through M as the losing strategies. In the subsequent description, we will focus on strategy 2 as an example of those strategies that always lose, for the purpose of simplifying the analysis.

From Algorithm 3, it can be deduced that in extreme cases, $s_k$ of the strategy 1 in Eq. (9) is LP/M, while $s_k$ of the strategy 2 in Eq. (9) is 0. The temporary results of the updated execution probabilities can be calculated with the equations as follows:

$$\begin{array}{cc}\hfill P_1^{\prime t}=& \delta P_1^{t-1}+(1-\delta )/M\hfill \end{array}$$ 12

$$\begin{array}{cc}\hfill P_2^{\prime t}=& \delta P_2^{t-1}\hfill \end{array}$$ 13

According to Eq. (10), the sum of all the execution probabilities must be normalized to 1 in each learning period. Thus, we have the iterative calculation equation for strategy 2 as follows:

$$\begin{array}{c}\hfill P_2^t=\delta P_2^{t-1}/(\delta +(1-\delta )/M)\end{array}$$ 14

From Eq. (14), it can be inferred that the execution probability of the strategy 2 decreases in the proportion of $\delta /(\delta +(1-\delta )/M)$ in each learning period. The iterative calculation equation as the relationship between the current execution probability and the execution probability of the previous n learning rounds can be rewritten as follows:

$$\begin{array}{c}\hfill P_2^t={\delta }^n\ast P_2^{t-n}/{(\delta +(1-\delta )/M)}^n=P_2^{t-n}/{(1+\frac{(1-\delta )}{\delta \ast M})}^n\end{array}$$ 15

The parameter analysis framework is based on Eq. (15). Suppose $P_2^t$ and $P_2^{(t-n)}$ are the final value and the initial value of the execution probability of strategy 2; it means the parameter $\tau$ equals to n. If we don’t wish the execution probability of the strategy 2 decrease to a very little value after $\tau$ rounds of learning periods, for example, 1/10 of the initial value, we have:

$$\begin{array}{c}\hfill \frac{1}{{(1+\frac{1-\delta }{\delta \ast M})}^{\tau }}\ge \frac{1}{10}\end{array}$$ 16

If $\tau$ is set to 50, then we have:

$$\begin{array}{c}\hfill 1+\frac{1-\delta }{\delta \ast M}\le {10}^{\frac{1}{50}}\Rightarrow 1+\frac{1-\delta }{\delta \ast M}\le 1.047\end{array}$$ 17

In our proposed SLSSL, M is 4, then we have:

$$\begin{array}{c}\hfill 1+\frac{1-\delta }{\delta \ast M}\le 1.047\Rightarrow \delta \ge 0.842\end{array}$$ 18

In the same way, if $\tau$ is set to 20, then we have:

$$\begin{array}{c}\hfill 1+\frac{1-\delta }{\delta \ast M}\le {10}^{\frac{1}{20}}\Rightarrow \delta \ge 0.672\end{array}$$ 19

In this manner, the lower boundary of $\delta$ can be calculated when the value of $\tau$ has been determined.

Likewise, this analytical method can be applied to determine a suitable upper limit for $\delta$. In this case, the execution probability of the best performing search strategy (strategy 1) for solving the given problem is increased such that it becomes the dominant search strategy after $\tau$ rounds of learning periods. For example, its probability could at least double from its initial value. Accordingly, regarding M is 4, the execution probability of each of the remaining strategies (strategy 2) would need to decrease to a value that is less than or equal to 2/3 of its initial value. This leads to the following equation:

$$\begin{array}{c}\hfill \frac{1}{{(1+\frac{1-\delta }{\delta \ast 4})}^{\tau }}\le \frac{2}{3}\end{array}$$ 20

With the values of $\tau$ set to 20 and 50, respectively, we have:

$$\begin{array}{cc}& 1+\frac{1-\delta }{\delta \ast 4}\ge 1.5^{\frac{1}{50}}\Rightarrow \delta \le 0.968\hfill \end{array}$$ 21

$$\begin{array}{cc}& 1+\frac{1-\delta }{\delta \ast 4}\ge 1.5^{\frac{1}{20}}\Rightarrow \delta \le 0.924\hfill \end{array}$$ 22

Then, these two lower boundaries are rounded up and taken one decimal place, 0.8 and 0.7, respectively. The upper boundaries are also rounded up to 1 and 0.9. According to the analysis, two ranges, [0.8, 1] and [0.7, 0.9], are deduced for setting the proper values of $\delta$ when $\tau$ is set to 50 and 20, respectively.

However, it is important to note that the analysis presented is based on extreme scenarios. In typical situations, the range of suitable values for the parameter $\delta$ should be more flexible. In practice, a reasonable approach is to lower the lower bound of $\delta$ appropriately and expand the range of $\delta$ to provide a more flexible range for selecting the proper value of $\delta$.

In the subsequent experimental section, we will present a case study to demonstrate how to determine suitable values for the parameters $\tau$ and $\delta$ based on the theoretical value range and the practical approach mentioned above.

In comparison to the conventional approach of setting minimum and maximum values for execution probabilities, the proposed parameter setting method offers several advantages. It not only determines a final feasible value for each execution probability but also effectively controls the rate of change for these probabilities. By leveraging the proposed parameter analysis framework, the parameter setting process for the SLSSA algorithm becomes more streamlined and eliminates the need for blind trial-and-error with multiple parameter combinations. Moreover, when setting different combinations of parameters M and $\tau$, the acceptable range for $\delta$ can be automatically and naturally obtained. This simplifies the application of the SLSSA algorithm in solving real-world optimization problems, making it a simpler and more efficient solution.

Time complexity analysis

Time complexity denotes the duration an algorithm takes to solve a problem when executed by its respective program. It serves as a vital measure for assessing the efficiency of an algorithm from a different perspective. By analyzing the pseudocodes outlined in Algorithms 2 and 1, the worst-case time complexity of the SLSSA algorithm is calculated and compared against that of the original SSA using big-O notation.

The computational complexity of SSA is determined by four subsequent steps: population initialization, population position updating, objective function evaluations, and elitist updating. In addition to these four steps, the computational complexity of SLSSA is further influenced by two additional steps: reward accumulator calculating and execution probability updating. Furthermore, due to the time complexity of an algorithm depends on the population size N and the maximum number of iterations T, the time complexity analysis of SSA and SLSSA is presented as follows:

1. In Algorithm 1, the time complexity of population initialization in line 1 is O(N) for SSA, while in Algorithm 2, it is also O(N) line 1 for SLSSA.

2. The time complexity of the population position updating operation in lines 8-14 of Algorithm 1 is O(TN) for SSA, while for Algorithm 2, the operation in lines 8-11 also maintains a time complexity of O(TN) for SLSSA.

3. In Algorithm 1, the objective function evaluations in line 4 incur a time complexity of O(TN) for SSA, while in Algorithm 2, the same operation in line 4 also has a time complexity of O(TN) for SLSSA.

4. The time complexity of the elitist updating operation in line 6 of Algorithm 1 is O(TN) for SSA, while the time complexity of the elitist updating operation in lines 5 and 7 of Algorithm 2 is O(TN) for SLSSA.

5. The time complexity of the reward accumulator calculating in line 12 of Algorithm 1 is O(TN) for SLSSA.

6. The time complexity of the execution probability updating in line 13 of Algorithm 1 is $O(\tau )$ for SLSSA, where $\tau$ is the learning rounds number and much small than T.

According to the analysis above, the total time complexity of SSA and SLSSA is the same and which is O(TN).

Despite SLSSA demanding more computational resources than SSA owing to its additional operations, it can be deduced that the increased computational load of SLSSA remains within a constant proportion of SSA’s computation time, as indicated by their identical time complexities in terms of big-O notation.

Experimental study on numerical optimization problems

In this section, the performance of the proposed SLSSA in solving the global numerical optimization problems, the CEC2014 benchmark suite with D=30, is investigated in comparison with other well-regarded and state-of-the-art algorithms. The CEC2014 benchmark suite is a general test standard of modern algorithms, which has strong test suitability for all kinds of metaheuristic algorithms. It contains 30 continuous test functions, which are categorized into four groups: unimodal functions, simple multimodal functions, hybrid functions, and composition functions. The test functions within the CEC2014 benchmark suite embody a diverse set of real-world optimization challenges, offering an opportunity to evaluate algorithmic performance in global optimization statistically. Consequently, the CEC2014 benchmark suite is widely accepted as a benchmark in many newly published studies. The detailed description and characteristics of the test functions can be found in the literature⁵⁹.

For a fair comparison purpose, all algorithms are coded with MATLAB 2015a, and all the experiments were run on a personal computer with 3.40 GHz Intel(R) Core (TM) i7 CPU and 8 GB RAM. Since the algorithms in the simulation experiment are stochastic optimization techniques, to generate meaningful statistical results, each algorithm is run 50 times with a random initial population for each test function. The median($\tilde{x}$) and interquartile range (IQR) of the corresponding performance metrics achieved by the competing algorithms in the 50 independent runs on each test function were recorded as the measures of location (or central tendency) and statistical dispersion. In the tables presenting the median and interquartile range of the respective metrics, the result with the highest median value is identified as the best-performing one. In cases where two algorithms achieve the same median value, the algorithm with a smaller interquartile range is considered superior. The best result for each test function is visually distinguished by highlighting it with a bold font. Moreover, to provide confident comparisons, the statistical analysis methods include Friedman test⁶⁰ and Wilcoxon rank sum (WRS) test⁶¹ were conducted on the experimental results.

In the experiment, our first objective was to evaluate the effectiveness of the proposed parameter setting method for $\delta$ and $\tau$. To achieve this, we compared the results obtained by running the SLSSA algorithm with different parameter settings. Following the parameter setting evaluation, we proceeded to compare the results obtained by SLSSA with proper parameter settings to those achieved by other well-regarded and state-of-the-art algorithms. By conducting this comparative analysis, we were able to assess the competitiveness of SLSSA and determine its performance relative to existing approaches.

Evaluation of parameter setting method

To evaluate the effectiveness of the proposed parameter setting method, we designed an experiment based on both the theoretical value range and practical considerations outlined in subsection The parameter setting method. Accordingly, we created 9 different parameter settings combinations denoted as PS1-PS9 (shown in Table 2), containing various values of both $\tau$ and $\delta$ for comparison. In these parameter setting combinations, the values of $\tau$ were set at 50 and 20, respectively, while the values of $\delta$ for both $\tau$ settings spanned the entire range from 0 to 1. To enhance the performance differences between different parameter configurations and better distinguish their advantages and disadvantages, we set the difference between each pair of adjacent $\delta$ values to 0.3.

Table 2.

The 9 different parameter setting combinations.

	$\tau$	$\delta$
PS1	20	0
PS2	20	0.1
PS3	20	0.4
PS4	20	0.7
PS5	50	0
PS6	50	0.2
PS7	50	0.5
PS8	50	0.8
PS9	N/A	1

In this experiment, for SLSSA with all the parameter settings, the population size was set to 40, the maximum iterations number was set to 5000, then the maximum number of FEs is $40\times 5000=2.0\times {10}^5$.

The median and interquartile range of the objective function values obtained by the SLSSA with the 9 parameter settings in the 50 independent runs are shown in Table 3. Obviously, a smaller median objective function value indicates higher solution quality, whereas a lower interquartile range represents better stability. It is clear that PS3 and PS7 have the two top performances among the 9 competitors. PS3 obtains the best results in 7 test functions and PS7 obtains the best results in 6 test functions. To further evaluate and compare the performance of these parameter settings, we computed the average Friedman ranking (AFR) ratios, which are presented in Table 4. The lower the rank value, the better the performance of the parameter setting. Based on the AFR results, PS3 was ranked as the best-performing parameter setting, while PS7 had the second-best rank, only slightly larger than that of PS3. The pairwise WRS test, as in the literature⁶², was conducted at the 5$\%$ significance level, and each parameter setting has a one-to-one test with each of the other parameter settings. Therefore, there is $C_9^2=36$ pairwise comparison results. The pairwise WRS test results of the best and the second-best parameter settings, PS3 and PS7, against the rest parameter settings are shown in Tables 5 and 6. The symbol “$+$” signifies that the result of the parameter setting mentioned on the left side of “vs. ” is significantly superior to that of the parameter setting on the right, whereas the symbol “−” denotes significant inferiority of the left-side parameter setting compared to the right-side setting. Meanwhile, the symbol “$\ast$” indicates an insignificant difference between the two settings. Obviously, both the PS3 and PS7 outperform the other parameter settings significantly in solving these test functions. In the direct comparison of PS3 and PS7, the WRS test results are very close, and the results on most of the test functions have no statistical differences.

Table 3.

Objective function values on CEC2014 for the 9 parameter settings of SLSSA.

TFs	PS1	PS2	PS3	PS4	PS5	PS6	PS7	PS8	PS9
f1	1.0742E+07 (8.8658E+06)	8.1122E+06 (7.4222E+06)	3.4198E+06 (1.7811E+06)	3.2908E+06 (2.1666E+06)	2.5085E+08 (4.4430E+08)	8.9234E+06 (5.9806E+06)	5.0508E+06 (2.0193E+06)	3.2147E+06 (1.5584E+06)	3.5353E+06 (2.1556E+06)
f2	1.0230E+04 (1.2696E+04)	8.8759E+03 (8.9864E+03)	8.0067E+03 (9.6964E+03)	8.6527E+03 (8.2759E+03)	6.1503E+10 (1.0161E+10)	7.8772E+03 (6.1437E+03)	1.0436E+04 (1.1402E+04)	8.3972E+03 (8.0436E+03)	1.1525E+04 (1.2407E+04)
f3	2.0277E+04 (1.2057E+04)	1.2905E+04 (6.8484E+03)	9.7019E+03 (5.1007E+03)	1.1194E+04 (4.6166E+03)	2.9560E+04 (1.3550E+04)	1.9619E+04 (9.9923E+03)	1.0320E+04 (5.0497E+03)	1.0198E+04 (3.2712E+03)	1.2643E+04 (5.8106E+03)
f4	5.1519E+02 (5.3597E+01)	4.8891E+02 (4.2493E+01)	5.0514E+02 (4.8545E+01)	5.0505E+02 (4.6098E+01)	1.0869E+04 (2.7100E+03)	4.8741E+02 (4.6377E+01)	5.1066E+02 (6.1780E+01)	5.0766E+02 (4.5246E+01)	4.9568E+02 (4.4423E+01)
f5	5.2005E+02 (9.3471E-02)	5.2002E+02 (1.4584E-01)	5.2003E+02 (9.5448E-02)	5.2001E+02 (6.4688E-02)	5.2003E+02 (8.0063E-02)	5.2003E+02 (1.0457E-01)	5.2002E+02 (1.2718E-01)	5.2006E+02 (1.2060E-01)	5.2001E+02 (8.8895E-02)
f6	6.1605E+02 (5.4584E+00)	6.1327E+02 (3.5635E+00)	6.0943E+02 (4.3140E+00)	6.0716E+02 (3.5401E+00)	6.4080E+02 (2.0608E+00)	6.1518E+02 (4.6952E+00)	6.1092E+02 (4.8979E+00)	6.0822E+02 (3.2938E+00)	6.0762E+02 (4.0163E+00)
f7	7.0001E+02 (1.2321E-02)	7.0001E+02 (2.2127E-02)	7.0001E+02 (1.9678E-02)	7.0001E+02 (1.2316E-02)	1.2719E+03 (1.0013E+02)	7.0001E+02 (1.7226E-02)	7.0001E+02 (1.4780E-02)	7.0001E+02 (1.7226E-02)	7.0001E+02 (1.7226E-02)
f8	9.6118E+02 (1.9134E+02)	9.2785E+02 (3.5818E+01)	8.8706E+02 (2.8854E+01)	8.8009E+02 (2.9848E+01)	1.1356E+03 (1.8261E+01)	9.3581E+02 (3.5818E+01)	9.1591E+02 (3.3828E+01)	8.7761E+02 (2.6864E+01)	8.7114E+02 (3.6813E+01)
f9	1.2587E+03 (1.8424E+01)	1.0557E+03 (5.4722E+01)	9.9800E+02 (3.4823E+01)	9.8208E+02 (3.2833E+01)	1.2699E+03 (1.5955E+01)	1.0672E+03 (3.4823E+01)	1.0403E+03 (4.2783E+01)	9.8407E+02 (2.9848E+01)	9.7114E+02 (2.0894E+01)
f10	4.3959E+03 (1.2544E+03)	3.9982E+03 (8.3682E+02)	3.9690E+03 (6.8735E+02)	4.1941E+03 (1.1049E+03)	4.2320E+03 (1.0345E+03)	4.3085E+03 (7.5625E+02)	4.2306E+03 (1.0052E+03)	4.0731E+03 (7.8376E+02)	4.1584E+03 (1.1875E+03)
f11	4.4556E+03 (9.7376E+02)	4.2852E+03 (9.9893E+02)	4.3493E+03 (9.5120E+02)	4.1460E+03 (1.0885E+03)	4.8352E+03 (9.6184E+02)	4.3928E+03 (1.0369E+03)	4.2759E+03 (1.0637E+03)	4.4900E+03 (8.8108E+02)	4.3023E+03 (1.0421E+03)
f12	1.2001E+03 (7.9776E-02)	1.2001E+03 (1.0275E-01)	1.2001E+03 (1.1122E-01)	1.2001E+03 (1.1501E-01)	1.2002E+03 (1.7447E-01)	1.2001E+03 (1.2567E-01)	1.2001E+03 (1.6513E-01)	1.2001E+03 (6.8901E-02)	1.2001E+03 (1.3268E-01)
f13	1.3005E+03 (1.4764E-01)	1.3004E+03 (1.0726E-01)	1.3004E+03 (1.2103E-01)	1.3004E+03 (1.0203E-01)	1.3061E+03 (9.4021E-01)	1.3004E+03 (1.2479E-01)	1.3004E+03 (1.2148E-01)	1.3004E+03 (1.4291E-01)	1.3004E+03 (1.2241E-01)
f14	1.4003E+03 (7.2672E-02)	1.4002E+03 (7.0123E-02)	1.4003E+03 (6.2011E-02)	1.4003E+03 (6.2931E-02)	1.5912E+03 (3.2315E+01)	1.4002E+03 (6.6589E-02)	1.4002E+03 (4.8189E-02)	1.4003E+03 (8.3006E-02)	1.4003E+03 (7.2390E-02)
f15	1.5074E+03 (2.8357E+00)	1.5054E+03 (2.1386E+00)	1.5043E+03 (1.7032E+00)	1.5044E+03 (1.5146E+00)	1.4235E+05 (1.3048E+05)	1.5085E+03 (4.3123E+00)	1.5045E+03 (2.2968E+00)	1.5047E+03 (1.8974E+00)	1.5041E+03 (1.2528E+00)
f16	1.6114E+03 (7.1292E-01)	1.6116E+03 (9.6407E-01)	1.6115E+03 (9.3330E-01)	1.6115E+03 (8.8726E-01)	1.6116E+03 (6.9022E-01)	1.6115E+03 (9.3045E-01)	1.6115E+03 (1.3263E+00)	1.6113E+03 (9.4897E-01)	1.6114E+03 (1.0232E+00)
f17	6.2109E+05 (3.5590E+05)	3.1207E+05 (3.8945E+05)	1.6230E+05 (1.5865E+05)	1.8887E+05 (2.7077E+05)	5.4445E+05 (6.2328E+05)	5.5423E+05 (6.1136E+05)	1.8336E+05 (2.0574E+05)	1.8072E+05 (1.6313E+05)	1.7532E+05 (2.4751E+05)
f18	2.6221E+03 (4.6003E+03)	2.9491E+03 (3.6830E+03)	6.0846E+03 (9.9567E+03)	6.9656E+03 (7.9417E+03)	1.0159E+09 (1.4020E+09)	3.4192E+03 (2.6793E+03)	4.9824E+03 (8.7907E+03)	5.5760E+03 (1.1245E+04)	7.1135E+03 (1.2016E+04)
f19	1.9158E+03 (2.9162E+00)	1.9128E+03 (2.0682E+00)	1.9123E+03 (2.4120E+00)	1.9123E+03 (2.4864E+00)	2.2027E+03 (1.3204E+02)	1.9146E+03 (2.4981E+00)	1.9123E+03 (2.7284E+00)	1.9118E+03 (3.3288E+00)	1.9129E+03 (2.9191E+00)
f20	3.1185E+03 (1.3445E+03)	3.2250E+03 (2.4057E+03)	3.0982E+03 (1.5401E+03)	3.0998E+03 (1.2413E+03)	3.2845E+03 (2.0404E+03)	3.0074E+03 (1.0041E+03)	3.3358E+03 (1.0782E+03)	3.5741E+03 (2.0786E+03)	4.2294E+03 (2.4203E+03)
f21	6.9294E+04 (7.7620E+04)	7.0046E+04 (7.0483E+04)	8.3359E+04 (7.2305E+04)	9.8333E+04 (1.0709E+05)	1.0022E+05 (9.4175E+04)	7.9302E+04 (7.8656E+04)	8.3349E+04 (9.4886E+04)	8.5015E+04 (5.5272E+04)	1.0223E+05 (9.2255E+04)
f22	2.4927E+03 (2.2548E+02)	2.4991E+03 (2.1114E+02)	2.4797E+03 (2.2513E+02)	2.4902E+03 (1.9877E+02)	2.4958E+03 (2.1834E+02)	2.4890E+03 (2.8486E+02)	2.5203E+03 (2.0530E+02)	2.5067E+03 (2.3526E+02)	2.4754E+03 (1.7711E+02)
f23	2.5000E+03 (0.0000E+00)	2.5000E+03 (0.0000E+00)	2.5000E+03 (0.0000E+00)	2.5000E+03 (2.2000E-05)	2.5000E+03 (0.0000E+00)	2.5000E+03 (0.0000E+00)	2.5000E+03 (0.0000E+00)	2.5000E+03 (1.6000E-05)	2.5000E+03 (9.0000E-05)
f24	2.6000E+03 (0.0000E+00)	2.6000E+03 (0.0000E+00)	2.6000E+03 (6.0000E-05)	2.6000E+03 (1.2900E-04)	2.6000E+03 (1.0000E-06)	2.6000E+03 (0.0000E+00)	2.6000E+03 (7.0000E-06)	2.6000E+03 (1.7000E-04)	2.6000E+03 (1.8500E-04)
f25	2.7099E+03 (1.2690E+01)	2.7000E+03 (4.3000E-05)	2.7000E+03 (0.0000E+00)	2.7000E+03 (1.0000E-06)	2.7303E+03 (2.4848E+01)	2.7000E+03 (3.9000E-05)	2.7000E+03 (0.0000E+00)	2.7000E+03 (0.0000E+00)	2.7000E+03 (2.0000E-06)
f26	2.7005E+03 (1.6503E-01)	2.7004E+03 (1.3970E-01)	2.7004E+03 (1.3382E-01)	2.7004E+03 (1.6624E-01)	2.7012E+03 (1.0121E+00)	2.7004E+03 (1.7892E-01)	2.7004E+03 (1.5687E-01)	2.7004E+03 (1.4552E-01)	2.7004E+03 (1.4560E-01)
f27	3.1045E+03 (2.5416E+00)	2.9000E+03 (1.6600E-04)	2.9000E+03 (7.0000E-06)	2.9000E+03 (1.3000E-05)	3.1072E+03 (6.5532E+00)	2.9000E+03 (1.0700E-04)	2.9000E+03 (1.0000E-06)	2.9000E+03 (1.2000E-05)	2.9000E+03 (1.2000E-05)
f28	3.7549E+03 (6.8216E+01)	3.0000E+03 (2.6200E-04)	3.0000E+03 (8.0000E-06)	3.0000E+03 (3.5000E-05)	3.8308E+03 (2.5699E+02)	3.3382E+03 (7.6325E+02)	3.0000E+03 (1.4000E-05)	3.0000E+03 (2.9000E-05)	3.0000E+03 (3.1000E-05)
f29	3.8905E+03 (1.0762E+03)	3.5570E+03 (3.9618E+02)	3.5027E+03 (1.8459E+02)	3.4662E+03 (1.3679E+02)	2.5030E+04 (5.1058E+05)	3.3846E+03 (4.8450E+02)	3.3653E+03 (2.8542E+02)	3.4198E+03 (1.7181E+02)	3.4648E+03 (1.4200E+02)
f30	1.0469E+04 (5.3806E+03)	3.2233E+03 (3.9451E+01)	3.2159E+03 (6.7500E+00)	3.2216E+03 (7.4890E+00)	1.9295E+04 (1.0386E+04)	1.1512E+04 (6.5924E+03)	3.2098E+03 (8.3538E+00)	3.2199E+03 (7.2541E+00)	3.2247E+03 (7.5406E+00)

Table 4.

Average Friedman ranking of objective function values on CEC2014 of the 9 parameter settings for SLSSA.

Settings	Ranking	Settings	Ranking
PS1	5.759667	PS6	5.544333
PS2	4.775000	PS7	4.003000
PS3	3.999667	PS8	4.263667
PS4	4.390667	PS9	4.698333
PS5	7.565667

Table 5.

Comparison results of PS3 with other 8 parameter settings for SLSSA on CEC2014 by the WRS test.

Comparison group	$+$/*/-	Comparison group	$+$/*/-
PS3 vs. PS1	15/13/2	PS3 vs. PS5	22/7/1
PS3 vs. PS2	11/17/2	PS3 vs. PS6	15/13/2
PS3 vs. PS7	5/20/5	PS3 vs. PS8	6/21/3
PS3 vs. PS4	6/21/3	PS3 vs. PS9	10/17/3

Table 6.

Comparison results of PS7 with other 8 parameter settings for SLSSA on CEC2014 by the WRS test.

Comparison group	$+$/*/-	Comparison group	$+$/*/-
PS7 vs. PS1	16/13/1	PS7 vs. PS5	23/7/0
PS7 vs. PS2	10/19/1	PS7 vs. PS6	15/13/2
PS7 vs. PS3	5/20/5	PS7 vs. PS8	7/19/4
PS7 vs. PS4	9/17/4	PS7 vs. PS9	12/13/5

Our analysis of extreme situations revealed that the theoretical ranges of the parameter $\delta$ for $\tau$=50 and $\tau$=20 are [0.8, 1] and [0.7, 0.9], respectively. However, considering the practical situation discussed in subsection The parameter setting method, it is reasonable to adjust the value of $\delta$. Specifically, by decreasing the lower boundary of $\delta$ by 0.3 and widening the range of $\delta$ by 0.3, the corresponding ranges of $\delta$ are set as [0.5, 0.8] and [0.4, 0.7] for $\tau$=50 and $\tau$=20, respectively. Importantly, we observed that parameter settings (PS3, PS4, PS7, and PS8) with $\delta$ values falling within the target ranges established through our parameter analysis yielded significantly better results than those with $\delta$ values outside the target ranges. This finding underscores the effectiveness and practicality of our proposed theoretical analysis method and adjustment approach for parameter settings.

Combining the results of the two statistical analysis methods, we choose PS3 as the parameter setting for SLSSA in solving CEC2014, and SLSSA shows SLSSA-PS3 in the rest of the experiments.

Comparison with other algorithms

In this subsection, the experimental result obtained by SLSSA is compared to the original SSA⁴ and four new proposed state-of-the-art improved versions of SSA, including SSA-COBL³³, CSSA⁴⁰, ESSA⁴¹, and LSC-SSA³⁰, on solution accuracy, stability, and computation efficiency. The main reason of selecting these SSA variants as compared algorithms is to examine the advantages of the self-learning mechanism and the newly proposed multiple food sources search strategy, given that the other search operators in SLSSA are consistent with those of the comparison algorithms. These four algorithms used as comparison baselines in the experiments exhibit significant competitiveness across both benchmark test functions and practical engineering problems. They outperform several well-regarded algorithms, including Particle Swarm Optimization (PSO), Whale Optimization Algorithm (WOA), Grey Wolf Optimiser (GWO), Sine Cosine Algorithm (SCA), Moth-Flame Optimization (MFO), Differential Evolution (DE), Covariance Matrix Adaptation Evolution Strategy (CMA-ES), Fruit Fly Optimization Algorithm (FOA), Chimp Optimization Algorithm (ChOA), Giza Pyramids Construction (GPC), Arithmetic Optimization Algorithm (AOA), and their respective state-of-the-art improved versions. These improvements include the Self-adaptive Differential Evolution (SaDE), the hybrid of Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA), the Fruit Fly Optimization Algorithm with Decreasing Step Size (SFOA), and the Chaotic Whale Optimization Algorithm (CWOA), among others.

According to the corresponding reference⁴, no parameters need to be set for SSA. The parameters for the SSA-COBL³³, CSSA⁴⁰, ESSA⁴¹ and LSC-SSA³⁰ were set as in their corresponding papers. Specifically, the generation jumping probability is set to 0.3 for SSA-COBL; $\delta$=8 for CSSA; Q=5, F=5, k=rand(0,1), and Limit=20 for ESSA; and a=0.01 for LSC-SSA. For a fair comparison, the maximum number of FEs is set at $2.0\times {10}^5$, and the population size is set at 40 for all the competing algorithms on each test function.

For population-based stochastic algorithms, the primary concern in performance comparison lies in their capability to search for solutions with favorable objective function values. Hence, the analysis of solution accuracy and stability of the proposed SLSSA is based on the objective function values obtained, in contrast to other algorithms, as presented in the first sub-subsection. Furthermore, the efficiency of an algorithm in reaching the attained optimum holds significant importance when comparing performance among optimization algorithms, and this aspect will be addressed in the second sub-subsection.

Solution accuracy and stability

In Table 7, the median and interquartile range of the best objective function values obtained in the 50 independent runs by each algorithm on the 30 benchmark test functions are shown to illustrate the solution accuracy and stability of the competing algorithms. The best result among all six competing algorithms on each test function is highlighted with a bold font. And to illustrate the median value comparison result clearly, even if its corresponding IQR value is worse than some other algorithms, the best median value for each test function is also highlighted with a bold font. The WRS test results are shown in the last column of Table 7. Symbols “$+$”, “$\ast$”, and “−” indicate that the obtained objective function values of the proposed SLSSA are better than, equal to, and worse than those of the corresponding compared algorithms in the WRS test, respectively. The symbols in the first column present the performance difference between SLSSA and SSA-COBL, while the symbols in the other four columns present the comparison between SLSSA and CSSA, ESSA, LSC-SSA, and SSA in the listed sequence. Further, Table  8 illustrates the average rankings of the objective function values achieved by the Friedman test for these competing algorithms.

Table 7.

Objective function values on CEC2014 of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, and SSA.

TFs	SLSSA	SSA-COBL	CSSA	ESSA	LSC-SSA	SSA	WRS
f1	3.4198E+06 (1.7811E+06)	1.9481E+07 (9.5254E+06)	1.4948E+07 (1.0851E+07)	3.0950E+07 (1.1347E+07)	2.1886E+07 (1.8916E+07)	2.1732E+07 (1.6893E+07)	+/+/+/+/+
f2	8.0067E+03 (9.6964E+03)	9.2304E+03 (6.7138E+03)	7.9752E+03 (1.9210E+04)	3.6194E+05 (7.6755E+05)	4.8286E+06 (2.1551E+06)	9.0896E+03 (1.7823E+04)	*/*/+/+/*
f3	9.7019E+03 (5.1007E+03)	4.3026E+04 (1.4417E+04)	2.5833E+04 (1.4126E+04)	2.4045E+04 (9.8864E+03)	2.8940E+04 (1.4338E+04)	1.9229E+04 (8.4508E+03)	+/+/+/+/+
f4	5.0514E+02 (4.8545E+01)	5.4334E+02 (5.2037E+01)	5.3255E+02 (5.5854E+01)	5.9775E+02 (4.7918E+01)	5.4084E+02 (3.8552E+01)	5.3992E+02 (4.7079E+01)	+/+/+/+/+
f5	5.2003E+02 (9.5400E-02)	5.2000E+02 (2.0000E-04)	5.2034E+02 (2.5720E-01)	5.2034E+02 (7.8220E-01)	5.2100E+02 (7.9700E-02)	5.2000E+02 (1.0000E-04)	-/+/+/+/-
f6	6.0943E+02 (4.3140E+00)	6.1717E+02 (4.3296E+00)	6.2818E+02 (7.4535E+00)	6.1749E+02 (2.8227E+00)	6.2598E+02 (5.6522E+00)	6.2584E+02 (4.8263E+00)	+/+/+/+/+
f7	7.0001E+02 (1.9700E-02)	7.0002E+02 (1.5300E-02)	7.0001E+02 (7.4000E-03)	7.0040E+02 (3.0580E-01)	7.0108E+02 (9.1900E-02)	7.0000E+02 (1.1800E-02)	+/*/+/+/*
f8	8.8706E+02 (2.8854E+01)	8.8159E+02 (1.7909E+01)	9.8556E+02 (8.3430E+01)	8.2147E+02 (1.5048E+01)	8.5994E+02 (2.4971E+01)	9.2835E+02 (4.8753E+01)	*/+/-/-/+
f9	9.9800E+02 (3.4823E+01)	9.8507E+02 (2.0894E+01)	1.1474E+03 (6.5667E+01)	9.9950E+02 (3.7809E+01)	9.8404E+02 (4.3339E+01)	1.0383E+03 (4.4772E+01)	-/+/*/-/+
f10	3.9690E+03 (6.8735E+02)	3.6715E+03 (8.2867E+02)	5.4033E+03 (9.2600E+02)	1.3831E+03 (3.4349E+02)	4.2704E+03 (8.9783E+02)	4.2524E+03 (8.2428E+02)	-/+/-/*/+
f11	4.3493E+03 (9.5120E+02)	4.5052E+03 (7.5243E+02)	5.8026E+03 (9.8661E+02)	5.0247E+03 (3.0940E+03)	5.8214E+03 (9.5352E+02)	4.6538E+03 (6.9455E+02)	*/+/*/+/+
f12	1.2001E+03 (1.1120E-01)	1.2002E+03 (1.3260E-01)	1.2008E+03 (3.6390E-01)	1.2007E+03 (1.4453E+00)	1.2020E+03 (8.2840E-01)	1.2007E+03 (3.1310E-01)	+/+/+/+/+
f13	1.3004E+03 (1.2100E-01)	1.3005E+03 (5.7700E-02)	1.3006E+03 (1.5000E-01)	1.3005E+03 (1.2120E-01)	1.3005E+03 (1.4360E-01)	1.3006E+03 (2.1390E-01)	+/+/+/+/+
f14	1.4003E+03 (6.2000E-02)	1.4002E+03 (6.1100E-02)	1.4003E+03 (8.1100E-02)	1.4003E+03 (4.7200E-02)	1.4004E+03 (4.1330E-01)	1.4003E+03 (7.5000E-02)	-/+/*/+/+
f15	1.5043E+03 (1.7032E+00)	1.5067E+03 (2.9963E+00)	1.5112E+03 (5.8492E+00)	1.5338E+03 (1.2451E+01)	1.5220E+03 (4.7462E+00)	1.5157E+03 (7.2963E+00)	+/+/+/+/+
f16	1.6115E+03 (9.3330E-01)	1.6118E+03 (5.8460E-01)	1.6129E+03 (5.6660E-01)	1.6123E+03 (7.0500E-01)	1.6128E+03 (4.4640E-01)	1.6119E+03 (8.7280E-01)	+/+/+/+/+
f17	1.6230E+05 (1.5865E+05)	7.7654E+05 (6.8371E+05)	6.0818E+05 (8.1046E+05)	1.7771E+06 (1.5251E+06)	1.7167E+06 (1.6656E+06)	6.6372E+05 (6.5447E+05)	+/+/+/+/+
f18	6.0846E+03 (9.9567E+03)	2.3603E+03 (1.0979E+03)	1.3413E+04 (1.9366E+04)	2.9404E+03 (1.6479E+03)	3.6545E+04 (2.7984E+04)	3.7646E+03 (5.9445E+03)	-/+/-/+/*
f19	1.9123E+03 (2.4120E+00)	1.9152E+03 (2.6127E+00)	1.9158E+03 (3.7909E+00)	1.9213E+03 (1.0057E+01)	1.9143E+03 (6.6796E+01)	1.9177E+03 (3.8029E+00)	+/+/+/+/+
f20	3.0982E+03 (1.5401E+03)	1.7705E+04 (1.0844E+04)	5.1964E+03 (5.5855E+03)	1.6931E+04 (6.3069E+03)	1.7309E+04 (1.6479E+04)	4.0470E+03 (2.9292E+03)	+/+/+/+/+
f21	8.3359E+04 (7.2305E+04)	1.8406E+05 (1.0426E+05)	1.9978E+05 (2.2172E+05)	2.9022E+05 (2.6362E+05)	2.8673E+05 (2.9357E+05)	1.5201E+05 (2.2152E+05)	+/+/+/+/+
f22	2.4797E+03 (2.2513E+02)	2.6440E+03 (1.8719E+02)	2.9062E+03 (2.1342E+02)	2.6083E+03 (1.0196E+02)	2.7798E+03 (2.3539E+02)	2.7477E+03 (2.6333E+02)	+/+/+/+/+
f23	2.5000E+03 (0.0000E+00)	2.6222E+03 (3.2299E+00)	2.6275E+03 (1.1998E+01)	2.6196E+03 (5.1125E+00)	2.6166E+03 (1.5777E+00)	2.6308E+03 (1.0831E+01)	+/+/+/+/+
f24	2.6000E+03 (1.0000E-04)	2.6274E+03 (2.9527E+00)	2.6025E+03 (2.6304E+00)	2.6000E+03 (0.0000E+00)	2.6398E+03 (1.2823E+01)	2.6423E+03 (1.2759E+01)	+/+/-/+/+
f25	2.7000E+03 (0.0000E+00)	2.7060E+03 (6.3318E+00)	2.7134E+03 (8.2163E+00)	2.7000E+03 (0.0000E+00)	2.7121E+03 (4.9217E+00)	2.7152E+03 (6.9237E+00)	+/+/*/+/+
f26	2.7004E+03 (1.3380E-01)	2.7005E+03 (9.4800E-02)	2.7005E+03 (1.9720E-01)	2.7005E+03 (1.3440E-01)	2.7005E+03 (1.8980E-01)	2.7005E+03 (2.0540E-01)	+/+/+/+/+
f27	2.9000E+03 (0.0000E+00)	3.3405E+03 (1.9265E+02)	3.8299E+03 (2.1265E+02)	2.9000E+03 (2.1179E+02)	3.6092E+03 (1.9673E+02)	3.5612E+03 (5.0946E+02)	+/+/-/+/+
f28	3.0000E+03 (0.0000E+00)	4.1879E+03 (8.4959E+02)	4.6265E+03 (5.1353E+02)	3.9215E+03 (2.7044E+02)	4.0495E+03 (5.5681E+02)	4.1346E+03 (6.2033E+02)	+/+/+/+/+
f29	3.5027E+03 (1.8459E+02)	2.5842E+05 (3.6219E+05)	9.6442E+06 (1.2124E+07)	4.6395E+03 (6.8998E+02)	2.0479E+04 (8.9206E+06)	2.0644E+04 (1.2843E+07)	+/+/+/+/+
f30	3.2159E+03 (6.7500E+00)	4.2746E+04 (2.5530E+04)	4.1109E+04 (6.5382E+04)	2.2150E+04 (1.8122E+04)	1.7795E+04 (1.5936E+04)	3.4108E+04 (2.4554E+04)	+/+/+/+/+

Table 8.

Average Friedman ranking of objection function values on CEC2014 of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, and SSA.

Algorithms	Ranking
SLSSA	1.899000
SSA-COBL	3.825667
CSSA	4.451333
ESSA	3.398000
LSC-SSA	4.485333
SSA	3.816667

The results in Table 7 reveal that the proposed SLSSA obtains the best median values in 22 out of the 30 benchmark test functions and demonstrates the least IQR in 14 out of the 30 benchmark test functions. Even in cases where SLSSA does not secure the first position, the majority of the median or IQR values obtained using SLSSA rank among the top results. This illustrates SLSSA’s superior performance in terms of solution accuracy and stability compared to other algorithms. The results of the WRS test reveal significant differences in performance between SLSSA and the other algorithms in most of the test functions. Specifically, SLSSA outperforms all the other compared algorithms in 18 out of the 30 cases and surpasses at least three compared algorithms in 25 out of the 30 cases. The total number of “$+$” in the WRS test results in Table 7 is 124 out of 150, while the total number of “−” is 13 out of 150. Furthermore, Table 8 demonstrates that the proposed SLSSA achieves the best average ranking value, significantly outperforming the ranking values of the compared algorithms. Clearly, the average ranking value of SLSSA is significantly lower than that of all other competing algorithms. In fact, when compared to ESSA, which ranks second, SLSSA’s value is 44% lower. Taking out some of the results from Tables 7 and 8 separately, that is, the comparison results of SLSSA with the original SSA, SSA-COBL, and ESSA, it can be seen that the proposed SLSSA demonstrates a marked improvement over the original SSA in terms of solution quality and stability. Additionally, the SLSSA also significantly outperforms SSA-COBL and ESSA, which employ the oppositional learning operator and the two original search strategies of SSA. This demonstrates the benefits of the proposed MFSS mechanism in reaching the final optimum. Taken together, the data from Tables 7 and 8 lead to the conclusion that SLSSA exhibits significant advantages in solution accuracy and stability across diverse problems. It can be inferred that the self-learning strategy, ensuring diverse search patterns, and the two supplementary new search strategies, enhancing both exploration and exploitation search abilities, contribute to the success of SLSSA in reaching highly accurate solutions compared to the competing algorithms. The multiple search operators offer diverse capabilities to effectively conquer various fitness landscapes, while the self-learning mechanism assigns computational resources to the most suitable operators, enhancing search efficiency. On the other hand, it should be noted that SLSSA does not perform optimally in some test cases because the four search operators cannot cover all the complex landscapes presented by the test suite. This indicates the potential for future improvements to SLSSA if additional search operators are included in the self-learning framework.

Computation efficiency

To evaluate the computation efficiency of SLSSA, the CPU computation time for the execution of each algorithm in each run is measured. Moreover, the speed of each algorithm in approximating the obtained best objective function value of each test function is also investigated.

CPU time

In addition to high performance, a short finite computation time is essential for potential applications in the real world.

In Table 9, the median and interquartile range of CPU time (measured in seconds) for 50 runs of each test function by the six competing algorithms are summarized. The last column of the table presents the results of the Wilcoxon rank sum test under the same significance level, using the symbols “$+/\ast /-$” to represent the same meanings as in Table 7. Table 10 illustrates the average rankings achieved by the Friedman test on CPU time for these competing algorithms.

Table 9.

CPU time on CEC2014 of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, and SSA.

TFs	SLSSA	SSA-COBL	CSSA	ESSA	LSC-SSA	SSA	WRS
f1	2.3622E+00 (2.9700E-02)	1.4185E+00 (1.3400E-02)	1.8522E+00 (1.1000E-02)	7.7870E+00 (3.4620E-01)	6.5276E+00 (4.7600E-02)	2.0024E+00 (9.6000E-03)	-/-/+/+/-
f2	2.0649E+00 (2.4900E-02)	1.1826E+00 (9.8000E-03)	1.6464E+00 (1.0400E-02)	7.1915E+00 (1.9800E-02)	6.2957E+00 (4.3100E-02)	1.7139E+00 (9.2000E-03)	-/-/+/+/-
f3	2.0621E+00 (1.9100E-02)	1.1777E+00 (1.0300E-02)	1.6498E+00 (4.5400E-02)	7.2035E+00 (7.7600E-02)	6.3350E+00 (3.5700E-02)	1.7021E+00 (9.0000E-03)	-/-/+/+/-
f4	2.2030E+00 (3.6900E-02)	1.1691E+00 (1.1700E-02)	1.6337E+00 (7.8000E-03)	7.1996E+00 (2.8100E-02)	6.2943E+00 (4.3200E-02)	1.6927E+00 (7.3000E-03)	-/-/+/+/-
f5	2.3203E+00 (1.8300E-02)	1.2666E+00 (1.2400E-02)	1.7169E+00 (2.3800E-02)	7.3098E+00 (3.8700E-02)	6.3787E+00 (4.5400E-02)	1.8132E+00 (1.2200E-02)	-/-/+/+/-
f6	1.4578E+01 (9.3400E-02)	1.0812E+01 (7.5580E-01)	1.1067E+01 (1.3380E-01)	1.6177E+01 (1.3530E-01)	1.5056E+01 (7.0400E-02)	1.4068E+01 (1.4620E-01)	-/-/+/+/-
f7	2.2854E+00 (2.7700E-02)	1.1833E+00 (1.0100E-02)	1.7882E+00 (7.9000E-03)	7.3030E+00 (3.6800E-02)	5.9243E+00 (3.5700E-02)	1.8459E+00 (1.0100E-02)	-/-/+/+/-
f8	1.9955E+00 (2.1500E-02)	1.0401E+00 (1.1200E-02)	1.6244E+00 (1.1500E-02)	7.1718E+00 (2.5600E-02)	5.7365E+00 (3.5300E-02)	1.6409E+00 (9.2000E-03)	-/-/+/+/-
f9	2.1823E+00 (1.7800E-02)	1.1479E+00 (1.2500E-02)	1.7342E+00 (1.1100E-02)	7.2711E+00 (2.8900E-02)	5.8446E+00 (3.1500E-02)	1.7827E+00 (8.9000E-03)	-/-/+/+/-
f10	2.6826E+00 (3.8300E-02)	1.4302E+00 (1.9200E-02)	2.0203E+00 (1.3400E-02)	7.6062E+00 (1.8600E-02)	6.1471E+00 (3.7900E-02)	2.1406E+00 (1.6950E-01)	-/-/+/+/-
f11	2.6534E+00 (6.6100E-02)	1.5548E+00 (2.0800E-02)	2.1367E+00 (1.6200E-02)	7.7230E+00 (2.4900E-02)	6.2628E+00 (3.9500E-02)	2.1475E+00 (2.4300E-02)	-/-/+/+/-
f12	4.1518E+00 (2.0000E-02)	2.7689E+00 (1.0100E-02)	3.3030E+00 (1.3700E-02)	8.9144E+00 (2.8700E-02)	7.4541E+00 (3.4300E-02)	3.6670E+00 (1.4800E-02)	-/-/+/+/-
f13	2.1725E+00 (1.7800E-02)	1.0892E+00 (6.7000E-03)	1.6267E+00 (8.6000E-03)	7.2134E+00 (2.3400E-02)	5.7755E+00 (3.4900E-02)	1.5687E+00 (5.7000E-03)	-/-/+/+/-
f14	2.1894E+00 (2.4400E-02)	1.1059E+00 (1.1800E-02)	1.6484E+00 (1.0400E-02)	7.2272E+00 (2.3100E-02)	5.7949E+00 (3.0900E-02)	1.5881E+00 (4.8000E-03)	-/-/+/+/-
f15	2.2587E+00 (2.7000E-02)	1.1722E+00 (1.2400E-02)	1.7833E+00 (8.8000E-03)	7.3107E+00 (4.4200E-02)	5.8905E+00 (4.0300E-02)	1.6796E+00 (7.0000E-03)	-/-/+/+/-
f16	2.2338E+00 (2.2700E-02)	1.2036E+00 (1.1000E-02)	1.7664E+00 (1.1900E-02)	7.3168E+00 (3.3500E-02)	5.9398E+00 (3.2950E-01)	1.7112E+00 (7.4000E-03)	-/-/+/+/-
f17	2.4517E+00 (2.2100E-02)	1.3312E+00 (1.4200E-02)	1.9142E+00 (1.5700E-02)	7.4177E+00 (4.3900E-02)	6.4181E+00 (4.3800E-02)	1.8769E+00 (1.2200E-02)	-/-/+/+/-
f18	2.1906E+00 (2.1500E-02)	1.1589E+00 (9.7000E-03)	1.7218E+00 (1.2800E-02)	7.2476E+00 (3.6900E-02)	6.2098E+00 (4.0700E-02)	1.6522E+00 (6.0000E-03)	-/-/+/+/-
f19	4.3313E+00 (3.2700E-02)	2.9754E+00 (2.1700E-02)	3.6784E+00 (7.5400E-02)	9.1344E+00 (6.8000E-02)	8.2689E+00 (8.4100E-02)	3.9717E+00 (2.9000E-02)	-/-/+/+/-
f20	2.2381E+00 (2.4100E-02)	1.1870E+00 (1.0600E-02)	1.7597E+00 (1.0800E-02)	7.3213E+00 (4.8100E-02)	6.2561E+00 (3.9200E-02)	1.6831E+00 (6.6000E-03)	-/-/+/+/-
f21	2.3693E+00 (2.0300E-02)	1.2972E+00 (1.0500E-02)	1.8617E+00 (1.2300E-02)	7.3835E+00 (2.3700E-02)	6.3793E+00 (3.6100E-02)	1.8283E+00 (8.4000E-03)	-/-/+/+/-
f22	2.6518E+00 (2.6500E-02)	1.4294E+00 (1.4700E-02)	1.9981E+00 (1.5500E-02)	7.5821E+00 (1.7500E-02)	6.5195E+00 (4.1400E-02)	2.0011E+00 (1.4200E-02)	-/-/+/+/-
f23	4.7380E+00 (1.6500E-02)	3.1788E+00 (8.5000E-03)	3.7695E+00 (1.5200E-02)	9.2742E+00 (3.2700E-02)	8.3883E+00 (4.6800E-02)	4.1782E+00 (1.0100E-02)	-/-/+/+/-
f24	3.9667E+00 (1.2500E-02)	2.5227E+00 (1.1700E-02)	3.1767E+00 (1.3300E-02)	8.6177E+00 (4.5900E-02)	7.7160E+00 (5.7300E-02)	3.3715E+00 (9.6000E-03)	-/-/+/+/-
f25	4.3958E+00 (8.8000E-03)	2.8553E+00 (1.4800E-02)	3.4896E+00 (2.6000E-02)	8.9464E+00 (5.3200E-02)	8.0701E+00 (7.3900E-02)	3.7802E+00 (1.2100E-02)	-/-/+/+/-
f26	1.7189E+01 (3.8600E-02)	1.2936E+01 (1.4460E-01)	1.3870E+01 (2.9600E-02)	1.9357E+01 (2.1160E-01)	1.9008E+01 (4.6300E-01)	1.6477E+01 (8.2600E-02)	-/-/+/+/-
f27	1.6998E+01 (4.1200E-02)	1.2612E+01 (1.4630E-01)	1.3461E+01 (2.6130E-01)	1.8533E+01 (8.5920E-01)	1.8968E+01 (3.2890E-01)	1.6360E+01 (9.8410E-01)	-/-/+/+/-
f28	5.5073E+00 (3.3100E-02)	3.8027E+00 (2.0500E-02)	4.3760E+00 (4.1200E-02)	9.9524E+00 (1.3080E-01)	9.0933E+00 (4.0500E-02)	4.9791E+00 (2.3500E-02)	-/-/+/+/-
f29	6.3701E+00 (3.0600E-02)	4.3644E+00 (3.9700E-02)	4.6199E+00 (1.8230E-01)	1.0490E+01 (7.2700E-02)	9.8487E+00 (7.0600E-02)	5.7041E+00 (4.6300E-02)	-/-/+/+/-
f30	4.1532E+00 (3.4100E-02)	2.7565E+00 (1.4700E-02)	3.3461E+00 (4.3300E-02)	8.9157E+00 (5.2800E-02)	8.0098E+00 (6.0300E-02)	3.6645E+00 (2.1000E-02)	-/-/+/+/-

Table 10.

Average Friedman ranking of CPU time on CEC2014 of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, and SSA.

Algorithms	Ranking
SLSSA	3.988000
SSA-COBL	1.002000
CSSA	2.314667
ESSA	5.969333
LSC-SSA	5.030000
SSA	2.696000

The self-learning strategy in SLSSA requires additional computation to update the search strategies’ execution probabilities, which results in longer computation times compared to the original SSA. As shown in Table 10, SLSSA ranks 4th among all the competitors in terms of overall computational speed. However, the additional computation time required by SLSSA is still manageable. In all test functions, the median CPU time cost by SLSSA is no more than 40$\%$ additional compared to the original SSA. In certain test functions such as , and , where the objective function values require more computation time, the median CPU time of SLSSA is comparable to that of the original SSA. Meanwhile, SLSSA requires significantly less computation time than ESSA and LSC-SSA. Moreover, SLSSA only incurs a small additional time cost in most test functions compared to the fastest algorithm in the experiment. All in all, the computation efficiency of the proposed SLSSA, in terms of CPU time, is acceptable when considering the ratio of performance and cost. The experimental results regarding CPU time align with the conclusions drawn from the time complexity analysis, confirming that the total time complexity of both SSA and SLSSA is identical in terms of big-O notation.

Convergence speed

The speed in approximating the optimum is another salient yardstick to evaluate the performance of an optimization algorithm. Since all the algorithms in the experiment are stochastic, comparing the convergence curves in a certain run is not sufficiently representative of the algorithms’ overall performance on convergence speed. Therefore, in each independent run, the best objective function value obtained so far is recorded as a data point in every 400 FEs. For each test function, the median values of the data points of the 50 independent runs are used to demonstrate the overall convergence speed. Each point of a convergence curve is calculated with the following equation:

23

where $f_k(i)$ refers to the i-th data point in the k-th independent run, TR is the total independent run number for a test function, and Median is the function that returns the median value of the input data.

The convergence curves of all six contenders in the 30 test functions are graphically presented in Figs. 2 and 3. It is evident from the graph that SLSSA initially exhibits a relatively slower convergence speed. However, as the optimization progresses, SLSSA demonstrates robust global convergence capabilities, surpassing previous leading competitors in many test functions. While SLSSA may not demonstrate the best early-stage convergence efficiency among its competitors, it consistently converges towards the optimal solution across most test functions throughout the entire evolutionary process. In contrast, the other competing algorithms were frequently trapped in stagnation situations. Therefore, the overall convergence accuracy of SLSSA outperforms that of the competing algorithms.

Fig. 2.

The convergence graphs of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, and SSA on f1 to f15 of CEC2014.

Fig. 3.

The convergence graphs of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, and SSA on f16 to f30 of CEC2014.

It is understandable that due to the inertia of the execution probabilities, search agents in the early stage of SLSSA tend to explore unknown areas with diverse search strategies, while in the later stage, the most efficient search strategy becomes the major search pattern, accelerating the convergence speed. The slower convergence speeds observed during the initial optimization phases can be regarded as a characteristic of the proposed self-learning strategy. However, this is an inevitable trade-off for achieving the high-quality final solution and the accelerated convergence speed in the later stages.

It can be summed up from all the experimental results in this section that the proposed SLSSA balances well between exploration and exploitation, thus can improve the solution quality, enhance robustness, and accelerate convergence speed for solving different optimization problems. Also, compared with the original SSA, SLSSA does not increase the computation time significantly. Therefore, SLSSA can be regarded as a reliable and promising technique for solving various kinds of global optimization problems.

Application on MLP classifier training

In this subsection, the proposed SLSSA is applied to solve the optimization problems in MLP training.

An artificial neural network (ANN) is a computational model inspired by the human brain, consisting of interconnected nodes that learn complex patterns from data, and it has been widely applied in the fields of engineering, such as computer vision and computer simulation, to address various problems^63,64. According to the literature^63,65, combining traditional methods with ANNs can efficiently address tasks across various applications. The MLP is a widely recognized ANN extensively employed for classification tasks in many domains, such as the semiconductor industry. Obtaining the best combination of the network parameters, including the connection weights and biases by minimizing the error between the desired and calculated network outputs, is vital in applications relying on MLPs. The process of training MLPs is to tune the network parameters delicately. In the literature¹⁰, a general framework has been designed to handle this MLPs training optimization problem. As demonstrated in Fig. 4, a common form of MLPs consists of three layers: the first layer is called the input layer, the last layer is known as the output layer, and the rest layer between the input and output layers is called the hidden layer. In the process of training an MLP using an optimization algorithm, also shown in Fig. 4, firstly, the weights and biases of MLP are set to the values of the decision variables of each search agent, then the cost by this MLP on all training samples is calculated and assigned to the corresponding search agent as its fitness value. With this fitness assignment method, the optimization algorithms iteratively update the decision variables of the search agents to minimize the cost on all training samples. More details of training an MLP using optimization algorithms can be found in the literature^10,66.

Fig. 4.

The MLP training.

Normally, the fitness function of the optimization for training MLPs refers to the cost function defined as the mean square error (MSE) or the average cross-entropy error (CEE) of the combination of network parameters on all training samples. In this study, CEE is adopted as the fitness function. Given a set of network parameters, the corresponding fitness value of the optimization for an MLP on a specified dataset can be calculated by Eq. (24) and Eq. (25) for binary and multi-class classification, respectively.

$$\begin{array}{cc}\hfill CEE=& -\frac{1}{U}\sum _{i=1}^U\left(y_i^{\prime },log\left(y_i\right),+,\left(1,-,y_i^{\prime }\right),l,o,g,\left(1,-,y_i\right)\right)\hfill \end{array}$$ 24

$$\begin{array}{cc}\hfill CEE=& -\frac{1}{U}·\frac{1}{Z}\sum _{i=1}^U\sum _{j=1}^Z\left(y_{i,j}^{\prime },l,o,g,\left(y_{i,j}\right)\right)\hfill \end{array}$$ 25

where $y_i^{\prime }$ and $y_i$ in Eq. (24) are the desired classification result and the actual output of the i-th training sample, $y_{i,j}^{\prime }$ and $y_{i,j}$ in Eq. (25) are the j-th desired classification result and the actual output of the i-th training sample, U is the number of training samples, Z is the number of classes.

In this simulation experiment, the proposed SLSSA is applied to train the MLP classifiers for handling 6 datasets with various numbers of features, outputs, and instances obtained from the UCI machine-learning repository. The detailed parameters of the classification datasets used in the experiment are shown in Table 11 . The datasets were selected under dedicated consideration to represent a wide range of optimization problems in training MLPs and vary between easy and complex. According to the literature, a universal guideline for determining the number of neurons within the hidden layer does not exist. In this study, we conform to the procedure of the literature⁶⁷, configuring the number of neurons in the hidden layer to $2\ast I+1$, where I denotes the total number of the input neurons. Therefore, the total number of network parameters of the MLPs for the corresponding datasets ranges from 55 to 462. Many researchers have adopted these datasets to evaluate their proposals^66,68. Since the features of the datasets have different scales, to eliminate this effect, normalizations are conducted on all datasets with the min-max function.

Table 11.

The classification datasets.

No.	Datasets	No. of samples	No. of attributes	No. of classes	MLP structure	Problem Dimension
1	Balloon	16	4	2	4-9-1	55
2	Iris	150	4	3	4-9-3	75
3	Scale	625	4	3	4-9-3	75
4	Seeds	210	7	3	7-15-3	168
5	Breast cancer	699	9	2	9-19-1	210
6	Wine	178	13	3	13-27-3	462

Unless otherwise specified, the algorithms codes and running environment settings for the experiments in this section and the meaning of the tables showing the experimental results are consistent with those given in Experimental study on numerical optimization problems section. It should be noticed that the experimental results obtained by SLSSA are compared to the same five competing algorithms, including SSA-COBL, CSSA, ESSA, LSC-SSA, and original SSA, as in the previous experiment on CEC2014. Moreover, to illustrated the performance of SLSSA in solving MLP training problems extensively, two other remarkable optimization algorithms, including Battle Royale Optimizer (BRO)⁶⁸, and Modified Sine Cosine Algorithm (MSCA)⁶⁶ are adopted as the algorithms for comparison in this experiment because these algorithms show excellent performance in the optimization for MLP training. The parameter settings for all the competing algorithms are set at the same values of the previous experiment on CEC2014 given in Comparison with other algorithms subsection. The population sizes of all algorithms for the Balloon dataset and the rest of the datasets are set at 50 and 200, respectively. And the maximum number of FEs is set at $2.0\ast {10}^4$ for the Balloon dataset and $8.0\ast {10}^4$ for the other datasets, respectively. For BRO, the initial value for $▵$ is configured to $\frac{\mathit{maxCicle}}{round({\mathit{log}}_{10}(maxCicle))}$ as indicated in the literature⁶⁸, where maxCicle denotes the maximum iterations. For MSCA, the parameter configuration of a = 2, b = 0.5, and c = 4, as indicated in the literature⁶⁶, are used.

In this study, the evaluation utilizes k-fold cross-validation, a widely used technique in machine learning for assessing model performance, is adopted. This method entails the random partitioning of the dataset into k mutually exclusive subsets or ’folds.’ During each iteration, one fold is designated as the validation set, while the remaining k-1 folds serve as the training data. The model undergoes training and validation k times, with each fold utilized exactly once for validation. For this research endeavor, a value of k=5 is selected, thereby implementing 5-fold cross-validation (5CV).

In the context of stochastic optimization techniques, the variability inherent in each algorithm’s execution can yield divergent outcomes across independent runs within a simulation experiment. To ensure the derivation of statistically significant findings, a rigorous approach is adopted wherein the empirical results are synthesized from the aggregation of 10 independent runs conducted for each dataset. Throughout each run, the average outcome of the 5-fold cross-validation process is recorded, thereby providing a robust basis for analysis and inference. In the following content, the evaluation of the solution accuracy and stability of the proposed SLSSA relies on the objective function values obtained, with the corresponding classification accuracies discussed in the first sub-subsection. Additionally, the efficiency of the SLSSA in achieving the optimal solution will be explored in terms of CPU time and convergence speed in the second sub-subsection.

Solution accuracy and stability

The median and interquartile range of the obtained objective function (CEE) values in the 10 independent runs of the competing algorithms are shown in Table 12. A lower median value of the objective function indicates higher accuracy, whereas a lower interquartile range represents better stability. The meanings of the contents in the last columns of Table 12 is are consistent with the definitions of the last column in Table 6. That is, the symbols “$+$ / $\ast$ / −” indicate that, in the WRS test, the corresponding metric values of proposed SLSSA are statistically better than, equal to, and worse than that of the corresponding compared algorithms in orders, respectively. Moreover, to reveal the performances of the MLPs trained by the proposed SLSSA in classification accuracy, the corresponding training classification accuracy and test classification accuracy of the solutions with the median CEE values in Table 12 are shown in Tables 13 and 14. Furthermore, average rankings achieved by the Friedman test for these compared algorithms on CEE are illustrated in Table 15.

Table 12.

Objective function values on MLP training of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, SSA, MSCA, and BRO.

Dataset	SLSSA	SSA-COBL	CSSA	ESSA	LSC-SSA	SSA	MSCA	BRO	WRS
Balloon	0.0000E+00 (0.0000E+00)	6.8000E-03 (7.3000E-03)	0.0000E+00 (2.0000E-04)	2.0000E-04 (5.0000E-04)	3.0000E-04 (6.0000E-04)	1.8000E-03 (1.5000E-03)	0.0000E+00 (0.0000E+00)	0.0000E+00 (0.0000E+00)	+/+/+/+/+/*/*
Iris	2.0800E-02 (2.5000E-03)	1.4370E-01 (2.6100E-02)	2.7000E-02 (2.9000E-03)	4.1500E-02 (2.5000E-03)	5.3100E-02 (6.7000E-03)	7.9600E-02 (2.8000E-02)	3.9300E-02 (2.6000E-03)	3.4200E-02 (2.7000E-03)	+/+/+/+/+/+/+
Scale	1.4130E-01 (8.1000E-03)	3.9080E-01 (3.8600E-02)	2.0400E-01 (1.7700E-02)	2.2270E-01 (1.3400E-02)	2.8060E-01 (1.8600E-02)	3.6950E-01 (7.1000E-03)	5.1480E-01 (1.5860E-01)	1.6980E-01 (9.1000E-03)	+/+/+/+/+/+/+
Seeds	2.0700E-02 (4.6000E-03)	2.0310E-01 (2.9700E-02)	5.0000E-02 (6.9000E-03)	1.0040E-01 (2.4500E-02)	9.7500E-02 (1.1000E-02)	1.7440E-01 (2.3200E-02)	9.5800E-02 (2.1100E-02)	4.7400E-02 (6.3000E-03)	+/+/+/+/+/+/+
Cancer	2.2900E-02 (5.2000E-03)	8.6000E-02 (2.7000E-03)	3.8500E-02 (9.3000E-03)	7.2400E-02 (3.4000E-03)	7.1600E-02 (7.1000E-03)	8.7300E-02 (4.5000E-03)	6.9500E-02 (3.5000E-03)	3.3200E-02 (2.3000E-03)	+/+/+/+/+/+/+
Wine	1.0000E-04 (0.0000E+00)	1.5470E-01 (4.9100E-02)	2.0000E-04 (2.0000E-04)	1.4300E-02 (7.2000E-03)	1.4700E-02 (6.8000E-03)	1.5930E-01 (6.3300E-02)	1.6020E-01 (7.9800E-02)	3.0000E-04 (2.0000E-04)	+/+/+/+/+/+/+

Table 13.

Training classification accuracy on MLP training of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, SSA, MSCA, and BRO.

Dataset	SLSSA	SSA-COBL	CSSA	ESSA	LSC-SSA	SSA	MSCA	BRO
Balloon	1.0000E+02	9.9500E+01	1.0000E+02	1.0000E+02	1.0000E+02	1.0000E+02	1.0000E+02	1.0000E+02
Iris	7.8133E+01	6.5267E+01	7.8000E+01	7.6533E+01	7.4867E+01	7.3467E+01	7.6667E+01	7.7200E+01
Scale	8.4480E+01	5.9020E+01	8.2920E+01	8.1020E+01	7.5220E+01	7.0000E+01	6.5120E+01	8.2640E+01
Seeds	9.0298E+01	4.3393E+01	8.6310E+01	7.8393E+01	7.8512E+01	6.3690E+01	8.1429E+01	8.4643E+01
Cancer	9.3294E+01	8.7911E+01	9.2794E+01	8.9574E+01	8.9056E+01	8.8823E+01	8.7893E+01	9.1201E+01
Wine	1.0000E+02	4.3745E+01	1.0000E+02	9.3329E+01	9.5074E+01	6.9449E+01	7.9070E+01	1.0000E+02

Table 14.

Test classification accuracy on MLP training of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, SSA, MSCA, and BRO.

Dataset	SLSSA	SSA-COBL	CSSA	ESSA	LSC-SSA	SSA	MSCA	BRO
Balloon	1.0000E+02	9.9500E+01	1.0000E+02	1.0000E+02	1.0000E+02	1.0000E+02	1.0000E+02	1.0000E+02
Iris	1.8733E+01	1.6733E+01	1.8600E+01	1.8667E+01	1.8200E+01	1.7667E+01	1.8533E+01	1.8733E+01
Scale	8.3040E+01	5.8720E+01	8.1040E+01	8.0640E+01	7.5120E+01	6.9520E+01	6.2160E+01	8.2160E+01
Seeds	8.6667E+01	5.0238E+01	8.0000E+01	7.9286E+01	7.5952E+01	6.6905E+01	8.0238E+01	8.3095E+01
Cancer	9.1562E+01	8.8701E+01	9.0987E+01	8.8699E+01	8.8769E+01	8.8486E+01	8.6621E+01	8.9418E+01
Wine	9.4643E+01	5.1103E+01	8.5944E+01	8.3643E+01	8.7087E+01	6.6016E+01	7.4722E+01	9.1016E+01

Table 15.

Average Friedman ranking on CEE for MLP training of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, SSA, MSCA, and BRO.

Algorithms	Ranking
SLSSA	1.100000
SSA-COBL	7.466667
CSSA	2.833333
ESSA	4.883333
LSC-SSA	5.150000
SSA	6.983333
MSCA	5.083333
BRO	2.500000

The results in Table 12 illustrate that, out of 6, the proposed SLSSA obtains 6 best median values and 4 least IQR values on CEE. Moreover, the total “+” of the WRS test results in Table 12 is 40 out of 42. Moreover, the results in Tables 13 and 14 indicate that the corresponding training classification accuracies and test classification accuracies of the solutions generated by the SLSSA are significantly better than those of the compared algorithm, for it obtains 6 best results out of 6 in both metrics. On the other hand, it is revealed in Table 15 that the proposed SLSSA has the best average rankings in CEE. Clearly, the average ranking of SLSSA is substantially lower than that of all other competing algorithms. Specifically, when compared to BRO, which ranks second, SLSSA’s value is 56% lower. It can infer from Tables 12, 13, 14, and 15 that the SLSSA significantly outperforms the other competing algorithms in CEE, training classification accuracy, and test classification accuracy. Since the difficulty of classification on those datasets is great, these results are strong evidence for the efficiency of SLSSA in training MLPs. The results verify that SLSSA has a more efficient search ability to approximate more accurate and stable solutions regardless of whether those datasets have a smaller number of features or a larger number of features and whether those datasets are involved with two classes or three classes.

Computation efficiency

To compare the computation efficiency of the competing algorithms in training MLPs, the corresponding CPU computation time cost and the convergence speed are measured.

Table 16 shows the median and interquartile range of CPU computation time (measured in seconds) for 10 runs of each dataset by the eight competing algorithms in this experiment. In addition, to give a more intuitive comparison between the competing algorithms, the average rankings achieved by the Friedman test on CPU computation time for these competing algorithms are illustrated in Table 17. The results in Tables 16 and 17 are consistent with the results in Tables 9 and 10 for the experiment on CEC2014. Compared with the original SSA, the SLSSA needs additional computation time to complete the extra operations on execution probabilities update. However, compared to the fastest algorithm in all those datasets, the additional median CPU time cost by the SLSSA is no more than 25$\%$ in all cases and less than 10$\%$ in most cases. It can be inferred that the computation efficiency of the SLSSA is competitive when handling the MLPs training optimization problems. Furthermore, it can be deduced that, in terms of CPU time, the stable computational efficiency of SLSSA on MLP training aligns with the computational efficiency exhibited in the CEC2014 benchmark suite. There is only a marginal increase in computation time when comparing SLSSA to the faster algorithm in the experiment. This difference is acceptable and makes SLSSA a potential solution for practical applications. It is apparent that the diverse problem dimensions and population sizes involved in MLP training, different from those found in the CEC2014 benchmark suite, lead to the issue of unstable computational efficiency for some of the compared algorithms, such as SSA-COBL.

Table 16.

CPU time on MLP training of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, SSA, MSCA, and BRO.

Dataset	SLSSA	SSA-COBL	CSSA	ESSA	LSC-SSA	SSA	MSCA	BRO	WRS
Balloon	3.0878E+00 (4.2700E-02)	2.6362E+00 (2.2800E-02)	2.4784E+00 (1.1900E-02)	3.3076E+00 (1.4000E-02)	3.0795E+00 (2.4600E-02)	2.5656E+00 (2.5100E-02)	1.0705E+01 (1.0460E-01)	1.2293E+01 (1.3100E-01)	-/-/+/* /-/+/+
Iris	1.6687E+01 (2.0500E-02)	1.5724E+01 (6.6200E-01)	1.5323E+01 (9.0000E-03)	1.9697E+01 (9.9800E-02)	1.9250E+01 (5.9500E-02)	1.5576E+01 (4.4900E-02)	1.6517E+01 (5.0100E-02)	1.7973E+01 (4.5700E-02)	-/-/+/+ /-/-/+
Scale	2.5173E+01 (3.3050E-01)	2.4158E+01 (7.2600E-02)	2.5706E+01 (1.1400E-01)	2.7999E+01 (1.3960E-01)	2.7558E+01 (2.4278E+00)	2.4267E+01 (6.4200E-02)	2.4752E+01 (4.6000E-02)	2.6655E+01 (6.2800E-02)	-/+/+/+ /-/-/+
Seeds	2.4031E+01 (2.0900E-02)	2.3174E+01 (5.0900E-02)	2.4404E+01 (5.4000E-02)	2.7521E+01 (6.6300E-02)	2.9464E+01 (3.9400E-02)	2.3064E+01 (3.3700E-02)	2.4630E+01 (3.9900E-02)	2.6268E+01 (6.2600E-02)	-/+/+/+ /-/+/+
Cancer	4.8663E+01 (1.4610E-01)	4.8062E+01 (8.9800E-02)	4.8647E+01 (2.3890E-01)	5.2803E+01 (1.3470E-01)	5.6630E+01 (3.3600E-02)	4.7814E+01 (3.1940E-01)	4.8873E+01 (6.8920E-01)	5.1406E+01 (1.5570E-01)	-/*/+/+ /-/+/+
Wine	4.6757E+01 (6.0500E-02)	4.4391E+01 (4.5800E-02)	4.5570E+01 (1.3500E-01)	4.9948E+01 (8.7200E-02)	6.3396E+01 (1.0300E-01)	4.5900E+01 (1.4000E-01)	4.8828E+01 (1.2990E-01)	5.1117E+01 (7.5200E-02)	-/-/+/+ /-/+/+

Table 17.

Average Friedman ranking on CPU time for MLP training of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, SSA, MSCA, and BRO.

Algorithms	Ranking
SLSSA	4.116667
SSA-COBL	2.016667
CSSA	2.716667
ESSA	6.950000
LSC-SSA	7.083333
SSA	1.883333
MSCA	4.766667
BRO	6.466667

In order to illustrate the optimum approximating process of the competing algorithms, Fig. 5 depicts the convergence curves of the competing algorithms in the 6 datasets. Figure 6 presents an enlarged view of the latter part of Fig. 5. Following the convergence curve generation method of the previous experiment on CEC2014, the median values of the data points of the 10 independent runs are used to demonstrate the overall convergence speed for each dataset. Similar to the convergence curves in the experiment on CEC2014, it is observed from Fig. 5 that SLSSA is relatively slow in the early stage. However, SLSSA shows a more powerful ability to approximate the optimum while the other competing algorithms were trapped in stagnation situations. As a result, in a later stage, SLSSA has surpassed the previous leading competitors in the datasets. Moreover, observing each convergence curve of the whole evolution process, SLSSA is one of the fastest algorithms among all the competing algorithms, suggesting the enhanced search efficiency of SLSSA.

Fig. 5.

The convergence graphs of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, SSA, MSCA, and BRO on MLP training.

Fig. 6.

The latter part convergence graphs of SLSSA, SSA-COBL, CSSA, ESSA, LSC-SSA, SSA, MSCA, and BRO on MLP training.

In conclusion, four advancements of the proposed SLSSA algorithm can be drawn from the experimental results: (1) The four search operators, each with distinct characteristics, enhance the algorithm’s ability to tackle various search challenges. (2) The self-learning mechanism improves searching efficiency by gradually allocating computational resources to the most suitable search pattern for a specific target problem. (3) A simple and straightforward parameter-setting method facilitates the easy implementation of the algorithm for various optimization problems. (4) The algorithm maintains the same level of time complexity as SSA, ensuring acceptable search efficiency in practical applications.

Conclusion and future work

This study introduces the self-learning salp swarm algorithm (SLSSA), a new method that enhances search accuracy and robustness. SLSSA tackles diverse problems using four concurrent search strategies in a self-learning mechanism that adjusts based on past experiences. One key strategy is the multiple food sources search strategy, which improves SSA’s existing methods. Additionally, an analysis framework and guidelines for easily configuring two key parameters in the self-learning mechanism are also provided.

To test SLSSA’s performance, we conducted experiments with the CEC2014 benchmark suite and compared it to five state-of-the-art algorithms. We also applied SLSSA to train multilayer perceptrons (MLPs) to see how it performs in real-world situations. The results show that the parameter setting method effectively finds the best combination of key parameters. Additionally, SLSSA outperforms the other algorithms in terms of accuracy, stability, and overall convergence speed, while only slightly increasing computation time compared to the original SSA.

In future work, the authors aim to integrate more efficient search strategies into SLSSA to improve its overall performance in approximating the optima of various optimization problems. On the other hand, to reduce the computational cost, a simpler and more efficient method for updating execution probabilities will be investigated. Moreover, a more in-depth examination of the self-learning approach will be conducted through both theoretical and experimental analyses.

Author contributions

Z.Y. : Conceptualization, methodology, software, investigation, writing—original draft preparation; Y.J. : investigation, Writing-review and editing; W.Y. : investigation, Writing-review and editing. All authors have read and agreed to the published version of the manuscript.

Funding

This work was funded in part by the Department of Education of Guangdong Province of China (Research Grant no. 2022ZDZX1065, 2023ZDZX1072, 2021ZDZX1137), in part by the Guangzhou Municipal Science and Technology Bureau of China (Research Grant no. 202002030133), in part by the Department of Science and Technology of Guangdong Province of China (Research Grant no. 2022A1515010999), and in part by the Guangzhou Panyu Polytechnic (Research Grant no. 2023KJ03).

Data availability

Data will be made available on reasonable request with the coorsponding author.

Declarations

Competing interests

The authors declare no competing interests.